Non-Gaussian Correlations between Reflected and Transmitted Intensity PatternsEmerging from Opaque Disordered Media

I. Starshynov,1 A. M. Paniagua-Diaz,1 N. Fayard,2 A. Goetschy,2 R. Pierrat,2 R. Carminati,2,* and J. Bertolotti1,†1University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom

2ESPCI Paris, PSL Research University, CNRS, Institut Langevin, 1 Rue Jussieu, F-75005 Paris, France

(Received 1 December 2017; revised manuscript received 8 March 2018; published 11 May 2018)

The propagation of monochromatic light through a scattering medium produces speckle patterns inreflection and transmission, and the apparent randomness of these patterns prevents direct imaging throughthick turbid media. Yet, since elastic multiple scattering is fundamentally a linear and deterministic process,information is not lost but distributed among many degrees of freedom that can be resolved andmanipulated. Here, we demonstrate experimentally that the reflected and transmitted speckle patterns arerobustly correlated, and we unravel all the complex and unexpected features of this fundamentally non-Gaussian and long-range correlation. In particular, we show that it is preserved even for opaque media withthickness much larger than the scattering mean free path, proving that information survives the multiplescattering process and can be recovered. The existence of correlations between the two sides of a scatteringmedium opens up new possibilities for the control of transmitted light without any feedback from the targetside, but using only information gathered from the reflected speckle.

DOI: 10.1103/PhysRevX.8.021041 Subject Areas: Complex Systems, Mesoscopics,Photonics

I. INTRODUCTION

In multiply scattering materials, the random inhomoge-neities in the refractive index scramble the incident wave-front, mixing colors and spatial degrees of freedom, resultingin a white and opaque appearance [1]. Under illuminationwith coherent light and for elastic scattering, interferenceproduces large intensity fluctuations that are not averagedout by a single realization of the disorder, resulting in aseemingly random speckle pattern [2]. In principle, thespeckle pattern encodes all the information on the sampleand the incident light [3], and complete knowledge of thescattering matrix allows one to reverse the multiple scatteringprocess and to recover the initial wavefront, thus permittingimaging through turbid materials [4,5].Speckle patterns are not as random as they appear at

first sight. Interference between the possible scatteringpaths in the medium is known to produce spatial correla-tions between the intensity measured at different positions

[6–8], and correlations of different ranges have beenidentified [9]. Short-range correlations determine the sizeof a speckle spot, while long-range correlations emerge as aconsequence of constraints such as energy conservation orreciprocity [10–12]. The idea of using spatial correlationsfor imaging has recently emerged [13,14], but it has been sofar limited to the optical memory effect [15], a correlationof purely geometrical origin.At first glance, as transmitted and reflected waves are

expected to undergo very different multiple scatteringsequences, correlations between transmitted and reflectedwavefronts are expected to quickly average to zero. Verylittle attention has been given to the cross-correlationbetween the intensities measured at two points on oppositesides of the scattering medium (i.e., in the reflected andtransmitted speckles patterns), and their existence has onlybeen mentioned in passing [16,17]. However, a recenttheoretical study suggested that a long-range correlationshould survive even for thick (opaque) scattering media[18]. The existence of this reflection-transmission (R-T)correlation suggests that one could noninvasively extractinformation on the transmitted speckle from a measurementrestricted to the reflection half-space. As the discovery ofnew speckle correlations, like the recently described shiftmemory effect [19], has been systematically translated intonovel imaging techniques in the past [20], we suggest thatthe reflection-transmission correlation we describe herewill be of fundamental importance for future developments

*Corresponding author.remi.carminati@espci.fr

†Corresponding author.j.bertolotti@exeter.ac.uk

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

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of applications based on reflection measurements, such asin vivo biological imaging [21].In this work, we identify the nontrivial conditions required

to detect the correlation between transmitted and reflectedspeckle patterns and report the first experimental proof of itsexistence. Furthermore, we show that this correlation isrobust and we provide a complete understanding of all itscomplex features, for scattering materials with thickness Land scattering mean free path l covering the entire rangefrom single scattering (L≲ l) to diffusive transport(L ≫ l). The data are supported by 3D numerical simu-lations and by a theoretical analysis of the line shape ofthe correlation function, as well as its dependence on theexperimental parameters. The experiments and the theoryembrace the complexity and the richness of the phenome-non, thus opening the way to its use as a basic ingredient inthe design of new approaches for sensing, imaging, orcommunicating through opaque scattering media.Although major properties of speckle patterns are well

captured by modeling wave propagation with Gaussianrandom processes [22], we will show that the cross-samplecorrelation emerges from a non-Gaussian correction tothe speckle properties. As a consequence, the correlationfunction is always small in amplitude but long range. Inparticular, it does not contain any feature oscillating at thewavelength scale. In addition, we will demonstrate that thecorrelation function is independent of the disorder strengthin the regime of large optical thickness in which themedium is opaque. This means that changing the scatteringmean free path l, for example, by changing the density ofscatterers, does not affect the correlation function. Hence,the dimensionless conductance (proportional to l) is notthe crucial parameter governing the cross-sample correla-tion measured here, contrary to the behavior of long-rangecorrelations measured in transmission, which have beenextensively studied [7,8]. Furthermore, we will show thatthe information content of the correlation function betweenreflection and transmission crucially depends on the dis-tance to the sample. This is also in sharp contrast withlong-range transmission correlations for which far-fieldand surface correlations are essentially related by Fouriertransforms so that they carry the same information. Finally,we will identify a regime of moderate optical thickness,where the correlation function becomes anisotropic andkeeps a memory of the illumination angle. This memoryeffect, due to long-range correlations, has never beendetected before and is fundamentally different from thewell-known memory effect observed in transmission orreflection and resulting from Gaussian statistics [15].

II. MEASUREMENT OF THE TRANSMISSION-REFLECTION CORRELATION

The experimental apparatus is shown in Fig. 1(a). Amonochromatic wave (2-mW He-Ne laser) is incident atan angle of approximately 45° on a suspension of TiO2

particles in glycerol, held between two microscope slides toform a scattering slab. The slab thickness L is controlledusing calibrated spacers, and the mean free path l iscontrolled by varying the TiO2 concentration (seeAppendix A). Typical samples with different optical thick-ness b ¼ L=l, from semitransparent to fully opaque, areshown in Fig. 1(b). For a set of given L and l, we recordthe intensity patterns RðrÞ and TðrÞ on the surface ofthe sample in reflection and transmission, respectively,with two identical imaging systems, each composed of a10× microscope objective, a planoconvex 150-mm lens,and a CCD camera (Allied Vision Manta G-146). As thesamples are liquid, the resulting speckle patterns change intime due to the Brownian motion of the scatterers, with adecorrelation time τ that depends strongly on the samplethickness. Choosing an integration time <τ, and a timeinterval between successive measurements >τ, allows us tomeasure speckle images RðrÞ and TðrÞ for a large ensembleof configurations of the disordered medium. For all ourexperiments, the integration time was set to 1 ms. Anexample pair of images measured for a given realization ofdisorder is shown in Figs. 2(a) and 2(b). For each pair of Rand T, we calculate the correlation function CRT , defined as

CRTðΔrÞ ¼ hδRðrÞδTðrþ ΔrÞihRðrÞihTðrþ ΔrÞi ; ð1Þ

FIG. 1. (a) Experimental setup. A scattering slab, formed by asuspension of TiO2 particles in glycerol, is illuminated by a laserbeam incident at an angle of approximately 45°. The specklepatterns on the two surfaces, Tðx; yÞ and Rðx; yÞ are recordedwith two identical imaging systems. (b) Examples of sampleswith thickness L ¼ 20 μm but different TiO2 concentrations:from left to right, 5 g=dm3, 10 g=dm3, and 40 g=dm3, whichcorrespond to a mean free path of (60, 20.4, and 9.8) �2.5 μm,respectively.

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where Δr ¼ ðΔx;ΔyÞ is a transverse shift between theimages, and δf ¼ f − hfi denotes the statistical fluctuationof a random variable f, with h·i the ensemble average. Inthe experiment, the averaging process is performed in twosteps (see Appendix B). First, a cross-correlation productbetween δR and δT, i.e., the integral

RδRðrÞδTðrþ ΔrÞdr,

is taken for each realization of disorder. Plotted as a 2Dmap, the correlation product appears random, with agranularity similar to that of a speckle image [Fig. 2(c)].Second, an ensemble averaging over the realizations of thedisorder is taken, resulting in the appearance of a clearpattern in CRTðΔx;ΔyÞ [Fig. 2(d)], and demonstratingthat the transmitted and reflected speckle patterns areindeed correlated.Speckle correlations are commonly divided into three

categories: short-range correlations (C1) that decay with theseparation between the observation points on the scale of thewavelength, long-range correlations (C2) that have a poly-nomial decay, and infinite-range correlations (C3) [8,9]. Theshort-range correlation C1 corresponds to the approximationof a field obeying Gaussian statistics [2], while C2 and C3

are non-Gaussian corrections. An additional infinite-rangecorrelation (C0) appears under illumination by a point sourcelocated inside the medium [23]. One can see in Fig. 2(d) thatthe line shape ofCRTðΔrÞ is much wider than a speckle spot,indicating that the dominant contribution to this correlationis long range in nature.In order to characterize the line shape of the correlation

function, and to probe its dependence on the sample

parameters, we measured CRTðΔrÞ for different values ofl and L, covering the full range from the single scattering(L≲ l) to the multiple scattering (L ≫ l) regime. Theresults are summarized in Fig. 3 (center and right columns),where both 2D maps CRTðΔx;ΔyÞ and cross sections alongthe line Δy ¼ 0 (indicated as a dotted line in the 2D maps)are displayed. It is interesting to note that both the shape andthe sign of the reflection-transmission correlation substan-tially depend on L and l. In the single scattering regime(optical thickness b ≲ 1), CRT is dominated by a narrowpeak (still much larger than a single speckle spot) with anegative side lobe. In themultiple scattering regime (b ≫ 1),CRT is dominated by a wide negative dip.The short-range contributions to CRT (C1) decay on the

scale of the wavelength [2] and are negligible in allmeasurements since, in the reflection-transmission geom-etry, the observation points are separated by a distanceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ jΔrj2

p, which is much larger than λ for even the

thinnest sample (see Appendix C for a detailed discussion).Hence,CRT is necessarily a long-range correlation of theC2

type. It is interesting to note that the reflection-transmissiongeometry naturally favors the observation of long-rangecorrelations, without requiring any postprocessing toremove the C1 contribution that dominates in the puretransmission geometry [10,11,24,25]. Another feature ofthe experiment is the illumination/detection geometry thatexcludes any contribution from specularly reflected andtransmitted fields. Indeed, in the geometry in Fig. 1(a), thedetectors do not collect the specularly reflected and trans-mitted averaged fields, but only the scattered light, i.e., theintensities of the fluctuating fields TðrÞ ¼ jδETðrÞj2 andRðrÞ ¼ jδERðrÞj2. This permits us to track various long-range correlations all the way from b≲ 1 to b ≫ 1 byavoiding spurious interference terms in the intensity corre-lation function that would not be negligible in the singlescattering regime. The contribution of the averaged fieldsto the full field correlation function is discussed inAppendix D.

III. NUMERICAL SIMULATIONS

To support the experimental data, we have performedfull numerical simulations of wave propagation in three-dimensional disordered media. In the simulations, thesamples consist of slabs of dipole scatterers with randompositions. The scalar wave equation is solved numericallyusing the coupled-dipole method [26]. Since the measure-ments are not resolved in polarization, and since the inputlight is expected to depolarize on a length scale on the orderof l [27], we neglect polarization and numerically solvethe scalar wave equation. To limit the number of scatterersand save computational time, the polarizability α of eachscatterer has been chosen to maximize the scattering crosssection σs ¼ k4jαj2=4π, leading to α ¼ 4iπ=k3, where k isthe wave number. Adjusting the number density of

FIG. 2. Typical measured speckle patterns in transmission (a)and reflection (b), for a sample with L ¼ 20 μm and l ≃ 20 μm.(c) Cross-correlation product between the speckle patterns in (a)and (b). (d) Correlation function CRTðΔx;ΔyÞ obtained afteradditional ensemble averaging from 104 realizations of thedisorder. The long-range character of the correlation function,which extends far beyond the size of a speckle spot, is clearlyvisible.

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scatterers ρ, we can vary the scattering mean free pathl ¼ 1=ρσs and simulate different kinds of samples. Solvingnumerically the coupled-dipole equations, we compute thescattered field at any point on the input and exit surfaces ofthe slab and deduce the correlation function CRTðΔrÞ. Theensemble averaging is performed by computing the fieldfor N realizations of the positions of the scatterers. Nshould be sufficiently large to satisfy CRT ≫ σ=

ffiffiffiffiN

p, where

σ is the standard deviation of the unaveraged correlationfunction. In the multiple scattering regime, where CRT ∼1=ðkLÞ2 (see below) and σ ∼ 1, we get N ≫ ðkLÞ4. As anexample, for kl ¼ 10 and b ¼ 1.5, we have used 2685dipoles and N ¼ 2.6 × 107 configurations. The results ofthe simulations are displayed in Fig. 3 (left column) and arein very good agreement with the experimental data. Thegeneral shape of the correlation in the regimes b < 1, b ≃ 1,and b > 1 is well reproduced in the simulations. It is alsoworth noting that, in the experiment, N must be replaced bythe effective number of realizations Neff ¼ NAL−2 ≫ Nthat takes into account the spatial averaging, with A the

integration area in the speckle images. Taking advantage ofthe small decorrelation time of the medium and the largefield of view of the camera, this effective number can bemade large at will, allowing us to probe the reflection-transmission correlation for sample thicknesses inacces-sible in the simulations.

IV. THEORETICAL ANALYSIS

In order to refine the analysis, and to get more physicalinsight, we have also used a formal perturbation theory, inwhich the correlation function CRTðΔrÞ defined in Eq. (1)is directly computed from a statistical ensemble averag-ing, without going through the intermediate cross-correlation product used for the experimental data.Both averaging processes coincide provided that l ≫ λ,a condition that is always satisfied in our experiments.Formal perturbation theory uses 1=kl as a small parameterand relies on a diagrammatic formalism that allowsone to derive explicit expressions of intensity correlationfunctions [6–8]. In the reflection-transmission geometry,

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

FIG. 3. Average reflection-transmission correlation function CRT for different values of L and l and the optical thickness b ¼ L=l.Left column: 3D numerical simulations of 2D maps of CRTðΔx;ΔyÞ. Center and right columns: Experimental results. For clarity, both2D maps of CRTðΔx;ΔyÞ and cross sections along the line Δy ¼ 0 (indicated as a dotted line in the 2D maps) are displayed. Tworegimes are identified. For moderate optical thickness (b≲ 1), the correlation function is dominated by a narrow peak with a negativeside lobe. For large optical thicknesses (b > 1), the correlation function is dominated by a wide negative dip.

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care must be taken to properly account for leadingcontributions [16,28].Let us first discuss the regime of large optical thickness

L ≫ l corresponding to Figs. 3(f)–3(i). Strikingly, weobserve in this regime that CRTðΔrÞ is negative, in agree-ment with the prediction in Ref. [18]. This means that, forevery bright spot in reflection (transmission), the corre-sponding area in transmission (reflection) is more likelyto be darker, and vice versa. This feature is consistentwith flux conservation arguments. Indeed, defining T ∝RTðrÞdr and R ∝

RRðrÞdr, energy conservation imposes

T þ R ¼ 1 for a nonabsorbing medium, from whichit follows that

RCRTðΔrÞdΔr ∝ hδTδRi ¼ −hδT2i < 0.

Note that the existence of negative long-range C2 corre-lations has been previously pointed out in Refs. [17,29].We stress that the previous simple flux conservation

argument imposes a global constraint on the correlation,but does not determine its sign for each position Δr. Only afull theoretical calculation accounting for the interferencesbetween crossing paths can give the full line shape ofCRTðΔrÞ. Nonetheless, an intuitive understanding of thelocal sign of the reflection-transmission speckle correlationcan be built in the following way. In a thick scattering slab,interferences create large fluctuations of the wave intensityat the scale of the wavelength (bulk speckle pattern). Theseintensity fluctuations create local fluctuations of the energyflux that act as a source term for the diffusive transport of theintensity towards the medium boundaries. This mechanismis formally analogous to that of particle diffusion driven byLangevin forces in a molecular fluid [30,31]. Because oflocal flux conservation, a positive flux fluctuation pointingtowards the transmission interface has to be compensated onaverage by a negative fluctuation pointing towards thereflection interface, giving rise to a local negative correla-tion. This analysis shows that the existence of R-T corre-lations does not rely on global energy conservation, so thatwe expect the correlation to be robust against absorption.In Appendix E, we have refined the theoretical analysis

performed in Ref. [18] in the regime L≳ l ≫ λ. We findthat both the amplitude and the width of the correlationfunction depend on L and l, as in the experimental data inFigs. 3(f)–3(i). For L ≫ l, the dominant diagrams belongto the class represented in Fig. 4(a), which are typical oflong-range C2 correlations. They predict a correlationfunction that is isotropic, independent of the angleof incidence, and scales as CRTðΔrÞ ¼ CRT

2 ðΔrÞ ¼−fðjΔrjÞ=ðkLÞ2, where f is a dimensionless function thatdecays on a range jΔrj ≃ L [18,32]. This long-rangecharacter of the correlation function originates from thecrossing of two diffusive paths that probe a transversedistance L before escaping, as represented in Fig. 4(a).Note that this path crossing is the analog of the source termin the Langevin picture mentioned above. Moreover, thecorrelation function in this regime is independent of thedisorder strength kl, which makes it strikingly different

from that observed in a pure transmission geometry, forwhich CTT

2 ∼ 1=½ðklÞðkLÞ� ∝ 1=g, where g is the dimen-sionless conductance of the sample [9]. Another importantdifference between CRT

2 and CTT2 is the evolution of their

information content with respect to the detection scheme.Although CTT

2 contains the same information whether it ismeasured on the sample surface or in the far field, this is notthe case for CRT

2 . Indeed, in the far field, we haveCRTðkb;kb0 Þ ∼

RCRTðΔrÞdΔr ¼ const for any pair of

observation directions kb, kb0 , as the information contentis spread uniformly over all degrees of freedom. We givea more detailed explanation of this phenomenon inAppendix E. For this reason, we focus our discussion onthe correlations measured on the sample surface.In the regime of moderate optical thickness l ∼ L ≫ λ,

where single scattering is expected to dominate, an inten-sity correlation extending far beyond the size of a singlespeckle spot is still observed [see Figs. 3(c)–3(e)], but witha positive peak appearing in the vicinity of the negativecontribution. The apparent relative position and amplitudebetween the peak and the dip depends on the angle ofincidence of the illumination (see Appendix F). Contrary tothe negative dip in the correlation function observed atlarge optical thickness, the line shape is anisotropic, withnegative side lobes (hardly visible in the experimentaldata in Figs. 3(c)–3(e), but visible in the calculationspresented in Appendix F) that are more pronounced alongthe direction of the projection of the incident beam onto thesample surface. Moreover, the amplitudes of both thepositive peak and the side lobes substantially depend onthe incidence angle. These two features of the correlationfunction (long-range extent and dependence on the angleof incidence) suggest a qualitative description based ondiagrams of the class represented in Fig. 4(b). Suchdiagrams satisfy both properties simultaneously. The fieldexchange, which creates the correlation, occurs in the firstscattering event and encodes a phase difference in the

FIG. 4. Diagrams contributing to the CRTðΔrÞ correlation. Anintensity correlation depends on two intensities (four fields) thatpropagate through the sample, therefore involving four inputs andfour outputs. Shaded tubes represent diffusive paths and opencircles stand for scatterers; single solid lines stand for averagedfields and single dashed lines for their complex conjugates. Thediagram in panel (a) is representative of the class of C2 diagramsdescribing the negative contribution of the correlation function atlarge optical thickness (L ≫ l). Panel (b) represents the class ofC0-type diagrams that contribute to the positive peak dominant inthe regime l ∼ L ≫ λ.

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subsequent diffusive paths (shaded tubes) that depends onthe angle of incidence. Moreover, the diffusive propagationprovides the long-range behavior. The theoretical evaluationof these diagrams is detailed in Appendix G. They lead to acontribution to the correlation function scaling as 1=ðkLÞ4for b ≫ 1. This is consistent with the fact that, according tothe measurements and the numerical simulations, thiscontribution has to be negligible at large optical thickness,where the CRT

2 correlation function discussed previouslyscaling as 1=ðkLÞ2 dominates. The observed anisotropy inthe correlation function is also well reproduced, supportingthe relevance of the analysis based on the diagrams inFig. 4(b). Interestingly, these diagrams are formally similar tothose leading to the infinite-range correlations C0 observedwhen the sample is excited by a point source [23,33]. In anonabsorbing medium, as a consequence of energy con-servation, theC0 contribution is related to the fluctuations ofthe local density of states at the source position [34,35]. In thepresent context, where a plane wave excitation is used, theC0-type contribution to the reflection-transmission correla-tion function is long range and satisfies

R hCRT0 ðΔrÞidΔr ¼

0. This property also leads to the conclusion that theC0-typecontribution is specific to speckle patterns measured on thesurface of the sample and vanishes in the case of far-fieldangular measurements (see Appendix G).Finally, in the quasiballistic regime l ≫ L ≫ λ, which is

not the focus of our experiment, we expect the correlationCRT to contain additional contributions to C2 and C0 (seethe discussion in Appendix G), which still result in anoverall positive peak. Note that this positive correlationdoes not contradict the flux conservation argument men-tioned earlier. Indeed, this argument rigorously applies forintensity correlations built from the total fields (includingthe averaged reflected and transmitted fields), whichcoincides with the measured correlation Eq. (1) at largeoptical thickness only (see Appendix D).

V. CONCLUSIONS

In summary, we have demonstrated experimentally theexistence of a cross-correlation between the speckle pat-terns measured in reflection and transmission on the surfaceof a disordered medium. The correlation persists in theregime of large optical thickness L ≫ l, in which thesample is opaque due to multiple scattering. The measure-ments are supported by 3D numerical simulations and havebeen analyzed using a perturbative theory (valid whenl ≫ λ). We have found that the reflection-transmissioncorrelation has two contributions: a positive peak dominantat moderate optical thicknesses L≲ l and a negative dipdominant in the multiple scattering regime L ≫ l, whichwe interpret as (respectively) C0- and C2-type scatteringsequences. In the regime L ≫ l, the amplitude of CRT

scales as 1=ðkLÞ2 in 3D, but, at the same time, the range overwhich the correlation has an effect grows as L2,

compensating the decrease in amplitude with the increaseof the number of speckle spots contributing to the crossinformation [32]. The possibility to extract information onthe transmitted speckle from a measurement limited to thereflection half-space offers new possibilities for the detec-tion of objects hidden behind opaque scattering media,including ghost imaging schemes, and for the control ofwave propagation bywavefront shaping techniques [36,37].

The research data supporting this publication are openlyavailable from the University of Exeter’s institutionalrepository [38].

ACKNOWLEDGMENTS

This workwas supported by the Leverhulme Trust’s PhilipLeverhulme Prize, and by LABEX WIFI (Laboratory ofExcellence within the French Program “Investments for theFuture”) under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*. I. S. and A.M. P.-D. acknowledgesupport from EPSRC (EP/L015331/1) through the Centreof Doctoral Training inMetamaterials (XM2). N. F. acknowl-edges financial support from the French “Direction Generalede l’Armement” (DGA).

I. S., A. M. P., N. F. are contributed equally.

APPENDIX A: MEAN FREE PATHCHARACTERIZATION

In order to determine the mean free path of the threesamples with different concentrations of the TiO2, we usedthe well-known Lambert-Beer law I ¼ I0e−L=le, where Iand I0 are the transmitted and initial intensities, respec-tively; L is the thickness of the sample; and le theextinction length. This law measures the attenuation of

FIG. 5. Experimental data to determine the mean free path ofthe three different sample concentrations we used. In red are thedata corresponding to the concentration of 50 mg of TiO2 in10 mL of glycerol, in blue the concentration of 150 mg of TiO2 in10 mL of glycerol, and in yellow the data corresponding to theconcentration of 400 mg of TiO2 in 10 ml of glycerol. The blackdashed lines represent the Lambert-Beer law fitting to thecorresponding experimental data.

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the ballistic light when going through the sample.Since absorption in the sample of TiO2 and glycerol isnegligible (albeit nonzero) compared to scattering, we canconsider le ≈ l. We measured the attenuation of theballistic beam for different thicknesses of the sample andobtained the scattering mean free path of the three differentsamples by fitting the Lambert-Beer law, as shownin Fig. 5.The scattering mean free paths for the samples with

concentrations of 50 mg of TiO2, 150 mg of TiO2, and400 mg of TiO2 in 10 ml of glycerol were found to be,respectively, 58.5�1.3μm, 18.0�0.5μm, and 6.9�0.7μm.

APPENDIX B: EXPERIMENTALDETERMINATION OF THECORRELATION FUNCTION

Experimentally, the determination of the intensitycorrelation function CRTðΔrÞ defined by Eq. (1) is per-formed in two steps. First, a spatially averaged functionCRTðΔrÞ ∝ R

δRðrÞδTðrþ ΔrÞdr is calculated for eachpair of reflected and transmitted speckle images. Then,an ensemble averaging is performed, leading to thecorrelation function hCRTðΔrÞi. In this appendix, wecomment on the definition of CRTðΔrÞ that allows us toremove some experimental artifacts, and we detail thenecessary conditions for the correlation functionhCRTðΔrÞi to coincide with the usual correlation functionCRTðΔrÞ that only involves ensemble averaging.Measuring images (including speckle patterns) with

coherent light inevitably leads to artifacts due to interfer-ence of the signal with scattered light. Figures 6(a) and 6(b)show an example of a raw measurement for the specklepatterns in transmission and reflection. From a singlemeasurement, it is almost impossible to notice, but allthe raw measurements sit on an irregular fringe patternbackground due to the reflections in the protective glasswindow in front of the CCD detector. In fact, if we averagethe raw measurements over disorder, the speckle patternsaverage out, and the inhomogeneous background becomesapparent [see Figs. 6(c) and 6(d)].If one were to use the raw measurements to find the

cross-correlation function, the correlation between theinhomogeneous background patterns would dominateand obscure any real speckle correlation [see Fig. 6(e)].These artifacts can be eliminated using the followingprocedure. Let us denote by Sm the measured specklepattern in reflection or transmission, and by F the unwantedfringes. We have Sm ¼ Sþ F, where S is the desiredspeckle pattern, and F is unknown but is also the samefor each realization of disorder. Thus, we can calculateSm − hSmi ¼ Sþ F − ðhSi þ FÞ ¼ S − hSi ¼ δS. Thisprocedure eliminates F and directly leads to a measurementof δS. For this reason, we have defined a spatially averagedfunction

CRTðΔrÞ ¼ N −11 δRðrÞδTðrþ ΔrÞ; ðB1Þ

with the normalization factor

N 1¼ ½δRðrÞ−δRðrÞ�21=2× ½δTðrþΔrÞ−δTðrþΔrÞ�21=2:ðB2Þ

Here, the overline represents the spatial average over thecoordinate r. This quantity is directly accessible from theexperimental data and is free of artifacts, as shown inFig. 6(f).In the experiment, the spatially averaged function

CRTðΔrÞ is averaged over an ensemble of realizations of

disorder, leading to a correlation function hCRTðΔrÞi.Assuming that spatial and ensemble averaging are equiv-

alent, we can write hCRTðΔrÞi ≃N −12 hδRðrÞδTðrþ ΔrÞi,

with

N 2 ¼ h½δRðrÞ − hδRðrÞi�2i1=2× h½δTðrþ ΔrÞ − hδTðrþ ΔrÞi�2i1=2: ðB3Þ

This correlation differs from Eq. (1), since the normaliza-tion factor N 2 involves intensity fluctuations in reflectionand transmission, and not averaged values. Since theexperiments are carried out in the weak scattering regime

FIG. 6. Typical raw average intensity measurement in reflection(a) and transmission (b), with the averaged intensity distributionover 2000 realizations of disorder in reflection (c) and trans-mission (d). The inhomogeneous background is clearly visible inboth hRi and hTi. (e) Direct cross-correlation of R and T (withoutsubtracting the background). (f) Cross-correlation between δRand δT as described in the main text.

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kl ≫ 1, we can assume that the intensity in each specklepattern follows a Rayleigh statistics to a very goodapproximation. This amounts to neglecting non-Gaussiancontributions to the field in each speckle pattern. In thiscase, we have hδR2i ¼ hRi2 and hδT2i ¼ hTi2, and

hCRTðΔrÞi ≃ CRTðΔrÞ ¼ hδRðrÞδTðrþ ΔrÞihRðrÞihTðrþ ΔrÞi : ðB4Þ

We illustrate the good agreement between hCRTðΔrÞi andCRTðΔrÞ in Fig. 7, where both correlations have beencalculated numerically for a 3D scattering medium withoptical thickness b ¼ 1.5 and scattering strength kl ¼ 10.

APPENDIX C: GAUSSIAN AND NON-GAUSSIANCONTRIBUTIONS TO THE CORRELATION

FUNCTION

The calculation of the correlation function given inEq. (1) requires us to evaluate

hRðrÞTðr0Þi ¼ hδERðrÞδERðrÞ�δETðr0ÞδETðr0Þ�i; ðC1Þ

where r0 ¼ rþ Δr. The Gaussian contribution, usuallydenoted C1, is obtained by pairing fields to form averagesof complex conjugate pairs. On the other hand, non-Gaussian contributions necessarily involve scattering pathsthat connect four fields, since hδERðrÞi ¼ hδETðrÞi ¼ 0.By noting h…ic as the non-Gaussian contributions, weobtain

CRTðΔrÞ ¼ jhδERðrÞδETðr0Þ�ij2hRðrÞihTðr0Þi

þ hδERðrÞδERðrÞ�δETðr0ÞδETðr0Þ�ichRðrÞihTðr0Þi : ðC2Þ

The first term of Eq. (C2) is the C1 contribution. Thiscontribution can be large, of the order of unity, if thedistance between observation points is smaller than thewavelength. This is always possible in the T-T configura-tion, where points r and r0 are measured in the same plane:The C1 contribution dominates the correlation CTTðΔrÞfor Δr≲ λ. On the contrary, in the R-T configuration, thedistance between observation points is

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ Δr2

p.

Therefore, the C1 contribution is always negligible forL ≫ λ, as is the case in our experiment. We have checkednumerically that this statement remains valid at moderateoptical thickness, where the diffusion approximation breaksdown. In Fig. 8, we compare the full correlation CRTðΔrÞwith the C1 contribution and the connected contribution[last term of Eq. (C2)], calculated from a microscopicwave propagation simulation in a three-dimensional slab ofoptical thickness L=l ¼ 1 and scattering strength kl ¼ 15.We observe that the full correlation [Fig. 8(a)] is wellapproximated by its connected part [Fig. 8(b)], indicatingthat the C1 contribution [Fig. 8(c)] is indeed negligible.Other simulations (not shown) have revealed that the C1

contribution becomes important for kL≲ 1, as expected.This analysis shows that the R-T configuration is particu-larly adapted to access and study non-Gaussian quantities

−

−

−

−

−

−

−

−

−− − − −

FIG. 7. Comparison between the two correlation functionshCRTðΔrÞi and CRTðΔrÞ. The transverse distance Δr ¼ Δx isvaried along the direction of the illumination plane. Theparameters of the 3D numerical simulation are L=l ¼ 1.5,kl ¼ 10, θa ≃ 45°.

FIG. 8. Correlation function CRT calculated from numerical simulation of the wave equation in a 3D slab. (a) Full correlation.(b) Connected part of the correlation [second term of the rhs of Eq. (C2)]. (c) C1 contribution of the correlation [first term of the rhs ofEq. (C2)]. The parameters are kl ¼ 15, L=l ¼ 1, θa ≃ 45°.

I. STARSHYNOV et al. PHYS. REV. X 8, 021041 (2018)

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in mesoscopic physics, such as the connected and long-range contributions hδERðrÞδERðrÞ�δETðr0ÞδETðr0Þ�ic.Indeed, the latter dominate the correlation in the full rangefrom the deep diffusive (L ≫ l) to the quasiballistic(L≲ l) regime. This is in sharp contrast with the T-Tconfiguration, for which it is difficult to extract the smallnon-Gaussian part of the correlation: This requires adelicate fitting procedure or an additional field measure-ment that can be cumbersome with optical waves; see, forinstance, Refs. [33,39].

APPENDIX D: R-T CORRELATION BUILTFROM THE TOTAL REFLECTED AND

TRANSMITTED FIELDS

The experiment is performed with a laser illuminationoriented with a nonzero angle with respect to the samplesurface, which allows us to measure the intensity of thefluctuating part of the fields, RðrÞ ¼ jδERðrÞj2 andTðr0Þ ¼ jδETðr0Þj2, only. We want to discuss here whatthe R-T correlation would be if the mean fields were alsomeasured. Let us note XðrÞ ¼ jEXðrÞj2 ¼ jhEXðrÞi þδEXðrÞj2, where X stands for R or T. By expressing Xin terms of X and using hXi ¼ hXi þ jhEXij2, we get

hδRðrÞδTðr0Þi¼ hδRðrÞδTðr0Þi þ 2Re½hETðr0ÞihδETðr0Þ�RðrÞiþ hERðrÞihδERðrÞ�Tðr0Þiþ hERðrÞihETðr0Þi�hδERðrÞ�δETðr0Þiþ hERðrÞihETðr0ÞihδERðrÞδETðr0Þi��: ðD1Þ

Hence, the R-T correlation built from the total reflected andtransmitted fields contains additional interferences betweenthe mean fields and the scattered fields. We illustrate theirrole in Fig. 9, where we compare CRT [panel (a)] and CRT

[panel (b)] calculated for a 3D disordered slab (L=l ¼ 1,

kl ¼ 15, and illumination angle θa ¼ 45°). On top of thelong-range component of CRT , CRT also exhibits tinyoscillating contributions and additional long-range contri-butions due to the four terms of Eq. (D1). These contri-butions, negligible in the deep diffusive regime L ≫ l,become important at moderate optical thickness, L ∼ l.This explains why the positive contribution to the long-range correlation discussed in this work was not detected inRef. [18], where only CRT was analyzed. In Fig. 10, wecompare both correlation functions, calculated from 2Dnumerical simulations such as those performed in Ref. [18],for two different optical depths. At large optical depth, thetwo correlations are equal, while it is no longer the case inthe regime L ∼ l, where interference terms dominate.

APPENDIX E: ANALYTICAL CALCULATIONOF CRT

2 ðΔrÞIn this appendix, we refine the calculation of CRT

2 thatwas previously performed by some of us in Ref. [18]. Let usfirst recall the physical picture that gives rise to the long-range C2 correlation. The correlator (B4) is the average offour fields (measured at the two detector positions) that canbe decomposed as sums over all scattering paths. The idea

FIG. 9. Comparison of two R-T correlation functions calculatedfrom numerical simulation of the wave equation in a 3D slab.(a) Correlation function CRTðΔrÞ ¼ hδRðrÞδTðrþ ΔrÞi=hRðrÞihTðrþ ΔrÞi built from fluctuating parts of the fields. (b) Corre-lation function CRT ¼ hδRðrÞδTðrþ ΔrÞi=hRðrÞihTðrþ ΔrÞibuilt from the full fields (see text for details). The parametersare kl ¼ 15, L=l ¼ 1, θa ¼ 45°.

FIG. 10. Comparison of two R-T correlation functions calcu-lated from numerical simulation of the wave equation in a 2Dslab. (a) Moderate optical thickness L ¼ l and shifted incidenceθa ≃ −35°. (b) Large optical thickness L=l ¼ 7 at normalincidence θa ¼ 0. The other parameter is kl ¼ 10.

NON-GAUSSIAN CORRELATIONS BETWEEN REFLECTED … PHYS. REV. X 8, 021041 (2018)

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is to select paths that give a nonvanishing contribution tothe average. The CRT

2 contribution is obtained by consid-ering pairs of fields that propagate diffusively in thedisordered medium until their paths cross at an arbitraryposition inside the medium. When this crossing occurs,pairs of fields exchange their partners to form new pairs thattravel again diffusively through the system until they reachthe output boundaries. There are two different ways to doso, as represented in Fig. 11. For the same reason that madeCRT1 negligible (see Appendix C), the diagram represented

in Fig. 11(b) is negligible for all positions r and r0 in theregime kL ≫ 1.Let us stress here the difference between CRT

2 and CTT2 .

CTT2 is made of the same diagrams as in Fig. 11, but with

the point r0 in the transmission plane. As a result, thediagram of Fig. 11(b) is not negligible for observationpoints in the same transmission plane. In particular, thisdiagram with observation points in the far field is theFourier transform of the one represented in Fig. 11(a), withpoints at the sample surface. Conversely, for CRT

2 , since thediagram of Fig. 11(b) is negligible, we measure in the farfield the contribution of Fig. 11 (a) only,

CRTðkb;k0bÞ ¼

ZdΔrA

CRTðΔrÞ; ðE1Þ

for any couple of observation directions kb, k0b. Here, A is

the transverse area covered by the input illumination.Although CTT

2 contains the same information whether itis measured at the sample surface or angularly in the farfield, this is not the case for CRT

2 .Mathematically, the contribution of Fig. 11(a) to the

correlator (C1) reads

hδERðrÞδERðrÞ�δETðr0ÞδETðr0Þ�iC2

¼Z

jhEðr1Þij2jhEðr2Þij2Lðr2; ρ2ÞLðr1; ρ1Þ

×Hðρ1; ρ2; ρ3; ρ4ÞLðr3; ρ3ÞLðr4; ρ4Þ× jhGðr0 − r3Þij2jhGðr − r4Þij2dr1…dr4dρ1…dρ4;

ðE2Þ

where hGðrÞi is the mean Green’s function of the waveequation; Lðr; r0Þ represents a diffusive pair of fields thatpropagate from r to r0; and the operator Hðρ1; ρ2; ρ3; ρ4Þstands for the diffusion partner exchange. The latter iscalled a Hikami vertex and reads

Hðρ1;ρ2;ρ3;ρ4Þ ¼h4δðρ1;ρ2;ρ3;ρ4Þ

× ðΔρ1þ� � �þΔρ4

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{1

þ 2∇ρ1:∇ρ2zfflfflfflfflfflffl}|fflfflfflfflfflffl{2

þ 2∇ρ3:∇ρ4zfflfflfflfflfflffl}|fflfflfflfflfflffl{3

Þ; ðE3Þ

where δðρ1; ρ2; ρ3; ρ4Þ meansRδðρ1−ρÞδðρ2−ρÞδðρ3−ρÞ

δðρ4−ρÞdρ and h is the weight of the vertex defined inRef. [6], for example. We have labeled three terms inEq. (E3). In the literature dedicated to mesoscopic physics,it is often argued that the term 1 is negligible (since it forcesthe crossing to occur at the sample surface; see Ref. [7]),while the two others give rise to equal contributions; seeRef. [8]. This was the approach adopted in Ref. [18], wherethe Hikami vertex was replaced by twice the term 3. We callCout2 the analytical form of the correlation calculated in this

way. Similarly, we call Cin2 the form obtained by keeping

twice the term 2. In fact, in the R-T configuration, there isno good reason to neglect the term 1 nor to assume thatterms 2 and 3 are of the same amplitude. For this reason, wecompute here all contributions explicitly. We write thecomplete correlator as

CRT2 ðr; r0Þ ¼ CΔ

2 ðr; r0Þ þCin2 ðr; r0Þ þ Cout

2 ðr; r0Þ2

; ðE4Þ

where the three contributions come from the three termslabeled in the vertex (E3). Following the same approach asin Ref. [18], we find in three-dimensional space

CΔ2 ¼ −3

4k2l2

ZJ0ðqΔr=LÞshðqz0=LÞ2qsh½qð1þ 2z0=LÞ�2

×

�27L10l

sh

�q

�1þ z0

L

��sh

�qz0L

�

þ�27

20þ 5l2

3L2

�qsh

�q

�1þ 2

z0L

���dq; ðE5Þ

FIG. 11. Typical diagrams contributing to the connected four-field correlations in R-T. Panel (a) corresponds to the case wherethe fields exchange at the entrance, propagate, and exchangeinside the medium. After this exchange, they can travel diffu-sively through long distances to eventually be measured at thedesired points. Panel (b) corresponds to the case where the twopairs of fields propagate first together, then exchange inside themedium, and, at the end, have to exchange again to eventually bemeasured at two different points. The output vertex of thisdiagram is identical to the output vertex of the C1 correlation.

I. STARSHYNOV et al. PHYS. REV. X 8, 021041 (2018)

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Cin2 ¼ −45

8k2l2

ZJ0ðqΔr=LÞshðqz0=LÞ2q2sh½qð1þ 2 z0=LÞ�2

×�−qch

�q�1þ z0

L

��þ shðqÞ

�dq; ðE6Þ

Cout2 ¼ −45

8k2l2

ZJ0ðqΔr=LÞshðqz0=LÞ2q2sh½qð1þ 2z0=LÞ�2

×

�−qchðqÞþ

�1þq2

�1þ 2z0

Lþ 2z20

L2

��shðqÞ

�dq;

ðE7Þ

and in two-dimensional space

CΔ2 ¼

−Lπkl2

ZcosðqΔr=LÞshðqz0=LÞ2q2sh½qð1þ2z0=LÞ�2

×

�16ð1þπ=2ÞLπ2ð1þπ=4Þl sh

�qz0L

�sh

�q

�1þz0

L

��

þ�4ð1þπ=2Þ2π2ð1þπ=4Þþ

ð1þπ=4Þl2

L2

�qsh

�q

�1þ2

z0L

���dq;

ðE8Þ

Cin2 ¼ −32L

πkl2

ð1þ π=4Þπ2

ZcosðqΔr=LÞshðqz0=LÞ2q3sh½qð1þ 2 z0=LÞ�2

×

�−qch

�q

�1þ z0

L

��þ shðqÞ

�dq; ðE9Þ

Cout2 ¼−32L

πkl2

ð1þ π=4Þπ2

ZcosðqΔr=LÞshðqz0=LÞ2q3sh½qð1þ 2z0=LÞ�2

×�−qchðqÞþ

�1þq2

�1þ 2z0

Lþ 2z20

L2

��shðqÞ

�dq;

ðE10Þ

where z0 is the extrapolation length (see Ref. [18] fordetails). We have represented these different contributionsin Fig. 12 in the case of wave propagation through a 2Ddisordered slab. The contributions of Cout

2 and CΔ2 are

negative whereas Cin2 is positive. In addition, Cin

2 and Cout2

do not have the same amplitude. However, the sum of allterms, given by Eq. (E4), turns out to be well approximatedby Cout

2 , as it was done in Ref. [18]. Both expressions are ingood agreement with simulations of microscopic wavepropagation. Hence, we conclude that all conclusionsof Ref. [18] remain qualitatively valid. In particular,at large optical depth L ≫ l, we find CRT

2 ðΔrÞ ¼−fðΔr=LÞ=ðkLÞd−1, where fðxÞ is a positive decayingfunction of range unity given by

fðxÞ¼Z

∞

0

qcosðqxÞshðqÞ

�ð1þπ=2Þ2þπð1þπ=2Þ4πð1þπ=4Þ

þ1þπ=4π

½−2qchðqÞþ shðqÞð2þq2Þ�q2shðqÞ

�dq ðE11Þ

in dimension 2 and by

fðxÞ¼Z

∞

0

q2J0ðqxÞshðqÞ

×

�21

20þ5

4

½−2qchðqÞþ shðqÞð2þq2Þ�q2shðqÞ

�dq ðE12Þ

in dimension 3. This means, in particular, that the R-Tcorrelation becomes independent of the disorder strengthkl in the deep diffusive regime.

APPENDIX F: DEPENDENCE ON THE ANGLEOF INCIDENCE IN THE REGIME L ∼ l

In this appendix, we discuss the angular dependence ofthe shape of the correlation CRTðΔrÞ in the regime ofmoderate optical thickness. In Fig. 13, we have representedthe result of 3D numerical simulations of the wavepropagation in a disordered slab of optical thickness L=l ¼1 and disorder strength kl ¼ 15. The horizontal axis isdefined as the intersection of the incidence plane with thesample surface (here, θa ≃ 75°). As discussed in the maintext, the correlation is positive for Δr≲ L and presentsnegative side lobes that are more pronounced along theillumination direction. We have analyzed the angulardependence of this shape along the direction Δy ¼ 0.The results are presented in Fig. 14. For θa ¼ 57°, thecorrelation CRTðΔxÞ is asymmetric. When the angle ofincidence θa increases, both the positive central peak andthe negative side lobes grow. In addition, the correlationbecomes more and more symmetric. We interpret the shape

−

−

−− − − − −

FIG. 12. Analytical predictions for the CRT2 correlation (black

solid line), Cout2 correlation (red dotted line), and Cin

2 correlation(green dashed-dotted line) compared with the simulation of 2Dwave propagation in a disordered medium (blue dashed line). Theparameters of the simulations are kl ¼ 10, L=l ¼ 10, θa ¼ 0.

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of this correlation function as the result of the superpositionof two contributions, CRT

2 and CRT0 . The contribution CRT

2 isa negative dip with a minimum located at Δx > 0. Asdiscussed in the main text, this contribution is almostindependent of θa. On the other hand, the contribution CRT

0

contains both a positive peak located at Δx ≃ 0 andsymmetric negative side lobes. It is also strongly dependenton the illumination angle (see Appendix G for details).Hence, the latter is responsible for the anisotropic shapeobserved in Fig. 13 and the evolution presented in Fig. 14.In particular, the correlation CRTðΔxÞ shown in Fig. 14becomes more and more symmetric for increasing θa,because the amplitude of the negative side lobes of CRT

0

gets larger than the CRT2 dip. A microscopic interpretation

of this phenomenon is proposed in the next appendix.

APPENDIX G: CRT0 CORRELATION WITH

PLANE WAVE ILLUMINATION

The C0 correlation has first been introduced inRefs. [23,34] in the case of a point source excitation.Here, we consider the same class of scattering processes,but generated by a plane wave excitation. As we will see,both the formal calculation and the qualitative consequencesare different from the case of a point source excitation. Themicroscopic representations of the CRT

0 diagrams are rep-resented in Fig. 15. Each diagram involves scattering pathsthat visit a common scatterer located near the front side of thesample. The symmetric diagrams (not shown) that involve acommon scatterer at the outputs can be neglected for thesame reason as the C1 correlation (see Appendix C). Usingthe same notations as in Appendix E, the four-field corre-lator CRT0 ðΔrÞ ¼ hδERðrÞδERðrÞ�δETðr0ÞδETðr0Þ�iC0

takes,in 3D, the form

CRT0 ðΔrÞ¼ 4π

l

ZVðr2;r3ÞLðr2;r4ÞLðr3;r5Þ

× jhGðr− r4Þij2jhGðr0− r5Þij2dr2dr3dr4dr5;ðG1Þ

where Vðr2; r3Þ is the sum of the four possibilities forconnecting the input plane wave to the ladder diagramsstarting in r2 and r3, as represented in Fig. 15. For example,the contribution of Fig. 15(a) to the vertex V is

VðaÞðr2; r3Þ ¼Z

hEðr2Þi�hEðr3Þi�jhEðr1Þij2

× hGðr2 − r1ÞihGðr3 − r1Þidr1; ðG2Þ

−

−

−− − − −

FIG. 14. Dependence of CRTðΔrÞ on the illumination angleθa, along the illumination direction Δy ¼ 0 (horizontal axisof Fig. 13). The parameters of the 3D simulation areL=l ¼ 1, kl ¼ 15.

FIG. 13. CRTðΔrÞ calculated from 3D numerical simulations ofthe wave propagation in a disordered slab of moderate opticaldepth. The direction Δy ¼ 0 is defined as the intersection of theincidence plane with the sample surface. The parameters areL=l ¼ 1, kl ¼ 15, θa ≃ 75°.

FIG. 15. Leading diagrams contributing to CRT0 . Shaded tubes

represent diffusive paths (ladders); single solid lines stand foraveraged fields, and single dashed lines for their complexconjugates. The extra scatterer located near the surface boundarycan connect the ladders in four different ways.

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where themean field hEðrÞi depends on the incidence angle.By integrating over the transverse coordinates, we get

CRT0 ðΔrÞ ¼ l16π3

Zð½VðaÞ

qa;qðz2; z3Þþ VðcÞqa;qðz2; z3Þ�eiðqa−qÞ:Δr

þ ½VðbÞqa;qðz2; z3Þþ VðdÞ

qa;qðz2; z3Þ�eiðq−qaÞ:ΔrÞ×Lq−qaðz2;0ÞLq−qaðz3;LÞdqdz2dz3; ðG3Þ

where z labels longitudinal coordinates; qa is the transversecomponent of the incident wave vector ka; and Lqðz; z0Þis the Fourier transform of Lðr; r0Þ with respect to thetransverse part of the coordinate r − r0. In addition, thecomponents V are given by

VðaÞqa;qðz2;z3Þ¼

ZGqað0;z1Þ2G2qa−qðz1;z3Þ

× Gqðz1;z2ÞGqað0;z2Þ�Gqað0;z3Þ�dz1; ðG4Þ

VðcÞqa;qðz2; z3Þ ¼

ZjGqað0; z1Þj2Gqðz1; z3Þ�

× Gqðz1; z2ÞGqað0; z2Þ�Gqa

ð0; z3Þ; dz1;ðG5Þ

VðbÞ ¼ VðaÞ�, and VðcÞ ¼ VðdÞ�. In these expressions,Gqðz;z0Þ¼ieikzðz0−zÞe−jz0−zj=2μl=ð2kzÞ, with kz¼

ffiffiffiffiffiffiffiffiffiffiffiffiffik2−q2

p ≡kμ, is the transverse Fourier transform of the meanGreen’s function of the Helmholtz equation. We now makethe approximations Lðz2; 0; q − qaÞ ≃ Lð0; 0; q − qaÞ andLðz3; L; q − qaÞ ≃ Lð0; L; q − qaÞ, and we integrateEq. (G3) over the longitudinal coordinates z1, z2, and z3.The correlator becomes

CRT0 ðΔrÞ ¼ l4

128π3k6

Zcos ½ðqa − qÞ:Δr�

× Lq−qað0; 0ÞLq−qað0; LÞFðμ; μa; klÞdq;ðG6Þ

with

Fðμ; μa; klÞ ¼2ðμa − μÞ

μað2μa − μÞðμþ μaÞ½9μ3 þ 18μ2μa

þ 11μμ2a þ 2μ3a þ 4μ3μ2aðμ − μaÞ2k2l2�=ð½9μ2 þ μ2a þ 6μμa þ 4μ2μ2aðμ − μaÞ2k2l2�× ½ðμþ μaÞ2 þ 4μ2μ2aðμ − μaÞ2k2l2�Þ;

ðG7Þ

where μa ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − q2a=k2

pand μ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − q2=k2

p.

Normalizing the correlator by the intensity producthjδERðrÞj2ihjδETðr0Þj2i ¼ l=6k4μ2aL and the integrationmomentum variable by the sample thickness L, we finally

obtain

CRT0 ðΔrÞ ¼ 27μ2akL

4πðklÞ3Z

cos

�ðq0a − q0Þ:Δr

L

�

× Pðq0a;q0ÞFðμ; μa; klÞdq0; ðG8Þ

where q0a ¼ qaL ¼ kL sin θa, and the ladder contributionPðq0

a;q0Þ is defined as

Pðq0a;q0Þ ¼ sh½jq0

a − q0jz0=L�3sh½jq0a − q0jð1þ z0=LÞ�

jq0a − q0j2sh½jq0

a − q0jð1þ 2z0=LÞ�2:

ðG9Þ

In order to obtain the result (G8) we used variousapproximations that are justified in the diffusive regimeL > l only. Therefore, we must be cautious not to use thisresult in the quasiballistic regime L≲ l. We note also thatin the deep diffusive regime L ≫ l, z0 and for small angleof incidence (μa ≃ 1), the CRT

0 correlation with plane waveillumination takes the compact form

CRT0 ðΔrÞ≃ 5

16πk4L4

Zcos

�ðq0a − q0Þ:Δr

L

�q02jq0

a − q0jsh½jq0

a − q0j�dq0;

ðG10Þ

which scales as CRT0 ∝ 1=ðkLÞ4. In this regime, it is, there-

fore, much smaller than CRT2 ∝ 1=ðkLÞ2 (see Appendix E).

This explains why it is not observed experimentally in thediffusive regime.Before analyzing the result (G8) in more detail, let us

comment on the differences with the correlation calculatedwith plane wave outputs or point-source inputs. For plane-wave outputs, we find

CRT0 ðkb;kb0 Þ ¼

ZdΔrA

CRT0 ðΔrÞ ¼ 0; ðG11Þ

for all observation directions kb, kb0 . This striking resultcomes from the fact that Fðμa; μa; klÞ ¼ 0. This meansthat, for plane wave outputs, the diagram of Fig. 15(a)[Fig. 15(b)] is compensated by the one in Fig. 15(c)[Fig. 15(d)]. This result turns out to be completely differentfrom the one obtained in the configuration involvingpointlike sources and detectors, where the diagrams ofFigs. 15(a) and 15(b) do not contribute to the correlation.Let us now discuss the strong dependence of the result

(G8) on the illumination angle θa. At the origin of thisdependence is the total momentum conservation duringthe interaction with the common scatterer of Fig. 15. Theinformation carried by the illumination plane wave istransmitted to the ladder diagrams that conserve momen-tum over long distances, so that the input informationfinally reaches the sample boundaries. The same propertyoccurs in the well-known memory effect introduced in

NON-GAUSSIAN CORRELATIONS BETWEEN REFLECTED … PHYS. REV. X 8, 021041 (2018)

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Ref. [15]. We illustrate the dependence of the correlation(G8) on θa in Fig. 16. For θa ¼ 0, the rotational symmetryis preserved so that CRT

0 ðΔrÞ depends on Δr only. Itpresents a positive peak centered in Δr ¼ 0 that extendsover a distance Δr ∼ L. Beyond this distance, the corre-lation presents small negative side lobes, which are suchthat the sum rule (G11) is satisfied. When the rotationalsymmetry is broken (θa ≠ 0), the correlation becomesanisotropic. It now presents two mirror symmetries withrespect to the intersection of the incidence plane with thesample surface. The direction of this intersection definesthe horizontal axis in Fig. 16. We observe that the negativeside lobes become more pronounced along this direction. Inaddition, the amplitude of both the central peak and the sidelobes gets larger for increasing θa, in agreement with thesum rule (G11), as well as the numerical observationsdiscussed in the previous appendix.The previous analysis shows that CRT

0 reproduces thefeatures observed experimentally in the R-T correlation inthe regime L ∼ l. As was observed in the experiment, CRT

0

is long range, keeps a memory on the incidence angle,becomes anisotropic for θa ≠ 0, and presents a central peakand negative side lobes. Both the peak and the side lobesbecome more pronounced when θa is increased. That said,it should be stressed that the formula (G8) does notreproduce quantitatively the amplitude of the positivecorrelation observed in the regime L ∼ l. This is not muchof a surprise since, as we explained above, the result (G8)was obtained in the diffusive regime L > l. In addition, it

is worth mentioning that the scattering processes describedby CRT

0 and CRT2 are not the only ones that contribute in the

quasiballistic regime L≲ l. For example, it is clear that, inthe regime L ≪ l, the scattering sequences where the fourfields interact with a common scatterer play an importantrole as well. We did not discuss such contributions to thecorrelation in the main text because the regime L ≪ l isnot probed experimentally and such scattering sequencesdo not possess the dependence on the illumination anglementioned above.

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FIG. 16. Analytical prediction for the CRT0 correlation (normalized to unity) represented in Fig. 15 for two different illumination

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