PHYSICAL REVIEW A 99, 033838 (2019) Controlling the intensity statistics of speckle patterns: From normal to subthermal or superthermal distributions Samuel B. Alves, 1 Hugo L. D. de S. Cavalcante, 2 Gilson F. de Oliveira Jr., 3 Thierry Passerat de Silans, 4 Itamar Vidal, 5 Martine Chevrollier, 6 , * and Marcos Oriá 6 1 Coordenação de Física, Instituto Federal do Sertão Pernambucano, Campus Salgueiro, BR232-Km504, 56000-000 Salgueiro, Pernambuco, Brazil 2 Departamento de Informática, Universidade Federal da Paraíba, Avenida dos Escoteiros s/n, Mangabeira VII, 58055-000 João Pessoa, Paraiba, Brazil 3 Instituto de Formação de Educadores, UFCA, 63260-000 Brejo Santo, Ceará, Brazil 4 Departamento de Física/CCEN, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900 João Pessoa, Paraiba, Brazil 5 LOQEF, 57039-739 Maceió Alagoas, Brazil 6 UACSA, Universidade Federal Rural de Pernambuco, Cabo de Santo Agostinho, 54518-430 Pernambuco, Brazil (Received 19 October 2018; published 19 March 2019) We study the propagation of speckle patterns in a tunable nonlinear medium. Starting from a spatial pattern of photons with a normal distribution (speckle field), we characterize the evolution of the radiation to either subthermal or superthermal distributions and demonstrate the control of this evolution through a single parameter. Our experimental control of the local degree of coherence of the output photons is well described by numerical simulations that consider both dispersive and absorptive nonlinear effects. Particularly, we analyze the role of the nonlinear absorption that either enhances or competes with the dispersive effects in the evolution of speckle fields. In some realizations, an asymptotic power law is observed in the photon spatial distribution. DOI: 10.1103/PhysRevA.99.033838 I. INTRODUCTION Many natural and artificial systems exhibit deviations from normal (Gaussian) statistics. Non-normal statistics have been observed, for instance, in biological [1,2], ecological [3], geological [4], behavioral [5,6], and economic [7] systems. Therefore, an increasing effort has been made to explain and eventually forecast such non-normal behaviors. Electronic [8] and optical [9] setups have been used as proxies for studying anomalous system properties. A large variety of issues in basic and applied sciences may be addressed by analyzing light propagation in different media. For example, rogue waves, an extreme phenomenon in sea waters, have been studied in opti- cal systems [9–12]; Anderson localization has been associated with statistical properties of radiation, signaling conditions in which to observe wave localization in disordered media [13,14]; the statistical regime in random lasers is related to the inherent disorder of their gain medium [15–17]. More generally, photon propagation appears to be a well-adapted framework for studies of non-normal statistical behavior, a major signature of complex systems [5,8,18]. Moreover, there is an intrinsic interest in the description of the propagation of radiation in inhomogeneous media with implications in biology [19], medicine [20], and communications [21]. Propagation of light in inhomogeneous media exhibits a diversity of statistical behaviors because of particular spatial geometry [22] or spectral response of the medium [23]. In * Corresponding author: martine.chevrollier@ufrpe.br the latter case, studies have been performed by observing the anomalous spatial statistics [23,24] or the spectral evolution [25] of resonant photons scattered in an atomic vapor. In these works, the medium response was linear with the light intensity. Here, we report on the propagation of speckle fields in nonlinear (NL) media where propagating photons are de- scribed as interacting particles. The nature of the interaction, whether attractive or repulsive, as well as its strength, can be adjusted through a single control parameter, the radiation frequency, for instance. Changes in the spatial correlation of the interacting photons modify the spatial distribution of the light and, therefore, the statistics of spatial photon density. A two-dimensional (2D) + 1 configuration is considered where the field E (  r ,ω) propagates along one dimension (1D) (Oz) and is scattered on the plane perpendicular to (Oz), following the paraxial equation: 2ik ∂ E ∂ z = −∇ 2 ⊥ E − k 2 χ E, (1) where k = ω/c is the vacuum wave number, ∇ 2 ⊥ E represents the transverse diffraction of the field in the medium, and χ = [Re(χ ) + i Im(χ )] is the complex electric susceptibility of the medium, governing its interaction with the electromagnetic field. The spatially random incident field is characterized through its intensity distribution P(I 0 ) and propagates through the NL medium whose response depends on the (local) in- tensity of the field, thus creating a spatial pattern of index of refraction that determines the direction of scattering of the field itself. These dispersive effects are produced by the real 2469-9926/2019/99(3)/033838(6) 033838-1 ©2019 American Physical Society