# Shaping the branched flow of light through disordered media

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 21, 2019 (received for review March 26, 2019)

## Significance

In the presence of a weakly fluctuating potential landscape, waves exhibit pronounced enhancements along so-called “branches.” The formation of these beautiful branches is a universal phenomenon occurring on vastly different length scales and for many types of waves, such as for tsunami waves traveling through the rough ocean sea bed or for light beams propagating through a soap film. Here, we show that wavefront shaping techniques can be used to control this phenomenon and to steer waves through the potential landscape along a single branch rather than along many of them in parallel (as has always been observed so far). Our numerical results show that this feat should be directly implementable with current-day technology.

## Abstract

Electronic matter waves traveling through the weak and smoothly varying disorder potential of a semiconductor show a characteristic branching behavior instead of a smooth spreading of flow. By transferring this phenomenon to optics, we demonstrate numerically how the branched flow of light can be controlled to propagate along a single branch rather than along many of them at the same time. Our method is based on shaping the incoming wavefront and only requires partial knowledge of the system’s transmission matrix. We show that the light flowing along a single branch has a broadband frequency stability such that one can even steer pulses along selected branches—a prospect with many interesting possibilities for wave control in disordered environments.

When waves propagate through a disorder landscape that is sufficiently weak and spatially correlated, they form branched transport channels in which the waves’ intensity is strongly enhanced. This phenomenon of “branched flow” was first discovered for electrons gliding through semiconductor heterostructures (1). Instead of an isotropic spreading into all possible directions, the electron density injected through a quantum point contact was observed to form esthetically very appealing branch patterns. This intriguing behavior can be attributed to ripples in the background potential that are always present in such structures (1, 2), which act like an array of imperfect lenses, giving rise to caustics (3) and thereby, to distinct intensity enhancements along branches (4⇓–6). Although first discovered as a nanoscale wave effect, branched flow was soon understood to occur on a wide range of length scales up to the formation of hot spots in tsunami waves as a result of the propagation through the rough ocean sea bed (7⇓⇓⇓⇓⇓–13).

Whereas a number of previous studies have already focused on the statistics of this phenomenon (3, 10, 14) and on its origins (2, 3, 15⇓⇓–18), the question of how branched flow can be controlled and thereby, put to use for steering waves through a complex medium has not been addressed so far. This is probably due to the fact that the possibilities to shape and manipulate electrons or ocean waves are, indeed, very limited. In other words, for the experiments where branched flow was observed so far, the incoming wavefront as well as the potential that the wave explores were considered as predetermined and immutable. These limitations are currently about to be overcome in a new generation of experiments, where coherent laser light was observed to exhibit branching when propagating through very thin disordered materials, such as the surface layer of a soap bubble (19). Specifically, we expect that the transfer of branched flow to the optical domain will open up the whole arsenal of photonics to shape the wavefront of such branched light beams (20, 21).

A particularly exciting question that we will explore here from a theoretical point of view will be whether optical wavefront shaping tools, like spatial light modulators (SLMs), can be used to manipulate an incoming light beam in such a way that it follows only a single branch through a disorder landscape rather than many of them in parallel. The protocol that we will introduce based on our analysis will open up ways of sneaking a beam of light through a disordered medium while maintaining its focus throughout the entire propagation distance—like a highway for light through a scattering medium. More generally, we expect our approach to be useful in a variety of different contexts, where steering waves through a complex environment to a predetermined target is a key goal, like in wireless communication (22), adaptive optics (23), underwater acoustics (24), wave focusing (25⇓⇓–28), and biomedical imaging (29, 30) as well as for wave control in disordered systems at large (20, 21, 31, 32).

## Results

### System.

The system that we consider is shown in Fig. 1*A* and consists of a rectangular scattering region of length L and width W that is attached to two straight semiinfinite waveguides (leads) of the same width W on the left and right (only the left lead is shown). In transverse direction, hard-wall boundary conditions are applied (i.e., the wavefunction is zero at these boundaries). In all of the calculations reported below, we choose the number of propagating open lead modes to be *A*). For simplicity, we set

In analogy to the first observation of branched flow, where electrons were injected through a constriction (quantum point contact) into a high-mobility electron gas (1), we also include such a constriction in the form of an aperture of width *A*. This correlated refractive index *SI Appendix*).

The scalar scattering problem in this two-dimensional setup is described by the two-dimensional Helmholtz equation*Methods* has more details) to efficiently evaluate the scattering states *Methods*.)

To observe the branched flow of light, we inject the different lead modes from the left into the constriction and superimpose the corresponding wave intensities that they give rise to. In the superposition, we consider only the first 100 lead modes (of 200) to avoid high-angle scattering and to ensure a high visibility of the individual branches. The branched structure in the propagation of waves through our setup is clearly visible in Fig. 1*A*.

The challenge that we rise to in the next step is to address these branches individually through a suitable coherent superposition of incoming modes in the left lead. The methods that we choose for this purpose involve only the transmission matrix t from Eq. **2**, which is available in optics through interferometric measurements involving an SLM (21, 35). As the branched flow in our system naturally leads to a concentration of intensity at certain spots at the output, we find here that the knowledge of the transmission matrix t for modes concentrated around these spots is sufficient for a clean separation of branches. In other words, we may restrict ourselves to those regions in space at the output where the branches arrive. These regions are determined from the intensity profile at the output facet of our system at *B*). Seven intensity maxima corresponding to the arrival of different branches are clearly visible in Fig. 1*B* and highlighted in light blue/magenta. For each intensity maximum, we manually set lower and upper boundaries, which are indicated by vertical lines in Fig. 1*B*, and define a reduced transmission matrix *SI Appendix* has more details on the transmission matrix in a spatial basis).

### First Approach: Transmission Eigenstates.

Our first approach to achieve clean branch separation is to use a singular value decomposition of *B*, *E*–*I*, *P*, and *T*), we find that a clean branch separation is, indeed, possible for a number of cases. We also find, however, that several among the highly transmitting eigenchannels follow two different branches in parallel instead of only one. Fig. 2*O* shows an example of such a state, where one can clearly see a mixing of one branch propagating directly into the selected region (marked with blue bars at the output) with another branch bouncing off the lower boundary. Demanding high transmission into a desired region by choosing high-transmission eigenchannels is thus clearly not enough to guarantee clean branch separation, since high transmission can also be obtained by propagating along multiple branches at once. In a possible optical experiment, such a mixing can be expected to be even more prevalent than in our numerical example, simply because optical implementations can typically involve a large number of branches (19).

### Second Approach: Time-Delay Eigenstates with Large Transmission.

To also be able to address such mixed branches individually, we now introduce a more efficient method. Specifically, our aim is to set up an approach in terms of the scattering matrix S and the Wigner–Smith time-delay operator (36⇓–38) derived from it:*O*) can be distinguished by their different time delays (as determined by the different branch lengths). To be specific, we only work with those N transmission eigenvalues **4**,**6** involves a quasi-inverse “**6** are complex (in contrast to the real eigenvalues of the Wigner–Smith time-delay operator Q in Eq. **4**). The imaginary parts of the complex eigenvalues are, however, very small, and the real parts can still be used as a good measure for the physical delay times (44).

To put this method directly to the test, we turn our attention to the state shown in Fig. 2*O* featuring a mixture of two branches with different path lengths and correspondingly, different time delays. A singular value decomposition of **6** and indeed, find among its eigenstates the desired wave fields that follow the two involved branches individually (Fig. 2 *R* and *S*).

Restricting the construction of time-delay eigenstates to the subspace of high transmission thus yields already very good results. Using this method, we, however, also observed a few time-delay eigenstates that mix two different branches as, for example, shown in Fig. 2*J*. These two branches, however, turn out to be individually addressable through those transmission eigenstates *G* and *I*) [the time delay of a transmission eigenstate

### Combined Method.

One may thus also decide to turn the above strategy on its head and look for transmission eigenstates in the subspace of short time delays. Since neither one of these opposite strategies seems to have an a priori advantage, we now combine them with each other in a synergistic way to improve our results even farther. (*i*) We evaluate all eigenstates *B*. (*ii*) We select those states that are identical in both eigenstate sets, since they turn out to be individual branch excitations in all of the observed cases. To do this, we project each eigenvector *iii*) In a last step, we deal with those eigenstates that consist of more than one contribution from the respective other eigenstate set (i.e., that have more than one nonzero element in the corresponding row/column of m). Our task here is to select those states that consist of only single branches and to discard those states that propagate along more than one branch at once. Since, however, the coefficients *Methods* has details).

Following the above three steps (*i*–*iii*), which notably rely only on the experimentally accessible transmission matrices *A–I*, *K–N*, and *P–U*) that stay collimated throughout the entire scattering region and that feature an average transmittance of over *C*, *F*, *H*, *L*, *N*, and *R*) (40).

## Discussion and Summary

### Injection into Empty Cavity.

To underscore the nontrivial nature of these collimated branch states that we identify here, we inject several of the states shown in Fig. 2 into a clean waveguide without any disorder. The results are displayed in Fig. 3, showing that these states feature a considerably reduced collimation compared with the case including the disorder (Fig. 3, *Insets*). This observation demonstrates that the states that we identify here do not just rely on a trivial injection with a narrow angular distribution at the input and that the disorder plays a crucial role for the states’ collimation.

### Pulse Propagation.

In the last part of this study, we also demonstrate explicitly that our collimated single-branch states can be sufficiently stable in frequency to allow for the transmission of pulses along a branch. Consider here, as an example, the time-delay eigenstate shown in Fig. 4*A* that propagates along a certain branch. Taking a superposition of this branch state at different frequencies to form a Gaussian wave packet, we obtain a pulse propagating along the selected branch as shown in Fig. 4 *B*–*D* at three different time steps (

### Summary.

In summary, this work demonstrates how to control the flow of waves through a correlated and weak disorder potential landscape. Such systems give rise to branches along which incoming waves travel through the disorder. We introduce a method that allows us to inject waves in such a way that almost all of the flow travels along a single branch alone. This nontrivial finding can even be extended to the temporal domain as we show by creating pulses that remain on a single branch throughout the entire transmission process. Implementing such concepts in optics requires only a small subpart of the transmission matrix and is thus within reach of present-day technology. We expect our work to be generalizable from scalar to vector waves and from two to three dimensions, where it may give rise to interesting applications in communication and imaging technology.

## Methods

### Numerical Method.

To solve the Helmholtz Eq. **1** numerically, we discretize the central scattering region on a Cartesian grid with a grid spacing

### Scattering Formalism.

Our scattering system is a waveguide as described. The semiinfinite asymptotic regions (also called leads) feature a constant refractive index of **1** in the asymptotic regions yields straightforward solutions—the so-called lead mode basis functions*A* in orange. The lead modes in Eq. **7** are orthogonal and complete, and they can thus be used as a basis to decompose any arbitrary wave such that the wave can be described by a corresponding coefficient vector **6**, the coefficient vector that corresponds to an eigenstate of this operator reads

### Spatial and Angular Profile of Eigenstates.

To find individual branch excitations among all eigenstates

To estimate the angular distribution of an eigenstate at the aperture, we work with the Hermitian operator

Fig. 5 *A* and *B* display the spatial and angular components of the transmission eigenstate shown in Fig. 2*O* (red) and the time-delay eigenstate in Fig. 2*R* (blue). We see that the spatial profile of the transmission state is broader and that it features more angular components than the time-delay state. We can, therefore, conclude that the transmission state is more likely to address multiple branches, whereas the time-delay state addresses only one single branch, which is confirmed by the wave plots shown Fig. 2 *O* and *R*. In Fig. 5 *C* and *D*, we plot the same distributions for the time-delay state shown in Fig. 2*J* and the transmission state shown in Fig. 2*I*. From Fig. 5*C*, we deduce that the time-delay state consists of more than one branch due to the larger spatial distribution, which is confirmed by the wave plots. We successfully applied this procedure to all eigenstates

## Acknowledgments

We thank Miguel A. Bandres for fruitful discussions on his recent experiments (19) and Florian Libisch for his help with the numerical code. We acknowledge support by European Commission Project Non-Hermitian Quantum Wave Engineering (NHQWAVE, MSCA-RISE 691209) and by the Austrian Science Fund (FWF) through the Project SFB-NextLite F49-P10. A.B. is a recipient of a DOC Fellowship of the Austrian Academy of Sciences at the Institute of Theoretical Physics of Vienna University of Technology. The computational results presented have been achieved using the Vienna Scientific Cluster.

## Footnotes

↵

^{1}A.B. and A.G. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: abrandstotter{at}hotmail.com or stefan.rotter{at}tuwien.ac.at.

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

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1905217116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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