Observation of Branched Flow of Light

Abstract

The evolution of coherent waves in two-dimensional systems with a random potential can give rise to an interesting phenomenon known as Branched Flow. In this process, a wave scatters from a weak random potential with correlation length longer than the wavelength, and forms focused channels that keep dividing as the wave propagates, creating a pattern resembling the branches of a tree. This phenomenon was observed for electrons [1–3], for a specific example of microwaves [4] and for ocean waves [5], but thus far never for light at optical frequencies. Furthermore, the statistical features of branched flow were predicted theoretically but were never observed in any experimental system. Here, we present the first observation of branched flow in optics, prove that the experiments represent branched flow, and study the statistical features. In our experiments, we couple an optical beam to a thin liquid soap film and observe its evolution within this thin membrane. The light experiences scattering from thickness variations in the soap film, which acts as a two-dimensional medium with a random potential. The beam propagates and scatters from the random thickness variations, forming focused branches that keep dividing, ending up in a pattern that resembles the branches of a tree, as shown in Fig.1a. To view the thickness variations directly, we construct a white light microscope illuminating the thin soap film from above, and observe the colorful map shown in Fig. 1b. The colors in Fig. 1b are true colors, and they emerge due to the reflection of white light from the thin soap film, indicating the local thickness (Fig.1c). The colors are mapped to the thickness map shown in Fig. Id, and from that to a two-dimensional map of effective refractive index landscape. The formation of the branches is the result of the appearance of caustics in the random field, following the variations in the refractive index. The theory of Branched Flow [6] gave rise to several predictions, most of which have never been studied in experiments. One example for such a prediction is the distance to the first caustic, d 0 , which was predicted by never observed. From the experiments, we extract d 0 using the Scintillation Index — the variance of the intensity of the fluctuations. It is a convenient notion, because caustics give rise to the highest intensity fluctuations, hence the scintillation index is a measure of the steepness of the caustics. The first caustic is the steepest one; hence, it corresponds to the peak in the scintillation index shown in Fig. 1e. The distance to the first caustic is ∼1 in units of ζe−2/3, correlation length and potential strength, as shown in that figure. In this regime, the phenomenon of branching of caustics perfectly matches the model of branched flow. To support our experiments, we also carry out simulations (Fig. 1e), of a coherent beam launched into a random 2D potential constructed from the actual experimental image. The beam splits and divides by the potential variations, displaying branched flow as in experiments. From the experiments, we extract additional statistical features of branched flow that were predicted but have never been observed, such as the statistics of the extreme events. In addition, we also study branched flow in curved space (where the soap membrane is curved) and the nonlinear behavior occurring when the light exerts forces on the soap bubble.