We provide a vectorial model to simulate second-harmonic generation (SHG) in birefringent, transparent media with an arbitrary configuration of non-linear ($χ^{(2)}$) crystalline grains. We apply this model on disordered assemblies of LiNbO$_3$ and BaTiO$_3$ grains to identify the influence of the birefringence on the random quasi-phase-matching process. We show that in monodispersed assemblies, the birefringence relaxes the grain-size dependence of the SHG efficiency. In polydispersed assemblies with sufficiently large grains, we find that the birefringence introduces an SHG efficiency enhancement of up to 54% compared to isotropic reference crystals, which is grain size independent. This enhancement increases linearly with the grain size, if the birefringent grains can be phase matched. These two different scaling behaviours are used in Kurtz and Perry's powder-technique to identify the phase-matchability of a material. We show on the example of LiNbO$_3$ and ADP that this technique cannot be applied when the grains get smaller than the coherence length, because the SHG scaling with the grain size becomes material specific.
A detailed theoretical and numerical analysis of the localization length in alternating metamaterial-birefringent random layered stacks, under uncorrelated thickness-disorder, has been performed. Similar structures have recently been reported to suppress the Brewster delocalization for $\it{p}$-polarized light, when "standard" isotropic layers (with positive index of refraction) are considered instead of metamaterial layers, providing a generic means to produce polarization-insensitive, broadband reflections. However, this enhancement of localization is valid for short wavelengths $λ$ compared…
Birefringent crystals are extensively used to manipulate polarized light. The generalized transfer matrix developed allows efficient calculation of the full polarization state of light transmitted through and reflected by a stack of arbitrarily-many discrete layers of isotropic and birefringent materials at any frequency and angle of incidence. The matrix of a uniaxial birefringent crystal with arbitrary rotation is calculated, along with its reduction to the matrix of an isotropic medium. This method is of great practical importance where tight control of systematic effects is needed in optic…
Based on a recent work by Schuller et al., a geometric representation of all skewonless, non-birefringent, linear media is obtained. The derived constitutive law is based on a "core", encoding the optical metric up to a constant. All further corrections are provided by two (anti-)selfdual bivectors, and an "axion". The bivectors are found to vanish if the optical metric has signature (3,1) - that is, if the Fresnel equation is hyperbolic. We propose applications of this result in the context of transformation optics and premetric electrodynamics.
A stochastic theory is developed for the light transmitting the optical media exhibiting linear and nonlinear birefringence. The starting point is the two--component nonlinear Schr{"o}dinger equation (NLSE). On the basis of the ansatz of "soliton" solution for the NLSE, the evolution equation for the Stokes parameters is derived, which turns out to be the Langevin equation by taking account of randomness and dissipation inherent in the birefringent media. The Langevin equation is converted to the Fokker--Planck (FP) equation for the probability distribution by employing the technique of functi…
We consider the paraxial propagation of nondiffracting singular beams inside natural biaxial and biaxially-induced birefringent media in vicinity of one of the optical axes in terms of eigenmode vortex-beams, whose angular momentum does not change upon propagation. We have predicted a series of new optical effects in the natural biaxial crystals such as the stable propagation of vector singular beams bearing the coupled optical vortices with fractional topological charges, the conversion of the zero-order Bessel beam with a uniformly distributed linear polarization into the radially-, azimutha…
A theoretical study is presented for the random aspect of an optical vortex inherent in the nonlinear birefringent Kerr effect, which is called the optical spin vortex. We start with the two-component nonlinear Schrödinger equation. The vortex is inherent in the spin texture caused by an anisotropy of the dielectric tensor, for which the role of spin is played by the Stokes vector (or pseudospin). The evolutional equation is derived for the vortex center coordinate using the effective Lagrangian of the pseudospin field. This is converted to the Langevin equation in the presence of the fluctuat…
The paper addresses the space-frequency correlations of electromagnetic waves in general random, bi-anisotropic media whose constitutive tensors are complex Hermitian matrices. The two-frequency Wigner distribution (2f-WD) for polarized waves is introduced to describe the space-frequency correlations and the closed form Wigner-Moyal equation is derived from the Maxwell equations. Two-frequency radiative transfer (2f-RT) equations is then derived from the Wigner-Moyal equation by using the multiple scale expansion. For the simplest isotropic medium, the result coincides with Chandrasekhar's tra…
We introduce a wavefront shaping protocol for focusing inside disordered media based on a generalization of the established Wigner-Smith time-delay operator. The key ingredient for our approach is the scattering (or transmission) matrix of the medium and its derivative with respect to the position of the target one aims to focus on. A specifc experimental realization in the microwave regime is presented showing that the eigenstates of a corresponding operator are sorted by their focusing strength - ranging from strongly focusing on the designated target to completely bypassing it. Our protocol…
Understanding the interaction of light with a highly scattering material is essential for optical microscopy of optically thick and heterogeneous biological tissues. Ensemble-averaged analytic solutions cannot provide more than general predictions for relatively simple cases. Yet, biological tissues contain chiral organic molecules and many of the cells' structures are birefringent, a property exploited by polarization microscopy for label-free imaging. Solving Maxwell's equations in such materials is a notoriously hard problem. Here we present an efficient method to determine the propagation …
We propose an birefringence route to perfect electromagnetic (EM) wave tunneling through a metallic film which relies on homogeneous birefringent coatings with moderate and positive parameters only. EM transparency is achieved in such an birefringent-metal-birefringent (BMB) structure for both polarizations and over nearly all incident angles. The stringent restrictions in conventional dielectric-metal-dielectric media, i.e., dielectrics with extremely negative permittivity, high magnetic field and polarization dependence (only for TE waves), are not required in our method. The criterion for p…
Optical scattering strength of fractal optical disordered media with varying fractal dimension is reported. The diffusion limited aggregation (DLA) technique is used to generate fractal samples in 2D and 3D, and fractal dimensions are calculated using the box-counting method. The degree of structural disorder of these samples are calculated using their light localization strength, using the inverse participation ratio (IPR) analyses of the optical eigenfunctions. Results show non-monotonous behavior of the disorder-induced scattering strength with the fractal dimension, attributed to the compe…
We present a new model for the propagation of polarized light in a random birefringent medium. This model is based on a decomposition of the higher order statistics of the reduced Stokes parameters along the irreducible representations of the rotation group. We show how this model allows a detailed description of the propagation, giving analytical expressions for the probability densities of the Mueller matrix and the Stokes vector throughout the propagation. It also allows an exact description of the evolution of averaged quantities, such as the degree of polarization. We will also discuss ho…
The dependent scattering effect (DSE), which arises from the wave nature of electromagnetic radiation, is a critical mechanism affecting the radiative properties of micro/nanoscale discrete disordered media (DDM). In the last a few decades, the approximate nature of radiative transfer equation (RTE) leads to a plethora of investigations of the DSE in various DDM, ranging from fluidized beds, photonic glass, colloidal suspensions and snow packs, etc. In this article, we give a general overview on the theoretical, numerical and experimental methods and progresses in the study of the DSE. We firs…
Transport in complex systems is characterized by a fractal dimension -- the walk dimension -- that indicates the diffusive or anomalous nature of the underlying random walk process. Here we report on the experimental retrieval of this key quantity, using light waves propagating in disordered media. The approach is based on measurements of the time-resolved transmission, in particular on how the lifetime scales with sample size. We show that this allows one to retrieve the walk dimension and apply the concept to samples with varying degree of fractal heterogeneity. In addition, the method provi…
Understanding light-matter interaction in disordered photonic media allows people to manipulate light scattering and achieve exciting applications using seemingly scrambled media. As the concentration of scattering particles rises, they are inclined to step into near fields of each other in deep subwavelength scale. The fundamental physics involving the interplay between disorder and near-field interaction (NFI) is still not fully understood. We theoretically examine the role of NFI by analyzing the underlying multiple scattering mechanism. We find NFI leads to a stronger collective behavior i…
We review our recent theoretical results on multiple scattering of waves in nonlinear disordered media. In particular, we discuss angular correlations of scattered waves, coherent backscattering cone, and the new phenomenon of temporal instability of multiply scattered waves in disordered media with sufficiently strong nonlinearity. Instability threshlod is derived, dynamics of speckle pattern beyond the threshold is discussed.
Spatial field correlation functions represent a key quantity for the description of mesoscopic phenomena in disordered media and the optical characterization of complex materials. Yet many aspects related to the vector nature of light waves have not been investigated so far. We study theoretically the polarization and coherence properties of electromagnetic waves produced by a dipole source in a three-dimensional uncorrelated disordered medium. The spatial field correlation matrix is calculated analytically using a multiple scattering theory for polarized light. This allows us to provide a for…
The paper addresses the two-point correlations of electromagnetic waves in general random, bi-anisotropic media whose constitutive tensors are complex Hermitian, positive- or negative-definite matrices. A simplified version of the two-frequency Wigner distribution (2f-WD) for polarized waves is introduced and the closed form Wigner-Moyal equation is derived from the Maxwell equations. In the weak-disorder regime with an arbitrarily varying background the two-frequency radiative transfer (2f-RT) equations for the associated $2\times 2$ coherence matrices are derived from the Wigner-Moyal equati…
We demonstrate order of magnitude coherent control of total transmission of light through random media by shaping the wavefront of the input light. To understand how the finite illumination area on a wide slab affects the maximum values of total transmission, we develop a model based on random matrix theory that reveals the role of long-range correlations. Its predictions are confirmed by numerical simulations and provide physical insight into the experimental results.
In this paper we develop a generalized mode-expansion scheme for the vector lightwaves propagating in 3D disordered media, and find the photonic phase transition from an extended-state (ES) phase to a mixed-state (MS) phase as a localization intensity parameter $G$ rises above $1/\sqrt{2}$. For the disordered media consisting of plenty of uniform spherical scattering particles, we formulate this phenomena at first in terms of actual optical and geometric parameters of the disordered media.
We develop a self-consistent theory of temporal fluctuations of a speckle pattern resulting from the multiple scattering of a coherent wave in a weakly nonlinear disordered medium. The speckle pattern is shown to become unstable if the nonlinearity exceeds a threshold value. The instability is due to a feedback provided by the multiple scattering and manifests itself in spontaneous fluctuations of the scattered intensity. The development of instability is independent of the sign of nonlinearity.
Electronic matter waves traveling through the weak and smoothly varying disorder potential of a semi-conductor show branching behavior instead of a smooth spreading of flow. By transferring this phenomenon to optics, we show how the branched flow of light can be controlled to propagate along a single branch rather than through 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 we can even st…
We develop a model based on a multiple scattering theory to describe the diffusion of polarized light in disordered media exhibiting short-range structural correlations. Starting from exact expressions of the average field and the field spatial correlation function, we derive a radiative transfer equation for the polarization-resolved specific intensity that is valid for weak disorder and we solve it analytically in the diffusion limit. A decomposition of the specific intensity in terms of polarization eigenmodes reveals how structural correlations, represented via the standard anisotropic sca…
Light scattering in a magnetic medium with uncorrelated inclusions is theoretically studied in the approximation of ladder diagram. Correlation between polarizations of electromagnetic waves that are produced by infinitely-distant dipole source is considered. Here white noise disorder model with Gaussian distribution is taken into account. In such a medium the magneto-optical interaction leads to correlation between perpendicular light polarizations. Spatial field correlation matrix with nonzero nondiagonal elements is obtained in the first order on gyration.
We consider the multiple scattering of a scalar wave in a disordered medium with a weak nonlinearity of Kerr type. The perturbation theory, developed to calculate the temporal autocorrelation function of scattered wave, fails at short correlation times. A self-consistent calculation shows that for nonlinearities exceeding a certain threshold value, the multiple-scattering speckle pattern becomes unstable and exhibits spontaneous fluctuations even in the absence of scatterer motion. The instability is due to a distributed feedback in the system "coherent wave + nonlinear disordered medium". The…
We theoretically and numerically investigate the capability of disordered media to enhance the optical path length in dielectric slabs and augment their light absorption efficiency due to scattering. We first perform a series of Monte Carlo simulations of random walks to determine the path length distribution in weakly to strongly (single to multiple) scattering, non-absorbing dielectric slabs under normally incident light and derive analytical expressions for the path length enhancement in these two limits. Quite interestingly, while multiple scattering is expected to produce long optical pat…
We formulate a Langevin description of dynamics of a speckle pattern resulting from the multiple scattering of a coherent wave in a nonlinear disordered medium. The speckle pattern exhibits instability with respect to periodic excitations at frequencies $Ω$ below some $Ω_{\mathrm{max}}$, provided that the nonlinearity exceeds some $Ω$-dependent threshold. A transition of the speckle pattern from a stationary state to the chaotic evolution is predicted upon increasing nonlinearity. The shortest typical time scale of chaotic intensity fluctuations is of the order of $1/Ω_\mathrm {max}$.
The statistical properties of the phases of several modes nonlinearly coupled in a random system are investigated by means of a Hamiltonian model with disordered couplings. The regime in which the modes have a stationary distribution of their energies and the phases are coupled is studied for arbitrary degrees of randomness and energy. The complexity versus temperature and strength of nonlinearity is calculated. A phase diagram is derived in terms of the stored energy and amount of disorder. Implications in random lasing, nonlinear wave propagation and finite temperature Bose-Einstein condensa…
A fundamental manifestation of wave scattering in a disordered medium is the highly complex intensity pattern the waves acquire due to multi-path interference. Here we show that these intensity variations can be entirely suppressed by adding disorder-specific gain and loss components to the medium. The resulting constant-intensity (CI) waves in such non-Hermitian scattering landscapes are free of any backscattering and feature perfect transmission through the disorder. An experimental demonstration of these unique wave states is envisioned based on spatially modulated pump beams that can flexi…
We demonstrate natural optical activity in disordered ensembles of non-chiral plasmonic resonators. We show that the statistical distributions of rotatory power and spatial dichroism are strongly dependent on the scattering mean free path in diffusive random media. This result is explained in terms of the intrinsic geometric chirality of disordered media, as they lack mirror symmetry. We argue that chirality and natural optical activity of disordered systems can be quantified by the standard deviation of both rotatory power and spatial dichroism. Our results are based on microscopic electromag…
In this work we present a method for generating random matrices describing electromagnetic scattering from disordered media containing dielectric particles with prescribed single particle scattering characteristics. Resulting scattering matrices automatically satisfy the physical constraints of unitarity, reciprocity and time reversal, whilst also incorporating the polarization properties of electromagnetic waves and scattering anisotropy. Our technique therefore enables statistical study of a variety of polarization phenomena, including depolarization rates and polarization-dependent scatteri…
We study the propagation of waves in a set of absorbing subwavelength scatterers positioned on a stealth hyperuniform point pattern. We show that spatial correlations in the disorder substantially enhance absorption compared to a fully disordered structure with the same density of scatterers. The non-resonant nature of the mechanism provides broad angular and spectral robustness. These results demonstrate the possibility to design low-density materials with blackbody-like absorption.
We analytically compute a localization criterion in double scattering approximation for a set of dielectric spheres or perfectly conducting disks uniformly distributed in a spatial volume which can be either spherical or layered. For every disordered medium, we numerically investigate a localization criterion, and examine the influence of the system parameters on the wavelength localization domains.
We propose and demonstrate a system for wavefront shaping which generates optical foci through complex disordered media and achieves an enhancement factor of greater than 100,000. To exploit the 1 Megapixel capacity of a digital micro-mirror device and its fast frame rate, we developed a fast and efficient method to handle the heavy matrix algebra computation involved in optimizing the focus. We achieved an average enhancement factor of 101,391 within an optimization time of 73 minutes. This unprecedented enhancement factor may open new possibilities for realistic image projection and the effi…
Light wave reflection from optical disordered media is always associate with its phase, and the phase statistics influence the reflection statistics. We report a detailed numerical study of the statistics of the reflection coefficient RR* and its associated phase(theta) for plane electromagnetic waves reflected from one dimensional (1D) Gaussian white-noise optical disordered media, ranging from weak to strong disordered regimes. We solve numerically the full Fokker-Planck (FP) equation for the joint probability distribution in the RR* - phase(theta) space for different lengths of the sample w…
We propose a concept of disordered resonant media, which are characterized by random variations of their parameters along the light propagation direction. In particular, a simple model of disorder considered in the paper implies random change of the density of active particles (two-level atoms). Within this model, the effect of disorder on self-induced transparency (SIT) is analyzed using numerical simulations of light pulse propagation through the medium. The transition from the SIT to localization regime is revealed as well as its dependence on the disorder level, atom density, medium thickn…
Time-resolved scattering experiments enable imaging of materials at the molecular scale with femtosecond time resolution. However, in disordered media they provide access to just one radial dimension thus limiting the study of orientational structure and dynamics. Here we introduce a rigorous and practical theoretical framework for predicting and interpreting experiments combining optically induced anisotropy and time-resolved scattering. Using impulsive nuclear Raman and ultrafast X-ray scattering experiments of chloroform and simulations, we demonstrate that this framework can accurately pre…
We develop a multiple scattering theory for the absorption of waves in disordered media. Based on a general expression of the average absorbed power, we discuss the possibility to maximize absorption by using structural correlations of disorder as a degree of freedom. In a model system made of absorbing scatterers in a transparent background, we show that a stealth hyperuniform distribution of the scatterers allows the average absorbed power to reach its maximum value. This study provides a theoretical framework for the design of efficient non-resonant absorbers made of dilute disordered mater…
We study the optimal diffusive transmission and absorption of broadband or polychromatic light in a disordered medium. By introducing matrices describing broadband transmission and reflection, we formulate an extremal eigenvalue problem where the optimal input wavefront is given by the corresponding eigenvector. We show analytically that a single wavefront can exhibit strongly enhanced total transmission or total absorption across a bandwidth that is orders of magnitude broader than the spectral correlation width of the medium, due to long-range correlations in coherent diffusion. We find exce…
We experimentally investigate the interplay between spatial shock waves and the degree of disorder during nonlinear optical propagation in a thermal defocusing medium. We characterize the way the shock point is affected by the amount of disorder and scales with wave amplitude. Evidence for the existence of a phase diagram in terms of nonlinearity and amount of randomness is reported. The results are in quantitative agreement with a theoretical approach based on the hydrodynamic approximation.
The transmission of a wave through a randomly chosen `pile of plates' typically decreases exponentially with the number of plates, a phenomenon closely related to Anderson localisation. In apparent contradiction we construct disordered planar permittivity profiles which are complex-valued (i.e. have reactive and dissipative properties) that appear to vary randomly with position, yet are one-way reflectionless for all angles of incidence and exhibit a transmission coefficient of unity. We contrast these complex-valued 'random' planar permittivity profiles with a family of real-valued, two-way r…
A fundamental challenge in physics is controlling the propagation of waves in disordered media despite strong scattering from inhomogeneities. Spatial light modulators enable one to synthesize (shape) the incident wavefront, optimizing the multipath interference to achieve a specific behavior such as focusing light to a target region. However, the extent of achievable control was not known when the target region is much larger than the wavelength and contains many speckles. Here we show that for targets containing more than $g$ speckles, where $g$ is the dimensionless conductance, the extent o…
The effect of gravitation field of spherically symmetric mass on the birefringent properties of crystals has been analysed. It has been shown that the gravitation field with spherical symmetry can lead to a change of birefringence in anisotropic media.
Coherent backscattering (CBS) arises from complex interactions of a coherent beam with randomly positioned particles, which has been typically studied in media with large numbers of scatterers and high opacity. We develop a first-principles scattering model for scalar waves to study the CBS cone formation in finite-sized and sparse random media with specific geometries. The results provide new insights into the effects of density, volume size, and other relevant parameters on the angular characteristics of the CBS cone emerging from bounded random media for various types of illumination. This …
Most applications of memory effects in disordered optical media, such as the tilt-tilt and shift-shift spatial correlations, have focused on imaging through and inside biological tissues. Here we put forward a metamaterial platform not only to enhance but also to tune memory effects in random media. Specifically, we investigate the shift-shift and tilt-tilt spatial correlations in metamaterials composed of coated spheres and cylinders by means of the radiative transfer equation. Based on the single-scattering phase function, we calculate the translation correlations in anisotropically scatteri…
The disorder induced feedback makes random lasers very susceptible to any changes in the scattering medium. The sensitivity of the lasing modes to perturbations in the disordered systems have been utilized to map the regions of perturbation. A tracking parameter, that takes into account the cumulative effect of changes in the spatial distribution of the lasing modes of the system has been defined to locate the region in which a scatterer is displaced by a few nanometers. We show numerically that the precision of the method increases with the number of modes. The proposed method opens up the po…
Imaging through scattering and random media is an outstanding problem that to date has been tackled by either measuring the medium transmission matrix or exploiting linear correlations in the transmitted speckle patterns. However, transmission matrix techniques require interferometric stability and linear correlations, such as the memory effect, can be exploited only in thin scattering media. Here we show the existence of a statistical dependency in strongly scattered optical fields in a case where first-order correlations are not expected. We also show that this statistical dependence and the…
We demonstrate experimentally that long-range intensity correlation for light propagating inside random photonic waveguides can be modified by changing the shape of the waveguide. The functional form of spatial correlation is no longer universal in the regime of diffusive transport and becomes shape-dependent due to the non-local nature of wave propagation. The spatial dependence of the correlation may be asymmetric for light incident from opposite ends of the waveguide. This work opens the door to control non-local effects in mesoscopic transport of waves by manipulating the geometry of rando…
We experimentally measure the monochromatic transmission matrix (TM) of an optical multiple scattering medium using a spatial light modulator together with a phase-shifting interferometry measurement method. The TM contains all information needed to shape the scattered output field at will or to detect an image through the medium. We confront theory and experiment for these applications and we study the effect of noise on the reconstruction method. We also extracted from the TM informations about the statistical properties of the medium and the light transport whitin it. In particular, we are …
We introduce a method to experimentally measure the monochromatic transmission matrix of a complex medium in optics. This method is based on a spatial phase modulator together with a full-field interferometric measurement on a camera. We determine the transmission matrix of a thick random scattering sample. We show that this matrix exhibits statistical properties in good agreement with random matrix theory and allows light focusing and imaging through the random medium. This method might give important insights into the mesoscopic properties of complex medium.
Two-frequency radiative transfer (2f-RT) theory is developed for geometrical optics in random media. The space-frequency correlation is described by the two-frequency Wigner distribution (2f-WD) which satisfies a closed form equation, the two-frequency Wigner-Moyal equation. In the RT regime it is proved rigorously that 2f-WD satisfies a Fokker-Planck-like equation with complex-valued coefficients. By dimensional analysis 2f-RT equation yields the scaling behavior of three physical parameters: the spatial spread, the coherence length and the coherence bandwidth. The sub-transport-mean-free-pat…
We investigate energy-leaky modes induced by incomplete measurements of the scattering matrices of complex media. Due to the limited numerical apertures of an optical system, it is experimentally challenging to access theoretically predicted perfect transmission channels in the diffusive regime. By conducting numerical simulations on scattering matrices, we demonstrate that leaky modes contributed from uncollected transmission in the TM provides an energy transmission that is more enhanced than that predicted by measurement. On the other hand, a leaky mode originating from the uncollected refl…
We study theoretically and numerically a new kind of spatial correlation for waves in disordered media. We define $C_Γ$ as the correlation function of the fluorescent decay rate of an emitter at two different positions inside the medium. We show that the amplitude and the width of $C_Γ$ provide decoupled information on the structural correlation of the disordered medium and on the local environment of the emitter. This result may stimulate the emergence of new imaging and sensing modalities in complex media.
