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# Invariance property of wave scattering through disordered media

Edited by Steven M. Girvin, Yale University, New Haven, CT, and approved October 28, 2014 (received for review September 15, 2014)

## Significance

The diffusion of particles and waves through disordered media encompasses a large variety of phenomena, from the motion of insects to the scattering of electrons or light in complex environments. One of the core features of diffusive transport is that the mean length of trajectories traversing a system depends only on the size of the system and of its boundary, which are both independent of the microscopic structure of the underlying medium. Here we show, based on insights from wave-scattering theory, that this fundamental invariance property can be significantly extended beyond the diffusive random walk picture. Our result not only provides an interesting link between all the diverse fields in which wave scattering plays a role but also holds promise for a number of practical applications.

## Abstract

A fundamental insight in the theory of diffusive random walks is that the mean length of trajectories traversing a finite open system is independent of the details of the diffusion process. Instead, the mean trajectory length depends only on the system's boundary geometry and is thus unaffected by the value of the mean free path. Here we show that this result is rooted on a much deeper level than that of a random walk, which allows us to extend the reach of this universal invariance property beyond the diffusion approximation. Specifically, we demonstrate that an equivalent invariance relation also holds for the scattering of waves in resonant structures as well as in ballistic, chaotic or in Anderson localized systems. Our work unifies a number of specific observations made in quite diverse fields of science ranging from the movement of ants to nuclear scattering theory. Potential experimental realizations using light fields in disordered media are discussed.

In the biological sciences it has been appreciated for some time now that the movement of certain insects (such as ants) on a planar surface can be modeled as a diffusive random walk with a given constant speed *v* (1⇓–3). Using this connection, Blanco and Fournier (4) proved that the time that these insects spend on average inside a given domain of area *A* and with an external boundary *C* is independent of the parameters entering the random walk such as, for example, the transport mean free path (MFP) *t* between the moments when an insect enters the domain and when it first exits it again is given by the simple relation *V* is the volume and Σ is the external surface of a given domain. Extensions of this result exist for trajectories beginning inside the domain (5) or for the calculation of averaged residence times inside subdomains (6). As a generalization of the mean-chord-length theorem (7) for straight-line trajectories with an infinite MFP, this fundamental theorem has numerous applications, for instance in the context of food foraging (8) and for the reaction rates in chemistry (9).

The surprising element of this result can be well appreciated when applied to the physical sciences and, in particular, to the transport of light or of other types of waves in scattering media. In that context it is well known that the relevant observable quantities all do depend on *L* scales with

To describe wave transport in a disordered scattering medium without solving the full wave equation numerically is a challenging task that can be approached from many different angles (10, 14, 15). As the first step, we will consider the radiative transfer equation (RTE), which describes the transport of an averaged radiation field through a disordered medium in the limit *g*, which measures the degree of forward scattering at a scattering event, **u**′ stands for integration over the solid angle. The specific intensity **r**, along direction **u**, at frequency *δ* and at time *τ*. Eq. **1** is Fourier-transformed with respect to *τ* (with Ω being the conjugate Fourier variable). The expressions for the extinction and scattering MFP are given by *δ* or the linewidth Γ. Note that the condition

On this basis we can evaluate the average time spent by light trajectories inside the medium by calculating the weighted temporal average *τ*, Σ is the medium boundary, and **n** the outward normal. In frequency domain, this expression can be cast in the following compact form (*Supporting Information*):**1**. In the numerical simulation, we consider a 3D slab geometry of length *L* with on-resonance optical thickness *A*). This corresponds to a situation where the incident specific intensity does not depend on the point and direction of incidence (Lambert’s cosine law is satisfied). Using a Monte-Carlo scheme (18), we solved Eq. **1** without approximation and obtained the results plotted in Fig. 2. By tuning the linewidth Γ of the scatterers, we can either simulate a nonresonant medium in which the intensity spends most of the time between the scatterers (

In the nonresonant case (Fig. 2*A*), we recover the results by Blanco and Fournier (4) and find an average time *δ* that determines the scattering properties in the present RTE calculation). Moreover, we clearly see that the times associated to the reflected and transmitted parts of the outgoing flux, which can be computed separately, strongly depend on *δ*, showing that the invariance of the average time *B* and *C*. Also note that by varying the detuning *δ* from 0 to 2 in Fig. 2*A* we effectively perform a cross-over from the diffusive to the single scattering regime. In the latter (

In case of a resonant medium (Fig. 2*B*) the situation is substantially different. The average time *δ*, and therefore on the scattering properties of the medium. Because this result clearly falls outside the scope of the invariance relation derived by Blanco and Fournier (4), the question arises whether a new quantity can be defined that remains invariant even in the limit of strongly dispersive scatterers. To address this issue we rewrite the average time **2** as the ratio of the total energy *U* stored in the system and the outgoing flux *Supporting Information*, this relation measures *U* takes to flow out of the medium with flux *U* in terms of the specific intensity *V* and Σ are the volume and the external boundary of the medium, and **n** is the outward normal. For a uniform and isotropic illumination on the surface (as assumed here), the specific intensity is uniform, isotropic, and independent on detuning inside the medium (a particular case of such a situation is blackbody radiation) (14). As a result, Eq. **4** can be drastically simplified into *L* gives *Supporting Information*):*C* as obtained by renormalizing the numerical results for *B* with the analytical expression (Eq. **5**) of the transport velocity *δ*, with a constant value

Whereas the above extension of the RTE allowed us to find a new invariant quantity for the case of scattering in a disordered medium with resonant scatterers, the ansatz of the RTE itself is intrinsically restricted to the limit *λ* is comparable to or even larger than the mean free path *V* of the system. Correspondingly, the volume *V* and the surface Σ appearing in the invariance relation *ξ*.

To explore this question in detail we will now work with the full wave equation in two dimensions which, for stationary light scattering, is given in terms of the Helmholtz equation:*k* and *ω* interchangeably. In the situations we study here, the disorder scattering is induced by the spatial variations of the static refractive index *k*) inside a given spatial region one can conveniently use the so-called Wigner–Smith time-delay operator^{†}:*ω*-dependent scattering matrix *S*, evaluated at the external boundary *C* of the considered region, contains all of the complex transmission and reflection amplitudes that connect in- and outgoing waves in a suitable mode basis. To obtain also here the average time associated with wave scattering we take the trace of *Q* and divide by the number

To evaluate the average time *Lower*). Accordingly, the correct number of scattering channels **6** on a finite-difference grid, using the advanced modular recursive Green’s function method (23, 24). In Fig. 3 we display our numerical results for different degrees of disorder. In Fig. 3*A* we show the results obtained for an empty scattering region, corresponding to the ballistic transport regime. In Fig. 3*B*, the case with altogether 13 scatterers is shown, for which already a strong reduction of transmission is observed. The distribution of the transmission eigenvalues *Supporting Information*). Finally, in Fig. 3*C* we increased the degree of disorder even more (placing altogether 211 scatterers) so as to enter the regime of Anderson localization. Here the distribution of transmission eigenvalues agrees very well with the predictions for the case when Anderson localization suppresses all but a single transmission eigenchannel (*Supporting Information*) (25, 26). To make all three cases easily comparable with each other, the different geometries all have the same scattering area *A*, which for ballistic scattering is the entire rectangular region between the leads, whereas for the other two cases the area occupied by the impenetrable scatterers is not part of *A*.

Based on the above identification of the different transport regimes that our model system can be in, we investigate now the corresponding results for the average time *A*) we see that the average time, plotted as a function of the incoming wavenumber *k*, shows pronounced periodic enhancements around the random walk prediction by Blanco and Fournier (4), *d*. To understand why these mode openings cause an increase in the scattering dwell time we resort to a fundamental connection between the average dwell time *A*. This demonstration also allows us to show that the time, averaged over an interval of *k* that is larger than the distance between successive mode openings, converges exactly to the prediction by Blanco and Fournier (4). Quite remarkably, we find in this sense that the estimate from the mean-chord-length theorem and, correspondingly, the random walk prediction also holds, on average, for ballistic wave scattering in a system without any disorder.

Moving next to the disordered system in Fig. 3*B* we see that the presence of the disorder strongly reduces the above mode-induced fluctuations, leaving the frequency-average value of time unchanged. To explain this result, the DOS clearly needs to be estimated differently here than in the ballistic case of uncoupled waveguide modes. Also, because the disorder leads to system- and frequency-specific fluctuations of the DOS, we are looking here for an estimate for the ensemble and frequency-averaged DOS. To obtain this quantity, we invoke a result first put forward by Weyl in 1911 (38), who estimated that the average DOS in the asymptotic limit of *ω*-dependent number of incoming channels is given as an integer-valued step-function

Does this invariance of the average scattering time also persist in the strongly scattering limit, when Anderson localization sets in? Our numerical results shown for this case in Fig. 3*C* display a small but apparently systematic frequency dependence of the average time *ω*. Because the numerical calculations are very challenging and the frequency derivative appearing in Eq. **7** can reach very large values for highly localized scattering states, we first tested the accuracy of our simulations by evaluating **3**, the expression for the dwell time in the case of the Helmholtz Eq. **6** is given by *m*-th scattering channel and *C*. To explain this robust deviation from the result by Blanco and Fournier (4) we thus have a more careful look on the Weyl estimate which, in addition to the leading-order term that we used above, also features a next-order correction proposed by Weyl (39, 40), *A* but also the internal boundary of the scattering region *B*, which is notably different from the external boundary *C* through which waves can scatter in and out. The internal boundary *B* in the case of our waveguide system under study is given by *C* were approximated with Neumann boundary conditions, which contribute with the opposite sign as the Dirichlet boundary conditions on the surface of the waveguide and of the scatterers. In systems with a small boundary-to-area ratio this next-order correction of the Weyl law is negligible. Because, however, the number of scatterers that we have placed inside the system (from 0 in the ballistic case, to 13 in the chaotic case, to 211 in the localized case) increases this ratio, the additional boundary term in the Weyl law may become important here. To check this explicity, we reevaluate the expression for the average dwell time *C*) yields excellent agreement and indicates that the observed deviation from the prediction by Blanco and Fournier (4) stems from the comparatively large boundary of the many small scatterers that we placed inside the scattering region. We emphasize at this point that this correction to the Blanco and Fournier estimate only contains the boundary values *B* and *C* as additional input and remains entirely independent of any quantities that characterize the scattering process itself, such as *ξ*. This insight is of considerable importance, because it means that Eq. **8** defines a new invariant quantity that is independent of the scattering regime we are in and thus accurately matches our numerical results for the average time in the ballistic, chaotic, and localized limit. This invariant quantity for waves deviates from the prediction by Blanco and Fournier (4) only through an additional term originating in the fact that waves feel the boundary of a scattering region already when being close to it on a scale comparable with the wavelength. We speculate that additional wave corrections to the result by Blanco and Fournier may arise when waves have access to a larger scattering area *A* than classical particles through the process of tunneling.

In summary, we have derived a universal invariance property for wave transport through disordered media. The invariance of the averaged path length or averaged time spent by a wave in an open finite medium has been established based on scattering theory. In the appropriate limit of diffusive and nonresonant media, the random walk picture is recovered, and the result coincides with the expression of the averaged path length initially established by Blanco and Fournier (4). Our work confers to this invariance property a degree of universality that extends its implications far beyond applications of random walk theory. This extension to waves opens up new possible applications in optics, acoustics, seismology, or radiofrequency technologies, where propagation in complex media is the subject of intense research (41). Indeed, in the context of wave transport through disordered media most spatial or temporal observables scale with **8**, to the internal surface *B* of scatterers embedded in a scattering medium through a time-resolved transport experiment. Such an approach could go as far as to measure the fractal dimension of the scatterer surface by linking our results with the Berry–Weyl conjecture (44, 45).

An extension of our findings to media with gain and loss (46, 47) should also be of interest, both from a theoretical and an applied standpoint. Our study should also be very relevant to the field of wave control, which has recently emerged as a powerful paradigm for light manipulation and delivery in complex media (48), showing for instance that suitably shaped wavefronts can deliver light at a specific time and position (49⇓–51). Finally, let us point out that although we only studied here 3D slab and 2D waveguide geometries with uncorrelated disorder, the invariance property, thanks to its connection to the DOS, is very general and should apply to a wide range of geometries and excitation strategies, as well as to nonuniform scattering properties, biological tissues, and correlated disorder, from partially ordered to entirely ordered systems such as Levy glasses or photonic crystals (52, 53). An experimental demonstration of the discussed invariance property should be within reach, in particular in optics, where time-resolved techniques and sensitive detectors are available.

## Acknowledgments

We thank Stéphane Hallegatte for pointing out the result of Blanco and Fournier and for an initial exchange of ideas. We also thank Jacopo Bertolotti, Florian Libisch, Romolo Savo, and Jolanda Schwarz for fruitful discussions as well as the administration of the Vienna Scientific Cluster for granting us access to computational resources. This work was supported by the Laboratory of Excellence ANR-10-LABX-24 Waves and Imaging from Fundamentals to Innovation (Labex WIFI) within the French Program “Investments for the Future” under reference ANR-10-IDEX-0001-02 PSL*. P.A., A.H., and S.R. are supported by the Austrian Science Fund (FWF) through Projects NextLite F49-10 and I 1142-N27 (GePartWave). S.G. is funded by European Research Council Grant 278025.

## Footnotes

↵

^{1}R.P. and P.A. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: stefan.rotter{at}tuwien.ac.at.

Author contributions: R.P., P.A., S.G., R.C., and S.R. designed research; R.P., P.A., A.H., and S.R. performed research; R.P., P.A., R.C., and S.R. analyzed data; and R.P., P.A., S.G., R.C., and S.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵

^{†}One can show that the quantity measured by the Wigner–Smith time-delay operator is equal to the dwell time (Eq.**3**) if the frequency dependence of the coupling between the scattering region and its surrounding becomes negligible (20). This is the case in the systems considered here.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1417725111/-/DCSupplemental.

Freely available online through the PNAS open access option.

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