ABSTRACT As the desire to explore opaque materials is ordinarily frustrated by multiple scattering of waves, attention has focused on the transmission matrix of the wave field. This matrix

gives the fullest account of transmission and conductance and enables the control of the transmitted flux; however, it cannot address the fundamental issue of the spatial profile of

eigenchannels of the transmission matrix inside the sample. Here we obtain a universal expression for the average disposition of energy of transmission eigenchannels within random diffusive

systems in terms of auxiliary localization lengths determined by the corresponding transmission eigenvalues. The spatial profile of each eigenchannel is shown to be a solution of a

Article Open access 23 April 2021 OHM’S LAW LOST AND REGAINED: OBSERVATION AND IMPACT OF TRANSMISSION AND VELOCITY ZEROS Article Open access 05 December 2024 INTRODUCTION The transmission

matrix (TM), whose elements are field transmission coefficients through disordered media, provides a powerful basis for a complete understanding of the statistics of

transmission1,2,3,4,5,6,7,8,9,10 and of the degree of control that may be exerted over the transmitted wave by manipulating the incident wavefront11,12,13,14,15,16,17,18,19,20,21,22. Control

of the transmitted wave has been demonstrated and characterized in recent years in tight focusing11,12,13,14,15, and in enhanced or suppressed transmission16,17,18,19,20,21,22 of sound11,

elastic waves22, light12,13,14,16,18,20,21 and microwave radiation15,19. The striking similarities in the statistics and scaling of transport of classical and electron waves in disordered

systems reflect the equivalent descriptions of transmission and conductance in terms of the TM, _t_ (refs 1, 2, 3, 4, 5, 6, 7, 8, 9, 10). The electronic conductance in units of the quantum

of conductance is equivalent to the optical transmittance, which is the sum over all flux transmission coefficients, , where the _t__ba_ are elements of the TM between _N_ incident and

outgoing channels, _a_ and _b_, respectively23. These channels may be the transverse modes of empty waveguides or momentum states of ideal leads connecting to a disordered sample at a given

frequency or energy. The eigenchannels or natural channels of transmission are linear combinations of these channels, which can be obtained from the singular value decomposition of the TM, .

Here U_n_ and V_n_ are unit vectors comprising the _n_th transmission eigenchannel and _τ__n_ are the corresponding transmission eigenvalues23. The transmittance may be expressed as the sum

over the _N_ transmission eigenvalues, . Moreover, many key statistical properties of transport through random media such as the fluctuations and correlations of conductance and

transmission may be described in terms of the statistics of the transmission eigenvalues2,3,4,5,6,7,8,9,10. Notwithstanding the success of the TM in describing the statistics of transmission

through disordered systems, it cannot shed light on the disposition of energy inside the sample. Recent simulation24,25,26,27,28 and measurements22 in single realizations of random samples

suggest that the peak of the energy density profile moves towards the centre of the sample and the total energy of the corresponding eigenchannel increases as the transmission eigenvalue _τ_

increases. This is consistent with the measurements of the composite phase derivative of transmission eigenchannels, which show that the integrated energy inside the sample increases with

_τ_ (ref. 29). The possible universality of the structure of each transmission eigenchannels within scattering media remains an unexplored question, which is the subject of this work. In

contrast, the average energy distribution over all _N_ eigenchannels can be obtained by solving the diffusion equation yielding a linear falloff of energy inside a diffusive sample governed

by Fick’s law for particle diffusion. For localized waves, the energy distribution inside the sample is governed by a generalized diffusion equation and deviates dramatically from a linear

fall. This deviation can be explained in terms of a position-dependent diffusion coefficient30,31,32,33,34,35,36,37. We seek to discover a universal expression for the average over a random

ensemble of samples of the spatial distribution of the energy density averaged over the transverse dimensions for eigenchannels with specified transmission eigenvalue _τ_. This would extend

our understanding of the universality in wave propagation from the boundaries to the interior of the sample. Aside from its fundamental importance, a universal description of the scaling of

energy density in eigenchannel allows for the control of the energy density profile within the sample. This could be exploited to obtain depth profiles of random media in measurements that

are sensitive to optical absorption, emission or nonlinearity. The possibility of depositing energy well below the surface would lengthen the residence time of the emitted photons in active

random systems and thus lower the lasing threshold of amplifying diffusive media below that for traditionally pumped random lasers. The pump threshold for random lasers is traditionally high

because the residence times of emitted photons are relatively short as a consequence of the shallow penetration of the pump laser due to multiple scattering38,39,40. We show that the

average of the energy density profile within the sample of an individual eigenchannel with transmission _τ_ is closely related to the generalized diffusion equation 30–37. The energy density

of the perfectly transmitting eigenchannel with _τ_=1 is essentially the sum of a spatially uniform background equal to the energy density of the incident and outgoing wave and the product

of this background energy density and the probability density for the wave to return to the cross section at given position within the sample. For _τ_<1, we find that the energy density

can be expressed as a product of the profile of the fully transmitting eigenchannel and a function governed only by the auxiliary localization length, which was previously used to

parameterize the corresponding transmission eigenvalue1,2. RESULTS AND DISCUSSION ENERGY DENSITY PROFILES The energy density profiles averaged over 500 configurations drawn from a random

ensemble with _N_=66 and _L_/_ξ_=0.05 for transmission eigenvalues _τ=_1, 0.5, 0.1 and 0.001 are shown in linear and semilog plots in Fig. 1a,b. The localization length _ξ_ is determined

from <_T_>=_πξ_/2_L_ (refs 8, 9), where <…> indicates the average over an ensemble of random samples. For ease of presentation, we normalize the energy density _W_τ(_x_) so that

it is equal to the transmission coefficient _τ_ at the output surface, _W__τ_(_L_)=_τ_. This is accomplished by dividing the energy density by the average longitudinal component of the wave

velocity at the output surface, _v_+. In general, _W__τ_(_x_) may be associated with the sum of the forward and backward flux divided by _v_+. The normalized energy density at the incident

surface is equal to 2−_τ_. For the Lambertian probability distribution for waves transmitted through random two-dimensional (2D) media without internal reflection at the boundaries,

_v_+=_πc_/4, where _c_ is the speed of the wave in the medium. A single peak in _W__τ_(_x_) is seen to increase and shift towards the middle of the sample as _τ_ increases. The average of

the energy density profiles over all the transmission eigenchannels, , is seen in Fig. 1c to fall linearly as expected25. Simulations for a shorter and wider sample with _L_/_ξ_ =0.03 are

shown in Fig. 1d. The peak values of the energy density are lower in this sample for all values of _τ_. The details of the simulations are given in the first part of the Methods section.

PERFECTLY TRANSMITTING EIGENCHANNEL We first consider the energy density profile within the sample of the perfectly transmitting eigenchannel. This is seen in Fig. 1 to be of the form, with

_F_1(_x_) a symmetric function peaked in the middle of the sample, which vanishes at the longitudinal boundaries of the sample. This is reminiscent of the probability density for a wave to

return to a cross section at depth _x_ in an open random medium. This probability density in the case where the surface internal reflection vanishes can be obtained from the correlation

function, _Y_(_x_, _x_′), which is up to an inessential overall factor that depends upon the wave frequency and the average density of states35, evaluated at _x_=_x′_. Physically, _Y_(_x,

x′_) is the ensemble average of local energy density averaged over the cross section at _x_ due to a unit plane source located at _x′_. By using exact microscopic methods, it was

found31,32,33 that _Y_(_x, x′_) is the solution of the generalized diffusion equation, for waves propagating in open random media. We emphasize that, as in conventional diffusion, _D_(_x_)

is an intrinsic observable independent of the sources and is determined by microscopic parameters of the system such as disorder strength and incident wavelength. The explicit expression of

_D_(_x_) can be obtained by either exact microscopic theories31,32,33 or approximate self-consistent treatments30. With _D_(_x_) given explicitly by either analytical or numerical methods,

the generalized diffusion equation leads to a generalization of Fick’s law giving the average flux, , and gives _Y_(_x_, _x_′) once the appropriate boundary conditions are implemented. We

solve the generalized diffusion equation with the boundary conditions, _Y_ (_x_=0, _x_′)=_Y_(_x_=_L_, _x_′)=1/_v_+. Upon setting _x_=_x_′ in the solution, we find _Y_(_x_,

_x_)=1/_v_+(1+_Y_1(_x_, _x_)), with being the generalized probability density of return to the cross section at _x_ (see the second part of the Methods section). In the diffusive limit,

_D_(_x_) reduces to the Boltzmann diffusion coefficient _D_0 and equation (2) yields _Y_1(_x_, _x_)/_v_+=_x_(_L_−_x_)/_D_0_L_. The quadratic form of _Y_1(_x_, _x_) is the return probability

density in the diffusive limit30,31,32,33. In 2D, the diffusion coefficient is _D_0=_cℓ_/2, where _ℓ_ is the transport mean free path. With the energy density rescaled by _v_+, we obtain

_Y_1(_x,x_)=_πx_(_L_−_x_)/2_Lℓ_. Plots of _F_1(_x_) for three diffusive samples are shown in Fig. 2a. When normalized to their peak values at _L_/2 in Fig. 2b, these profiles collapse to a

single curve, 4_x_(_L−x_)/_L_2, showing that _F_1(_x_) equals _Y_1(_x_,_x_) for diffusive waves. The peak value _F_1(_L_/2) is seen in Fig. 2c to increase linearly with _L_/_ℓ_ with a slope

found from the expression above for _F_1(_x_), namely _F_1(_L_/2)=_πL_/8_ℓ_. The relationship between _F_1(_x_) and _Y_1(_x_, _x_) is further explored by comparing these functions for

localized waves for which _Y_1(_x_, _x_) depends upon _L_/_ξ_. The profiles of _F_1(_x/L_)/_F_1(_L_/2) obtained in simulations for samples with _N=_10 and in one-dimensional (1D) samples

(the first part of the Methods section) are seen in Fig. 3a to narrow as _L/ξ_ increases. _Y_1(_x,x_) is computed from equation (2) with _D_(_x_) determined from the gradient of _W_(_x_)

found in simulations as shown in Fig. 3b for the same samples as in Fig. 3a. Profiles of _Y_1(_x_, _x_)/_Y_1(_L_/2, _L_/2) and _F_1(_x_)/_F_1(_L_/2) are seen in Fig. 3a to match for each of

the random ensembles studied, thereby establishing the result, _F_1(_x_)=_Y_1(_x,x_) and _W__τ_=1(_x_) =1+_Y_1(_x_, _x_). In Supplementary Note 1 and Supplementary Figures 1 and 2 we show

that this result has a counterpart even in single random samples. FACTORIZATION OF THE ENERGY DENSITY PROFILE We find from simulations for diffusive waves that the energy density profile can

be written as a product in which _S__τ_(_x_) is independent of the value of _L_/_ξ_ and _L_/_ℓ_, as demonstrated in Fig. 4a. This factorization is proved for both diffusive and localized

waves by a non-perturbative diagrammatic technique, and the diagrammatic meaning of _S__τ_(_x_) in terms of the interactions caused by dielectric fluctuations within the medium is given in

Supplementary Note 2 and Supplementary Figures 3 and 4. In the case of the perfectly transmitting eigenchannel, _W__τ_=1(_x_)=1+_Y_1(_x_, _x_) consists of two parts. One is a spatially

uniform background energy density. The other, _Y_1(_x_, _x_′=_x_), is associated with the probability of return of waves that have reached the cross section at _x_, with the source of

uniform strength, _S__τ_=1(_x_)=1, provided by the background energy density that enters into the right-hand side of the generalized diffusion equation as the coefficient of the Dirac delta

function. For eigenchannels with _τ_<1, the strength of the source term _S__τ_(_x_) falls with increasing depth into the sample since waves do not readily penetrate into the bulk. This

modifies the generalized diffusion equation to and the boundary conditions to _Y_(_x_=0, _x_′)=_Y_(_x_=_L_, _x_′)=_S__τ_(_x_)/_v_+. Solving this equation (see the second part of the Methods

section) and setting _x_=_x_′ in the solution gives _W_τ(_x_)=_S__τ_(_x_)+_S__τ_(_x_)_Y_1(_x_, _x_), which is the same as equation (3). Thus, we establish the relation between the

factorization and the generalized diffusion equation. We have not found the analytical form for _S__τ_(_x_) for arbitrary _τ_<1. Instead, we could conjecture the form by considering the

distribution of transmission eigenvalues. Dorokhov1,2 showed that for quasi 1D scattering media the variation of average conductance with sample length _L_ (ref. 41) could be understood in

terms of the correlated scaling of the transmission eigenvalues in terms of a set of auxiliary localization lengths _ξ__n_, which gives the average transmission eigenvalue of the _n_th

eigenchannel as cosh−2(_L_/_ξ__n_). For _n_<_N_/2, varies linearly with channel index _n_ (ref. 8). For diffusive waves, this corresponds to ∼<_T_> open eigenchannels with the

average transmission eigenvalue >1/_e_,1,2,3 while for localized waves, this implies that the transmission is dominated by a single eigenchannel and the probability distribution of

_L_/_ξ_′ exhibits peaks at _L_/_ξ__n_ (refs 42, 43). Here the auxiliary localization length _ξ_′ is associated with the transmission eigenvalue _τ_=cosh−2(_L_/_ξ_′). To find an expression

for _S__τ_(_x_), we hypothesize that average intensity profile in each eigenchannel is related to _ξ_′. For diffusive waves, _S__τ_(_x_) is governed only by _L_/_ξ_′. A natural assumption is

that the analytic continuation of _S__τ_(_x_) inside the sample at the boundaries is _S__τ_(_L_)=_τ_ and _S__τ_(0)=2−_τ_ so that _W__τ_(_L_)=_τ_ and _W__τ_(0)=2−_τ_. Thus, _S__τ_(_x_) at

the boundaries must reduce to _S__τ_(_L_)=cosh−2(_L_/_ξ_′) and _S__τ_(0)=2−cosh−2(_L_/_ξ_′). Perhaps the simplest expression consistent with these conditions is . However, this expression

does not agree with the results of simulations shown in Fig. 1. Rather, agreement with simulations is only found once we incorporate an empirical function _h_(_x_/_L_) into the expression

above, giving To find the function _h_(_x_/_L_), we solve equation (5) with _S__τ_(_x_) found in simulations for a given value of _τ_. Surprisingly, we find that _h_(_x_/_L_) obtained in

this way for a single value of _τ_ in one diffusive sample gives good agreement for all _τ_ in all diffusive samples (Fig. 4a). _h(x/L)_ is plotted in Fig. 4b. _W__τ_(_x_) found from

equations (1, 3, 5) using this form for _h_(_x_/_L_) are plotted as dashed black curves in Fig. 1a,b,d. These are in excellent agreement with the profiles found in simulations for diffusive

waves. The universal form of energy density profiles of transmission eigenchannels found here apply to all classical and quantum waves in homogeneously disordered samples. The auxiliary

localization lengths for transmission eigenvalues proposed by Dorokhov are seen to determine the average energy density profiles of the transmission eigenchannels for samples of specified

length and transmission for random ensembles and so provide approximate profiles in individual samples. Aside from extending the knowledge of the average disposition of energy from the

boundary into the bulk, the expression for _W__τ_(_x_) provides a window on eigenchannel dynamics and the density of states. The integral of _W__τ_(_x_) is the delay time for the

transmission eigenvalue _τ_ and gives the contribution of the eigenchannel to the density of states of the sample29. The sum of the integrals of _W__τ_(_x_) over all channels is the density

of states of the medium. For waves such as light and sound for which it is possible to control the incident waveform, these results make it possible to tailor the distribution of excitation

within a random medium. This may allow depth profiling of random media and make it possible to deliver radiation deep into multiply scattered media. METHODS NUMERICAL SIMULATIONS The

profiles of the energy density are found from recursive Green’s function simulations44,45,46. We consider the propagation of a scalar wave in a quasi 1D strip, which is locally 2D, with

length _L_ much greater than the transverse dimension _L__t_. The precise geometry of the cross section, which is linear in our study, does not influence the results since waves are mixed in

the transverse direction within the bulk of the quasi 1D sample1,2,3,4,5,6,7,8,9,10. The results are expected to remain valid for locally 3D samples. We consider a model of the disordered

region stripped of inessential nonuniversal elements. The random sample is index matched to its surroundings with index of refraction of unity so that both the internal and external

reflections are negligible. Fluctuations _δɛ_(_x_, _y_) in the position-dependent dielectric constant, _ɛ_(_x_, _y_)=1+_δɛ_(_x_, _y_), are drawn from a rectangular distribution, the width of

which determines the strength of disorder and so the average transmittance of the sample. The wave equation is discretized on a square grid and solved using the recursive Green’s function

method. The product of the wave number in the leads _k_0 and the grid spacing is set to unity. The Green’s function _G_(_r,r′_) is calculated between grid points _r=_(0_,y_) and

_r′=_(_x′,y′_) on the incident plane _x_=0 and in the interior of the sample at depth _x′_, with _y_ being the transverse coordinate. The field transmission coefficient at _x′_ between the

incoming mode _a_ and outgoing mode _b_ is calculated by projecting the Green’s function onto the empty waveguide modes _φ__n_(_y_), , where _v__a_ is the group velocity of the empty

waveguide mode _a_ at the frequency of the wave. The product of _t_(_x′_) and V_n_ yields the field at _x′_ for different channels due to the _n_th incoming eigenchannel. In particular,

summing the square of the field amplitude at _x′_=_L_ over all _N_ channels gives the transmission coefficient _τ__n_. Averages are taken over 500 statistically equivalent samples for each

ensemble so that precise comparisons can be made with potential forms for energy density profiles. For all the values of _τ_ in all quasi 1D samples, simulations of _W__τ_(_x_) are averaged

over a subensemble of eigenchannels with transmission between 0.98 _τ_ and 1.02 _τ_. We also carry out scattering matrix simulations in a 1D disordered system to explore the profiles of

_W__τ_=1(_x_) and _D_(_x_). An electromagnetic plane wave is normally incident on a layered system with fixed number of layers _L_ with alternating indices of refraction between a fixed

value _n_ and 1. The layer thickness is drawn from a distribution that is much greater than the wavelength. The mean free path in this system is equal to the localization length, _ℓ_=_ξ_

(ref. 47). _W_τ=1(_x_) is computed from the average of the energy density over a subset of samples with transmission _τ_ ranging from 0.995 to 1. SOLUTION OF THE GENERALIZED DIFFUSION

EQUATION Let us multiply both sides of equation (4) by _D_(_x_). With the change of variable, which gives d_z_=d_x_/_D_(_x_), we rewrite equation (4) as The boundary condition is

correspondingly modified such that _Y_(_z_(0), _z_′)=_S__τ_(_z_′)/_v_+. This is the normal diffusion equation and is readily solved to give We note that since _D_(_x_)>0, the function

_z_(_x_) is strictly monotonic and is therefore invertible. Taking this into account and substituting the identity, , into the expression of _Y_(_z_, _z_′), we may rewrite _Y_(_z_, _z_′) as

Setting _x_=_x_′, we find with the generalized return probability density given by equation (2). ADDITIONAL INFORMATION HOW TO CITE THIS ARTICLE: Davy, M. _et al_. Universal structure of

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references ACKNOWLEDGEMENTS We thank Pier Mello and Boris Shapiro for stimulating discussions. The research was supported by the National Science Foundation (DMR-1207446), by the National

Science Foundation of China (No. 11174174) and by the Tsinghua University ISRP. AUTHOR INFORMATION Author notes * Jongchul Park Present address: Present address: Chiral Photonics, Inc. 26

Chapin Road, Pine Brook, New Jersey 07058, USA, AUTHORS AND AFFILIATIONS * Institut d’Electronique et de Télécommunications de Rennes, University of Rennes 1, Rennes, 35042, France Matthieu

Davy * Department of Physics, Queens College of the City University of New York, Flushing, 11367, New York, USA Zhou Shi, Jongchul Park & Azriel Z. Genack * The Graduate Center, CUNY,

365 Fifth Avenue, New York, 10016, New York, USA Zhou Shi, Jongchul Park & Azriel Z. Genack * Institute for Advanced Study, Tsinghua University, Beijing, 100084, China Chushun Tian

Authors * Matthieu Davy View author publications You can also search for this author inPubMed Google Scholar * Zhou Shi View author publications You can also search for this author inPubMed 

Google Scholar * Jongchul Park View author publications You can also search for this author inPubMed Google Scholar * Chushun Tian View author publications You can also search for this

author inPubMed Google Scholar * Azriel Z. Genack View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS A.Z.G. initiated the research and

brainstormed with M.D. and Z.S. regarding the forms of the intensity profiles, M.D. and Z. S. carried out the numerical simulations for multichannel samples, M.D. and J. P. carried out the

numerical simulations for 1D samples, M.D. and Z.S. analysed the data, C.T. and M.D. suggested the link between _W__τ_(_x_) and the generalized diffusion equation, which was demonstrated by

C.T., A.Z.G., M.D., C.T. and Z.S. wrote the manuscript. CORRESPONDING AUTHOR Correspondence to Azriel Z. Genack. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing

financial interests. SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION Supplementary Figures 1-4, Supplementary Notes 1-2 and Supplementary References (PDF 1101 kb) RIGHTS AND PERMISSIONS

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons

license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to

reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Davy, M., Shi, Z.,

Park, J. _et al._ Universal structure of transmission eigenchannels inside opaque media. _Nat Commun_ 6, 6893 (2015). https://doi.org/10.1038/ncomms7893 Download citation * Received: 24

November 2014 * Accepted: 10 March 2015 * Published: 20 April 2015 * DOI: https://doi.org/10.1038/ncomms7893 SHARE THIS ARTICLE Anyone you share the following link with will be able to read

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