Light fields in complex media: mesoscopic scattering meets wave control
LLight fields in complex media: mesoscopic scattering meets wave control
Stefan Rotter * and Sylvain Gigan † Institute for Theoretical Physics,Vienna University of Technology (TU Wien),Wiedner Hauptstraße 8–10/136,1040 Vienna, Austria,EU Laboratoire Kastler Brossel,UMR8552 Universit´e Pierre et Marie Curie,Ecole Normale Sup´erieure,Coll`ege de France, CNRS,24 rue Lhomond,75005 Paris, France,EU (Dated: February 20, 2017)
The newly emerging field of wave front shaping in complex media has recently seenenormous progress. The driving force behind these advances has been the experimentalaccessibility of the information stored in the scattering matrix of a disordered medium,which can nowadays routinely be exploited to focus light as well as to image or totransmit information even across highly turbid scattering samples. We will providean overview of these new techniques, of their experimental implementations as wellas of the underlying theoretical concepts following from mesoscopic scattering theory.In particular, we will highlight the intimate connections between quantum transportphenomena and the scattering of light fields in disordered media, which can both bedescribed by the same theoretical concepts. We also put particular emphasis on howthe above topics relate to application-oriented research fields such as optical imaging,sensing and communication.
PACS numbers: 42.25.Bs, 42.25.Hz, 42.25.Fx
CONTENTS
I. Introduction 2II. Scattering theory for complex media 3A. Basic formalism 41. Wave equations 42. Continuity equation and flux 53. Green’s function 54. Scattering matrix 65. Random Matrix Theory (RMT) 106. DMPK equation 14B. Open transmission eigenchannels and shot noise 17C. Time-delay 19III. Mesoscopic effects in optical systems: theoretical andexperimental analogies 22A. Conductance quantization 22B. Conductance fluctuations 23C. Weak localization 25D. Memory effect 26E. Distribution of transmission eigenvalues 27IV. Optical wave front shaping in complex media 28A. Wavefront shaping concepts and tools 291. Classical optical elements 292. Active and adaptive optics 30 * [email protected] † [email protected] 3. From aberrating layers to aberrating volumes 314. Optical phase conjugation 31B. Digital tools for wave manipulation in optics 321. Matricial detector arrays 322. Matricial spatial light modulators 333. Other types of spatial light modulators 344. Temporal modulators 34C. The thin disordered slab: an opaque lens 351. Transmission matrix in the spatial domain 362. Temporal and spectral aspects 383. Accessing the monochromatic transmission matrixof an opaque lens 384. Accessing the temporally or spectrally-resolvedtransmission matrix 39D. Light manipulation through the opaque lens 391. Time reversal, analog and digital phase conjugationthrough the opaque lens 392. Focusing and iterative optimization 413. Imaging 434. Deterministic mixing 445. Polarization control 446. Temporal and spectral control 45E. Other complex scattering systems 461. Multimode optical fibers 462. Biological tissues 49V. Mesoscopic physics and wave front shaping 50A. Memory effect 501. Imaging using the memory effect 502. Beyond the conventional memory effect 51B. Bimodal distribution of eigenchannels 521. Accessing the bimodal distribution 53 a r X i v : . [ phy s i c s . op ti c s ] F e b
2. Unraveling and exploiting open and closedchannels 54C. Time delay eigenstates 561. Principal modes in a fiber 562. Particle-like scattering states 58D. Wavefront shaping in media with gain or loss 591. Absorbing media 592. Amplifying media 60VI. Conclusions and outlook 62A. Wavefront shaping for unraveling mesoscopicphenomena 62B. New systems 62C. Applications of mesoscopic concepts in optics 63Acknowledgments 64References 64
I. INTRODUCTION
Recent years have witnessed enormous conceptual andexperimental progress in the ability to manipulate lightfields both spatially and temporally. On the experimen-tal side these advances have largely been enabled by theavailability of highly tunable digital arrays, known also asspatial light modulators (Savage, 2009), which are mean-while being used to create arbitrarily complex light fields.In this sense the current state-of-the-art in the field of op-tical wave front shaping is reminiscent of the situation inrelated areas like acoustics and seismology, which weresimilarly promoted by antenna or transducer arrays thatcan retrieve information from a complex environment.The availability of such versatile tools now also in opticsopens up the way to address a topic where conventionaloptical techniques are hard to apply, like the control oflight propagation in turbid media such as in amorphousor disordered materials, in biological tissues, complexphotonic structures, plasmonic systems, multimode fibersetc. Starting points for these activities were a number ofproof-of-principle experiments that have recently demon-strated that a disordered material can be used to focuslight (van Putten et al. , 2011; Vellekoop and Mosk, 2007)and that its transmission matrix can be measured in de-tail (Popoff et al. , 2011a, 2010b) such as to reconstructthe transmission of images across highly scattering sam-ples (Popoff et al. , 2010a). Beyond spatial control, ex-plicitly time-dependent measurements were able to showthat a wave scattered in a disorder region can not only befocused in space but also in time (Aulbach et al. , 2011;Katz et al. , 2011; McCabe et al. , 2011; Mounaix et al. ,2016a). Following the pioneering concepts introduced in(Freund, 1990a), further work successfully demonstratedthat the information stored in the scattering matrix of adisordered system can be used for turning a disorderedsample into a perfect mirror (Katz et al. , 2012) or into ahigh resolution spectral filter or spectrometer (Redding et al. , 2013a; Small et al. , 2012). These insights can beexpected to have impact on a very broad range of fields like on biology and medicine (Cox, 2012), where imagingthrough disorder is a major challenge; on nanophotonics(Kawata et al. , 2002), where the challenge is to addressand control quantum systems in a disordered environ-ment (Sapienza et al. , 2011); on quantum information(Defienne et al. , 2016; Ott et al. , 2010; Wolterink et al. ,2016), where entangled states could be guided and trans-formed; as well as on communication technology (Tse andViswanath, 2005), where the principal goal is to securethat information sent through a complex environmentends up at a desired receiver.A sound theoretical basis required to describe all of theabove phenomena is given in terms of scattering theory.In the specific context of disorder scattering it is mostlythe work in mesoscopic physics (Akkermans and Mon-tambaux, 2007; Imry, 2002; Sebbah, 2001; St¨ockmann,2006), quantum transport (Datta, 1997; Ferry and Good-nick, 1997; Mello and Kumar, 2004) and Random MatrixTheory (Alhassid, 2000; Beenakker, 1997; Mitchell et al. ,2010) that has been the principal driving force behindtheoretical progress. This is because electron scatter-ing through disordered or chaotic systems has been andcontinues to be one of the paradigms in the mesoscopicphysics community. In spite of the progress made, manyof the results obtained for the situation on the meso-scopic scale do, however, remain unknown to the newlyemerging scientific communities working on wave frontshaping in complex media. The reason why many in-sights penetrated only weakly outside the community ofmesoscopic physics is probably due to the vastness of thefield, which makes it difficult to overlook, and due to aspecific scientific jargon which scientists working outsidethis community are typically not familiar with.The intended goal of our review article will be to bridgethis knowledge gap. Our strategy will be to demonstratehow theoretical insights from mesoscopic scattering the-ory have direct relevance for the recent wave control ex-periments and vice versa. We will start in section II witha brief review of mesoscopic transport theory in whichbasic concepts like the scattering matrix and its statisti-cal properties following from Random Matrix Theory orrelated approaches are introduced. We will discuss herethe concept of transmission eigenchannels as well as theirconnection to electronic shot noise, which provides indi-rect access to the distribution of transmission eigenvaluesin measurements of electronic current. A particular em-phasis will also be put on the concept of time-delay inscattering and its relation to the density of states as wellas to coherent wave absorption and to the quantum-to-classical crossover. With such a solid theoretical basisbeing established, we move on in section III to review anumber of mesoscopic transport effects that have mean-while been observed in optical experiments without thehelp of any wavefront shaping tools. As we will show,quite a number of theoretical concepts first studied in amesoscopic context could be successfully transferred and,indeed, observed with light fields in complex media. Ex-amples which we highlight here are those related to thequantization of the conductance and its universal fluctu-ations, weak localization, the memory effect etc. Whilethese observations are very encouraging for the appli-cability of the theoretical tools introduced in II, muchmore can be done in these experiments with the toolsof wavefront shaping. These tools will then be reviewedin section IV, starting with a historical perspective onwhere such experimental techniques were first employedsuch as in adaptive optics. Once spatial light modulatorsare introduced, we will discuss how they can be used formeasuring and modulating the light transmitted throughthe paradigmatic case of a thin disordered slab. As willbe discussed in our review, such an “opaque lens” canbe used for focusing, imaging as well as for controllingthe polarization and the temporal shape of the transmit-ted light. In section V we will review how the predictionsfrom mesoscopic transport theory can be fully brought tobear using the wavefront shaping tools introduced earlier.First, we will focus on effects that were already realized incorresponding experiments such as those related to openand closed transmission channels, the memory effect etc.In a next step we will provide a collection of many in-teresting predictions that still await an implementationin the laboratory. This outlook also serves the purposeof indicating future directions of research and of demon-strating how much “uncharted territory” is yet to be de-veloped in this increasingly active field of research. Ourreview is rounded off with a summary in section VI.We have intentionally restricted the scope of this re-view to the interface between mesoscopic scattering the-ory and the recent advances in wavefront shaping. Withthis focus we hope to provide some added value to both ofthe corresponding two communities that we are trying tobetter connect with our piece. At this point we emphasizethat excellent reviews and books are already available foreach of these two separate fields: Regarding mesoscopicscattering, our readers may find a wealth of informationon specific topics, like on random matrix theory (Ake-mann et al. , 2011; Beenakker, 1997; Guhr et al. , 1998),or on the maximum entropy approach (Mello and Kumar,2004), as well as on the many interesting connections be-tween electronic transport and light scattering (Akker-mans and Montambaux, 2007; Dragoman and Dragoman,2004; Lagendijk and van Tiggelen, 1996). Also for wave-front shaping first short reviews have recently becomeavailable (Mosk et al. , 2012; Shi et al. , 2015; Vellekoop,2015; Vos et al. , 2014). The niche we intend to fill withour own contribution is to highlight the underappreci-ated connection between the above topical areas and itspotential for future research.
Figure 1 (color online). Schematic of different methods usedfor describing wave scattering in disordered media on differentlevels of accuracy and simplicty, adapted from (Bertolotti,2011).
II. SCATTERING THEORY FOR COMPLEX MEDIA
The scattering of waves through disordered or other-wise complex media is a problem that can be approachedfrom many different angles. In particular, a whole hi-erarchy of different methods have been developed thatprovide insights on different levels of accuracy, typicallyanti-correlated with the complexity of a specific method(see Fig. 1). For the purpose of this review, we willmostly be interested in those approaches, which retainthe wave nature of the scattering process, such as to in-corporate effects due to interference. A full solution ofthe corresponding wave equation is, however, very costlynumerically and often does not provide much insight intothe general features underlying a whole class of relatedproblems. To overcome such limitations, much work hasbeen invested into “mesoscopic scattering theory”, whichwe will provide a short review of. This term refers to aset of theoretical tools that were largely developed in thecontext of mesoscopic electron transport, in which thephase coherence of electrons, the finite number of modesthrough which they can scatter as well as the correlationsbetween these modes play a significant role.Whatever the formalism chosen to describe wave prop-agation in disordered media, there are a few commonparameters to quantify the scattering properties of themedium. The most important one is probably the trans-port mean free path, usually referred to as ℓ tr or ℓ ⋆ inthe literature (we will use the notation ℓ ⋆ in this review).This length scale measures after which distance the prop-agation direction of an incoming photon is randomizedand thus governs the macroscopic transport propertiesof the medium. At a more microscopic level, the scat-tering mean free path ℓ (or ℓ s in the literature) mea-sures the average distance traveled between two scatter-ing events and thus quantifies the scattering strength ofthe medium. The link between ℓ ⋆ and ℓ is through theanisotropy of scattering, ℓ ⋆ = ℓ/ (1 − 𝑔 ), quantified bythe anisotropy factor 𝑔 = ⟨ cos 𝜃 ⟩ , i.e., the average of thecosine of the scattering angle 𝜃 . An important case is theone of isotropic scattering, for which ⟨ cos 𝜃 ⟩ = 0, 𝑔 = 0and ℓ ⋆ = ℓ . A. Basic formalism
1. Wave equations
A good starting point for setting up the formalism formesoscopic scattering is the observation that electromag-netic waves in a dielectric medium behave similarly aselectrons in a potential (see here also the correspondingreferences listed in the last paragraph of the introduc-tion). Since this analogy will also be the bridge acrosswhich many of the results from mesoscopic transport the-ory can be carried over to the domain of optics, we shallstart here by elucidating this connection.Consider first the Schr¨odinger equation for the evolu-tion of a particle of mass 𝑚 in a potential 𝑉 ( r ), [︂ p 𝑚 + 𝑉 ( r ) ]︂ 𝜓 ( r , 𝑡 ) = 𝑖 (cid:126) 𝜕 𝑡 𝜓 ( r , 𝑡 ) . (1)For stationary states with a well-defined real energy 𝐸 = (cid:126) 𝜔 and a corresponding time-evolution exp( − 𝑖𝜔𝑡 )the Schr¨odinger equation reduces to the following formwithout a time-derivative, {︂ Δ − 𝑚 (cid:126) [ 𝑉 ( r ) − 𝐸 ] }︂ 𝜓 𝐸 ( r ) = 0 , (2)where we have used the standard definition for the mo-mentum operator p = − 𝑖 (cid:126) ∇ . For appropriate boundaryconditions the differential operator in Eq. (2) will be Her-mitian such that the eigenstates, 𝜓 𝑚 ( r ) (labeled by theirmode index 𝑚 ) satisfy the conventional orthogonality re-lations ∫︁ 𝑑 r 𝜓 𝑚 ( r ) * 𝜓 𝑛 ( r ) = 𝛿 𝑚𝑛 , (3)and form a complete basis of states.To find equivalent relations also for light scattering,consider first the wave equation for the electric field E ( r , 𝑡 ) in a source-free, linear and frequency-independentdielectric medium with dielectric function 𝜀 ( r ), whichis directly derived from Maxwell’s equations (Jackson,1998), − ∇ × ∇ × E ( r , 𝑡 ) = 𝜀 ( r ) 𝑐 𝜕 𝑡 E ( r , 𝑡 ) . (4)To make the analogy to the Schr¨odinger equation, we firstrestrict ourselves to monochromatic states, which, in per-fect analogy to the stationary states of the Schr¨odingerequation, feature a harmonic time-dependence E ( r , 𝑡 ) = E 𝜔 ( r ) exp( − 𝑖𝜔𝑡 ). Unless explicitly stated otherwise, wewill work with this complex notation in the followingwith the understanding that the real (physical) electricfield is extracted as the real part of this complex quan-tity, Re[ E ( r , 𝑡 )]. As it turns out, even for appropriateboundary conditions, the curl operator above is not aHermitian operator when the dielectric function 𝜀 ( r ) is spatially varying and when the conventional inner prod-uct is used (Viviescas and Hackenbroich, 2003). We thusrewrite the electric field through the vector-valued func-tion 𝜑 𝜔 ( r ) = √︀ 𝜀 ( r ) E 𝜔 ( r ), such that Eq. (4) can berewritten with the Hermitian differential operator ℒ , ℒ 𝜑 𝜔 ( r ) ≡ √︀ 𝜀 ( r ) ∇ × [︃ ∇ × 𝜑 𝜔 ( r ) √︀ 𝜀 ( r ) ]︃ = 𝜔 𝑐 𝜑 𝜔 ( r ) . (5)With appropriate boundary conditions, the eigenstates, 𝜑 𝑚 ( r ) are then orthogonal to each other and provide acomplete basis of states. As a consequence, the electricfield satisfies the following orthogonality relation, ∫︁ 𝑑 r 𝜑 𝑚 ( r ) 𝜑 𝑛 ( r ) = ∫︁ 𝑑 r 𝜀 ( r ) E 𝑚 ( r ) E 𝑛 ( r ) = 𝛿 𝑚𝑛 . (6)To make the analogy to the stationary Schr¨odinger equa-tion even more apparent, we simplify the curl opera-tor by the following vector identity ∇ × ( ∇ × A ) = ∇ ( ∇ · A ) − Δ A , such that we arrive at what is termedthe vectorial Helmholtz equation, {︂ Δ − 𝜔 𝑐 [1 − 𝜀 ( r )] + 𝜔 𝑐 }︂ E 𝜔 ( r ) = . (7)We need to emphasize at this point, however, that theabove equation holds only under the assumption that ∇ · E = 0 as for for source-free media which are linear,homogeneous and isotropic (in the linear regime gain andloss in the medium may be included through a complexdielectric 𝜀 ( r )). For inhomogeneous media, where thedielectric permittivity is position dependent, the waveequation Eq. (4) as well as the above Helmholtz equationEq. (7) do not hold. One may just consider the approx-imation of a locally homogeneous medium for which thevariation of 𝜀 ( r ) is slow as compared to the wavelength 𝜆 (Lifante, 2003). Alternatively, when the medium consistsof piecewise homogeneous constituents, one may use theHelmholtz equation for each sub-part, but different field-components get mixed at the boundaries between them.We also mention here, that it may computationally bemore efficient to consider the magnetic field, rather thanthe electric field in a nonmagnetic medium (Joannopou-los et al. , 2008).Even the vector Helmholtz equation itself is very diffi-cult to solve for most cases and closed solutions only existin very special limits. In practice, one therefore often re-duces the Helmholtz equation to a scalar form, in whichthe scalar quantity 𝜓 𝜔 ( r ) stands for one of the three com-ponents of the electric or magnetic field. Implicit in thisstrategy is the assumption that the coupling of differentvectorial components does not contain important physics– a point which in many publications remains open [see,e.g., a corresponding analysis in (Bittner et al. , 2009)].Certainly, the description of light as a scalar field maylead to quite different results than those based on a fullsolution of the Maxwell equations (Lagendijk and vanTiggelen, 1996; Skipetrov and Sokolov, 2014), such thata careful analysis for each individual case at hand mustbe recommended.In all cases where the scalar Helmholtz equation pro-vides a good approximation of the real physics (Kragl,1992), {︂ Δ − 𝜔 𝑐 [1 − 𝜀 ( r )] + 𝜔 𝑐 }︂ 𝜓 𝜔 ( r ) = 0 , (8)all the quantities in this scalar equation for light fieldscan be directly compared with those of the Schr¨odingerequation for electrons, Eq. (2). For the case of the dielec-tric constant of vacuum, 𝜀 ( r ) = 1, or, equivalently, for thecase of vanishing potential 𝑉 ( r ) in the Schr¨odinger case,we can see immediately, that the resulting two equations,Eq. (2) and Eq. (8), are the same if we identify the “lightenergy” as follows, 𝐸 light = ( (cid:126) 𝜔 ) / (2 𝑚𝑐 ) (Lagendijk andvan Tiggelen, 1996). For the case of free space propa-gation, both equations also have the same fundamentalplane wave solutions, 𝜓 𝐸 ( r ) = 𝜓 𝜔 ( r ) = 𝜓 k ,𝜔 exp( 𝑖 kr − 𝑖𝜔𝑡 ), characterized by a single frequency 𝜔 and a singlewave vector k , where | k | = 𝑘 = 𝜔 √ 𝜀𝜇 = 𝑛𝑘 with 𝑛 therefractive index and 𝑘 = 𝜔/𝑐 . In the case of a spatiallynon-uniform dielectric function, 𝜀 ( r ), or potential land-scape, 𝑉 ( r ), the scalar Helmholtz and the Schr¨odingerequation can still be mapped onto each other for anygiven frequency 𝜔 , when we identify the following rela-tion for the “light potential” 𝑉 light ( r ) = 𝐸 light [1 − 𝜀 ( r )]and keep in mind that the Helmholtz equation is validfor locally homogeneous media only (Lifante, 2003).The equivalence between the fundamental equations,which describe electronic and light scattering, will beessential for many of the effects discussed in this re-view and for their occurrence in both of the differentresearch fields. Note, however, that this analogy alsohas well-defined limits, as, e.g., when attempting to de-scribe the microscopic details of the scattering field ina disordered medium, which goes beyond the capacityof the Helmholtz equation and requires a full treatmentbased on Maxwell’s equations. As we will see below, formany other quantities, in particular for those related tothe statistical properties of scattering amplitudes, manycommon aspects in electron and light scattering can beidentified.Fundamental differences between the scattering of elec-trons and light do, however, remain: These become ap-parent, e.g, when going away from the stationary pictureat a given scattering energy 𝐸 or frequency 𝜔 . Due tothe difference between the linear dispersion relation forlight ( 𝜔 = 𝑘𝑐 , or, equivalently, 𝐸 ∝ 𝑝 ) and the quadraticdispersion for matter ( 𝐸 ∝ 𝑝 ), the temporal dynamicsin scattering will be very different for these two cases.Consider here, e.g., the free motion of a wave packet in one dimension which satisfies Eq. (4) with the linear dis-persion relation 𝜔 = 𝑘𝑐/𝑛 . In free space, where 𝑛 = 1,both the group velocity 𝑣 𝑔 = 𝜕𝜔/ ( 𝜕𝑘 ) = 𝑐 and the phasevelocity 𝑣 𝜑 = 𝜔/𝑘 = 𝑐 are independent of 𝜔 or 𝑘 suchthat wave packets of light preserve their shape in vac-uum. In contrast, for electronic matter waves the cor-responding velocities 𝑣 𝑔 = (cid:126) 𝑘/𝑚 and 𝑣 𝜑 = (cid:126) 𝑘/ (2 𝑚 ) dodepend on 𝑘 , such that different frequency componentsof the wave packet travel with different speeds, leadingto wave packet spreading even in vacuum. Furthermore,since the relation 𝜀 ( r ) > 𝑉 light < 𝐸 light , a dielec-tric medium can never form a tunneling barrier for lightin the same way as an electrostatic potential can do forelectrons. Also any effects related to the vectorial charac-ter of the electric field (such as the polarization of light)have no simple analogy to the electronic case. Whenconsidering stationary scattering problems in which thepolarization does not play an important role, the anal-ogy between electron and light scattering can, however,be used extensively. At points where this analogy breaksdown, this will be mentioned explicitly.
2. Continuity equation and flux
The scattering of electrons and the scattering of lightin a lossless, static and linear dielectric medium have incommon that a conservation relation applies, 𝜕 𝑡 𝑊 ( r , 𝑡 ) + ∇ · J ( r , 𝑡 ) = 0 . (9)This so-called “continuity equation” states that any tem-poral change of the density 𝑊 must be compensatedby a corresponding flux J . For the electronic casethese two quantities are given by the probability density 𝑊 ( r , 𝑡 ) = | 𝜓 ( r , 𝑡 ) | and by the probability current den-sity J ( r , 𝑡 ) = Re [ 𝜓 ( r , 𝑡 ) * p 𝜓 ( r , 𝑡 )] /𝑚 , respectively. Thecorresponding quantities for light are the electromagneticenergy density 𝑢 ( r , 𝑡 ) and the Poynting vector S ( r , 𝑡 ),which fulfill the relation in Eq. (9), now termed “Poynt-ing theorem”, when the following replacements are made: 𝑊 ( r , 𝑡 ) → 𝑢 ( r , 𝑡 ) = [︀ 𝜀 E ( r , 𝑡 ) + 𝜇 − B ( r , 𝑡 ) ]︀ and J ( r , 𝑡 ) → S ( r , 𝑡 ) = E ( r , 𝑡 ) × B ( r , 𝑡 ) /𝜇 (for which defi-nitions we used real-valued fields E , B ) (Griffiths, 1999).Note that these quantities are of particular importancein experiments, since what detectors typically measure isthe integrated flux, counted in terms of the number ofelectrons or of photons that hit the detector surface (La-gendijk and van Tiggelen, 1996; van Tiggelen and Kogan,1994).
3. Green’s function
A central issue that we will address in this reviewis the question of how the radiation emitted by a setof given sources is scattered to a set of receivers. Forthis purpose it is convenient to resort to the conceptof the Green’s function (Morse and Feshbach, 1953).We start here again with Maxwell’s equations for anon-magnetic medium described by a dielectric function 𝜀 𝜔 ( r ) = 𝜀 𝑟𝜔 + 𝜀 𝑠𝜔 ( r ) that is embedded in an infinite homo-geneous reference medium 𝜀 𝑟𝜔 . In the presence of externalcurrent sources J 𝜔 ( r ) we end up with an inhomogeneousvector Helmholtz equation of the following form (Tsang et al. , 2004) −∇×∇× E 𝜔 ( r )+ (︁ 𝜔𝑐 )︁ 𝜀 𝜔 ( r ) E 𝜔 ( r ) = 𝑖𝜇 𝜔 J 𝜔 ( r ) , (10)with 𝜔/𝑐 = 𝑘 being the vacuum wavenumber. Whensimplifying the notation in the following way −∇×∇× → 𝒟 , 𝑘 𝜀 𝑟𝜔 ( r ) → e 𝑟 , 𝑘 𝜀 𝑠𝜔 ( r ) → e 𝑠 , the above Eq. (10) iswritten as ( 𝒟 + e 𝑟 + e 𝑠 ) E = 𝑖𝜇 𝜔 J 𝜔 . The desired Green’sfunction G (which is actually a dyadic tensor) satisfiesthe corresponding equation ( 𝒟 + e 𝑟 + e 𝑠 ) G = 𝛿 ( r − r ′ ) ,where we have used the following simplified notation: G ( r , r ′ , 𝜔 ) → G and is the unit tensor. With the helpof the tensorial Green’s function, we may relate vecto-rial current sources with vectorial electric fields througha convolution, E 𝜔 ( r ) = 𝑖𝜇 𝜔 ∫︁ 𝑑 r ′ G ( r , r ′ , 𝜔 ) J 𝜔 ( r ′ ) . (11)In the case that no current sources are present in amedium, the inhomogeneity in Eq. (10) vanishes. Anelectric field can still be present, however, when an in-cident field is considered. Such a scattering problemcan be treated by setting up equivalent relations forthe incident field E that satisfies ( 𝒟 + e 𝑟 ) E = 0in the homogeneous and source-free reference system.With the corresponding Green’s function G satisfying( 𝒟 + e 𝑟 ) G = 𝛿 ( r − r ′ ) one finds the so-called Dysonequation G = G − G e 𝑠 G and E = ( − Ge 𝑠 ) E (Martin et al. , 1995). One thus has a generalized field propagator K = ( − Ge 𝑠 ) at hand that connects the incident fieldwith the full field distribution (including the scatteredpart) again through a convolution, E 𝜔 ( r ) = ∫︁ 𝑑 r ′ K ( r , r ′ , 𝜔 ) E 𝜔 ( r ′ ) . (12)Note that both Eq. (11) and Eq. (12) are valid inde-pendently of the form of the current sources or of theincident field. The central piece of information necessaryto solve these equations is the system response encap-sulated in the Green’s tensor G . To obtain this quan-tity, one may proceed through direct inversion of theDyson equation ( G is known analytically (Morse andFeshbach, 1953)), or through iteration (corresponding it-eration schemes have been put forward both for electro-magnetic wave propagation (Martin et al. , 1995) as wellas for mesoscopic electron scattering (Datta, 1997; Ferryand Goodnick, 1997; Rotter et al. , 2000)). Figure 2 (color online). Illustration of the scattering systemconsidered in the text: A rectangular disordered region oflength 𝐿 fills the middle part of an infinite wave of height 𝐷 . Here the flux injected from the left through transversewaveguide modes can be transmitted (to the right) or reflected(to the left). In the general case of a non-uniform medium, whichmay also change the polarization of the electric field, thetensorial character of the Green’s function is essential.The reduction to a scalar Green’s function is allowed,however, when considering the emission and detection inwell-defined polarization directions only (Papas, 2011).
4. Scattering matrix
A primary goal in scattering theory is to connect theincoming flux that is impinging on the system of interestto the outgoing flux scattered away from this system. Aconvenient tool for carrying out the corresponding book-keeping is the scattering matrix which connects all theincoming and outgoing flux “channels” to be defined indetail below. The scattering matrix, in turn, is intimatelyrelated to the Green’s function since the latter connectsall points in space with each other.To illustrate this in detail, we introduce as a model sys-tem a two-dimensional slab geometry of uniform height 𝐷 with a disordered dielectric medium of length 𝐿 in themiddle and lossless semi-infinite waveguides of the sameheight attached on the left and right (see Fig. 2). Thismodel system will serve as a convenient tool to many ofthe features which we want to explain below for scatter-ing through waveguides, fibres and disordered media ingeneral. To simplify matters, we will assume that we canuse the scalar Helmholtz equation [see Eq. (8) above], asfor a transverse magnetic polarized electro-magnetic fieldmode in a three-dimensional medium which is invariantin 𝑧 -direction. The relevant scalar field component whichwe thus describe is the 𝑧 -component of the electric field, 𝐸 𝑧 , assuming hard-wall (Dirichlet)-boundary conditionsat the upper and lower boundaries of the scattering do-main.In the asympotic regions (far away from the disorderedpart), the field in the left ( 𝛼 = 𝑙 ) and right ( 𝛼 = 𝑟 ) waveguide will naturally be decomposed into different waveg-uide modes 𝜒 𝑛 ( 𝑦 ) = √︀ /𝐷 sin( 𝑛𝜋𝑦/𝐷 ), as determinedby the boundary conditions in transverse direction, 𝜓 𝜔 ( x ) = 𝑁 ∑︁ 𝑛 =1 𝑐 + 𝛼,𝑛 𝜒 𝑛 ( 𝑦 ) 𝑒 𝑖𝑘 𝑥𝑛 𝑥 √︀ 𝑘 𝑥𝑛 + 𝑐 − 𝛼,𝑛 𝜒 𝑛 ( 𝑦 ) 𝑒 − 𝑖𝑘 𝑥𝑛 𝑥 √︀ 𝑘 𝑥𝑛 . (13)For fully defining the above scattering state in the asymp-totic region we have summed over all 𝑁 = ⌊ 𝜔𝐷/ ( 𝑐𝜋 ) ⌋ flux-carrying modes for which the propagation constant 𝑘 𝑥𝑛 = √︀ 𝜔 /𝑐 − ( 𝑛𝜋/𝐷 ) is real (evanescent modes withan imaginary propagation constant have died out asymp-totically). The complex expansion coefficients 𝑐 ± 𝑛 corre-spond to right-moving (+) and left-moving waves (–),respectively. The terms in the denominators ∼ √︀ 𝑘 𝑥𝑛 arerequired to make sure that all the basis states on whichwe expand the field, have the same flux in longitudinaldirection 𝐽 ‖ (see definitions in section II.A.2). Basedon the above representation of the field in the asymp-totic region, we can define the scattering matrix as thecomplex matrix which connects the incoming expansioncoefficients with the outgoing coefficients, c out = S c in with c in ≡ (︂ c + 𝑙 c − 𝑟 )︂ , c out ≡ (︂ c − 𝑙 c + 𝑟 )︂ . (14)The 2 𝑁 × 𝑁 complex coefficients in the scattering ma-trix can be subdivided into four block matrices, S = (︂ r t ′ t r ′ )︂ , (15)where the quadratic blocks on the diagonal contain thereflection amplitudes for incoming modes from the left( 𝑟 𝑚𝑛 ) and from the right ( 𝑟 ′ 𝑚𝑛 ), respectively. The off-diagonal blocks contain the transmission amplitudes forscattering from left to right ( 𝑡 𝑚𝑛 ) and from right to left( 𝑡 ′ 𝑚𝑛 ), respectively. Note that in the case that the num-ber of modes is different on the left ( 𝑁 ) and right side( 𝑀 ), the reflection matrices r , r ′ remain quadratic (ofsize 𝑁 × 𝑁 and 𝑀 × 𝑀 , respectively), whereas the trans-mission matrices t , t ′ are then just rectangular (of size 𝑀 × 𝑁 and 𝑁 × 𝑀 , respectively). In this general casethe total transmission 𝑇 𝑛 and reflection 𝑅 𝑛 associatedwith a given incoming mode 𝑛 on the left read as follows, 𝑇 𝑛 = ∑︀ 𝑀𝑚 =1 | 𝑡 𝑚𝑛 | and 𝑅 𝑛 = ∑︀ 𝑁𝑚 =1 | 𝑟 𝑚𝑛 | (equivalentrelations also hold for the quantities 𝑇 ′ 𝑛 , 𝑅 ′ 𝑛 , 𝑇 ′ , 𝑅 ′ withincoming modes from the right). We choose here theconvention to label the incoming (outgoing) mode withthe second (first) subindex in the matrices such as to co-incide with the convention of matrix multiplication, i.e., c out = S c in is 𝑐 out ,𝑚 = ∑︀ 𝑛 𝑆 𝑚𝑛 𝑐 in ,𝑛 .In electronic scattering the modes cannot be addressedindividually, such that the relevant quantity in this con-text is the total transmission 𝑇 = ∑︀ 𝑁𝑛 =1 𝑇 𝑛 , correspond-ing to the transmission through all equally populated in-coming modes from the left (similarly the total reflection 𝑅 = ∑︀ 𝑁𝑛 =1 𝑅 𝑛 ). Neglecting the smearing effect of a finitetemperature and counting each of the spin polarizations separately, this total transmission 𝑇 can be directly re-lated to the electronic conductance 𝐺 = (2 𝑒 /ℎ ) 𝑇 = (2 𝑒 /ℎ ) ∑︁ 𝑚,𝑛 | 𝑡 𝑚𝑛 | , (16)a quantity which is directly measurable in the experi-ment. The above connection between the conductanceand the transmission was first derived in (Economou andSoukoulis, 1981; Fisher and Lee, 1981), and is commonlyknown as the “Landauer formula” (Landauer, 1957). Itsgeneralization to multi-terminal systems (B¨uttiker, 1986)is referred to as the “Landauer-B¨uttiker formalism”.Since in a scattering process without gain and loss thecombined value of the transmission and reflection for eachmode must be one, we can write 𝑇 𝑛 + 𝑅 𝑛 = 1 and 𝑇 ′ 𝑛 + 𝑅 ′ 𝑛 = 1 or, more generally, 𝑇 + 𝑅 = 𝑁 and 𝑇 ′ + 𝑅 ′ = 𝑀 .These relations, together with the flux-normalization ofmodes in Eq. (13), demonstrate the conservation of fluxin systems without sources or sinks. In other words, theincoming flux in a scattering process, ∑︀ 𝑛 | 𝑐 in ,𝑛 | = | c in | ,must be equal to the outgoing flux ∑︀ 𝑛 | 𝑐 out ,𝑛 | = | c out | ,such that, c † out c out = c † in c in → c † in ( S † S − ) c in = 0 , (17)which relation can only be fulfilled if the scattering ma-trix is unitary, S † S = . Inserting the block-matrix formof the scattering matrix, Eq. (15), into this unitarity con-dition, we arrive at the corresponding relations which thetransmission and reflection matrices have to satisfy: t † t + r † r = t ′† t ′ + r ′† r ′ = , r † t ′ + t † r ′ = t ′† r + r ′† t = (18)as well as tt † + r ′ r ′† = t ′ t ′† + rr † = , rt † + t ′ r ′† = tr † + r ′ t ′† = (19)following from the alternative formulation of the unitar-ity condition, SS † = .The above Hermitian matrices t † t , r † r , tt † , rr † andtheir primed counterparts play an important role in thetheoretical description of multi-mode scattering prob-lems. This is because they can be used to convenientlyexpress several of the scattering quantities of interest.Consider, e.g., the total transmission 𝑇 and reflection 𝑅 for all incoming modes from the left (see above),which can also be written as 𝑇 = Tr( t † t ) = ∑︀ 𝑛 𝜏 𝑛 and 𝑅 = Tr( r † r ) = ∑︀ 𝑛 𝜌 𝑛 , where 𝜏 𝑛 and 𝜌 𝑛 are the realeigenvalues of t † t and r † r , respectively. Also the trans-mission 𝑇 𝑛 or reflection 𝑅 𝑛 of a given mode 𝑛 on the leftcan be expressed as follows: 𝑇 𝑛 = ( t † t ) 𝑛𝑛 , 𝑅 𝑛 = ( r † r ) 𝑛𝑛 .Note that the transmission of a lead mode 𝑇 𝑛 is dif-ferent from that of a “transmission eigenchannel” 𝜏 𝑛 ,only their sum is the same 𝑇 = ∑︀ 𝑛 𝑇 𝑛 = ∑︀ 𝑛 𝜏 𝑛 , asrequired by the invariance of the trace. From the re-quirement that t † t + r † r = we can further deduce thatthe quadratic matrices t † t and r † r are simultaneouslydiagonalizable and that their eigenvalues are related asfollows: 𝜏 𝑛 = 1 − 𝜌 𝑛 .To better understand the relation between the matri-ces t † t and tt † we can write the non-Hermitian and notnecessarily quadratic transmission matrix t in its singu-lar value decomposition, t = U Σ V † , where the unitarymatrices U (of size 𝑀 × 𝑀 ) and V (of size 𝑁 × 𝑁 ) con-tain, as their columns, the left and right singular vectorsof t , respectively. The matrix Σ in the center containsthe real and non-negative singular values 𝜎 𝑖 on its diag-onal. These quantities are now elegantly connected tothe quadratic and Hermitian matrices t † t and tt † : Theleft singular vectors of t (contained in U ) are the or-thogonal eigenvectors of tt † and the right singular vec-tors of t (contained in V ) are the orthogonal eigenvectorsof t † t . The non-zero singular values 𝜎 𝑖 of t (containedin Σ ) are the square roots of the non-zero eigenvaluesof t † t and tt † , i.e., 𝜎 𝑖 = √ 𝜏 𝑖 (if these two matricesare different in size, 𝑀 ̸ = 𝑁 , the larger matrix has atleast | 𝑀 − 𝑁 | zero eigenvalues). With the help of theseidentities, we can write t † t = V Σ V † = V 𝜏 V † and tt † = U Σ U † = U 𝜏 U † , where the diagonal matrix 𝜏 contains the transmission eigenvalues from above on itsdiagonal 𝜏 = diag ( 𝜏 , . . . , 𝜏 𝑀 ) . From these identitieswe can conclude that t † t and tt † share the same eigen-values (except for | 𝑀 − 𝑁 | zero eigenvalues) and due tothe identities Eq. (18) and Eq. (19) these eigenvalues arealso the same as those of the matrices t ′† t ′ , t ′ t ′† , − r † r , − rr † , and − r ′† r ′ , − r ′ r ′† .Additional, so-called reciprocity relations (also calledOnsager relations) can be obtained for the scattering ma-trix (Jalas et al. , 2013). In terms of the transmission andreflection matrix elements from above, reciprocity trans-lates into an identity between the amplitude for scatter-ing from mode 𝑚 to another mode 𝑛 and the amplitudefor the reverse process (i.e., from mode 𝑛 to mode 𝑚 ): 𝑟 𝑛𝑚 = 𝑟 𝑚𝑛 , 𝑟 ′ 𝑛𝑚 = 𝑟 ′ 𝑚𝑛 , 𝑡 𝑛𝑚 = 𝑡 ′ 𝑚𝑛 , corresponding toa transposition-symmetric scattering matrix, S = S T .Similar reciprocity relations can also be derived for gen-eralized transmission and reflection coefficients (Nieto-Vesperinas and Wolf, 1986). Generally speaking, theserelations tell us that if one can scatter from a mode 𝑚 to another mode 𝑛 then the reverse process also hap-pens with the same amplitude. One may be temptedto associate this property with time-reversal symmetry,which is, however, misleading. Time-reversal symmetryimplies reciprocity, but not the other way around. Thebest example to illustrate this fact is a medium with ab-sorption, for which time-reversal symmetry is obviouslybroken but the above reciprocity relations may still hold(van Tiggelen and Maynard, 1997). Breaking the reci-procity of a medium typically requires a time-dependentdielectric function, non-linear effects or a magnetic field(Jalas et al. , 2013).Our choice to evaluate the scattering matrix S in thelead-mode basis 𝜒 𝑛 is arbitrary and other basis sets can be more useful for addressing particular problems. Anatural basis, in which the scattering matrix is diagonalis, of course, its eigenbasis, S = Ω diag ( 𝑒 𝑖𝜑 , . . . , 𝑒 𝑖𝜑 𝑁 ) Ω † . (20)Being a unitary matrix, the eigenvalues of S lie on theunit circle in the complex plane and can be parametrizedas above, using the so-called scattering phase shifts 𝜑 𝑛 .The transformation to the eigenbasis is mediated by theunitary matrix Ω which contains the eigenvectors of S . Inthe presence of time-reversal symmetry where the scat-tering matrices are transposition-symmetric, Ω can bechosen real and is then an orthogonal matrix Ω T Ω = .The above parametrization of the scattering matrix hasthe disadvantage that the modes on the left and right ofthe sample are strongly mixed as eigenvectors of S typi-cally feature components from all modes, irrespective oftheir asymptotic behavior. An alternative parametriza-tion which disentangles the modes on the left and rightside was proposed in (Martin and Landauer, 1992; Mello et al. , 1988). This so-called “polar decomposition” isbased on the singular value decomposition of the scatter-ing matrix blocks discussed above and reads as follows S = (︂ V 00 U )︂ (︂ −√ − 𝜏 √ 𝜏 √ 𝜏 √ − 𝜏 )︂ (︂ V ′
00 U ′ )︂ . (21)In the general case the primed matrices satisfy U ′ = U † , V ′ = V † and in the presence of time-reversal symme-try one has U ′ = U T , V ′ = V T . The above transforma-tion from the lead modes to the transmission eigenchan-nels of t † t and tt † has the advantage that the scatteringamplitudes on either side of the medium stay well sep-arated but their interrelation becomes maximally trans-parent.The scattering matrix S which relates the flux-amplitudes of incoming to outgoing modes at a fixedscattering frequency 𝜔 shares a very close relationship(Fisher-Lee relation) with the corresponding retardedGreen’s function 𝐺 + at the same value of 𝜔 (Fisher andLee, 1981) (see (Datta, 1997; Ferry and Goodnick, 1997)for a review). This close connection is well exemplified byconsidering the scattering matrix elements correspondingto incoming modes from the left, 𝑡 𝑛𝑚 ( 𝜔 ) = − 𝑖 √︀ 𝑘 𝑥𝑚 𝑘 𝑥𝑛 × (22) × ∫︁ 𝐷 𝑑𝑦 𝑙 ∫︁ 𝐷 𝑑𝑦 𝑟 𝜒 𝑚 ( 𝑦 𝑙 ) 𝐺 + ( 𝑦 𝑙 , 𝑦 𝑟 , 𝜔 ) 𝜒 𝑛 ( 𝑦 𝑟 ) 𝑟 𝑛𝑚 ( 𝜔 ) = 𝛿 𝑛𝑚 − 𝑖 √︀ 𝑘 𝑥𝑚 𝑘 𝑥𝑛 × (23) × ∫︁ 𝐷 𝑑𝑦 ′ 𝑙 ∫︁ 𝐷 𝑑𝑦 𝑙 𝜒 𝑚 ( 𝑦 ′ 𝑙 ) 𝐺 + ( 𝑦 ′ 𝑙 , 𝑦 𝑙 , 𝜔 ) 𝜒 𝑛 ( 𝑦 𝑙 )where the appearing integrals are evaluated along atransverse section in the left ( 𝑦 𝑙 ) and in the right ( 𝑦 𝑟 )lead. The corresponding relations for incoming modesfrom the right lead is fully equivalent. The flux normal-ization factors √︀ 𝑘 𝑥𝑛 , which are necessary to convert fieldamplitudes into flux amplitudes, correspond to the direc-tion cosines in Fresnel-Kirchhoff diffraction theory (Bornand Wolf, 1999).Based on the above, it is interesting to take note of thedifference in the information content between the Green’sfunction and the scattering matrix: Whereas the Green’sfunction along the considered sections contains an infi-nite set of propagation amplitudes from any point on thetransverse section to any other point, the scattering ma-trix relates only the finite and discrete set of flux-carryingmodes to one another. The reduced information contentin the scattering matrix is due to the neglect of evanes-cent modes which carry no flux and thus also do notcontribute to transport. Note, however, that evanescentmodes play a crucial role in the near field of the scat-tering region where they need to be taken into accountif the field distribution close to the scattering region isof interest. Correspondingly, extended definitions of thescattering matrix as well as of their unitarity and reci-procity relations that also include evanescent modes havebeen put forward in (Carminati et al. , 2000).Since Eq. (22) and Eq. (23) provide a relation betweenthe scattering matrix S and the scalar Green’s function 𝐺 + ( r , r ′ , 𝜔 ) and the latter is, in turn, related to theHelmholtz operator ℒ defined in Eq. (5), there shouldalso exist a direct link between S and ℒ . To uncoverthis relation one first subdivides space into an interior(scattering) region 𝒬 , where 𝜀 ( r ) may vary in space, andan exterior (asymptotic) region 𝒫 with a constant 𝜀 ( r ),where the scattering matrix S is evaluated. Following theso-called Feshbach projection operator technique (Fesh-bach, 1958, 1962) (see (Zaitsev and Deych, 2010) for areview), such a sub-division is carried out with corre-sponding projection operators, 𝒬 = ∫︁ r ∈𝒬 | r ⟩⟨ r | , 𝒫 = ∫︁ r ∈𝒫 | r ⟩⟨ r | , (24)which project onto the corresponding regions and satisfy[ 𝒫 , 𝒬 ] = 0 , 𝒫 + 𝒬 = . With these operators, we canwrite Eq. (5) in the equivalent form as, (︂ ℒ 𝒬𝒬 ℒ 𝒬𝒫 ℒ 𝒫𝒬 ℒ 𝒫𝒫 )︂ (︂ 𝜇 𝜔 𝜈 𝜔 )︂ = 𝜔 𝑐 (︂ 𝜇 𝜔 𝜈 𝜔 )︂ , (25)where the Hermitian diagonal matrix blocks ℒ 𝒬𝒬 = ℒ †𝒬𝒬 , ℒ 𝒫𝒫 = ℒ †𝒫𝒫 are the projections of ℒ into thescattering and asymptotic region, respectively, and thenon-Hermitian off-diagonal blocks ℒ †𝒬𝒫 = ℒ 𝒫𝒬 are thecoupling operators between these two regions. The re-duced Hermitian operators are now used to define eigen-value problems in the spaces 𝒬 and 𝒫 , ℒ 𝒬𝒬 𝜇 𝑚 = 𝜔 𝑚 𝑐 𝜇 𝑚 , and ℒ 𝒫𝒫 𝜈 𝑛,𝜔 = 𝜔 𝑐 𝜈 𝑛,𝜔 . (26) Due to the confinement of states 𝜇 𝑚 in 𝒬 the correspond-ing eigenvalues 𝜔 𝑚 are discrete, whereas the eigenvalues 𝜔 in the unconfined asymptotic regions are continuous ( 𝑛 is just a channel index in this case). In both regions 𝒫 , 𝒬 the eigenfunctions form a complete set and can thus beused to expand modes of arbitrary complexity in the re-spective subspaces. The interface between 𝒫 and 𝒬 canbe chosen anywhere in the asymptotic region and alsothe boundary conditions on 𝒬 are arbitrary but shouldbe such that the operator ℒ 𝒬𝒬 is Hermitian (with Dirich-let or Neumann boundary conditions being the standardchoices). If needed, the boundary between 𝒫 and 𝒬 canalso be placed in the direct vicinity of the scattering re-gion, in which case the coupling to evanescent modesneeds to be properly taken into account (Viviescas andHackenbroich, 2003).To make the connection with the scattering matrix S we place the boundary between 𝒫 and 𝒬 outsidethe sections where the scattering matrix is being eval-uated in Eq. (23). It can be shown in this case (Ma-haux and Weidenm¨uller, 1969) (see (Datta, 1997; Guhr et al. , 1998; Rotter, 2009) for a review) that the retardedGreen’s function 𝐺 + ( r , r ′ , 𝜔 ) appearing in Eq. (23) with r , r ′ ∈ 𝒬 can then again be written as a resolvent, G 𝒬𝒬 = [ 𝜔 − ℒ eff ] − , with the help of an effective non-Hermitian operator ℒ eff = ℒ 𝒬𝒬 + Σ ( 𝜔 ). The so-called self-energy Σ ( 𝜔 ) can be written as follows Σ ( 𝜔 ) = ℒ 𝒬𝒫 ( 𝜔 ) 1 𝜔 − ℒ 𝒫𝒫 ( 𝜔 ) + 𝑖𝜀 ℒ 𝒫𝒬 ( 𝜔 ) , (27)with 𝜀 being here an infinitesimal positive number.Through this self-energy the Green’s function G 𝒬𝒬 =[ 𝜔 − ℒ 𝒬𝒬 ] − of the closed region 𝒬 (the superscript0 denotes the absence of coupling to 𝒫 ) turns intothe Green’s function of the corresponding open system G 𝒬𝒬 = [ 𝜔 − ℒ 𝒬𝒬 + Σ ( 𝜔 )] − (where the coupling to 𝒫 is included). Due to its restriction to the interfacebetween 𝒫 and 𝒬 the self-energy is nothing else than anon-Hermitian boundary condition which parametricallydepends on the real scattering frequency 𝜔 in the outsidedomain 𝒫 . This boundary condition is known under thename of Kapur-Peierls or constant-flux boundary condi-tion (Kapur and Peierls, 1938; T¨ureci et al. , 2006) withthe latter terminology being motivated by the fact thatthe outgoing flux in 𝒫 is conserved (corresponding toa real value of 𝜔 ). In the context of quantum scatter-ing, the operator ℒ eff is also known as the “effective” ornon-Hermitian Hamiltonian. Being non-Hermitian, theoperator ℒ eff has complex eigenvalues, corresponding toeigenstates which decay through the system boundariesand thus have only a finite lifetime.To establish the link between ℒ eff and the scatteringmatrix S we need to express the above operators in thebasis vectors of the regions 𝒫 and 𝒬 , respectively. Forthe closed system operator ℒ 𝒬𝒬 the matrix elementsare 𝐻 𝜆𝑚 = ⟨ 𝜇 𝜆 |ℒ 𝒬𝒬 | 𝜇 𝑚 ⟩ = 𝜔 𝑚 𝛿 𝜆𝑚 and for the self-0energy we get Σ 𝜆𝑚 ( 𝜔 ) = − 𝜆𝑚 ( 𝜔 ) − 𝑖𝜋 ( WW † ) 𝜆𝑚 ( 𝜔 ),where the real matrix elements Δ 𝜆𝑚 contain frequencyshifts and the Hermitian matrix WW † contains dampingterms resulting from the coupling between the boundedregion 𝒬 with the continuum region 𝒫 , i.e., 𝑊 𝜆𝑚 ( 𝜔 ) = ⟨ 𝜇 𝜆 |ℒ 𝒬𝒫 | 𝜈 𝑚 ( 𝜔 ) ⟩ √︀ 𝑘 𝑥𝑚 . These matrix elements whichcan be calculated analytically for simple systems (Vivi-escas and Hackenbroich, 2003) or numerically for com-plex geometries (Sadreev and Rotter, 2003; St¨ockmann et al. , 2002), allow us to write the effective operator ℒ eff = ℒ 𝒬𝒬 − Δ − 𝑖𝜋 𝑊 𝑊 † and with it the scat-tering matrix in mode representation (Mahaux and Wei-denm¨uller, 1969) 𝑆 𝑚𝑛 ( 𝜔 ) = 𝛿 𝑚𝑛 − 𝑖 [︂ 𝑊 † ( 𝜔 ) 1 𝜔 − ℒ eff ( 𝜔 ) 𝑊 ( 𝜔 ) ]︂ 𝑚𝑛 , (28)which represents the desired relation between the scat-tering matrix and the differential operator ℒ introducedat the beginning of this section, see Eq. (5).An interesting correspondence that can be establishedbased on Eq. (28) is that between the poles of the Green’sfunction G 𝒬𝒬 and the resonances in the transmissionand reflection amplitudes in the scattering matrix S ( 𝜔 ).The complex frequency values 𝜔 at which these polesare located are implicitly defined through the eigenval-ues Ω 𝑘 ( 𝜔 ) of ℒ eff ( 𝜔 ), which have to satisfy the relation 𝜔 − Ω 𝑘 ( 𝜔 ) = 0. To find the solutions of this equationone can iteratively track the values of 𝜔 from the real axisto the desired fixed point for each specific eigenvalue Ω 𝑘 .The complex resonance values found in this way play animportant role for scattering problems, as their real partsspecify the positions of scattering resonances and theirimaginary parts fix the corresponding resonance widths,which, in turn, are inversely proportional to the decaytime of a resonant state in this open system. Due totheir finite life-time the resonances are also often referredto as quasi-bound states or quasi-modes of the systemand starting from the original work by Gamow (Gamow,1928) many theoretical studies are based on these states(Moiseyev, 2011). In contrast to the constant-flux statesmentioned above, the quasi-bound states do, however,have the problem that they diverge to infinity outside ofthe system boundaries, which requires much care whenusing them to expand a field in this set of states (Ching et al. , 1998). On the other hand, quasi-bound states donot feature a parametric dependence on the frequencyoutside the system (as the constant-flux states do), sincefor the quasi-bound states both of the involved frequen-cies are equal: 𝜔 𝑘 = Ω 𝑘 .
5. Random Matrix Theory (RMT)
A very convenient tool to understand the statisticalrather than the system-specific properties of scattering processes in disordered media is Random Matrix The-ory (RMT). The basic assumption of RMT is that thestatistical properties of a sufficiently chaotic or disor-dered system are the same as those of those of a suitablychosen ensemble of random matrices. This idea, whichwas originally introduced by Wigner to model the dis-tribution of energy spacings in nuclei (Wigner, 1955a),has meanwhile found a broad range of applications, notonly in nuclear physics (see (Mitchell et al. , 2010; Wei-denm¨uller and Mitchell, 2009) for a review), but alsoin mesoscopic physics (see (Beenakker, 1997) for a re-view) and increasingly so in disordered photonics (see(Beenakker, 2011) for a review). The broad applicabilityof RMT (see (St¨ockmann, 2006) for a review) is stronglylinked to the so-called Bohigas-Giannoni-Schmitt (BGS)conjecture (Bohigas et al. , 1984) according to whichRMT describes well the spectral statistics of any “quan-tum” or “wave” system (governed by a wave equation)whose “classical” counterpart (governed by a correspond-ing Hamiltonian equation of motion) is chaotic . Clas-sically, such chaotic systems are characterized by hav-ing more degrees of freedom than constants of motion.Quantum mechanically this translates into having moredegrees of freedom than “good” quantum numbers. Find-ing a proof for the BGS conjecture has turned out to bevery difficult (proofs in certain limits have meanwhilebeen proposed (M¨uller et al. , 2004)). Extensive theoret-ical and experimental work (see (Beenakker, 1997) fora review) has, however, shown that the BGS conjectureis very well satisfied in many different physical scenar-ios not only for mesoscopic quantum systems and thecorresponding matter waves, but for many other typesof waves as well (like optical, acoustic and micro-wavesetc.) (Dietz and Richter, 2015; Ellegaard et al. , 1995;Gr¨af et al. , 1992; Guhr et al. , 1998; St¨ockmann, 2006).Furthermore, based on the analogies (see section II.A.1)between quantum systems (described by a Schr¨odingerequation) and optical scattering systems (described by aHelmholtz equation) many of the results that have beenexplored in the field of mesoscopic physics can now becarried over to the domain of optical scattering. Beforedemonstrating this explicitly by means of concrete ex-amples (see section III), we will first review the basictheoretical concepts of RMT.Our starting point for applying RMT to the systemsconsidered in this review will be the approach by Wignerand Dyson (Wigner, 1955a), which consists in replac-ing the matrix representation of the differential opera-tor ℒ 𝒬𝒬 in Eqs. (25,26) for a specific closed system 𝒬 by a random matrix H . The latter contains as each ofits elements 𝐻 𝑚𝑛 a randomly generated number froman ensemble with the following Gaussian distribution 𝑃 ( 𝐻 𝑚𝑛 ) = ( 𝑤 √ 𝜋 ) − exp[ − 𝐻 𝑚𝑛 / (2 𝑤 )] and zero aver-age ⟨ 𝐻 𝑚𝑛 ⟩ = 0 (the value of 𝑤 determines the meanlevel spacing of the corresponding eigenvalues). Sincethe matrix elements 𝐻 𝑚𝑛 can, in general, be complex, we1may choose both the real and imaginary parts from thisGaussian ensemble independently. The only additionalconstraint that is imposed on the matrix elements 𝐻 𝑚𝑛 is that they are those of a Hermitian matrix H † = H ,i.e., 𝐻 * 𝑚𝑛 = 𝐻 𝑛𝑚 . For a time-reversal-symmetric sys-tem, the matrix elements are real and symmetric, i.e., 𝐻 𝑚𝑛 = 𝐻 𝑛𝑚 ∈ R . Having replaced the differential op-erator ℒ 𝒬𝒬 by a random matrix H , all system-specificinformation about 𝒬 is lost and only statements on thestatistical properties of a whole class of systems can bemade that can be associated with the same Gaussian en-semble of matrices (Porter, 1965) (see (Mehta, 2004) fora review). For the Hermitian matrices with complex,Gaussian-distributed elements this ensemble is called theGaussian Unitary Ensemble (GUE). The term unitary refers here to the unitary matrices containing the eigen-vectors of these Hermitian matrices. Similarly, the sym-metric matrices with real, Gaussian-distributed elementsare referred to as the Gaussian Orthogonal Ensemble(GOE), where orthogonal refers to the corresponding or-thogonal eigenvector matrix. It can now be shown thateach of these matrix ensembles has a very specific distri-bution of eigenvalues. In particular, when we take Gaus-sian random matrices of very large size ( 𝑁 → ∞ ) andcompute the set of eigenvalues 𝐸 𝛼 , then their distribu-tion will be universal in the average over many matrixrealizations. The corresponding distribution function forthe eigenvalues, 𝑃 ( { 𝐸 𝑛 } ) ≈ 𝑐𝑜𝑛𝑠𝑡. × 𝑁 ∏︁ 𝑚<𝑛 | 𝐸 𝑛 − 𝐸 𝑚 | 𝛽 × 𝑁 ∏︁ 𝑛 exp[ − 𝐸 𝑛 / (2 𝑤 )](29)is known as the Wigner-Dyson distribution, followingthe original work by these two authors (Dyson, 1962a;Wigner, 1957, 1967). The parameter 𝛽 here is assignedthe value 𝛽 = 1 for GOE and 𝛽 = 2 for GUE. An interest-ing result contained in this distribution is the repulsionof nearby levels, i.e., 𝑃 ( 𝛿 ) → 𝛿 →
0. More specif-ically, when considering the normalized spacing betweennearest eigenvalues 𝛿 = ( 𝐸 𝛼 +1 − 𝐸 𝛼 ) / Δ with the meanlevel spacing Δ = ⟨ 𝐸 𝛼 +1 − 𝐸 𝛼 ⟩ then one finds that thelevel repulsion scales like 𝑃 ( 𝛿 ≪ ∝ 𝛿 𝛽 . Comparingthis result with experimental data both for time-reversalinvariant systems with 𝛽 = 1 (see Fig. 3(a)) as well asfor systems without time-reversal symmetry and 𝛽 = 2(see Fig. 3(b)), shows good agreement. Note that theresults shown in Fig. 3 stem from very different physicalsystems, like an atomic nucleus in (a) and a microwavebilliard with an attached isolator in (b).To investigate how these results for bounded systemscarry over to the case of unbounded scattering systems,we follow the so-called “Heidelberg approach” (Mahauxand Weidenm¨uller, 1969) (see (Guhr et al. , 1998) for a re-view). Here, the random Hamiltonian matrix H describ-ing the bounded region ( 𝒬 ) is coupled to the unboundedoutside domain ( 𝒫 ) by way of the frequency-dependent (b) (a) δ P ( δ ) P ( δ ) δ Poisson GOE GUE (adapted) Figure 3 (a) Comparison of the nearest neighbor level spac-ing distribution 𝑃 ( 𝛿 ) in a set of nuclear scattering resonances(histogram) with the corresponding RMT prediction from theGaussian Orthogonal Ensemble (GOE) for time-reversal in-variant systems. The very good agreement found confirmsthat the statistical property of nuclei can be approachedthrough RMT. For comparison also the prediction from aPoisson distribution is shown, corresponding to uncorrelatedlevels. Image adapted from (Bohigas et al. , 1983). (b) Levelspacing distribution in a quasi two-dimensional microwave bil-liard with an isolator attached. Here the data is well describedby the Gaussian Unitary Ensemble (GUE). The solid line isadapted to account for missing levels and the inset shows thequadratic increase of the level repulsion for small values ascharacteristic for GUE. Image adapted from (Stoffregen et al. ,1995). coupling matrices 𝑊 𝜆,𝑚 ( 𝜔 ) introduced in section II.A.4.Using Eq. (28) then yields the corresponding scatteringmatrix for transmission and reflection through the re-gion described by the Hamiltonian H . Note that in thisapproach no approximation is introduced by the subdi-vision of space into 𝒫 and 𝒬 . When one is interested inthe statistical properties of the scattering matrix, a lessrigorous calculation is usually sufficient. A common ap-proximation is, e.g., to neglect the frequency-dependenceof the coupling matrix elements 𝑊 𝜆,𝑚 ( 𝜔 ), which are thendrawn from an ensemble of random numbers, just like thematrix elements of H .In this approach, which has found interesting appli-cations in nuclear scattering (Iida et al. , 1990a,b; Ver-baarschot et al. , 1985), the only frequency dependence inthe scattering matrix comes from the 𝜔 -term in Eq. (28).This explicit frequency dependence is essential as it canbe used to study frequency correlations in the scatter-ing matrix elements (Guhr et al. , 1998). Consider here,in particular, that the bound eigenstates of the Hamil-tonian H are coupled by the matrix elements 𝑊 𝜆,𝑚 ( 𝜔 )to the waveguide modes, which turns these states intoquasi-bound resonances (as discussed in the last para-graph of section II.A.4). Depending on whether the cou-pling strength (as determined by the matrix elements 𝑊 𝜆,𝑚 ( 𝜔 )) is smaller or larger than the mean level spac-ing of the Hamiltonian eigenstates (as determined bythe variance of the matrix elements 𝐻 𝑚𝑛 ) these reso-nances will be isolated (weak coupling) or overlapping(strong coupling), resulting in very different frequencycorrelations in the scattering matrix elements (see also(Brouwer, 1995) for more details on the situation with2non-ideal waveguide coupling).A different strategy to set up a random matrix the-ory for coherent scattering, also known as the “Mex-ico approach”, starts not with the Hamiltonian H , butwith the scattering matrix S as the fundamental quantity(Baranger and Mello, 1994; Friedman and Mello, 1985a,b;Jalabert et al. , 1994; Mello et al. , 1985). In this approach(see (Alhassid, 2000) for a review), which was developedindependently of the “Heidelberg approach”, one replacesthe scattering matrix elements by random complex num-bers. In analogy to the random Hamiltonian matrix el-ements from above, which had to be chosen such as torespect the Hermiticity of the Hamiltonian, the randomscattering matrix elements have to respect the unitar-ity of the scattering matrix. In the case of time-reversalsymmetry, the scattering matrix additionally has to besymmetric (see discussion in section II.A.4). The corre-sponding matrix ensembles are referred to as Dyson’s cir-cular ensemble (Dyson, 1962b) with the parameter 𝛽 = 1assigned to unitary symmetric and 𝛽 = 2 for generalunitary matrices. Assuming such a distribution leads tovery specific correlations in the scattering phase shifts inEq. (20), 𝑃 ( { 𝜑 𝑛 } ) ∝ ∏︁ 𝑛<𝑚 | exp( 𝑖𝜑 𝑛 ) − exp( 𝑖𝜑 𝑚 ) | 𝛽 , (30)which were found by Bl¨umel and Smilansky (Bl¨umel andSmilansky, 1990) to describe the phase shifts in chaoticscattering very well. To link the circular ensemble to ex-perimentally more accessible quantities like the statisticsof transmission and reflection, the corresponding distri-bution of the transmission eigenvalues 𝜏 𝑛 of the matrices t † t and tt † needs to be evaluated (Baranger and Mello,1994; Jalabert et al. , 1994). The corresponding jointprobability density of transmission eigenvalues is givenas follows, 𝑃 ( { 𝜏 𝑛 } ) ∝ ∏︁ 𝑛<𝑚 | 𝜏 𝑛 − 𝜏 𝑚 | 𝛽 × ∏︁ 𝑝 𝜏 − 𝛽/ 𝑝 . (31)The product between neighboring transmission eigenval-ues leads to a level repulsion similar to the repulsion ofenergy eigenvalues of the random Hamiltonian H . Note,however, that the above Eq. (31) also applies for the caseof just a few scattering channels 𝑁 down to the singlechannel case where 𝑁 = 1.The above result already contains very interestingphysics. Consider, e.g., the limiting case of the abovedistribution for a very large number of scattering chan-nels ( 𝑁 → ∞ ), broken down to the one-point probabil-ity density of transmission eigenvalues 𝑃 ( 𝜏 ). The latteris given as the mean value of the microscopic density 𝜌 ( 𝜏 ) = ∑︀ 𝑁𝑛 𝛿 ( 𝜏 − 𝜏 𝑛 ) with respect to the ensemble aver-age according to the probability distribution from above, 𝑃 ( 𝜏 ) ≡ ⟨ 𝜌 ( 𝜏 ) ⟩ = ∫︁ 𝑑𝜏 . . . ∫︁ 𝑑𝜏 𝑁 𝑃 ( { 𝜏 𝑛 } ) 𝜌 ( 𝜏 ) , (32) τ 1 0 τ 1 0 P(τ) P(τ) chao,c diffusive (a) (b) Figure 4 (color online). Distribution of transmission eigenval-ues 𝜏 for the case (a) of chaotic scattering and (b) of diffusivescattering. The corresponding analytical expressions for thesefunctions (following from random matrix theory) are spelledout in Eq. (33) and in Eq. (40), respectively. resulting in the following bi-modal distribution for thelimit 𝑁 ≫ et al. , 1994), 𝑃 ( 𝜏 ) = 1 𝜋 √︀ 𝜏 (1 − 𝜏 ) . (33)Results for small 𝑁 or for an asymmetric number of chan-nels on the left and right are provided in (Savin and Som-mers, 2006). Note that in the case of broken time-reversalsymmetry ( 𝛽 = 2) for which Eq. (33) was derived, thedistribution 𝑃 ( 𝜏 ) is symmetric around 𝜏 = 1 /
2. Forthe time-reversal symmetric case ( 𝛽 = 1), however, thesecond product in Eq. (31) induces an asymmetry intothis distribution which leads to the following results forthe average transmission (Baranger and Mello, 1994; Jal-abert et al. , 1994) ⟨ 𝑇 ⟩ = 𝑁 (︂ − 𝛽 )︂ + 𝒪 (1 /𝑁 ) . (34)The reduction in transmission for 𝛽 = 1 is called the weaklocalization correction, which in a semi-classical picturecan be partially associated with the presence of time-reversed path pairs that enhance the reflection 𝑅 (hencethe name “localization”). The term “weak” refers hereto the fact that the correction is of order 𝒪 (1), which ismuch smaller than the leading term 𝒪 ( 𝑁 ). Experimentaldemonstrations of this effect will be discussed in sectionIII.C.The first product in Eq. (31) involving pairs of trans-mission eigenvalues suppresses the likelihood of neigh-boring transmission eigenvalues approaching each othervery closely. This eigenvalue repulsion leads also to aspacing distribution between nearest neighbor transmis-sion eigenvalues which scales like 𝑃 ( 𝛿 ≪ ∝ 𝑠 𝛽 wherethe normalized spacing is given by 𝛿 = ( 𝜏 𝑚 +1 − 𝜏 𝑚 ) / Δ,with Δ = ⟨ 𝜏 𝑚 +1 − 𝜏 𝑚 ⟩ . Note the similarity here withthe spacing distribution obtained earlier for the eigen-values of a random Hamiltonian, see Eq. (29). Due tothis “spectral rigidity” the transmission eigenvalues only3fluctuate between the limits imposed by their neighbor-ing values, which is obviously much less than for uncor-related transmission eigenvalues that would be describedby a Poisson distribution 𝑃 ( 𝛿 ≪ ∝ exp( − 𝛿/ Δ). Thissuppression of fluctuations is so strong that the varianceof the fluctuations in the total transmission (as definedbelow Eq. (15)) approach a universal, but 𝛽 -specfic value(Lee and Stone, 1985), which is of order 𝒪 (1) and thusindependent of 𝑁 , var 𝑇 = 18 𝛽 . (35)This prediction for universal conductance fluctuations ina chaotic cavity was obtained with the “Heidelberg ap-proach” for the Hamiltonian by (Iida et al. , 1990a,b; Ver-baarschot et al. , 1985) and with the “Mexico approach”for the scattering matrix by (Baranger and Mello, 1994;Jalabert et al. , 1994). An extended discussion of uni-versal conductance fluctuations can be found in sectionIII.B.Another highly non-trivial aspect of coherent chaoticscattering is contained in the shape of the distributionfunction of transmission eigenvalues (see Fig. 4a): Con-trary to what one would naively expect, the transmissioneigenvalues 𝜏 𝑛 are not uniformly distributed between thelimiting values 0 and 1; instead, Eq. (33) predicts thatthe 𝜏 𝑛 are peaked near 0 and 1, corresponding to trans-mission channels that are almost closed (near 𝜏 ≈ 𝜏 ≈
1) – a phe-nomenon also known under the name “maximal fluctu-ation theorem” (Pendry et al. , 1990). These open andclosed transmission eigenchannels (see Fig. 4) which werefirst discovered by Dorokhov (Dorokhov, 1984) will playan important role for many of the effects which we aregoing to discuss in this review. To understand the ori-gins of the bi-modal distribution consider the repulsionof transmission eigenvalues inherent in Eq. (31). Thecloser a given transmission eigenvalue is to 0 (to 1), themore it is repelled by its higher (lower) neighbors (sim-ply because there are more of them). Together with therestriction to the inverval 𝜏 ∈ [0 , 𝑃 ( 𝜏 )with the distribution which one would get for the scat-tering of classical particles rather than of waves. Sinceparticles (like billiard balls) that enter a scattering re-gion connected to a left and right port, can either befully reflected or fully transmitted, but nothing in be-tween, the corresponding classical distribution functionhas two delta peaks, 𝑃 cl ( 𝜏 ) = 𝛼 𝛿 ( 𝜏 )+(1 − 𝛼 ) 𝛿 (1 − 𝜏 ), with 𝛼 = 1 / (a) (d) (b) (c) Figure 5 (color online). Different experimental realizationsof chaotic quantum dots with figures adapted from (Marcus et al. , 1992) (a), (Chang et al. , 1994) (b), (Oberholzer et al. ,2002) (c), and (Marcus et al. , 1997) (d). In all four cases theseelectronic billiards are fabricated based on a semi-conductorheterostructure with high mobility. The current between thesource and drain enters and exits through slits (quantum pointcontacts), which are small compared to the overall dimensionof the chaotic scattering region in between. sion in the classically forbidden region around 𝜏 ≈ . Figure 6 Experimental results from (Huibers et al. , 1998) onthe distribution of the conductance 𝑃 ( 𝑔 ) in electrostaticallydefined quantum dots (see schematic on the top). Both theinput as well as the output point contacts only feature a sin-gle open transverse mode such that these distributions areequivalent to the distribution of transmission eigenvalues inthe single-channel limit. The corresponding theoretical pre-dictions from RMT (dashed lines) do not reproduce the exper-imental data (connected symbols). Only when effects due tofinite temperature ( 𝑇 ) and dephasing (through the dephasingrate 𝛾 𝜑 ) are taken into account (see solid lines) good agree-ment is found. Both the situation with and without time-reversal symmetry are considered with the latter case beingrealized through the application of a finite magnetic field ( 𝐵 )applied perpendicular to the scattering area. in Fig. 6 that mesoscopic transport experiments usu-ally suffer from several imperfections (decoherence pro-cesses, finite temperature, etc.) which severely spoil theagreement between the measurement data and an RMTprediction (even for well-engineered electron billiards asshown in Fig. 5). Figure 6 also shows that a decentagreement can be found when the influence of these realworld effects is taken into account in the correspondingRMT model. As it turns out, also the optical scatteringthrough a disordered medium (as discussed in the lattersections of this review) is not well described by the abovesimple models. The amendments to RMT that are neces-sary in these cases are, however, different from the onesemployed for electron transport through quantum dotsas in Fig. 6.
6. DMPK equation
The starting point for this section is the insight thata disordered system is inherently more complex than achaotic quantum dot and its RMT description. This isbecause a disordered medium does more than just ran-domize all incoming waves in equal measure and let-ting them escape again symmetrically on either side. Inparticular, depending on how long the incoming wavesremain inside the medium the degree of disorder scat- tering that they will suffer from will be very different.Also transmission and reflection will of course dependon the thickness of this medium as compared to thetransport mean free path ℓ ⋆ . To cope with this situa-tion the simple RMT models from section II.A.5 wereextended by concatenating many random scattering ma-trices from the appropriate RMT ensemble in series. Thisapproach has first been used in (Altland, 1991; Iida et al. ,1990a,b; Weidenm¨uller, 1990) to describe electronic scat-tering through a disordered wire and corresponds to thedescription of a system consisting of a series of chaoticcavities, see Fig. 7(a). Whereas this ansatz allows one toconveniently extend RMT to such more complicated sce-narios, the approach also has several limitations, in par-ticular, as the different transport regimes in a wire (bal-listic, diffusive, localized) and their respective crossoversare hard to treat with it (see (Dembowski et al. , 1999)for a microwave experiment on coupled cavities).To properly describe all of these regimes Dorokhov,Mello, Pereira and Kumar (DMPK) already earlier pro-posed a model (Dorokhov, 1982; Mello et al. , 1988),which subdivides the scattering region into a series ofweakly scattering segments rather than using the fullyrandomized matrices from RMT, see Fig. 7(b). Choos-ing each segment of length Δ 𝑧 shorter than the transportmean free path, Δ 𝑧 ≪ ℓ ⋆ , but longer than the wave-length, Δ 𝑧 ≫ 𝜆 , has the advantage that adding a newsegment can be described as a perturbative correction.Assuming, in addition, that in each segment all incom-ing channels are scattered by the disorder into all of theavailable channels isotropically (i.e., with equal weight),one can derive a Fokker-Planck equation for the “Brown-ian motion” of the transmission eigenvalues 𝜏 𝑛 (with con-stant diffusion coefficient). The corresponding evolutionequation for the distribution of transmission eigenvalues 𝜏 𝑛 as a function of the wire length 𝐿 is known as theDMPK equation, 𝜕𝜕𝑠 𝑃 ( { 𝑥 𝑛 } , 𝑠 ) = 12 𝛾 𝑁 ∑︁ 𝑚 =1 𝜕𝜕𝑥 𝑚 [︂ 𝜕𝑃𝜕𝑥 𝑚 + 𝛽𝑃 𝜕𝜕𝑥 𝑚 Ω( { 𝑥 𝑛 } ) ]︂ (36)where we have substituted the transmission eigenvalues 𝜏 𝑛 by new variables 𝑥 𝑛 according to 𝜏 𝑛 = 1 / cosh 𝑥 𝑛 andused 𝑠 = 𝐿/ℓ ⋆ , 𝛾 = 𝛽𝑁 + 2 − 𝛽 as well asΩ( { 𝑥 𝑛 } ) = − ∑︁ 𝑚<𝑛 ln | sinh 𝑥 𝑛 − sinh 𝑥 𝑚 |− 𝛽 ∑︁ 𝑚 ln | sinh 2 𝑥 𝑚 | . (37)The variables 𝑥 used above for simplifying the equationscan be interpreted such that 𝐿/𝑥 𝑚 is the channel-specific“localization length” for the transmission channel 𝑚 inthe disordered region (we will see in a later part of thissection what localization is).Note that in real disordered wires the isotropy assump-tion, which corresponds to an ergodicity assumption in5 (a) (b) Figure 7 (color online). (a) Stacking chaotic quantum dotsbehind each other leads to an effective wire geometry withstatistics that can be well described by a corresponding ran-dom matrix approach. (b) In the more refined DMPK ap-proach weakly scattering segments are recursively added tothe wire geometry. the transverse direction, is in general not well fulfilledfor short lengths 𝐿 . This is because any specific disor-der profile typically features a non-uniform differentialscattering cross-section that leads to preferential cou-pling between specific mode pairs. Since the length scalefor transverse diffusion is not taken into account in theDMPK equation, its validity is restricted to “quasi-one-dimensional” (quasi-1D) wire geometries which are muchlonger than their transverse width, 𝐿 ≫ 𝑊 . For verylong systems (with 𝐿 ≫ ℓ ⋆ ) the solutions to the DMPKequation have equivalent statistics as those of the con-catenated random scattering matrix model from above.In a similar spirit, one can also set up an alternativemodel where not the scattering matrix but the Hamilto-nian is the central quantity of interest. As demonstratedby Efetov and Larkin (Efetov and Larkin, 1983) such anapproach can be mapped onto a so-called supersymmet-ric nonlinear 𝜎 -model, which was shown to be equiva-lent to the DMPK equation (Brouwer and Frahm, 1996)in the “thick wire limit” with many scattering channels 𝑁 . Since all of these models were extensively discussedalready in several reviews and books (Beenakker, 1997;Brouwer, 1997; Janssen, 2001), we will not review themagain here. Rather, we will present in the following asummary of the main results of the DMPK model thatwill be useful for later chapters. For this purpose we willrely on the exact solutions of the DMPK equation. As thecase of broken time-reversal symmetry ( 𝛽 = 2) is mucheasier to treat, we will discuss it first and then point outcorrections for the case when time-reversal symmetry isrestored ( 𝛽 = 1).Focusing on the case of a thick wire with many trans-verse channels ( 𝑁 ≫
1) the DMPK equation makes spe-cific predictions for three characteristic regimes:(i) In the ballistic regime the scattering is very weak,such that the system length is smaller than the mean freepath, 𝐿 (cid:46) ℓ . In this limit the waves can be thought of astravelling ballistically on straight lines, rather than beingmultiply scattered by the disorder. Correspondingly, thetransmission eigenvalues are all very close to one and no reflection occurs. In optics the ballistic regime is veryimportant since most imaging techniques only work inthe ballistic limit. Correspondingly, in systems throughwhich X-rays propagate ballistically and visible light doesnot – a situation commonly encountered in biomedicalimaging – the former type of radiation is better suitedfor imaging purposes.(ii) In the diffusive regime, where the system lengthis in between the mean free path and the localizationlength, ℓ ⋆ (cid:46) 𝐿 (cid:46) 𝜉 (with 𝜉 ≈ 𝛽𝑁 ℓ ⋆ in quasi-1D sys-tems), the transmitted waves have already undergonemany scattering events. This translates into a random-ization of the transmission eigenvalues 𝜏 𝑛 , which is quan-tified by solving the DMPK equation starting with a“ballistic initial condition” 𝜏 𝑛 = 1 for all 𝑛 imposed at 𝑠 = 𝐿/ℓ ⋆ = 0 up to the length 𝐿 at which 𝑠 = 𝐿/ℓ ⋆ ≫ 𝑃 ( { 𝑥 𝑛 } , 𝑠 ) ∝ ∏︁ 𝑖<𝑗 [︀(︀ sinh 𝑥 𝑗 − sinh 𝑥 𝑖 )︀ ( 𝑥 𝑗 − 𝑥 𝑖 ) ]︀ × ∏︁ 𝑖 [︁ exp( − 𝑥 𝑖 𝑁/𝑠 )( 𝑥 𝑖 sinh 2 𝑥 𝑖 ) / ]︁ . (38)Distilling out of this result the one-point probability den-sity of transmission eigenvalues (by integration) one findsthat in the regime of very long systems 𝑠 ≫ 𝒪 ( 𝑁 ) (which is independent of 𝛽 ) featuresa uniform distribution of the transformed transmissioneigenvalues 𝑥 𝑛 , 𝑃 ( 𝑥, 𝑠 ) ≡ ⟨ 𝜌 ( 𝑥 ) ⟩ 𝑠 = 𝑁𝑠 Θ( 𝑠 − 𝑥 ) . (39)Note that the only length dependence that remains hereis the upper cut-off of this uniform distribution intro-duced by the Heaviside-Theta function Θ. This cut-offsets the transmission of all modes to zero for which 𝑥 (cid:38) 𝐿/ℓ ⋆ and it keeps the normalization at ∫︀ ∞ 𝑃 ( 𝑥, 𝑠 ) 𝑑𝑥 = 𝑁 . If we translate this result back to the transmissioneigenvalues 𝜏 𝑛 we find that out of the 𝑁 transmissionchannels about 𝑁 ℓ ⋆ /𝐿 have a finite transmission with 𝜏 > − 𝐿/ℓ ⋆ ), which again follow a bi-modal butasymmetric distribution, see Fig. 4(b) (Dorokhov, 1984;Imry, 1986; Pendry et al. , 1992), 𝑃 ( 𝜏 ) = 𝑁 ℓ ⋆ 𝐿 𝜏 √ − 𝜏 . (40)The remaining 𝑁 (1 − ℓ ⋆ /𝐿 ) channels are closed (i.e., 𝜏 ≈ ∫︀ 𝜏 𝑃 ( 𝜏 ) 𝑑𝜏 = 𝑁 with the lower integration limit 𝜏 = 4 exp( − 𝐿/ℓ ⋆ ). In practice the closed transmissioneigenvalues are smeared over several eigenvalue spacings;since, however, they only contribute weakly to transportthis cut-off is usually not specified in more detail. Most6importantly, we have thus obtained the spectacular re-sult that even in transmission through a highly scatteringquasi-1D system open transmission channels with 𝜏 ≈ 𝐿 ≪ 𝑊 (Goetschy and Stone, 2013).The above results obtained for 𝛽 = 2 are subject tocorrections of next to leading order 𝒪 (1) for the time-reversal symmetric case of 𝛽 = 1. These corrections weremeasured in the experiment (Mailly and Sanquer, 1992)and also appear in the average transmission and the vari-ance of the fluctuations of the transmission as induced bychanging the disorder configuration or an external pa-rameter (like the scattering wavenumber 𝑘 ), ⟨ 𝑇 ⟩ = 𝑁 𝑙𝐿 + 𝛽 − 𝛽 + 𝒪 (1 /𝑁 ) , var 𝑇 = 215 𝛽 + 𝒪 (1 /𝑁 )(41)Note the interesting analogy of the leading order termin transmission ∝ /𝐿 to the Ohmic behavior of a clas-sical wire whose resistance ( ∝ /𝑇 ) scales linearly withthe length 𝐿 . Given the fact that we have used here awave picture of transport rather than a classical trajec-tory picture (as in the Drude model), this analogy is farfrom obvious, in particular, in view of the bi-modal distri-bution of transmission eigenvalues. Another interestingobservation based on Eq. (41) is that the application ofa mechanism that breaks time-reversal symmetry (likea magnetic field for electrons) leads to a slight increaseof the average transmission (weak localization) and to atwo-fold decrease of the transmission fluctuations var 𝑇 .The latter are apparently also universal in the case ofdiffusive scattering, with an 𝑁 -independent value for theleading order term var 𝑇 = 2 / (15 𝛽 ). Note that theseresults agree exactly with those from an independentcalculation using a diagrammatic perturbation theory(Altshuler, 1985; Anderson et al. , 1979; Gorkov et al. ,1979; Lee and Stone, 1985). Extensive reviews of thediagrammatric framework can be found in (Akkermansand Montambaux, 2007; Dragoman and Dragoman, 2004;Montambaux, 2006). Diagrammatic techniques have thedownside that they do not provide access to the full dis-tribution of transmission eigenvalues, which is why wereview them only very briefly in section III.B(iii) In the localized regime we are in the situationwhere the system length is larger than the so-called “lo-calization length”, 𝐿 (cid:38) 𝜉 , which, in quasi-1D systemsis connected to the mean free path ℓ ⋆ by the relation, 𝜉 ≈ 𝛽𝑁 ℓ ⋆ . The effect of localization, originally proposedby P.W. Anderson in (Anderson, 1958), exponentiallysuppresses transmission due to multiple interference andis thus entirely due to the wave nature of the scattered flux (the effect is nonexistent in a trajectory picture asfor classical particles). As reviewed in (Abrahams, 2010;Lagendijk et al. , 2009)), several experiments have mean-while successfully demonstrated localization of differentkinds of waves (as for sound, microwaves, light and coldatomic gases). Solving the DMPK equation in the lo-calized limit ( 𝑠 ≫ 𝑁 ) (Dorokhov, 1982, 1983; Pichard,1991) yields a joint probability density of the transmis-sion eigenvalues which nicely factorizes into a product ofGaussian distributions, 𝑃 ( { 𝑥 𝑛 } , 𝑠 ) = (︁ 𝜋𝑠𝑁 )︁ − 𝑁/ 𝑁 ∏︁ 𝑛 =1 exp [︀ − ( 𝑁/𝑠 )( 𝑥 𝑛 − ¯ 𝑥 𝑛 ) ]︀ (42)centered around the regularly spaced mean values ¯ 𝑥 𝑛 =( 𝑠/𝑁 )(2 𝑛 − /
2. Since, in the limit 𝑠 ≫ 𝑁 , thewidth of the Gaussians is much smaller than the spac-ing to the nearest neighbors, the transmission eigenval-ues 1 ≪ 𝑥 ≪ 𝑥 ≪ . . . ≪ 𝑥 𝑁 “crystallize” on a regularlattice with a lattice spacing of 𝛿𝑥 = 𝑁/ ( 𝐿𝑙 ) (for 𝑁 ≫ et al. , 1990; Stone et al. , 1991). It is interesting to compare this crystal-likebehavior in the localized regime with the liquid-like be-havior found for the diffusive regime, where the 𝑥 𝑛 areuniformly distributed, see Eq. (39). In the transition re-gion between these two regimes also the distribution func-tion is intermediate between a constant function with acut-off and a series of Gaussians as expected for a par-tially melted solid (see also experimental data in Fig. 14of section III.E).If we translate this result for the 𝑥 𝑛 to the conventionaltransmission eigenvalues 𝜏 𝑛 = 1 / cosh 𝑥 𝑛 , we can usethe fact that 𝑥 𝑛 ≫ 𝜏 𝑛 ≈ − 𝑥 𝑛 ). Thetransmission eigenvalues thus have a log-normal distribu-tion. Since for the total transmission 𝑇 the first transmis-sion eigenvalue 𝜏 then dominates over all others, we alsofind that the total transmission, 𝑇 ≈ − 𝑥 ) takeson a log-normal distribution with the following meanvalue and variance ⟨ 𝑇 ⟩ = − 𝑠/𝑁 + 𝒪 (1) , var ⟨ 𝑇 ⟩ = − ⟨ ln 𝑇 ⟩ = 4 𝐿𝜉 . (43)These results were calculated for the case of broken time-reversal symmetry, 𝛽 = 2. The connection between themean and the variance of the conductance, however, staysvalid for other values of 𝛽 (Beenakker, 1994; Pichard,1991). Note, how, based on this relationship, the vari-ance of the transmission increases as the transmissionitself is reduced, a result which nicely contrasts the con-stant value of the transmission fluctuations in the diffu-sive case. To arrive at this result one uses the fact thatthe crystallization of transmission eigenvalues in the lo-calized regime reduces the multi-channel scattering prob-lem effectively to a one-channel problem.This single-channel regime of transport occurring inthe deeply localized limit is also very instructive for re-lating the transmission eigenchannels with the internal7modes in the system – a connection that is already in-herent in the definition of the scattering matrix, seediscussion below Eq. (28). The internal “quasi-boundstates” or “resonances” are responsible for mediating thetransmission from one side of the medium to the other.In most circumstances, like in the diffusive scatteringregime, these modes will have a resonance width 𝛿𝜈 whichexceeds their mean level spacing Δ 𝜈 , resulting in a ratio(called the Thouless number) (Edwards and Thouless,1972; Thouless, 1977) 𝛿 ≡ 𝛿𝜈/ Δ 𝜈 > et al. , 2008; Persson et al. , 2000).The opposite limit of well-resolved modes and with it theregime of Anderson localization itself, is characterizedby 𝛿 < et al. , 1979; Thouless, 1977) that in the localizedlimit the transmission 𝑇 through the system (or, equiva-lently, the dimensionless conductance 𝑔 = 𝐺 ℎ/ (2 𝑒 )) andthe Thouless number become the same 𝑔 = 𝛿 (wherebythe spectral and the transport properties become inti-mately connected). The Thouless number also turns outto govern all statistical properties of Anderson localiza-tion (Abrahams et al. , 1979).As was meanwhile demonstrated successfully in anexperimental micro-wave study, the good resolution ofmodes in the localized limit ( 𝛿 <
1) allows to decomposea speckle pattern of radiation transmitted through a dis-ordered sampled into a sum of only a few individual modepatterns (Wang and Genack, 2011). Not only do veryfew localized modes dominate transmission in the local-ized regime, but also just a few transmission channels areopen. As has meanwhile been demonstrated explicitly,these modes and channels are not merely strongly cor-related (Choi et al. , 2012a), but, in fact, directly linkedwith each other (Pe˜na et al. , 2014): In the deeply local-ized regime the single dominant transmission eigenchan-nel is given either by a single localized mode or by a so-called “necklace state”, which is a highly transmittingsuperposition of overlapping localized modes (Pendry,1987). Another curious observation in this deeply local-ized limit is that due to the dominance of a single trans-mission transmission eigenchannel, the entire scatteringsystem can be mapped onto a strictly one-dimensionalsystem with the same statistical properties, provided thatits localization length is properly renormalized (Pe˜na et al. , 2014). Such a mapping can also be carried out formedia with multiple open transmission channels that canthen be mapped onto a sum of several one-dimensionalsystems, not only in terms of the transmission statistics,but also in terms of the density of states in the mediumand the corresponding time-delay (Davy et al. , 2015a,b).
B. Open transmission eigenchannels and shot noise
One of the most spectacular predictions of RMT andof the DMPK equation is the existence of “open trans-mission eigenchannels” which have been discovered firstby Oleg Dorokhov in 1984 (Dorokhov, 1984) (see thecorresponding distribution of transmission eigenvalues inFig. 4). Due to the absence of wave front shaping toolsfor coherent electron scattering, directly probing thesechannels with electrons is, however, only possible for verysimple geometries like quantum point contacts. As notedfirst by Yoseph Imry (Imry, 1986), the open transmissioneigenchannels do, however, leave very conspicuous statis-tical signatures on the transport properties of electrons.In particular, as we will explain in the following, the pres-ence of open transmission eigenchannels is detectable inthe electronic shot noise (not to be confused with pho-tonic shot noise or conductance fluctuations).The term “shot noise” was originally introduced byWalter Schottky who was measuring the temporal fluc-tuations of the electric current in a vacuum tube (Schot-tky, 1918). As he first pointed out, these time-dependentfluctuations around the mean current value are due tothe granularity of the electronic charge. In other words,since electrons come in discrete charge packets (i.e., theelementary charge) they don’t produce a fluid-like flow ofcurrent but rather a random succession of discrete chargeimpact events. Comparing this situation to the (acous-tic) noise produced by the small metal pellets from a“shot gun” when impinging on a solid surface, Schottkypredicted shot noise to be a convenient tool to measurethe value of the electron charge. Specifically, he proposeda relationship between the so-called shot noise spectraldensity 𝑆 ( 𝜔 ) = ⟨ 𝛿𝐼 ( 𝜔 ) ⟩ /𝛿𝜔 based on the frequency-dependent current fluctuations around the mean currentvalue, 𝛿𝐼 ( 𝜔 ) = 𝐼 ( 𝜔 ) − ⟨ 𝐼 ( 𝜔 ) ⟩ and the mean current, 𝑆 ( 𝜔 ) = 2 𝑒 ⟨ 𝐼 ( 𝜔 ) ⟩ . (Note that the factor of 2 comes fromthe contribution of positive and negative frequencies andthat the formula only holds in the limit where contri-butions from thermal or 1 /𝑓 -noise can be disregarded.)Since both 𝑆 ( 𝜔 ) and ⟨ 𝐼 ( 𝜔 ) ⟩ can be measured in an exper-iment, one should be able to determine the elementarycharge 𝑒 according to Schottky, who assumed electrons tobe completely uncorrelated (Poisson distributed) to de-rive this relation. Due to the residual correlations amongelectrons (even in a vacuum tube), Schottky’s prediction,however, failed to reach the accuracy of the seminal Mil-likan experiment using oil droplets (see (Beenakker andSch¨onenberger, 2003) and (Blanter and B¨uttiker, 2000)for a review).Contrary to the expectation from the famous Franck-Hertz experiment, the shot noise produced in Schot-tky’s vacuum tube can be understood completely clas-sically (Sch¨onenberger et al. , 2001). In the mescoscopiclimit, however, where electrons behave as quantum mat-ter waves (as in ultra-thin wires at a few milli-Kelvin)8the correlations that lead to deviations from the Schot-tky formula become dominant. These deviations are typ-ically quantified in terms of the so-called “Fano factor”, 𝐹 = 𝑆/𝑆 𝑃 , which is the ratio of the noise spectral density 𝑆 for a given system (with correlations), as compared tothe uncorrelated value of Schottky for Poissonian statis-tics 𝑆 𝑃 = 2 𝑒 ⟨ 𝐼 ⟩ . In the mesoscopic limit these quanti-ties can be conveniently evaluated using the Landauer-B¨uttiker framework (see section II.A.4) to estimate boththe current (B¨uttiker, 1988), ⟨ 𝐼 ⟩ = (2 𝑒 /ℎ ) 𝑉 ∑︀ 𝑁𝑛 =1 𝜏 𝑛 , aswell as the noise spectral density (B¨uttiker, 1990; Khlus,1987; Lesovik, 1989), 𝑆 = 2 𝑒 (2 𝑒 /ℎ ) 𝑉 ∑︀ 𝑁𝑛 =1 𝜏 𝑛 (1 − 𝜏 𝑛 ).Note that the latter prediction relates the magnitudeof the time-dependent current fluctuations ( 𝑆 ) with thetime-independent transmission eigenvalues 𝜏 𝑛 . An intu-itive interpretation for this expression can be given asfollows (Beenakker and Sch¨onenberger, 2003): Since ac-cording to the Pauli principle, at zero temperature (asconsidered here) all levels up to the Fermi energy 𝐸 𝐹 arefilled with electrons and above 𝐸 𝐹 all levels are empty, allthermal fluctuations are suppressed. The quantum shotnoise thus comes from the electrons in a given transmis-sion eigenchannel, attempting to transmit from source todrain with transmission probability 𝜏 𝑛 . Since the elec-tron in channel 𝑛 can either pass or not pass, one getsbinomial statistics as in a sequence of statistically inde-pendent yes/no experiments, each of which has a proba-bility of 𝜏 𝑛 to give “yes” as an answer. Correspondingly,the fluctuations in the transmitted current will be pro-portional to 𝜏 𝑛 (1 − 𝜏 𝑛 ) for channel 𝑛 ; since, furthermore,all channels are statistically independent, the total fluc-tuations will be proportional to ∑︀ 𝑁𝑛 =1 𝜏 𝑛 (1 − 𝜏 𝑛 ), justas predicted in the above formula for the noise spectraldensity. Note that in contrast to classical electronic shotnoise which is due to the randomness associated with theemission of electrons, for quantum electronic shot noisethe randomness in emission is completely suppressed bythe Pauli principle. Instead, the noise is here due to theintrinsic indeterminism inherent in any quantum trans-mission problem to which only a transmission “proba-bility” can be assigned. Due to the different statistics(Bose-Einstein vs. Fermi-Dirac) the shot noise will alsobe different when replacing electrons with photons – evenwhen considering systems with the same scattering ma-trix. Loosely speaking, photons are more “noisy” due tobunching, whereas electrons are more “quiet” due to anti-bunching. As a consequence, the results from above cannot be directly mapped from the electronic to the pho-tonic case, where primarily amplification and absorption,rather than scattering, shift the Fano factor away fromits Poissonian value (Beenakker and Patra, 1999). Morerecent work, however, has also found that mesoscopicfluctuations influence the photocount statistics of coher-ent light scattered in a random medium (Balog et al. ,2006). (a) (b) current I (µA) current I (nA) Fano factor F=1/3 Fano factor F=1/4 Figure 8 Shot noise power (a) in a metallic diffusive wire and(b) in a chaotic quantum cavity. Figures adapted from (Henny et al. , 1999) for (a) and (Oberholzer et al. , 2002) for (b).The linear rise of the experimentally obtained noise powerwith increasing current has a slope that follows the theoreticalpredictions for the universal Fano factors 𝐹 = 1 / 𝐹 =1 /
4, respectively. For small currents the noise deviates fromthe linear increase due to finite temperature effects.
To make contact with the open and closed transmis-sion eigenchannels, consider that the noise spectral den-sity 𝑆 for electrons is a very sensitive measure of thedistribution of transmission eigenvalues 𝑃 ( 𝜏 ) studied insections II.A.5 and II.A.6. To understand this point, con-sider that, for many scattering channels, 𝑁 ≫
1, the ex-pression for 𝑆 can be conveniently rewritten as follows, 𝑆 = 2 𝑒 (2 𝑒 /ℎ ) 𝑉 ∫︀ 𝑃 ( 𝜏 ) 𝜏 (1 − 𝜏 ) 𝑑𝜏 , from which we mayconclude that the distribution function 𝑃 ( 𝜏 ) enters thespectral density 𝑆 through its first and second moment(i.e., ∫︀ 𝑃 ( 𝜏 ) 𝜏 𝑛 𝑑𝜏 with 𝑛 = 1 , 𝑃 ( 𝜏 ), a quantity which will be stongly influenced bythe presence of open and closed channels. In particular, ifwe replace 𝑃 ( 𝜏 ) with the bi-modal distributions obtainedfrom RMT and from the DMPK equation, one finds veryspecific values for the Fano factor.Consider first the case where we assumed the scatter-ing matrix to be distributed according to Dyson’s circularensemble with 𝑃 ( 𝜏 ) = 1 / [ 𝜋 √︀ 𝜏 (1 − 𝜏 ) ] [see Eq. (33)]. Inthis case the shot noise Fano factor can be calculated byhand to take on the universal value 𝐹 = 1 / et al. , 1994), corresponding toa shot noise spectral density 𝑆 which is reduced to onefourth of the Poissonian value 𝑆 𝑃 of Schottky (Schottky,1918). When taking, instead, the transmission eigen-value distribution which we found for the wire in thediffusive regime, 𝑃 ( 𝜏 ) ∝ / ( 𝜏 √ − 𝜏 ) [see Eq. (40)] onefinds the shot noise to be suppressed to one third witha corresponding Fano factor of 𝐹 = 1 / ℓ ⋆ and of the system length 𝐿 . In the transition from the diffusive to the ballisticlimit (where all eigenchannels open up) the Fano factorvanishes, 𝐹 → 𝐹 → et al. , 2002; Steinbach et al. , 1996) have mean-while confirmed the theoretical predictions (see Fig. 8),thereby providing a convincing proof for the existence ofopen transmission channels in transport through chaoticand disordered media, respectively. As we will see below,going beyond this statistical evidence by accessing trans-mission eigenchannels in optics directly will only becomepossible through the techniques of wave-front shaping,see section IV.The above universal values for the Fano factor relyon the assumption that waves entering in a scatteringregion get perfectly randomized before exiting this re-gion. In fact, this assumption is the starting point forRMT and in weaker form, also enters the DMPK equa-tion through the approximation of transverse isotropy.There are of course many ways in which a specific scat-tering system can fail to fulfill these assumptions: Firstof all, a scattering region might neither be fully chaoticnor disordered (Agam et al. , 2000; Aigner et al. , 2005;Oberholzer et al. , 2002), or its disorder might featurespatial correlations which lead to very specific transmis-sion statistics (Izrailev and Makarov, 2005). Also anyeffects like absorption (Brouwer, 1998; M´endez-S´anchez et al. , 2003) and dephasing (Baranger and Mello, 1995;Brouwer and Beenakker, 1995; Huibers et al. , 1998) havea significant influence (see Fig. 6). Consider also thatthe way in which one couples to a disordered region (as,e.g., by barriers or point contacts) can lead to the sit-uation that part of the incoming flux is immediatelybackreflected, rather than being randomized. Such non-universal contributions to the transport statistics can,however, be suitably described with tools like the Pois-son kernel (Brouwer, 1995). Alternatively, one might alsobe confronted with systems like thin disordered interfaceswhich, on the one hand, scatter incoming waves stronglybut which are shorter than the wavelength, on the otherhand, such that they fall outside of the predictions forballistic, diffusive or localized samples; see (Schep andBauer, 1997) for a successful treatment of such cases.A particularly interesting challenge to conventionaltheories arises for the case when the randomization ina given scattering system affects only a sub-part of thescattered flux. This situation occurs, e.g., when “direct”scattering processes are able to penetrate the randommedium in a time that is below the time scale necessaryfor randomization to set in (Agam et al. , 2000; Goparand Mello, 1998). For conventional strongly scatteringmedia the fraction of such “ballistic” scattering states de-creases exponentially with the system size. In imaging,this strong suppression of “ballistic light” in turbid mediais in fact one of the key challenges for techniques based on light in the visible part of the spectrum, which is scat-tered significantly, e.g., in biological tissue (Ntziachristos,2010). Also in the field of quantum shot noise a wholebody of work exists in which the influence of such non-universal contributions is investigated in detail (Agam et al. , 2000; Aigner et al. , 2005; Jacquod and Sukhorukov,2004; Marconcini et al. , 2006; Nazmitdinov et al. , 2002;Oberholzer et al. , 2002; Rotter et al. , 2007; Schomerusand Jacquod, 2005; Silvestrov et al. , 2003; Sukhorukovand Bulashenko, 2005). Generally speaking, one findsthat ballistic scattering contributions reduce the Fanofactor below the universal values found above. This isbecause the fully closed or fully open transmission eigen-channels (with 𝜏 = 0 ,
1) associated with ballistic scatter-ing are “noiseless” in terms of their contribution to shotnoise (Silvestrov et al. , 2003). We will see in the nextsection and in section V.C.2 that such ballistic noiselessstates in electronic quantum transport correspond to ge-ometric optics states in light scattering, i.e., light rays towhich the eikonal approximation applies and which havea well-defined time-delay (Rotter et al. , 2011).
C. Time-delay
When speaking of dynamical aspects of scatteringproblems, well-defined time scales are required to pro-vide an estimate for the duration of a scattering pro-cess. Whereas many different definitions of such timescales are available in the literature, the most rigorouslydefined and most commonly used ones are the time de-lay (also called delay time or group delay ) and the dwelltime , which quantities measure the duration of a scatter-ing process and the time spent inside a designated region,respectively. As can be expected, these two time scaleswill turn out to be quantitiatively similar for many prac-tical purposes, but also the subtle differences betweenthem provide instructive insights.The foundations for work on time-delay were laidby Eugene Wigner and his student Leonard Eisenbudwho studied the single-channel scattering phase shifts inresonant quantum scattering (Eisenbud, 1948; Wigner,1955b). Their fundamental insight was that the time-delay 𝜏 𝜑 ( 𝐸 ) that an incident wave accumulates duringa resonant scattering event (in one specific channel) ascompared to non-resonant free propagation can be esti-mated by taking the energy derivative of the scatteringphase shift 𝜑 ( 𝐸 ) (see Fig. 9). The corresponding Wigner(or Wigner-Eisenbud) phase delay time is then given as 𝜏 𝜑 ( 𝐸 ) = (cid:126) 𝜕𝜑 ( 𝐸 ) /𝜕𝐸 = 𝜕𝜑 ( 𝜔 ) /𝜕𝜔 where 𝐸 is the scat-tering energy of a quantum particle and 𝜔 its angularfrequency. Since the energy derivative at a sharp scat-tering resonance can be very large, also the correspond-ing time delay will, correspondingly, take on very largepositive values at such resonant energy values. Note,however, that the value of the time-delay can, in prin-0 Wigner-‐Smith -me delay Sca3ering phase V(x) ⌧ = @ @! in = e ikx i!t ! out = e ikx i!t + ( ! ) e + i ( ! ) Figure 9 (color online). The Wigner-Smith time delay is cal-culated as the derivative of the scattering phase accumulateddue to the presence of a scattering potential, here given as 𝑉 ( 𝑥 ). The blue (solid) and the red (dashed) lines displayschematically the scattering wave function in the presenceand absence of this potential, respectively (two different scat-tering energies are shown and reflections by the potential areignored). Figure courtesy of R. Pazourek (Pazourek et al. ,2015). ciple, also be negative as, e.g., for the case of scatteringthrough a repulsive potential. In this case one speaks of a“time-advance” the value of which is limited by causalityconstraints.Subsequent work (Ili´c et al. , 2009; Jauch et al. , 1972;Martin, 1976) showed that the above definition of thetime-delay can be reformulated as follows, 𝜏 𝜑 ( 𝐸 ) = 1 𝜎 scat 𝑣 ∫︁ 𝑑 𝑟 [︀ | 𝜓 𝐸 ( r ) | − ]︀ . (44)This integral contains the single-channel scattering states 𝜓 𝐸 ( r ) and extends over all of space. 𝜎 scat is the scatteringcross-section and 𝑣 is the velocity of the incident flux.Multiplied together, 𝜎 scat 𝑣 is equal to the incident flux onthe scatterer, 𝐽 in (see section II.A.2). Neglecting a self-interference term outside the scattering region Ξ (whichusually averages out (Smith, 1960; Winful, 2003)), thetime-delay 𝜏 𝜑 thus measures an excess in the “dwell time” 𝜏 𝑑 = 1 𝐽 in ∫︁ Ξ 𝑑 𝑟 | 𝜓 𝐸 ( r ) | . (45)inside the scattering region Ξ as compared to propaga-tion in free space. Both the scattering states 𝜓 𝐸 ( r ) andthe incoming flux associated with them are normalizedhere such that the integral ∫︀ Ξ 𝑑 𝑟 | 𝜓 𝐸 ( r ) | = 1 when thescattering region Ξ is replaced by free space (for whichcase the excess dwell time is zero).To understand intuitively why Eq. (45) represents adwell time, consider that the integral which appears theremeasures the intensity stored inside the scattering regionΞ. To obtain the time, which this intensity stays insidethe scattering region, one has to divide it by the out-going flux 𝐽 out , which, for a stationary scattering statelike 𝜓 𝐸 ( r ), is equal to the incoming flux 𝐽 in (the absence of gain and loss in the medium is assumed here). Theidentity in Eq. (45) thus corresponds to what one wouldexpect from a simple classical picture. To emphasize thisanalogy we mention, parenthetically, that also the dwelltime of water molecules in a bath tub can be estimatedequivalently based on the knowledge of the water vol-ume contained in the tub and the incoming water flux(provided that the latter is equal to the outgoing flux).The above expressions have originally been derivedfor the scattering of matter waves as described by theSchr¨odinger equation. Using the analogy with theHelmholtz equation (see section II.A.1) these results cannow be carried over to light scattering. Note, how-ever, that in this case the electromagnetic energy density 𝑢 ( r , 𝜔 ) defined in section II.A.2 enters the definition ofthe dwell time 𝜏 𝑑 , which gives rise to additional termsrelated to a potential energy density stored in the di-electric medium. This contribution is particularly largewhen the dielectric constant of the medium is much largerthan the vacuum value. Details on these terms as well ason their relation to the energy-dependence of the opticalpotential 𝑉 light introduced in section II.A.1 are providedin (Lagendijk and van Tiggelen, 1996), see section 3.2.3there.The reason why the concept of time-delay has beenso successful and widely used in a variety of differentcontexts is because it allows one to extract temporal in-formation out of spectrally resolved scattering quantitieslike the scattering phase shift. A second major asset ofthe time-delay concept is its close relation to other phys-ically relevant quantities, of which the stored intensityinside a scattering region is just one. Another one isthe absorption time 𝜏 a , which measures the exponentialdecay of the light intensity in an absorbing medium (see(Lagendijk and van Tiggelen, 1996) for a review). Specif-ically, if we consider a light ray in a uniformly absorbingmedium of constant refractive index 𝑛 = 𝑛 𝑟 + 𝑖𝑛 𝑖 then thecorresponding wave amplitude along the ray path can bewritten as follows, 𝜓 𝜔 ( 𝑥, 𝑡 ) = 𝐴 exp( 𝑖𝑘𝑛𝑥 − 𝑖𝜔𝑡 ), with 𝐴 being an overall amplitude and 𝑥 the spatial coordinatealong the ray trajectory. The incoming wave intensity | 𝜓 in | = | 𝐴 | will have decreased exponentially due toabsorption, | 𝜓 out | = | 𝐴 | exp( − 𝑛 𝑖 𝑘𝐿 ) when leaving themedium. The trajectory length 𝐿 inside the medium isnow easily related to a corresponding time, 𝜏 = 𝐿𝑛 𝑟 /𝑐 ,which is now both the dwell time inside the medium (dueto its relation with 𝐿 ) and the absorption time (due toits relation with the decreased intensity | 𝜓 out | ). Thesimple physical reasoning behind this correspondence isthat a wave suffers the more from absorbtion the longerit stays inside an absorbing medium. In the limit of smallabsorption, where 𝑛 𝑖 𝑘𝐿 ≪
1, this time can be estimatedas follows (Lagendijk and van Tiggelen, 1996), 𝜏 = lim 𝑛 𝑖 → 𝑛 (1 − | 𝜓 out | / | 𝜓 in | )2 𝑛 𝑖 𝜔 . (46)1Note that in this relation the ratio of outgoing to incom-ing wave intensity emerges, which is known as the albedoof a scatterer, 𝑎 = ⟨| 𝜓 out | ⟩ / ⟨| 𝜓 in | ⟩ . For the visible partof the light spectrum the albedo (measured in reflection)ranges from values below 10% for very dark substances(like coal) to almost 90% for very bright substances (likesnow). According to the above simple derivation, suchalbedo measurements provide very accurate informationabout the time light stays inside a given medium (Fesh-bach, 1962; Tiggelen et al. , 1993). Note that for very in-homogeneously absorbing media the dwell time and theabsorption time may also be quite different from eachother as the absorption time then depends significantlyon whether regions of high absorption are visited by ascattering wave or not.In a seminal paper (Smith, 1960) it was shown that thetime delay concepts can be straightforwardly extended tomultiple channels. In particular, for flux-conserving sys-tems without loss or gain a corresponding multi-channeltime-delay matrix Q can be defined based on the unitaryscattering matrix S in the following way: Q = − 𝑖 (cid:126) S † 𝜕 S 𝜕𝐸 . (47)This Wigner-Smith (or Eisenbud-Wigner-Smith) time-delay matrix Q generalizes the concept of the “phasedelay time” from above to multiple channels. Care must,however, be taken with respect to the definition of theasymptotic states that are related by the scattering ma-trix: Depending on whether the asymptotic states incor-porate the free space propagation between incoming andoutgoing asymptotic region, the Wigner-Smith matrix ei-ther measures the times associated with the phase delaysor with the phases themselves. The matrix Q has thesame number of 2 𝑁 × 𝑁 complex elements as the scat-tering matrix itself (see section II.A.4) and it is Hermi-tian by construction. Its real eigenvalues 𝑞 𝑛 are referredto as the “proper delay times” which, when assumingan RMT distribution for the matrix elements of S can beshown to follow very specific distribution functions in thechaotic (Brouwer et al. , 1997) as well as in the diffusivelimit (Ossipov et al. , 2003) (see also (M´endez-Berm´udezand Kottos, 2005) and the following review on this topic(Kottos, 2005)).The Wigner-Smith time-delay matrix, in turn, shares avery deep connection to the density of states (DOS) 𝜌 ( 𝜔 )of an open scattering system. Specifically, for a finiteopen medium the DOS can be defined as the sum overall quasi-bound states or “resonances”, evaluated at thefrequency 𝜔 (Breit and Wigner, 1936), 𝜌 ( 𝜔 ) = 1 𝜋 ∑︁ 𝑚 Γ 𝑚 / 𝑚 / + ( 𝜔 − 𝜔 𝑚 ) , (48)where each of the Lorentzian mode profiles in this sumis spectrally normalized. Following the work of Gamow (Gamow, 1928) the resonance energies 𝜔 𝑚 and theirwidths Γ 𝑚 are the real and imaginary parts of com-plex resonance eigenvalues at which the scattering ma-trix S ( 𝜔 ) has its poles (see last paragraph in sectionII.A.4). Based on this connection Krein, Birman, Lyu-boshitz and Schwinger (Birman and Yafaev, 1992; Krein,1962; Lyuboshitz, 1977; Schwinger, 1951) showed thatthe DOS is directly expressible through the scatter-ing matrix, 𝜌 ( 𝜔 ) = [ − 𝑖𝑐/ (2 𝜋 )]Tr S † 𝜕 S /𝜕𝜔 , and thusthrough the trace of the Wigner-Smith time-delay matrix 𝜌 ( 𝜔 ) = 𝑐/ (2 𝜋 )Tr Q , a connection which has meanwhilebeen verified also numerically (Yamilov and Cao, 2003)and experimentally (Davy et al. , 2015b). From this re-lation we conclude that the DOS is directly proportionalto the sum of the time delays associated with all the2 𝑁 channels described by the scattering matrix (Smith,1960; Wigner, 1955b). Since, in addition, the local DOS 𝜌 ( r , 𝜔 ) (where the DOS 𝜌 ( 𝜔 ) = ∫︀ 𝑑𝑟 𝜌 ( r , 𝜔 )) is also con-nected to the imaginary part of the Green’s function, 𝜌 ( 𝜔, r ) = − 𝜔 Im[ 𝐺 ( r , r ′ , 𝜔 )] /𝜋 (assuming a scalar field,see (Wijnands et al. , 1997), chap. 4), the above relationsalso uncover a direct connection between the time-delayand the Green’s function (see also Krein-Friedel-Lloydformula as discussed, e.g., in (Faulkner, 1977)).The close connection between the time-delay and theDOS also has a very fundamental insight in store thatcan be obtained through a result derived by HermannWeyl (Arendt and Schleich, 2009; Weyl, 1911). This so-called “Weyl law” states that the average DOS in a finitedomain asymptotically (for increasing eigenfrequencies)follows a universal function (in frequency) that just de-pends on the volume and the surface area of the system,but not on the specific geometric details of the scatter-ing potential. Through the equivalence between the DOSand the time-delay, the latter is thus also just determinedby the volume and the surface of the scattering domain –a universal result that holds independent of whether theunderlying medium leads to ballistic, diffusive scatter-ing or even Anderson localization (Pierrat et al. , 2014).One very interesting consequence of this result is that,through the connection between the time-delay and thedwell-time (see Eqs. (44,45)) this “universal” time-delaycan also be directly linked with the energy stored in themedium for unit incident flux in each of the scatteringchannels.To show this explicitly we, however, first need to gen-eralize the definition of the dwell-time in Eq. (45) from asingle to multiple scattering channels – in a similar way aswe did earlier for the time-delay. For the correspondingdefinition of a “dwell time operator” to be meaningful,we demand that the expectation value of this operator,for a given multi-channel incoming state, yields the corre-sponding dwell time 𝜏 𝑑 of this state. Since, according toEq.(45), the dwell time involves the integral of the cor-responding scattering state over the scattering volume,the definition of the dwell time operator needs to incor-2porate the knowledge on the scattering states. Followingsection II.A.3 we know that any scattering state can beconnected to its incoming waves by way of the Green’sfunction G , with the result that the dwell time opera-tor Q 𝑑 is given as follows (Ambichl, 2012; Sokolov andZelevinsky, 1997), Q 𝑑 = (cid:126) W † ( G ) † GW , (49)where W is the energy-dependent coupling matrix fromthe scattering to the exterior region introduced inEq. (28). We emphasize here that the above expressionfor Q 𝑑 involves the knowledge of the Green’s functionson all points inside the scattering medium such that anevaluation of Q 𝑑 based on Eq. (49) is very complex (i.e.,numerically very costly and experimentally close to im-possible). Alternatively, one can connect this dwell-timematrix with the definition of the Wigner-Smith time-delay matrix (which only involves the knowledge of thescattering matrix): When restricting the action of theenergy (frequency) derivative of the scattering matrixin Eq. (47) to only the explicit energy-dependence inEq. (28) and neglecting the energy-dependendence of thecoupling matrix W , the time-delay and the dwell-timeoperators for unitary scattering systems are the same(Sokolov and Zelevinsky, 1997). In this sense, the time-delay and the dwell-time differ only by the above men-tioned “self-interference” term which involves also theevanescent modes in the near-field of the scatterer (Am-bichl, 2012).In a similar way, the connection between the dwell time 𝜏 𝑑 and the absorption time 𝜏 a for the single-channel casesuggests that such a relation might also exist on the moreformal operator level. Following this idea, it was shownin (Savin and Sommers, 2003) that for any uniformly ab-sorbing medium ( 𝑛 𝑖 ( r ) is uniform in space) with arbitraryspatial complexity ( 𝑛 𝑟 ( r ) varying in space) the followingrelation holds between the scattering matrix S and thedwell-time operator Q 𝑑 , − S † S = Γ 𝑎 Q 𝑑 , (50)where the parameter Γ 𝑎 is a phenomenological absorp-tion rate and both S as well as Q 𝑑 are evaluated in thepresence of absorption. The above relation suggests thatin a uniformly absorbing medium the time-delay opera-tor is nothing else but the operator which measures theunitarity deficit or the “sub-unitarity” of the scatteringmatrix. Since, in addition, the scattering matrix con-nects the incoming with the outgoing states in a scat-tering problem, which, in turn, are related to each otherthrough the albedo 𝑎 , Eq. (50) is in fact nothing else butthe multi-channel generalization of Eq. (46).As a last point in this section, we mention that theabove concepts on time-delay may also be used to workout appropriately defined velocities. This is particularlyrelevant for the case of resonant wave scattering in a disordered medium, for which the conventionally usedgroup and phase velocities fail to satisfy the causality re-lations required from special relativity (Lagendijk andvan Tiggelen, 1996). A viable alternative is here thetransport or energy velocity 𝑣 𝐸 , which determines thespeed of energy transport (Brillouin, 1960) and is thusstrictly causal also near the resonances of scatterers inthe medium. Characteristic differences between 𝑣 𝐸 forlight and for electrons have been discussed in (Lagendijkand van Tiggelen, 1996). III. MESOSCOPIC EFFECTS IN OPTICAL SYSTEMS:THEORETICAL AND EXPERIMENTAL ANALOGIES
The theoretical framework presented in section II hasbeen and continues to be applied to a whole host ofdifferent questions arising in the context of mesoscopicscattering. Many experiments, in particular for coherentelectronic transport through mesoscopic conductors likequantum point contacts, “quantum billiards”, nanowires,etc. have meanwhile been carried out in which many ofthe above predictions could be studied in detail. Our em-phasis here will be on predictions from mesoscopic scat-tering theory, which could be realized both in electronictransport as well as in optical experiments that will herebe compared with each other. We will review the firstgeneration of such experiments, where also in the opticalcontext “mesoscopic physics” effects have been revealedwithout resorting to wavefront shaping techniques. Letit be clear that when we speak of “mesoscopic effects” inoptics, we do not refer to the signatures of the quantum(optical) nature of the electromagnetic field; rather, werefer here to signatures of light scattering that are intrin-sically related to the finite mode number of a medium aswell as to correlations between these modes.
A. Conductance quantization
One of the foundational experiments in mesoscopictransport was the demonstration of conductance quan-tization. By varying the opening width of a so-calledquantum point contact (through electronic gates on topof a hetero-junction) the conductance was observed tochange in quantized steps of height 2 𝑒 /ℎ , see Fig. 10a(Houten and Beenakker, 2008; van Wees et al. , 1988).The origin of this effect is the quantization of the trans-verse momentum in the quantum point contact; in otherwords, the electrons do get transmitted through indi-vidual transverse “modes”, which can be labeled witha discrete quantum number 𝑚 or 𝑛 (see section II.A.4where we introduced this concept already). Since each ofthese modes has a specific threshold that depends on thewidth of the quantum point contact, the conductance in-creases in a step-wise fashion whenever such a threshold3 (i) (ii) gate voltage (V) slit width (μm) t r a n s m i e d p o w e r ( a . u . ) (a) (b) c o ndu c t a n c e ( e / h ) -2 -1.8 -1.6 -1.4 -1.2 -1 0 5 10 15 20 Figure 10 (a) Conductance quantization in coherent electrontransport through a quantum point contact (see upper insetand similar point contacts at the openings of the quantumbilliards in Fig. 5). The conductance shows steps at integervalues of the conductance quantum 2 𝑒 /ℎ , adapted from (vanWees et al. , 1988). (b) Power of light transmitted througha slit with variable width (see upper inset). Similar steps asin (a) occur here when the slit width corresponds to integermultiples of 𝜆/ 𝜆 being the light wavelength. Beforepropagating through the slit the light is sent through a dif-fuser realized (i) with a piece of paper and (ii) with an arrayof parallel glass fibers. Figure adapted from (Montie et al. ,1991). is crossed. The experiment by van Wees et al. was thuscrucial to lend credibility to the Landauer formula (seeEq. (16) in section II.A.4) that describes the conductance 𝐺 as a problem of coherent transmission 𝑇 through mul-tiple modes. As this description is, however, solely due tothe wave nature of electrons, one should observe it alsowith other types of waves, including electromagnetic ra-diation. This idea was picked up in (Montie et al. , 1991),where a corresponding experiment was realized with lightwaves that were sent through a slit of tunable width, seeFig. 10b. To mimic the way in which electrons impingeon the quantum point contact, the optical experimentfeatured a diffuse light source to distribute the incomingflux equally over all available transverse modes. Withthis type of illumination the light intensity transmittedthrough the slit was observed to follow the same step-likepattern as the electrons do in the mesoscopic analogue,see Fig. 10b.Surprisingly, this optical experiment, which is mucheasier to carry out than its preceding electronics counter-part, was performed only after the corresponding physicswas first understood in a mesoscopic context. This pointnicely underlines the main message of this review andindicates that a strategy along these lines may havemany more interesting insights and surprises in store.Note here, in particular, that in optics, the transmissionthrough and the reflection from a scattering object aretypically accessible in a mode-resolved way, which is notthe case for electrons. This is also the case when study-ing the scattering through an extended disordered regionwhere both the incoming as well as the outgoing modes are quantized. To probe the total optical transmission 𝑇 = ∑︀ 𝑚 𝑇 𝑚 = ∑︀ 𝑚𝑛 𝑇 𝑛𝑚 as inherent in electronic con-ductance, one can use a diffuser (as in Fig. 10b) to en-sure a nearly isotropic spatial illumination, that excitesall modes equally. In contrast, an illumination througha collimated laser beam with a well-defined incoming an-gle would probe the transmission through a suitable de-fined incoming mode 𝑇 𝑚 = ∑︀ 𝑛 𝑇 𝑛𝑚 . The information onthe transmission 𝑇 𝑛𝑚 = | 𝑡 𝑛𝑚 | through the outcouplingmodes 𝑛 is contained in the speckle-pattern appearing be-hind the scattering region, which contains the spatiallyresolved transmission pattern. B. Conductance fluctuations
Experiments that were crucial for uncovering the non-trivial correlations in these different types of coherenttransmission amplitudes were those reporting on conduc-tance fluctuations in small metallic wires and rings (Um-bach et al. , 1984; Washburn et al. , 1985; Webb et al. ,1985). These fluctuations observed in low-temperaturemeasurements as a function of an applied magnetic field(see Fig. 11b) were actually unexpected. Their originwas first believed to be “finite size effects” of the conduc-tor; it was, however, soon revealed (Lee and Stone, 1985)that these fluctuations are due to the multiple disorder-scattering and the corresponding multi-path interference,which sensitively depends on all of the system parame-ters (like the disorder configuration, the Fermi-energy,the magnetic field etc.). In this sense the conductancefluctuations are like a fingerprint of the medium, whichis highly complex but fully reproducible when measur-ing the conductance again a second time. An intrigu-ing aspect of these conductance fluctuations is that theirvariance has a universal value of the order of 𝑒 /ℎ (atzero temperature), which is independent of the degree ofdisorder (in the diffusive regime) as well as of the sam-ple size, hence the name “universal conductance fluctua-tions (UCF)”. As we demonstrated in sections II.A.5 andII.A.6, this surprising result as well as the exact value ofthe universal fluctuations can be understood based onthe spectral rigidity of the transmission eigenvalues. Al-ternatively, one can also obtain very instructive insightsinto this phenomenon based on so-called diagrammatictechniques (Berkovits and Feng, 1994; Feng et al. , 1988;Lee and Stone, 1985) (see (Akkermans and Montam-baux, 2007; Dragoman and Dragoman, 2004; Montam-baux, 2006) for reviews of these techniques).Conceptually speaking, the universal value of the elec-tronic conductance fluctuations is a clear signature of thequantum coherence in the scattering process. We shouldthus expect to observe similar effects also with coher-ent disorder scattering of light. Since the optical specklepatterns contain sizable fluctuations as well, one may betempted to think that UCF are just a different aspect of4speckle fluctuations. As it turns out, this is, however, notthe case. To understand this in more detail consider therelation for the variance of the fluctuations, which for thetransmission of light is given by 𝜎 = ⟨ 𝑇 ⟩ − ⟨ 𝑇 ⟩ . Writ-ing 𝑇 = ∑︀ 𝑚𝑛 𝑇 𝑚𝑛 and 𝛿𝑇 𝑚𝑛 = 𝑇 𝑚𝑛 − ⟨ 𝑇 𝑚𝑛 ⟩ , we obtain 𝜎 = ∑︀ 𝑚𝑛𝑚 ′ 𝑛 ′ 𝐶 𝑚𝑛𝑚 ′ 𝑛 ′ , where 𝐶 𝑚𝑛𝑚 ′ 𝑛 ′ = ⟨ 𝛿𝑇 𝑚𝑛 𝛿𝑇 𝑚 ′ 𝑛 ′ ⟩ .For evaluating this expression for coherent scattering pro-cesses, a classical diffusion equation is clearly insufficient;instead, one can employ a perturbative approach in thelimit of weak but multiple scattering, where the pertur-bation parameter is 1 / ( 𝑘ℓ ⋆ ) ≪ ℓ ⋆ ≪ 𝐿 ( ℓ ⋆ isthe transport mean free path as discussed at the begin-ning of section II and 𝐿 the medium thickness). In thecorresponding expansion (Feng et al. , 1988) the follow-ing contributions to the correlation function 𝐶 𝑚𝑛𝑚 ′ 𝑛 ′ = 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ + 𝐶 (2) 𝑚𝑛𝑚 ′ 𝑛 ′ + 𝐶 (3) 𝑚𝑛𝑚 ′ 𝑛 ′ + . . . can be distinguishedbased on their different contributing scattering diagrams,see Fig. 11a (only the first three terms in this expansionwill be considered in the following).The first term 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ is always present (also in theabsence of phase coherence) and of order zero in the ex-pansion parameter 1 / ( 𝑘ℓ ⋆ ) ≪
1. It corresponds to contri-butions from scattering paths that do not intersect whiletransmitting through the medium, see Fig. 11a left panel.In the absence of such intersections, also correlations be-tween modes are largely absent (the only correlations re-maining in rather thin media give rise to the so-calledmemory effect discussed in sections III.D and V.A). In 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ the most dominant contributions to the trans-mission arise when the difference between both the in-coming and the outgoing transverse momenta is zero,Δ 𝑞 𝑛 = 𝑞 𝑛 − 𝑞 𝑛 ′ = Δ 𝑞 𝑚 = 𝑞 𝑚 − 𝑞 𝑚 ′ = 0. (Note thatwe have implicitly used here that our modes 𝑚, 𝑛 have awell-defined transverse momentum 𝑞 𝑛 .) As a result onefinds that the fluctuations in the speckle pattern are oforder of the average, ⟨ 𝛿𝑇 𝑚𝑛 ⟩ = ⟨ 𝑇 𝑚𝑛 ⟩ . This fact, whichis also known as the Rayleigh law, is directly reflected inthe granularity of a speckle pattern which features strongfluctuations between dark and bright spots. Why dothese large fluctuations not translate to correspondinglylarge fluctuations of the total transmission/conductance?The answer is that the correlation term 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ only hascontributions for the above very specific mode combina-tions and thus, although being formally of the largestscale, their relative contribution to fluctuations dimin-ishes with the number of modes considered. Overall, 𝐶 (1) correlations yield only a sub-dominant contributionto the total transmission fluctuations.This is where the additional correlation functions 𝐶 (2) 𝑚𝑛𝑚 ′ 𝑛 ′ and 𝐶 (3) 𝑚𝑛𝑚 ′ 𝑛 ′ come into play, see Fig. 11a mid-dle and right panel, respectively. In the diagrammaticexpansion those two contributions come from scatteringpaths with one and two quantum crossings in the trans-mission process. Corresponding to the reduced likelihoodfor such crossings to occur, the scale of these contribu- (c) c o ndu c t a n c e fl u c t u a - o n s ( e / h ) (b) -me (ms) (a) C (1) C (2) C (3)
0 2 4 6 8 magne-c field (Tesla) -1 1 0 C ( ) - c o rr e l a - o n x a a' a a b b b' b b' Figure 11 (a) Schematic of the different scattering paths ina disordered medium and their correlations due to crossings,adapted from (Scheffold and Maret, 1998). Non-intersectingpaths ( 𝐶 (1) ) as in the left panel give rise to short-range specklefluctuations. Those paths with a single crossing ( 𝐶 (2) ) as inthe middle panel are already correlated with each other. Theuniversal conductance are, however, induced by paths withtwo crossing ( 𝐶 (3) ) as in the right panel. (b) Fluctuations ofthe electronic conductance 𝐺 with respect to its mean value ⟨ 𝐺 ⟩ , measured in a 310 nm long and 25 nm wide gold wireat 10 mK as a function of a perpendicular magnetic field 𝐵 ,adapted from (Washburn and Webb, 1986). The variance ofthe conductance Δ 𝐺 ≈ . 𝑒 /ℎ corresponds well to the the-oretical prediction (1 / 𝑒 /ℎ spelled out in Eq. (41). (c)Universal conductance fluctuations of light 𝐶 (3) ( 𝑡 ), measureddynamically in a turbid colloidal suspension, adapted from(Scheffold and Maret, 1998). The experimental data is com-pared with a theoretical prediction using a dimensionless con-ductance of 𝑔 = 89 and a sample thickness of 𝐿 = 13 . 𝜇𝑚 . tions is reduced. In the sum for the total correlation, thisreduction is, however, compensated by a less restrictiveangular selection in terms of the differences Δ 𝑞 𝑛 , Δ 𝑞 𝑚 :Whereas the 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ term features only short-range cor-relations, the 𝐶 (2) 𝑚𝑛𝑚 ′ 𝑛 ′ and 𝐶 (3) 𝑚𝑛𝑚 ′ 𝑛 ′ terms feature longand infinite range correlations, respectively. It turns out,however, that not the long-range angular correlations in-herent in 𝐶 (2) 𝑚𝑛𝑚 ′ 𝑛 ′ , but only the infinite-range correla-tions in 𝐶 (3) 𝑚𝑛𝑚 ′ 𝑛 ′ yield the desired universal contributionto UCF, 𝜎 = ∑︀ 𝑚𝑛𝑚 ′ 𝑛 ′ 𝐶 𝑚𝑛𝑚 ′ 𝑛 ′ ≈ ∑︀ 𝑚𝑛𝑚 ′ 𝑛 ′ 𝐶 (3) 𝑚𝑛𝑚 ′ 𝑛 ′ ≈
1. Optical experiments can go much further than simplyre-measuring the universal value of UCF found alreadyearlier in mesoscopic transport. In (Scheffold and Maret,1998) the time-dependent correlation functions 𝐶 (2) ( 𝑡 )and 𝐶 (3) ( 𝑡 ) were recorded in transmission through asmall pinhole filled with a turbid colloidal suspension.The temporal fluctuations of the transmitted light weredue to the Brownian motion of scattering particles. Byfinding quantitative agreement with the theoretical pre-dictions based on the above diagrammatic terms, it thusbecame possible to not only establish UCF in light scat-tering, but also to verify their microscopic origins; seealso corresponding experiments with microwaves (Gehler5 et al. , 2016; Shi and Genack, 2012a). As laid out in a re-view by Berkovits and Feng (Berkovits and Feng, 1994)the above concepts on the correlation functions can beconveniently extended to describe also correlations in fre-quency as well as in the angle or in the spatial positionof the emission from the disordered medium. Note alsothe possibility offered in optics to probe other quantities,such as the 𝐶 (0) correlations (Caz´e et al. , 2010; Shapiro,1999), that arise when the source is embedded inside thescattering medium, a setup that could be used for imag-ing purposes (Carminati et al. , 2015; Skipetrov and May-nard, 2000). C. Weak localization
Another fundamental phenomenon for which the an-gular correlations play an important role and which alsorelies on specific quantum crossings of scattering pathsis the so-called “weak localization effect’ (Abrahams et al. , 1979; Akkermans and Maynard, 1985; Bergmann,1983; Khmel’nitskii, 1984; Wolf and Maret, 1985) (seeBergmann in (Abrahams, 2010) and (Akkermans andMontambaux, 2007; Dragoman and Dragoman, 2004;Montambaux, 2006) for reviews). The term “weak” refersto the overall weakness of this effect as compared to the“strong” (Anderson) localization for which weak local-ization is a precursor. Our starting point here is theobservation from mesoscopic transport theory (see sec-tions II.A.5 and II.A.6) that the transmission througha chaotic scattering system or a disordered wire is re-duced by a small amount (again of the order of 𝑒 /ℎ )as compared to the value expected from a classical (i.e.,incoherent) estimate. This suggests already that inter-ference effects which are based on the coherence of thescattering process, are at the heart of this phenomenon.To properly capture this effect, we employ a similardiagrammatic picture as in the previous section in whichthe total transmission 𝑇 = | ∑︀ 𝛼 𝑠 𝛼 | can be written asa sum over all paths 𝛼 with a corresponding complexamplitude 𝑠 𝛼 (the phase of which is given by the classi-cal action). This expression does not only contain theincoherent summation over all individual probabilities ∑︀ 𝛼 | 𝑠 𝛼 | for scattering paths to go from one side of thedisordered medium to the other (as inherent in the clas-sical Drude formula), but also features interference terms ∑︀ 𝛼 ̸ = 𝛽 𝑠 𝛼 𝑠 * 𝛽 (corresponding to products of scattering am-plitudes for different paths). One might argue that theseinterference terms average out to zero, since their ran-dom phases will lead to constructive and destructive in-terference with equal measure. On close inspection, thisargument turns out to be incorrect, however; this is be-cause certain path-pairs have the same or very similarphase due to reciprocity (see section II.A.4) and maythus lead to a certain bias away from the classical result,as we will see in the following. Consider here, in par- ticular, those paths that emanate from a source (outsideof the medium) and return to it after scattering in thedisordered medium (see illustration in Fig. 12a). Sincethese loops can be traversed in two possible directions,we end up with two paths in the loop which have exactlythe same phase as well as amplitude and thus always in-terfere constructively | 𝑠 𝛼 + 𝑠 𝛼 | = 4 | 𝑠 𝛼 | , independentlyof the disorder configuration. Since this contribution ofsuch time-reversed path pairs to the reflection is largeras compared to the classical result, where we would have | 𝑠 𝛼 | + | 𝑠 𝛼 | = 2 | 𝑠 𝛼 | , they enhance the portion of thewaves that are “coherently back-scattered” to the sourceby a factor of 2 and thus increase the reflection 𝑅 . Inorder to conserve the unitarity of the entire scatteringprocess this increased reflection must be compensated bya corresponding decrease of the transmitted waves – inperfect correspondence with our earlier observation (seesections II.A.5 and II.A.6). The transmitted scatteringpaths that are responsible for this reduction can be shownto be self-crossing paths which feature loops in their scat-tering patterns which can be traversed both in a clock-wise and a counter-clock-wise direction (Akkermans andMontambaux, 2007).Although it turns out that the coherent backscatter-ing contribution (resulting from time-reversed path pairs)explains the weak-localization effect only partially (Hast-ings et al. , 1994), a controlled breaking of reciprocity willeventually destroy weak-localization entirely and restorethe classical (i.e., incoherent) transport result. For elec-tronic scattering problems this can easily be done by ap-plying an external magnetic field perpendicular to thescattering region. As shown in Fig. 12b the resistivity ofan array of chaotic scatterers is, indeed, reduced whenthe magnetic field is applied (note that the reduction isindependent on the sign of the field, due to the Onsagerrelations, see section II.A.4).For light waves, implementing a reciprocity-breakingmechanism is not so straightforward (adding absorptionto the medium is, e.g., not sufficient as it only breakstime-reversal symmetry, but not reciprocity). A viablealternative is provided here by the access to the spa-tial degrees of freedom of light. Specifically, to measurethe weak-localization of light its dependence on the an-gle of the light backscattered from the medium has beenused as a key signature (see the angle 𝜃 in Fig. 12a andits influence on the backscattered intensity in Fig. 12c).Since only those paths that return directly to the sourcefind a partner with the same phase, the enhanced re-flection is concentrated in a very narrow back-reflectioncone which is rather difficult to measure. Employing aFraunhofer diffraction analysis, Akkermans (Akkermans et al. , 1986) could show that the phase difference be-tween two time-reversed reflected scattering path is givenas ( k 𝑖 + k 𝑓 )( r − r ′ ), where k 𝑖 and k 𝑓 are the initialand the final wavenumber, respectively, that impinge onthe disordered medium at the first scatterer position r (c) r e l a ) v e i n t e n s i t y magne)c field (Gauss) r e s i s t a n c e ( k Ω ) (b) (a) -0.5 0 0.5 angle (degrees) -20 -60 -40 0 20 40 60 0 1 2 2.15 2.20 2.25 2.30 2.35 Figure 12 (a) Electron or light paths associated with the weak-localization effect, from (Akkermans et al. , 1986). Due todisorder scattering, the reflected wave vector ^ k 𝑓 is rotated by an angle 𝜃 with respect to the incoming wave vector ^ k 𝑖 . (b)Resistance of an array of stadium-shaped quantum dots as a function of a perpendicular magnetic field, adapted from (Chang et al. , 1994). Different solid curves show experimental data at different temperatures (from top to bottom: 𝑇 = 50mK, 𝑇 = 200mK, 𝑇 = 400mK, 𝑇 = 800mK, 𝑇 = 1 . 𝑇 = 2 . 𝑇 = 4 . et al. , 1988). Differentcurves show experimental results for samples with different degree of absorption, as measured by the absorption mean free path ℓ 𝑎 . From top to bottom: ℓ 𝑎 = ∞ , ℓ 𝑎 = 810 𝜇 m, ℓ 𝑎 = 190 𝜇 m. The intensity is normalized with respect to the value at onedegree, measured in the non-absorbing sample ( ℓ 𝑎 = ∞ ). The dots are predictions from diffusion theory. and leave it again at the last encountered scatterer po-sition r ′ (see a corresponding illustration in Fig. 12a).When the backscattering is perfect, k 𝑖 = − k 𝑓 , we obtainthe enhancement by a factor of exactly two (as foundabove). This number can be reduced slightly when re-current scattering events occur in the strong scatteringlimit (Wiersma et al. , 1995). This maximum, however,degrades when the angle 𝜃 between k 𝑖 and k 𝑓 satisfies | 𝜃 − 𝜋 | > 𝜆/ | r − r ′ | , where 𝜆 is the wavelength. Forthe shortest possible loop involving just two scatteringevents, the typical value of | r − r ′ | is given by the meanfree path ℓ ⋆ , resulting in a typical angular width of thebackscattering cone 𝜃 ≈ 𝜆/ℓ ⋆ . Larger excursions of lightpaths in the medium with increased differences | r − r ′ | areresponsible for the peak of the cone, which was predictedto take on an approximately triangular shape (Akker-mans et al. , 1986). In the presence of absorption thisshape gets rounded as dissipative mechanisms affect pri-marily longer paths (see Fig. 12c).Sophisticated optical experiments could meanwhilemeasure the details of this cone lineshape, from whichnot only the value of ℓ ⋆ but essentially the entire pathlength distributions in the scattering medium can be ex-tracted (Akkermans et al. , 1986) (compare also with sim-ilar studies for electronic scattering (Chang et al. , 1994)).Note that in the optical regime the weak-localization ef-fect also depends on the polarization of light, as has beenpointed out both theoretically (Akkermans et al. , 1986)and experimentally (Albada and Lagendijk, 1985; Drago-man and Dragoman, 2004). Recent experiments haveeven been able to go a step further in that they could ac-tively suppress the coherent backscattering of light by ex-posing the medium to an ultra-fast pump pulse (Muskens et al. , 2012), which opens up interesting opportunities foractively controlling mesoscopic interference phenomena. D. Memory effect
In section III.B on the universal conductance fluctu-ations we have already analyzed the different correla-tions that exist between modes involved in the scatter-ing process across a disordered medium. Using diagram-matic scattering techniques, it was discovered in the samecontext that the correlation function 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ points tothe existence of correlations between incoming modesthat have a similar transverse momentum, i.e., for whichΔ 𝑞 < /𝐿 (Feng et al. , 1988). The inverse proportional-ity with respect to the thickness of the medium 𝐿 meansthat the angular range over which such correlations existbecome smaller for increasing medium thickness – a resultthat is notably independent of the value of the transportmean free path ℓ ⋆ , and on the exact realization of disor-der. Quite interestingly, this so-called “memory effect”was discovered based on mesoscopic transport theory, al-though in mesoscopic electron transport an experimen-tal study of this effect is not possible, since no angularresolution is available in electron scattering. It was re-alized very quickly, however, that the memory effect canbe directly mapped to optical scattering setups wherea laser beam with well-defined transverse momentum issent onto a disordered medium (Freund et al. , 1988). Wewill also see in section V.A how this phenomenon hasdeveloped into a very useful and practical effect withconsequences for imaging through or inside a disorderedmedium (with and without wavefront shaping).Within the angular range discussed above, a small an-gular rotation in the input beam then leads to a rota-tion of the output speckle pattern by the same angle (seeFig. 13), and correspondingly to a shift at a distance fromthe medium. For larger angles, the transmitted specklepattern will rapidly become totally uncorrelated with the7original (un-shifted) speckle image. This behavior can beintuitively understood by considering that shifting theangle of incident light corresponds to a linear phase shiftbetween the Huygens-spots at which the incoming beamhits the disordered medium (see Fig. 13b). In the dif-fusive regime, each of these spots will produce a coneof scattering pathways which reach the back side of themedium as a circular speckle halo, the diameter of whichis about 2 𝐿 , where 𝐿 is the medium thickness. To suc-cessfully transfer the phase ramp from the input beamonto these output speckle patterns, the speckle disks atthe backside of the medium may not acquire a phasethat is larger than about 𝜋 from one disk to its nearestnon-overlapping neighbor (see Fig. 13c). Using simpletrigonometry, this condition, which is intuitively neces-sary to prevent phase mixing, can be translated to therequirement Δ 𝑞 < /𝐿 found already earlier using di-agrammatic techniques (Feng et al. , 1988). Practically,this restriction limits the thickness of the disordered opti-cal medium to a few tens of micrometers in order to havea measurable effect. First experiments (Freund et al. ,1988) on the optical memory effect have been performedsoon after the theoretical proposal (Feng et al. , 1988), inwhich the predicted shift of the speckle pattern could beunambiguously identified (see Fig. 13a).When measuring correlations in reflection from a disor-dered medium, 𝐶 𝑚𝑛𝑚 ′ 𝑛 ′ = ⟨ 𝛿𝑅 𝑚𝑛 𝛿𝑅 𝑚 ′ 𝑛 ′ ⟩ rather than intransmission, one ends up in the interesting situation thatboth the memory effect as well as weak-localization cor-rections come into play. It turns out that, to first order,the angular width of the coherent backscattering coneand the memory effect angle are the same in reflection.This is because both effects rely on the diffuse spot size,i.e., the width of both peaks is now related to the meanfree path, Δ 𝑞 < /ℓ ⋆ , which may be quite different fromthe memory effect angle in transmission, that is relatedto the thickness of the disordered sample, Δ 𝑞 < /𝐿 .The interplay between both effects has been studied in(Berkovits and Kaveh, 1990), again through the differentcontributions 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ , 𝐶 (2) 𝑚𝑛𝑚 ′ 𝑛 ′ , 𝐶 (3) 𝑚𝑛𝑚 ′ 𝑛 ′ . It turns outthat due to reciprocity these correlation functions getadditional peaks as compared to the corresponding ex-pressions for transmission. Consider, e.g., the first con-tribution without quantum crossings, 𝐶 (1) 𝑚𝑛𝑚 ′ 𝑛 ′ , which, inreflection, not only has a single peak at Δ 𝑞 = 0 (corre-sponding to 𝑞 𝑛 = 𝑞 𝑛 ′ and 𝑞 𝑚 = 𝑞 𝑚 ′ ), but also a secondone at Δ 𝑞 = 𝑞 𝑛 + 𝑞 𝑚 (corresponding to 𝑞 𝑚 = − 𝑞 𝑛 ′ and 𝑞 𝑛 = − 𝑞 𝑚 ′ ) which is due to the time-reversed contribu-tions. Similar arguments can also be made for the nextcontribution 𝐶 (2) 𝑚𝑛𝑚 ′ 𝑛 ′ (see (Berkovits and Kaveh, 1990)for details and (Dragoman and Dragoman, 2004) for a re-view). Corrections to these results may be necessary dueto internal surface reflections, as pointed out in (Freundand Berkovits, 1990). Very recent acoustical measure-ments use the memory effect in reflection to obtain in- L (c) (b) L L (a) Figure 13 (color online). (a) Speckle pattern as recordedbehind a thin disordered medium for different incident angles(0, 10 and 20 mdeg from left to right), adapted from (Freund et al. , 1988). The arrows below the three panels point toa specific pattern (arc above a bright spot) that serves asa convenient visual reference for seeing the rightward moveof the speckle pattern with increasing tilt of the input laser.(b),(c) Schematic illustration to explain this “memory effect”:(b) A plane wave, represented here by three focal spots withthe same phase, impinges on a disordered slab of thickness 𝐿 and creates a speckle pattern in transmission. The distanceof 2 𝐿 between the input spots is the minimal one for whichthe output speckle do not yet significantly overlap. (c) Whentilting the incoming laser a phase gradient is imposed on theincoming wave. Provided that the tilt is smaller than a criticalangle, 𝜃 (cid:46) 𝜆/ (4 𝐿 ), this gradient is faithfully mapped onto thetransmitted wave, resulting eventually in a shift of the speckleimage recorded at a screen in the far field as in (a). formation on the path distribution in a random mediumclose to the transition to Anderson localization, finding astrong recurrence of scattering paths at the point wherethey enter the medium (Aubry et al. , 2014). E. Distribution of transmission eigenvalues
As we saw in the theoretical calculations presented insection II of this review, many interesting transport ef-fects have their origin in the statistical distribution ofthe so-called “transmission eigenvalues” 𝜏 𝑛 , which are theeigenvalues of the Hermitian matrix t † t (or the squaredsingular values of t itself). In principle, also the weak-localization correction to the conductance and the UCFcan be expressed through the distribution of the 𝜏 𝑛 andtheir correlations, respectively. Mesoscopic experiments,however, went much further in addressing also the inter-esting consequences of the bi-modal distribution of thetransmission eigenvalues, 𝑃 ( 𝜏 ), which we derived earlierboth for chaotic cavities, see Eq. (33), as well as for diffu-sively scattering waveguides, see Eq. (40). Since the first8 Figure 14 (color online). Experimental results from (Shi andGenack, 2012b) on the distribution of individual transmis-sion eigenvalues 𝜏 𝑛 (labeled by different colors) in microwavescattering through a strongly disordered quasi-1D geometry.As predicted theoretically, Anderson localization leads to a“crystallization” of transmission eigenvalues, correspondingto an equidistant spacing (on a logarithmic scale) betweenneighboring peaks of the corresponding distribution functions 𝑃 (ln 𝜏 𝑛 ). The top curve in black shows the distribution of theoverall transmission 𝑃 (ln 𝜏 ), where 𝜏 = ∑︀ 𝑛 𝜏 𝑛 . moment of this distribution, ⟨ 𝜏 ⟩ = ∫︀ 𝑑𝜏 𝑃 ( 𝜏 ) 𝜏 , corre-sponding to the average transmission per incoming chan-nel, is basically unaffected by the bi-modal shape of 𝑃 ( 𝜏 ),measurements of the electronic conductance alone (whichis just proportional to ⟨ 𝜏 ⟩ ) do not reveal any signaturesof the bi-modality. This is different for other experimen-tal observables, which depend on the higher moments ofthis distribution, ⟨ 𝜏 𝑛 ⟩ = ∫︀ 𝑑𝜏 𝑃 ( 𝜏 ) 𝜏 𝑛 , as, e.g., the quan-tum shot-noise power of electrons which probes the sec-ond moment ⟨ 𝜏 ⟩ in addition to the first (see section II.Bfor more details). The corresponding mesoscopic trans-port measurements with cavities and nano-wires (Stein-bach et al. , 1996) (see (Blanter and B¨uttiker, 2000) fora review) could not only observe the shot-noise suppres-sion below the Poissonian value, corresponding to differ-ent predictions for the values of the Fano factor 𝐹 (seeFig. 8); in fact, even the deviations from such univer-sal behavior could be measured (Oberholzer et al. , 2002)and understood in detail (Agam et al. , 2000; Aigner et al. ,2005; Jacquod and Sukhorukov, 2004; Marconcini et al. ,2006; Rotter et al. , 2007; Sukhorukov and Bulashenko,2005).Whereas a fair amount of convincing evidence for thebi-modal law has thus been put forward, no direct mea-surement of the transmission eigenvalues or of their dis-tribution could be achieved in the mesoscopic context.Also for electromagnetic waves such measurements arealso challenging, as the knowledge of the complete trans-mission matrix t is required to have access to its eigenval-ues and eigenvectors, which determine the correspondingtransmission channels. In particular in optics, where thenumber of such channels is huge (as being proportional tothe sample cross-section and to the inverse squared of the wavelength, 𝑁 ∝ 𝐴/𝜆 ) measuring the entire transmis-sion matrix of a disordered sample is currently still outof reach (Popoff et al. , 2010b; Yu et al. , 2013). This con-dition is, however, much more relaxed for waves with alonger wave-length, as for micro-waves or also for acousticwaves. First micro-wave measurements (Shi and Genack,2012b) on the full transmission matrix through metallictubes, filled with randomly placed and strongly scatter-ing aluminium spheres, confirmed already many of theinteresting predictions which we discussed in the theorysection II: In the diffusive regime, theory predicts thatthe largest transmission eigenvalue 𝜏 is close to unityand thus corresponds to an open channel. In the local-ized regime, in turn, this largest transmission eigenvaluedominates the total transmission 𝑇 = ∑︀ 𝑛 𝜏 𝑛 , such that 𝜏 ≫ 𝜏 𝑛> . The cross-over between these two regimesinvolving a “crystallization of transmission eigenvalues”(see section II.A.6) with an equidistant spacing betweenthe “crystal sites” (ln 𝜏 𝑛 ) was, indeed, observed in a mi-crowave experiment (see Fig. 14).In spite of this very good theory-experiment corre-spondence, the elusive bi-modality of the transmissioneigenvalue distribution could so far not be verified withmicrowaves. An alternative strategy has recently beenput forward based on the propagation of elastic Lambwaves in a two-dimensional macroscopic metal stripe intowhich holes were drilled to emulate disorder (G´erardin et al. , 2014). Although the scattering matrix recordedhere with laser interferometry was also not fully unitary,the bi-modality of 𝑃 ( 𝜏 ) could be verified with this setup.An interesting advantage of this experimental setup isthat it not only allows one to measure all transmissionand reflection amplitudes, but, in fact, also the scatteringwave functions inside the disordered medium in analogyto similar scanning techniques used for electrons (Top-inka et al. , 2001), micro-waves (H¨ohmann et al. , 2010)and optical fields (Fallert et al. , 2009).We will discuss in section V the optical experimentsdedicated to unraveling or exploiting open and closedchannels. IV. OPTICAL WAVE FRONT SHAPING IN COMPLEXMEDIA
Progress in semi-conductor and electronic engineeringhas led to the emergence of a now vast range of techniquesand devices to actively manipulate light, in particularspatial light modulators (SLMs). SLMs are mostly basedon either liquid crystal technology (Lueder, 2010) oron microelectromechanical systems (MEMS) (Cornelis-sen et al. , 2012; Gehner et al. , 2006; Gad-el Hak, 2010;Hornbeck, 2001), see Fig. 15. They are nowadays offer-ing control of up to a few millions of spatial degrees offreedom (pixels) of light, in phase or amplitude (Conkey et al. , 2012b; Goorden et al. , 2014a; van Putten et al. ,92008) and are meanwhile widely used in imaging and mi-croscopy (Maurer et al. , 2011). The advent of digitalimage sensors (mainly CCD and CMOS) also allows todetect a correspondingly large number of degrees of free-dom in intensity or in amplitude with the help of digitalholography (Cuche et al. , 2000; Leith et al. , 1965; Yam-aguchi and Zhang, 1997).In the last 50 years, deformablemirror technology (Babcock, 1953) and adaptive opticsconcepts have revolutionized imaging through the atmo-sphere (Lee and Harp, 1969) and thereby also earth-basedastronomy (Roddier, 1999).The aim of the first two parts IV.A and IV.B of thissection is to review these experimental techniques. Inpart IV.C and IV.D, we will show how these methodsand concepts have been applied successfully to complexmedia. Starting point will be the paradigmatic case ofthe so-called “opaque lens” concept (Cartwright, 2007),which has a large range of applications, in particular inimaging. We will detail a few more specific systems ofparticular interest in part IV.E.While many techniques described here are nowadaysstandard in optics, the reader from the mesoscopicphysics community might not be familiar with them. Wewill therefore take a pedagogical and historical approach,and show how optical elements have evolved from pas-sive to active and finally to digital elements, which canmeanwhile not only compensate weak perturbations ofthe wave front, but also strong perturbations as in themultiple scattering regime.
A. Wavefront shaping concepts and tools
1. Classical optical elements
Classical optics relies on a variety of linear optical el-ements such as mirrors, lenses, prisms, to convenientlymanipulate light for bending, deflecting or diffracting toachieve a particular goal. In the paraxial approxima-tion fields propagate at low incidence in a given direction(with 𝑧 being the propagation axis) and most optical el-ements can be modeled as transforming the optical fieldby modulating it spatially in phase and amplitude in athin layer in the ( 𝑥, 𝑦 ) plane. For instance, a lens adds aquadratic phase to the beam, thus focusing a plane waveto a diffraction limited spot. A grating will periodically(in space) modulate the phase or the amplitude of thetransmitted or reflected wave, thus diffracting light inwell-defined directions. An amplitude mask, be it a slit,a hole, or a complex image, will transmit and diffract apartial field. In all these cases, we can write the effect ofthe optical element formally as a well-defined and fixedspatial mask in phase and amplitude. In between thethin optical elements, one usually considers homogeneousmedia (glass, air, etc.) where the propagation can be de-scribed using a whole variety of approximations ranging Figure 15 (color online). Widely used types of digital spa-tial light modulators (SLMs). (top) MEMS-based binary am-plitude modulators from Texas Instruments (adapted from(Rabinovitz, 2011)). (middle) MEMS-based phase-only SLM,available both with isolated pixels or with a deformable mem-brane (adapted from (Bifano, 2011)). (bottom) Liquid crys-tal based phase-only SLM. left: photo (courtesy of HoloeyePhotonics AG), right: working principle in phase-only mode(courtesy of Monika Ritsch-Marte). from paraxial rays to a full electromagnetic description.In the propagation through thin optical elements and ho-mogeneous media spectral dispersion can be easily takeninto account.Formally, if we consider a monochromatic paraxial field E 𝜔 ( 𝑥, 𝑦, 𝑧 ) at frequency 𝜔 , its transmission through anoptical element will be described by a transmission func-tion 𝑡 𝜔 ( 𝑥, 𝑦 ), which can be a phase-only (complex of normunity), amplitude-only (real), or a phase and amplitudefunction simultaneously. If 𝑡 𝜔 ( 𝑥, 𝑦 ) depends explicitly on 𝜔 (such as a prism or a grating), then the optical ele-ment is dispersive. For conventional passive optical ele-ments, this transmission is less or equal to one in absolutemagnitude, and does not change over time (in contrastto temporal modulators discussed in subsection IV.B.4).Obviously, the polarization of light and polarizing ele-ments can also be taken into account, by writing thetransmission as a tensor.All imaging systems, such as microscopes, telescopes,0etc. can be modeled by such a formalism. Getting a sharpimage on a detector means carefully designing the opti-cal system to ensure low chromatic or spatial aberrations.This requires high quality optical components (planeityof the mirrors, curvatures of the optics, dispersion) toensure stigmatism, i.e., diffraction limited images. Care-fully designed optics can compensate for natural disper-sion and simple aberrations. For instance, high numeri-cal microscope objectives can be pre-compensated for thespherical aberrations and the chromatic dispersion intro-duced by a glass cover slide. We will see how this imagingparadigm is modified when the propagation occurs in acomplex medium.
2. Active and adaptive optics
While well-determined transformations, such as thoseintroduced by a homogeneous medium or by imperfectoptical elements, can be readily compensated in an op-tical system, the problem of imaging through a mediumwith unpredictable inhomogeneities of the refractive in-dex has remained a major challenge. Even weak spatialvariations of the refractive index can introduce signifi-cant local phase variations when adding up during prop-agation (Lee and Harp, 1969). In particular, these localphase variations will degrade the so-called point spreadfunction, i.e., the resolution of the optical system. Oneobtains a blurred image, and we say the system is aber-rating. An example of high practical importance is atmo-spheric turbulence that appears on many different tem-poral and spatial scales (Kolmogorov, 1941).Most importantly, these fluctuations cannot be cor-rected by a passive element since they are by nature un-predictable and often time-dependent. Accordingly, theyhave posed a limit for the spatial resolution of conven-tional passive optical systems in imaging, as well as forthe bandwidth in free-space telecommunications. A so-lution to this problem was proposed by Babcock in the1950s (Babcock, 1953), who suggested the use of an ac-tive deformable multi-element that, when coupled witha so-called wavefront sensor, can correct in real time forthese spatial inhomogeneities. This was the base for thefield of adaptive optics, that is now routinely used interrestrial astronomical telescopes (Roddier, 1999). Thetechnique requires fast measurement of the wavefront,then analog feedback over multiple elements to compen-sate for the aberrations faster than the medium evolves(see Fig. 16).In essence, if the aberrating medium can be modeledas a single thin layer of thickness 𝐿 with local change inrefractive index 𝑛 ( 𝑥, 𝑦 ), then its effect on light is to addan additional length Δ 𝑧 ( 𝑥, 𝑦 ) = 𝐿𝑛 ( 𝑥, 𝑦 ). This pertur-bation can be canceled by measuring the spatial phaseof the light (the wavefront, see subsection IV.B.1 for adescription of techniques to measure it) at the output of Figure 16 (color online). Principle of astronomical adaptiveoptics. The aberrations from a star can be measured with awavefront sensor, and corrected in real time on a deformablemirror, restoring sharpness of the image on a camera. (Image:Center for Adaptive Optics, University of Santa Cruz.) the layer, then correcting this wavefront to re-obtain theoriginal unperturbated wave by inserting a deformablemirror of shape − Δ 𝑧 ( 𝑥, 𝑦 ) at a later stage in the opticalsystem, in a plane that images the original plane (called“conjugate plane”). This correction method necessitatesthe following two remarks: (i) As long as the index isindependent of the wavelength, the correction is by na-ture broadband because the path difference is corrected,rather than the phase: all wavelengths are corrected si-multaneously. (ii) Since the mirror completely correctsthe aberrations of the layer, any incoming wavefront willbe corrected: the correction is said to be widefield . Notethat the quality of the correction depends on the abilityto perfectly map the path length distribution Δ 𝑧 ( 𝑥, 𝑦 ) onthe deformable mirror, which is limited by the numberof actuators and by their achievable displacement. De-formable mirrors for astronomy typically have 100-1000actuators with relatively long displacement range (10-100microns or more), that are optimized for atmospheric tur-bulence of low transverse spatial frequency. Also, sincepropagation is invariant by time-reversal, it was realizedearly that the correction not only compensates the in-coming light in order to form a sharp image on a detector(for astronomy for instance), it also corrects a backwardpropagating wave, that would traverse the layer in the op-posite direction. This feature can be used in the contextof long distance bidirectional free-space communications(Zhu and Kahn, 2002). Adaptive optics for turbulencemitigation was the first instance in a whole series of digi-tal tools for optical wavefront shaping that we will reviewin the following parts.1
3. From aberrating layers to aberrating volumes
One major limitation of the above approach stems fromthe fact that aberrating layers like the atmosphere arein reality aberrating volumes , meaning that they haveto be described by a three-dimensional distribution ofthe refractive index 𝑛 ( 𝑥, 𝑦, 𝑧 ). The transmission thus hasto be modeled by a more complex transmission matrix t than the one defined in section II and it cannot befully corrected with a deformable mirror. If one mea-sures the wavefront issued from a point source (a starfor instance), then the wavefront corrections applied onthe mirror will perfectly correct the wavefront originatingfrom this source, as if the aberrations were very thin, butwill only correct a small angle (or a small field of view)around it. This so called “isoplanetic patch”, depends onthe thickness and on the distance of the aberrating vol-ume to the detector, and bears some strong similaritieswith the memory effect in the multiple scattering regime(discussed in sections III.D and V.A). In order to correctmore accurately for such a complex propagation, the con-cept of multiconjugation has been introduced (Beckers,1988). There, multiple deformable mirrors in series areplaced in conjugated planes of multiple depths inside thescattering volume, leading to a correction over of a largerfield of view.Another limit of the adaptive optics approach above isthat it works only in the weak aberration regime. Thismeans that locally the variations of the refractive indexmust be small in amplitude, and of low spatial frequency,such that the perturbation of the wavefront is small lo-cally and backscattering is limited. In addition, also theoverall perturbation must be of low spatial frequency inorder to be corrected with a few actuators on the de-formable mirror. When the index of refraction varieswith larger amplitude and over smaller scales, then scat-tering becomes stronger and the spatial frequencies of thefluctuations to be compensated (as well as their ampli-tude) exceed the capabilities of deformable mirrors: theadaptive optics approach breaks down.
4. Optical phase conjugation
The arsenal of techniques for compensating a wave-front distortion found an interesting extension in the1960s when it was realized that one can generate thephase-conjugate of a wave. Such a transformation, thatcorresponds to changing the sign of the complex phaseof the field, makes the wave propagate backward and“undo” the distortion, thanks to the time reversal in-variance and reciprocity of the wave equation. This con-cept was first realized with a holographic method (Leithand Upatnieks, 1966) (see Fig. 17). A hologram can berecorded by interfering both the distorted wave and areference plane wave on a photosensitive plate. By illu-
Figure 17 Two implementations of optical phase conjugation(OPC). (top) OPC via holographic recording (from (Leithand Upatnieks, 1966)). The hologram is the interference ofthe signal with a reference beam and is stored on a photo-graphic plate. The phase conjugate beam can be emitted ata later time by shining the reference beam on the plate. (bot-tom) OPC by four wave mixing. The signal beam is incidenton a non-linear crystal. Two counter-propagating waves (thepump) interfere with the signal beam and generate automati-cally and instantly the phase-conjugate wave via a non-linearprocess (He, 2002). minating the plate with the same reference plane wave,the scattering of this reference beam by the hologramcarries the phase-conjugate of the incident beam, andcan repropagate through the medium (however complexit may be) and reform the initial object or focus (Leithand Upatnieks, 1966). Based on the emergence of non-linear optics in the 1960s and 1970s, it was suggested byYariv (Yariv, 1976) that this holographic optical phaseconjugation could be performed in real time, using vari-ous non-linear processes. An implementation of this con-cept was first realized via four-wave mixing (Bloom andBjorklund, 1977; Yariv, 1978; Yariv and Pepper, 1977),and via stimulated Brillouin scattering (Kr´alikov´a et al. ,1997), in liquid crystals (Karaguleff and Clark, 1990), orusing three wave mixing (Ivakhnik et al. , 1980; Voronin et al. , 1979). Many more details on optical phase conju-gation (OPC) can be found in (Fisher, 2012).In the presence of gain or loss without saturation, themedium is still reciprocal (as discussed in section II.A.4),which is a sufficient condition for a phase conjugatedwave to effectively retrace its path. We will discuss wave-front shaping in presence of gain or loss in section V. Also,even partial phase conjugation will refocus, albeit with alower efficiency. In the monochromatic domain, optical2phase conjugation has led to numerous applications, inparticular for lasers (Brignon and Huignard, 2004). Inthe time-domain, this has also led to spectacular demon-strations in acoustics (Fink et al. , 2000), where transduc-ers naturally provide amplitude detectors and arbitrarywave generators.In a nutshell, all these techniques work independentlyof the complexity of a medium, provided it is reciprocal.Although it had been developed primarily for aberratinglayers, OPC even works for a strongly scattering medium.An important limitation of OPC is, however, that it can-not by itself generate a given wavefront; it requires in-stead an input field or, in other terms, a physical sourcethat will emit a wave. We will see in section IV.D.1 howdigital tools nowadays allow not only for digital-OPC,but also more advanced OPC-like operations, in particu-lar focusing behind a complex medium without a source.
B. Digital tools for wave manipulation in optics
In the optical realm, there is no generic or universalspatial and temporal modulator for light that is ableto generate an arbitrary spatial and temporal waveform,quite in contrast to what is possible for acoustics usinga transducer array, or in the radio-frequency domain us-ing an antenna array. Whereas for these latter types ofwaves the modulators, sources, and detectors can also allbe the same device, this is unfortunately not the casein the optical domain. As we are mostly interested incoherent manipulation in this review, we will focus ourdiscussion on coherent sources such as lasers, that can beeither broadband (for pulsed operation) or narrowband(for monochromatic emission). Other sources will be rel-evant as well, such as superluminescent diodes (SLED)or supercontinuum sources, which are spatially coherentbut temporally incoherent. These light sources can thenbe controlled spatially and temporally by a wide rangeof modulators, that can either be analog or digital. Theaim of this section is to give a flavor to the non-opticianreader of what can and what cannot be done in optics.The emergence of microelectronics has pushed thegeneration and detection of signals into the digitalage. Instead of recording an image on a photographicfilm, and transmitting an image through a fixed op-tical element, multi-element arrays are now routinelyused for spatial modulation or spatial detection in cam-eras (charged coupled device (CCD), or complementarymetal–oxide–semiconductor (CMOS) technologies in par-ticular). We will focus our description on the modulationpart, which is less well-known and more critical for thepurpose of controlling light in complex media. For spatialmodulation we can distinguish between amplitude modu-lators, i.e., display devices (such as the liquid crystal dis-lay (LCD) of modern televisions) and phase-modulators(that are also based on liquid crystal technologies). These modulators are usually referred under the generic termof spatial light modulators (SLMs). While it is by nomeans the purpose of this review to enter into the tech-nical details of these devices, we will try to review theirdesign principle, as well as the current state of the artof their performance. We will leave out of this reviewthe deformable mirrors that have been mentioned in theprevious section for adaptive optics, but are in little usein complex media studies. Two aspects of SLM devicesand digital cameras make them invaluable tools for meso-scopic physics investigation. The first one is that the sizeof individual actuators or detectors can be of the order ofa few microns, which means that in a microscopy systemthe diffraction limit can easily be reached, both for de-tection and for modulation. The second advantage is themassively parallel nature of these arrays, which can inprinciple give easy access simultaneously to up to a fewmillions of degrees of freedom, at a very small cost. Thisfeature provides an unprecedented technological toolsetfor manipulating or detecting optical waves in complexmedia.
1. Matricial detector arrays
CCD and CMOS cameras are now in everyday use forimaging in the visible range. Other types of sensor ar-rays for different wavelength domains also exist, such asInGaAs detector arrays for the near infrared range, forinstance. Typically all such electronic devices are tooslow to access the waveform of an optical field in real-time, in contrast to what is possible with acoustics andwith radiofrequency signals. Whatever the conversionprocess, photodetectors are sensitive only to the inten-sity of the field, i.e., they integrate the local energy overa certain duration that is much longer than the period.As a result, the duration of an optical pulse cannot bemeasured when it is shorter than the integration time ofthe photodetector. Another essential limitation in opti-cal detection is that the phase of the field is not directlyaccessible.Fortunately, for monochromatic fields, several tech-niques allow to retrieve the complex amplitude and phaseof an image. The most widely used method is digitalholography, that relies on recording a stationary interfer-ence pattern (or “hologram”) on the camera, between themonochromatic image to record (called the signal) and aplane wave at the same frequency (called the reference).The complex field of the image can be inferred either bytaking several holograms (at least three) while adding aglobal phase to the reference, resulting in what is calledphase-shifting digital holography (Yamaguchi and Zhang,1997). The complex field can also be retrieved in a sin-gle image, using a tilted plane wave as a reference fol-lowed by a digital filtering, a technique called off-axisholography (Cuche et al. , 2000; Leith et al. , 1965). Fi-3nally, some techniques allow complex field recovery bytaking images in different planes and solving an inverseproblem, e.g., based on the transport of intensity equa-tions (Teague, 1983). Alternatively, computational tools,such as phase-retrieval of phase-diversity algorithms can,in some instances, allow to retrieve the phase of an un-known field from intensity images (see (Fienup, 1982) fora review).When the signal is smooth and slowly varying in space,and when one in interested only in measuring the wave-front, other sensing techniques provide easy access tothe local phase gradient without a reference, even whenthe illmumination is incoherent. This is the domain ofwavefrond sensing, dominated by the so-called Shack-Hartmann wavefront sensor (Platt and Shack, 2000),which is widely used in adaptive optics experiments, butother types of wavefront sensors exist (Bon et al. , 2009;Ragazzoni, 1996). Note that all wavefront sensors com-bine a diffractive element (a lenslet array for the Shack-Hartmann sensor for instance) followed by a CCD orCMOS sensor.
2. Matricial spatial light modulators
In section IV.A.2, we have already described de-formable mirrors, that could be used for a digital andsmooth deformation of a wavefront as needed for adap-tive optics. Thanks to the speed and the long courseof the actuators, deformable mirrors are well adapted tofast and broadband compensation of atmospheric pertur-bations, but they have a limited number of actuators, andan inherent cross-talk between the pixels due to the factthat the actuators deform a common membrane. Whilethis is an advantage for smooth compensation of aberra-tions, these features are impractical when highly complexwavefronts are needed; in this case SLMs with very highspatial resolution and independence of the spatial controlare crucial.Matricial SLMs (see Fig. 15) satisfy these stringentconditions. They comprise an array of pixels that can lo-cally modulate the light. In practice, a wave incident onthe SLM is transmitted or reflected off each pixel with alocally determined partial transmission or reflection (foramplitude modulation) or with a path retardation or ad-vance (i.e., a phase delay). The resulting spatially mod-ulated wave can be designed at will, in general by settingthe voltage applied to each pixel. There is unfortunatelyno “ideal” SLM that can generate arbitrary phase andamplitude modulations. Most SLMs are usually eitherphase-only or amplitude-only, depending on the tech-nology used. Note also that SLMs are typically rela-tively broadband: the path retardation or attenuationaffects all wavelengths the same way. Still, for phase-modulation, a given path retardation means that thephase-retardation is wavelength-dependent, which can be problematic for broadband light. Another feature ofthese light modulators is that they are typically slow (afew kHz at most) such that they cannot perform anyfast temporal modulation. This limitation usually comesfrom the physical mechanism at the heart of the mod-ulation process such as a mechanical displacement or amirror or the slow rotation of a liquid crystal. In thissense, the control is truly spatial only (with a slow evolu-tion in time). We will detail in the next section how thetemporal or spectral degrees of freedom of the light canbe controlled, when true temporal control is required.The most straightforward way to spatially modulatelight is to reflect it off a reflective array of micro-actuators(i.e., a set of movable mirrors) that are electrically ormagnetically actuated (Gad-el Hak, 2010). These are theso-called digital micromirror devices, and they belong tothe very general class of MOEMS (micro-opto-electro-mechanical systems). A linear (piston-like) movementof each mirror will result in a pure path length changeΔ 𝑧 , resulting in a modulation of the phase 𝜑 = 2 𝜋 Δ 𝑧/𝜆 .Pure phase modulators are not very widespread but ex-ist, up to 64 ×
64 pixel arrays (Cornelissen et al. , 2012;Gehner et al. , 2006), and are used for adaptive optics.Spatially modulating the intensity of light, on the otherhand, would require a continuous change of reflectivity,which is actually very difficult to achieve in practice withMOEMS. Still, MOEMS amplitude modulators are nowmass-produced for display applications, but they are dy-namic binary modulators based on the technology de-veloped by Texas instruments Corp., where the mirrorcan switch very rapidly between two angular positions(one where light is reflected, and one where light is de-flected towards an absorber). Intermediate intensity lev-els can be achieved, but only on average, thanks to a veryfast mechanical switching time. The level of reflected in-tensity is then obtained by controlling the ratio of on-and off-time (Hornbeck, 2001). To be more concrete,we provide some typical numbers in the following: TheDLP 0.95 1080p chipset from Texas Instruments has a1920 × 𝜇 m with a tilt angle of 12 ∘ betweenthe two switching positions, and a modulation speed of23 kHz.Another widespread technology for spatial light mod-ulators relies on liquid crystals. In essence, a modulationof the local polarization, and pure-phase control of thelight can be locally achieved by modifying the orienta-tion of a liquid crystal based on its birefringence. A localorientation of the liquid crystal can be achieved electron-ically or optically (Lueder, 2010). With additional po-larization optics, amplitude modulation can also be re-alized, a feature which led to the development of liquidcrystal displays (LCD). Some devices work in transmis-sion, but most devices work in reflection. We will focusour discussion on phase modulators, that are used in thevast majority of the experiments covered in this review.4Since phase modulation is achieved by a small changeof the refractive index, only limited retardation can beachieved. With at least 2 𝜋 phase retardation, it allows forarbitrary phase form generation at a given wavelength.However, unlike deformable mirrors, it does not permitbroadband retardation patterns because of the problemof “wrapping” of the phase for path delays of more thana wavelength. In applications where speed is required,liquid crystal-based SLMs are sometimes too slow: thelimiting speed factor is the unavoidable time requiredfor the liquid crystals to orientate. Typical twisted ne-matic liquid crystal SLMs have around 10 pixels, likedigital micromirror devices (DMDs). They allow purephase modulation over 2 𝜋 in the visible to near-infraredrange, at the cost of a refresh rate of a few tens of Hztypically. Other types of liquid crystals are also used asSLMs, e.g., based on the ferroelectric effect. These de-vices are usually limited to binary phase modulation andare to date much less used for wavefront shaping in com-plex media, despite having a very fast switching time ofonly 100 microseconds, and a refresh rate of up to severalkHz.There are a number of techniques that allow the useof these SLMs to generate any wavefront shape in bothphase and amplitude. One possibility is to combine aphase and an amplitude modulator in series, but this ap-proach is expensive, complex to implement, and thereforerarely used. Thanks to the propagation law of Fourieroptics, it is also possible to compute the wavefront froma single modulator (amplitude or phase, binary or not)that will generate an arbitrary object in a Fourier plane(Conkey et al. , 2012b; Lee, 1978) (for instance in the focalplane of a lens). It is also possible, by appropriate filter-ing in the Fourier space, to generate an arbitrary phaseand amplitude object from a phase-only modulator (vanPutten et al. , 2008), and even from a binary modulator(Goorden et al. , 2014a). These techniques can be usefulwhen one wants to generate a specific input state withhigh accuracy, as we will see in the next section. Notealso that SLMs are two-dimensional masks: while theycan generate complex spatial fields, they cannot performan arbitrary transform that would be necessary to per-fectly compensate for a volume scattering or an arbitrarytransmission matrix t . Some works, however, have sug-gested theoretically and experimentally that it is possibleto use several SLMs in series, or multiple reflections offa single SLM, to generate an arbitrary spatial unitarytransform (Morizur et al. , 2010). While promising, theseimplementations of unitary transforms are not yet ma-ture enough to have found applications in complex me-dia, since the number of degrees of freedom that can beprogrammed is still limited to a few tens of modes max-imum. First successful applications to few-modes fibershave emerged recently (Labroille et al. , 2014).In terms of applications, SLMs are now widely used inoptical laboratories as reconfigurable diffractive optical elements. They are extremely versatile tools to generatecomplex spatial states of light. In particular, the gen-eration of high orbital angular momentum states (Gib-son et al. , 2004), of states with multiple focii for opticalmanipulation (Grier, 2003), and of complex illuminationspatterns for microscopy (Maurer et al. , 2011) should herebe mentioned.
3. Other types of spatial light modulators
In addition to SLMs, several other devices allow spatialcontrol of an optical beam. The most relevant ones in thecontext of this review are deflectors, i.e., devices that areable to deflect and rapidly scan angularly a laser beam,thus producing a translation of the beam in the far field.Two orthogonal deflectors can be used in series for 2D de-flection. They are commonly used in such a configurationfor scanning microscopes (Mertz, 2004) to raster scan a2D region with a focused laser beam in order to recoveran image. There are two main types of such deflectors:The first one are acousto-optic deflectors (AODs), thatuse diffraction of a laser beam on an acoustic wave. Byvarying the acoustic frequency, the deflection angle canbe varied (Gottlieb et al. , 1983). The second type arescanning mirrors, or galvanometers: A lightweight mirroris placed on a fast rotating stage, that produces angularrotation of the mirror. While deflectors cannot generatearbitrary wavefronts, they are well adapted to raster scanthe different input modes of a medium. Both device typeswork similarly, but each has specific advantages. Mirrorscan be large (several millimeters) and therefore may re-flect and deflect complex wavefronts, which can be usefulwhen one wants to exploit the memory effect (see discus-sion in V.A). We will see examples of such devices usedin conjunction with matricial SLMs. AODs on the otherhand, require diffraction limited beams to pass througha small aperture, but can be orders of magnitude faster(in the MHz regime versus kHz for mechanical scanners).
4. Temporal modulators
We have seen in the previous section how to spatiallyshape an arbitrary wave. We will now review the differentways in which one can achieve temporal control in optics.Note that this control is normally achieved on a singlespatial mode only.For a laser source (e.g., a monochromatic laser at fre-quency 𝜔 𝐿 ), it is possible to use an electro-optic mod-ulator (EOM) or an acousto-optic modulator (AOM) totemporally modulate a beam (in amplitude or in phase)by modifying the index of refraction of the propagationmedium. The effect of this modulation is best understoodas a modulation of the transmission in the frequency do-main, 𝑇 ( 𝜔 ). Since the frequency domain that is avail-5able for this kind of modulation ranges from the MHzrange (AOM) to a few GHz (EOM), the effect of thismodulation on an optical signal can be understood as aslow envelope modulation. In practice, a modulation at 𝜔 𝑀 will create new sideband frequencies in the spectrumat 𝜔 𝐿 ± 𝜔 𝑀 (Wooten et al. , 2000). In the context ofscattering through complex media, temporal control onthis spectral range would primarily be interesting if themedium exhibits a temporal response on a similar time-scale (nanoseconds, to microseconds), which is importantfor either very large scattering systems or for long opticalfibers.The second kind of temporal modulation is spectralshaping, where one tries to directly control the opticalwaveform. Since the optical frequencies involved preventthe direct use of electronics, one needs to manipulateinstead the spectral components of the light to controlits temporal profile. For this technique, one requires abroadband coherent source such as a mode-locked laser.This type of source consists of a laser with broad inho-mogeneous gain where several frequencies can lase simul-taneously. A mode locking mechanism (usually a Kerreffect, or a spatial filter) favors emission from all thesesmodes with a fixed relative phase. In such a situation, thelaser emits ultrashort pulses with a Fourier limited dura-tion, and a repetition rate given by the round-trip timein the cavity. A very typical system, the Titane-Sapphirefemtosecond laser, shows gain in the 700-1100 nm band.Most sources emit with a central frequency in the near-infrared around 800nm, with a bandwdith of 10-100 nm,thus providing pulses in the 10-100 femtoseconds range,typically. The repetition rate is usually around 80 MHzfor these lasers, meaning that around 12 ns separates twosuccessive pulses.Starting from such Fourier-limited pulses, it is possi-ble to modify at will the temporal shape of the pulsesby controlling their spectral phase. For this domain ofpulse shaping we refer the reader to several detailed re-views (Cundiff and Weiner, 2010; Monmayrant et al. ,2010; Weiner, 2000). While there are different ways toshape temporally an ultrashort pulse, we will focus in ourdiscussion on the most common implementation, i.e., theFourier-transform pulse shaping (see Fig. 18). In essence,a dispersive element (prism or grating) disperses the dif-ferent spectral components. In a Fourier plane of thedispersive element (e.g., in the focal plane of a lens),each spectral component is focused at a different position.A linear spatial light modulator in this plane then addsan arbitrary phase 𝜑 ( 𝜔 ). When recombining all spectralcomponents, a pulse is formed with the chosen tempo-ral shape governed by the spectral phase imposed by theSLM. The pulse shape is arbitrary, but the maximumarbitrary pulse duration is determined by the resolutionof the apparatus (i.e., limited by the pixel size and thespectral resolution of the dispersion line), and the refreshrate is limited by the SLM speed. ! Figure 18 (color online). Principle of Fourier transform pulseshaping. An ultrashort pulse is incident on a dispersive el-ement (here grating), that separates spatially the differentspectral components, which are then individually addressedby means of an SLM, thus generating a controllable spectralphase function. The different spectral components are laterrecombined into a single spatial mode to form a pulse with awell-defined temporal shape. (Figure adapted from (Weiner,2000).)
C. The thin disordered slab: an opaque lens
In section II, we have treated the important case of adisordered wire, and introduced in this context the con-cepts of ballistic wave scattering, transport mean freepath, modes, the scattering matrix, etc. These conceptsfrom the mesoscopic formalism will now be mapped tothe optical domain. However, there is no exact opti-cal equivalent of the disordered wire. In particular, theimpenetrable and lossless boundaries of a wire cannotbe easily reproduced in optics. Multimode optical fiberscould be considered a stricly bounded complex systemwith a limited number of modes (and will be described insection IV.E.1), but they behave very differently from thedisordered wire since there is no significant bulk disorder(scattering mostly comes from the boundaries), and al-most no backscattering. Rather than a multimode fiber,the paradigmatic system in optics is the slab geometry:a disordered slab of finite thickness 𝐿 and of infinite lat-eral extension, with a transport mean free path ℓ ⋆ that isshort enough to push the system into the multiple scat-tering regime, ℓ ⋆ ≪ 𝐿 . From now on we will refer to thissystem as the “opaque lens” (Cartwright, 2007).A common experimental realization of the opaque lensis typically a layer of dielectric scatterers of micrometeror sub-micrometer size, randomly packed, deposited ona transparent holder, such as a glass slide. To ensuresufficient scattering, the layer should be thick enough (afew micrometers to a few tens of micrometers), produc-ing a white and opaque appearance, provided that thematerial is non-absorptive (see Fig. 19). This realizationhas the specific advantage of being easy to fabricate andextremely stable. Wavefront shaping techniques are bynature relatively slow, making the stability of a partic-ular realization of disorder an essential requirement. Ofcourse, this simple system can be mapped to several prac-tical situations and materials. Snow, biological tissues,6 RESULTS
TURBIDITY SUPPRESSION BY OPTICAL PHASE CONJUGATION
The salient features of TSOPC are well illustrated in the firstexperiment. We used a photorefractive 45 -cut 0.075% Fe-dopedLiNbO crystal as the OPC light-field generator or phase-conjugatemirror (PCM). The recording and playback scheme (Fig. 1a) isdetailed in the Methods. In this study, our target was a 0.46-mm-thick chicken breast tissue section. Light at l ¼
532 nm, 3.5 mWpower and with a 1 / e beam size of 0.7 mm was transmittedthrough a standard negative United States Air Force (USAF) target,that is, a resolution test slide with a clear pattern on a chromebackground. The patterned light was then imaged onto the frontface (face 1, Fig. 1b) of the tissue section (S) using a 1:1 imagingrelay lens. The forward-scattered light traversed the tissue sample,exited from face 2, and arrived at the photorefractive crystal forholographic recording. The recording geometry is examined moreclosely in Fig. 1b. In the experiment, the separation between thecrystal facet and the tissue section was 0.5 mm. The 7.8-mm 1 / e diameter, 6.5-mW reference beam used during the recordingprocess crossed the crystal at a 1-mm distance from the crystalfacet facing the tissue section. This implies a nominal maximumrecording angle range of 66 . The hologram recording time was ! a a ! " with a scattering medium can be expressed as (Fig. 1d) b b ! " ¼ ! S ! S ! S ! S ! " a a ! " ð Þ where ! S ! S ! S ! S ! " is the scattering matrix associated with themedium, b b ! " is the output light field and a ¼
0. Thesubscript denotes the terminal face 1 or 2 of the scatteringmedium. In this case, the light field impinging on the PCM isgiven by S¯ a . The OPC light field travelling back towards face 2of the tissue section can be expressed as c phase conjugate ¼ ! A ! S % a % ð Þ where ! A represents the reduction in angular range of the reconstructedwave owing to the incomplete recording and playback of thetransmitted wave. The reconstructed light field on face 1 of the tissuesection can be written as c reconstructed ¼ ! S ! A ! S % a % : ð Þ As the relationship between any two points on the medium’ssurface is symmetrical we obtain ! S ¼ ! S T .In an ideal case, the capture of the initial light transmission iscomplete, which leads to a reduction of ! A to a unitary matrix.Also, the medium in such a case is lossless and backscattering isabsent—this leads to ! S ! S y ¼ I by energy conservation, where ! S y is the complex conjugate of ! S T . The reconstructed light-fieldexpression can then be written as c ideal ; reconstructed ¼ ! S ! S y a % ¼ a % : ð Þ The extent to which an experimental realization approaches this idealis verified in our experiment (Fig. 2). Figure 2a shows USAF targetimaging through a 0.46-mm-thick agarose section and Fig. 2b fora tissue section of the same thickness, using plane-waveillumination. At l ¼
532 nm, the chicken breast tissue scatteringcoefficient was 38 mm (quantified through interferometricmeasurement of ballistically propagating transmission throughtissue) and the sum of the reduced scattering coefficient and the P1M1 USAFtargetLaser λ = 532 nm WP2at 45º Referencebeam LiNbO CCD M2c-axisBS SL3 L4L5Collimatedbeam RL P2 M3M4 ConjugatereferencebeamWP1 WP3at 45ºL1 L2 ReferencebeamRL TissueUSAFtarget LiNbO Face 1 Face 2 Conjugatereference beamRL TissueCP LiNbO a Scatteringmediumb a b Face 1 Face 2
Figure 1
Schematics of TSOPC set-up and scattering medium. a , Experimental set-up to confirm the TSOPC phenomenon in biological tissues. The concentric(black) dots and circles represent vertical polarization, and the double-ended arrow in the plane of the paper symbolizes horizontal polarization. b , c , Schematics forrecording of tissue turbidity information ( b ) and reconstruction of the OPC light field ( c ), respectively. d , Schematic of a scattering medium. a i and b i are the complexincident and scattered fields, respectively, at the i th face of the scattering medium. L i , i th spherical lens; RL, relay lens; CP, compensation plate; M i , i th mirror; WP i , i th half-wave plate; P i , i th polarization beam splitter; BS, 50 /
50 beam splitter; S, scattering sample.
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36 Experimental apparatus
Figure 2.13: a )Scanning electronmicroscopy imageof aZnO sample(side view). Boththesample-airinterface (top) and the sample-glass interface (bottom) are visible. b ) Photograph of aZnO sample (on a background of millimeter paper). Various other suspensions were used to make samples. The most homogeneoussamples were obtained with a mixture of 2.5 parts airbrush paint to 1 part acrylicmedium . Wealsocreatedsamples usingasuspension of5gofrutileTiO pigment in10 mlairbrushthinner. Boththesamples withairbrushpaint andthe samples withrutile pigment are weakly fluorescent, which makes it hard to distinguish betweenthe probe signal and the background fluorescence. Therefore, for the experiments inChapter 5 we used ZnO samples, which do not fluoresce in the relevant wavelengtharea. Allelementsoftheexperimentarecontrolledandsynchronizedbyacontrolprogram.This programmanagesframe grabbing,detectortriggering,videohardwareaccelera-tion, optimization algorithms, user input, realtime visual feedback, data storage andmore. To keep a program of this scope maintainable we used a component baseddevelopment strategy. Instead of thinking of the program as a whole, we separatedit into self-contained components that offer a predefined service. An excellent intro-duction into the ideas behind of component based programming and agile softwaredevelopment can be found in Ref. 26.Each component implements one or more interfaces. An interface is a specifica-tion of a set of properties (publicly accessible data) and methods (executable func-tions) with acertainfunctionality. Forexample, the interface ofacamera componentdefinesamethodfortriggeringthecamera,anditdefinespropertiessuchastheshut- Hansa pro-color opaque white. Pigmented, water-based acrylic polymer paint. Pigment: TiO PW 6,CI Schminke50602acrylicairbrushthinner SachtlebenRutilR210 / Figure 19 (color online). The opaque lens in optics: a thindisordered slab of scattering material. (left) Schematic model(adapted from (Yaqoob et al. , 2008)). (center) Scanning elec-tron microscope (SEM) side view. (right) Photograph of aZnO opaque lens. (Middle and right panels adapted from(Vellekoop, 2008).) white paper, egg shell, bones, are just a few examples ofthin or thick materials that can be understood with thesame formalism. An important difference between thesesystems and electronic mesoscopic systems is the verylarge number of modes supported: Since objects in op-tics are usually macroscopic, often in the millimeter scaleor larger, the number of optical modes, which scales as
𝐴/𝜆 (with 𝐴 being the transverse area and 𝜆 being thewavelength), can easily be in the 10 range or higher.Most of the results extend also to dynamical systems(such as milk, fog, clouds etc.) although the geometryand the fast dynamics makes them very challenging forwavefront shaping.To describe an opaque lens more formally, let us nowconsider a three-dimensional slab of finite thickness 𝐿 ,that features complex inhomogeneities of the refractiveindex, such as to be fully disordered. Such a case is re-alized for a slab composed of a random packing of parti-cles of high refractive index of different sizes, in a matrixof low refractive index (as, e.g., in air). These inhomo-geneities scatter light in a very complex way, but as wehave seen in section II, this does not prevent us fromdescribing such a system with the formalism of the scat-tering or transmission matrix, as long as the system islinear (see Fig. 19).We will consider the case when the light that is trans-mitted through the sample has been multiply scattered(i.e., ℓ ⋆ ≪ 𝐿 ) and absorption can be neglected (althoughabsorption or gain do not necessarily break the linearityassumption). We also need to make sure that the scatter-ing strength is not too strong ( 𝑘ℓ ⋆ ≫ 𝐶 (1) (defined in section III.B).
1. Transmission matrix in the spatial domain
In this section, we consider the transmission matrix ofa disordered slab, i.e., of an opaque lens. This transmis-sion matrix will be labeled as ˜ t , and links the fields ofthe 𝑁 input to the 𝑀 output pixels of the SLM and ofthe detector, respectively.We have seen in Eq. (14) of section II that the trans-mission matrix t connects the incoming field modes fromthe left to the outgoing field modes on the right, written c + r = t c + l . The difference between the experimentallymeasured ˜ t and the full transmission matrix t of the dis-ordered medium, will be that ˜ t also comprises the propa-gation of the field from the SLM to the medium and fromthe medium to the detector. In addition ˜ t only containsa small part of the full transmission matrix t , usually de-composed in pixels, which do not constitute a completebasis. Since additionally the pixels, both in modulationand detection, are typically illuminated at close to nor-mal and constant incidence, we can usually neglect inpractice the problem of flux normalization discussed inII.A.4. For simplicity, we will note these outgoing (in-coming) modes as ˜ E out ( ˜ E in ), in which notation the fieldon the 𝑚 -th output pixel is ˜ 𝐸 out 𝑚 = ∑︀ 𝑛 ˜ 𝑡 𝑚𝑛 ˜ 𝐸 in 𝑛 , or equiv-alently ˜ E out = ˜ t ˜ E in . We will explain in section IV.C.3how to retrieve ˜ t .In the same way as the singular values 𝜎 𝑖 of t and7the corresponding transmission eigenvalues 𝜏 𝑖 = 𝜎 𝑖 giveaccess to the physics of wave propagation through thedisordered wire (section II), we will now work with thesingular values and singular vectors of the transmissionmatrix ˜ t for the opaque lens, first in the monochromaticpicture, and then including the temporal and spectralaspects.In the limit defined above, where one has access onlyto a small number of well separated modes of the opensystem, i.e., where each input mode 𝑛 gives rise to an in-dependent speckle uncorrelated with the others, the sin-gular values of the transmission matrix ˜ t are expectedto follow the so-called Marˇcenko-Pastur law (Marˇcenkoand Pastur, 1967) that describes the singular value dis-tribution (SVD) of rectangular random matrices withoutcorrelations. In essence, this law states that, for a 𝑁 × 𝑀 random matrix ( 𝑀 ≥ 𝑁 ) of uncorrelated identically dis-tributed elements, the distribution of singular values ̃︀ 𝜎 (normalized to the average transmission) depends onlyon the ratio 𝛾 = 𝑀/𝑁 ≥ 𝑃 𝛾 ( ̃︀ 𝜎 ) ≈ 𝛾 𝜋 ̃︀ 𝜎 √︁ ( ̃︀ 𝜎 − ̃︀ 𝜎 𝑚𝑖𝑛 )( ̃︀ 𝜎 − ̃︀ 𝜎 𝑚𝑎𝑥 ) ∀ ̃︀ 𝜎 ∈ [ ̃︀ 𝜎 min , ̃︀ 𝜎 max ] ≈ ̃︀ 𝜎 𝑚𝑖𝑛 = 1 − √︀ 𝛾 − and ̃︀ 𝜎 𝑚𝑎𝑥 = 1 + √︀ 𝛾 − . Cor-respondingly, the normalized SVD is bounded in the do-main [1 − 𝛾 − , 𝛾 − ]. We have supposed 𝑀 ≥ 𝑁 ,but the SVD remains the same if we reverse the role of 𝑁 and 𝑀 . The interesting case of the square matrix 𝑁 = 𝑀 gives rise to a circular distribution of the singu-lar values in the interval [0 ,
2] and is usually referred to asthe “quarter-circle law”. Still, for
𝑁, 𝑀 finite, Eq. (51)is an approximation and the eigenvalue density outsidethe indicted interval is exponentially small, but not zero.This result is routinely used in telecommunications toassess the bit rate and the error rate of data transmission(Chizhik et al. , 2003). The Marˇcenko-Pastur distributionhas also been measured experimentally through a multi-ply scattering medium in acoustics (Aubry and Derode,2009; Sprik et al. , 2008). In the next section, we will showhow the transmission matrix ˜ t can be measured and thesame distribution can be experimentally recovered fromit also in optics (Popoff et al. , 2011b) (see Fig. 20). Thequestion that immediately arises at this point is how thisresult relates to the bimodal distribution of transmissioneigenvalues derived in section II. The crucial point to ob-serve here is that in the experimental conditions we haveconsidered, we access only a small fraction of the modes.Under these conditions, the bimodal distribution reducesto the Marˇcenko-Pastur law when decreasing the numberof input and output channel considered, whereby corre-lations get increasingly lost (Goetschy and Stone, 2013).To approach the regime where mesoscopic correlationsstart playing a role, particular care has to be taken tocontrol and measure a large fraction of the modes, alsofor large solid angles (e.g., by using high numerical aper- Figure 3.
Normalized SVDs for different asymmetric ratios =
1, 1.5, 2, 3, 4, 5and 6. Dashed line: experimental results; solid line: theoretical prediction fromRMT. Calculations on experimental TM were performed using the filtered matrix K fil .different values of , we record a large TM with =
11 and then create submatrices by takinglines selected randomly among the lines of the original TM. By increasing we increase theprobability of having neighboring pixels in the TM and then are more sensitive to correlationsbetween nearby pixels. This modifies the statistics and decreases the number of independentoutput segments. This explains the deviation from the theory. Other effects that could bringabout correlations in certain experimental conditions are discussed is section 4.4.
3. Phase conjugation
It was demonstrated in [20, 35] that one can take advantage of multiple scattering to focus ontight spots, thanks to the reversibility of the wave equation.In such an experiment, the responses to short temporal signals emitted by an array ofsources at a receiver at the output of a disordered medium are recorded. Those signals,linked to the Green’s functions associated with the couples (source / receiver), are sent reversedin time. The waves generated converge naturally toward the targeted spot. The waves arefocused in space and time. Phase conjugation is the monochromatic equivalent of the previousconcept named time reversal. Those techniques are robust methods for achieving focusing andimaging. In this section, we will experimentally and theoretically study the efficiency of phaseconjugation on focusing.When using phase conjugation for focusing, it corresponds to shaping the input wavefrontto put in phase the contributions of each input pixel at a desired output target. Denoting by E targetout New Journal of Physics Figure 20 (color online). Normalized distribution of singu-lar values ˜ 𝜎 of transmission matrices of opaque lenses as afunction of the ratio 𝛾 between the number of output to in-put modes 𝑀/𝑁 . (Values of ˜ 𝜎 are normalized to the aver-age transmission and can thus be larger than 1.) Dashedblue line: experimental data,; red line: independent identi-cally distributed random matrices. The discrepancy can beattributed to residual correlations in the measured matrices.For the value 𝛾 = 1 one obtains the so-called “quarter circlelaw”. Adapted from (Popoff et al. , 2011b). ture optics), and for both polarizations. This particulartopic will be covered in section V.B. For the time being,we conclude that in most practical cases, we can considerthe disordered slab as a “perfect” mixer for light, obeyingMarˇcenko-Pastur’s law.So far we have not discussed what happens in reflectionfrom the disordered slab. In principle, the reflection ma-trix can also be defined in the same way as for transmis-sion. However, reflected light comprises not only multiplyscattered light, but also singly scattered components, aswell as all components in between. The reflection willtherefore not be as perfectly “mixed” as the transmis-sion, in particular in terms of the polarization of lightthat is conserved for single scattering and can be par-tially conserved for few scattering events (MacKintosh et al. , 1989). Weak localization effects such as the coher-ent backscattering cone described in section III are alsopresent, although their signature has not been observeddirectly in the optical reflection matrix (see (Aubry et al. ,2014; Aubry and Derode, 2009) for realizations in acous-tics). Still, most of the results and experiments describedbelow translate almost perfectly from transmission to re-flection. We will see how these deviations from perfectmixing can be retrieved and exploited for imaging in sec-tion IV.E.2.8
2. Temporal and spectral aspects
In the temporal domain, light enters a disordered slabof thickness 𝐿 , diffusely propagates in it, and exits oneither side, or is absorbed. Knowing the diffusion param-eters for light, such as the diffusivity 𝐷 = 𝑣 𝐸 ℓ ⋆ / ℓ ⋆ is the transport mean-free-path and 𝑣 𝐸 the energy ve-locity defined at the end of section II.C), it is possibleto recover the so-called Thouless time 𝜏 𝐷 of the medium(Thouless, 1977). The parameter 𝜏 𝐷 corresponds to theaverage time that a photon, already in the medium, takesto reach the medium boundaries, and is related to theThouless number defined in section II.A.6. The time 𝜏 𝐷 scales with 𝐿 / ( 𝜋 𝐷 ), and corresponds to a spectralbandwidth Δ 𝜔 𝐷 = 1 /𝜏 𝐷 . While this time intrinsicallydescribes the photon lifetime in the medium, it is notexactly the relevant quantity for transmission and reflec-tion, where a photon first needs to enter the medium,before exiting on either side. For the thin but multi-ply scattering slab geometry, where ℓ ⋆ ≪ 𝐿 , the reflec-tion time 𝜏 𝑅 is typically much shorter than the transmis-sion time and than the Thouless time 𝜏 𝐷 , since a pho-ton typically only explores a small volume of depth ℓ ⋆ before exiting on the same side; as such, 𝜏 𝑅 is of the or-der of ℓ ⋆ /𝑣 𝐸 . Meanwhile, the transmission time 𝜏 𝑇 willbe on average slightly longer than 𝜏 𝐷 since the photonmust first enter the medium before exiting (Landauerand B¨uttiker, 1987; Vellekoop et al. , 2005), but the dis-tribution of transmission times will have an exponentialtail of exponent − 𝑡/𝜏 𝐷 . A rigorous way to define andto assess these scattering times is through the conceptsof “time delay” and “dwell time”, discussed in sectionII.C. For practical purposes it is often convenient to useMonte Carlo simulations (Patterson et al. , 1989) or tomeasure the times experimentally (Curry et al. , 2011;McCabe et al. , 2011; Vellekoop et al. , 2005) (see Fig. 21for a spatiotemporal speckle and its spatial and tempo-ral average). To these transmission and reflection timesspectral bandwidths Δ 𝜔 𝑇 and Δ 𝜔 𝑅 are associated, whichwill in turn correspond to a spectral correlation of the re-spective transmission and reflection matrices of the slab.
3. Accessing the monochromatic transmission matrix of anopaque lens
The monochromatic transmission matrix of a complexmedium can indeed be measured (Popoff et al. , 2010b).In essence, it is possible to send a set of input spatialmodes, to record for each of these modes the transmittedamplitude, and to determine directly from these input-output measurements the transmission matrix elements˜ 𝑡 𝑚𝑛 linking the input mode pixels of the SLM to theoutput pixels on a CCD camera.In an initial implementation (see Fig. 22), the inputspatial modes were generated using a liquid crystal SLM, ARTICLE (cid:21)(cid:1)
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1434 © Macmillan Publishers Limited. All rights reserved. may result in the onset of thermally induced damage. As shown in time-reversal and wavefront shaping experiments, another striking feature is that the more scattering the medium is, the more e ffi cient the focusing will be. Indeed, the signal-to-noise ratio is governed by the ratio of the Th ouless time of the medium over the initial dura-tion of the pulse, that is, the number of independent spectral degrees of freedom. Furthermore, in contrast with conventional phase com-pensation techniques, here all shaping imperfections a ff ect the sig-nal-to-noise ratio; the temporal duration and spatial localization are limited by the Fourier-limit duration and autocorrelation size of the speckle, respectively .In conclusion, we have characterized the speckle spatio-tempo-ral electric fi eld of a multiply scattered ultrafast pulse. Furthermore, we have implemented an open-loop correction of the spectral phase to produce a naturally Fourier- and di ff raction-limited pulse a ft er the medium at a chosen position. Th e results show the bene fi ts of a spatially resolved measurement: typical control experiments con-sider a spatial average, but for such systems that lack large-scale correlations such averaged measurements are valueless. Moreover, we have demonstrated that these correlations permit spatial control without spatial resolution in the spectral pulse shaper. Our study branches out previous spatial speckle shaping techniques to the temporal domain and bridges the gap with time-reversal experi-ments in acoustics and electromagnetism. Th is capacity to recover a short pulse on a selected spatial speckle point has important poten-tial applications for quantum control and photonics, as well as for the fundamental studies of complex media. It suggests that ultrafast diagnostic techniques, including nonlinear microscopy and time-resolved spectroscopy, may be performed deep within or beyond biological tissue, beyond the range of ballistic photons. Methods
Experimental system . Th e ultrafast source used for this experiment is an 80 MHz oscillator that delivers 4 nJ pulses at 800 nm with a spectral bandwidth of 80 nm. Th e oscillator pulse train is divided to form the two arms of an interferometer. In one arm is placed both a spectral pulse shaper and the scattering sample, the other arm acts as a reference and is recombined with an adjustable relative angle and delay at the entrance slit of a home-built imaging spectrometer . Th e spectral cali-bration was performed using a calibration lamp and veri fi ed against a commercial spectrometer; to ensure consistency the same calibration was used for the pulse shaper. Th e spectrometer response—calibrated against a white-light source—var-ied by < 10% over the bandwidth of the pulse. Th e spectrometer has spatial and spectral optical resolutions of 2.0 (cid:77) m and 0.4 nm, respectively; the two-dimensional detector permits the speckle spectrum to be spatially resolved along the slice that falls on the entrance slit (see inset of Fig. 1). Th e spatio-spectral intensity adhered to the well-known exponential decay probability distribution , con fi rming that the speckle pattern was fully resolved. For a typical measurement, 8 × oscillator pulses are integrated. Th e phase and amplitude pulse shaper comprises a pair of liquid–crystal spatial light modulators in the Fourier plane of a folded double-pass 4 f zero-dispersion line . To optimize the trade-o ff between bandwidth and shaping resolution, the 4 f -line grating is chosen so as to over fi ll the spatial light modulators; thus, the output bears a clipped 30 nm hyper-gaussian spectrum and may be shaped arbitrarily within a 23 ps time window. Th is output is then focused to a waist of approximately 15 (cid:77) m onto the surface of the sample. Th e scattered, transmitted light is collected by a lens with a numerical aperture of 0.25 and imaged onto the spectrometer slit with a magni fi cation factor of 14. SSI technique . Th e SSI technique performs a relative measurement of the spec-tral phase between a reference and unknown pulse. Th e interference between the sample image and reference beams causes interference fringes on the spectrometer, whose spectral and spatial periods are determined by the relative delay and angle of the two beams, respectively. Th ese fringes are additionally modulated by the rela-tive phase between the two beams. S x A y e A y eA s i s y r i r y ky ys ( , ) = ( , ) ( , )= ( ( , ) ( , ) 2 (cid:88) (cid:88) (cid:88)(cid:88) (cid:71) (cid:88) (cid:71) (cid:88) (cid:88)(cid:85) | || [ ] (cid:11) (cid:11) (cid:11) ,, ) ( , )( , ) ( , ) ( , ) ( , ) y A yA y A y y y rs r s r | | || || | cos[ (cid:11)(cid:11) (cid:13) (cid:13) (cid:88)(cid:88) (cid:88) (cid:71) (cid:88) (cid:71) (cid:88) (cid:88)(cid:85) (cid:13)(cid:13) k y y ]. Here (cid:84) is the time delay between the two pulses and k y is the di ff erence between the transverse components of the propagation vectors (such that their subtended angle is (cid:81) = k y /| k |)). A s , A r , (cid:70) s and (cid:70) r denote the spatio-spectral amplitude and phase of the scattered (s) and reference (r) pulses, respectively.To recover the amplitude and phase of the unknown pulse, the raw interfero-gram is thus Fourier transformed along both the spatial and spectral axes, and the relative spatio-spectral phase may be extracted through the fi ltering of an a.c. term followed by the inverse Fourier transform. Th is isolates one of the complex expo-nential terms that correspond to the cosine of the fi nal summand of equation (1); the argument of this exponential reveals the relative spectral phase modulo (cid:111) (cid:80) . Sample preparation . Th e scattering medium is thick layer of ZnO powder (a widely used white paint component) deposited homogeneously on a microscope slide by sedimentation. Th e thickness has been measured to be L = 35 (cid:111) (cid:77) m, the transport mean-free path has been measured as l* = 2.1 (cid:111) (cid:77) m, and absorption is known to be negligible. As L > 10 l *, the multiple scattering regime applies and virtually no ballistic light traverses the medium. References
1. Sebbah, P.
Waves and Imaging through Complex Media (Springer, 2001). (1)(1) I n t en s i t y
210 21043 I n t en s i t y t (fs)0 2,000 –2,000 t (fs)0 2,0004 3 y ( m m ) y ( m m ) Figure 3 | Spatio-temporal focusing.
Reconstructed spatio-temporal intensities | E out ( t, y )| ( a ) before and ( b ) after compensation of the phase at y = 2.66 mm. An intense peak emerges from the background with a contrast ratio of 15. One-dimensional ‘lineouts’ at the location of this peak (projections onto walls) and integrated signal (top) show that the peak is focused in time (black) and localized in space (red). The temporal and spatial widths of the peak are the Fourier-limit pulse duration and the spatial phase correlation distance, respectively. S p a $ a l & i n t e g r a $ o n & I ( t ) & S i n g l e & p o s i $ o n & I ( y , t ) & I(y,t)&
Figure 21 (color online). Representation of a spatiotempo-ral speckle, resulting from the propagation of a focussed ul-trashort pulse through a thin ZnO Sample. The speckle ismeasured along one spatial dimension and as a function oftime. One observe a complex spatiospectral structure 𝐼 ( 𝑦, 𝑡 )with speckle statistics, that are apparent when looking at atemporal or a spatial section for a given time or position, i.e., 𝐼 ( 𝑦 , 𝑡 ) and 𝐼 ( 𝑦, 𝑡 ). When looking at a projection (integra-tion) along the temporal or spatial coordinate, i.e., 𝐼 ( 𝑦 ) or 𝐼 ( 𝑡 ), one retrieves respectively the diffuse halo, and the aver-age temporal broadening of the pulse. (Figure adapted from(McCabe et al. , 2011).) and the output amplitudes were obtained using phase-shifting holography (Yamaguchi and Zhang, 1997), i.e.,by recording several images on the CCD, resulting fromthe interference of the output wave to be measured with areference wave with different phase shifts. Later on, sev-eral variants were used to either modulate or detect theamplitude of the field and recover the transmission ma-trix. In (Choi et al. , 2011b), the medium was illuminatedwith a plane wave, with an angle of illumination thatwas varied using a galvanometer-mounted tilting mirror(Choi et al. , 2007). While this method allows to mea-sure every input angle directly, thanks to the movablemirrors, it does not permit to generate a given arbitrarywavefront directly. On the detection side, in order torecord the amplitude hologram of the output speckle ina single image rather than in a sequence, off-axis holog-raphy was implemented (Akbulut et al. , 2013; Kim et al. ,2012). Using a set of polarization beamsplitters and po-larization optics, it is also possible to control or to de-tect both polarizations at the same time, thus access-9 tions, we achieve an almost phase-only modulation [19]with a ! modulation in phase and a maximum residualintensity modulation below 10%. The surface of the SLMis imaged on the pupil of a 20x objective with a numericalaperture (NA) of 0.5, thus a pixel of the SLM matches awave vector at the entrance of the scattering medium. Thebeam is focused at one side of the sample and the outputintensity speckle is imaged 0.3 mm from the surface of thesample by a 40x objective ( NA ¼ : ) onto a CCD cam-era (AVT Dolphin F-145B). The speckle is stationary wellover the measurement time (several minutes).We choose the size of the input and output independentpixels to have a perfect matching in size between a pixeland a mode, in particular, a CCD pixel has the size of aspeckle grain. Thus, in our setup, the input and outputmodes are the SLM and the CCD pixels, respectively,and the TM corresponds to the system comprised of boththe scattering sample and the optical system between theSLM and the CCD camera. From now on, we fixed both thenumber of controlled segments on the SLM and the num-ber of subdivisions measured on the observation window ofthe CCD to N ¼ .To access the complex optical field, we used interfer-ences with a known wave front and a full field ‘‘four phasesmethod’’ [20]. For any input vector, if the relative phase isshifted by a value " , the intensity in the m th output mode isgiven by I " m ¼ j E out m j ¼ !!!!!!!! s m þ X n e i " k mn E in n !!!!!!!! ¼ j s m j þ !!!!!!!! X n e i " k mn E in n !!!!!!!! þ < " e i " ! s m X n k mn E in n ; (1)where s m is the complex amplitude of the optical field usedas reference in the m th output mode.Thus, if we inject the n th input mode and we measure I m , I ! = m , I ! m and I ! = m , respectively, the intensities in the m thoutgoing mode for " ¼ , ! = , ! and ! = , and wecompute ð I m $ I ! m Þ = þ i ð I ! = m $ I ! = m Þ = ¼ ! s m k mn .For practical reasons, we choose the Hadamard basis asinput basis over the canonical one, whose elements areeither þ or $ in amplitude. It perfectly fits with the useof a phase-only SLM and it also maximizes the measuredintensity and consequently improves the experimental sig-nal to noise ratio (SNR) [21]. For all Hadamard basisvectors, the intensity is measured on the canonical basisof the pixels on the CCD camera and an observed trans-mission matrix K obs is acquired, which is related to the realone K by K obs ¼ K & S ref , where S ref is a diagonal matrixrepresenting the static reference. Ideally, the referencewave front should be a plane wave to directly have accessto K . In this case, all s m are constant and K obs is directlyproportional to K . However this requires the addition of areference arm to the setup, as well as interferometricstability. To have the simplest experimental setup and ahigher stability, we modulate only 65% of the wave front going into the scattering sample (this corresponds to thesquare inside the pupil of the microscope objective as seenin Fig. 1), the speckle coming from the 35% static partbeing our reference. S ref is now unknown and no longerconstant along its diagonal. Nevertheless, since S ref isstationnary over time, we can measure the response of allinput vectors on the m th output pixel as long as thereference speckle is bright enough on the consideredmodes. We will quantify the effect of the reference speckleand show that neither does it impair our ability to focus orimage using the TM, nor does it affects the statisticalproperties of the TM.A good way to confirm the physical relevance of the TMis to use it to focus light on any desired outgoing mode. In[11,14], it was demonstrated that using time reversal, onecan take advantage of multiple scattering to focus on tightspots. TR being a matched filter [11], the energy is maxi-mized both temporally and spatially at the intended loca-tion. The monochromatic equivalent of TR is phaseconjugation, which can be straightforwardly done usingour acquired TM. We expect similar focusing results as in[16], which were obtained through a procedure that ensuresan optimum phase-modulated wave front maximizing theenergy of a given mode. Denoting E target the output targetvector, the input vector that approximates the desiredpattern at the output for a perfect phase conjugated focus-ing is given by E in ¼ K y E target , where y stands for theconjugate transpose. Thus, the theoretical effective outputvector is E eff ¼ O foc E target where O foc ¼ KK y is the so-called time reversal operator [22]. Since our setup is lim-ited to phase modulation only, and given the fact that we donot acquire K but K obs , the input vector for a given outputtarget reads E in ¼ K y obs E target = j K y obs E target j : (2)We use the setup shown in Fig. 1 to record the trans-mission matrix of our system, which is done in approxi-mately 3 min (and requires N measurements), comparableto the time needed to perform an iterative focusing as FIG. 1 (color online). Schematic of the apparatus. The laser isexpanded and reflected off a SLM. The phase-modulated beam isfocused on the multiple-scattering sample and the output inten-sity speckle pattern is imaged by a CCD camera: lens (L),polarizer (P), diaphragm (D).
PRL week ending12 MARCH 2010
Figure 22 (color online). Setup of the first measurement ofthe transmission matrix. The laser is expanded and reflectedoff an SLM. Part of the SLM is unmodulated and serves as areference for the interferometric measurement of the outputfield. The phase-modulated beam is focused on the multiple-scattering sample and the output intensity speckle pattern isimaged by a CCD camera. Additional elements: lens (L),polarizer (P), diaphragm (D). (Figure adapted from (Popoff et al. , 2010b).) ing a polarization-resolved transmission matrix (Tripathi et al. , 2012). Finally, using phase-retrieval algorithms, itis even possible to infer the phase and amplitude of thefield from intensity measurements, without the need ofthe reference (Dr´emeau et al. , 2015). Once the trans-mission matrix has been measured, one can either use itto study the medium, e.g., by looking at the modes (seeFig. 20 and the discussion in section V.B), or to controlthe light transmitted through the medium, as we will seein section IV.D.
4. Accessing the temporally or spectrally-resolved transmissionmatrix
In the monochromatic approach, one characterizes thebehavior of the medium at a specific wavelength, at theexpense of ignoring the richness of the spectral and tem-poral behavior of light in the medium. This additionalinformation can, in turn, be extremely useful when tryingto either control spectrally or temporally the transmittedlight. It also provides additional insights into the modesof the medium.Two approaches have been introduced in order to ex-plore this additional dimension. The first one is basedon accessing a spectrally-resolved transmission matrix,which amounts to measuring a monochromatic transmis-sion matrix at many frequencies. In this way the spec-tral behavior of the medium can be fully determined,provided the measurement is done with a spectral reso-lution comparable to, or better than the spectral corre-lation of the medium. This was achieved, e.g., by usinga tunable continuous-wave laser, and measuring several monochromatic transmission matrices for a set of closely-spaced wavelengths (Andreoli et al. , 2015; Mounaix et al. ,2016a).Another possibility, complementary to the first one,consists in measuring a time-resolved matrix, which canbe conveniently achieved when using broadband light vialow-coherence interferometry. Since the interference be-tween the transmitted light and the reference beam onlytakes place when their path length difference lies withinthe (short) coherence length of the source, it meansthat the recorded interferogram only contains informa-tion about a given fraction of the light, which had atime of flight defined by the path delay of the refer-ence beam. By varying the length of the reference arm,it is therefore possible to achieve a time-resolved mea-surement. This technique was implemented in reflection(Choi et al. , 2013; Kang et al. , 2015) as well as in trans-mission (Mounaix et al. , 2016b).
D. Light manipulation through the opaque lens
Digital tools have provided a way to change the config-uration of the light incident on an opaque lens in a con-trolled way. We will see that, even prior to the measure-ment of the transmission matrix of a complex medium,wavefront shaping tools have allowed some light controlthrough the opaque lens. We take here a didactic ratherthan a historic approach to introduce the different tech-niques and concepts that have been applied to this prob-lem.
1. Time reversal, analog and digital phase conjugation throughthe opaque lens
The concepts of phase conjugation and time reversaltell us that, thanks to the reversibility and reciprocity ofthe wave equation, an initial input wave can be recov-ered, when the wave resulting from the scattering of theincident wave by the slab is phase-conjugated and sentback through the medium. In practice, such a procedurerequires perfect phase conjugation, and thus a collectionof all the scattered light on both sides of the slab. In realexperiments with the slab geometry, however, we gen-erally have access only to one side of the medium andto a limited fraction of the incident light as well as ofthe scattered light. As was shown both in optics andin acoustics, in the case of multiply scattering materi-als (and also in the case of chaotic cavities) even limitedphase conjugation or incomplete time reversal can par-tially reconstruct the initial wave (Calvo and Pastawski,2010; Derode et al. , 1995; Draeger and Fink, 1997). Inessence, the different modes that are phase-conjugatedcontribute in a constructive way to re-inject energy intothe initial mode. If the initial mode originates from a0particular position, the wave will refocus to this posi-tion, with an efficiency that depends on the fraction ofthe energy that is phase-conjugated.Optical Phase Conjugation (OPC) was first performedby recording a hologram on a photographic plate (Leithand Upatnieks, 1966). However, based on the emergenceof non-linear optics in the 1960s and 1970s, it was sug-gested by Yariv (Yariv, 1976) that this holographic op-tical phase conjugation could be performed in real time,using various non-linear processes. An implementationof this concept was first realized via four-wave mixing(Bloom and Bjorklund, 1977; Yariv, 1978; Yariv andPepper, 1977), and via stimulated Brillouin scattering(Kr´alikov´a et al. , 1997), in liquid crystals (Karaguleff andClark, 1990), or using three wave mixing (Ivakhnik et al. ,1980; Voronin et al. , 1979). For a review on OPC, see(Fisher, 2012).For complex media investigations, OPC suffers, how-ever, from several shortcomings that are mainly due tothe limitations of the physical effects giving rise to thephase conjugate of an optical wave. Non-linear wave mix-ing (Gower and Proch, 1994) is usually complex to im-plement, as it requires non-linear crystals, specific wave-lengths and often intense laser sources. Nonetheless,phase conjugation has been employed since its early daysto refocus through a complex medium (Yariv et al. , 1979).Photorefractive crystals are another alternative for OPC,which, albeit being slow, was successfully used to refocusthrough thick biological tissues (Yaqoob et al. , 2008). Re-cently, new photorefractive materials have provided veryhigh conjugation speeds comparable to fast SLMs (Farahi et al. , 2012; Liu et al. , 2015b). Gain Media (such as lasercrystals) also typically provide very fast OPC but workonly for a narrow spectral range. They allow for amplifi-cation of the phase-conjugated wave (Feinberg and Hell-warth, 1980) and have been used for imaging in turbidmedia (Jayet et al. , 2013). Three wave mixing is fast andbroadband, but is only effective over a very small angu-lar range, yet was used for imaging through turbid media(Devaux et al. , 1998). Despite all its constraints, OPChas the advantage that it can conjugate a very large num-ber of modes simultaneously over the surface of the OPCmaterial (Xu et al. , 2011), typically one or two ordersof magnitude larger than what is currently achievable bydigital means. OPC therefore remains a very competitivetechnique, especially for biomedical applications.Thanks to the emergence of digital SLMs, it is nowpossible to envision a digital counterpart of optical phaseconjugation (DOPC). Provided one can measure thecomplex amplitude of a field, an SLM can in principe gen-erate its phase conjugate. First experimental demonstra-tions of this concept were performed to co-phase severalbeams through a fiber bundle (Paurisse et al. , 2009), thenapplied to a thin scattering slab (Cui and Yang, 2010)and later to multimode fibers (Lhermite et al. , 2010; Pa-padopoulos et al. , 2012). In all cases, an input beam systems, DOPC has two significant advantages. First, as an adaptive optics method, the power of the generated OPC wave is independent of the input signal and can be freely adjusted. Second, the same DOPC system can in principle work with both CW and pulsed laser systems at any power levels. Both of these two properties are highly desired for biomedical applications. In addition, the optical degrees of freedom handled by DOPC are significantly greater than conventional adaptive optics methods [17,18] and is capable of achieving phase conjugation through highly turbid samples. As DOPC processes the entire wavefront simultaneously, it is inherently a fast wavefront optimization process and is potentially suitable for in vivo biomedical applications. Since the wavefront is digitally controlled in the DOPC system, we can also use the DOPC system to study the fundamental properties of phase conjugation through random scattering media. One specific problem that the DOPC system is uniquely suited to tackle is the question of how tolerant the process of optical phase conjugation (OPC) through a random scattering media is to phase errors in the phase conjugation wavefront. The DOPC system allows us to introduce phase errors into the wavefront in a well-controllable fashion. Using our prototype, we experimentally found that, counter to intuition, OPC through random scattering media is surprisingly robust in the presence of significant phase errors. Our experimental findings are in good agreement with predictions derived from transmission matrix formulism [3]. In Section 2, we introduce the design of the DOPC system. In Section 3, we discuss the experimental implementation of the DOPC system and its calibration method. In Section 4, we show the experiment that was used to test the accuracy of the DOPC system. In Section 5, we discuss the experiments, in which we used the DOPC system to reconstruct an optical mode through a turbid medium ( µ s l ~13). In Section 6, we analyze the influence of phase errors on the phase conjugation signal through random scattering media and present experimental results. Fig. 1. The two elements of the DOPC system, a wavefront measurement device (sensor) and a spatial light modulator (actuator), are optically combined with a beam splitter. They function as a single system which can both measure an input wavefront and generate a phase conjugate output wavefront. (a) shows the wavefront measurement process wherein a reference wave interferes with the input signal. Their relative phase is controlled by an EO phase modulator. (b) shows the phase shaping process wherein the SLM modulates the incident reference wave.
2. Design
To generate phase conjugate wave digitally, we simply need a device which can be used both as a sensor and as an actuator. The piezo transducer employed in acoustic time reversal experiments is a good example [21,22]. Unfortunately, such a device does not currently exist for optical processing. We can potentially implement an equivalent system by combining a wavefront measurement device (sensor) with a spatial light modulator (SLM, actuator) in an optical arrangement as shown in Fig. 1. Such a composite system should work if the two components are exactly aligned with respect to each other so that each device forms a virtual image on the other device. In other words, we want every pixel of the sensor to form a virtual image on a corresponding pixel of the actuator, and vice versa. We name this approach digital optical phase conjugation (DOPC) because it is optoelectronics in nature. (C) 2010 OSA 15 February 2010 / Vol. 18, No. 4 / OPTICS EXPRESS 3446 ~0.12 micron, which is small compared to the 0.23 micron focus diameter. Considering the 1.8 mm focal length of the 100x objective used in the experiment, we achieved a ~3.2 degree phase conjugation range with a ~3.9x10 − degree accuracy. Figure 6(d) shows the reconstructed focus size as Lens 1 was shifted axially from −
100 to 100 microns. We noticed that the spot size variation was asymmetric. In the case of negative axial displacement, the beam exiting Lens 2 was a diverging beam that could fill the entire SLM. The phase variation near the center was slow and could be accurately sampled and compensated by the DOPC system. The phase variation near the outer area of the SLM could exceed the sampling rate of the DOPC system and cause error. In the case of positive axial displacement, the beam exiting Lens 2 was a converging beam that could only fill the center area of the SLM. The high spatial frequency components were truncated by the objective lens, which caused the asymmetry in Fig. 6(d). As a comparison, we used a mirror to replace the DOPC system in Fig. 4 and performed the lateral and axial displacement experiments. Figure 6(e) shows the lateral displacement results. Linear fitting shows a slope of 1.99, which is expected since the angle deviation doubled upon reflection by the mirror. Figure 6(f) shows the axial displacement result. Without DOPC compensation, one micron displacement caused the reflected spot size to increase by more than tenfold to ~3 micron.
The optical degrees of freedom of the DOPC system are limited by the pixel numbers of the SLM and the CCD camera. Given the number of pixels, we can estimate the amount of lateral displacement that can be compensated by the DOPC system. The SLM employed in our experiments has 768 x 1024 pixels that were mirrored onto an area on the CCD camera which contains slightly more pixels. In the experiments, 634 x 634 SLM pixels were imaged to the back aperture (~5mm in diameter) of the 100x objective lens. The maximum phase difference between adjacent pixels is π (Nyquist frequency), such that the total phase variation across the 634 pixels is 634 π (317 λ ). The maximum angle deviation that can be compensated is therefore 317 λ /5mm and the maximum lateral deviation at the focal plane is then 317 λ f/5mm, where f is the focal length of the objective. For an Olympus 100x objective, the focal length is ~1.8 mm and the theoretical maximum deviation from the center is therefore ~61 micron ( λ = 532 nm). The experimentally achieved lateral compensation range in our system is ~50 micron.
5. Evaluation with a random scattering medium
A potential application of DOPC is to reconstruct an optical mode through a highly turbid medium. To demonstrate such a capability, and as a stringent test of our method, we apply the DOPC system to return an OPC wave through a scattering medium of µ s l ~13. Fig. 7. (a) The DOPC measured phase profile. (b) DOPC reconstructed signal. The field of view is ~12 µ m. (c) Control measurement with the phase of the SLM set to 0. We employed the setup shown in Fig. 4 for the demonstration. To prepare a random scattering sample, we first dried a mixture of polystyrene microspheres of different diameters (0.2, 0.5, 1, 3, 5, 10 µ m) with equal weight percentage in a water suspension on a cover glass. (C) 2010 OSA 15 February 2010 / Vol. 18, No. 4 / OPTICS EXPRESS 3452 (c Figure 23 (color online). Principle of digital optical phaseconjugation (DOPC): (a) in a recording step, the interferencepattern between a reference wave and the signal to be phaseconjugated is recorded on a CCD, phase-shifting their relativephase difference with an electro-optic modulator (EO). In adigital playback step (b), the recorded pattern is displayedon the SLM, and the same reference beam, diffracting on theSLM, generates the phase-conjugate beam: (c) the DOPCmeasured phase profile and (d) the measured signal at theOPC output before the opaque lens, showing a strong focusin the center, on the original signal mode. (Figure adaptedfrom (Cui and Yang, 2010).) is incident on the medium and the transmitted light isrecorded on a camera, that has to be matched pixel topixel to a spatial light modulator situated in a conju-gated plane by means of a beamsplitter (see Fig. 23).One then needs to recover the field on the camera, dis-play the corresponding pattern on the SLM so that a laserbeam reflected off this SLM carries the phase-conjugatedwavefront. In its simplest implementation, off-axis digi-tal holography provides the necessary tool for both oper-ations: A reference plane wave interferes at an angle withthe unknown input wave, producing an interferogram onthe camera that contains the phase and amplitude in-formation of the unknown input field. It can be shownthat the same tilted reference plane wave diffracting offthe same interferogram now displayed on the SLM willgenerate the phase conjugate of the unknown wavefield,thus producing a refocusing on the source.Although it cannot match conventional OPC in termsof number of modes, DOPC has several advantages com-pared to its analog counterpart, in particular to conjugatea speckle field. Specifically, since the complex wavefrontsare actually recorded in a memory, it is possible to recordthe output for different waves and then replay them at1a later time in any order, which is not possible in ana-log phase conjugation, where the hologram engraved in acrystal is transient by nature. Even more interestingly, itis possible to modify the interferogram before displayingit on the SLM, or combine several hologram together,which also brings an additional flexibility compared toits analog counterpart, a feature that will become par-ticularly important as we will see in later examples re-lated to imaging. Still, rather cumbersome alignmentsare required (Cui and Yang, 2010), although simplifiedimplementations were proposed (Hillman et al. , 2013).A promising direction to simplify DOPC is to design aunique monolithic device that would play the role of thedetector and of the modulator simultaneously (Laforest et al. , 2012).
2. Focusing and iterative optimization
The seminal experiment by the group of Allard Moskin Twente published in 2007 (Vellekoop and Mosk, 2007)expanded this concept of OPC to a new level by remov-ing the need for a source. In essence, instead of record-ing a wavefront from a source and then re-emitting itsphase conjugate, thus achieving refocusing, the authorsproposed an iterative optimization technique to find theoptimal wavefront at the input of a disordered slab, thatwould maximize the intensity at a given position at theother side of the slab. In this experiment, each pixelcontrolled on the SLM is assumed to generate an inde-pendent speckle on the far side of the disordered slab, ona camera. The resulting speckle on the camera thereforecorresponds to the coherent sum of all the speckle contri-butions from all input pixels, which by itself is also a fullydeveloped intensity speckle pattern, since each specklegrain at the output is the result of a sum of differentcontributions with uncorrelated phases. In the formal-ism of the transmission matrix, it means that the fieldon pixel 𝑚 is given by 𝐸 𝑚 = ∑︀ 𝑁𝑛 =1 𝐴 𝑛 𝑡 𝑚𝑛 𝑒 𝑖𝜑 𝑛 , where 𝐴 𝑛 is the field incident on input pixel 𝑛 , 𝜑 𝑛 is the phasedelay (or advance) imposed by the SLM at the pixel, and 𝑡 𝑚𝑛 is the transmission matrix of the complex mediumbetween the pixels of the SLM and those of the CCD.By optimizing the phase 𝜑 𝑛 at each input pixel (i.e., bymodifying the spatial wave front) to maximize the inten-sity on a given output position, it is possible to convergeto a constructive interference at this target position. Asimple qualitative way to understand this process is to re-member that the intensity distribution of a speckle is theconsequence of the fact that each speckle grain is a sumof phasors (complex amplitudes) with uncorrelated andevenly distributed phases. When optimizing the phase of 𝑁 input pixels in the above way, the situation where the 𝑁 contributions add with uncorrelated phases is changedto a situation where 𝑁 contributions all add in phase.This corresponds to an increase of the final amplitude of the order of √ 𝑁 , and accordingly to an increase of thefinal intensity that scales with 𝑁 (Vellekoop and Mosk,2007), where 𝑁 is the number of pixels controlled. In thefirst realization (Vellekoop and Mosk, 2007), a focus morethan 2000 more intense than the average of the unopti-mized speckle background was observed (see illustrationin Fig. 24).This important result deserves extensive comments.Firstly, the above methodology assumes full indepen-dence between the different speckles generated by eachpixel, i.e., no correlations must be present, which isone of the main assumptions for the opaque lens. Thismakes the optimization process very simple, since thereis a unique optimum (up to a global phase), that anyalgorithm can find. Only when noise or decorrelationcomes into play will different algorithms perform differ-ently (Vellekoop and Mosk, 2008a). We will detail in sec-tion V.B what happens when correlations are present andhow the results are modified. Secondly, it is interestingto link this approach with phase conjugation. Indeed, itcan be shown that the final wavefront is very close to thephase-conjugate solution or, more precisely, to the phaseconjugate of the field emitted by a source placed at thetarget position, with the important advantage, however,that no source is required. An important difference withrespect to OPC is that the SLM is phase-only, so whilethe spatial phase corresponds to the phase-conjugate so-lution, the amplitude cannot be controlled and dependson the illumination: it is constant for a plane wave inci-dent on the SLM. Still, the wave produces a focus, corre-sponding to the earlier insight in acoustics that the phaseis the most important parameter when the aim is to put amaximum energy at a given point: In terms of signal-to-noise ratio, it was even shown that the focusing efficiencyis nearly equivalent to the one expected for perfect phaseconjugation (Derode et al. , 1999, 2001a,b). Additionally,with the assumption of independence of the input modesand of uncorrelated elements of the transmission matrix,the background speckle is not statistically modified whenthe wavefront is optimized, nor is the energy of the totaltransmission. We will study deviations from this behav-ior in more details in section V.B.Another important feature of the focusing effect is thespatial size of the focus. The output speckle field hasa well-defined grain size which corresponds to its 𝐶 (1) spatial intensity correlation (see section III.B). The opti-mization procedure can create locally a constructive in-terference, and the spatial extent of this focus is givenby the correlation distance, i.e., of the size of a specklegrain. This has two important consequences that haveled to the concept of the “opaque lens” (Vellekoop et al. ,2010): (i) The optimized focus is perfect, in the sensethat it is diffraction-limited, without aberrations, and itsits on a speckle background that can be orders of mag-nitude lower in intensity. (ii) The size of the focus isonly given by the 𝐶 (1) correlation of the medium in this2 Focusing coherent light through opaque stronglyscattering media
I. M. Vellekoop * and A. P. Mosk Complex Photonic Systems, Faculty of Science and Technology and MESA ! Research Institute,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands * Corresponding author: [email protected]
Received March 6, 2007; revised June 14, 2007; accepted June 21, 2007;posted June 28, 2007 (Doc. ID 80762); published August 2, 2007
We report focusing of coherent light through opaque scattering materials by control of the incident wave-front. The multiply scattered light forms a focus with a brightness that is up to a factor of 1000 higher thanthe brightness of the normal diffuse transmission. © 2007 Optical Society of America
OCIS codes: . Random scattering of light is what makes materialssuch as white paint, milk, or human tissue opaque.In these materials, repeated scattering and interfer-ence distort the incident wavefront so strongly thatall spatial coherence is lost [1]. Incident coherentlight diffuses through the medium and forms a vol-ume speckle field that has no correlations on a dis-tance larger than the wavelength of light. The com-plete scrambling of the field makes it impossible tocontrol light propagation using the well-establishedwavefront correction methods of adaptive optics (seee.g., [2]).We demonstrate focusing of coherent light throughdisordered scattering media by the construction ofwavefronts that invert diffusion of light. Our methodrelies on interference and is universally applicable toscattering objects regardless of their constitution andscattering strength. We envision that, with such ac-tive control, random scattering will become benefi-cial, rather than detrimental, to imaging [1] and com-munication [3–5].Figure 1 shows the principle of the experiment.Normally, incident light is scattered by the sampleand forms a random speckle pattern [Fig. 1(a)]. Thegoal is to match the incident wavefront to the sampleso that the scattered light is focused in a specifiedtarget area [Fig. 1(b)]. The experimental setup forconstructing such wavefronts is shown in Fig. 2.Light from a 632.8 nm HeNe laser is spatially modu-lated by a liquid-crystal phase modulator and focusedon an opaque, strongly scattering sample. The num-ber of degrees of freedom of the modulator is reducedby grouping pixels into a variable number ! N " ofsquare segments. A CCD camera monitors the inten-sity in the target focus and provides feedback for analgorithm that programs the phase modulator.We performed first tests of inverse wave diffusionusing rutile ! TiO " pigment, which is one of the moststrongly scattering materials known. The sampleconsists of an opaque, 10.1- " m-thick layer of rutile[6] with a transport mean free path of 0.55± 0.10 " mmeasured at = 632.8 nm. Since in this sample thetransmitted light is scattered hundreds of times,there is no direct relation between the incident wave-front and the transmitted image [7,8]. In Fig. 3 we show the intensity pattern of thetransmitted light. In Fig. 3(a) we see the pattern thatwas transmitted when a plane wave was focused ontothe sample. The light formed a typical randomspeckle pattern with a low intensity. We then opti-mized the wavefront so that the transmitted light fo-cused to a target area with the size of a singlespeckle. The result for a wavefront composed of 3228individually controlled segments is seen in Fig. 3(b),where a single bright spot stands out clearly againstthe diffuse background. The focus was over a factor of1000 more intense than the nonoptimized specklepattern. By adjusting the target function used asfeedback it is also possible to optimize multiple focisimultaneously, as is shown in Fig. 3(c) where a pat-tern of five spots was optimized. Each of the spotshas an intensity of approximately 200 times theoriginal diffuse intensity. In Fig. 3(d) we show thephase of the incident wavefront corresponding to Fig.3(c). Neighboring segments are uncorrelated, whichindicates that the sample fully scrambles the inci-dent wavefront.The algorithm that constructs the inverse diffusionwavefront uses the linearity of the scattering process.The transmitted field in the target, E m , is a linearcombination of the fields coming from the N differentsegments of the modulator: E m = n =1 N t mn A n e i $ n , ! " where A n and $ n are, respectively, the amplitude andphase of the light reflected from segment n . Scatter- Fig. 1. (Color online) Design of the experiment. (a) A planewave is focused on a disordered medium, and a speckle pat-tern is transmitted. (b) The wavefront of the incident lightis shaped so that scattering makes the light focus at a pre-defined target.August 15, 2007 / Vol. 32, No. 16 / OPTICS LETTERS ing in the sample and propagation through the opti-cal system is described by the elements t mn of the un-known transmission matrix. Clearly, the magnitudeof E m will be the highest when all terms in Eq. (1) arein phase. We determine the optimal phase for asingle segment at a time by cycling its phase from 0to 2 ! . For each segment we store the phase at whichthe target intensity is the highest. At that point thecontribution of the segment is in phase with the al-ready present diffuse background. After the measure-ments have been performed for all segments, thephase of the segments is set to their stored values.Now the contributions from all segments interfereconstructively and the target intensity is at the glo-bal maximum. A preoptimization with a small num-ber of segments significantly improves the signal-to-noise ratio. This method is generally applicable tolinear systems and does not rely on time reversalsymmetry or absence of absorption. Although math-ematically this algorithm is the most efficient, innoisy experimental conditions adaptive learning al-gorithms [9] might be more effective, and an investi-gation of such algorithms is on its way.The maximum intensity enhancement that can bereached is related to the number of segments that areused to describe the incident wavefront. For a disor-dered medium the constants t mn are statistically in-dependent and obey a circular Gaussian distribution[8,10–12], and the expected enhancement " , definedas the ratio between the optimized intensity and theaverage intensity before optimization, can be calcu-lated: " = ! ! N − " + 1. ! " It was assumed that all segments of the phase modu-lator contribute equally to the total incident inten-sity. We expect the linear scaling behavior to be uni-versal as Eq. (2) contains no parameters. Also, since we are free to choose the basis for Eq. (1), we expectto find the same enhancement regardless of whetherthe target is a focus or a far-field beam and regard-less of how the shaped wavefront is projected ontothe sample. Interesting correlations between thetransmission matrix elements, which will cause cor-rections on Eq. (2), are predicted when N approachesthe total number of mesoscopic channels [11,12].With our current apparatus we are far from this re-gime and no deviation from Eq. (2) is expected.We tested the universal scaling behavior impliedby Eq. (2) by changing N . Using the same TiO sample as before, the algorithm was targeted to con-struct a collimated beam. In Fig. 4 the enhancementis plotted as a function of the number of segments fordifferent focusing conditions. The linear relation be-tween the enhancement and the number of segmentsis evident until the enhancement saturates at " = 1000. All measured enhancements were slightly be-low the theoretical maximum. This is understand-able since all perturbations move the system awayfrom the global maximum. The main reason for de-viations from the optimal wavefront is residual am-plitude modulation in the phase modulator, which in-troduced an uncontrolled bias in the field amountingto 14% of the total intensity.The saturation of the enhancement is the result ofslow changes in the speckle pattern. This instabilityeffectively limited the number of segments for whichthe optimal phase could be measured. We estimatethat the effective enhancement decreases to " eff = " / ! NT / T p " , where T = 1.2 s is the time needed forone measurement and the persistence time T p = 5400 s is the time scale at which the speckle pat-tern of the TiO sample remains stable. Dependingon the environmental conditions, T p can be consider- Fig. 2. (Color online) Schematic of the apparatus. A632.8 nm HeNe laser beam is expanded and reflected off aHoloeye LR-2500 liquid crystal spatial light modulator(SLM). Polarization optics select a phase mostly modula-tion mode. The SLM is imaged onto the entrance pupil ofthe objective with a 1:3 demagnifying lens system (notshown). The objective is overfilled; we use only segmentsthat fall inside the pupil. The shaped wavefront is focusedon the strongly scattering sample (S), and a CCD cameraimages the transmitted intensity pattern. /4, quarter-wave plate; /2, half-wave plate; M, mirror; BS, 50% non-polarizing beam splitter; P, polarizer. Fig. 3. Transmission through a strongly scattering sampleconsisting of TiO pigment. (a) Transmission micrographwith an unshaped incident beam. (b) Transmission afteroptimization for focusing at a single target. The scatteredlight is focused to a spot that is 1000 times brighter thanthe original speckle pattern. (c) Multibeam optimization.The disordered medium generates five sharp foci at the de-fined positions. (a)–(c) are presented on the same logarith-mic color scale that is normalized to the average transmis-sion before optimization. (d) Phase of the incidentwavefront used to form (c). OPTICS LETTERS / Vol. 32, No. 16 / August 15, 2007 (c
Figure 24 (color online). Principle of wavefront optimiza-tion through a complex medium: A plane wave (a) incidenton an opaque multiply scattering layer of white paint givesrises to a speckle field on the far side (c). After optimiza-tion of the wavefront, an optimally shaped wave (b) gives risea speckle field that has a strong focus at a chosen position(d). (e) A wave front can also be focused on several positionsat the same time. (f) Typical phase mask on the SLM af-ter optimization, showing the apparent randomness and highcomplexity of the obtained solution. (Figure adapted from(Vellekoop and Mosk, 2007).) plane, which is independent of the entire optical systemlocated in front of the slab and its possible imperfections.Hence one can overcome the diffraction limit imposedby the limited angular apertures and the imperfectionsof the optical system. It is interesting to note that thesame concept was previously proposed in adaptive opticsto maximize a focus intensity by dithering the phase ofmultiple elements (Bridges et al. , 1974), albeit with onlya few degrees of control. In contrast to this last work,the optimization through a multiply scattering mediumrequires a very large number of degrees of freedom to beeffective, but takes advantage of the statistical propertiesof the speckle to have a well-defined focusing efficiencyand focus size. We also refer the reader to a review onoptimization methods (Vellekoop, 2015).All the techniques described above can be used to focuslight to a single speckle grain. Digital phase conjugationand optimization techniques readily provide the wave-front that focuses light to one or multiple targets, withthe difference that optimization provides a phase-only ap-proximation of the exact phase-conjugated field. In thecase that the transmission matrix is recorded, the corre- sponding wavefront can also be directly computed, anddisplayed using an SLM. As described in (Popoff et al. ,2010b), the input field ˜ E in that approximates the desiredtarget ˜ E target can be deduced from the matrix as follows,˜ E in = ˜t † ˜E target , (52)where ˜ E target is set to 1 at the desired focus (or focii)position, and 0 elsewhere. To understand this, let usgo back to the fundamental relation ˜ E out = ˜ t ˜ E in , whichwould suggest that to get a desired output field, one needsto invert the transmission matrix. Inversion is, however,rather unstable; an inversion is also sub-optimal for fo-cusing since it would try to match the output as closelyas possible, including minimizing the field outside of thefocus. Taking instead the Hermitian conjugate of ˜ t as inEq. (52) amounts to a time-reversal – or phase conjuga-tion – of the transmitted field (Prada and Fink, 1994).Since, however, the reflected field and the unmeasuredmodes are here not part of the time-reversal (for a unitary˜ t inversion and Hermitian conjugation would be equiv-alent), this reconstruction is not perfect, but turns outto be stable to measurement noise. Depending on themodulation scheme, a phase-only approximation (Popoff et al. , 2010b) or a more exact phase and amplitude input(see, e.g., (Kim et al. , 2012)) can be generated. All thesestechniques basically provide the same phase-conjugatedfield as a solution and share a comparable efficiency (i.e.,proportional to the number of controlled input pixels).For imaging purposes also more advanced operators canbe useful, as discussed in section IV.D.3.Several general remarks can be made at this point:Phase-only and full modulation both provide comparablefocusing efficiencies, up to a factor 2 in intensity. Phase-only modulation does not diminish the overall speckle in-tensity, and is optimal for delivering the maximal amountof energy to a given point (Vellekoop and Mosk, 2007).Amplitude-only modulation has also been shown to per-mit focusing (Akbulut et al. , 2011; Dr´emeau et al. , 2015),by essentially turning off a fraction of the input pixels toleave only pixels that contribute constructively at the fo-cus, thus simultaneously reducing the background. Jointphase and amplitude modulation provides a compromisebetween signal-to-noise ratio and focusing efficiency, bydiminishing the contribution of pixels that contribute lit-tle to the focus, but significantly to the background. Ageneral comparative discussion of the focusing efficiencyin the context of acoustics can be found in (Tanter et al. ,2000) and is fully applicable in optics. In transmission,an additional control of the polarization state does notchange the overall performance, except by doubling thenumber of modes effectively controlled.Extensive studies in acoustics and in the radiofre-quency domain are dedicated to the minimum attainablesize of the focus (Lerosey et al. , 2007). As we recalledalready before, far away from the scattering region thefocus is limited by the speckle grain size (i.e., by the3 𝐶 (1) correlation function), and as such it can surpassthe diffraction limit of conventional optics (Choi et al. ,2011b; Vellekoop et al. , 2010) and can get very close ofthe limit of 𝜆/ (2 𝑛 ) allowed by diffraction. For instancein (van Putten et al. , 2011) it was shown that the focuscould be made smaller than 100 𝑛𝑚 for a monochromaticlaser at 561 𝑛𝑚 and a refractive index of 𝑛 = 3 .
41. Thepossibility to modify the size of the focus by optimiz-ing on a sub-part of the momentum space forming thespeckle (using a spatial mask) was proposed and realizedin (Di Battista et al. , 2015): By exploiting a fraction ofthe speckle with a smaller 𝐶 (1) correlation, a focus no-tably smaller than the average speckle grain was demon-strated, albeit at the cost of a much lower efficiency. Thiswas extended to arbitrary point spread functions with atransmission matrix approach in (Boniface et al. , 2017).However, to break the diffraction limit and generate a fo-cus smaller than 𝜆/ (2 𝑛 ), it is necessary to have access toevanescent waves (Carminati et al. , 2007; Lerosey et al. ,2007; Pierrat et al. , 2013), such as just above the surfaceor inside the medium itself. Another approach is to usean active sink, as proposed in (Carminati et al. , 2000)and realized experimentally in acoustics (de Rosny andFink, 2002). The possibility of using a passive sink inoptics has also been put forward in (Noh et al. , 2013).Different ways of focusing inside the medium have beenconsidered. One of them proceeds with a source for dig-ital phase conjugation (Hsieh et al. , 2010b), another oneworks with a probe for iterative optimization (Vellekoop et al. , 2008). In both cases, an unambiguous proof thatthe focus is sub-Rayleigh is difficult to obtain. In con-trast, focusing at the surface of the scattering mediumcan be achieved using a scanning near field optical tech-nique (Park et al. , 2013). Note that when using res-onant systems, such as structured metallic layers thatexhibit plasmon resonances, a sub-wavelength focus hasalso been demonstrated (Gjonaj et al. , 2011).The ideal signal-to-background ratio 𝜂 in focusing iseasy to calculate and depends essentially on the numberof modes controlled, e.g., for phase-only optimization itwas calculated to be 𝜂 = 𝜋 ( 𝑁 −
1) + 1 (Vellekoop andMosk, 2007). However, the effective enhancement de-pends on the experimental conditions as well as on thealgorithm used and on the number of iterations steps forthe optimization. It was also shown that most meth-ods are affected by the signal-to-noise ratio of the detec-tion and are ultimately limited by shot noise fluctuations(Yılmaz et al. , 2013). Another limiting factor is the sta-bility of the medium, which is affected by small changesover time that decrease the efficiency of the process. Inpractice, most reported enhancement factors range be-tween 20% and 80% with respect to the ideal case. Ge-netic algorithms have been shown to be particularly ef-ficient in low signal-to-noise situations (Conkey et al. ,2012a).Focusing to multiple points or to areas larger than a speckle grain is possible with all the techniques describedabove. The total energy deposited via phase conjugationonly depends on the number of degrees of control. Asa consequence, the energy distributed over one or manytargets is the same, but the signal-to-noise ratio is re-duced by the number of modes that one seeks to control(Tanter et al. , 2000). This insight was evident alreadyfrom the first optimization experiment (Vellekoop andMosk, 2007), where focusing to five spots was realized(see Fig. 24e), and in (Popoff et al. , 2010b), where focus-ing on three spots yielded a three-fold reduction in thefocus to background intensity ratio. Interestingly, if thetransmission matrix of a medium is known, it is possiblenot only to perform digital phase conjugation by usingthe conjugate transpose operator as in Eq. (52), but alsoto go beyond the theoretical limits of phase conjugation,or to optimize a different metric by using a more ad-vanced operator, such as inversion for instance (Popoff et al. , 2010a) (see next section). Depending on the de-sired goal, the figure of merit can not only be the inten-sity, but also the contrast, or any measurable quantity tobe optimized. Note also that different algorithms, whenoptimizing multiple points, might perform differently inachieving equal intensity on every target.One interesting application of light focusing is opti-cal trapping (Dholakia and ˇCiˇzm´ar, 2011; Grier, 2003),where optical gradient forces are exploited to “trap” anano- or micro-particle at the focus of a tightly focusedbeam. In this domain, SLMs have been largely exploitedto generate multiple foci for trapping and moving aroundmultiple particles (Curtis et al. , 2002). While adaptiveoptics has long been proposed as a means to correct thefocus quality (Wulff et al. , 2006), it was only recentlyrealized that such techniques could be used for opticaltrapping in complex media ( ˇCiˇzm´ar et al. , 2010).
3. Imaging
Focusing or re-focusing a wave behind a scatteringmedium is indeed an important milestone also for imag-ing. In particular, the ability to scan a focus is at thebasis of many imaging techniques (e.g., multi-photon mi-croscopy), since focusing at different points would al-low in principle to form an image, e.g., by fluorescencemeasurement, as described in (Vellekoop and Aegerter,2010b). We will see in section V.A that in some occur-rences, the so-called “memory-effect” allows one to scanan optimized focus over a narrow angular range, withoutthe need to run an optimization algorithm for every pointor to measure a transmission matrix.Directly recovering a spatial distribution of intensityor phase from an object (i.e., direct imaging) from itstransmitted speckle pattern is more challenging. Thephase conjugation operation is by nature limited in thesignal-to-noise ratio when trying to form a complex shape4(as determined by the ratio of input to output degrees offreedom). Optical phase-conjugation, such as in (Yaqoob et al. , 2008), benefits from a very large optical etendue:Phase-conjugation is effective for a very large number ofspeckle grains within the non-linear crystal, thus allow-ing to reform a complicated image. In a similar way andin perfect analogy to Eq. (52), the transmission matrixallows to reconstruct an object field ˜ E obj from the outputfield ˜ E out through the phase conjugation operation,˜ E obj = ˜ t † ˜ E out . (53)First reconstructions were limited to very simple objects(one or two pixels turned on), using a square transmis-sion matrix (Popoff et al. , 2010b). Later, a more com-plex object (a resolution target) of 𝑁 = 20 .
000 pixelscould be retrieved in (Choi et al. , 2011b), by exploit-ing a transmission matrix measured over a very largenumber 𝑀 of output pixels (the full camera). An ad-ditional advantage of the transmission matrix is thatit is not restricted to phase conjugation. Other op-erators than phase-conjugation were successfully imple-mented in (Popoff et al. , 2010a), to demonstrate imagerecovery using, e.g., the so-called Tikhonov regularization(Tikhonov, 1963). This regularized inversion operation,has a much better performance than phase-conjugationand is robust to experimental noise.
4. Deterministic mixing
Another feature that can be exploited is the strong andoptimal mixing produced by the opaque lens, linked tothe fact that its transmission singular values follow theMarˇcenko-Pastur distribution. Specifically, the complexmixing of light by a multiply scattering material, that istoo complex to be copied or mimicked, has been consid-ered for cryptography and security (Pappu et al. , 2002).Important implications for the information capacity ofsuch a medium have also been discussed for communica-tion (Skipetrov, 2003). In the context of wavefront shap-ing, this natural optimal mixedness can be exploited innumerous ways and has probably many more potentialapplications. Among the emerging ideas that directlyexploit this deterministic and efficient mixing, one cancite the generation of quantum-secure classical keys (Go-orden et al. , 2014b) relying on the one-to-one associationbetween an optimized wavefront and a medium for few-photon states. Another interesting application is com-pressive imaging that can provide the reconstruction ofa sparse object with only a few measurements, providedeach local measurement carries global information aboutthe object (Candes and Tao, 2006; Donoho, 2006), whichis the case for a detector behind a disordered medium, asdemonstrated in (Liutkus et al. , 2014).
5. Polarization control
When considering the vectorial nature of light, a ques-tion that naturally arises is that of polarization control.We have seen earlier that at each elastic scattering event,the polarization of the scattered wave is modified in a de-terministic way. In a classical picture, during a scatteringevent from incident wavevector k to a scattered wavevec-tor k ′ , the input and output polarization vectors n and n ′ are related by n ′ ∝ n − ( k ′ . n ) k ′ (MacKintosh et al. ,1989). For this reason, forward and backward scatteringevents tend to maintain polarization, while large anglescattering tends to modify polarization more strongly.In the opaque lens case, all transmitted light has beenscattered a sufficient number of times to ensure a fullymixed polarization. As a consequence, the speckle result-ing from the propagation through the opaque lens alsohas a complex polarization state at any point, which isin general elliptic. In essence, it is the sum of two orthog-onally polarized speckle, uncorrelated to each other. Thisfeature has been proposed as a way to exploit the opaquelens for polarimetric measurements (Kohlgraf-Owens andDogariu, 2008, 2010).It is important to point out that the polarization be-havior of the opaque lens, which is universal in transmis-sion, is clearly system-dependent in reflection. Firstly, asingle-scattering contribution is always present. Just likefor a reflection off a mirror at normal incidence, it has thesame polarization as the input in backreflection directionfor a linear input polarization, and a reversed helicity foran elliptical input polarization. Secondly, the reflectedlight also has contributions from multiply scattered lightwith few scattering events, that retain partial polariza-tion memory. This second contribution depends stronglyon the scattering properties of the medium (MacKintosh et al. , 1989). When working in reflection, it is possibleto eliminate the single scattering contribution and a frac-tion of the light that endured few scattering events by se-lecting a specific polarization of the reflected light (e.g.,the orthogonal polarization for linearly polarized inputlight), a trick used for measuring the coherent backscat-tering cone (van Albada et al. , 1990).In the context of wavefront shaping, early experiments(Popoff et al. , 2010b; Vellekoop and Mosk, 2007) were re-alized by placing a polarizer between the opaque lens andthe detector, so as to obtain on the camera an intensitypattern corresponding to a single scalar speckle ratherthan the sum of two uncorrelated speckles (Goodman,1976). Only later it was demonstrated experimentallythat this additional degree of freedom can be turned toan advantage and allows to generate an arbitrary polar-ization state at the focus (Guan et al. , 2012; Tripathi et al. , 2012) (see Fig. 25). Since the total speckle isthe sum of two orthogonally polarized speckles, one cangenerate any polarization state (linear, circular, elliptic)at will. The final quality of the polarization state fol-5 (a) Unoptimized patternsontalonent
A BC (b) calonent (d)
Optimized to vertical polarization (e) (g)
Optimized to horizontal polarization (h) (j)
Optimized to circular polarization (k) -1 -0.5 0 0.5 1-1-0.500.51 E x E y (c) zation X500 display gain
ABC -1 -0.5 0 0.5 1-1-0.500.51 E x E y (f) -1 -0.5 0 0.5 1-1-0.500.51 E x E y (i) -1 -0.5 0 0.5 1-1-0.500.51 E x E y (l) H o r i z on t a l c o m ponen t V e r t i c a l c o m ponen t P o l a r i z a t i on s t a t e Figure 25 (color online). Illustration of the full control of theoutput polarization state. Since the output polarized speckeis composed of two uncorrelated speckle patterns of orthog-onal polarizations (left column), it is possible to focus inde-pendently on one or the other polarization (second and thirdcolumn). Combining the two wavefronts that focus at thesame position for either polarization state, one can generateat this point a focus with arbitrary polarization (here, circu-lar, last column). (Figure adapted from (Guan et al. , 2012).) lows the usual signal-to-noise restrictions common to allphase-conjugation techniques discussed above.
6. Temporal and spectral control
While all focusing and imaging experiments throughthe opaque lens discussed so far have considered a spe-cific wavelength only, i.e., a monochromatic light sourcein conjunction with the spatial degrees of freedom of themedium at this wavelength, we have seen that the behav-ior of the opaque lens is strongly wavelength-dependent– a feature has been exploited for a long time to retrievediffusion properties of the medium (Curry et al. , 2011;Vellekoop et al. , 2005). Correspondingly, in the contextof focusing, it has been shown that if a phase pattern gen-erates a focus for a given wavelength, then the focus willbe resilient to a small wavelength variation. The corre-sponding bandwidth is exactly the frequency-bandwidthof the medium (van Beijnum et al. , 2011), a propertythat has been exploited to use the medium as a spec-tral filter (Small et al. , 2012). Performing optimizationwith polychromatic light is possible, and has been shownto result in a narrowing of the spectrum (Paudel et al. ,2013). When measuring a multispectral transmission ma-trix (Andreoli et al. , 2015), it is also possible to focusseveral spectral components at a single or at multiplepositions (see Fig. 26).In the acoustic time-reversal community, it had beenrealized from the early days on that spatial and tempo-ral degrees of freedom of a complex medium could becoupled (Fink, 1997). In particular, it was understood
Figure 26 (color online). Spatiospectral control of broadbandlight, adapted from (Andreoli et al. , 2015). Using the infor-mation gathered from the multispectral transmission matrixof an opaque lens, it is possible to (a) spectrally focus dif-ferent spectral components of a broadband pulse at arbitrarypositions. (b-d) Scanning in continuous mode the same laserdemonstrates that each focus corresponds to a different wave-length. In this way the opaque lens is turned into a gener-alized grating with a spectral resolution given by its spectralcorrelation bandwidth. that time reversing the signal received at a given loca-tion would allow spatiotemporal focusing on the sourceat a specific time. Also, thanks to the reciprocity of thepropagation, re-emitting at the source the time-reversedsignal from the detector would yield spatiotemporal fo-cusing on the detector. Nonetheless, when performingsuch a single-channel time reversal experiment in an opensystem, the observed spatiotemporal focusing is truly atemporal focusing only: When integrating the energy atthe source position over time a significant energy increaseis not observed. To truly enhance the total energy at thesource position either requires multiple detectors to betime-reversed simultaneously, or a closed system such asa chaotic cavity (Draeger and Fink, 1997).An analogue of such a single-channel time reversal inoptics was performed in (McCabe et al. , 2011). A shortpulse from a femtosecond laser was sent through a layerof paint and the complex spatiotemporal speckle figurewas recorded on the far side using an imaging spectrom-eter (see Figs. 21 and 27). Note that due to the difficultyof measuring and controlling an optical signal directly inthe time domain, both the temporal measurement andthe temporal emission were performed in the spectraldomain: The temporal speckle was measured using spa-tially and spectrally resolved Fourier-transform interfer-ometry (SSI) (Tanabe et al. , 2002), and the pulse wastime-shaped using a spectral shaper (Monmayrant et al. ,2010). In this experiment, the time-reversed signal (mea-sured via SSI) was sent from the source (the pulse-shaperat the output of the femtosecond laser) to the detector.At the output position, it was shown that the pulse wascompressed temporally close to the initial pulse duration,thanks to the reciprocity of the wave equation. Still, in-tegrated over the pulse duration, the total intensity atthe target spot was not increased. Just like spatial fo-cusing can be seen as an extension of adaptive optics tomultiply scattering material, this work can be seen as an6 spatial coordinate. To fix the time, the speckle pulse isoverlapped with a reference pulse in a heterodyne detec-tion scheme in a configuration similar to [26]. The hetero-dyne signal exactly corresponds to the cross correlation ofthe speckle pulse and the reference pulse [27]. Effectively,it is an instantaneous measurement of the transmitted fieldamplitude at the delay time of the reference pulse ! . Thissignal serves as feedback for an optimization algorithm,which controls the incident wave front via the SLM.The principle of the experiment can be described asfollows. Light reflected from a single segment on theSLM is transmitted through the sample, giving rise to thefield E i ð t Þ at the detector. Its phase can be modified by atime-independent phase shift !" i via the SLM. The totalfield scattered into the detector E out ð t Þ is therefore given bythe sum over all segments E out ð t Þ ¼ X Ni ¼ E i ð t Þ e i !" i : (1)Multiple scattering allows us to assume that the contri-butions E i ð t Þ from the different segments at every singlepoint in time t are uncorrelated random variables withRayleigh distributed amplitudes j E i ð t Þ j and uniformly dis-tributed phases " i ð t Þ [28]. For the nonoptimized case, theresulting field E out ð t Þ can be viewed as the result of arandom walk in the complex field plane. After the optimi-zation, all contributions are in phase, adding up construc-tively. The average amplitude enhancement is givenby [17] h " i ¼ hj E opt ji rms hj E rnd ji rms ¼ ! ð N $ Þ þ " = & ! N " = : (2)Since the instantaneous field amplitude is optimized, theinstantaneous intensity accordingly increases, with an av-erage intensity enhancement $ ¼ " . The simple modelleading to Eq. (2) does not provide a prediction for theresulting pulse duration. We address this point later in thisLetter.The nonoptimized data were obtained by setting randomphase values to the SLM segments. The optimizationalgorithm adjusts the phase shifts !" i such that theamplitude of the heterodyne signal is maximized. Weperformed the optimization at 20 equidistant time delays ! opt between $ :
05 ps to þ : . For each ! opt , the optimization was performed four times, with N ¼ , 48,192, and 300 segments, respectively, each time startingfrom a new random phase pattern.Our main result is displayed in Fig. 2, showing theamplitudes of both the nonoptimized and the optimizedpulses for different time delays ! opt and N ¼ segmentson the SLM. The long time-tail of the average nonopti-mized transmission reflects the broad path length distribu-tion which has been observed in similar earlier studies [12].The optimized amplitudes show sharp, distinct peaks withdramatically increased amplitudes at the desired time de-lay. We can control the amount of time the optimizedpulses stay in the sample by the time delay ! opt , and bythat we control the path length of the pulses through thesample. Note that the heterodyne signal is proportional tothe field amplitude, the intensities exhibit even more pro-nounced optimized peaks.The enhancement " versus time delay ! opt is shown inFig. 3. Its magnitude, depending on the number of seg-ments on the SLM, is constant from zero to several pico-seconds time delay. This result shows that our methodworks for short light paths as well as for light paths morethan 10 times longer than the sample thickness.For long time delays a continuous decrease of " isobserved, which is related to the noise level of the experi-ment. We include a quantitative analysis of this effect inthe supplemental material [27].Figure 4 shows the average enhancement in the constantregime in Fig. 3 versus N together with the enhancementexpected from theory [Eq. (2)], h " i ¼ % ð N Þ = . The pre-factor % ¼ : corrects for the nonuniform illumination FIG. 2 (color online). Optimized and random speckle pulses.(a) Amplitude of heterodyne signal of a nonoptimized pulse as afunction of time delay, averaged over 50 random speckle pulses.(b) Typical single random speckle pulse. (c)–(g) Amplitudes ofsingle pulses after optimization at different time delays whichare indicated by the dashed arrows. The optimization has beenperformed by dividing the SLM into 300 segments. The opti-mization generates strong, short pulses from diffuse light. Thezero delay position is at the maximum amplitude with no sample.The plotted curves have been normalized to the maximum of theaverage nonoptimized heterodyne signal (factor :
53 mV $ ).FIG. 1 (color online). Experimental setup (see text). PRL week ending11 MARCH 2011
ARTICLE (cid:19)(cid:1)
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1434 © Macmillan Publishers Limited. All rights reserved. T he multiple scattering of coherent light is a problem of both fundamental and applied importance . In optics, phase conju-gation allows spatial focusing and imaging through a multiply scattering medium ; however, temporal control is nonetheless elu-sive , and multiple scattering remains a challenge for femtosecond science. When light propagates through a thick multiply scattering medium, the large number of scattering events exponentially deplete the ballistic photons with propagation distance, and the transmitted light is dispersed in a highly complex manner. When illuminated by a single-frequency laser, such a medium is known to give rise to a spatial speckle . Furthermore, this speckle is wavelength dependent: illuminated by a broadband laser, the medium therefore also pro-duces a spectral speckle whose characteristic size is inversely related to the con fi nement time in the medium, or Th ouless time. It corre-sponds to a severe temporal broadening of the pulse . Nonetheless, multiple scattering is deterministic and coherence is not destroyed. In the spatial domain, recent experiments have thus demonstrated that wavefront shaping is able to generate an intense focus through an opaque medium, and can be interpreted as phase conjugation . For the acoustic and GHz-electromagnetic regimes, time-reversal experiments are the counterpoint to this principle in the time domain : a short pulse incident on a medium generates a long coda, which may be time-reversed and retransmitted through the medium to regain the initial short pulse. Th ese experiments have not only demonstrated that multiple scattering can be compensated but—more importantly—that this can actually improve addressing, imaging or communication . Owing in particular to the inability to measure electric fi elds directly in the temporal domain at higher frequencies, an optical domain time-reversal experiment remains elusive; yet the ability to measure and shape femtosecond electric fi elds in the spectral domain nevertheless o ff ers the opportunity of a route to the same goal.From a quantum-control perspective, the use of optimally shaped pulses to control light–matter interactions has been a fertile fi eld of research . Th e absence of predesigned control mechanisms in complex systems may be obviated by a closed-loop feedback scheme to fi nd e ffi cient pulse shapes at the expense of insight into the physical mechanism . By contrast, theoretically tractable model systems permit an open-loop approach where the appropriate control pulse is calculated directly . In parallel, the propagation of ultrafast pulses through strongly dispersive media has been a topic of research for many years. Atmospheric aberrations have been successfully compensated as well as the dispersion due to thick optical media . However, despite some insight into the multiple scattering of ultrafast pulses , measurement of the scattering medium transfer function at a single speckle grain via scanning the frequency of a continuous-wave laser and some two-photon excitation experiments in scattering samples , the e ff ects of multiple scattering have ultimately been considered too complicated to be compensated.In this paper, we initially demonstrate a full spatio-temporal characterization of a femtosecond speckle fi eld using spatially and spectrally resolved Fourier-transform interferometry (SSI). Owing to the linearity of the scattering process, knowledge of the spec-tral phase facilitates active temporal focusing of the speckle via the open-loop feedback of the measured phase to a spectral pulse shaper placed before the sample. Here we give the fi rst experimental dem-onstration of this e ff ect. Th is experiment relies on a spatially resolved phase measurement, as the lack of large-scale spatial homogeneity in the speckle fi eld prevents the spatial integration typical of control experiments. An e ff ective phase compensation is spatially localized according to the correlation length; thus, a spatially resolved spec-tral shaper is not a prerequisite for a degree of spatial control. Results
Spatio-temporal characterization of the speckle fi eld . Th e schematic experimental layout is shown in Figure 1 (see Methods for experimental details). In essence, an ultrashort pulse passes through a pulse shaper and is focused on an opaque sample. Th e sample is a scattering ZnO layer that transforms the ultrashort pulse into a complex spatio-temporal speckle. Th is speckle is stationary on a timescale much longer than the experiment duration. Th e spatio-temporal speckle is reimaged and spatio-spectrally characterized in phase and amplitude via SSI (Methods).Figure 2a shows the intensity and Figure 2b shows the phase of a typical spatio-spectral speckle fi eld E out ( (cid:87) , y ) as measured by SSI. Th e spatio-spectral speckle is clearly demonstrated and the complex structure of E out ( (cid:87) , y ) is fully resolved in both phase and intensity—a prerequisite for quantum control. Th is structure is also visible in the one-dimensional spatial (Fig. 2a, red, solid) and spectral (Fig. 2a, green, solid) ‘lineout’ slices indicated. Th e integrated projections of the speckle intensity (Fig. 2a, dotted lines), however, show that a spectrally unresolved speckle image, as measured on a camera, would yield a strongly reduced contrast, while a non-imaging spec-tral measurement would only yield the initial source spectrum. Th is further motivates the necessity of a spatially resolved phase mea-surement for the temporal focusing experiment described below. Meanwhile the spectral phase reveals a similar complex structure (Fig. 2b); as a consequence, it is clear that a spatially averaged phase measurement o ff ers no utility whatsoever for pulse recompression.A Fourier transform of the complex fi eld along the spectral axis gives the spatially resolved temporal behaviour of the speckle E out ( t , y ; Fig. 3a). Th is spatio-temporal fi eld exhibits the same complex speckle structure as before, as evinced by both the three-dimensional plot and the spatial (red) and temporal (black) lineouts L1L2SBS (cid:1) P o s i t i on Position P o s i t i on Figure 1 | Experimental setup.
The output of a laser oscillator (Osc) is divided by a beamsplitter (BS) to form an interferometer. One arm passes through a pulse shaper (PS) and is focused onto the sample (S) by lens L1. The output speckle field is imaged by lens L2 onto the slit of a two-dimensional spectrometer (2D-SM), which performs a measurement of the spectral intensity spatially resolved along the slit (upper inset). The reference arm is combined with an adjustable delay (cid:84) and angle. The spectral pulse shaper is a folded 4 f line, and comprises a grating (G), cylindrical mirror (CM), plane mirror (M), folding mirror (FM) and spatial light modulator (SLM). scattering 1-mm-thick rat brain tissue and through a turbid 500- m m-thick bone sample. Results
Simultaneous focusing and pulse compression by wavefrontshaping.
The experimental set-up for spatiotemporal focusing ispresented in Fig. 1a. An ultrashort pulse was focused through arandom scattering medium to a 2PF screen. As in the work ofVellekoop and colleagues, the incoming wavefront phase wasadaptively optimized using a two-dimensional SLM . However,instead of maximizing the optical intensity, we maximized anonlinear 2PF signal at a selected point on the screen, imaged byan electron-multiplying charge-coupled device (EMCCD; seeMethods). Imaging the 2PF after optimization reveals that theoptimized 2PF is enhanced and refocused at the optimized spot(Fig. 1b,c). The refocused 2PF is not only localized in thetransverse ( x – y ) dimensions, but is confined along the axial ( z )dimension, as verified by imaging different depths in the 1-mm-thick 2PF screen (Fig. 1d,e). The axial confinement of the 2PF isobtained in the same manner by which optical sectioning isachieved in 2PF microscopy . Most importantly, as we showbelow, this spatial focusing is accompanied by significanttemporal focusing, even though no special attempt has been madeto control the temporal degrees of freedom. To investigate the temporal properties of the scattered fields andto prove temporal compression, the characterization set-up pre-sented in Fig. 2a was devised. The key element in this set-up is aMichelson interferometer inserted before the SLM, which producesspatially resolved 2PF autocorrelation . With this set-up we couldthen characterize the temporal profile of the light fields at eachand every point in the image. By scanning the pulse separation t while collecting 2PF images with the EMCCD, we extracted thepulse intensity autocorrelation at each point in the image simul-taneously (Fig. 2b–d). Note that the set-up presented in Fig. 2a isrequired only for characterization purposes, and the simple set-upof Fig. 1a is sufficient for spatiotemporal focusing.To demonstrate that our optimization scheme could correct forboth the distortions induced by scattering and also for existing tem-poral distortions of the input pulse, we pre-chirped the input laserpulses by passing them through a 152-mm-long slab of F3 glass.Figure 2b shows that before any optimization, the autocorrelationof the light emerging from the scattering medium is .
800 fs longat all points (1 / e width after background subtraction). After apply-ing the 2PF optimization algorithm (with interferometer delay armset to zero), we repeated the temporal characterization with the opti-mized field. Inspection of these results (Fig. 2c,d) confirms that bysimply optimizing the 2PF using only the spatial degrees ofcontrol, we obtained a refocused pulse in both space (Fig. 1b–e)and time (Fig. 2b–d). The optimized pulse autocorrelation 1 / e width is 370 fs, which is not only significantly shorter than thenon-optimized scattered pulse, but is shorter than the originalchirped input pulse (715 fs 1 / e width, Fig. 2d), proving spatiotem-poral focusing and compression by wavefront-controlled randomscattering. Careful inspection of the background-subtracted auto-correlation traces (Fig. 2d) reveals that the scattered pulse (plottedin green) has a similar full-width at half-maximum (FWHM) asthe chirped input pulse (plotted in red), but a narrower peak.This is the well-known signature of the ‘coherence spike’ of an inten-sity noise burst (temporal speckle), set by the pulse bandwidth . Asthe random medium transmission matrix has a short spatial corre-lation length, the pulse at the optimization point is of shorter dur-ation than any other point in the sample (Fig. 2c). We have used thedata collected by our experimental system to render a spatiotem-poral reconstruction of the scattered and optimized light fields auto-correlations, which reveals a dispersed spatiotemporal speckle in thenon-optimized case (Fig. 2e), and a localized 2PF in both space andtime in the optimized case (Fig. 2f; Supplementary Video 1). Themeasured optimized 2PF intensity is 15 times more intense thanthe average 2PF before optimization (Fig. 1b,c). However, theactual enhancement of the axially confined 2PF is much larger, asfor the non-optimized case off-plane fluorescence contributionsfrom the 1-mm-thick 2PF screen are significant. Taking intoaccount the off-plane contributions, we estimate the localized 2PFenhancement to be close to 800. This is consistent with a measured20-fold gain of the excitation laser field intensity, and further exper-iments using thin fluorescent screens (Supplementary Fig. 2).To understand how the scattered light field focuses in both spaceand time, even though no effort is made to explicitly minimize thewidth of the focus or the pulse temporal width, one needs to considerthe dependence of the optimized 2PF signal on the spatial and tem-poral distributions of the light field. Assuming a two-photon absorp-tion spectrum that is wider than the pulse bandwidth, the 2PFintensity at any point ( x , y , z ) in space is given by: I PF ( x , y , z ) / ! I ( x , y , z , t ) d t , where I ( x , y , z , t ) is the excitationfield intensity at the point ( x , y , z ) at time t . Thus, as the input pulsetotal energy E = ! I ( x , y , z , t ) d t d x d y is fixed, the maximum 2PF isobtained for the field that is focused to the smallest possible spatialextent and the shortest temporal duration. This approach is in thespirit of previous works optimizing a nonlinear signal for temporalcompression alone and the recent works optimizing a linear z xy z xy b cd e xy xy Initial 2PF Optimized 2PF 10.80.60.40.20SLM 2PFscreen × × a EMCCDF
Figure 1 |
Spatiotemporal focusing by optimizing 2PF: spatialcharacterization. a , Experimental set-up. An ultrashort pulse is focusedto a 2PF screen placed behind a scattering medium. An SLM controls theincident wavefront, optimizing the 2PF at a selected point imaged by anEMCCD (F-bandpass filter). b , c , 2PF images before ( b ) and after ( c )optimization at the optimized plane ( x – y ). d , e , Depth-resolved 2PF imagesbefore ( d ) and after ( e ) optimization, showing the axial ( z ) confinement ofthe optimized 2PF. The localized 2PF enhancement is estimated to be ! m m. Rendered x – y – z field in d and e is 190 × × m m. NATURE PHOTONICS
DOI: 10.1038/NPHOTON.2011.72
ARTICLES
NATURE PHOTONICS (a)$(b)$(c)$
Spectral$pulse$shaper$ Spa0ospectral$$interferometry$(SSI)$
Gra0ng$ $ fs$laser$ spectrometer$femtosecond$$pulse$$$$$ SLM$ Sca=ering$$medium$ EMCCD$$camera$2Bphoton$screen$Sca=ering$$sample$ pinhole$feedback$ Figure 27 (color online). Schemes for temporal focusing viaspatial-only shaping. (a) Spatial shaping and optimization onthe intensity at a given time-delay (Aulbach et al. , 2011). (b)Spatial shaping and optimization of a nonlinear signal (Katz et al. , 2011). extension of temporal pre-compensation of dispersion ofan ultrashort pulse (Delagnes et al. , 2007) to the opaquelens.While temporal control provides temporal focusing,and spatial control spatial focusing, spatial and tempo-ral degrees of freedom are coupled in a complex medium.We will now describe this spatiotemporal coupling andrelated experiments exploiting this effect in the opaquelens, i.e., a diffusive slab in the multiply scatteringregime. As shown in (McCabe et al. , 2011), a spatiotem-poral speckle along a line, as measured behind an opaquelens by SSI, has a scalar field distribution 𝐸 ( 𝑥, 𝜔 ), char-acterized by a short range spatial correlation function ⟨ 𝐸 ( 𝑥, 𝜔 ) 𝐸 ( 𝑥 ′ , 𝜔 ) ⟩ that has a well-defined width that isgiven by the speckle grain size, and by a spectral correla-tion function ⟨ 𝐸 ( 𝑥, 𝜔 ) 𝐸 ( 𝑥, 𝜔 ′ ) ⟩ , whose width is directlyrelated to the traversal time of the medium 𝜏 𝑇 (defined insection IV.C.2). These spectral correlation functions canbe retrieved from the wavelength correlation within thetransmission matrix ˜ t ( 𝜔 ). In the time domain, a giveninput time 𝑡 correspondingly couples to all times 𝑡 ′ > 𝑡 within a few 𝜏 𝑇 , and that coupling strongly depends onthe input and output position. This in turn, means thatthere can be spatiotemporal couplings within this range.This behavior can also be understood within the modeformalism: The existence of a well-defined traversal timeand of this spatiotemporal coupling can be linked tothe fact that in the diffusive regime, the mean spacingbetween the modes is much smaller that their averagelinewidth, as characterized by the Thouless number 𝛿 defined in section II.A.6. If one sends an optical pulseof duration shorter than the average transmission time of the medium 𝜏 𝑇 , its spectrum is therefore broadbandcompared to the average distance between the modes. Asa consequence, it couples to many different transmissionmodes, at different frequencies, that recombine in a com-plex way after the medium, thus producing a complexspatiotemporal pattern (Wang and Genack, 2011). Thisbehavior is modified in the localization regime, wheremodes are spectrally isolated and where a short pulsemight only couple to one of a few transmission channelsonly (Pe˜na et al. , 2014).A consequence of the spatiotemporal coupling is thatspatial shaping can generate a temporal focus at a givenposition, by setting a constructive temporal interferenceat a given time between different frequency components,as pointed out in (Lemoult et al. , 2009). This was ex-ploited in two seminal works in optics, where spatial-only phase control over a broadband pulse was shownto be able to induce temporal focusing as well as time-integrated spatial focusing. Both approaches were basedon a 2D-SLM optimization algorithm, but using differ-ent signals as a feedback. In (Aulbach et al. , 2011), theoptimization was performed on the intensity at a givenposition and time using a heterodyne pulsed detection,while (Katz et al. , 2011) used a two-photon absorptionsignal at a given position. As this signal depends onthe square of the intensity, it is proportional not only tothe total energy integrated in time, but also to the av-erage pulse duration. Both approaches are summarizedin Fig. 27. As an alternative to optimization techniques,also a matricial approach can be taken. Measuring themultispectral transmission matrix (or MSTM) (Andreoli et al. , 2015), it is possible to demonstrate arbitrary pulseshaping, provided the spectral phase can be addressed.This was demonstrated in (Mounaix et al. , 2016a), wherenot only pulse recompression was shown, but also moreadvanced temporal functions were realized such as twopulses with a controllable delay.Another possible approach for temporal control thathas been investigated is a time-resolved matrix mea-surement (Choi et al. , 2013; Kang et al. , 2015). Theseapproaches have been performed in reflection geometry,mainly to achieve depth-sectioning and light delivery ata certain depth, in analogy with optical coherence to-mography, using ballistic light. However they also allowfor temporal focusing at the detection plane and are analternative to spectral or non-linear measurements. E. Other complex scattering systems
1. Multimode optical fibers
A very interesting system has recently emerged as acomplex medium in optics: multimode optical fibers. Op-tical fibers are composed of a core material, where lightis guided, and of a cladding, the function of which is to7confine light within the core, so that the fiber behaveslike a wave guide. The confinement can be achieved invarious ways: total internal reflection on an index-step,index gradient, or photonic bandgap confinement. In allcases, the guiding occurs within a certain angular cone,that defines an effective numerical aperture. Dependingon the diameter of the core and on the wavelength, oneor many transverse modes can be supported. If only onetranverse mode propagates within the core, then the fiberis called a single-mode fiber (SMF). When many coresare implemented in a single fiber, one speaks of a multi-core fiber (MCF). If the diameter of the core is increased,then the fiber can support a few modes and is called afew mode fiber (FMF), while for many modes one speaksof a multimode fibers (MMF). The available number ofmodes scales with the diameter 𝐷 of the core and withthe numerical aperture NA as ( 𝐷 × NA /𝜆 ) , resultingin up to thousands of modes. FMFs and MMFs are in-creasingly considered in the context of high bitrate fibercommunications for space-division multiplexing (SDM),which means using several transverse modes as indepen-dent transmission channels (Berdagu´e and Facq, 1982)(see (Richardson et al. , 2013) for a review). Spatial de-grees of freedom thus come as extra degrees of freedom forinformation transmission, complementary to wavelengthand polarization, to carry more than one channel of infor-mation on a single optical fiber. While multicore fibers(i.e., fiber bundles) have long been used as independentspatial modes (despite some cross coupling), using a sin-gle core FMF or MMF remains a challenge because ofthe complex nature of the modes.An ideal MMF should support well defined modes,where the linearly polarized (LP) mode family is gen-erally used (Snyder and Love, 1983) to describe eigen-modes of the fiber. However, due to fabrication imper-fections and bendings, the ideal linearly polarized (LP)modes are not in general the eigenmodes of the system:The eigenmodes of a large-core MMF tend to be differentfrom the ideal case. Nonetheless, it is possible to gener-ate the so-called “principal modes” of an MMF (Fan andKahn, 2005; Shemirani et al. , 2009), which are unaffectedby modal dispersion to first order of frequency variation.Their design principle bears very strong similarities withthe eigenstates of the Wigner-Smith time-delay matrixdescribed in section II.C (a detailed comparison betweenprincipal modes and so-called “particle-like states” willbe provided in section V.C). Only in the case of an idealstraight fiber where all modes have different group veloc-ities do the principal modes coincide with the LP fibermodes. In practice, the exact phase delay is very sensi-tive to experimental conditions, and each mode has itsown group delay, which means that a short (broadband)pulse will be stretched. Interestingly, the distribution ofdelay times follows a semi-circle law as was shown basedon a suitable random matrix model (Ho and Kahn, 2011).Even for bent or long fibers, principal modes tend to re- This imaging system allows the mode decomposition performed by the receiver using the SLM [5] to be compared with the beam actually observed on the camera.
Fig. 1. The mode decomposition and mode generation setup. The asterisk marks the place at which the beam is sampled by the polarization diverse imaging system.
The mode decomposition process is similar to that discussed previously in [5]. For each mode in each polarization as launched by the transmitter, the receiver SLM displays phase masks for each basis mode and polarization one at a time to measure the amplitude of each mode. The phase of the modes within a polarization is then found by adding together the phase masks for the different basis modes with varying phase shifts and measuring the variation on a power meter. When the phase mask is conjugated with the incoming beam, the power will be maximized. In previous experiments using a similar system [5], the phase difference between the two polarizations was not a parameter of interest and hence was not measured. In this demonstration, the two polarizations are interfered by adjusting the relative phase between the two polarizations on the SLM and observing the interference between the polarizations using an external fiber polarizing beam splitter (PBS) with its polarization axis aligned to 45 degrees with respect to the polarization axis of the SLM receiver. Given the measurements thus far, the relative phases between all the spatial and polarization modes at the receiver are defined for each of the basis modes launched at the transmitter to within an absolute phase offset. That is, the relative phases between modes along a column of the mode transfer matrix are defined, but the relative phases between columns is not meaningful as no interference has occurred along that axis. For many purposes locking the columns of the matrix to a common phase reference is not required as this has no effect on the measured characteristics of the fiber such as mode dependent loss and is irrelevant for MDM where there is no meaningful phase relationship between independent channels. However in order to know how different launched modes will interfere it is necessary to define all the phases of the mode transfer matrix relative to the same phase reference. To achieve this, as a final step each mode at the transmitter is excited in superposition with a reference mode and the corresponding phase masks for the mode of interest and the reference mode are interfered on the SLM at the receiver to measure their relative phase. Now the phases of the entire matrix are defined relative to the reference mode. Theoretically, the choice of reference mode is arbitrary and could be any basis mode or superposition of modes. However there is some practical advantage to using the fundamental mode as its simplicity makes it straightforward to excite accurately and it is the mode with the least degeneracy in the fiber. The fundamental mode is only degenerate between its two polarizations, in contrast to the other modes of a graded-index multimode fiber which have more complicated degeneracies which in turn lead to more complicated output patterns which are more sensitive to the environment and hence less stable over time. In contrast to Swept-Wavelength Interferometry (SWI) [6,7] which uses an external phase reference arm of an interferometer to define the measured phases, this approach sends the phase reference along the fiber under test itself as the reference mode. Hence all the light for all the measurements travels along the same fibers and as a consequence, the requirements on the coherence of the light source is greatly reduced. There is no need to approximately match path lengths between a phase reference arm and the fiber under test [6] as they are the same fiber in this case. Another advantage of practical significance is that the phases are defined with reference to the plane of the SLMs at the transmitter and receiver for all modes. The reference plane is a part of the mode coupling system itself rather than being located in external splitters, where each mode is fed in using a different input fiber which is likely to not be path length matched and may or may not have the same polarization axis relative to the fiber under test. Although theoretically all such path lengths and polarization rotations could be calibrated out, in practice, doing so for a large number of spatial modes would be very unwieldy and difficult to keep temporally stable in fiber. This system is also convenient in that the apparatus of Fig. 1 is the same regardless of the number of spatial modes being characterized.
3. Mode transfer matrix
Fig. 2. (a) Amplitude of the mode transfer matrix for all 110 modes. (b) Singular values of that mode transfer matrix representing mode dependent loss.
The fiber under-test theoretically supports 110 spatial and polarization modes. This consists of 55 modes in each polarization. As the fiber has an approximately parabolic refractive index profile these modes can be organized into approximately degenerate mode-groups, with 10 groups in total where all LP l,m modes that share the same value of 2 m + l have the same propagation constant and hence will mix heavily. The amplitude of the measured mode transfer matrix is shown in Fig. 2(a) The x and y axes run from mode 1 (LP Horizontally polarized) to mode 110 (LP vertically polarized) and the white lines demarcate the different degenerate groups. Mode coupling occurs mostly between modes within a degenerate group which corresponds to the square white boxes that lie along the diagonal of the matrix in Fig. 2(a). Performing the singular value decomposition (SVD) of the measured matrix yields the eigenvalues and the corresponding eigenvectors for each of the 110 orthogonal channels the fiber supports. The eigenvalues, or singular values, of Fig. 2(b) represent the relative loss, sorted in increasing order, for each of these channels represented by the corresponding eigenvector superposition of modes. It can be seen in Fig. 2(b) that there is a steady increase in mode dependent loss as the number of channels is increased until approximately the 90th channel at which point the loss increases sharply. The large loss of these final channels signifies that they are beyond the cutoff of the fiber. Theoretically, a 50 μ m core graded-index multimode fiber could support 10 degenerate modes groups (110 modes total), however the 10th group is very close to cutoff even in theory and in practice appears to be beyond cutoff in this particular fiber. The steady accumulation of mode dependent loss for approximately the first 90 channels is likely more a measurement of the quality of the SLM based measurement Figure 28 (color online). Measurement of a 110 ×
110 trans-mission matrix of a multimode fiber in the LP modes basis.(top) Experimental setup: two SLMs on each side to controltwo polarizations are recombined on a polarizing beam split-ter (PBS), allowing the near-perfect detection and injectionof a well-defined mode at the input and output of the fiber tomeasure the corresponding coefficient on the matrix. (bottomleft) Amplitudes of the transmission matrix in the LP modesbasis. (bottom right) Singular value decomposition, show-ing the mode-dependent losses. In the considered weak modecoupling limit, the transmission matrix is relatively block di-agonal as subfamilies of LP modes are preferentially coupled.(Figure adapted from (Carpenter et al. , 2014b).) main well isolated from each other. Measurements arechallenging (Milione et al. , 2015), but have meanwhilebeen reported (see (Carpenter et al. , 2014b, 2015; Xiong et al. , 2016) and Fig. 28). Without these techniques, amonochromatic input is typically injected into more thanone principal mode and will give rise at the output to acomplex superposition of these modes, i.e., a speckle pat-tern.Many of the concepts developed to take advantage ofopaque lenses for imaging and spatial and temporal con-trol can therefore be translated to MMFs. A first setof experiments in this context is related to MCFs andFMFs, that were studied in terms of their potential ashigh power fiber lasers and fiber amplifiers. For this classof problems, the difficulty is to exploit several transversemodes to achieve higher intensity. Unfortunately, due tothe dispersion between the modes, the output laser modeis typically very multimode spatially, which is detrimen-tal when high spatial quality is required. In the contextof multicore fiber arrays, maintaining or retrieving a com-mon phase between the different output modes is neces-sary in order to maintain a high transverse spatial quality(i.e., a beam quality factor close to unity). Correspond-ing cophasing methods, similar to phase-conjugation, canbe either passive (Lhermite et al. , 2007) or active using8
Figure 29 (color online). Digital Optical Phase conjugationthrough a multimode fiber. (top) Experimental setup withDOPC (right-hand side of the fiber) and injection plus imag-ing (left hand side of the fiber). (bottom) A DOPC experi-ment: (a) Injection of a focused spot. (b) Output speckle. (c)Hologram used for digital phase conjugation. (d) Refocusedspot. (Figure adapted from (Papadopoulos et al. , 2012) and(Farahi et al. , 2013).) piezo-electric fiber stretchers (Yu et al. , 2006) or spatiallight modulators (Bellanger et al. , 2008; Lhermite et al. ,2010).Following the progress made in scattering media, thiskind of monochromatic phase-conjugation has been ex-tended to imaging. In particular, it was realized that,just like in a scattering medium, wave front shapingcould allow the formation of a sharp focus on the farside of a fiber, be it by optimization (Di Leonardo andBianchi, 2011; ˇCiˇzm´ar and Dholakia, 2011) or via dig-ital phase-conjugation (Caravaca-Aguirre et al. , 2013;Morales-Delgado et al. , 2015; Papadopoulos et al. , 2012;ˇCiˇzm´ar and Dholakia, 2012), and even by way of mea-suring the transmission matrix of the fiber (Bianchi andDi Leonardo, 2012; Choi et al. , 2012b). One example ofsuch focusing techniques is shown in Fig. 29.Thanks to the ability to focus to single or multiplepoints, or reconstruct an image, a wide variety of imagingmodalities were proposed and realized, in particular fluo-rescence microscopy (Papadopoulos et al. , 2013a; ˇCiˇzm´arand Dholakia, 2012) and photoacoustic microscopy (Pa-padopoulos et al. , 2013b). In all these endoscopic appli-cations images are retrieved in depth, with a diffractionlimited resolution given by the numerical aperture of thefiber. Beyond imaging, the possibility to create one orseveral foci at the tip of an MMF was also shown to al-low optical trapping of dielectric particles (Bianchi andDi Leonardo, 2012).Of course, the exact mode-mixing in the fiber stronglydepends on its specific configuration. The stability ofthe fiber can be extremely good when left untouched,but movements, temperature drifts etc. may degrade the stability of the focusing considerably. Overall, this sensi-tivity scales with the numerical aperture of the fiber, itslength and its diameter. For example, the resilience ofthe transmission matrix to bending of the fiber was inves-tigated in (Choi et al. , 2012b), where it was shown thatit remained exploitable when moving the tip of the fiberby one centimeter, for a 1 m long, 200 𝜇 m diameter fiberof numerical aperture 0.48. Different methods were pro-posed to compensate for fiber movements, e.g., fast opti-mization to a point (Caravaca-Aguirre et al. , 2013). An-other approach relies on storing a set of phase-conjugatepatterns for different fiber positions, and determiningthe fiber position at any time using a so-called “virtualholographic beacon”(Farahi et al. , 2013) and correlatingthe emission from this beacon with the set of measure-ments to recover the fiber position and use it for imag-ing. A more recent approach relies on predicting the TMby evaluating the effect of propagation and bending onthe phase retardation of each principal mode (Pl¨oschner et al. , 2015).In practice, many of the results of the opaque lens onimaging and focusing apply to MMFs as well, but theyalso have several unique features that arise from theirparticular propagation properties. In the weak couplinglimit the correlations in the spatial pattern on the farside of the fiber strongly depend on the injection mode,because the major contributions in the transmission ma-trix are centered around the diagonal (see Fig. 28). Thedistribution of the input mode in 𝑘 -space (i.e., the angu-lar range) matters: In particular, injecting a plane waveat low incidence will populate preferentially the low or-der fiber modes, while a strongly focused wave will re-sult in a decomposition over higher order modes, thatwill partially survive propagation (at least for short dis-tance) and result at the output in variable speckle grainsizes. Symmetries are also important: A focused beamwill produce qualitatively very different speckle patternsdepending on the input position.Moreover, since the number of modes is well-definedand the numerical aperture is limited, the TM can becompletely measured, as in (Carpenter et al. , 2015; Choi et al. , 2012b; Xiong et al. , 2016). In addition, since mostof the light is transmitted in the forward direction, mostsingular values of the transmission value are of modu-lus close to unity (although, in practice, absorption andimperfect injection degrades the flatness of the distribu-tion, see Fig. 28 (bottom right)). This means that, incontrast to scattering systems, imaging is much more ro-bust to noise and image reconstruction can be straight-forwardly achieved using phase-conjugation (Choi et al. ,2012b; ˇCiˇzm´ar and Dholakia, 2012). Polarization mixingis present during propagation in MMFs (Shemirani et al. ,2009) and can be compensated via phase-conjugatedtechniques (McMichael et al. , 1987). In the endoscopicworks with digital wavefront control, a control of bothpolarizations has been achieved ( ˇCiˇzm´ar and Dholakia,92011). Finally, in contrast to the opaque lens, the numberof modes in an MMF is limited, therefore it is possibleto control near-perfectly the ouput pattern with an SLM(with 95% fidelity reported in (Loterie et al. , 2015)). Aconsequence for focusing is that the fraction of light in-tensity that can be brought to the focus can be closeto unity, thereby strongly diminishing the backgroundspeckle. This in turn means that the speckle grains arenot completely independent as in the opaque lens, butare correlated, due to energy conservation and due to thefact that one can achieve almost complete modal control.(We will see how speckle correlations affect the opaquelens in the next section V.)As discussed above, the temporal or spectral behav-ior of MMFs is highly complex and of immediate rele-vance for telecommunications. Like opaque lenses, spa-tiotemporal coupling is present and can in principle beexploited. For instance, temporal focusing of an ultra-short pulse by DOPC has been demonstrated (Morales-Delgado et al. , 2015). An interesting application thathas also been proposed is to use an MMF as a high res-olution spectrometer (Redding and Cao, 2012; Redding et al. , 2013b). In essence, a fixed spatial input (a SMF)serves to inject a well-defined spatial mode, that gener-ates on the far side a complex speckle that depends on thewavelength, with a sensitivity proportional to the fiberlength. The system must first be calibrated with a tun-able monochromatic source. In a second step, a complexspectrum is injected and it produces on the distal side asuperposition of many different speckle patterns that addincoherently, i.e., in intensity rather than in amplitude.The spectrum responsible for this pattern can finally beretrieved by inversion. As shown in Fig. 30, the resolutioncan be extreme for long fibers (the authors demonstrate8pm resolution for a 20 m fiber of 105 𝜇 m core, with 0.22NA).
2. Biological tissues
A complex system of high interest for imaging andwave front shaping is obviously biological tissue. Whileoptical imaging in biological tissues is a vast field, in par-ticular with an endless variety of coherent or incoherentimaging techniques able to retrieve ballistic informationfrom the multiple scattering background and depth res-olution (see (Ntziachristos, 2010; Wang and Hu, 2012;Wang and Wu, 2012; Yu et al. , 2015) for reviews). Acommonly accepted order of magnitude for the scatteringmean free path ℓ of tissues is of the order of 100 𝜇 m in thevisible range, but due to the high forward anisotropy ofthe scattering, the transport mean free path ℓ ⋆ is usuallyof the order of a millimeter. Obviously, there is a strongvariability from tissue to tissue, and a vast amount of lit-erature exists on measurements of scattering propertiesof tissues (see, e.g., (Cheong et al. , 1990)). Figure 30 (color online). Multimode fiber based spectrom-eter. (top) Experimental setup. (bottom) Example of laserline determination with 8pm accuracy. Two laser lines sep-arated by 8pm can be resolved through the reconstructionfrom the speckle pattern. (Figure adapted from (Redding et al. , 2013b).)
While most of the concepts developed in the frameworkof a multiple scattering slab can be adapted to biologicaltissues, such as focusing or imaging with wavefront shap-ing, there are some specific questions to be addressedwhen dealing with imaging in biological tissues. Thefirst one is the problem of decorrelation, that is inher-ent to soft media: The distribution of the refractive in-dex changes relatively rapidly with time, similar to whatis encountered in atmospheric adaptive optics. Measur-ing the transmission matrix or running an optimizationalgorithm must thus be performed on a timescale compa-rable with the decorrelation of the medium, typically ofthe order of a millisecond for in-vivo tissues. Another im-portant aspect is that tissues are typically very thick (afew centimeters) and the distal side of it might not be ac-cessible. At a more fundamental level, one usually wantsto image or focus inside rather than through a scatteringmedium. For this reason, the concept of a thin slab isnot directly relevant: although it is reasonable to con-sider that the medium up to the depth 𝐷 at which onewants to image is an opaque slab of thickness 𝐷 to betraversed, this is, in fact, only partially true. Considerhere, e.g., that light can also propagate deeper than 𝐷 before diffusing back to the plane at depth 𝐷 that is ofinterest.Workarounds for this problem of accessing the regionof interest have meanwhile been developed. One of themis to try to get access to the local light intensity deepinside the tissue using a complementary technique suchas acoustics. Particularly promising progress has beenmade in acousto-optics (Si et al. , 2012; Wang et al. , 2012;Xu et al. , 2011), and in photo-acoustics (Chaigne et al. ,2014a,b; Kong et al. , 2011) (and for a review (Bossy andGigan, 2016). In all these techniques, the resolution is0governed by the acoustic wavelength, which is typicallymuch larger than the optical wavelength. Nonetheless,it is possible to overcome the acoustic resolution andget closer to optical speckle scale resolution, either bycareful spatial coding (Judkewitz et al. , 2013) or by ex-ploiting non-linearities (Conkey et al. , 2015; Lai et al. ,2015). Another option is to rely on reflection measure-ments only. For instance, optimizing the wavefront tomaximize the total non-linear reflected signal can forcelight to focus at depth (Katz et al. , 2014b; Tang et al. ,2012). It is also possible to measure the reflection matrix(Choi et al. , 2013) and exploit its statistical propertiesfor imaging. Another possible idea is to use differentialmeasurements to focus on a moving target (Zhou et al. ,2014). Most of these techniques are detailed in a recentreview (Horstmeyer et al. , 2015). V. MESOSCOPIC PHYSICS AND WAVE FRONTSHAPING
Wavefront shaping techniques have led to remarkableprogress for imaging in or through complex media. Wewill see in this chapter how these techniques can be usedto unravel and exploit mesoscopic effects. We will explainin section V.A how the memory effect has emerged as apowerful tool for imaging, and describe in section V.Brecent optical experiments and theoretical works, wherefirst evidence on the existence of open and closed chan-nels have been discussed. In the next part V.C we willdescribe the properties of time-delay eigenstates in dif-ferent contexts and in V.D new avenues for wave frontshaping in media with gain and loss will be discussed.
A. Memory effect
As we have seen in section III, the information on theincident wavefront is not lost when traversing a multi-ply scattering medium. A special role in this contextplays the memory effect, where spatial variations of theincident wavefront are partially mapped onto easily pre-dictable changes in the transmitted speckle. In transmis-sion, we have seen in section III.D that the thickness ofthe medium 𝐿 determines the typical transverse spatialfeatures that are conserved, which implies that this effectis independent of the strength of the scattering or of theexact scattering properties of the medium.While the mechanism behing the memory effect is verygeneral, we have seen that a particularly important caseis the one of a linear phase ramp on the input wave-front (see Fig. 13). This angular rotation is transferredto the far side of the medium, provided that the trans-verse wavevector 𝑞 of the phase-ramp is changed onlyslightly, Δ 𝑞 < /𝐿 , corresponding to a varation of the an-gle of rotation below the so-called memory effect angle, 𝜃 < 𝜆/ (2 𝜋𝐿 ). More precisely, the transmitted speckledecorrelates over a characteristic angle determined bythe memory effect. The shape of the angular correla-tion function was predicted in (Feng et al. , 1988) to be asinh, but the first experimental realization (Freund et al. ,1988), showed that this function is closer to an exponen-tial decrease. The phase shift of the speckle pattern cor-responds to a rotation by the same angle as the incidentone, When the field propagates away from the sample.Far away from the sample (as on a distant screen) thespeckle field will thus be translated. As shown already in(Freund et al. , 1988), the memory effect is also present inreflection, but the corresponding angle is then not givenby the thickness but by the transport mean free path ℓ ⋆ ,which measures the extent of the diffuse spot in reflection(as discussed in section III.C).A very fundamental insight is that, based on the mem-ory effect, not only linear phase ramps, but actually anyarbitrary modification of the input wave front can betransferred through the medium (with a cut-off spatialfrequency determined by the medium thickness 𝐿 ). Aquadratic phase can, e.g., generate a longitudinal shiftof the resulting speckle far away from the sample, etc.In a visionary paper as early as 1990 (Freund, 1990a),Isaac Freund realized that this means that an opaquelayer could be used for several functions, such as a lens, agrating, a mirror, as well as for imaging. Unfortunately,the possibilities of shaping the speckle to a focus werenot yet available at that time, and the proposal relied onimage correlations between the speckle of interest and areference speckle.
1. Imaging using the memory effect
While most early work considered a plane wave input,the memory effect is also effective for an arbitrary ini-tial input wavefront. In particular, it also works whenthe wavefront has been shaped to obtain a speckle thatcontains a bright focus. If the medium is thin and ifthe focus has been achieved at a distance, it means thatthe focus can be translated. This approach is particu-larly interesting for imaging since, using a single focusand raster scanning it around, it is possible to recover animage, for instance of a fluorescent object, as was firstdemonstrated in (Vellekoop and Aegerter, 2010b) (seeFig. 31). In such a setup, it does not matter how thefocus has been initially obtained, as, e.g., by optimiza-tion of the intensity on a CCD camera (van Putten et al. ,2011; Vellekoop and Aegerter, 2010b). When using thetransmission matrix method, one naturally has the abil-ity to focus at any measurement position. If the mem-ory effect is present, it can be retrieved from correlationsbetween lines of the transmission matrix correspondingto neighboring positions, as demonstrated in (Chaigne et al. , 2014b; Popoff et al. , 2011b). Still, the ability to1
Scattered light fluorescence microscopy:imaging through turbid layers
Ivo M. Vellekoop * and Christof M. Aegerter Physik Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland Fachbereich Physik, Universität Konstanz, Universitätsstrasse 10, 78457 Konstanz, Germany * Corresponding author: [email protected]
Received January 4, 2010; revised March 11, 2010; accepted March 12, 2010;posted March 16, 2010 (Doc. ID 122270); published April 15, 2010
A major limitation of any type of microscope is the penetration depth in turbid tissue. Here, we demonstratea fundamentally novel kind of fluorescence microscope that images through optically thick turbid layers.The microscope uses scattered light, rather than light propagating along a straight path, for imaging withsubwavelength resolution. Our method uses constructive interference to focus scattered laser light throughthe turbid layer. Microscopic fluorescent structures behind the layer were imaged by raster scanning thefocus. © 2010 Optical Society of America
OCIS codes: . The fluorescence microscope has become an indis-pensable tool in any biological or medical laboratory.The development of fluorescent genetic constructs [1]has revolutionized cell and developmental biology,and in vivo fluorescent markers are playing an in-creasing role in biomedical imaging [2]. Currently,one of the main limitations of even the most ad-vanced microscopes is the penetration depth in tur-bid materials [3]. This limitation is fundamentallydue to the fact that inside a turbid medium smallparticles and imperfections scatter the light before itreaches the desired image plane.There are tremendous ongoing efforts in improvingthe imaging depth and resolution of fluorescence mi-croscopes. Historically, two approaches can be identi-fied. The first is to form an image using the fractionof the light that is not scattered. This so-called bal-listic light propagates along a straight line and con-verges to a sharp focus. The difficulty lies in rejectingor reducing the undesired contribution of the scat-tered light. Notable examples of this category areconfocal fluorescence microscopy and multiphotonmicroscopy [4].The second approach is to record the scatteredlight and then use advanced inversion schemes to re-construct the fluorescent structure. This way it ispossible to image up to tens of centimeters deep in,for instance, human tissue [5], or resolve mesoscopicdetails in developing fruit flies [6]. However, the verynature of these methods limits the resolution: detailswith sizes comparable with the wavelength of thelight cannot be resolved.Parallel to these research efforts, methods were de-veloped to focus laser light through turbid materials.Recent examples include turbidity suppressionthrough optical phase conjugation [7], time reversalof electromagnetic radiation [8], and spatial wave-front shaping [9,10]. These methods have in commonthat the incident wave is shaped spatially and/ortemporally to match the exact scattering behavior ofthe material. The shaped wave scatters in such a waythat it interferes constructively at the desired point,effectively creating a focus. Since it is interference of the scattered light that is forming the focus, thesemethods can be summarized as “interferometric fo-cusing,” as opposed to the geometric focusing of alens.Here we demonstrate experimentally the firstscanning fluorescence microscope based on interfero-metric focusing. Our technique, called scattered lightfluorescence microscopy (SLFM) uses scattered lightto form an image with subwavelength resolution. Theprinciple of SLFM is depicted in Fig. 1. Given a fluo-rescent structure that is hidden behind a turbidlayer, conventional imaging fails because all incidentlight is scattered by the layer [Fig. 1(a)]. First, one ofthe above mentioned methods is used to interfero-metrically focus the light through the scatteringlayer [Fig. 1(b)]. Once the focus is formed, the inci-dent wave uniquely matches the microscopic struc-ture of the layer. Thus the focus is lost when the layer
Fig. 1. (Color online) Principle of SLFM. (a) A turbid layerblocks a fluorescent structure from sight; all incident lightis scattered. (b) By use of interferometric focusing (e.g.,phase conjugation or wavefront shaping), scattered light ismade to focus through the layer. (c) Imaging: the focus fol-lows rotations of the incident beam. The hidden structureis imaged by scanning the focus. (d) Experimental setup. Alaser beam is raster scanned, and its wavefront is shapedwith a spatial light modulator (SLM). Dotted lines are con-jugate planes. For ease of view, the SLM is drawn as atransmissive device, and folding mirrors are omitted.April 15, 2010 / Vol. 35, No. 8 / OPTICS LETTERS
Figure 31 (color online). Principle of imaging using wave-front shaping and the memory effect. (Figure and captionadapted from (Vellekoop and Aegerter, 2010b).) (a) A thinscattering layer blocks a fluorescent structure from sight; allincident light is scattered. (b) By use of interferometric focus-ing (e.g., phase conjugation or wavefront shaping), scatteredlight is made to focus through the layer. (c) Imaging: the fo-cus follows rotations of the incident beam over a short angularrange. (d) Simplified schematic of the experimental setup for2D imaging. A laser beam is raster scanned, and its wavefrontis shaped with a spatial light modulator (SLM). Dotted linesare conjugate planes. move a focus by adding an angular tilt is interesting, par-ticularly because it means fast scanners can be used torapidly raster scan a focus, possibly orders of magnitudefaster than a pixellated SLM.It is also possible to use the memory effect withouthaving access to the focus region, as was demonstratedby phase-conjugating the second harmonic signal gener-ated by an implanted probe (Hsieh et al. , 2010a), usingthe photoacoustic effect to remotely monitor the lightintensity (Chaigne et al. , 2014b), or after optimizing anon-linear signal to a focus (Katz et al. , 2014b; Tang et al. , 2012). Also, by adding not only a linear phaseshift but also a quadratic phase ramp, the technique canbe extended to the third dimension both for scanningand imaging (Ghielmetti and Aegerter, 2012, 2014; Yang et al. , 2012).Once a wavefront has been shaped by an SLM to fo-cus at a given position, it means that a source placedat this position would be transformed by the same SLMinto a plane wave, and therefore can be conjugated toa focal spot by a subsequent imaging system. Basedon this concept, it was then realized in (Katz et al. ,2012) that scanning the focus is not the only way torecover an image. As discussed by Isaac Freund (Fre-und, 1990a), the memory effect features an isoplaneticangle over which the speckle remains correlated. In otherwords, the correction of the wavefront compensates thescattering medium for a small angle and for a small rangeof frequency, irrespectively of the illumination. In (Katz et al. , 2012), a point source generates a speckle after ascattering medium, after which an SLM is placed and the wavefront is optimized to form a focus on a CCD camera,therefore performing the analog of a focusing experiment,except that the SLM is placed after the medium ratherthan before. The SLM is conjugated with the outputplane of the scattering medium in order to maximize thememory angle range. If the point source is displaced, sois the focus image on the camera, provided the displace-ment is smaller than the one allowed by the memory ef-fect. The point source is then replaced by an extendedsource (an object), and an image is directly obtained onthe camera. Crucially, the correction even works if theobject is illuminated by spatially and temporally inco-herent light, but since the correction of the wavefront isonly valid within a given angle around the focus and fora given spectral bandwidth related to the spectral corre-lation of the medium around the calibration frequency,only this fraction of the light is well conjugated.Finally, several works reverted to the original idea ofFreund (Freund, 1990a) of using the memory effect with-out shaping or focusing to image behind a turbid layer.All these approaches exploit the fact that a speckle, de-spite being a very complex pattern, has a well-definedpeaked autocorrelation function. In (Bertolotti et al. ,2012), a fluorescent object placed at a distance behind ascattering layer is illuminated by a speckle that is trans-lated (by scanning the illumination angle), and its flu-orescence is collected as a function of the shift of thespeckle, thus forming an image (see Fig. 32). Whilethe resulting image is very complicated and speckle-like,its autocorrelation is actually the product of the auto-correlation of the object and of the autocorrelation ofthe speckle, with the latter having a diffraction-limitedpeaked function. Therefore, one has access to the au-tocorrelation of the object with a resolution given bythe speckle grain size. Using a reconstruction techniqueknown as phase-retrieval (Fienup, 1982), the image of theobject can be retrieved from its autocorrelation function.In (Yang et al. , 2014), the same technique was used to re-cover the image of blood cells behind a scattering layer oftissues. In (Katz et al. , 2014a), another case was studied,where a semi-transparent object, illuminated by spatiallyincoherent light, could be retrieved from the autocorrela-tion of the speckle it produced behind a scattering layer.In all these approaches, no calibration is required, sincethe exact scattering properties of the scattering layer arenot important. However, in most of these work, a scat-tering layer, rather than a scattering volume was used, toensure light transmission and a very pronounced memoryeffect.
2. Beyond the conventional memory effect
While the memory effect is mostly considered in trans-mission, it is also present in reflection (Freund et al. ,1988) and can be exploited. As mentioned in section2
LETTER doi:10.1038/nature11578
Non-invasive imaging through opaque scatteringlayers
Jacopo Bertolotti * , Elbert G. van Putten { * , Christian Blum , Ad Lagendijk , Willem L. Vos & Allard P. Mosk Non-invasive optical imaging techniques, such as optical coher-ence tomography , are essential diagnostic tools in many disci-plines, from the life sciences to nanotechnology. However, presentmethods are not able to image through opaque layers that scatterall the incident light . Even a very thin layer of a scattering ma-terial can appear opaque and hide any objects behind it . Althoughgreat progress has been made recently with methods such as ghostimaging and wavefront shaping , present procedures are stillinvasive because they require either a detector or a nonlinearmaterial to be placed behind the scattering layer. Here we reportan optical method that allows non-invasive imaging of a fluore-scent object that is completely hidden behind an opaque scatteringlayer. We illuminate the object with laser light that has passedthrough the scattering layer. We scan the angle of incidence ofthe laser beam and detect the total fluorescence of the object fromthe front. From the detected signal, we obtain the image of thehidden object using an iterative algorithm . As a proof of con-cept, we retrieve a detailed image of a fluorescent object, compar-able in size (50 micrometres) to a typical human cell, hidden6 millimetres behind an opaque optical diffuser, and an imageof a complex biological sample enclosed between two opaquescreens. This approach to non-invasive imaging through stronglyscattering media can be generalized to other contrast mechanismsand geometries. As experienced on a foggy day, scattering of light severely impairsour ability to see. A strongly scattering medium allows light to pass inthe form of a diffuse halo, but completely scrambles all the spatialinformation . A strategy that has proved very successful in imagingthrough scattering materials is to separate the small amount of lightthat did not change direction owing to random scattering (ballisticlight) from the scattered background using a gated technique such asoptical coherence tomography . In this way it is possible to obtainsharp images through semi-transparent media, but for stronger scat-tering the medium appears opaque to the eye and prevents presentnon-invasive optical imaging techniques from obtaining detailedimages . Absorptive objects deep inside a scattering medium can belocated using diffuse wave tomography, which does not allow one toresolve details much smaller than the depth .Speckle correlations can be used to transmit highly detailed imageinformation through scattering media . To demonstrate non-invasive imaging with speckle correlations, we constructed the set-up illustrated in Fig. 1a. A 50- m m-wide fluorescent object made ofdye-doped polymer (Supplementary Information) is placed a distance d O ( r ), and the speckle intensity, S ( r ), where r is thevector of spatial coordinates.In our measurement procedure, we scan the angle of incidence, h ( h x , h y ), of the laser beam using a pair of scanning galvanic mirrors(Supplementary Information). Although the speckle illuminating theobject might appear random, it contains correlations that can beexploited. In particular, the angular correlation known as the memoryeffect means that rotating the incident beam over small angles h does not change the resulting speckle pattern but only translates it overa distance D r < h d . Therefore, up to a proportionality constant that wewill set to 1, the total measured fluorescence as a function of theincident angle is given by * These authors contributed equally to this work. Complex Photonic Systems (COPS), MESA Institute for Nanotechnology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. University of Florence, Dipartimento di Fisica, 50019Sesto Fiorentino, Italy. Nanobiophysics (NBP), MESA Institute for Nanotechnology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. FOM Institute for Atomic and MolecularPhysics, Science Park 104, 1098 XG Amsterdam, The Netherlands. { Present address: Philips Research Laboratories, 5656 AE Eindhoven, The Netherlands. ab N o r m a li z e d i n t e n s i t y
101 mm c Detector θ d = 6 mm μ m Illuminatingspeckle xyz
Figure 1 | Schematic of the apparatus for non-invasive imaging throughstrongly scattering layers. a , A monochromatic laser beam illuminates anopaque layer of thickness L at an angle h . A fluorescent object is hidden adistance d b , Photographof the scattering layer a distance d c , Intensity of fluorescenceemitted by the hidden object, as measured in front of the scattering layer. Asingle fluorescent image contains no information on the shape of the object. Macmillan Publishers Limited. All rights reserved ©2012 I h ð Þ ~ ð ? { ? O r ð Þ
S r { h d ð Þ d r ~ O S ½ % h ð Þ where denotesthe convolution product. Owing to the random natureof the speckle pattern, the measured intensity, I ( h ) (Fig. 2a), does notdirectly resemble that of the original hidden object. Instead, the imageinformation is encoded in the correlations of the measured signal.To separate the shape of the object from the random speckle, wecalculate the autocorrelation product of the measured intensity andobtain I ? I h i D h ð Þ ~ O S h i ? O S h i ~ O ? O h i S ? S h i ~ O ? O ½ % S ? S h i where ? is the cross-correlation product and angle brackets denote theaverage over speckle realizations (that is, the average over differentscans). Because the average autocorrelation of a speckle pattern, Æ S ? S æ ,is a sharply peaked function , we are effectively measuring the auto-correlation of the object O ? O with a resolution given by the averagespeckle size. For a circular illumination beam of width W , we find that I ? I h i D h ð Þ ~ O ? O ½ % J k j h j W ð Þ k j h j W " " D h ð Þ | k j D h j L sinh k j D h j L ð Þ " ð Þ where J is a first-order Bessel function of the first kind, L is the layerthickness and k is the wavenumber . The second term in the convolu-tion represents the average speckle size and can be made arbitrarilyclose to the diffraction limit by increasing W . The final (multiplicative)factor accounts for the fact that when the change in the angle ofincidence of the laser is not small enough the speckle pattern is notonly rotated but also decorrelates, effectively limiting the memoryrange and, thus, the available field of view. We note thatground-glass diffusers are effectively single scattering layers and thushave a very large memory range while being completely opaque (Supplementary Information). Equation (1) does not depend on thedetailed scattering properties of the scattering layer, allowing us tomeasure O ? O for objects hidden behind any completely opaque layer.The average autocorrelation of nine subsequent scans is shown inFig. 2b. We obtained the independent measurements of I ( h ) needed toaverage S ? S by starting each scan at a different incidence angle. Infact, if the difference between the starting angles is larger than theangular size of the object, the speckle realizations are independent.Comparing the measured autocorrelation with a microscope imageof the object (Fig. 3a), we recognize some features such as the presenceof two vertical legs. Yet an autocorrelation contains only informationon the relative distance between the various parts of an object, and notdirectly on the object itself. Furthermore, the autocorrelation of a real object is always centred and centrosymmetric. To obtain an image ofthe object we need to invert the autocorrelation.In two and three dimensions, autocorrelations can be numericallyinverted using a Gerchberg–Saxton-type iterative algorithm byexploiting some manifest properties of the measured signal as con-straints . In our case, we used the fact that a fluorescent image isalways real and positive. Other common choices are the fact that O is areal function, as in stellar speckleinterferometry , and the positivenessof both the real and the imaginary part of O , as in X-ray scattering .We used a standard version of this algorithm, which can performthe inversion in a few seconds on a normal desktop computer (Sup-plementary Information). The results of the inversion are shown inFig. 3. In Fig. 3a we show a fluorescence microscope image of the objectbefore it is placed behind the scattering layer, and in Fig. 3b we showthe object retrieved from the measured autocorrelation presented inFig. 2b. The two images of the symbol p show an excellent resemblanceto each other. Small features such as the flattening of the left ‘foot’of the symbol or the inhomogeneities in the intensity are faithfullyrecovered, demonstrating successful imaging of the object through anopaque layer.To test our method on a complex biological sample, we placed a sliceof the stem of Convallaria majalis between two diffusers (4.5 cmbehind the front one and 6 mm in front of the back one), effectivelyenclosing the sample. The structure presents intracellular autofluor-escence and did not require staining. The light emitted when thesample was illuminated with the speckled light was collected frombehind the back diffuser (Fig. 4a), and after averaging over five scanswe obtained the autocorrelation shown in Fig. 4b. Figures 4c and 4drespectively show an image of the sample taken with the back diffuserremoved (thus allowing free optical access) and the reconstructedimage obtained starting from the measured autocorrelation. The bluelines in Fig. 4c are contours of the reconstructed image at 20% of themaximum intensity, showing that all the high-intensity features of the C o rr e l a t i o n c o e f fi c i e n t –0.5 0.50 θ y ( ° ) θ x ( ° ) θ x – θ x ,0 ( ° ) θ y – θ y , ( ° ) N o r m a li z e d i n t e g r a t e d i n t e n s i t y ba Figure 2 | Experimental retrieval of the hidden object’s autocorrelation.a , Integrated fluorescent intensity, I , as a function of the incident angle, h ( h x , h y ). b , Autocorrelation I ? I averaged over nine scans taken at different values of the starting incidence angle, h , to average over the differentrealizations of the speckle, S . μ m a Hidden object 1 10 0 N o r m a li z e d i n t e n s i t y b Retrieved object
Figure 3 | Comparison of the retrieved image with the hidden object.a , Fluorescence microscope image of the object taken without the scatteringlayer in place. b , The retrieved object that we found from the measuredautocorrelation in Fig. 2b. Even small details such as the intensityinhomogeneities of the original object are recovered in the retrieved image. LETTER RESEARCH
Macmillan Publishers Limited. All rights reserved ©2012 b c
Figure 32 (color online). (a) Schematic of the apparatusfor non-invasive imaging through strongly scattering layers.A monochromatic laser beam illuminates an opaque layer ofthickness 𝐿 at an angle 𝜃 . A fluorescent object is hidden adistance of 6 mm behind the layer. The fluorescent light isdetected from the front of the scattering layer by a camera.(b) Integrated fluorescent intensity on the camera, as a func-tion of the incident angle, 𝜃 = ( 𝜃 𝑥 , 𝜃 𝑦 ). (c) Autocorrelationof the intensity, averaged over nine scans taken at differentvalues of the starting incidence angle, 𝜃 , to average over thedifferent realizations of the speckle. From the autocorrela-tion, the original image (here the letter 𝜋 ) can be retrievedby phase-retrieval. (Figure adapted from (Bertolotti et al. ,2012).) III.D the limiting memory effect angle in reflection de-pends on the transport mean free path ℓ * , which deter-mines the size of the diffusive halo for focused incidentlight. For a sufficiently strongly scattering material suchas a paint layer, where the mean free path can be on theorder of a micrometer, the angular range of the mem-ory effect can be of several degrees. The relatively largefields of view resulting from this estimate have been ex-ploited for instance in (Katz et al. , 2012) and in (Katz et al. , 2014b) to image “around corners”. The memoryeffect has also been studied in the context of time-reversal(Freund and Rosenbluh, 1991) and polarization (Freund,1990b).An important point in the quest towards exploitingthe memory effect for biological imaging (see also sec-tion IV.E.2) is to assess whether some memory effect canbe present inside a biological tissue. For some biologicalsystems the scattering occurs mainly on a thin layer atthe surface, and the rest of the sample is mostly trans-parent. This is, e.g., the case for the drosophilia puppaat its early development stages, where the fly embryois covered in a thin (8 micrometers) but very scatteringlayer of cells (Vellekoop and Aegerter, 2010a). Inside a volumic scattering medium, which is the case of inter-est for deep biomedical imaging, one would expect frommesoscopic theory that the memory effect is not present,since in the derivation it is supposed to be only observ-able at a distance from the scattering layer. However,some works indicate that this view is conservative. In-deed, tissues typically have a scattering mean free path ℓ of 50-100 micrometers (Cheong et al. , 1990) and more im-portantly, they scatter mostly forward with g-parameters(the average of the cosine of the mean scattering angle)often larger than 0 .
9. This means that, at small depth(millimeters), there are still forward scattered photons,and the diffuse halo is narrower than the one given by afully diffusive model. Experimentally, in some instances,a thin scatterer (onion layer, or chicken breast slice, fixedbrain slice) could be used for imaging within its memoryeffect range, which was characterized to be larger thanpredicted by a diffusive model (Katz et al. , 2014b; Schott et al. , 2015), a strong indication that the memory effectshould be present inside a medium. In (Tang et al. , 2012),a focus was obtained at depth inside brain tissues (800micrometers) and was scanned over a few micrometers.In forward scattering media such as biological tissues, a“translational” memory effect was identified: A lateralshift of the input wavefront resulted in a lateral shift ofthe focus (Judkewitz et al. , 2015), an effect valid insidethe medium rather than at a distance.Analogues of the angular memory effect were alsodemonstrated in MMFs. Here the memory effect is nottransverse, but longitudinal. This comes from the factthat a plane wave with a given 𝑘 -vector that is injected,is mixed angularly but not radially, provided the fiberis not too long or twisted, producing at the output anarrow cone of speckle with the same transverse angleof incidence. The width of this output cone correspondsto an azimutal correlation of the speckle. This meansthat any radial curvature to the initial wavefront canbe transferred to the output. In ( ˇCiˇzm´ar and Dholakia,2012) light is brought to a focus by wavefront-shaping atthe distal end of an MMF. Using those azimutal correla-tions, the focus was shifted axially, but also elongated toproduce a Bessel beam (McGloin and Dholakia, 2005),a doughnut-shaped focus for stimulated emission deple-tion (STED) (Willig et al. , 2007) and more generally forengineering the point-spread-function. There is also arotational memory effect, coming from the fact that aMMF conserves some rotational symmetry, which canbe used to rotate a focus around its center of symmetry(Amitonova et al. , 2015; Rosen et al. , 2015). B. Bimodal distribution of eigenchannels
We have seen in the previous section how the trans-mission matrix (TM) of a disordered slab, a multimodefiber, or of any linear optical system can be measured.3However, it is very important to stress the difference be-tween the TM ˜ t as measured in experiments and the fullTM t as described in sections II and III. The experimen-tal TM ˜ t takes as input modes the different modes thatcan be generated and detected by modulating the inputbeam, i.e., typically pixels on an SLM and on a cam-era, respectively. In the opaque lens, the number of con-trolled modes is typically much smaller than the numberof available input modes of the medium. In addition, thenumber of detected modes is much smaller than those be-ing populated at the ouput of the slab. One of the moststriking features of mesoscopic transport, the bimodaldistribution of eigenchannels (see Fig. 4), is however elu-sive when only measuring a sub-part of the full TM ofthe system. In all the different works on the subject,the ratio of modes that are controlled, detected or illu-minated, is always the limiting parameter. There is noconsensus on notation or even on the definition of thisratio, that is defined and labeled differently in every pa-per, and that depends on the specific situation. In thissection, we chose to leave the different definitions (andnotations) as they were used in the literature, and pointout when they differ.
1. Accessing the bimodal distribution
Since a complete channel control for accessing openand close channels is currently not available, it is veryhelpful to resort to simulations, as in (Choi et al. , 2011a),where the full monochromatic transmission matrix of adisordered slab is numerically evaluated. The numericaldata is in good agreement with RMT (Nazarov, 1994),and the resulting modes when injecting open and closedchannels are evaluated. A very striking result is shownin Fig. 33, where the intensity distribution inside themedium and the scattered fields are computed for a planewave input as well as when injecting the optimal wave-front for exciting an open or closed channel. One can seea very dramatic difference in the intensity distributionalong the longitudinal direction when comparing thesedifferent cases. When injecting a plane wave, which ex-cites all available transmission channels very broadly, theaveraged intensity diminishes linearly with depth, as pre-dicted by Ohm’s law. When the wavefront correspondsto injecting a closed channel, the decay is much fasterand exponential, while when injecting an open channelthe intensity first increases with depth until the centerof the slab, and only diminishes thereafter. As a result,the intensity is almost symmetric with a maximum inthe center of the slab, as necessary in order to transmita significant amount of energy through the medium. An-alytical expressions for these distribution functions havebeen proposed in (Davy et al. , 2015a).Note that the above description considers a simplifiedsystem, which serves as a good starting point to under-
TRANSMISSION EIGENCHANNELS IN A DISORDERED MEDIUM PHYSICAL REVIEW B , 134207 (2011)FIG. 4. (Color online) Field distributions of eigenchannels insidethe medium. (a)–(c) Field distributions of a plane wave whoseincident angle is 11.5 ◦ , open eigenchannel and closed eigenchannel,respectively. The incident field is subtracted on the left-hand sideof the medium. Color bar: amplitude normalized to the input wave.Scale bar: 10 µ m. (d) Average intensity along the x direction as afunction of the depth in the z direction. The disordered medium fillsthe space between 0 and 16 µ m in depth. The intensity is normalizedto that of a normally incident plane wave. thickness of the medium is much larger than the transport meanfree path.For the same disordered medium, we obtain eigenchan-nels from singular value decomposition of the transmissionmatrix, and we solve their propagation through the medium.Figure 4(b) displays the propagation of an open eigenchannelwhose eigenvalue is 0.955. We find that the field strength insidethe medium is enhanced such that its average intensity is higherthan that of the input [blue curve in Fig. 4(d)]. This is in analogywith a Fabry-Perot cavity in which internal field strengthincreases at the resonance condition due to the constructiveinterference. Likewise, the constructive interference enhancesthe internal energy in open eigenchannels, which leads tostrongly enhanced transmission. It is interesting to note thatspatial mode coupling alone can enhance the field strength inthe case of the disordered medium regardless of the source fre-quency. In the case of a closed mode whose eigenvalue is close to zero [Fig. 4(c)], a steep decrease of intensity is observedalong the z direction as soon as the wave enters the medium.This suggests strong destructive interference as the wavereaches the end of the medium. The intensity profile follows theexponentially decaying curve [red curve in Fig. 4(d)]. Overall,the location of the peak intensity inside the medium shifts fromthe center to the input side with the decrease of the eigenvalue.The internal energy of the single-channel optimizing mode isclose to the open eigenchannel in connection with its enhancedtransmission.Open eigenchannels in a passive medium may have aconnection to photon diffusion in a medium with gain. AsPayne and Yamilov have analyzed, the propagation profilehas an enhanced peak in the middle of the medium whenthere is gain and it is exponentially decaying when thereis absorption. These profiles have some similarities withthe open and closed eigenchannels of the passive medium,respectively. This may be due to the fact that the openeigenchannels preferentially elongate photon residence time,and gain in the photon diffusion emphasizes the importance ofa very long light path. VII. CONCLUSION
In this study, we numerically explored the properties ofeigenchannels in their field distribution inside a disorderedmedium and observed that open eigenchannels significantlyenhance field energy in the medium in accordance with theirhigh transmission. Furthermore, by constructing a transmis-sion matrix in experimentally feasible open slab geometrywith full coverage of input and output channels, we validatethat the open slab geometry can still be modeled as waveguidegeometry used in analytic theory as long as the thickness ofthe medium is 0.18 times smaller than the sampling width.Finally, we confirmed that the single-channel optimizationprocess is equivalent to emphasizing the open eigenchannels.Our method will pave the way for an exploration of theexperimental implementation of open eigenchannels and theirpotential use for imaging through turbid media and randomlasers.
ACKNOWLEDGMENTS
This research was supported by the Basic Science ResearchProgram through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Tech-nology (Grant No. 20100011286), by the Korea government(MEST) (No. 2010-0028713, 2010-0019171), and by a KoreaUniversity Grant. C. Vanneste, P. Sebbah, and H. Cao, Phys. Rev. Lett. , 143902(2007). J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, andH. Kalt, Nat. Photon. , 279 (2009). P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys.France , 63 (1988). I. M. Vellekoop and A. P. Mosk, Phys. Rev. Lett. , 120601(2008). I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, Nat. Photon. , 320(2010). S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara,and S. Gigan, Phys. Rev. Lett. , 100601 (2010).134207-5
Figure 33 (color online). Simulation of scalar field distri-butions of transmission eigenchannels inside a 2D disorderedslab with 299 channels. The medium of height 130 𝜇 m andthickness 16 𝜇 m is shown in the middle of each of the three toppanels (a)–(c). Field distributions (a) of a plane wave whoseincident angle is 11 . ∘ , (b) of an open transmission eigen-channel (transmission of 0.955) and (c) of a closed transmis-sion eigenchannel (close to zero transmission), respectively.The incident field is subtracted on the left-hand side of themedium. Color bar: amplitude normalized to the input wave.Scale bar: 10 𝜇 m. (d) Averaged intensity along the 𝑥 di-rection as a function of the depth in the 𝑧 direction for thesame three different input wavefronts, plus the wavefront cor-responding to a focusing optimization, as in (Vellekoop andMosk, 2008b). The disordered medium fills the space between0 and 16 𝜇 m in depth. The intensity is normalized to that ofa normally incident plane wave. (Figure adapted from (Choi et al. , 2011a).) stand the difficulty to measure this distribution and in-ject the corresponding modes in practice: First, it is as-sumed that all modes are accessible (from both sides ofthe slab, including also the polarization degrees of free-dom), and that the system is two-dimensional only (asin section II for the waveguide geometry). First experi-ments which could satisfy these demanding requirementswere reported in acoustics (G´erardin et al. , 2014) and inoptics (Sarma et al. , 2016). In both setups the dramaticchange in the internal energy distribution (see Fig. 33)could, indeed, be observed. In most experiments, how-ever, these conditions are difficult to meet in practice.An analytical model and numerical simulations in thewaveguide geometry (Goetschy and Stone, 2013) werededicated to what happens to the distribution of mea-sured transmission eigenvalues in case of partial chan-nel access (considering both control and detection), andhow it affects the maximal transmission 𝑇 max that can be4 computed using the recursive Green’s function method[22], and members of the filtered ensemble are then gen-erated by random projection.When m is further reduced, the correlations containedin the transmission matrix are progressively lost and p ~ t y ~ t ð T Þ evolves such that the distribution of X ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi T= h T i p converges to the quarter circle law, p X ð x Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi $ x p = ! ,independent of ‘ . Thus, universal uncorrelated behavior isreached when m & ! T (or M & G ), in agreement withmeasurements reported in [13]. This loss of correlationsin the limit of small degree of channel control remainswhen m ! m , but in a more subtle form. For example, for m % m & ! T and m ¼ , we find that p X approaches theMarcˇenko-Pastur (MP) law [14], describing rectangularrandom matrices with uncorrelated Gaussian matrixelements, but for a matrix ensemble with a disorder-dependent, effective value of m ! ~ m . Specifically p X ð x Þ ’ ! ~ mx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x þ $ x Þð x $ x $ Þ q ; (7)where x ’ ¼ ð ’ ffiffiffiffi ~ m p Þ and ~ m % m ð = ! T $ Þ [20]; thiscorresponds to a MP distribution for ~ N ( M matrices, with ~ N ¼ ! TN= ð $ ! T Þ .The statistics of lossless reflection with ICC can beobtained similarly to those of transmission, with qualita-tively similar results (i.e., suppression of extremal valuesand convergence to an effective MP distribution).However, in the case of an absorbing disordered medium within a waveguide, one has a distinct statistical ensemble[23,24] from that in the lossless case. The extremal eigen-value statistics of this ensemble were recently studied byChong and Stone [10] and lead to the phenomenon of CEA.Here the (nonunitary) S matrix and the reflection matrixcoincide, and $ R n represents the absorbed fraction ofan incident eigenchannel, where f R n g are the eigenvaluesof r y r . Let ‘ a ) ‘ be the ballistic absorption length andconsider the regime in which ‘ * ffiffiffiffiffiffiffiffi ‘‘ a p < L , ‘ a , so thatelastic scattering is strong, absorption is weak, but trans-mission is negligible. In Ref. [10], it was found that when N ð ‘=‘ a Þ ) , the smallest R n (reflectivity of the mosthighly absorbed eigenchannel) was orders of magnitudesmaller than the mean reflectivity, ! R h R min i ! R ’ N a ; a % ‘‘ a * : (8)As N ! 1 , h R min i ! while ! R remains + , which wouldnot be true, e.g., for the MP law, and is the essence ofCEA; in addition, the density p r y r ð R Þ diverges at R ¼ (see Fig. 2). The =N scaling of h R min i holds even whenthe absorption is nonuniform, e.g., for a buried absorberbehind an ‘‘opaque’’, lossless layer. The effect of ICC inthis case is again found by solving Eq. (3), now with A ¼ r .We will specialize to the case where a fraction m ¼ M=N of the input channels can be excited, while the field in alloutput channels is collected, m ¼ m , m ¼ . We find theeigenvalue density p ~ r y ~ r , from the known density p r y r [23,24] and the associated resolvent, FIG. 1 (color online). Transmission eigenvalue density of adisordered slab placed in a waveguide with N ¼ channels(length L ¼ =k , width W ¼ =k ), for different fractionsof controlled channels m ¼ M=N . Numerical results (dots) areobtained from solving the wave equation for 120 realizationsof the slab, with dielectric function " ð r Þ ¼ n þ ð r Þ ; n ¼ : and ð r Þ is uniformly distributed between ½$ : ; þ : - in theslab, and ð r Þ ¼ in the empty waveguide. The solid lines arethe theoretical prediction based on Eqs. (2), (3), and (6), where ! T ¼ h P Nn ¼ T n i =N ¼ : is found from the simulation withcomplete channel control ( m ¼ ). Inset shows the maximaltransmission enhancement possible for a given m , where h T i ¼ m ! T , and h T max i is calculated by the method given in [19]. FIG. 2 (color online). Reflection eigenvalue density of a dis-ordered absorbing slab with the same geometry and dielectricfunction as in Fig. 1 except for the addition to " ð r Þ of a constantimaginary part, 0.003i, representing absorption. A fraction m ¼ M=N of the input channels are excited while all output reflectionchannels are collected. Numerical results are based on 100realizations of the disordered slab. Solid lines are the theoreticalprediction based on Eqs. (2), (3), and (9), where a ¼ : isdetermined by the numerical value of ! R ¼ : with m ¼ . Theballistic and diffusive absorption lengths are ‘ a ¼ ‘=a ¼ : L and ffiffiffiffiffiffiffiffi ‘‘ a p ¼ : L . PRL week ending9 AUGUST 2013
Figure 34 (color online). Transmission eigenvalue densityof a disordered slab placed in a waveguide with 𝑁 = 485channels (length 𝐿 = 150 /𝑘 , width 𝑊 = 900 /𝑘 ), for differentfractions of controlled channels 𝑚 = 𝑀/𝑁 . Numerical results(dots) are obtained from solving the wave equation for 120realizations of the slab for fixed disorder strength. The solidlines are the theoretical prediction based on free probabilitytheory. The inset shows the maximal transmission enhance-ment possible for a given 𝑚 . (Figure adapted from (Goetschyand Stone, 2013).) achieved by wavefront shaping when injecting the maxi-mally open channel that can be measured. For this pur-pose the control parameter 𝑚 = 𝑀/𝑁 was introduced,where 𝑀 is the number of channels controlled and 𝑁 thetotal number of channels. The most striking result is thateven a small degree of imperfect control ( 𝑚 (cid:46)
1) abruptlysuppresses the mode of unit transmission, and the mea-sured distribution rapidly deviates from the bimodal one,with the disappearance of the peaked distribution aroundunity (see Fig. 34). An increase of the transmission rel-atively to the mean transmission (i.e., 𝑇 𝑚𝑎𝑥 ≫ 𝑇 ) cannonetheless be achieved with partial control, until thedistribution converges to the Marˇcenko-Pastur distribu-tion (as in Eq. (51) and in Fig. 20). Interestingly, thecrossover to uncorrelated Gaussian matrices occurs whenthe number of modes controlled is lower than the to-tal transmission 𝑇 . Similar results are derived in reflec-tion, where the perfect reflection expected when injectingclosed modes is suppressed with imperfect control anddetection. In particular, detection or control of a sin-gle polarization immediately results in loss of half of themodes. The result by Goetschy and Stone (Goetschy andStone, 2013) was extended in (Popoff et al. , 2014) to in-clude not only the waveguide geometry, but also the slabgeometry with partial illumination. In particular, the ef-fective control parameter 𝑚 is extended for the case of anillumination zone 𝐷 smaller than the thickness 𝐿 of theslab. Due to the fact that the area of the diffusive halo onthe far side is larger than the injection area, the numberof modes at the output is automatically larger than thenumber of input modes, which results in a diminution ofthe maximal transmission achievable.
2. Unraveling and exploiting open and closed channels
Despite these difficulties in accessing the full bimodalnature of the transmission eigenchannels of a disorderedslab, several experiments have managed to reveal somefeatures of bimodality by means of wavefront shaping.Historically, the first experimental result was reportedby Vellekoop and Mosk as early as 2008 (Vellekoop andMosk, 2008b), where an optimization through a slab wasperformed in the limit where a noticeable fraction of themodes is controlled (up to approximately 30% at the in-put). Experimentally, this was achieved in two ways; firstby designing a relatively thin multiply-scattering sam-ple (down to 5 . 𝜇 m) to minimize the number of modesto be controlled; secondly by controlling and detectingboth polarizations using polarization separators, and us-ing high numerical aperture objectives to access high an-gles of incidence. The result, when performing the samepoint optimization as in (Vellekoop and Mosk, 2007) wasa spectacular deviation from the opaque lens predictions(see Fig. 35). While optimizing a single speckle spot,an increase of the overall transmitted intensity was ob-served, not only in the focus, but also in the surround-ing speckle, which meant that strong spatial correlationsmust be present in the speckle, due to the fact that onlya few modes contribute to the transmitted speckle. Theincreased transmission was compared with RMT predic-tions, and could be well-interpreted as a redistribution ofthe input energy from closed to open channels.The authors further derive that perfect optimizationto a single point should, on average, increase the totaltransmission to the universal value of 2 /
3. This valueof 2/3 is directly linked to the electronic quantum shotnoise for the case of a bimodal distribution of trans-mission eigenvalues (Beenakker, 2011), which we havefound to be characterized by a sub-Poissonian shot noiseFano factor 𝐹 = tr[ tt † ( − tt † )] / tr[ tt † ] = 1 / E in = t † E target (we neglect here for the momentthat in the experiment only ˜ t , i.e., the partial trans-mission matrix restricted to the measured and controlledmodes is available). The output field, in turn, is givenas follows, E out = tt † E target , and the total intensity is 𝐼 out = | E out | . When averaging over realizations or overtarget positions, one can show that the average trans-mission is 𝑇 = 𝐼 𝑜𝑢𝑡 /𝐼 𝑖𝑛 = tr( tt † tt † ) / tr( tt † ). Since thisrelation is directly related to the shot noise Fano factorby 𝑇 = 1 − 𝐹 , we simply get the result 𝑇 = 2 /
3. Thisresult was numerically confirmed in (Choi et al. , 2011a),where it was further noted that this result is connectedto the fact that the contribution of each eigenchannel 𝑛 to the optimized wavefront was, on average, proportionalto its eigenvalue | 𝜏 𝑛 | , a well-known property of the timereversal operator tt † (Tanter et al. , 2000). This theoreti-5 We now proceed to introduce a quantitative measure ofthe control we exert over a shaped wave front. Our algo-rithm maximizes the intensity in a diffraction limited spot,which is exactly one of the transmitted free modes. Welabel this special target mode with the index ! . The ideallyshaped incident wave front E opt ; ! a for maximizing theintensity in ! is given by [16] E opt ; ! a ¼ ð T ! Þ $ = t % ! a ; T ! & X Na j t ! a j ; (2)where T ! normalizes the total incident power. Our opti-mization algorithm proceeds as follows: the matrix ele-ments t ! a are measured up to a constant prefactor bycycling the phase of the light in the incident mode a whileobserving the intensity in target mode ! [15]. After N phases have been measured, the optimized incident wavefront is constructed according to Eq. (2). This optimizedwave front couples to a superposition of eigenchannels,mostly to channels with high transmission eigenvalues.In any experiment, the resolution and the spatial extentof the generated field are finite. Therefore, it will never bepossible to exactly construct the wave front described byEq. (2). To quantify how well the actual incident field E act ; ! a matches the optimal incident field E opt ; ! a , we introduce theoverlap coefficient " as " & X Na ð E opt ; ! a Þ % E act ; ! a : (3)The degree of intensity control is j " j . We can now writeany incident wave front as a linear superposition of theperfect wave front and an error term E act ; ! a ¼ " E opt ; ! a þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $ j " j q ! E a ; (4)where the error term ! E a is normalized. For ideal controlover the incident wave front j " j ¼ . In earlier experi-ments [15] the degree of control was relatively low( j " j ( : ), and total transmission did not increase meas-urably. In this Letter, we discuss experiments at muchhigher values of j " j , up to 0.33.The experimental apparatus (see Fig. 1) is designed toapproach the optimal wave front as closely as possible, bycontrolling the largest possible fraction of the incident freemodes. An expanded beam from a 632.8 nm HeNe laser isrotated to a 45 ) linear polarization by a half wave plate andimpinges on a polarizing beam splitter cube. Horizontallyand vertically polarized beams are modulated with separatereflective liquid crystal displays (Holoeye LC-R 2500) andthen recombined, to provide control over modes with bothpolarizations. We used a 4-pixel macropixel modulationmethod [17] to control the phase of the light withoutresidual amplitude modulation. The modulator is dividedinto 3816 independently programmable segments. A com-puter programs the modulators using feedback from a camera as discussed below. A sequential optimizationalgorithm [15] was used to optimize the wave front [16].A high numerical aperture objective ( NA ¼ : , ZeissAchroplan * ) projects the shaped wave front onto thesample.Each sample consists of a layer of spray-painted ZnOparticles on a standard glass microscope cover slip. Theparticles have an average diameter of 200 nm, whichmakes them strongly scattering for visible light. Themean free path was determined by measuring the totaltransmission and equals : + : m at a wavelengthof 632.8 nm. We used samples with thicknesses of 5.7 and : m . The samples were positioned in the focal planeof the microscope objective to minimize the size of thediffuse spot, and thereby the number of contributingmodes. The number of such contributing free modes wasestimated from the intensity profile of the transmitted light[12] to be : * and : * modes, for the thin andthe thick samples, respectively. The samples were mountedon a motorized stage to translate them in the focal plane.A high NA oil-immersion objective (Nikon TIRF * = NA ¼ : ) collects the transmitted light. The transmittedlight is split into horizontal and vertical polarizations by abeam splitter cube. A second polarizer improves the ex-tinction ratio for reflected light. The magnification of thedetection system is * , enough to well-resolve indi-vidual speckles. A camera measures the power of thehorizontally polarized light in a disk with a diameter of : m at the sample, which is smaller than a singlespeckle, to provide feedback for the optimizationalgorithm.After optimization, a calibrated neutral density filterwith a transmission of : * $ is placed in front ofthe camera to measure the high intensity in the target. Asecond camera images the intensity of the vertically polar-ized light.Optimizing the incident wave front caused the intensityin the target to increase dramatically. In Fig. 2 we plot the HeNe
SLM
PBS L L A λ /2 S L M NA =0.95 S CCD
63x 60xNA =1.49 L PBS
CCD
P ND PC feedback FIG. 1 (color). Experimental setup. HeNe, expanded 632.8 nmHeNe laser; $ = , half wave plate; PBS, polarizing beam splittercube; SLM, spatial light modulator. A, iris diaphragm; * ,microscope objective; * , oil-immersion microscope objec-tive; S, sample; P, polarizer; ND, neutral density filter; L , L , L , lenses with focal length of, respectively, 250, 200, and600 mm. ND and S are translated by computer controlled stages. PRL week ending19 SEPTEMBER 2008 transmitted intensity through a : ! m -thick sample for anonoptimized wave front and for the optimized wave front[16]. Before optimization, the transmitted intensity forms adiffuse spot on the back surface of the sample. Afteroptimization, a strong peak emerges in the target focus.The intensity increase in the center of the target was afactor of ! . After optimization, 2.3% of the inci-dent power is transmitted into the target focus.More importantly, the intensity in an area with a radiusof approximately ! m around the target also increased,even though the algorithm did not use this intensity asfeedback. This observation indicates that we have redis-tributed the incident light from closed eigenchannels toopen eigenchannels. As a result of optimizing a singletarget point, the total angle-integrated transmission in-creased from 0.23 to 0.31. This change amounts to arelative increase of 35%.For a quantitative analysis we need to know the degreeof control j " j . Factors like measurement noise and ther-mal drift result in a different degree of control for eachsingle run of the experiment. Fortunately, it is possible tomeasure j " j directly for each run by observing the inten-sity in the target. Only the controlled fraction of the inci-dent wave front contributes to the intensity in the target.The transmission to the target mode equals j E act j ¼ !!!!!!!! X Na t a E act ; a !!!!!!!! (5) ¼ j " j !!!!!!!! X Na t a E opt ; a !!!!!!!! ; (6)where we used Eq. (4) and the fact that the error term isorthogonal to the ideal wave front. By substituting Eq. (2)we obtain j E act j ¼ j " j T : (7)Equation (7) allows us to obtain the degree of control bymeasuring the intensity in the target focus j E act j and T . Inthe experimental procedure it is very impractical to mea-sure T . Therefore, we approximate T ¼ T tot N=M . Here, T tot is the ensemble averaged total transmission of anunoptimized wave front, and M is the number of trans-mitted free modes. Since our samples are sandwichedbetween a glass substrate on one side and air on the otherside, the number of modes on the back of the sample islarger and M ¼ n N , with n ¼ : the refractive index ofthe substrate. This approximation neglects the C fluctua-tions [18] in the total transmission, which are in the orderof 2% for our samples.For the experimental run that is shown in Fig. 2, we finda degree of control of j " j ¼ : . This means that theincident field is a linear superposition of the perfectlyshaped wave front (carrying 23% of the incident power)and a noise term (carrying the rest of the power). The total, angle-integrated transmission T acttot contains contributionsboth from the perfectly shaped wave front and from thenoise term, T acttot ¼ T c þ ð % j " j Þ T tot ; (8)where T c is the part of the transmission resulting from theperfectly shaped fraction of the incident wave front. Bysubstituting Eq. (2) into Eq. (1) and summing the power inall transmitted free modes, we find T c ¼ j " j X Mb T !!!!!!!! X Na t ba t ’ a !!!!!!!! ( j " j C ; : (9)We evaluate C ; theoretically by averaging over all pos-sible target modes . We assume that C ; is self-averaging, which is verified by our experiment.Neglecting small correlation terms between numeratorand denominator we find C ; ¼ h T i " X Mb !!!!!!!! X Na t ba t ’ a !!!!!!!! ¼ Tr t y tt y t Tr tt y : (10)From Eq. (10), it becomes clear that C ; is a measure forthe width of the distribution of the transmission eigenval-ues [9]. By measuring the total transmission after optimiz-ing the incident wave front, we have direct experimentalaccess to this value for each single sample. Since wemeasured j " j separately, we can use Eqs. (8) and (9) toobtain C ; from a single, nonideal experimental run. In theparticular run in Fig. 2, we find C ; ¼ : .The ensemble averaged value for C ; was derived usingRMT. RMT [19] predicts h C ; i ¼ = for a nonabsorbingsystem far away from the localization transition, regardlessof the original transmission coefficient of the system. For asingle realization of disorder, we found C ; ¼ : . To FIG. 2 (color). Intensity distribution of horizontally polarizedlight in a ! m ) ! m square area at the back of the sample.(a) For a nonoptimized incident wave front. (b) For an optimizedwave front. (c) Intensity summed in the y direction to averageover speckle. Dashed curve, transmission of nonoptimized wavefront; solid curve, transmission of optimized wave front. PRL week ending19 SEPTEMBER 2008
Figure 35 (color online). Coupling to open channels by wave-front optimization to a focus spot. (top) Schematic of theexperiment: two SLMs are used to control both polarizationdirections; high numerical aperture objectives ensure maximalcoverage of incident angles; both polarizations are detected ontwo CCD cameras on the far side. (bottom) Intensity of thetransmitted speckle figures at the output plane (a) before and(b) after optimization of the wavefront. (c) Intensity summedover the 𝑦 direction to average the speckle: dashed curve forunoptimized and solid curve for optimized wavefront. Thetotal transmission is increased by 35%. (Figure adapted from(Vellekoop and Mosk, 2008b).) cal result was later verified experimentally in (Kim et al. ,2013). Of course, imperfect control of the wavefront leadsto a reduced transmission compared to the ideal case.The authors introduce in (Vellekoop and Mosk, 2008b) aparameter 𝛾 that represents the overlap between the in-jected mode and the perfect optimized mode, the differ-ence coming both from imperfect channel control, phase-only operation, and noise in the optimization process.The expected total transmission is 𝑇 𝑐 = | 𝛾 | 𝑇 , and ex-cellent agreement in different experimental conditions isfound.In contrast to the opaque lens case, optimization andimaging through a medium gives quite different resultsif the number of open channels becomes lower than thenumber of degrees of freedom that one has access to.As shown in (Vellekoop and Mosk, 2008b), when opti-mizing a single point, the background increases, whichmeans that the signal to background ratio is lower thanexpected from the opaque lens analysis. This effect hasbeen discussed in (Davy et al. , 2012), in the context of microwaves but the result remains valid for optics. Inessence, the signal to background in a point optimiza-tion experiment is bounded by the number of transmit-ting modes and can be down to one (no optimization)in the single open channel regime (Pe˜na et al. , 2014).Of course, in this limiting case, the intensity at the focushas been indeed increased, but since there is only a singlemode that dominates the transmission, the backgroundhas also increased correspondingly.Several works have also reported on measuring a TMand subsequently injecting the mode with the largesttransmission. In (Kim et al. , 2012), the limiting case ofvery sparse measurement was explored: A square trans-mission matrix was measured over a single polarizationand a very limited angular view (covering a numericalaperture of 𝑁 𝐴 ≃ . et al. , 2010b), the authors showthat when sending the input vector corresponding to thehighest transmission mode given by the measured trans-mission matrix, they recover a higher transmission by afactor 3 .
99 relative to the mean transmission within thedetection angle, in good agreement with the fact that thedistribution of eigenvalues is bounded to twice the meanamplitude transmission in the case of a square randommatrix (corresponding to a factor four in intensity). In(Kim et al. , 2014), the authors reported on the promis-ing use of a binary amplitude modulator to measure theTM and to inject eigenchannels. They show that such abinary modulator is able to match the calculated modefor single mode injection with 40% fidelity, and demon-strate a two-fold increase of transmission over the meantransmission in the detection angle of their apparatus.Measurements of very large TMs in strongly scatteringmedia to approach the bimodal distribution are reportedin (Yu et al. , 2013) and (Akbulut et al. , 2016). In bothstudies, deviations from the Marˇcenko-Pastur distribu-tion were observed, and could be qualitatively modeled,but due to limited control, the direct observation and in-jection of open channels in optics has so far not yet beenreported in the literature (see (G´erardin et al. , 2014) fora first realization of open channels in acoustics).Another possibility of high practical interest, is to mea-sure the reflection matrix. When sending the mode withthe lowest reflection, a high transmission should be ob-tained. This idea was experimentally implemented in(Kim et al. , 2015) where a three-fold increase in trans-mission is reported when injecting the mode with thesmallest eigenvalue in reflection.An alternative to single-point optimization was re-ported experimentally in (Popoff et al. , 2014), see Fig. 36for a corresponding illustration. Instead of measuring asingle position and observing an increase of the transmis-sion, here the total transmitted intensity was optimized6directly. In order to detect all modes at the output andnot to be limited by collection optics, a large-area pho-todetector is directly placed on the backside of the sam-ple, thus collecting all modes with the same efficiency.This is in contrast with previous results where a limitedcollection angle was inherent to all implementations, andtherefore only partial transmission enhancements werereported, from which changes in total transmission couldonly be inferred. When optimizing the wavefront to max-imize or minimize the total transmitted intensity, an ap-proximately 3 . . et al. , 2013, 2014), ora photocurrent (Liew et al. , 2016). As was shown bothexperimentally and theoretically (Hsu et al. , 2017) long-range-correlation effects as discussed in section III.A andin (Garc´ıa-Mart´ın et al. , 2002; Scheffold et al. , 1997) in-crease the dynamic range of light-delivery into regionscontaining multiple speckles. C. Time delay eigenstates
In this section we will discuss the interesting propertiesof the eigenstates of the time-delay operator or matrix Q = − 𝑖 S † ( 𝜕 S ) / ( 𝜕𝜔 ) defined in sec. II.C, see Eq. (47).Thanks to the techniques of wavefront shaping discussedin section IV, it is now within reach not only to mea-sure the time delay matrix of a system, but also to excitean eigenvector at the input, thus generating a time-delayeigenstate. We will first describe these states in all theirgenerality, then discuss their applications in two specificcases relevant for optics: principal states and particle-likestates (see sections V.C.1 and V.C.2). The eigenstates q 𝑖 of Q are defined as follows, Q q 𝑖 = 𝑞 𝑖 q 𝑖 , and they are as-sociated with a well-defined scattering time delay 𝑞 𝑖 (alsocalled proper delay time), which measures the time ac-cumulated between entering and exiting the scatteringregion. When the scattering matrix S of a problem isunitary, the associated time delay operator Q is Hermi-tian, such that the proper delay times are real and thetime delay eigenstates q 𝑖 form a complete and orthogonalset at the input to the scattering region.The question we address in the following is, whichpractical consequences can be associated with the abovedefinitions and how such states can be determined andgenerated experimentally. Generally speaking, a wavethat enters a disordered slab will have components thatexit the slab rather quickly, while other components willstay inside the slab for longer (as discussed in section mesoscopic correlations, and can be calculated microscopi-cally with no fitting parameter.To control total transmission through a disordered slab,we designed an experiment to achieve a high degree ofcontrol of the phase of the input light with both polar-izations. The illumination area on the slab surface is muchlarger than the wavelength. The experimental apparatus ispresented in Fig. 1(a) and detailed in the SupplementalMaterial [22]. The modulated wave front is projected ontothe pupil of a microscope objective of numerical aperture0.95. Adjacent pixels of the SLM are grouped to form “ macropixels, ” whose size determines the illuminationarea on the sample. In order to collect light in all outputchannels, we place the sample directly onto a largephotodetector. This allows us to measure the total trans-mitted light without being limited by the numerical apertureof the collecting optics. Two additional photodetectors areused to measure the incident light intensity before themicroscope objective and the reflection from the sample.We then perform a feedback optimization procedure similarto the sequential algorithm developed in Ref. [10] toincrease or decrease the total transmission. The value tomaximize or minimize is the ratio of the total integratedtransmitted intensity over the input intensity, henceforthtermed the total transmission T . It is crucial to optimizethe ratio, because wave front shaping by the SLM modifiesnot only the transport of light through the sample, but alsothe transmission of the optical system that delivers light from the SLM to the sample, and is, hence, vulnerable tosystematic errors or artifacts [23].The scattering samples are slabs of randomly packedpolydisperse TiO microparticles of median diameter410 nm, deposited on glass cover slips by evaporation.The mean free path, measured from a coherent back-scattering experiment, is l ¼ . " . μ m. To demon-strate coherent control we both maximize and minimize T .In Fig. 1(b) we show results for a sample of averagetransmission h T i ∼ that demonstrate an enhancement of T ∼ . , and a reduction ∼ . . Thus, the total transmissionof a single realization of a scattering medium can be tunedby more than a factor of 11 between 1.6% and 18%. Thediameter of the illumination area on the sample surface is . μ m, and the number of macropixels of the SLM, whosephases are optimized, is N in ¼ . We also measure thechange in reflection R (ratio of the reflected light intensityover the incident intensity), and compare it to the changeestimated from the transmission using the relation R= h R i ≃ ð − T Þ = ð − h T i Þ [Fig. 1(b)]. The good agreement con-firmed that the variations of the measured total transmissionare due to changes of the total transmission through thescattering sample (further verifications are considered inthe Supplemental Material [22]).We show in the following that mesoscopic correlationsare essential to the significant variation of total trans-mission. We compare our data to the predictions of theuncorrelated random matrix ensemble. For an uncorrelatedTM described by the MP law, the mean maximum trans-mission satisfies [20] h T max ih T i ¼ ð þ ffiffiffi γ p Þ ; (1)where γ is the ratio of the number of controlled inputchannels to the number of excited output channels. Areasonable estimate is γ ≃ ð D=D out Þ , where D out is thetypical size of the diffusive output spot. The maximumpossible transmission (1) is monotonically decreasing withthe sample thickness L because D out increases with L forfixed input illumination diameter D . In Fig. 2(a) we plot T max = h T i measured versus L for a fixed D , finding thatinstead of decreasing, it increases and then saturates atthe largest L shown. The value of the enhancement at thelargest L is more than twice that of the MP law. Similarly,for fixed L and variable D , the data in Fig. 2(b) show muchhigher enhancements than the predictions of the uncorre-lated model, implying that significant correlations in theTM enable larger coherent control of transmission.To further confirm this, we intentionally spoil thecorrelations by increasing the illumination diameter, whichincreases the total number of input channels N tot , butwithout increasing the number of controlled input channels N in . This should reduce the transmission enhancementtowards the MP value. We first use an illumination diameter FIG. 1 (color online). (a) Schematic of the experimental setupfor the control of total transmission. As detailed in theSupplemental Material [22], the two polarizations of aNd:YAG laser, λ ¼ nm, are modulated by two differentareas of a phase-only SLM. The scattering sample is placed at thefocal plane of the objective. Three photodetectors, PD , PD ,and PD , measure, respectively, the intensities of transmitted,incident, and reflected light. (b) Measured T= h T i (left panel)and R= h R i (right panel) versus the optimization step for enhance-ment (increasing blue curve) and reduction (decreasing redcurve) of the total transmission. The sample is μ m thick,and the average transmission h T i ∼ . The dotted linerepresents the reflection estimated from the transmissionusing R= h R i ¼ ð − T Þ = ð − h T i Þ . PRL week ending4 APRIL 2014
Figure 36 (color online). Experimental setup and results fortotal transmission optimization. (a) Schematic of the exper-iment for the control of total transmission. The two polar-izations of the laser are modulated by two different areas ofa phase-only SLM, with 1740 macropixels controlled. Thescattering sample is placed at the focal plane of the objective.Three photodetectors PD1, PD2, and PD3 measure, respec-tively, the intensities of transmitted, incident, and reflectedlight, and the optimization is performed on the total trans-mission 𝑇 , measured as the intensity on detector PD1 normal-ized by the incident intensity measured on detector PD2. (b)Measured 𝑇 / ⟨ 𝑇 ⟩ (left panel) and 𝑅/ ⟨ 𝑅 ⟩ (right panel) versusthe step number of optimization for enhancement (increasingblue curve) and reduction (decreasing red curve) of the totaltransmission. The sample is a 20 m thick ZnO layer, andthe average transmission is ⟨ 𝑇 ⟩ ∼ 𝑅/ ⟨ 𝑅 ⟩ = (1 − 𝑇 ) / (1 − ⟨ 𝑇 ⟩ ). (Figure adapted from (Popoff et al. , 2014).) IV.C.2). This is different when injecting a state definedby the time-delay coefficient vector q 𝑖 into the slab (e.g.,through an SLM), since this state is characterized by justa single and well-defined time, i.e., its proper delay time 𝑞 𝑖 . This feature leads to very advantageous propertiesrelated to the fact that a well-defined time-delay can belinked to a suppression of frequency dispersion and to astrong collimation of ballistic scattering states.
1. Principal modes in a fiber
Consider a multi-mode fiber (MMF) which transmitslight almost perfectly, i.e., it has very little reflectionand absorption. In this case the transmission matrix t associated with this fiber is close to unitary such thatall the transmission eigenvalues 𝜏 𝑛 are near unity. Cor-respondingly, the associated transmission eigenchannelsstudied in section II.B will not be in any way special,since the massive degeneracy in the linear subspace asso-ciated with 𝜏 ≈ v ,which is transmitted by the fiber to an output vector u = tv , where we will assume the vectors and thematrix to be given in the mode basis (polarization de-grees of freedom will be neglected). If we now demandthat the transverse profile at the fiber output shouldbe dispersion-free this means that the output vector u should not change, when changing the input frequency 𝜔 slightly while keeping the input vector v the same (Fanand Kahn, 2005). To be more specific, we demand thatthe orientation of the output vector u stays invariant(which is equivalent to demanding that the output fielddistribution stays unchanged up to a prefactor). Decom-posing u into an amplitude and the corresponding unitvector which contains this orientation u = 𝑢 ^ u , we obtainthe following relations, 𝑑 u 𝑑𝜔 = 𝑑𝑢𝑑𝜔 ^ u + 𝑢 𝑑 ^ u 𝑑𝜔 = 𝑑𝑢𝑑𝜔 𝑢 − tv + 𝑢 𝑑 ^ u 𝑑𝜔 ≡ 𝑑 t 𝑑𝜔 v . (54)Requiring that ^ u is dispersion free, 𝑑 ^ u /𝑑𝜔 ≡
0, and mul-tiplying from the left with − 𝑖 t − , we end up with thefollowing relation, − 𝑖 t − 𝑑 t 𝑑𝜔 v = − 𝑖𝑢 − 𝑑𝑢𝑑𝜔 v . (55)which tells us that those input states v which are trans-mitted without transverse dispersion (to first order) areeigenstates of the matrix ˜ Q = − 𝑖 t − 𝑑 t /𝑑𝜔 . For unitarytransmission matrices, for which t − = t † , this expres-sion for ˜ Q is perfectly equivalent to the expression forthe Wigner-Smith time-delay operator Q which we hadfound before, see Eq. (47). For non-unitary transmissionmatrices as for fibers with finite reflection or loss, onecan further modify the right hand side of Eq. (55) [using 𝑢 = | 𝑢 | exp( 𝑖𝜑 )] to find the following expression for thecorresponding eigenvalue of this new operator, 𝑞 = − 𝑖 𝑑 ln | tv | 𝑑𝜔 + 𝑑 arg( tv ) 𝑑𝜔 . (56)The first term on the right-hand side of Eq. (56) is ameasure for the losses due to reflection or outcouplingthat depends only on the norm of the output; the sec-ond term is the derivative of the scattering phase at theoutput, i.e., the time delay.Due to their superior properties, the modes associatedwith the eigenvectors of ˜ Q have been termed “princi-pal modes (PMs)” (Fan and Kahn, 2005). Note in this (a) (b) C o rr e l a t i o n s ( d B ) -314 -157 157 3140-2-4-6 PrincipalmodeRandominput Δω (GHz) ω - ω = (a) (b) Random input C o rr e l a t i o n s ( d B ) λ (nm) LG PM H V H V ω -20 -15 -10 -5 Figure 37 (color online). Experimental data on principalmodes (PMs) in fibers with (a) weak and (b) strong modemixing, adapted from (Carpenter et al. , 2015) for (a) andfrom (Xiong et al. , 2016) for (b). In both figure parts themain panel shows the spectral correlation of the output fieldpattern measured relative to the center frequency for a PMas compared to a Laguerre-Gaussian (LG) mode in (a) and ascompared to a random superposition of LP modes in (b). Inboth cases the PM features a considerably increased stabilityof the field configuration at the fiber output. This output isshown in the color images: (a) Top: PM, bottom: LG mode.Horizontal (H) and veritcal (V) polarization directions areshown separately. (b) Top left: PM, top right: random in-put. Images recorded at Δ 𝜔 = − context, that for a perfectly straight fiber without mode-coupling, the PMs and the fiber modes become the same(in the absence of degeneracies). In this sense, the advan-tages of the PMs assert themselves fully in the presence ofa finite crosstalk between the ideal fiber modes (Ho andKahn, 2014). First observations of PMs in MMFs havebeen reported in (Carpenter et al. , 2014a, 2015; Xiong et al. , 2016). In Fig. 37a we show data from a measure-ment on fibers with weak mode coupling, which demon-strates the increased stability of PMs as compared to con-ventional linearly polarized (LP) fiber modes. WhereasPMs feature already by design a frequency-stability tofirst order, the weak coupling of modes enhances theirstability further. In the regime of strong mode couplingthe frequency stability of PMs is reduced, but still farsuperior compared to arbitrary input configurations (seeFig. 37b). The same can be expected for a disorderedslab geometry, for which PMs can also be constructed,but for which case no experiments have been reportedso far. We will see in the following that also anotherclass of time-delay eigenstates can be found in ballisticor quasi-ballistic scattering structures with a frequencyrobustness that goes beyond the above first-order stabil-ity.8
2. Particle-like scattering states
Consider the simple case of a resonator geometry,which, for reasons of simplicity, is assumed to be justtwo-dimensional. The scalar waves, which are injectedthrough a waveguide on the left, can be reflected throughthe same waveguide or transmitted through a secondwaveguide attached on the right (see illustration inFig. 38). One can now try to steer waves through theresonator such that they will follow the path of a classicaltrajectory throughout the entire scattering process ratherthan being diffractively scattered. To select the “geomet-ric optics” states from the full set of scattering states that“wave optics” will produce in this setup, we further as-sume that only the scattering matrix S of the systemis available (but no information on its interior scatter-ing landscape). The presence of such ballistic scatter-ing states leads to non-universal contributions to the dis-tribution of transmission eigenvalues 𝑃 ( 𝜏 ) at the values 𝜏 ≈ 𝜏 ≈
1, corresponding to fully closed and opentransmission channels, respectively. In the mesoscopicregime (see section II.B) it was exactly such system-specific contributions which were responsible for the sup-pression of electronic shot-noise below the universal limit(Agam et al. , 2000; Aigner et al. , 2005; Jacquod andSukhorukov, 2004; Sukhorukov and Bulashenko, 2005),which itself was already reduced below the Poissonianlimit by Schottky (Beenakker and Sch¨onenberger, 2003;Schottky, 1918).In analogy to the situation found for the MMF inthe previous section V.C.1, all the fully transmitted (re-flected) waves are completely mixed in the degeneratesubspace corresponding to 𝜏 ≈ 𝜏 ≈ Q = − iS † 𝜕 S / ( 𝜕𝜔 ) introduced in sectionII.C; this is because its eigenvalues, i.e., the “proper delaytimes” allow one to sort all the different ballistic scatter-ing contributions by way of their different time-delays(Rotter et al. , 2011). Specifically, one determines thoseeigenstates q of Q that only have incoming componentsin the left wave guide, i.e., q = ( q 𝑙 , ) T . Writing thetime-delay operator with its four sub-blocks (in corre-spondence to the subdivision of the scattering matrix it-self, see Eq. (15)), one finally obtains the following eigen-value problem, ⎛⎝ Q Q Q Q ⎞⎠ ⎛⎝ q 𝑙 ⎞⎠ = ⎛⎝ Q q 𝑙 Q q 𝑙 ⎞⎠ = 𝑞 ⎛⎝ q 𝑙 ⎞⎠ . (57)From the last equality the following two conditions can bededuced: (i) Q q 𝑙 = 𝑞 q 𝑙 and (ii) Q q 𝑙 = ⃗
0. Since Q is a Hermitian matrix ( Q and Q are not), condition(i) yields an orthogonal and complete set of eigenstates inthe incoming waveguide. Out of this set of states, thosewhich, according to condition (ii), lie in the null-space(kernel) of Q are the desired time-delay eigenstateswith a well-defined input port. One can show (Rotter et al. , 2011) that both conditions can only be fulfilled bywaves which are either fully transmitted or reflected; inother words these states are simultaneously eigenstates of Q and of t † t with deterministic transmission eigenvalues 𝜏 close to zero or one. Practically, the degree to whichcondition (ii) is fulfilled can be evaluated by a measure 𝜒 = ‖ Q q 𝑙 ‖ which should be the closer to zero the bet-ter condition (ii) is fulfilled. This measure can thus beconveniently used for assessing how well a given state willbe able to follow a classical bouncing pattern. Typicallythose states with a small value of 𝜒 are also those with asmall time-delay eigenvalue 𝑞 , in agreement with the ex-pectation that only states which stay inside the scatteringregion for a time shorter than the Ehrenfest time 𝑞 < 𝜏 𝐸 will be able to behave “particle-like” (Agam et al. , 2000).In Fig. (38)b-d three examples of states are shown whichfeature very small values of 𝜒 for different resonator ge-ometries. It can clearly be seen how these states tendto follow a (short) ray from geometric optics that avoidsany diffractive scattering throughout its propagation (as,e.g., at the sharp corners of the input and output facets).A first experimental demonstration of particle-like scat-tering states as well as of corresponding wave packetshas been reported based on acoustic wave scattering ina metal plate studied by laser interferometry (G´erardin et al. , 2016).When comparing these particle-like states to the PMsfrom the previous section V.C.1, the following commentscan be made: PMs can be constructed for arbitrary scat-tering media (including strongly disordered samples) andtheir frequency stability to first order will be assuredin all of these systems by construction. Particle-likestates, on the other hand, can only be found in systemswhere waves can propagate along sufficiently stable bal-listic scattering pathways. Due to their collimation onthese pathways, particle-like states feature, in turn, amuch higher frequency stability than principle modes (ina similar way as geometric optics states are frequency-independent by default). Both sets of states have very ad-vantageous properties for communication purposes, likethe dispersion-free transmission as well as the high direc-tionality of the particle-like states that seems well suitedfor steering a signal to a well-defined target. In sectionV.D.1 we will also see that the time-delay eigenstatesoptimally avoid or enhance the effect of dissipation in amedium.9 (a) (d) (a) (b) (c) Figure 38 (color online). (a) Wave intensity of a transmissioneigenchannel with transmission close to unity in scatteringthrough a resonator with two waveguides attached on the leftand right (the flux is incoming from the left). Whereas trans-mission eigenchannels typically lead to highly complex inter-ference patterns inside the resonator (as in wave optics), theparticle-like states shown in (b,c,d) follow a classical bouncingpattern throughout the entire scattering process (as in geo-metric optics). (b) Transmitted particle-like state in a cleanrectangular resonator. (c,d) Reflected particle-like states ina geometry of the same dimensions as in (a,b), from which,however, a quarter-circular piece was removed in (c) and asmooth and weak disorder potential was added in (d) (seebottom part of this panel). (Figures partially adapted from(Rotter et al. , 2011).)
D. Wavefront shaping in media with gain or loss
In this section we will discuss the application of wavefront shaping techniques in systems with gain or loss,with a focus on disordered media.
1. Absorbing media
Consider the case of an absorbing disordered medium,which, for simplicity, we will assume to be uniformly ab-sorbing (i.e., the absorption rate is independent of thespatial position in the medium). For this situation weknow from our analysis in section II.C that the absorp-tion is directly proportional to the time spent insidethe absorbing medium. Since, in turn, the time associ-ated with a stationary scattering state can be measuredthrough the dwell time operator Q 𝑑 , minimal or maxi-mal absorption of waves in a medium can be achieved byinjecting those eigenvectors of Q 𝑑 , which are associatedwith the smallest or largest eigenvalue, respectively. (Werecall here the result from section II.C that in the limitof vanishing absorption the dwell time operator Q 𝑑 andthe time-delay operator Q coincide up to mostly negligi-ble self-interference terms.) The procedure to obtain thestates with minimal absorption is thus equivalent to theapproach we presented in section V.C.2 for the genera-tion of particle-like scattering states, which are associated with the smallest values of the time-delay. The stateswith maximal absorption have been explicitly studied in(Chong and Stone, 2011), where it was shown how in aweakly scattering medium a suitably chosen input wavefront can increase the degree of absorption from a fewpercent to more than 99%. Such a coherent enhance-ment of absorption (CEA) can, in principle, be realizedat any frequency for which the input wave is shaped ap-propriately.An interesting point to observe here is that in thetheoretical approach put forward in (Chong and Stone,2011) these maximally absorbed states were not iden-tified through the help of the dwell-time operator, butrather as those states, which are minimally reflectedfrom an absorbing disordered medium. In the consid-ered single-port systems, where the reflection matrix isequivalent to the scattering matrix, we know, however,from Eq. (50), that the dwell time operator is equivalentto the unitary deficit of the scattering matrix such thatthese two different concepts to evaluate maximum ab-sorption perfectly coincide. In (Chong and Stone, 2011)the analysis was also extended to the case of a spatiallylocalized absorber buried behind a layer of lossless scat-tering medium – a situation that has also been studiedexperimentally by Vellekoop et al. in an attempt to focuslight on a fluorescent nanoscopic bead inside a disorderedmedium to increase the fluorescence (Vellekoop et al. ,2008). In this case the degree of optimal absorption wasfound to be more strongly bounded as compared to thecase where the entire medium is absorbing. Further workalso shows how the long-range spectral correlations inher-ent in the transmission and reflection matrices can helpto achieve enhanced absorption in a broadband frequencyrange (Hsu et al. , 2015). A first experimental realizationof a variant of CEA has been reported in (Liew et al. ,2016).In another numerical study the effect of absorptionwas investigated in scattering systems with more thanone port, like a 2D disordered wave guide connectedto two perfect leads on the left and right (Liew et al. ,2014a). In such systems both the maximally transmit-ted and the minimally reflected channels were studied.For weak absorption these two types of channels werefound to be dominated by diffusive transport and to beequivalent (as following from the connection between thetransmission and reflection matrices in unitary systems,see section II.A.4). For increasing absorption, however,the behavior of these two different channels decouples,as the reflection can then be minimized not only by in-creased transmission, but also by enhanced absorption:at a given absorption strength, the maximum transmis-sion channel was found to display a sharp transition toa quasi-ballistic transport regime. This transition doesnot occur for the minimal reflection channel, which getsincreasingly dominated by the absorption when the ab-sorption strength is increased. A very interesting aspect0that was also found in this context is that the shape ofthe transmission and reflection eigenvalue distributionsin disordered and dissipative media depends on the con-finement geometry (Yamilov et al. , 2016) - a fact thatmay be used for controlling this distribution at will.Whereas the above concepts relating to coherent en-hancement of absorption (CEA) can be implemented atany input frequency, it was shown in (Chong et al. , 2010)that at well-defined frequencies and at a carefully chosenamount of dissipation, certain incoming channels of lightcan be fully absorbed. Such a coherent perfect absorber(CPA) of light corresponds to the multi-mode generaliza-tion of a critically coupled oscillator, with the differencethat at least two input beams are required, which have tohave the correct amplitude and phase to interfere appro-priately. As a result, the relative phase between the inputbeams can be used to sensitively tune the degree of ab-sorption, as was done in the first CPA experiment (Wan et al. , 2011). From the conceptual point of view a CPAis a time-reversed laser, in the sense that a gain mediumat its first lasing threshold will emit coherent radiationat a well-defined frequency and with a well-defined phaserelationship, e.g., between beams emitted on either sideof the laser. The time-reversed process corresponds toan absorbing medium which perfectly absorbs the coher-ent illumination which impinges on it. If one considersa simple 1D edge-emitting laser that emits to the leftand right, the coherently absorbed light field of the cor-responding CPA features two beams (incoming from theleft and right), which share a specific phase relationshipto each other. At the points where this phase relationshipis satisfied, maximal absorption occurs. It is interestingthat, prior to these theoretical and experimental devel-opments, a seminal experiment in the field of plasmonicsshowed extraordinary absorption for a gold grating un-der specific incident illumination (Hutley and Maystre,1976), which was explained using a similar reasoning.The concept of a CPA can also be extended to othersystems (Noh et al. , 2012; Zanotto et al. , 2014), to 2D or3D as well as to disordered media. In the latter case theCPA would be the time-reverse of a random laser, corre-sponding to an absorbing random medium, which sucksin incoming waves that have exactly the same complexwave front as the emission profile of the random laser.To generate such a complex wave pattern one would ofcourse have to resort to wave front shaping techniquesusing SLMs or equivalent tools.Generally speaking, the theoretical concept behindCPAs builds on the analytical properties of the scatteringmatrix S ( 𝜔 ) (see section II.A.4 for more information). Inthe absence of loss or gain, this matrix is unitary and fea-tures poles (zeros) at complex frequencies with negative(positive) imaginary parts, respectively, located in thecomplex plane as mirror-symmetric pairs with respectto the real axis. When adding gain to the system thepoles and zeros move upwards in the complex plane un- til the point where the first pole reaches the real axisand lasing sets in. Alternatively, when adding loss tothe system, the poles and zeros move downwards untilthe first zero hits the real axis, at which point coherentperfect absorption can be realized (Chong et al. , 2010).Adding even more loss drags additional zeros across thereal axis, creating a CPA at each new intersection. Sub-sequent work also demonstrated how a laser and a CPAcan be combined in a single device (Chong et al. , 2011;Longhi, 2010). Such a laser-absorber (or CPA-laser) canbe realized based on the concept of 𝒫𝒯 -symmetric op-tical systems (Bender and Boettcher, 1998; El-Ganainy et al. , 2007; Makris et al. , 2008; R¨uter et al. , 2010) inwhich gain and loss are carefully balanced and poles andzeros of the scattering matrix can be brought to meeton the real axis. Realizing such concepts in the opticalexperiments is challenging as the CPA-lasing points atthe pole-zero crossing are affected by the noise due toamplified spontaneous emission. The first realization ofa CPA with a 𝒫𝒯 -phase transition has been reportedwith a pair of coupled resonators coupled to a microwavetransmission line (Sun et al. , 2014), followed by the firstsuccessful demonstration of lasing and anti-lasing in thesame 𝒫𝒯 -symmetric device (Wong et al. , 2016) .
2. Amplifying media
In the previous section we discussed how waves thatare injected into a certain disordered medium with ab-sorption can be shaped such as to be maximally or min-imally absorbed. Such an approach can, of course, alsobe considered with an amplifying medium, where one isnaturally concerned with maximal or minimal amplifica-tion. Work in this direction has, e.g., dealt with the non-trivial transient dynamics in photonic waveguide struc-tures composed of a combination of materials with bothloss and gain. Contrary to conventional expectation, spe-cific initial conditions for the incoming wave can lead topower amplification by several orders of magnitude evenif the waveguide is, on average, lossy (Makris et al. , 2014).Systems with gain and loss have also been proposed forthe realization of a special family of waves that havethe curious property of featuring a constant intensityin the presence of a non-homogeneous scattering land-scape (Makris et al. , 2015) – a feature that can not berealized with Hermitian scattering potentials. Extendingthis concept allows one to achieve perfect transmissioneven through strongly scattering disorder (Makris et al. ,2016). A realization of these curious wave solutions re-quires a careful shaping of the incoming wave front andof the medium’s gain-loss profile.As a medium with a sufficient amount of gain canemit coherent radiation on its own when crossing thelaser threshold, work on amplifying media has also al-ways had a very strong focus on engineering the gain1for a desired lasing action. In principle, optimizing thegain profile for a medium is a well-studied problem. Con-sider here, e.g., the case of a distributed feedback laser inwhich a so-called “gain grating”, consisting of a periodicarrangement of purely amplifying components, can effi-ciently pump lasing modes with the same periodicity asthe grating (Carroll et al. , 1998). Whereas these conceptsfor quasi-1D laser structures (like ridge or ring lasers)have meanwhile reached the level of industrial applica-tions, more advanced concepts on lasers with a quasi-2D(planar) geometry have just been explored very recently.Here, the main focus was on reducing the laser thresh-old of specific modes by increasing the spatial overlapbetween these selected modes and an externally appliedpump profile. Practical implementations of this con-cept include, e.g., electrically pumped devices for whichthe electrodes were patterned appropriately (Fukushima et al. , 2002; Kneissl et al. , 2004; Shinohara et al. , 2010).All of these implementations require, however, the priorknowledge of the spatial pattern of the selected mode.With the availability of wave front shaping tools, thepump profile as exerted on an optically pumped laser can,however, be tuned in a manner which is flexible enough toselect a given mode based on a simple feedback loop. Thisfeedback can be set up between the pump profile (as de-termined by the pixel configuration in an SLM) and, e.g.,the light spectrum of the laser pumped with this profile.Combining the feedback with an optimization algorithmhas, e.g., been suggested as a means to make the multi-mode emission spectrum of a random laser single-moded(Bachelard et al. , 2012). In an experimental realization,realized shortly after the theoretical proposal (Bachelard et al. , 2014) a mode-specific pump-selection and a cor-responding single-mode operation could be successfullydemonstrated for the challenging case of a weakly scat-tering random laser, see Fig. 39. For such a system noa-priori knowledge of the lasing mode is available andfinding the appropriate pump-grating is thus only possi-ble through optimization. As in this case the laser modesare also strongly overlapping both spectrally and spa-tially, the pump profiles obtained as a result of the op-timization process do not just follow the intensity of thelaser modes, but instead display a highly complex pat-tern only remotely related to the mode profiles. Theconnection between the pump profiles and the modesthey select was elucidated in subsequent theoretical work(Bachelard, 2014; Cerjan et al. , 2016) based on a non-Hermitian perturbation theory analysis. Alternatively,a mode-selection approach was proposed based on in-sights from gain saturation of interacting laser modes(Ge, 2015).A remarkable feature of the above feedback-basedpump optimization is its flexibility in terms of the op-timization goals that it can be employed for. Specifi-cally, it has been proposed that not only the multi-modespectrum of a random laser can be “tamed” with it, but
Pump sourceSpatial lightmodulator Dielectric scatterers
Pump profile Posi-on (nm) Posi-on (nm) N o r m a li z e d i n t e n s i t y ( a . u . ) N o r m a li z e d i n t e n s i t y ( a . u . ) Wave length (nm) Wave length (nm) (b)
Pump profile
Pump Pump SLM (a)
Ac-ve random medium SLM
Figure 39 (color online). (a) (left) Schematic to control theemission spectrum of a random laser consisting of a quasi-one-dimensional sequence of different dielectric layers. Tun-ing the incident pump beam through the spatial light modu-lator allows here to change how many lasing modes are activeand at which frequencies they emit. (right) A correspond-ing setup proposed to control the directionality of the emis-sion. (b) The spectral control of a random laser (left panel in(a), was realized in an experiment using opto-fluidic randomlasers (Bachelard et al. , 2014), where the laser emitted in twomodes for uniform pumping (see dotted lines in lower panels).By shaping the pump profile in the way shown in the toppanel, single-mode emission through either one of these twomodes was achieved (see solid lines in lower panels). (Figuresadapted from (Bachelard et al. , 2012) for (a) (left panel), from(Hisch et al. , 2013) for (a) (right panel) and from (Bachelard et al. , 2014) for (b).) also its highly irregular emission profile, see Fig. 39a(right panel) (Hisch et al. , 2013). A corresponding pump-shaping strategy for tuning the emission profile of a laserhas meanwhile been successfully implemented for micro-cavity lasers (Liew et al. , 2014b) (for random lasers acollimated output beam was achieved with other means(Sch¨onhuber et al. , 2016)). A next step could be to usethe spatial control of the applied pump to enhance thepower efficiency of lasers (Ge et al. , 2014).2
VI. CONCLUSIONS AND OUTLOOK
Wave front shaping techniques will become faster,more accurate and will involve an increasing number ofcontrolled pixels. Loosely speaking, this development canbe connected to the exponential increase in efficiency ofcomputer hardware (known as Moore’s law). With thisprojection in mind, one can foresee that experiments us-ing light modulation technology will soon be able to domuch more than they can do already now. In this lastsection a few ideas will be provided on where the furthertechnological developments could bring us or on what webelieve could be promising research topics in the next fewyears.
A. Wavefront shaping for unraveling mesoscopicphenomena
Whereas our review has already highlighted quite afew mesoscopic effects that were brought to light withwavefont shaping tools, we believe that many more fun-damental phenomena can still be explored with this newtechnology. Prominent examples could be, e.g., an un-ambiguous proof for Anderson localization in a three-dimensional medium (Lagendijk et al. , 2009; Scheffoldand Wiersma, 2013; Segev et al. , 2013; Skipetrov andSokolov, 2014; Sperling et al. , 2013) or the direct ex-perimental demonstration of the bimodal distribution oftransmission eigenvalues in optical scattering through adisordered medium (for which several indications havealready been put forward (Goetschy and Stone, 2013;Popoff et al. , 2014; Vellekoop and Mosk, 2008b)). In thiscontext it would also be of high interest to directly injectthe fully open transmission eigenchannels and to observetheir superior transmission characteristics, in a similarway to what has been done in acoustics (G´erardin et al. ,2014). Likewise, one could try to directly observe thepropagation of time-delay eigenstates (such as principalmodes (Fan and Kahn, 2005) or particle-like states (Rot-ter et al. , 2011)) through a weakly or strongly disorderedoptical medium, similar to what as recently achieved inoptical fibers (Carpenter et al. , 2015; Xiong et al. , 2016)and in acoustic waveguides (G´erardin et al. , 2016). Anexperimental demonstration that also still remains to bedone is that of a multi-channel coherent perfect absorber(Chong et al. , 2010), which could go as far as to demon-strate the time-reverse process of random lasing. Alsoother exotic effects, like “rogue waves” (Liu et al. , 2015a;Solli et al. , 2007) or “branched flow” (Metzger et al. ,2010; Topinka et al. , 2001), that have also been studiedin the context of ocean acoustics, could now be enhancedand tuned in various ways through wave front shaping.Optical scattering of course also brings in many new as-pects as compared to its electronic counterpart, in par-ticular through non-linearities (Bortolozzo et al. , 2011; Wellens and Gr´emaud, 2008) and the breaking of reci-procity (Muskens et al. , 2012; Peng et al. , 2014), whichfeatures have not yet been fully explored with wavefrontshaping tools.
B. New systems
We have seen that mesoscopic physics theory and ex-periments have mostly been focused on a handful ofcanonical systems in electronics: the disordered wire,the quantum billiard, the quantum point contact, etc.In the optical domain, a large fraction of the theoreti-cal and experimental work was focused on scattering inthree-dimensional bulk disorder, and restricted to a fewgeometries as well. Here, we want to comment on theopportunities of new optical systems to be studied withthe mesoscopic physics concepts already developed.Two systems have been discussed extensively alreadyin sections IV and V: biological tissues and multimodefibers. We have tried to highlight their specific featuresin terms of scattering and in which way these can beappropriately described by adapting the existing mesco-scopic physics concepts. We have also tried to highlighthow these system-specific aspects could lead to new op-portunities, e.g., for imaging (see here, in particular, thediscussion on the memory effect in chapters II and V).Plasmonic systems are also an interesting playground tostudy the effect of disorder and of wavefront shaping, inparticular due to the capability of the metal to localizelight well-below the diffraction limit. This includes notonly metallic hole arrays with disorder (Gjonaj et al. ,2011, 2013; Seo et al. , 2014), but also metal-dielectricfractal structures (Bondareff et al. , 2015; Gaio et al. ,2015).New optical systems are meanwhile emerging due tothe exciting possibility in photonics to tailor the prop-agation medium, e.g., to vary the amount of order anddisorder, or to change the dimensionality of the problem.Light transport has been studied in 1D stacks (Bertolotti et al. , 2005), in 1D waveguides (Sapienza et al. , 2010;Topolancik et al. , 2007), and in 2D disordered struc-tures (Garc´ıa et al. , 2012; Riboli et al. , 2011). A wholecommunity works on photonic crystals, trying to obtainperfectly regular structures, but the very small amountof residual disorder has been known to strongly affecttransport within these structures. Careful engineeringof the amount of disorder can, in turn, allow to con-trol the transport properties of the system (Garc´ıa et al. ,2013; Topolancik et al. , 2007). In 3D, self-organizationallows the fabrication of near-perfect 3D photonic crys-tals (Galisteo-Lopez et al. , 2011), for which the amountof disorder and its effect on the transport properties oflight can again be controlled. Another potential revolu-tion has been initiated by the possibility of engraving on amedium a completely designed refractive index distribu-3tion via direct laser writing (Deubel et al. , 2004; Kawata et al. , 2001), which meanwhile allows to create new kindsof disorder, such as hyper-uniform structures (Florescu et al. , 2009; Froufe-P´erez et al. , 2016; Haberko et al. ,2013; Muller et al. , 2014). Finally, we also mention op-tically reconfigurable structures where a refractive indexdistribution is engraved from an intensity pattern, suchas photorefractive crystals (Levi et al. , 2011; Schwartz et al. , 2007), an optical valve (Bortolozzo et al. , 2011)or integrated silicon-on-insulator multi-mode interferencedevices (Bruck et al. , 2015, 2016). These structures arepromising platforms on which not only the wavefront, butalso the disorder can be controlled using a spatial lightmodulator.
C. Applications of mesoscopic concepts in optics
A field where wavefront shaping concepts have alreadyhad a large impact is the field of optical imaging in tur-bid tissues (see also section IV). Several proof-of-conceptexperiments have shown that the conventional paradigmof ballistic imaging could be extended to the deep mul-tiple scattering regime (see section IV.E.2). While themultiscale nature and variability of these media makesthem very difficult to model, they are also a challenge forwavefront shaping due to a potentially very short decor-relation time, inhomogeneous absorption, and due to thefact that one typically has access to one side of themonly. In this respect, the exploitation of the memoryeffect (see section V.A) has already overcome this lim-itation, by providing an order of magnitude increase inspeed for scanning a point (Tang et al. , 2012) or evenfor single shot imaging (Katz et al. , 2014b). Other meso-scopic concepts, such as coherent perfect absorption orthe generation of time-delay eigenstates (see section V.Cand V.D), could be exploited in the future for a fur-ther improvement of imaging or light delivery, e.g., toavoid or to address regions in a tissue that are absorbingor where movements induce decorrelations. Due to theenormous complexity involved, calculating the transmis-sion matrix of a medium from the characterization of itsthree-dimensional shape, or worse, recovering the shapeof a medium from its transmission matrix, currently re-mains out of reach for disordered systems. Only in verysimple cases such demanding tasks can be achieved, suchas, e.g., in short straight multimode fibers (Pl¨oschner et al. , 2015).The perspective of better controlling the transmissionthrough a medium has tremendous potential in commu-nication technology, not only in optics but also in the ra-dio frequency domain, where increased transmission andbandwith could be realized as well as more secure andwell-isolated channels (Kaina et al. , 2014). In this con-text, the direct access to open channels or time-delayeigenstates would be particularly useful (see section V). Wavefront shaping has also moved multimode fibers(MMFs) into the focus of attention, both for imagingand for telecommunication purposes. Imaging throughan MMF is meanwhile a direct competitor to the bulkierfiber bundles (Gigan, 2012). In fiber communications,where the spatial degrees of freedom in MMFs are thelast ones remaining to be exploited for higher data rate(Richardson et al. , 2013), wavefront shaping has alreadyopened the possibility to physically decouple the trans-mission modes, rather than unmixing the transmittedinformation a posteriori using multiple-input multiple-output technology.Another domain of application is nanophotonics andquantum optics, where complex systems have increas-ingly been considered as a platform for light matter in-teraction. In a first step, a modification of the emis-sion properties of isolated single emitters has been dis-cussed and was connected to the local density of states(Birowosuto et al. , 2010; Krachmalnicoff et al. , 2010;Sapienza et al. , 2010, 2011), which in turn can be linkedto the modal structure of the medium as described bymesoscopic theory. More recently, the concepts of quan-tum networks have been studied, where multiple emittersare distributed and connected for quantum computingor simulations (Plenio and Huelga, 2008; Vedral et al. ,1996) as well as to understand quantum phenomena inbiology such as photosynthesis (Hildner et al. , 2013). In adisordered system, the connections between emitters canagain be understood through the underlying modal struc-ture and the associated Green’s function of the medium(Caz´e et al. , 2013). If mesoscopic theory can help tounderstand and better design such a network, wavefrontshaping can also be a particularly useful tool to interro-gate such a system for computation or simulation. Evena linear disordered system can be interesting in the con-text of quantum random walks of single or multiple pho-tons (Defienne et al. , 2014, 2016; Goorden et al. , 2014b;Huisman et al. , 2014; Ott et al. , 2010; Wolterink et al. ,2016). Mesoscopic effects such as Anderson localizationhave been studied with non-classical states in waveguidearrays (Crespi et al. , 2013; Schreiber et al. , 2011) and itwould be very interesting to study these effects in gen-uine disordered systems. Again, wavefront shaping couldserve here as an indispensable ingredient to make thesesystems useful for applications.Finally, we mention the following proofs of conceptthat have already been given for new devices, be it forcompact spectrometers (Redding et al. , 2013a), ultrafastswitches (Strudley et al. , 2014), tunable random lasers(Bachelard et al. , 2014; Hisch et al. , 2013) as well as forlight harvesting (Riboli et al. , 2014; Vynck et al. , 2012).We expect this list to be significantly extended in thevery near future.4
ACKNOWLEDGMENTS
The authors would like to thank all members of thecommunity that were kind enough to share the rightsto reprint their figures and the following colleagues forvery fruitful discussions: Philipp Ambichl, AlexandreAubry, Nicolas Bachelard, Hui Cao, R´emi Carminati,Adrian Girschik, Michel Gross, Thomas Hisch, Ori Katz,Florian Libisch, Matthias Liertzer, Stefan Nagele, Re-nate Pazourek, Romain Pierrat, and Patrick Sebbah. SRgratefully acknowledges the generous support of Insti-tut Langevin and Ecole Normale Sup´erieure through in-vited professor positions, during which part of this re-view was written. The authors’ own research presentedin this review was supported by the following fundingagencies: Austrian Science Fund (FWF) projects SFB-ADLIS F16, P17359, SFB-IR-ON F25, SFB-NextLiteF49 and I1142-N27 (GePartWave); Vienna Science Fund(WWTF) project MA09-030 (LICOTOLI); EuropeanResearch Council (ERC) grant 278025.
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