Transport of localized waves via modes and channels
11 Transport of localized waves via modes andchannels
Zhou Shi and Azriel Z. Genack
Queens College of the City University of New York, Flushing, NY, 11367, USA
Suppressed transport and enhanced fluctuations of conductance and trans-mission are prominent features of random mesoscopic systems in which thewave is temporally coherent within the sample [1–4]. The associated break-downs of particle diffusion and of self-averaging of flux were first consideredin the context of electronic conduction and for many years thought to bean exclusively quantum phenomena [1, 4–12]. Independently, however, wavelocalization was demonstrated theoretically for radio waves in a statisticallyinhomogeneous waveguide [13]. Over time, it became increasingly apparentthat localization and mesoscopic fluctuations reflected general wave proper-ties and might therefore be observed for classical waves as well [3, 13–38]. Inparticular, the level and transmission eigenchannel descriptions proposed,respectively, by Thouless [6, 7] and Dorokhov [10, 11] to describe the scal-ing of conductance in electronic wires at zero temperature, are essentiallywave descriptions involving the character of quasi-normal modes of excita-tion within the sample and speckle patterns of the incident and transmittedfield. Quasi-normal modes, which we will refer to as modes, are resonances ofan open system. These modes decay at a constant rate due to the combinedeffects of leakage from the sample and dissipative processes. Eigenchannelsof the transmission matrix, are obtained by finding the singular values ofthe field transmission matrix and represent linked field speckle patterns atthe input and output of the sample surfaces. Eigenchannels are linear com-binations of phase coherent channels impinging upon and emerging from thesample. Examples of such channels may be propagating transverse modes ofan empty waveguide, or transverse momentum states in the leads attachedto a resistor. In measurements, input and output channels are often com-binations of source and detectors at positions on the incident and output a r X i v : . [ c ond - m a t . d i s - nn ] M a y planes, respectively. Whereas modes are biorthogonal field speckle patternsover the volume of an open sample [39, 40], eigenchannels are orthogonal fieldspeckle patterns at the input and output planes of the sample. When thereis no risk of confusion, we will refer to eigenchannels as channels. Thoughlevels and channels have not been observed directly in electronic systems,these approaches have served as powerful conceptual guides for calculatingthe statistics and scaling of conductance [1].Recent measurements of spectra of transmitted field patterns and of thetransmission matrix of microwave radiation propagating through randommultimode waveguides have made it possible to determine the eigenvaluesof modes and channels as well as their speckle patterns in transmissionin mesoscopic samples [41, 42]. These experiments were carried out in amultimode copper tube filled with randomly positioned dielectric elements,which is directly analogous to a resistive wire in the zero-temperature limit,in which dephasing vanishes. The study of modes and channels promises toprovide a comprehensive description of transport and to clarify long-standingpuzzles regarding steady state and pulsed propagation.In this chapter, we will discuss studies of wave localization and strongfluctuations of the electromagnetic (EM) field, intensity, total transmissionand transmittance, also known as the “optical” conductance from the per-spectives of modes and channels. These approaches are useful in numerousapplications. The mode picture is of particular use in considering emission,random lasing, and absorption, while the channel framework is indispensablein optical focusing, imaging and transmission fluctuations. We will describelasing in disordered liquid crystals [43] and in random stacks of glass coverslips [44] in which the mode width falls below the typical spacing betweenmodes. The lasing threshold is then suppressed by the enhancement of thepump intensity and by the lengthening of the dwell time of emitted lightwithin the sample. Measuring the transmission matrix allows us to studythe fluctuations of transmittance over a random ensemble. The statistics oftransmittance can be described using an intuitive “Coulomb charge” model[45]. Measurements of the transmission matrix make it possible to obtain thestatistics of transmission in single samples at a particular incident frequency[46], which are crucial for focusing and imaging applications [46–49]. Thesestatistics, as well as the contrast in focusing, are given in terms of the par-ticipation number of transmission eigenvalues and the size of the measuredtransmission matrix [46, 49].In the next section (section 1.2), we discuss instructive analogies betweenthe transport of electrons and classical waves. Spectra of intensity, totaltransmission and transmittance are presented based on measurements of ransport of localized waves via modes and channels field transmission coefficients. A modal analysis of spectra of field specklepatterns on the output surface of a multimode waveguide and along thelength of a single mode waveguide is described in section 1.3. Pulse prop-agation in mesoscopic samples is discussed in terms of the distribution ofmode decay rates and the correlation between modal speckle patterns intransmission. The role of modes in lasing in nearly periodic liquid crystalsand in random slabs is described in section 1.4. In section 1.5, we describethe statistics of transmission eigenvalues and their impact on the statisticsof transmission for ensembles of random samples and in single instances ofthe transmission matrix. The manipulation of transmission eigenchannels tofocus radiation is described in section 1.6. We conclude in section 1.7 with adiscussion of the prospects for a complete description of transport in termsof modes and channels. Anderson [5] showed over 50 years ago that the electron wave function wouldnot spread throughout a disordered three dimensional crystal once the ratioof the width of distribution of the random potential at different sites relativeto the coupling between sites passes a threshold value. At lower levels ofdisorder, electrons diffuse in the band center but are localized in the tailof the band. Ioffe and Regel [50] pointed out shortly thereafter that anelectron wave function could not be considered to be properly propagatingif it were scattered after traveling less than a fraction of a wavelength sothat for traveling waves, (cid:96) > λ/ π , where (cid:96) is the mean free path. Thisgives the criterion for localization in three dimensions, k(cid:96) <
1, where k isthe wave number. Though not explicitly noted at the time, this criterion forlocalization applies equally to classical and quantum waves.In subsequent work, Thouless [6] considered the electronic state in thesystem as a whole rather than the strength of scattering within the medium.He argued that in bounded samples, the weight of electron states at theboundaries of the sample relative to points in the interior would be a usefulmeasure of the extension of electron states within the sample. Since localizedstates would be peaked within the sample remote from the boundaries, theirenergies could be expected to be insensitive even to substantial changesat the boundary such as are engendered in a periodic system when theboundary conditions for repeated sections of a random system are changedfrom periodic to antiperiodic [6]. When the energy shift is less than the typical spacing between states, the state is localized within the sample. Anassociated measure of electron localization, which relates to the propertiesof the states and not to the impact of some hypothetical manipulation ofthe sample, is the typical width of an electron level relative to the averagespacing between levels. When the electron wave function is exponentiallypeaked within the sample, electrons are remote from the boundary and theirescape from the sample is slow. The linewidth of the level is then smaller thanthe spacing between neighboring levels, which is the inverse of the density ofstates of the sample as a whole. This indicates that electron localization isachieved when the dimensionless ratio of the level width to level spacing, theThouless number, δ = δE/ ∆ E , falls below unity. Pendry [51] has describedthe electron localization process as one in which “electrons can be forced toabandon their predilection for momentum” in favor of space as the definingcharacteristic. The inhibition in transport manifested in localization in spacethen leads to a lengthened escape from the sample and narrow linewidthmanifested in terms of sharp spikes in energy “one per electron” [51] asopposed to a continuous spectrum. For diffusing electrons, the wave functionextends throughout the sample. Energy then readily leaks out of the sampleand levels are consequently short lived with line widths greater than thetypical spacing between levels. Thus the electron localization threshold liesat δ = 1. The Thouless number may equally well be defined for classicalwaves as the ratio of the typical frequency width to the spacing of quasi-normal modes, δ = δω/ ∆ ω , where ω is the angular frequency. The levelwidth is the inverse of the Thouless time τ T h in which a mode leaks outof the sample. Wave localization is signaled by exponentially long dwelltimes for the wave. Such long decay times contribute little to the averagelinewidth which could be dominated by spectrally broad modes peaked nearthe sample boundary with short decay times. To most meaningfully capturethe dynamics of a mode, it is therefore natural to identify δω with theaverage of the inverse Thouless time, δω ≡ (cid:104) τ − T h (cid:105) = (cid:104) / Γ n (cid:105) − , where (cid:104) . . . (cid:105) indicates an average taken over modes for an ensemble of samples and Γ n isthe leakage rate of energy in the n th mode of the sample [6, 41]. δ < δ as an indicator of the chang-ing character of the electron states with sample size. However δ cannot beeasily measured in electronic systems and has been measured only recentlyfor classical waves [41]. Using the Einstein relation, which gives the conduc-tivity in terms of a product of the electron diffusion coefficient D and thedensity of states, which is 1 / ∆ E divided by the sample volume, Thouless ransport of localized waves via modes and channels [7] showed that δ was equal to the conductance G in units of the quantumof conductance, δ = G/ ( e /h ) = g . Thouless [7] argued therefore that thedimensionless conductance would scale exponentially for localized waves aswould be expected for δ . He showed that the resistance of a wire at T = 0behaves ohmically [7], with the resistance increasing linearly with length L and the dimensionless conductance varying as g = N (cid:96)/L , only up to a length ξ = N (cid:96) , at which δ = 1. Here, N is the number of independent channelsthat couple to the resistor, N ∼ Ak F / π , where A is the cross sectional areaof the sample, k F is the electron wave number at the Fermi level, (cid:96) is theelectron mean free path, and ξ is the localization length. For L < ξ , electronsdiffuse with residence time within the conductor of τ T h ∼ L /D [6, 7]. Thelevel width would then be δE ∼ ¯ h/τ T h ∼ ¯ hD/L , while the level spacing∆ E would scale inversely with volume of the wire as 1 /L . As a result, for L < ξ , δ scales as 1 /L . For L > ξ , electrons would be localized and δ and g would fall exponentially while the resistance would increase exponentiallywith L . Abrahams et al. [8] showed that only above two dimensions is it pos-sible for transport to be diffusive at all length scales. For lower dimensions,localization always sets in as the size of the sample increases, independent ofthe scattering strength, so that a transition between diffusive and localizedtransport can only occur above two dimension [8].The scaling of average conductance and fluctuation in conductance mayalso be calculated within the framework of random matrix theory [10, 11, 45,52, 53]. The field in outgoing channel b is related to the field in all possibleincident channels a via the field transmission matrix t , E b = (cid:80) Na =1 t ba E a .Taking the two independent polarization states into account, the number ofpropagating modes in the empty waveguide leading to the sample is N =2 πA/λ , where A is the illumination area and λ is the wavelength of theincident wave. Summing over all possible incoming and transmitted channelsyields the transmittance T = (cid:80) na,b =1 | t ba | = (cid:80) Nn =1 τ n [54], where the τ n are the eigenvalues of the matrix product tt † . The transmission eigenvaluescan be found using the singular value decomposition of the transmissionmatrix t = U Λ V † . Here, U and V are unitary matrices and Λ is a diagonalmatrix with elements λ n = √ τ n . The ensemble average of T is equal to thedimensionless conductance, (cid:104) T (cid:105) = g [55]. Random matrix theory predictsthat, for diffusive waves, the transmission eigenvalue follow the bimodaldistribution, ρ ( τ ) = g τ √ − τ [11, 45, 52, 56]. Most of the contributions to T comes from approximately g eigenvalues that are larger than 1 /e , whilemost of the transmission eigenvalues are close to zero. The characteristicsof these“open” [57] and “closed” channels were first discussed by Dorokhov [10, 11]. He considered the scaling of each of the transmission eigenvalueswhich he expressed in terms of the auxiliary localization length ξ n . He foundthat the average spacing between inverse auxiliary localization lengths ofadjacent eigenchannels in a sample made up of N parallel chains with weaktransverse coupling to neighboring chains was constant and equal to theinverse of the localization length 1 /ξ [10, 11].Localization of quantum and classical waves in quasi-one-dimensional (Q1D)samples with lengths much greater than the transverse dimensions occursat a length at which even the highest transmission eigenvalue τ falls be-low 1 /e . Thus localization will always be achieved as the sample length isincreased in Q1D samples [7]. It is difficult, however, to localize EM wavesin three-dimensional dielectric materials. As opposed to s-wave scatteringprevalent in electronic systems, EM waves experience p-wave scattering andcannot be trapped by a confining potential. The scattering cross section onlybecomes appreciable once the size of the scattering element becomes compa-rable to the wavelength. But once the scattering length is comparable to thewavelength, the mean free path cannot fall significantly below the scatterersize and so it is hard to satisfy the Ioffe-Regel condition for localization inthree dimensions, k(cid:96) <
1. For smaller scattering elements such as spheresof radius a , the Rayleigh scattering cross section is proportional to a whilethe density of spheres is proportional to 1 /a . As a result, the inverse meanfree path for fixed volume fraction of particles is proportional to a . Forhigh particle density and a (cid:28) λ , the sample acts as an effective mediumwith mean free path (cid:96) ∼ /a . It is therefore not possible to achieve strongscattering with k(cid:96) < ransport of localized waves via modes and channels tures [62, 63] and highly anisotropic samples [64–66], particularly samplesin which the longitudinal structure along the direction of wave propagationis uniform. Anderson localization can be expected to occur for EM radiationat the edge of the conduction or pass band in nearly periodic three dimen-sional systems. Transport near the Anderson threshold has been observedfor ultrasound in a slab of brazed aluminum beads [67].An experimentally important difference between classical and quantumtransport is that coherent propagation is the rule for classical waves such assound, light and microwave radiation in granular or imperfectly fabricatedstructures, whereas electrons are only coherent at ultralow temperaturesin micron-sized samples. For classical waves in static samples, the wave isnot inelastically scattered by the sample so that the wave remains tempo-rally coherent throughout the sample even as its phase is random in space.In contrast, mesoscopic features of transport arise in disordered electronicsystems only in samples with dimensions of several microns at ultralow tem-peratures. Mesoscopic electronic samples are intermediate in size betweenthe microscopic atomic scale and the macroscopic scale. Electrons are typi-cally multiply scattered within conducting samples so their dimensions arelarger than the electron mean free path, which is on the scale of or largerthan the microscopic atomic spacing and electron wavelength. At the sametime, electronic samples are typically smaller than the macroscopic scale onwhich the wave function is no longer coherent. In contrast, monochromaticclassical waves are generally temporally coherent over the average dwelltime of the wave within large samples. It is therefore possible to explorethe statistics of mesoscopic phenomena with classical waves. Such studiesmay also be instructive regarding the statistics of transport in electronicmesoscopic samples. Measurements can also be made in both the frequencyand time domains. The impact of weak localization can be investigated inthe time domain by measuring transmission following an excitation pulse orby Fourier transforming spectra of field multiplied by the spectrum of theexciting pulse.The connection between electronic and classical transport emerges as wellfrom the equivalence proposed by Landauer [54, 55] of the dimensionless con-ductance g and the transmittance T , known as the“optical” conductance.The transmittance is the sum over all incident and outgoing channels of thetransmission coefficient of flux. The phase of electrons arriving from a reser-voir in different channels is randomized over the time of the measurementand the conductance is related to the incoherent sum of transmission coeffi-cients over all channels, g = (cid:104) T (cid:105) = (cid:104) (cid:80) Na,b =1 T ba (cid:105) = (cid:104) (cid:80) Na =1 T a (cid:105) . Measurementshave been made of the statistics of transmission coefficients of the field, t ba , intensity, I ba = | t ba | , and total transmission, T a = (cid:80) Nb =1 T ba , for a singleincident channel, a and for the transmittance, T .The experimental setup for measurements of microwave transmission inthe Q1D geometry described in this chapter is shown in Fig. 1. Measurements m m mm Figure 1.1 Copper sample tube containing a random medium with mi-crowave source and detector antennas. Intensity speckle pattern is producedwith a single source location. are carried out in ensembles of random samples contained in a copper tube.The samples are random mixtures of alumina spheres with diameter of 0.95cm and index of refraction of n = 3 .
14 at a volume fraction of 0.068. Sourceand detector antennas may be translated over a square grid of points cover-ing the incident and output surfaces of the sample and rotated between twoperpendicular orientations in the planes of the sample boundaries. Spectraof the field transmission coefficient polarized along the length of a shortantenna are obtained from the measurement of the in- and out-of-phasecomponents of the field with use of a vector network analyzer. The intensityfor a single polarization of the wave is the sum of the squares of the in-and out-of-phase components of the field. The sum of intensity across theoutput face for two perpendicular orientations of the detector antenna givesthe total transmission. The field speckle pattern for each antenna positionon the sample input is obtained by translating the detection antenna overthe output surface. An example of an intensity speckle pattern formed intransmission is shown at the output of the sample tube in Fig. 1. The tubeis rotated and vibrated momentarily after measurements are completed foreach sample configuration to create a new and stable arrangement of scatter-ing elements. In this way, measurements are made over a random ensembleof realizations of the sample. Field spectra can be Fourier transformed toyield the temporal response to pulsed excitation. ransport of localized waves via modes and channels Spectra of intensity, total transmission and transmittance normalized bythe ensemble average values s ba = T ba / (cid:104) T ba (cid:105) , s a = T a / (cid:104) T a (cid:105) , and s = T / (cid:104) T (cid:105) in a single random configuration in two different frequency ranges are shownin Fig. 2. Fluctuations of relative intensity are noticeably suppressed in thehigher frequency range as the degree of spatial averaging increases. For theensemble represented in Fig. 2, var( s a ) = 0.13 in the high frequency rangeand 3.88 in the low frequency range. Since waves are localized for var( s a ) > / Frequency (GHz) s ba s a s Frequency (GHz) (b)(d)(c) (f)(e)(a)
10 10.06 10.12 10.18 10.24
Frequency (GHz)
10 10.06 10.12 10.18 10.24
Frequency (GHz)
Frequency (GHz)
Frequency (GHz)
Figure 1.2 Spectra of transmitted microwave intensity, total transmission,and transmittance relative to the ensemble average value for each in a singlerandom configuration. The wave is localized in (a), (c),and (e) and diffusivein (b), (d),and (f).
It is instructive to consider the spectra in Fig. 2 from both the modeand channel perspectives. When the wave is localized, distinct peaks appearwhen the incident radiation is on resonance with a mode. The resonancecondition holds for all source and detector positions and therefore sharppeaks remain even when transmission is integrated over space. When thewave is diffusive, many modes contribute to transmission at all frequenciesand for all source and detector positions. The relative coupling strengthsof a single polarization component of the intensity into and out of eachof these modes has a negative exponential distribution and phases of thefield transmission coefficient are random so that relative fluctuations will besuppressed with increased spatial averaging. From the channel perspective, many orthogonal transmission channels contribute to transmission for dif-fusive waves and the coupling to channels varies with source and detectorpositions. Fluctuations in the incoherent sum of this random jumble of or-thogonal eigenchannels are therefore suppressed upon averaging over space.This suppresses the variance of transmission by a degree related to the num-ber of channels that contribute substantially to transmission. This may beexpressed quantitatively in term of the participation number of eigenvalueof the transmission matrix, M ≡ ( (cid:80) Nn =1 τ n ) / (cid:80) Nn =1 τ n . For diffusive waves,var( s a ) ∼ /M and relative fluctuations are enhanced since the number ofeffective channels M is smaller than the number of independent channels N . We will see below that for diffusive waves the spectrum of transmissioneigenvalues is rigid so that the number of transmission eigenvalues above1 /e fluctuates by approximately unity and fluctuations of conductance T are of order unity [1, 4, 57, 69–71]. This results in universal conductancefluctuations which are independent of the sample size for Q1D sample [1, 4].The localization transition may be charted in terms of a variety of relatedlocalization parameters, all of which can be measured for classical waves.In addition to δ and the average over a random collection of samples ofthe dimensionless conductance, g = (cid:104) T (cid:105) , measurements of fractional fluctu-ations of intensity or total transmission characterize the nature of the wavein random systems. In the diffusive limit, the variance of total transmissionrelative to the average value of total transmission over a random ensemble ofstatistically equivalent samples is inversely proportional to g , var( s a ) = 2 / g [3, 30–32, 36]. Since the wave is localized for g <
1, localization occurs whenvar( s a ) > /
3. Perhaps the most easily accessible experimental localizationparameter is the variance of fractional intensity, which can be expressed asvar( s ab ) = 1 + 4/3 g [31]. The localization threshold at g = 1 correspondsto var( s ab )=7/3. var( s a ) and var( s ab ) remains useful localization parameterseven for localized waves. Fluctuations are relatively insensitive to absorp-tion as compared to measurements of absolute transmission [36]. Mesoscopicfluctuations are directly tied to intensity correlation within the sample [20–22, 24, 72, 73]. The fractional correlation of intensity at two points on theoutput surface or between two transmission channels, b and b (cid:48) , is equal tothe variance of relative total transmission, κ = (cid:104) δs ba δs b (cid:48) a (cid:105) =var( s a ). It isequal to (cid:104) M − (cid:105) in the diffusive limit, which is enhanced over the value of1 /N that would be expected if mesoscopic correlation were not present.The relationships between key localization parameters mentioned abovearise since the nature of propagation in disordered Q1D samples, in whichthe wave is thoroughly mixed in the transverse directions, depends only on asingle dimensionless parameter [8]. For diffusive waves, δ = g = 2 / s a ) = ransport of localized waves via modes and channels / κ = 2 / (cid:104) M − (cid:105) . The relationships var( s a ) = κ and var( s ab )=2var( s a )+1hold through the localization transition, but the relationship between theother variable does not. However, we anticipate these relationships willchange in a manner that can be described in terms of a single parame-ter. Other classical wave measurements that indicate the closeness to thelocalization threshold are coherent backscattering [17–19, 29] and the trans-verse spread of intensity in steady state or in the time domain [64–67, 74].The width of the coherent backscattering peak gives the transverse spreadof the wave on the incident surface and hence the transport mean free path (cid:96) , from which the value of k(cid:96) can be found. We find that the fields at any point in the sample may be expressed as asuperposition of the field associated with the excitation of all the modes inthe sample. This superposition is a sum of products for each mode of the j polarization component of the spatial variation of the mode, a n,j ( r ), andthe frequency variation of the mode, which depends only upon the centralfrequency of the mode ω n and its linewidth, Γ n , E j ( r , ω ) = (cid:88) n a n,j ( r ) Γ n / n / i ( ω − ω n ) = (cid:88) n a n,j ( r ) ϕ n ( ω ) . (1.1)The frequency variation of the n th mode ϕ n ( ω ) is given by the Fourier trans-form of exp( − Γ n t/
2) cos ω n t for t >
0. Equation (1) can be fit simultaneouslyto the field at a large number of points on the output speckle patterns ina single configuration since all spectra share a common set of ω n and Γ n .Armed with the values of ω n and Γ n , we find a n,j ( r ) and hence the specklepattern for each of the modes.The transmission spectrum is determined by the variation with positionof the field amplitudes | a n,j ( r ) | and phases over the transmitted specklepatterns for the modes. The contribution of individual modes to transmissioncan be seen in the spectrum of total transmission near the single strong peakat 10.15 GHz shown in Fig. 3(a) in a random sample of length L = 61 cm.The asymmetrical shape for the line in both intensity and total transmissionindicates that more than a single mode contributes to the peak. The modalanalysis of the field spectra shows that three modes contribute substantiallyto transmission over this frequency range. Spectra of the total transmissionfor the three modes closest to 10.15 GHz acting independently are plottedin Fig. 3(a). The integrated transmission for the 28th and 29th mode found in the spectrum starting at 10 GHz are each greater than for the measuredpeak indicating that these modes interfere destructively. The intensity andphase patterns for these two modes are shown in Figs. 3b-e. Aside from adifference in the average value of transmission, the intensity speckle patternsof the two modes are nearly the same. The distributions of phase shift at10.15 GHz for the two modes are also similar except for a constant phasedifference between them of ∆ ϕ = 1 . π rad. The similarity between thespeckle patterns for these overlapping modes suggests that these modes areformed from coupled resonances within the sample which overlap spatiallyand spectrally. We expect that such resonances peaked at different locationswill hybridize to form modes of the system. Such modes may be close tosymmetric and antisymmetric combinations of the two local resonances. Thiswould produce similar intensity speckle patterns at the output with a phaseshift of ∼ π rad between the modes. The similarity in the intensity specklepatterns of these adjacent modes and the uniformity of the phase shift acrossthe patterns of these modes allows for interference between modes acrossthe entire speckle pattern. The similarity between modes is most evidentin a sample configuration such that a pair of nearest neighbor modes areparticularly close in frequency. This is a point of anticrossing which arisesbecause of level repulsion inside the sample [75, 76]. The magnitude of thefield inside a 1D sample is seen to be the same throughout the sample. Modesare orthogonal by virtue of a change of phase of π rad along the length ofthe sample. At the anticrossing, the fields at the sample output for the twomodes are the same except for a change in phase close to π rad. The Thoulessnumber, which equals the dimensionless conductance for diffusive waves,provides a key measure of the dependence of transmission on the underlyingcharacteristics of modes. The modal decomposition of transmission spectrafor the ensemble from which the configuration is analyzed in Fig. 3 is drawngives δ = 0.17 [41].The statistics of level spacing was first considered by Wigner [77] in thecontext of nuclear levels probed in neutron scattering. He conjectured theeigenvalues of the Hamiltonian matrix would have the same statistics as thespacing of eigenvalues of a large random matrix with Gaussian elements.Agreement was found between the spacing between peaks in the scatteringcross section and Wigner’s surmise for the spacing of eigenvalues of randomHamiltonian matrices. However, the analysis of spectra of nuclear scatteringcross sections was done in samples with relatively sharp spectral lines. Wehave seen above that even when δ <
1, a number of lines may coalesceinto a single peak. A comparison of level spacing statistics in samples withdifferent values of modal overlap δ in which the phase of the scattered wave ransport of localized waves via modes and channels T o t a l t r a n s m i ss i o n MeasuredFittedMode 28Mode 29Mode 30 x (mm)Intensity y ( mm ) b x (mm)Intensity y ( mm ) c x (mm)Phase y ( mm )
070 35 70035 – π – π /20 ππ /2 x (mm)Phase y ( mm ) π –0 π /2 ππ /2 x (mm) y ( mm ) Phase difference70 0 35 70035 0 ππ /23 π /2 a π efd Figure 1.3 (a) Three modes contribute to the asymmetric peak in the totaltransmission spectrum. Modes 28, 29 and 30 are in order of increasingfrequency. Intensity speckle patterns for modes 28 and 29 are shown in (b)and (c) and the corresponding phase patterns are shown in (d) and (e). (f)The phase in mode 29 is shifted by nearly a constant of 1.02 π rad relativeto mode 28. (Ref. [41]) can be measured is therefore of interest in forming a picture of the statisticsof transmission. Progressively stronger deviations from the Wigner surmiseare found for decreasing values of δ . Figure 1.4 (a) Spectra of the field amplitude at each point along a randomsample with spectrally overlapping peaks normalized to the amplitude ofthe incident field. (b) Top view of (a) in logarithmic presentation. (Ref.[61])
It has not been possible to access the field distribution within the interiorof multiply scattering three-dimensional samples, but spatial distributes canbe examined in one- and two-dimensional samples [59, 61, 75]. The presence of both isolated and overlapping modes within the same frequency rangehas been observed in measurements of field spectra along the length of slot-ted single-mode random waveguides. The waveguides contained randomlypositioned binary dielectric elements and a smaller number of low indexStyrofoam elements. Measurements were carried out in the frequency rangeof a pseudogap associated with the first stop band of a periodic structureof consecutive binary elements. The density of states is particularly low inthe frequency range of the band gap so that δ <
1. When spectrally iso-lated lines are found, they are strongly peaked in space and their intensityspectra at each point in the sample is Lorentzian with the same width at allpoints within the sample. When modes overlap spectrally, however, spectralpeaks have complex shapes which vary with position within the sample andthe spatial intensity distribution is multiply peaked. Mott [78] argued thatinteractions between closely spaced levels in some range of energy in which δ < π rad, each time the frequency istuned though a mode. The decomposition of field spectra inside the waveg-uide within the pseudogap into the modes and a background which variesslowly in frequency is shown in Fig. 5. The slowly varying background shownin Fig. 5(a) is the fit of a polynomial in the difference in frequency from apoint in the middle of the spectral range considered. This background ispresumably related to off-resonance excitation of many modes on either sideof the band gap. The mode structure within the single-mode waveguide sam-ple changes when a spacing is introduced between two parts of the sampleand is increased gradually. A succession of mode hybridizations is observedwith increasing spacing as a single mode tends to shift in frequency until itencounters the next mode. As the spacing is increased, the mode that had ransport of localized waves via modes and channels been moving becomes stationary and the next mode begins to move [76].Simulations have shown that changing the index of a single scatterer leadsto mode hybridization in 2D random systems, in which a single peak maybe transformed to multiple peaks [75]. Figure 1.5 Decomposition of field pattern in Fig. 4 into a slowly varyingpolynomial term and five modes. (Ref. [61])
One-dimensional localization has also been observed in optical measure-ments in single-mode optical fibers [60] and in single-mode channels thatguide light within photonic crystals [80–82]. When the structure bracket-ing the channel is periodic, the velocity of the wave propagating down thechannel experiences a periodic modulation so that a stop band is created.When disorder is introduced into the lattice, modes with spatially varyingamplitude along the channel are created. Modes near the edge of the bandgap are long lived and readily localized by disorder. An example of spec-tra of vertically scattered light versus frequency for light launched down achannel through a tapered optical fiber is shown in Fig. 6. The inset showsthe disordered sample of holes with random departure from circularity insilicon-on-insulator substrates at a hole filling fraction of f ∼ . ∼ The modal decomposition method described above can be applied to local-ized waves for which modal overlap is relatively small. The impact of modescan be seen in the changing decay rate of transmission following pulsed ex-citation, even for diffusive waves. The slowing down of the decay rate withtime will become more pronounced in samples in which the wave is morestrongly localized [83–85]. In the diffusive limit, the transverse extent of themodes is large and the wave is coupled to its surroundings through a largenumber of speckle spots. One expects therefore that the decay rate of allmodes will approach the decay rate of the lowest diffusion mode [86, 87],1 /τ = π D/ ( L + 2 z ) , after a time τ in which higher order modes withdecay rates 1 /τ n = n π D/ ( L + 2 z ) have largely decayed. Here, n is theorder of the diffusion mode and z is the length beyond the boundary of thesample at which the intensity inside the sample extrapolates to zero. Wefind that pulsed transmission deviates increasingly from the diffusion modelin nominally diffusive samples with g > g decreases andthe measured value of κ increases. Strong deviations from pulsed transmis-sion in diffusing samples far from the localization threshold are observed formicrowave radiation, light and ultrasound [67, 74, 88].Measurements of pulsed transmission through a random sample of alu-mina spheres at low-density in samples of different length and absorptionwith values of κ = 0.09, 0.13, 0.25, and 0.125 are shown in Fig. 7 in samplesA-D, respectively [88]. The decay rate of intensity is seen to deviate from theconstant rate of the diffusive limit and is seen in Fig. 7(b) to decreases at a ransport of localized waves via modes and channels t (ns)t (ns) -200 0 200 400 600 800 10000.00 - d l n < I ( t ) > / d t (b)(a) -200 0 200 400 600 800 100010 < I ( t ) > -12 -11 -8 -4 CA D B A D B C
Figure 1.7 (a) Average pulsed transmitted intensity in samples of aluminaspheres with lengths, L = 61 cm (A), 90 cm (B and D), and 183 cm (C).Sample D is the same as sample B except for the insertion of a titaniumfoil inserted along the length of the sample tube D to increase absorption.(b) Temporal derivative of the intensity logarithm gives the rate γ of theintensity decay due to both leakage out of the sample and absorption. (Ref.[88]) nearly constant rate. A linear falloff of the decay rate would be associatedwith a Gaussian distribution of decay rates for the modes of the medium[83]. A slightly more rapid decrease of the decay rate is associated with aslower than Gaussian falloff of the distribution of mode decay rates. Theslowing down of the decay rate at long times reflects the survival of moreslowly decaying modes [83, 88]. The distribution of modal decay rates isrelated to the Laplace transform of the transmitted pulse intensity.Sample D is the same as sample B except for the increased absorptiondue to a titanium foil inserted along the length of the sample tube. Thevariation with time of the decay rate in sample D is the same as that insample B except for an additional constant decay rate in sample D due toabsorption. This shows that, at the low level of absorption in these samples,scattering rates are not affected by absorption and that the effect of ab-sorption simply introduces a multiplicative exponential decay, which is thesame for all trajectories at a given time. Thus the degree of renormalizationof transport due to weak localization involving the interference of wavesfollowing time-reversed trajectories that return to a point in the mediumis not affected by absorption. We note that the fractional reduction of thedecay rate is greater at a given time delay in shorter samples with highervalues of g . This is because the length of trajectories of partial waves withinthe medium is the same for all samples at a given delay but the numberof crossings of trajectories is greater when the paths are confined within asmaller volume.The temporal variation of transmission can also be described in terms ofthe growing impact of weak localization on the dynamic behavior of waves,which can be expressed via the renormalization of a time-dependent diffu- sion constant or mean free path [89]. The decreasing decay rate has alsobeen explained using a self-consistent local diffusion theory for localizationin open media [90, 91]. The theory uses a one loop self-consistent calculationof an effective diffusion coefficient that falls with increasing depth inside thesample. The spatial variation of the local diffusion coefficient reflects theincreasing fraction of returns of wave trajectories to a point with greaterdepth due to the lengthened dwell time in the sample. This theory gives ex-cellent agreement with recent measurements of non-diffusive decay of pulsedultrasound transmission through a sample of aluminum spheres as seen inFig. 8 [67]. The value of g is just beyond the localization threshold as deter- –5 –6 –7 –8 –9 N o r m a l i z e d i n t e n s i t y t ( µ s)200 300 400 f = 2.4 MHzExperimentSelf-consistent theoryDiffusion theory Figure 1.8 Averaged time-dependent transmitted intensity I ( t ) normalizedso that the peak of the input pulse is unity and centered on t = 0, atrepresentative frequencies in the localized regimes. The data are fit by theself-consistent theory (solid curve). For comparison, the dashed line showsthe long-time behavior predicted by diffusion theory. (Ref. [67]) mined from measurements of var( s ba ), which is slightly above the value of7/3 predicted as the localization threshold. The intensity distribution on theoutput face of the sample is found to be multifractal as predicted near thethreshold for Anderson localization [83, 92]. However, the value of k(cid:96) in thissample is ∼ .
5, which would indicate the wave is diffusive. Measurementsof the spread of intensity on the sample output with increasing time delayshow a trend towards an exponential decay of intensity on the output planeat later times supporting the localization of the wave. A similar approach toan exponential decay of intensity with transverse displacement on the sam-ple output from the point of injection of the pulse on the incident surfacehas been observed by Sperling et al. [74] in optical measurements through aslab of titania particles. When k(cid:96) < .
5, the variance of the spatial intensitydistribution reaches a peak value and then actually falls. This is taken assupport of localization of the wave at k(cid:96) >
1. However, this criterion forlocalization is directly tied to the Thouless criterion for localization δ = 1. ransport of localized waves via modes and channels It might also be that at later times, longer lived modes, which are moreconfined in space are more heavily represented and dominate the spatialdistribution. Though each of these modes will not spread in time, the modesthat survive with increasing time delay would be the more strongly confinedmodes and would lead to a falling variance of the spatial intensity distribu-tion in time. Such states may be prelocalized with a slower falloff in spacethan exponential but still faster than for diffusive waves [93, 94].The slowing of the spread of the transmitted wave in the transverse direc-tion can also be seen in transverse localization in samples which are uniformin the longitudinal direction. This has been observed in a 2D periodic hexag-onal lattice with superimposed random fluctuations [65]. The structuredsample is created by first illuminating the photorefractive sample with ahexagonal optical pattern and then with a random speckle pattern of varyingstrength. A transition from a diffusive to a localized wave in the transverseplane is seen in the output plane with the ensemble average of the spatialintensity distribution changing from a Gaussian to an exponential functioncentered on the input beam as the thickness of the sample increases. Sincethe wave incident upon the sample, which is uniform along its length, isparaxial, it is not scattered in the longitudinal direction and travel timethrough the sample is proportional to the sample thickness. Transverse lo-calization is also observed in an array of disordered waveguides lattices [66].Measurements of pulsed microwave transmission of more deeply localizedwaves transmitted through a Q1D sample of random alumina spheres ofthickness approximately 2.5 times the localization length, are shown in Fig.9 [95]. The impact of absorption was removed statistically by multiplying t (ns)
100 200 300 400 0 t (ns)
500 1000 1500 < I ( t ) > < I ( t ) > -7 -6 -5 -4 -3 -2 -1 - (a) (b)L=61 cm L=61 cm experimentSCLTdiffusion theory Figure 1.9 (a) Fit of self-consistent localization theory (SCLT) (dashedcurve) at early times to the average intensity response (solid curve) to aGaussian pulse with σ ν = 15 MHz in a sample of length L = 61 cm and theresult of classical diffusion theory (dotted curve); (b) semilogarithmic plotof (cid:104) I ( t ) (cid:105) reaching to longer times; In (a) and (b), the curves are normalizedto the peak value. (Ref. [95]) the average measured intensity distribution by exp( t/τ a ) [36, 96]. For times near the peak of the transmitted pulse, diffusion theory corresponds wellwith the measurements of (cid:104) I ( t ) (cid:105) . For times up to 4 times τ D = L /π D ,the decay rate of the lowest diffusion mode, pulsed transmission is in ac-cord with self-consistent localization theory, but transmission decays moreslowly for longer times. This indicates the inability of this modified diffusiontheory to capture the decay of long-lived localized states. Such states areincluded in a position-dependent diffusion theory [97] that is in accord withsimulations of the steady-state intensity profile within random systems [98].The difference between self-consistent localization theory and the theory forposition-dependent diffusion is seen to be precisely in the ability of the latterto include the impact of long-lived resonant states [97, 98].Destructive interference between neighboring modes together with the dis-tribution of mode transmission strengths and decay rates can explain thedynamics of transmission. The average temporal variation of total transmis-sion due to an incident Gaussian pulse is found from the Fourier transformof the product of the field spectrum and the Gaussian pulse. The progres-sive suppression of transmission in time by absorption may be removed bymultiplying (cid:104) T a ( t ) (cid:105) by exp( t/τ a ) to give, (cid:104) T a ( t ) (cid:105) = (cid:104) T a ( t ) (cid:105) exp( t/τ a ) [36, 96].With the influence of absorption upon average transmission removed, decayis due solely to leakage from the sample. The measured pulsed transmissioncorrected for absorption is shown as the solid curve in Fig. 10 and is com-pared to the incoherent sum of transmission for all modes in the randomensemble corrected for absorption, (cid:80) n T an ( t ), shown as the dashed curve inthe Fig. 10. (cid:80) n T an ( t ) is substantially larger than (cid:104) T a ( t ) (cid:105) at early times,but converges to (cid:104) T a ( t ) (cid:105) soon after the peak. Though transmission associ-ated with individual modes rises with the incident pulse, transmission atearly times is strongly suppressed by the destructive interference of modeswith strongly correlated field speckle patterns such as those shown in Fig.3. At later times, random frequency differences between modes leads to ad-ditional random phasing between modes and averaged pulsed transmissionapproaches the incoherent sum of decaying modes. The decay of (cid:104) T a ( t ) (cid:105) ,shown as the solid curve in Fig. 10, is seen to slow considerably with timedelay reflecting a broad range of modal decay rates.Measurements by Bertolotti et al. [62] of pulsed infrared transmissionthrough random layers of porous silicon with different porosity producedby controlled electrochemical etching of silicon show that the pulse profilesdepend on the degree of spectral overlap of excited modes. As the numberof layers increases, spectra become sharper since propagation of a paraxialbeam in the structure is essentially one dimensional and δ falls with samplethickness. When the pulse excites an isolated resonance peak, the decay rate ransport of localized waves via modes and channels n < Σ T (t) > n −
200 0 200 400 600 800 1,000 1,20010 − − − − − − Time (ns) T o t a l t r a n s m i ss i o n σ = 3 MHz
43 is shown in Fig. 12 [95]. The decay is slow when the spectrum of thepulse overlaps a single narrow mode and fast when two peaks fall within thespectrum of the incident pulse. In the latter case, the transmitted intensityis significantly modulated at the frequency difference between the modes. (GHz) I () / < I > t (ns)
500 100010 -9 I ( t ) -8 -6 -5 -4 -3 (a) (b) Figure 1.12 (a) Transmitted intensity spectrum in a random sample of L = 40 cm and Gaussian spectra of incident pulses peaked at the centerof the isolated line and overlapping lines. (b) Intensity responses to theGaussian incident pulses with spectral functions shown in (a). (Ref. [95]) In addition to the reduction of the leakage rate with increasing delayobserved in diffusive samples, the variance of relative intensity fluctuationsand the degree of intensity correlation also increase with time delay [99–101].The field correlation function with displacement and polarization rotationin pulsed transmission is the same as in steady state [99]. This reflects theGaussian statistics within the speckle pattern of a given sample configura-tion and time delay. The intensity correlation function at a given time delaydepends on the square of the field correlation function and the degree ofintensity correlation in the same way as in steady state, but the degree ofextended range correlation k σ ( t ) depends on delay time and on the spectralbandwidth of the pulse σ . The probability distribution functions of intensityat various delay times have the same form as for steady state propagationand depend upon a single parameter, which is the variance of the total trans-mission relative to its average over the ensemble, which equals the degreeof intensity correlation at that time, var( s a ( t ))= κ σ ( t ). The time variationof κ σ ( t ) reflects the number of modes and the degree of correlation in thespeckle pattern of the modes. Since strong correlation in the speckle pat-terns of a number of modes tends to produce a single transmission channel ransport of localized waves via modes and channels formed from these modes while correlation at any time is directly relatedto the number of channels contributing to transmission at that time, modalspeckle correlation tends to increase the degree of intensity correlation.For narrowband excitation, κ σ ( t ) first falls before increasing at later timessince transmission at early times is dominated by a subset of short-livedmodes among all the modes overlapping the spectrum of the pulse thatpromptly convey energy to the output [101]. At later times, only the long-lived modes contribute to transmission and so the degree of correlation in-creases with time. But at intermediate times, when both short- and long-lived modes contribute to transmission, the number of modes and hence thenumber of channels contributing is relatively high. This is inversely propor-tional to the degree of correlation so that κ σ ( t ) reaches a minimum. T i m e ( n s ) T o t a l t r a n s m i ss i o n , T a log[ T a ( t, ν )] –7–6 ν ( G H z ) –5–4–3–21–0 Figure 1.13 Logarithm of time-frequency spectrogram of total transmissionplotted in the x-y plane following the color bar. The central frequency ν ofthe incident Gaussian pulse of linewidth σ =50.85 MHz is scanned. Each ofthe four spectra of total transmission at different delay times are normalizedto the total transmission at that time. (Ref. [41]) The changing distribution of modes contributing to transmission is seen inthe time-frequency spectrogram for a sample with L = 61 cm and δ = 0 . σ of itsGaussian spectrum as the central frequency of the pulse is tuned. The decayrate of the peak intensity of the mode at long times when isolated modesemerge in the time-frequency spectrogram is equal to the linewidth of themode Γ n to within experimental error of 10%. The nature of propagation divides according to the character of the modes ofthe medium. For δ >
1, transport can be described in terms of diffusing par-ticles of the wave with transmission falling inversely with sample thickness,while for δ < δ .For δ (cid:29) δ ∼
1, which can arise in strongly scattering but stilldiffusive samples which are not more than a few wavelengths thick [105, 106]and in 2D samples [107], in which the laser beam is tightly focused to createa small excitation volume [108, 109]. A small number of spectral peaks maythen be observed in emission. These peaks sharpen up in the presence ofgain due to enhanced stimulated emission in longer lived, spectrally narrowmodes [105].Letokhov [110] considered the lasing threshold in a spherical sample withuniform gain which is directly analogous to the critical condition for a nu-clear chain reaction. Lasing occurs when on average more than one newphoton is created for each photon that escapes the medium. Lasing wassubsequently considered in granular media and in colloidal samples com-posed of dielectric particles in dye solution. Lasing in amplifying colloidsreported by Lawandy et al. [111] is of particular interest since the strengthof scattering and amplification can be controlled independently. A narrow-ing of emission and a shortening of the emitted pulse was observed abovea threshold in pump power. A comparison of the emission spectrum in aneat dye solution and in colloidal solutions is shown in Fig. 14. The originalstudies were carried out in weakly scattering samples excited over trans-verse dimensions much greater than the sample thickness, which itself wasnot much thicker than the mean free path. Wiersma et al. [112] suggestedthat the observations reported could be due to scattering of light into trans-verse directions and its subsequent redirection out of the sample by anotherscattering event.The lasing threshold is typically not suppressed substantially below the ransport of localized waves via modes and channels . × − M solution of R640perchlorate in methanol pumped by a 3-mJ (7 ns) pulses at 532 nm. (b)and (c) Emission spectra of the TiO particles (2 . × − colloidaldye solution pumped by 2.2 µ J and 3.3 mJ pulses, respectively. Emission:(b) scaled up 10 times, (c) scaled down 20 times. (Ref. [111]) threshold for amplified stimulated emission in a neat dye solution. Lightpenetrates a depth into the sample equal to the absorption length of thepump radiation L a = √ Dτ a = (cid:112) (cid:96)(cid:96) a / /τ a is the absorption rate, (cid:96) the transport mean free path, (cid:96) a = vτ a the length of the trajectory in which the intensity falls to 1 /e due to absorp-tion, and v is the transport velocity [114]. The excitation region illustratedin Fig. 15 is near the boundary so that the typical length of the paths ofemitted photons would be comparable to those of the pump photons of ∼ (cid:96) a [115]. But this is the length over which stimulated emission occurs in a neatsolution. So the lasing threshold may not be lowered below the value atwhich appreciable amplified spontaneous emission occurs in a neat dye so-lution. The lasing threshold could be lowered by increasing the residencetime inside the medium in samples with a shorter mean free path at theemission frequency than at the pump frequency or by internal reflection atthe boundary. Above threshold, the optical transition pumped may be sat-urated so that absorption is suppressed and the wave can penetrate deeperinto the sample.The lasing threshold can be dramatically suppressed, however, for local-ized waves. When δ <
1, the intensity within the sample may grow expo-nentially when on resonance with a localized state far from the boundary.The excited region may then be in the middle of the sample and so emissionwill be into modes that overlap the excited mode and are similarly peakedin the middle of the sample and so are long lived. This was demonstratedin low-threshold lasing excited by a beam incident normally upon stacks of glass cover slips with thickness of approximately 100 µm and intervening airlayers of random thickness and with Rhodamine 6G dye solution betweensome of the slides [44]. A plane wave incident upon parallel layers of randomthickness is a one-dimensional medium and will be localized in the medium.In the present circumstance, the layers are not perfectly parallel and so lightis scattered off the normal. This leads to a delocalization transition with acrossover at a thickness at which the transverse spread of an incident rayis equal to the size of the speckle spots formed [63]. Beyond this thickness,the sample becomes three dimensional with regard to propagation of theinitially normally incident beam. The spread of the beam is abetted by thethick layers used and could be reduced dramatically if layers with thickness ∼ λ/ ransport of localized waves via modes and channels baaa )) ss tt nnuuoo cc DCC
DCC ( ( yy gg rr eenne e nnoo ii ssss ii mm EE ccc wavelength (nm) Figure 1.16 (a) Spectra of spontaneous emission in neat solution and ofspontaneous emission and near threshold lasing from Rhodamine 6G placedbetween layers of a glass slide stack at different laser pump energies.(b)Lasing in a single line above threshold and (c) in multiple lines well abovethreshold. The inset in (b) shows the sharp onset of lasing above thresholdfor excitation at a particular portion of the glass stack. (Ref. [44])
The role of resonance with localized modes at both the pump and emis-sion wavelengths is seen in the strong correlation of pump transmission andoutput laser power. Such strong correlation is opposite to what would beexpected for a nonresonant random laser in which peak emission would cor-respond to maximal absorption and so with reduced transmission.Low-threshold lasing via emission into long-lived modes excited by a pumplaser which penetrates deeply into a sample can be realized in periodic andnearly periodic structures. For 1D samples or for layered structures in whichthe dielectric function is modulated only along a single direction, stop bandare seen in the transmission spectrum perpendicular to layers. This is thecase even when the layers are anisotropic with an orientation that varieswith depth. The states at the edge of the band are long lived and can beexcited via emission from excited states of dopants in the periodic structureor of the structure itself when pumped by an external beam falling withinthe frequency range of the pass band. A coherent beam perpendicular to thelayers then emerges without special alignment.In an infinite structure, the group velocity vanishes as the band edge is approached. This leads to the expectation of a lowered lasing threshold atthe edge of a photonic band gap [116]. But in periodic structures of finitethickness, states at the band edge are standing Bloch waves with a lownumber n of anti-nodes in the medium rather than traveling waves [117].The intensity of the wave in each of these states is modulated by an envelopefunction sin nπx/L , where x is the depth into the sample of thickness L .Lasing properties are determined by the modes with increasingly narrowlinewidths and intensity as the band edge is approached. The width of modesincreases as n away from the band gap.Band edge lasing was demonstrated in dye-doped cholesteric liquid crys-tals (CLCs) [43]. Lasing from dye-dope CLCs was observed earlier and at-tributed to lasing at defect sites in the liquid crystal [118]. Roughly parallelrod-shaped molecules in CLCs with average local orientation of the longmolecular axis in a direction called the director rotate with increasing depthinto the sample. This periodic helical structure can be either right- or left-handed. The indices for light polarized parallel and perpendicular to thedirector are the extraordinary and ordinary refractive indices, n e and n o ,respectively. For sufficiently thick films, the reflectance of normally incident,circularly polarized light with the same sign of rotation as the CLC structureis nearly complete within a band centered at vacuum wavelength λ c = nP where n = ( n e + n o ) / P is the pitch of the helix equal to twice thestructure period. The reflected light has the same sign of rotation as theincident beam. The bandwidth is ∆ λ = λ c ∆ n/n , where ∆ n = n e − n o . Forcircularly polarized light of opposite circular polarization, the wave is freelytransmitted. In measurements on dye-doped cholesteric liquid crystal (CLC) R e l a t i v e i n t en s i t y Figure 1.17 Left and right circularly polarized emission spectra from aright handed dye-doped CLC sample as well as lasing emission at the short-wavelength edge of the reflection band. The height of the lasing lines is ∼ ransport of localized waves via modes and channels films, spontaneous emission is inhibited within the band and the density ofstates is enhanced at the band edge for light polarized with the same hand-edness as the chiral structure. Light of opposite chirality is unaffected bythe periodic structure. This makes it possible to make a direct measurementof the density of photon states by comparing the emission spectra of oppo-sitely polarized radiation. The observed suppression of the density of stateswithin the band and the sharp rise at the band edge are shown in Fig. 17and seen to be in good agreement with the calculated density of states in a1D structure. The left circularly polarized (LCP) emission spectrum in thisright handed structure is due to the spontaneous emission of the PM-597dye. RCP emission is suppressed in the stop band and peaked at the bandedges. The RCP light seen within the reflection band does not vanish be-cause the emitted LCP light is converted to RCP light in Fresnel reflectionfrom the surfaces of the glass sample holder. Multiple lasing lines are seenat the short-wavelength band edge.The lasing peaks in Fig. 17 do not correspond precisely to the modes of aperfectly periodic CLC structure. These modes are seen in transmission spec-tra in Fig. 18 in a dye doped CLC sample which was carefully prepared andallowed to equilibrate. A comparison between transmission measured witha tunable narrowband dye laser in a 37- µm thick CLC sample with moder-ate absorption and simulations for a periodic system is shown below. Thesimulated spectrum is displaced vertically for visibility. In a nondissipativesample, the resonance transmission of all modes modes reaches unity. In Fig.18, transmission through modes closest to the band edge is most suppressedby absorption since these modes are longest lived. Since the modes closestto the band edge are longest lived in nearly periodic systems, these statesare most susceptible to being localized by disorder. Such localized states areoften longer lived than the corresponding states of a periodic system and sodisorder can help as well as hinder lasing.Simulations in random amplifying systems show that it is possible tomaximize the lasing intensity at a particular frequency in the spectrum of arandom laser by iteratively feeding back the intensity at a selected frequencyto vary the intensity distribution of the pump beam [119]. The modes ofthe sample are not substantially modified in the lasing transition, but thespectral properties of the modes excited by the pump beam are selected bythe spatial profile of the pump beam. T¨ureci et al. have shown that modesof passive diffusive systems interact via the gain medium to create a uniformspacing in the laser spectrum [120]. In contrast, isolated modes of localizedlasers interact weakly and emit at a frequencies pegged to the modes of thepassive systems [121]. Wavelength (nm) A b s o l u t e T r a n s m i tt a n ce Wavelength (nm)Wavelength (nm) A b s o l u t e T r a n s m i tt a n ce A b s o l u t e T r a n s m i tt a n ce ExperimentSimulation
Figure 1.18 Comparison of measurements of transmission spectra in dye-doped CLC taken by Valery Milner in the lower curve with simulations inthe upper curve. The linewidth narrows as the index n of the mode awayfrom the band edge decreases. Differences between the frequencies of laserlines seen in Fig. 17 and frequencies of lines in high quality CLC samplesin this figure are due to disorder in the sample of Fig. 17. Transmission through a disordered medium is fully determined by the trans-mission matrix t [11, 45, 52, 53]. The optical transmission matrix was mea-sured by Popoff et al. [47] with use of a spatial light modulator (SLM) andan interference technique to find the amplitude and phase of the optical field.Measuring the transmission matrix allows one to focus the transmitted lightat a desired channel at the output surface by phase conjugating the trans-mission matrix [47, 49]. In this way, the transmitted field from different inputchannels arrives in phase at the focal spot and interferes constructively. Thepresence of a random medium can increase the number of independent chan-nels that illuminate a point so that the focused intensity and resolution areenhanced [122–124]. Because of the enormous number of channels in opti-cal experiments, only a small portion of the transmission matrix is typicallymeasured. The distribution of the singular values of the transmission matrixmay then follow the quarter circle law which is characteristic of uncorrelatedGaussian fluctuations of the elements of the transmission matrix [125, 126].Measurements of microwave radiation propagation through random me-dia confined in a waveguide allow us to measure the field on a grid of pointsfor the source and detector [42]. The closest spacing between points is ap-proximately the distance at which the field correlation function vanishes sothat the fields at different points on the gird are only weakly correlated. ransport of localized waves via modes and channels The number of independent channels N supported in the empty waveguideis ∼
66 in the frequency range of 14.7-14.94 GHz in which the wave is diffu-sive and ∼
30 from 10-10.24 GHz in which the wave is localized within thesample. To construct the transmission matrix, N /2 points are selected fromeach of two orthogonal polarizations. A representation of intensity patternsin typical transmission matrices for both diffusive and localized waves at agiven frequency is presented in Fig. 19. Each column presents the variationof intensity across the output surface at points b for a source at points a withtwo orthogonal polarizations. The intensity in each column shown in Fig. 19is normalized by its maximum value. For localized waves, intensity patternsin each column are similar indicating that transmission is dominated by asingle channel. In contrast, no clear pattern is seen for diffusive waves sincemany channels contribute to the intensity at each point. b a b a Figure 1.19 Intensity normalized to the peak value in each speckle patterngenerated by sources at positions a are represented in the columns with in-dex of detector position and polarization b for (a) diffusive and (b) localizedwaves. (Ref. [46]) In Fig. 20, we show a spectrum of the optical transmittance and the un-derlying transmission eigenvalues from a single random realization for bothlocalized and diffusive waves. This confirms that the highest transmissionchannel dominates the transmittance for localized waves while several chan-nels contribute to transmission for diffusive waves. Thus for localized waves,the incident wave from different channels couples to the same eigenchan-nel and excites the same pattern in transmission as seen in Fig. 19(b). Incontrast, the transmission patterns for incident waves for different incidentchannels are the sums of many orthogonal eigenchannels so that the trans-mitted patterns are weakly correlated.Dorokhov [10, 11] showed that, the spacing between the inverse of thelocalization length for adjacent eigenchannels is equal to the inverse of thelocalization length of the sample, ξ n +1 − ξ n = ξ . For localized waves, thisis equivalent to (cid:104) ln τ n (cid:105) − (cid:104) ln τ n +1 (cid:105) = L/ξ = 1 / g , where g is the bareconductance that one would obtain in the absence of wave interference andthe transport can be described in terms of diffusion of particles. In Fig. 21, T and τ n g = 6.914.7 14.76 14.82 14.88 14.9410 −1 (a) Frequency (GHz) Frequency (GHz) g = 0.3710 10.06 10.12 10.18 10.2410 −3 −2 −1 (b) Figure 1.20 Spectra of the transmittance T and transmission eigenvalues τ n for (a) diffusive sample of L = 23 cm with g =6.9 and (b) localizedsample of L = 40 cm with g =0.37. The black dashed line gives T and thesolid lines are spectra of τ n . (Ref. [42]) we show that (cid:104) ln τ n (cid:105) falls linearly with respect to the channel index n forboth diffusive and localized waves. We denote the constant spacing betweenadjacent values of (cid:104) ln τ n (cid:105) as 1 / g (cid:48)(cid:48) , (cid:104) ln τ n (cid:105) − (cid:104) ln τ n +1 (cid:105) = 1 / g (cid:48)(cid:48) . This supportsthe conjecture that g (cid:48)(cid:48) is the bare conductance. l n τ n > Channel index n < Figure 1.21 Variation of (cid:104) ln τ n (cid:105) with channel index n for sample lengths L = 23 (circle), 40 (square) and 61 (triangle) cm for both diffusive (greenopen symbols) and localized (red solid symbols) waves fitted, respectively,with black dashed lines. (Ref. [42]) We expect that the bare conductance should be influenced by the waveinteraction at the sample interface [127–129]. The wave interaction at thesample boundary can be described by a diffusion model [129] in which theincident wave is replaced by an isotropic source at a distance z p from theinterface in which the wave direction is randomized and with a length z ,which is the length beyond the sample boundary at which the intensityinside the sample extrapolates to zero. z was found by fitting the time offlight distribution of wave through random media [88, 130]. Once the surfaceeffect is taken into account, the bare conductance is given as: g = ηξ/L eff , ransport of localized waves via modes and channels where η is a constant of order of unity and L eff = L + 2 z is the effectivesample length. The constant value of g (cid:48)(cid:48) L eff seen in Fig. 22 is consistentwith g (cid:48)(cid:48) being the bare conductance and gives the localization length for thesamples at two frequency ranges. The absolute values of the transmittance T and of the underlying transmission eigenvalues τ n are obtained by equating (cid:104) T (cid:105) = C g (cid:48)(cid:48) for the most diffusive sample of length L = 23 cm, at whichthe renormalization of dimensionless conductance due to wave localizationis negligible. The normalization factor C is used to determine the values of g for other samples.
20 35 50 650 23 24 25 26 336337338339340
L (cm) × L e ff ' g ' // //// // g L eff = ~
339 cm ξ '' g L eff = ~
24 cm ξ '' Figure 1.22 The constant products of g (cid:48)(cid:48) L eff for three different lengths forboth diffusive and localized samples give the localization length ξ in thetwo frequency ranges. (Ref. [42]) The probability density of ln τ n of the first few eigenchannels and their con-tribution to the overall density ln τ is shown in Fig. 23 for the most diffusivesample with g =6.9. Aside from the fall of the probability distribution P (ln τ ) −1.2 −0.8 −0.4 0 ln τ n P ( l n τ n ) P ( l n τ ) Figure 1.23 Probability density of ln τ n (lower curves) and the density ofln τ (top dashed curve), P (ln τ ) = (cid:80) n P (ln τ n ) for the diffusive samplewith g =6.9. (Ref. [42])4 near ln τ ∼
0, which reflects the restriction τ ≤ P (ln τ ) is nearly constantwith ripples spaced by 1 / g (cid:48)(cid:48) . The nearly uniform density P (ln τ ) of corre-sponds to a probability density P ( τ ) = P (ln τ ) d ln τdτ = g /τ . This distributionhas a single peak at low values of τ , in contrast to the predicted bimodaldistribution, which has a second peak nearly unity [11, 45, 52, 131]. Thismay reflect the fundamental difference of measuring the transmission matrixbased on scattering between independent discrete points instead of waveg-uide modes. In theoretical calculation in which scattering between waveguidemodes is treated, all the transmitted energy can be captured. However, onlya fraction of energy transmitted through the disordered medium is capturedwhen the TM is measured on a grid of points. As a result, full informationis not available and the measured distribution of transmission eigenvaluesdoes not accurately represent the actual distribution in the medium. In par-ticular the bimodal distribution of transmission eigenvalues is not observed.This has been suggested in recent simulation of a scalar wave propagationin Q1D samples based on recursive Greens function method. Goetschy andStone [132] have recently calculated the impact of degree of control of thetransmission channels on the density of transmission eigenvalues. The ma-trix t is mapped to t (cid:48) = P tP , where P , P are N × M and M × N matrices which eliminate N − M columns and N − M rows, respectively,of the original random matrix t . Therefore, only M ( M ) channels are undercontrol on the input (output) surface, respectively, and the degree of controlon the input and output surfaces is measured by M /N ( M /N ). As a result,the density of transmission eigenvalues for diffusive samples changes from abimodal distribution to a distribution characteristic of uncorrelated Gaus-sian random matrices, when the degree of control is reduced [42, 47, 133].Nevertheless, key aspects of the statistics of wave propagation and the limitsof control of the transmitted wave can be explored using measurements ofthe transmission matrix.Measuring the transmission matrix allows us to explore the statistics oftransmittance, the most spatially averaged mesoscopic quantity. The im-portance of sample-to-sample fluctuation of conductance in disordered con-ductors was first recognized in conduction mediated by localized states, butfluctuations in transmission were first observed in the constant variance offluctuations of conductance in diffusive samples known as universal conduc-tance fluctuations [9, 57, 69, 70, 134]. For diffusive waves, for which a numberof transmission eigenchannels contribute substantially to the transmittance,the probability distribution of T is Gaussian with variance independent ofthe mean value of T and of sample dimensions. In the localization limit, L/ξ (cid:29)
1, in which transmittance T is dominated by the largest trans- ransport of localized waves via modes and channels mission eigenvalues, T ∼ τ , the single parameter scaling (SPS) theory oflocalization predicts that the probability distribution of the logarithm oftransmittance in 1D samples is a Gaussian function with a variance equalto the average of its magnitude, var(ln T )= −(cid:104) ln T (cid:105) . Therefore, the scalingof average of conductance and the entire distribution of conductance is de-termined by the single parameter |(cid:104) ln T (cid:105)| /L = 1 /ξ . In recent work, the ratio R ≡ − v ar (ln T ) / (cid:104) ln T (cid:105) , is found to approach unity in Q1D samples showingthat propagation in Q1D in this limit is one-dimensional [68].In the Q1D geometry, there is no phase transition between localization anddiffusion as L increases for samples with equivalent local disorder. Instead,there exists a crossover from the diffusive to localized regime. For samplesjust beyond the localization threshold, in which only a few transmissioneigenchannels contribute appreciably to the transmittance, numerical simu-lation [135–139] and random matrix theory calculation [140] by Muttalib andW¨olfle found a one-sided log-normal distribution for the transmittance. Thesource of this unusual probability distribution of conductance can be under-stood with the aid of the charge model proposed by Stone, Mello, Muttalib,and Pichard [45]. The charge model was first introduced by Dyson [141] tovisualize the repulsion between eigenvalues of the large random Hamiltonian.In this model, transmission eigenvalues τ n are associated with positions ofparallel line charges at x n and their images at − x n embedded in a com-pensating continuous charge distribution. The transmission eigenvalues arerelated to the x n via the relation, τ n = 1 / cosh x n . The repulsion betweentwo parallel lines of charges of the same sign with potential ln | x i − x j | mimics the interaction between eigenvalues of the random matrix. The op-positely charged jellium background provides an overall attractive potentialthat holds the structure together. The repulsion between charges for diffu-sive waves is the origin of universal conductance fluctuation. For localizedwaves, the charges are separated by a distance greater than the screeninglength due to the background charges so that the “Coulomb” interaction isscreened. The repulsion between the first charge at x associated with thehighest transmission eigenvalues τ and its image placed at − x providesceiling of unity.We have recently reported microwave measurements of the probabilitydistribution of the“optical” transmittance T in the crossover from diffusiveto localized waves [68]. A Gaussian distribution is found for diffusive wavesand a nearly log-normal distribution for deeply localized waves. Just beyondthe localization threshold, a one-sided log-normal distribution is observedfor an ensemble with g =0.37. In this ensemble, an exponential decay of P ( T ) is found for high values of transmittance as was found in simulations and calculations [142]. The rapid falloff of P ( T ) for T > T is therefore greatly suppressed due to the repulsionbetween these charges.Measurements of the transmission matrix provide the opportunity to in-vestigate the statistics in single disordered samples as opposed to the statis-tics of ensembles of random sample. Such statistics are essential in appli-cations such as imaging and focusing through a random medium. In theQ1D geometry, in which the wave is completely mixed within the sample,the statistics of the intensity relative to the average over the transmittedspeckle pattern, T ba / ( (cid:80) Nb =1 T ba /N ) = N T ba /T a , is independent of source ordetector positions [143, 144]. Because of the Gaussian distribution of thefield in any single speckle pattern, the probability distribution of relativeintensity is P ( N T ba /T a ) = exp( − N T ba /T a ). Since the statistics of relativeintensity are universal, the statistics of the transmission in a sample withtransmittance T would be completely specified by the statistics of totaltransmission T a relative to its average ( T /N ) within the sample.We find in random matrix calculations that the variance of normalizedtotal transmission within a single instance of a large transmission matrix isequal to the inverse eigenchannel participation number [46],v ar ( N T a /T ) = M − . (1.2)These results can be compared to measurements in samples of small N by grouping together measurements in collections of samples with simi-lar values of M . We show in Fig. 24 that the average of var( N T a /T ) insubsets of samples with given M − is in excellent agreement with Eq. 2.var[var( N T a /T )/ M − ] is seen in the insert of Fig. 24 to be proportional to1 /N indicating that fluctuations in the variance over different subsets areGaussian with a variance that vanishes as N increases.The central role played by M can be appreciated from the plots shownin Fig. 25 of the statistics for subsets of samples with identical values of M but drawn from ensembles with different values of g . The distributions P ( N T a /T ) obtained for samples with M − in the range 0 . ± .
01 selectedfrom ensembles with g =3.9 and 0.17 are seen to coincide in Fig. 25(a) andthus to depend only on M − . The curve in Fig. 25(a) is obtained from anexpression for P ( s a ) for diffusive waves given in Refs. [30, 31], in terms of asingle parameter g =2/3var( s a ) but with the substitution of 2 / M − for g .The dependence of P ( N T a /T ) on M − alone and its independence of T isalso demonstrated in Fig. 25(b) for M − over the range 0 . ± .
005 from ransport of localized waves via modes and channels v a r( N T / T ) g =3.9 g =0.17 g =0.37 g =0.61 a −1 M M −1 v a r[ V / M ] - g =6.9 Figure 1.24 Plot of the var(
N T a /T ) computed within transmission ma-trices over a subset of transmission matrices with specified value of M − drawn from random ensembles with different values of g . The straight lineis a plot of var( N T a /T ) = M − . In the inset, the variance of V /M − isplotted vs. M − , where V = v ar ( N T a /T ). (Ref. [46]) measurements in samples of different length with g =0.37 and 0.17. Sincea single channel dominates transmission in the limit, M − →
1, we have
N T a /T = | v a | , where v a is the element of the unitary matrix V whichcouples the incident channel a to the highest transmission channel. TheGaussian distribution of the elements of V leads to a negative exponentialdistribution for the square amplitude of these elements and similarly to P ( N T a /T ) = exp( − N T a /T ), which is the curve plotted in Fig. 25. In Fig.25, we plot the relative intensity distributions P ( N T ba /T ) correspondingto the same collection of samples as in Fig. 25, respectively. The curvesplotted are the intensity distributions obtained by mixing the distributionsfor P ( N T a /T ) shown in Fig. 25 with the universal negative exponentialfunction for the intensity of a single component of polarization.In addition, we find in microwave measurements in Q1D samples that theSPS ratio R is equal to average of M weighted by T , (cid:104) M T (cid:105) / (cid:104) T (cid:105) , whichapproaches unity for L/ξ (cid:29) M while the transmittance T serves as an overall normalization factor.Therefore, the statistics of intensity and total transmission over randomensemble is given by the joint probability distribution of T and M . −3 −2 −1 −3 −2 −1 (a) (b) NT / T a NT / T a P ( N T / T ) a P ( N T / T ) a g =3.9 g =0.17 -1 M =0.17±0.01 −6 −4 −2 −6 −4 −2 N T / T ba2 N T / T ba2 P ( N T / T ) ba2 P ( N T / T ) ba2 (c) (d) g =3.9 -1 M =0.17±0.01 g =0.17 g =0.37 -1 M =0.995±0.005 g =0.17 g =0.37 -1 M =0.995±0.005 g =0.17 Figure 1.25 (a) P ( N T a /T ) for subsets of transmission matrices with M − = 0 . ± .
01 drawn from ensembles of samples with L = 61 cm in twofrequency ranges in which the wave is diffusive (green circles) and localized(red filled circles). The curve is the theoretical probability distribution of P ( s a ) in which var( s a ) is replaced by M − in the expression for P ( N T a /T )in Ref. [30, 31]. (b) P ( N T a /T ) for M − in the range 0 . ± .
005 computedfor localized waves in samples of two lengths: L = 40 cm (black circles) and L = 61 cm (red filled circles). The straight line represents the exponentialdistribution, exp( − N T a /T ). (c,d) The intensity distributions P ( N T ba /T )are plotted under the corresponding distributions of total transmission in(a) and (b). (Ref. [46]) Focusing waves through random media was first demonstrated in acousticsby means of time reversal [145]. The amplitude and phase of the transmittedsignal in time for an incident pulse from a source is picked up by arrays oftransducers. The recorded signal is then played back in time and a pulseemerges at the location of the source. Recently, Vellekoop and Mosk [146]focused monochromatic light through opaque media by shaping the incidentwavefront. Employing a genetic algorithm with a feedback from the intensityat the target point to adjust the phase of the incident wavefront, the intensityat the focus was enhanced by three orders of magnitude. The wavefrontshaping method has been extended to focus optical pulses through randommedia at a spatial target at selected time delay [147, 148].In order to focus a wave at a target channel β once the field transmissionmatrix has been measured, one simply conjugates the phase of the incidentfield relative to the transmitted field at β , yielding t ∗ βa / (cid:112) T β for the normal- ransport of localized waves via modes and channels ized incident field. Here, the incident field is normalized by T β = (cid:80) Na =1 | t βa | so that the incident power is set to be unity. In this way, the field fromdifferent incident channels a arrives at the target in phase and interfereconstructively. Random matrix calculations confirmed by microwave mea-surements show that the contrast between the average intensity at the focalspot (cid:104) I β (cid:105) and the background intensity (cid:104) I b (cid:54) = β (cid:105) , µ = (cid:104) I β (cid:105) / (cid:104) I b (cid:54) = β (cid:105) , dependsupon the eigenchannel participation number M and size of the measuredtransmission matrix N [46], µ = 11 /M − /N . (1.3) M C on t r a s t MesurementCalculation N = 30 N = 66 N = 30 N = 66 N = 30 ' N >> M Figure 1.26 Contrast in maximal focusing vs. eigenchannel participationnumber M . The open circles and squares represent measurements fromtransmission matrices N = 30 and 66 channels, respectively. The filledtriangles give results for N (cid:48) × N (cid:48) matrices with N (cid:48) = 30 for points se-lected from a larger matrix with size N = 66. Phase conjugation is appliedwithin the reduced matrix to achieve maximal focusing. Equation (3) isrepresented by the solid red and dashed blue curves for N = 30 and 66,respectively. In the limit of N (cid:29) M , the contrast given by Eq. (3) is equalto M , which is shown in long-dashed black line. (Ref. [46]) This expression for the contrast is confirmed in measurements shown inFig. 26. This expression is still valid when the size of measured transmissionmatrix N (cid:48) is smaller than N and the corresponding M (cid:48) is correspondinglysmaller than M . This is demonstrated by constructing a matrix of size N (cid:48) =30 from the measured transmission matrix of size N = 66 and calculatingthe contrast by phase conjugating the transmission matrix of size N (cid:48) . Thecontrast computed falls on the curve for N = 30 for different values of M (cid:48) .These results may be applied to measurements of the optical transmission matrix in which the size of the measured matrix N (cid:48) is generally much smallerthan N . In the limit N (cid:29) M , the contrast approaches M .These results indicate that localized waves cannot be focused via phaseconjugation because the value of M is close to one. This is shown in Fig. 27in which phase conjugation has been applied to focus the transmitted waveat the center of output surface for both diffusive and localized waves. Onlyfor diffusive waves does a focal spot emerge from the background. We haverecently demonstrated the use of phase conjugation to focus pulsed trans-mission through random media. By phase conjugating a time-dependenttransmission matrix at a selected time delay, a pulse can be focused in spaceand time [149]. −30−20−10 0 10 20 30 00.511.522.53−30−20−100102030 −30−20−10 0 10 20 30 0102030405060−30−20−100102030 −30−20−10 0 10 20 30 050100150200−30−20−100102030 −30−20−10 0 10 20 30 012345−30−20−100102030 (a) (b)(d)(c) Figure 1.27 Intensity speckle pattern generated for L = 23 cm for diffusivewaves (a) and for L = 61 cm for localized waves (b) normalized to theaverage intensity in the respective patterns. Focusing at the central pointat the same frequency as in (a) and (b) via phase conjugation is displayedin (c) and (d) with 66 and 30 input points, respectively. (Ref. [49]) In this chapter, we have explored the mode and channel approaches to wavesin random media. We believe that each of these approaches has the poten-tial to provide a full description of transmission and its relation to the wavewithin the sample and that this will be of use in a wide variety of applica-tions.In recent work, we have considered four statistical characteristics of modesthat have proven to be particularly promising for explaining steady-state andpulsed transmission and will be reported elsewhere. These characteristics arethe statistics of the spacing and widths of modes, the degree of correlationin the speckle patterns of modes, and the mode transmittance. Correlationbetween speckle patterns includes correlation between the intensity patterns ransport of localized waves via modes and channels of modes as well as the average phase difference and the standard deviationof phase shift between these patterns. The mode transmittance representsthe transmittance integrated over frequency for a particular mode and isobtained from a modal decomposition of the transmission matrix based onmeasurements of field transmission spectra between sets of points on the in-cident and output surfaces. The analysis of waves into modes is of particularinterest in emission and lasing since it gives the density of states which isa key factor in the emission cross section as well as the lifetimes of modes.The relationship between modes and transmission eigenchannels can be elu-cidated by expressing the transmission eigenchannels at each frequency asthe sum of projections of the eigenchannels of the transmission matrix forthe individual modes upon the transmission eigenchannel [150]. Precisely ex-citing a particular mode with a desired spatial distribution provides promisefor control over energy deposition and collection within random media.The relationship between modes and transmission eigenchannels can beseen from the equality of the density of states obtained from the sum of thecontributions of modes and of eigenchannels. The density of quasi-normalmodes or resonances of a region per unit angular frequency, is the sum overLorentzian lines, ρ ( ω ) = π (cid:80) n Γ n / n / +( ω − ω n ) . This is found from the cen-tral frequencies and linewidths determined from a modal decomposition offields at any points in the medium. The density of states can also be obtainedfrom the sum of the contributions of each transmission eigenchannel, whichare the derivatives with angular frequency of the composite phase shift ofthe eigenchannel, ρ ( ω ) = π (cid:80) n dθ n dω . The phase derivative is the intensityweighted phase derivative between all channels on the incident and outputsurfaces [151]. dθ n dω is the transmission delay time for the n th transmissioneigenchannel. When a complete measurement of the transmission matrix ismade, dθ n dω is the integral of intensity inside the sample for the correspondingeigenchannel. The eigenchannel delay time and the associated intensity in-tegral inside the sample increases with the transmission eigenvalue τ n . Thedensity of states may be accurately measured from the transmission matrixas long as N (cid:48) (cid:29) M .We have also explored the distribution of transmission eigenvalues andseen that in a particular transmission matrix, the statistics of relative trans-mission depend only upon M . The absolute distribution within a singlematrix then depends upon these two parameters M and T . Thus the dis-tribution over a random ensemble of all transmission quantities dependsonly upon the joint distribution of M and T . This represents a consider-able simplification from the joint distribution of the full set of transmissioneigenvalues τ n . Manipulation of the incident beam with knowledge of the transmission matrix makes it possible to achieve maximal focusing in a sin-gle transmission matrix with the peak intensity depending only upon T andthe contrast depending upon the value of M in the measured matrix andthe dimension of this matrix. Knowledge of the spectrum of both modes andchannels may advance control over the wave projected within and throughopaque samples for applications in imaging, and energy collection and de-livery. Acknowledgments
We would like to thank Jing Wang, Matthieu Davy, Patrick Sebbah,Valery Milner, Victor Kopp, Andrey Chabanov, Zhao-Qing Zhang, XiaojunCheng and Jerry Klosner for many stimulating discussions and for contribu-tions to many of the results reviewed here. We thank the National ScienceFoundation for support under Grant Number DMR-1207446.
References [1] Altshuler, B. L., Lee, P. A., and Webb, R. A. (eds), 1991.
Mesoscopic phe-nomena in solids.
Elsevier, Amsterdam.[2] Akkermans, E., and Montambaux, G. 2007.
Mesoscopic physics of electronsand photons.
Cambridge university press.[3] van Rossum, M. C. W., and Nieuwenhuizen, T. M. 1999. Multiple scatteringof classical waves: microscopy, mesoscopy, and diffusion.
Rev. Mod. Phys.,
Phys.Rev. Lett.,
Phys.Rev.,
Phys. Rep.,
Phys. Rev.Lett., , 1167-1169.[8] Abrahams, E., Anderson, P. W., Licciardello, D., and Ramakrishnan, T. V.1979. Scaling theory of localization: absence of quantum diffusion in two di-mensions. Phys. Rev. Lett.,
Phys. Rev. B,
Pisma Zh. Eksp. Teor. Fiz.,
JETP Lett.,
Solid State Commun.,
50 years of Anderson localization.
World ScientificPublishing Co. Pte. Ltd. ransport of localized waves via modes and channels
Theory ofprobability and its applications., Phys. Rev. B,
Phys. Rev. B,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett., , 459-462.[22] Feng, S., Kane, C., Lee, P. A., and Stone, A. D. 1988. Correlations andfluctuations of coherent wave transmission through disordered media. Phys.Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Nature,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev.B,
R3813.4[32] Stoytchev, M., and Genack, A. Z. 1997. Measurement of the probability dis-tribution of total transmission in random waveguides.
Phys. Rev. Lett.,
Nature,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Nature,
Phys. Today,
Phys. Today,
Phys. Rev. A,
Rev. Mod. Phys.,
Nature,
Phys. Rev. Lett.,
Opt. Lett.,
Phys.Rev. Lett.,
Mesoscopicphenomena in solids.
Elsevier, Amsterdam.[46] Davy, M., Shi, Z., Wang, J., and Genack, A. Z. 2013. Transmission statisticsand focusing in single disordered samples.
Opt. Express,
Phys. Rev.Lett.,
Nat. Photon., Phys. Rev. B,
Prog. Semicond., ransport of localized waves via modes and channels Nature,
Ann. Phys., (N.Y.) , 290.[53] Beenakker, C. W. J. 1997. Random-matrix theory of quantum transport.
Rev.Mod. Phys.,
Rev.Mod. Phys.,
S306.[55] Landauer, R. 1970. Electrical resistance of disordered one-dimensional lat-tices.
Philos. Mag.,
Physica A
Europhys. Lett., Phys. Rev. Lett.,
Phys. Rev. Lett.,
J. Opt. Soc. of Am. B,
Phys. Rev. Lett.,
Phys. Rev.Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Nature,
Phys. Rev. Lett., Proceedings of the National Academy of Sciences (PNAS) , 2926.[69] Altshuler, B. L. 1985. Fluctuations in the extrinsic conductivity of disorderedconductors.
Pis’ma Zh. Eksp. Teor. Fiz., , 530. [ JETP Lett., , 648].[70] Lee, P. A., and Stone, A. D. 1985. Universal conductance fluctuations inmetals. Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev.Lett.,
NatPhoton., Opt. Lett.,
Phys. Rev. Lett.,
Proc. Cambridge Phil. Soc.,
Philos. Mag.,
J. Phys. C,
Phys.Rev. Lett.,
Appl. Phys. Lett.,
New J. Phys.,
Phys. Rep.,
Phys. Rev. E,
J. Phys. A,
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. Lett., Phys. Rev. Lett., ransport of localized waves via modes and channels
Phys. Rev. Lett.,
Phys. Rev. Lett.,
Phys. Rev. B,
Phys. Rev. Lett.,
Phys. Rev. B,
Phys. Rev. B,
Phys. Rev. Lett.,
Phys. Rev. B,
Phys. Rev. Lett.,
Phys. Rev. B,
Phys. Rev. B,
Phys. Rev. E,
Handbook of Optical Properties.
CRC Press, Boca Raton,FL.[104] Berger, G. A., Kempe, M., Genack, A. Z. 1997. Dynamics of stimulated emis-sion from random media.
Phys. Rev. E,
Phys. Rev. Lett.,
Phys.Rev. B,
Phys. Rev. Lett.,
Opt. Lett.,
Adv. Opt. Photonics, Sov. Phys. JETP,
Nature,
Nature,
Phys. Rev.Lett.,
Phys.Rev. Lett.,
Nature,
Jour. Appl. Phys.,
Prog. In Quant. Electron.,
Mol. Cryst. Liq. Cryst.,
Phys. Rev. Lett.,
Science,
Nature Photon., Nat Photon., Phys. Rev. Lett.,
Phys. Rev. Lett.,
Physics, Random matrices,
Phys. Lett. A.,
Phys. Rev. A,
Europhys.Lett.,
Phys. Rev. B,
Phys.Rev. Lett., , 134.[132] Goetschy, A., and Stone, A. D. 2013. Filtering random matrices: the effect ofimperfect channel control in multiple-scattering. arXiv:1304.5562. ransport of localized waves via modes and channels Nat. Photon., Adv. Phys., , 147.[135] Slevin, K., and Ohtsuki, T. 1997. The Anderson transition: time reversalsymmetry and universality. Phys. Rev. Lett.,
Phys. Rev.B,
Phys. Rev. Lett.,
Phys. Rev. Lett., , 588.[139] Garc´ıa-Mart´ın, A., and S´aenz, J. J. 2001. Universal conductance distributionsin the crossover between diffusive and localization regimes. Phys. Rev. Lett., , 116603 (2001).[140] Muttalib, K. A., and W¨olfle, P. 1999. “One-sided” log-normal distribution ofconductances for a disordered quantum wire. Phys. Rev. Lett., , 3013.[141] Dyson, F. J., and Mehta, M. L. 1962. Statistical theory of the energy levelsof complex systems. I-V. J. Math. Phys., Phys. Rev. B, , 174204.[143] Zhang, S., Lockerman, Y., and Genack, A. Z. 2010. Mesoscopic speckle. Phys.Rev. E,
Phys. Rev. E,
IEEE,
Opt. Lett.,
Phys. Rev. Lett.,
Nat Photon., Opt. Lett.38,