Optical Quadratic Measure Eigenmodes
Michael Mazilu, Joerg Baumgartl, Sebastian Kosmeier, Kishan Dholakia
aa r X i v : . [ m a t h - ph ] J u l Optical Quadratic Measure Eigenmodes
M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia
SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, Fife,KY16 9SS, [email protected]
Abstract:
We report a mathematically rigorous technique which facilitatesthe optimization of various optical properties of electromagnetic fields. Thetechnique exploits the linearity of electromagnetic fields along with thequadratic nature of their interaction with matter. In this manner we maydecompose the respective fields into optical quadratic measure eigenmodes(QME). Key applications include the optimization of the size of a focusedspot, the transmission through photonic devices, and the structured illu-mination of photonic and plasmonic structures. We verify the validity ofthe QME approach through a particular experimental realization where thesize of a focused optical field is minimized using a superposition of Besselbeams. © 2018 Optical Society of America
OCIS codes: (000.3860) Mathematical methods in physics; (260.1960) Diffraction theory;(090.1970) Diffractive optics;
References and links
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1. Introduction
The decomposition of fields into eigenmodes is a well established technique to solve variousproblems within physical sciences. The most prominent example refers to Schr¨odinger’s equa-tion, within the field of quantum mechanics, where energy spectra of atoms are determined viathe eigenvalue spectra and associate wavefunctions of the Hamiltonian operator. Indeed, elec-tron orbits are eigenmodes of the energy, angular momentum, and spin operators [1] and assuch they deliver fundamental insights into the physics of atoms. Within classical mechanics,modes of vibration of music instruments give, for example, their resonant frequencies whiletheir spectrum is associated with the shape of the instrument [2]. In the optical domain, modedecomposition is used in order to describe light propagation within waveguides [3], photoniccrystals [4], and optical cavities [5]. In the case of waveguides and photonic crystals, for ex-ample, eigenmodes describe electromagnetic fields that are invariant in their intensity profile asthey propagate along the fibre or crystal. Additionally, these modes are orthogonal and as suchlight coupled to one of these modes remains, in theory, in this mode forever. This optical modedecomposition can be expanded to include additional operators such as orbital and spin angularmomentum [6].In this paper, we report a novel method which we term “Quadratic Measure Eigenmodes(QME)” which represents a generalization of the powerful concept of eigenmode decomposi-tion within the field of optics. Crucially, it is shown that eigenmode decomposition is applicableto the case of any quadratic measure which is defined as a function of the electromagnetic field.Prominent examples of optical quadratic measures include the energy density and the energyflux of electromagnetic fields. The QME method makes it possible to describe an optical systemand its response to incident electromagnetic fields as a simple mode coupling problem and todetermine the optimal “excitation” for the given measure considered. Intuitively, a superposi-tion of initial fields is optimized in a manner that the minimum/maximum measure is achieved.For instance, the transmission through a pinhole is optimized by maximizing the energy fluxthrough the pinhole.From a theoretical perspective, the QME optimization method is mathematically rigorousand may be distinguished from the multiple techniques currently employed ranging from ge-netic algorithms [7] and random search methods [8] to direct search methods [9]. The majorchallenge encountered in any such approximate optimization and engineering of optical prop-erties is the fact that electromagnetic waves interfere. As such the interference pattern not onlymakes the search for an optimum beam problematic but crucially renders the superpositionfound unreliable, as the different algorithms may converge on different local minima whichare unstable with respect to the different initial parameters in the problem. In contrast, ourproposed QME method yields a unique solution to the problem and directly determines theoptimum (maximal/minimal) measure possible.In the first part of the paper, we introduce the background of the QME theory and show itsproperties in a general context of optimizing the quadratic measures of interfering waves. Inthe second part, we apply the QME formalism to maximize the transmission through aperturesand to minimize the focal spot size. For these applications, we describe, respectively, the elec-tromagnetic field as a superposition of scalar Laguerre-Gaussian beams and vectorial Besselbeams. In the final part of the paper, we report a particular experimental implementation of theME method using computer controlled spatial light modulators to squeeze the spot size of asuperposition of Bessel beams. The paper concludes with a discussion of the particular resultsobtained and with general comments on the versatility of the QME method to a wide range ofproblems.
2. Fundamental concepts
The QME method is based upon two fundamental properties of the electromagnetic field andits interactions. Firstly, the approach relies on the linearity of the electromagnetic fields, i.e. ,the sum of two solutions of Maxwell’s equations is itself a solution of them. As we considerfree space propagation, this criteria is satisfied. The second property relates to the interaction ofthe electromagnetic field with its environment. All such interactions can be written in the formof quadratic expressions with respect to the electric and magnetic fields. Examples includethe energy density, the energy flow, and Maxwell’s stress tensor. This allows us to designateappropriate QME to various parameters ( e.g. spot size) and subsequently ascertain the optimaleigenvalue which, in the case of a spot size operator, yields a sub-diffraction optical spot. Inthis section, we present the details of the theory underpinning our approach.
Electromagnetic waves
To demonstrate our method, we consider monochromatic solutionsof the free space Maxwell’s equations: (cid:209) · e E = , (cid:209) · m H = , (cid:209) × E = − i m w H , (cid:209) × H = i e w E , (1)where E and H are the spatial part of the electric and magnetic vector fields and where e and m denote the vacuum permittivity and permeability. The time dependent carrier wave is given byexp ( i w t ) . These monochromatic solutions of Maxwell’s equations can be written in an integralform linking the electromagnetic fields on the surface A with the fields at any position r , F i ( r ) = Z A P i j ( r , r ′ ) F j ( r ′ ) dS ′ (2)where √ F = ( √ e E , √ m H ) is a shorthand for the two electromagnetic fields having sixscalar components F i . The integration kernel P i j corresponds to a propagation operator givingrise to different vector diffraction integrals such as Huygens, Kirchhoff [10], and Stratton-Chu [11]. Quadratic measures
Crucially all “linear” and measurable properties of the electromagneticfield can be expressed as quadratic forms of the local vector fields and are therefore termed quadratic measures . For instance, the time averaged energy density of the field is proportionalto F ∗ · F = / ( e E ∗ · E + m H ∗ · H ) while the energy flux to 1 / ( E ∗ × H + E × H ∗ ) . Theasterisk ∗ stands for the complex conjugate. Integrating the first quantity over a volume deter-mines the total electromagnetic energy in this volume, and integrating the normal energy fluxacross a surface yields the intensity of the light field incident on this surface. All the quadraticmeasures m k can be represented in a compact way by considering the integral m k = Z V F ∗ i k i j F j d r = h F | k | F i V (3)where the kernel k i j = k † ji is Hermitian with † the adjoint operator including boundary effectsfor finite volumes. Table 1 enumerates some operators associated to common quadratic mea-sures. The integrand part of most of these quadratic measures corresponds to the conservingensities, which together with the associated currents are Lorentz invariant [6]. The volume,over which the integral is taken, does not need to be the whole space and can be a region ofspace, a surface, a curve, or simply multiple points. To account for this general integrationvolume, we broadly term it the region of interest (ROI) in the following.Operator 2 F ∗ i k i j F j EO e E ∗ · E + m H ∗ · H IO ( E ∗ × H + E × H ∗ ) · e k SSO r ( E ∗ × H + E × H ∗ ) · e k LMO e E ∗ · ( i ¶ k ) E + m H ∗ · ( i ¶ k ) H OAMO e E ∗ · ( i r × (cid:209) ) k E + m H ∗ · ( i r × (cid:209) ) k H CSO i ( E ∗ · H − H ∗ · E ) Table 1. Common quadratic measure operators including the energy operator (EO), inten-sity operator (IO), spot size operator (SSO), linear momentum operator (LMO), orbitalangular momentum operator (OAMO), and circular spin operator (CSO). The vector oper-ators include the subscript k indicating the different coordinates and e k the associated unitvectors. Quadratic measures eigenmodes
Finally, using the general definition (3) of the quadraticmeasure it is possible to define a Hilbert sub-space, over the solutions of Maxwell’s equa-tions, with the energy operator defining the inner product. Furthermore, any general quadraticmeasure defined by (3) can be represented in this Hilbert space by means of its spectrum ofeigenvalues and eigenfunctions defined by: l F i ( r ) = Z V k i j ( r , r ) F j ( r ) d r = k | F i V . Depending upon the kernel k i j or operator k , the eigenvalues l form a continuous or discretereal valued spectrum which can be ordered. This gives direct access to the solution of Maxwell’sequations with the largest or smallest measure. The eigenfunctions are orthogonal to each otherensuring simultaneous linearity in both field and measure.In the following, we study the case of different quadratic measure operators and their spec-tral decomposition into the QME. The convention for operator labeling we adopt is to use theshorthand QME followed by a colon and a shorthand of the operator name. In the following,we discuss some examples of different quadratic measure operators. QME: Intensity operator (QME:IO)
The quadratic measure corresponding to the QME:IOmeasures the electromagnetic energy flow across a surface ROI: m ( ) = Z ROI ( E ∗ × H + E × H ∗ ) · n dS (4)where k is chosen such that it corresponds to the projection of the Poynting vector on the normal n to the surface. The eigenvector decomposition of this operator can be used to maximize theoptical throughput through a pinhole or to minimize the intensity in dark spots. Considering aclosed surface ROI surrounding an absorbing particle, the QME approach gives access to thefield that either maximizes or minimizes the absorption of this particle. ME: Spot size operator (QME:SSO)
One way to define the spot size of a laser beam isby measuring the second order intensity moment (SOIM) w [12]. w can be expressed as w = s m ( ) m ( ) , (5)with m ( ) as defined in Eq. (4) and the QME:SSO defined by m ( ) = Z ROI | r − r | ( E ∗ × H + E × H ∗ ) · n dS , (6)where r is the position vector and r the centre of the beam. Accordingly, the square root ofthe QME:SSO eigenvalues multiplied by two gives direct access to the respective beam sizeprovided that the intensity is normalized to one within the ROI, i.e., m ( ) = QME: Optical force operator (QME:OFO)
The optical force acting on a scattering objectcan be calculated by considering the momentum flux, given by Maxwell’s stress tensor, across asurface ROI, surrounding the scattering particle. There are three quadratic measures associatedwith the optical force acting on the particle, one for each orthogonal direction: m ( F ) = Z ROI e E ∗ ( E · n ) + m H ∗ ( H · n ) − ( e E ∗ · E + m H ∗ · H ) n dS (7)where m ( F ) = (cid:16) m ( F ) x , m ( F ) x , m ( F ) x (cid:17) corresponds to the measure of the force. The eigenvectordecomposition of this operator can be used to maximize the optical scattering and optical trap-ping force on microparticles. This approach can be extended through the use of the angularmomentum operator, defined locally by i r × (cid:209) .
3. Applications
In this section, we put the QME concept into practice and provide a couple of examples whichdemonstrate the striking applicability of the QME formalism to answer different questionswithin the field of optics. One such question is determining the largest intensity that may betransmitted through a given optical structure such as metallic apertures [13]. Another questionis what is the smallest optical spot one may achieve. Before we discuss these questions, weshow how the operator formalism presented in section 2 is rendered into a practical formalismwhich we have also applied in the experiments described in section 4.
For practical purposes, in particular in terms of the experimental realization of the QME con-cept, the optimization procedure is spatially separated; that is we consider 1) an initial plane atthe propagation distance z = z where we can superpose a set of i = . . . N fields E i ( x , y , z ) and H i ( x , y , z ) ( N >
1) shaped both in amplitude and phase and 2) a target plane at the propagationdistance z = z where we have the ROI within which the optimization is actually carried out.Due to linearity a superposition of fields in the initial plane is rendered into a superposition inthe target plane both featuring the same set of superposition coefficients: E ( x , y , z ) = N (cid:229) i = a i E i ( x , y , z ) H ( x , y , z ) = N (cid:229) i = a i H i ( x , y , z ) −→ E ( x , y , z ) = N (cid:229) i = a i E i ( x , y , z ) H ( x , y , z ) = N (cid:229) i = a i H i ( x , y , z ) . (8)ased on this, the QME:IO defined in (4) can be rewritten as m ( ) = a ∗ M ( ) a . (9)The matrix M ( ) is a N × N matrix with the elements given by the overlap integrals M ( ) i j = Z ROI (cid:0) E ∗ i ( x , y , z ) × H j ( x , y , z ) + E i ( x , y , z ) × H ∗ j ( x , y , z ) (cid:1) · n dS . (10)This matrix is equivalent to the QME:IO on the Hilbert subspace defined by the fields { E i ( x , y , z ) , H i ( x , y , z ) } . M ( ) is Hermitian and positive-definite which implies that its eigen-values l ( ) k ( k = . . . N ) are real and positive and the eigenvectors v ( ) k are mutually orthogonal.Accordingly, the largest eigenvalue l ( ) max = max (cid:16) l ( ) k (cid:17) and the associated eigenvector v ( ) max willdeliver the initial plane ( z = z ) or target plane ( z = z ) superposition ( E max ( x , y , z ) = N (cid:229) i = v ( ) max , i E i ( x , y , z ) , H max ( x , y , z ) = N (cid:229) i = v ( ) max , i H i ( x , y , z ) ) , (11)which maximizes the intensity within the ROI. We remark, that the case of a superposition ofscalar fields u i ( x , y , z ) , the matrix operator (10) takes on the simpler form M ( ) i j = Z ROI u ∗ i ( x , y , z ) u j ( x , y , z ) dS , (12)an approach used in section 3.2 to maximize the transmission through a pinhole.Similar to the QME:IO, the QME:SSO defined in Eq. (6) can be rewritten as m ( ) = b ∗ M ( ) b , (13)where M ( ) and b must be represented in the intensity normalised base e E k ( x , y , z ) = N (cid:229) i = v ( ) k , i q l ( ) k · E i ( x , y , z ) , e H k ( x , y , z ) = N (cid:229) i = v ( ) k , i q l ( ) k · H i ( x , y , z ) (14)in order to fulfill the requirement of unity intensity within the ROI ( m ( ) ! =
1, see paragraph“QME: Spot size operator (QME:SSO)” in section 2.1). M ( ) is a N × N matrix with the ele-ments given by M ( ) i j = Z ROI | r − r | (cid:16)e E ∗ i ( x , y , z ) × e H j ( x , y , z ) + e E i ( x , y , z ) × e H ∗ j ( x , y , z ) (cid:17) · n dS . (15)After transforming back to the original base we denote the eigenvalues of M ( ) as l ( ) k andthe eigenvectors as v ( ) k . Then, the eigenvector v ( ) min associated with the smallest eigenvalue l ( ) min = min (cid:16) l ( ) k (cid:17) corresponds to the smallest spot achievable within the ROI. The respectivesuperposition is ( E min ( x , y , z ) = N (cid:229) i = v ( ) min , i E i ( x , y , z ) , H min ( x , y , z ) = N (cid:229) i = v ( ) min , i H i ( x , y , z ) ) , (16)for z = z and z = z . We employed this vectorial definition to minimize the size of a focusedspot in section 3.4 using a superposition of vector Bessel beams. The spot size minimizationperformed both in section 3.3 using a superposition of LG beams and in the experimental sec-tion 4 is based on the simpler scalar version ot the QME:SSO matrix defined as M ( ) i j = Z ROI | r − r | e u ∗ i ( x , y , z ) e u j ( x , y , z ) dS . (17) .2. Maximizing transmission through apertures using Laguerre Gaussian beams Fig. 1. 2D intensity profiles together with the ROI highlighted by the red circle. The trans-mittance through the ROI is described by the coefficient T . (a) Transversal and (b) longitu-dinal cross section of a Gaussian beam ( L = P = R = w ). In this subsection, we consider a superposition of LG beams propagating in the z -directionand modulating the carrier wave u c = exp ( − i ( k z − w t )) . In cylindrical coordinates, LG beamsare defined by [14]: u LP ( r , f , z ) = i C LP z r q ( z ) (cid:18) i k w r √ q ( z ) (cid:19) | L | (cid:18) − q ∗ ( z ) q ( z ) (cid:19) P L | L | P (cid:18) r w ( z ) (cid:19) exp (cid:18) − i k r q ( z ) − i L f (cid:19) (18)with z r = k w / q ( z ) = z + i z r , and w ( z ) = + z / z r where w , k , w are the Gaussianbeam waist, vacuum wave vector, and optical frequency, respectively. This beam profile is asolution of the paraxial equation for integer values of P and L corresponding respectively to theradial and azimuthal index of the beam. Within the paraxial approximation, the intensity of the ig. 2. (a) Total transmittance through the ROI for the QME intensity optimized beamas a function of the ROI relative radius R / w and for different numbers N of LG modesconsidered. (b) Relative intensity | v ( ) max , P | of the LG modes ( P = .. N ) decomposing theQME intensity optimized beam. beam transmitted through a planar ROI defined by a disk centered on r = R R p r | u LP ( r , f , z ) | dr where R is the radius of the disk. The coefficient C LP = p P ! / ( P + | L | ) ! isthe normalization factor such that the total intensity of the beam, for an infinite ROI, is unity.The transmission is maximized using the practical implementation of the QME concept de-scribed above in section 3.1; that is the QME:IO was assembled according to Eq. (12) usingthe respective representation in cylindrical coordinates. We only considered the radial familyof LG beams ( L =
0) and performed the optimization in the plane at z =
0. Figure 1 shows thefinal superposition u ( x , y , ) = (cid:229) NP = v ( ) max , P u P ( x , y , ) in the case of a ROI with a radius equalto the waist of the Gaussian envelope.In Fig. 2, we observe that the maximal transmission achievable via the QME intensity opti-mized beams depends on the number of LG beams considered in the superposition and on thesize of the ROI. Indeed, the QME for a smaller ROI needs a larger number of LG modes toachieve 100% transmission. Using a superposition of LG beams we have also minimized the size of a focal spot usingthe representation of the QME:SSO (17) in cylindrical coordinates. It is important to note atthis point that we only retain the intensity eigenmodes whose eigenvalues are within a chosenfraction of total intensity. This is equivalent to considering only beams that have a significantintensity contribution in the ROI. Intuitively, the optimization procedure may be performingso well that a spot of size zero is finally obtained if no intensity threshold is applied. Figure 3shows the smallest spot superposition where we observe the appearance of sidebands just out-side the ROI. These sidebands are a secondary effect of squeezing the light below its diffractionlimit. It is these sidebands that decrease the efficiency of the squeezed spot with respect to themaximal possible intensity in the ROI as calculated via the QME:IO. Using the ratio betweenthese two intensities we can define the intensity Strehl ratio [15] for the QME:SSO (see Fig. 4b).We remark that both, the spot size and the Strehl ratio, show resonances as a function of theROI size. This can be explained considering the number of intensity eigenmodes used for thespot size operator. Indeed, as the ROI size decreases, so does the number of significant inten-sity eigenmodes. Each time one of these modes disappears (step in Fig. 4), we have a suddenincrease in the minimum spot size achievable accompanied with an enhanced Strehl ratio aswe drop the most intensity inefficient mode. Overall, the Strehl ratios determined in our studiespredominantly exceeded values of 1% even when spots were tightly squeezed. Therefore, the ig. 3. (a) Transversal and (b) longitudinal 2D intensity cross sections of the QME super-position delivering the smallest focal spot in the ROI ( R = l ) considering 25 LG modes. w / w is the relative SOIM measured according to Eq. (5). The Strehl ratio in (a) is 4 . N ( ) fulfilling the intensity criteria for the N =
11 case. The arrows indicatethe corresponding scales. (b) Ratio between the ROI intensity of the smallest spot sizeeigenmode and the largest intensity achievable in the ROI (Strehl ratio). observed decrease of intensity is not to severe in terms of potential applications of squeezedbeams for optical manipulation and imaging.Analogously to the approach to calculate the smallest spot size in a planar ROI, we candetermine the QME:SSO for a volume. In this case, the sidebands appear in the outside of theROI in both the lateral and longitudinal directions. The different volumes considered in Fig. 5suggest that there is an intrinsic link between squeezing light in the lateral direction and in thelongitudinal direction.On a final note, we remark that squeezing light below its diffraction limit may be associatedwith the effect of super-oscillations [16]. This refers specifically to the ability to have a local k -vector (gradient of the phase) larger than the spectral bandwidth of the original field. To visu-alize this effect, in the case of QME spot size optimized beams, we have calculated the spectraldensity of the radial wave-vector for the smallest planar spot [17]. As shown in Fig. 6, this spec-tral density clearly identifies a spectral bandwidth (white background in Fig. 6). Regions of the ig. 5. (a,c) Transversal and (b,d) longitudinal intensity cross sections of the QME:SSO fortwo different 3D ROI. Strehl ratio: (a) 5 . . beam which exhibit locally larger wave-vectors than the ones supported by this spectral bandwidth correspond to super-oscillating regions. The local wave vector is defined as ¶ r arg ( u ( r )) where arg ( u ) defines the phase of the analytical signal u . We observe, that super-oscillationsoccur only in the dark region of the beam. Additionally, when the ROI is large compared to theGaussian beam waist w , there are no super-oscillating regions. These only appear when thebeam starts to be squeezed. The paraxial approximation employed above in the case of LG beams can be used to describesub-diffracting beams but breaks down when beams are tightly focused. As a consequence wemust consider full vectorial solutions of Maxwell’s equations. Here, we have chosen Besselbeams as a base-set and determined the superposition of Bessel beams which minimized thespot size in a planar finite ROI. Note that the problem of the finite intensity of Bessel beams [18]is easily circumvented here due to the finite ROI size considered. The monochromatic electric ig. 6. (a) Radial wavevector spectral density. Yellow highlights regions outside the spectralbandwidth. (b) Transversal cross section of the QME spot size optimized field intensity withyellow showing super-oscillating regions.Fig. 7. Intensity cross sections: (a) Airy disk for the maximum numerical aperture consid-ered NA = sin ( q max ) = .
1. The yellow dashed circle shows the position of the smallestzero-intensity circle taken as the ROI inside which the spot size is calculated. The spot sizeis normalized to the spot size of the reference Bessel beam. (b) Reference Bessel beamcorresponding to the largest cone angle q max . The SOIM of the reference Bessel beam isdenoted as w B . (c) QME spot size optimized beam for a superposition of Bessel beams( q ∈ [ , q max ] ) for a large ROI highlighted by the dashed yellow circle. Strehl ratio: 2%. (d)QME spot size optimized beam for a small ROI. Strehl ratio: 0.2%. The gray-scaled regionshows the sidebands while the color range the ROI. Notice that the two scales are different. vector field of the vectorial Bessel beam may explicitly be expressed as [19] E = E exp ( iL f + ik t z )) ( a e x + b e y ) J L ( k t r )+ ik t k z (( a + i b ) exp ( − i f ) J L − ( k t r ) − ( a − i b ) exp ( i f ) J L + ( k t r )) e z ! (19)where k t = k sin ( q ) and k z = k cos ( q ) are the transversal and longitudinal wave vectors with q the characteristic cone angle of the Bessel beam. e x , e y and e z are the unit vectors in theCartesian coordinate system. The parameter L corresponds to the azimuthal topological chargeof the beam while a and b are associated with the polarization state of the beam. The magneticfield H was deduced according to Maxwell’s equations and the QME:IO and QME:SSO oper-ators are assembled according to the expressions (10) and (15), respectively. Figure 7 shows a ig. 8. (a) Relative spot size D r / w B of the Bessel beam superposition as a function of therelative ROI radius R / R B . The SOIM w B and the ROI radius R B are associated with thereference Bessel beam shown in Fig. 7(b), where the ROI is indicated as dashed circle. Forcomparison, the red dot indicates the location of the reference beam in the D r / w B vs. R / R B plot. (b) Strehl ratio vs relative ROI radius R / R B . comparison between Airy disk, Bessel beam, and QME optimized spot considering a numericalaperture of NA = .
1. As in the case of the LG beams, squeezing the focal spot is accompaniedby side bands and a loss in efficiency shown by the Strehl ratio (see Fig. 8).
4. Experimental QME
Fig. 9. Experimental setup. FP = focal plane, L = Lens. Focal widths: f =
50 mm, f =
500 mm, f = f =
400 mm, f = P max =
10 mW, l =
633 nm, SLM: Holoeye HEO 1080 P dual display system, resolution = × = × . =
648 pixel ×
488 pixel, pixel size = . µ m × . µ m. According to the theoretical foundations of the QME concept, the successful experimentalimplementation within the field of optics requires a linear optical system along with the abilityto shape laser fields in both amplitude and phase. We have achieved this by using the exper-imental setup shown in Fig. 9. A HeNe laser beam is expanded and subsequently amplitudemodulated by a spatial light modulator (SLM) display operating in conjunction with a pair ofcrossed polarizers. Analogously to a liquid crystal display on a computer or laptop monitor, theiquid crystal SLM display rotates the polarization of the incident light by an angle dependingupon the voltage applied to the display pixels. The amplitude modulated beam is then imagedonto a second SLM display through a pair of lenses. This second SLM display along with a sub-sequent Fourier lens and aperture served to modulate the phase of the laser beam in the standardfirst order configuration [20]. The field modulations of interest were encoded as RGB imageswhere the blue channel represented the amplitude and the green channel the phase modulation.The SLM controller extracted these information and applied the two channels to the respec-tive panel. We have performed calibration measurements to ensure that both the amplitude andphase modulation exhibited a linear dependence on the applied 8-bit color value between 0 and255. A CCD camera allowed us to record images of laser fields in the Fourier plane of lens 5.To conform this experimental section to the conventions introduced in section 3.1 we remarkthat we shaped a set of test fields both in amplitude and phase in the initial plane at z = z which coincided with the two SLM panels and subsequently minimized the size of a focalspot in the target plane at z = z which coincided with the CCD camera chip. For our proof-of-principle experiments we ignored polarization effects and considered a set of scalar fields E i ( x , y , z ) = A i e i f i ( x , y , z ) ( i = . . . N ) where the CCD camera detected the associated intensity I i ( x , y , z ) (cid:181) | E i ( x , y , z ) | . Both the QME:IO and the QME:SSO were assembled from thesefields according to the scalar expressions (12) and (17), respectively.The amplitudes A i ( x , y , z ) were determined by simply recording an associated intensity im-age I i ( x , y , z ) and subsequently taking the square root. We used the three-step phase retrievalalgorithm described in Ref. [21] to retrieve the phase modulations f i ( x , y , z ) . This algorithmis based on interference with a reference field E R ( x , y , z ) = A R ( x , y , z ) e i f R ( x , y , z ) . The referencefield’s intensity was distributed uniformly over the ROIs considered in our experiments byadding a square phase f R (cid:181) ( x + y ) to the reference field using the SLM phase panel. More-over, a constant phase term f k = p k / k = , ,
2, and the superimposed fields E i ( x , y , z ) + E R ( x , y , z ) e i f k were then encoded onto the SLM. The associated intensity distribu-tions explicitly read I i , k = | A i | + | A R | + p A i A R cos ( f R − f i + p k / )= : I bg , i + g i cos ( D f i + p k / ) , (20)where the spatial coordinates ( x , y , z ) were omitted for brevity. These three intensity distribu-tions represent a 3-dimensional equation system with three unknowns I bg , i , g i and D f i . The latteris explicitly obtained as D f i mod 2 p = atan2 { ( − cos ( p k / )) ( I i , − I i , ) , sin ( p k / ) ( I i , − I i , − I i , ) } , (21)where atan2 { z , x } is the two argument arctangent function corresponding to the argument ofthe complex number x + i z . Note that the reference phase f R cancels out when calculatingthe operator elements due to multiplication of a complex conjugate field (cid:181) e − i f R ( x , y , z ) with acomplex field (cid:181) e i f R ( x , y , z ) . Therefore Eq. (21) yields the adequate phase modulation required toassemble the QME operators.During the course of our experiments we verified the linearity of our optical system by per-forming a comparison between what we term the “experimental superposition (Exp-S)” andthe “numerical superposition (Num-S)”. The Exp-S refers to the case where the set of QMEoptimized superposition coefficients a i is used to encode the optimized superimposed fieldonto the SLM. The CCD camera then detected the intensity I Exp-S ( x , y , z ) corresponding tothis encoded optimized field. The Num-S utilizes the fields E i ( x , y , z ) , which were individ-ually measured to assemble the QME operators, in order to numerically determine the in-tensity distribution as I Num-S ( x , y , z ) (cid:181) (cid:12)(cid:12) (cid:229) Ni = a i E i ( x , y , z ) (cid:12)(cid:12) . Crucially, linearity is verified if Exp-S ( x , y , z ) = I Num-S ( x , y , z ) . This is indeed observed in our experiments as demonstrated inthe following subsection which features a comparison of experimental and numerical intensitydistributions. Fig. 10. SLM encoded field modulations and resulting beam profiles. (a) Ring mask RGBimage as encoded onto the dual panel SLM. (b) Associated Bessel beam created in theCCD camera plane. (c) Aperture RGB image as encoded onto the dual panel SLM. (d)Associated Airy disk as detected by the CCD camera. The yellow bar in (b) represents 2times the SOIM w B of the Bessel beam’s central core. w in (d) is the SOIM of the Airydisk. In our experiments, we used N =
11 non overlapping amplitude ring masks with a constantphase modulation as fields of interest E i ( x , y , z ) . After propagation through the Fourier lens 5(see Fig. 9) the resulting fields E i ( x , y , z ) form a set of Bessel beams. Figure 10(a) shows thelargest ring modulation encoded onto the SLM with the resulting Bessel beam shown in Fig. (b).As this particular Bessel beam comes along with the highest NA compared to the Bessel beamscreated with smaller ring modulations, the beam shown in Fig. (b) exhibits the smallest centralspot of all beams realized in our experiments. The SOIM of the Bessel beam featuring thesmallest core is denoted as w B and used as reference for the measurements presented below.For comparison Figure 10(c) depicts a circular aperture which is encoded onto the SLM in orderto observe the Airy disk (see Fig. (d)). The SOIM of the Airy disk is approximately 1 . I Num-S ( x , y , z ) (top row) and the Exp-S intensity distributions I Exp-S ( x , y , z ) (bottom row)clearly reveals good agreement and thus verifies the linearity of our optical system as eluci-dated above. For completeness, the central row shows the Exp-S superposition in RGB formatas encoded onto the SLM. The color code features a blue channel representing the amplitudemodulation from 0 (black) to 1 (blue) and a green channel corresponding to phase modula-tions from 0 (black) to 2 p (green). Next, we conclude from the measured relative SOIM w / w B that the spot size decreases if the ROI size is reduced. The reduced spot size is achieved atthe expense of the spot intensity which is redistributed to a ring outside of the ROI similarto the theoretical results presented in section 3.4 and Fig. 7. Referring to the Exp-S data, for R = R = ig. 11. Experimental QME spot size minimization. Top row : I Num-S ( x , y , z ) for differentROI radii in pixel as indicated in the top left corner of all graphs shown. The ROI is exem-plary indicated as a dashed ring in the left hand side intensity distribution. The number inthe bottom left corner represents the SOIM w in units of the reference SOIM w B . Centralrow : Optimized experimental distribution as RGB encoded onto the SLM.
Bottom row : In-tensity distributions I Exp-S ( x , y , z ) . The relative SOIM w / w B is indicated in the lower leftcorner. predicted ones: The relative SOIM w / w B was evaluated for the Num-S for ROI radii rangingfrom R = R =
50 pixel in steps of one pixel (see Fig. 12). In comparison to the cor-responding graph shown in Fig. 8 we observe agreement in terms of both the general decreaseof w / w B with decreasing R and the peaks occurring periodically along the R -axis. Surprisingly,for R ≥
10, we do not observe values of w / w B ≈
1; that is the QME spot minimization does notdeliver, as one might naively expect, the reference Bessel beam featuring the smallest SOIM w B of all Bessel beams considered. This is due to the ring structure of Bessel beams whichsignificantly adds to the SOIM and therefore to the spot size in the case of large ROIs. As aconsequence, the QME optimization aims to superimpose the set of Bessel beams in a man-ner that the rings are destructively interfered within the ROI, only retaining the central core.Given that the ring structure is essential for the reduced core size of Bessel beams compared tothe diffraction limited Airy disk we observe values of w which are larger than w B . Indeed, thevalues of w are fairly close to the size of the Airy disk which was determined as w ≈ . w B .This strongly suggests that for large ROIs the QME spot minimization yields a superpositionof Bessel beams which closely resembles the diffraction limited Airy disk. Finally, we remarkthat for R < w / w B becomes very small and has to be taken with care dueto the possible truncation of the spot mediated by the CCD detector’s sensitivity threshold asmentioned above. w / w B R [pixel]0 10 20 30 40 50 Fig. 12. Relative SOIM w / w B within the ROI depending on the ROI radius R in pixel fornumerical superposition of the fields E i ( x , y , z ) in the CCD plane.
5. Discussion and Conclusion
We have theoretically and experimentally demonstrated a novel approach based on quadraticmeasures eigenmodes that enables the optimization of different optical measures. The theorythat we employ is rigorous and based on considering the light-matter interaction as a quadraticmeasure originating from the fields described by Maxwell’s equations. Excitingly, we can de-fine many quadratic measure operators to which our approach is applicable (see Table 1). Themethod is thus very powerful and the generic nature of our approach means that it may be ap-plied for example to optimize the size and contrast of optical dark vortices, the Raman scatte-ring or fluorescence of any samples, the optical dipole force, and the angular/linear momentumtransfer in optical manipulation. In the present paper we have verified the rigor of the methodby demonstrating experimental spot size operator and intensity operator optimization usingLaguerre-Gaussian and Bessel light modes using a dual SLM approach to implement the tech-nique. We envisage the QME approach as providing a powerful and versatile theoretical andpractical toolbox. Our generic approach is applicable to all linear physical phenomena wheregeneralized fields interfere to give rise to quadratic measures.