Self-Induced Diffusion in Disordered Nonlinear Photonic Media Yonatan Sharabi, 1 Hanan Herzig Sheinfux, 1 Yoav Sagi, 1 Gadi Eisenstein, 2 and Mordechai Segev 1 1 Solid State Institute, Technion—Israel Institute of Technology, Haifa 32000, Israel and Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel 2 Department of Electric Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel (Received 24 May 2018; published 7 December 2018) We find that waves propagating in a 1D medium that is homogeneous in its linear properties but spatially disordered in its nonlinear coefficients undergo diffusive transport, instead of being Anderson localized as always occurs for linear disordered media. Specifically, electromagnetic waves in a multilayer structure with random nonlinear coefficients exhibit diffusion with features fundamentally different from the traditional diffusion in linear noninteracting systems. This unique transport, which stems from the nonlinear interaction between the waves and the disordered medium, displays anomalous statistical behavior where the fields in multiple different realizations converge to the same intensity value as they penetrate deeper into the medium. DOI: 10.1103/PhysRevLett.121.233901 Anderson localization is a universal phenomenon, occur- ring in all linear disordered wave systems. It was proposed to describe the absence of diffusion of electrons in the presence of disorder. Rather than exhibiting diffusion, as expected from particles in a scattering medium, the electron wave becomes localized with an exponentially decaying wave function [1]. Anderson localization is now known to occur in a variety of systems, including electromagnetic (EM) waves [2–7], acoustic waves [8], water waves [9], and ultracold atoms [10]. Among these, localization of light became a popular experimental platform because photons have long coherence times, and unlike electrons, they do not interact with one another [11]. Importantly, disorder in one- and two- dimensional settings always leads to localization, whereas in three dimensions the transport can also be diffusive [12–15]. One of the assumptions Anderson made was that the system is linear; i.e., the waves in the random potential do not interact. For transverse localization of optical waves in dielectrics [5,6,16], as well as for matter waves in the mean- field regime, interactions are manifested as a nonlinear potential term in the Schrödinger-type equation, which is mathematically equivalent to self-focusing of paraxial optical beams [11]. Indeed, localization in the presence of nonlinearities has been observed for optical beams [5,6,11], although its asymptotic behavior is still not fully resolved [17–20]. Despite this extensive research, all studies on waves in disordered systems investigated set- tings in which the linear potential is disordered, and the nonlinearity is an additional effect. To our knowledge, there have been no studies about a system that is disordered only in its nonlinear (NL) properties, e.g., a system that has a spatially random Kerr coefficient, but is homogeneous in its linear properties. Here, we study a system that is linearly homogeneous but contains disorder in its NL coefficients and find that the nonlinearly induced disorder gives rise to a unique type of diffusion with unusual statistics and characteristic wave functions. Namely, waves decay in a fundamentally differ- ent fashion than the exponential decay characteristic of localization and have completely different statistics. We study a 1D multilayer dielectric system, where each layer has a nonlinear coefficient drawn randomly, with zero mean. Here, the propagating field induces the disorder in the refractive index, and consequently, the disorder level becomes dependent on the (local) intensity. We show that the EM field decays as it penetrates into the disordered medium, but exhibits diffusivelike behavior instead of becoming localized as expected for 1D linear disordered systems. We analyze the statistics of an ensemble of disorder realizations and find that the transmission in all realizations tends to converge to a single value, unlike linear disordered systems which display an increasing variance in transmission [21]. Finally, we examine the case of a saturable NL material and find that it exhibits a distinct transition from exponential decay to diffusivelike transport. Our system is a 1D multilayer structure [Fig. 1(a)], where the medium has the same linear refractive index n 0 every- where, but a spatially random n 2 Kerr coefficient drawn from a uniform distribution, n 2 ∈ ½-Δ; Δ. The refractive index in the mth layer is nðzÞ¼ n 0 þ n 2 ðmÞI ðzÞ, with I ðzÞ as the EM field intensity. While all layers are of equal width, the system is not periodic because the refractive index varies in a random fashion (with a zero mean) in the presence of light, due to the nonlinearity. The zero mean of the NL coefficient ensures that the effects arise from the nonlinearly induced disorder and not from a change of the average refractive index. The electric field of the light propagating in a linear 1D multilayer can be found through the transfer matrix method PHYSICAL REVIEW LETTERS 121, 233901 (2018) 0031-9007=18=121(23)=233901(5) 233901-1 © 2018 American Physical Society