Scattering of electromagnetic waves by ensembles of particles and discrete random media

Current problems of the theory of multiple scattering of electromagnetic waves by discrete random media are reviewed, with an emphasis on densely packed media. All equations presented are based on the rigorous theory of electromagnetic scattering by an arbitrary system of non-spherical particles. The main relations are derived in the circular-polarization basis. By applying methods of statistical electromagnetics to a discrete random medium in the form of a plane-parallel layer, we transform these relations into equations describing the average (coherent) field and equations for the sums of ladder and cyclical diagrams in the framework of the quasi-crystalline approximation. The equation for the average field yields analytical expressions for the generalized Lorentz-Lorenz law and the generalized Ewald-Oseen extinction theorem, which are traditionally used for the calculation of the effective refractive index. By assuming that the particles are in the far-field zones of each other, we transform all equations asymptotically into the well-known equations for sparse media. Specifically, the equation for the sum of the ladder diagrams is reduced to the classical vector radiative transfer equation. We present a simple approximate solution of the equation describing the weak localization (WL) effect (i.e., the sum of cyclical diagrams) and validate it by using experimental and numerically exact theoretical data. Examples of the characteristics of WL as functions of the physical properties of a particulate medium are given. The applicability of the interference concept of WL to densely packed media is discussed using results of numerically exact computer solutions of the macroscopic Maxwell equations for large ensembles of spherical particles. These results show that theoretical predictions for spare media composed of non-absorbing or weakly absorbing particles are reasonably accurate if the particle packing density is less than ∼30%. However, a further increase of the packing density and/or absorption may cause optical effects not predicted by the low-density theory and caused by near-field effects. The origin of the near-filed effects is discussed in detail.     

Forward Monte Carlo computations of fully polarized microwave radiation in non-isotropic media

A completely forward Monte Carlo radiative transfer code has been developed with biasing techniques to efficiently solve the polarized radiative transfer equation for the full Stokes vector. The code has been adapted to accommodate plane parallel/3-D vertically/horizontally inhomogeneous scattering atmospheres in Cartesian geometries. Particular attention has been paid in stochastically treating the propagation, the emission and the scattering through anisotropic media particularly suited for clouds containing perfectly or partially oriented particles. Our modelling is very appealing because all its biasing techniques do not introduce unphysical Stokes vector. Numerical results and comparisons with benchmark tests are presented for verification.     

On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media

Light transport in graded index media follows a curved trajectory determined by Fermat's principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.     

Radiative properties of densely packed spheres in semitransparent media: A new geometric optics approach

This contribution presents a new Ray-tracing method for calculating effective radiative properties of densely packed spheres in non-absorbing or semitransparent host medium. The method is restricted to the geometric optic objects and neglects the wave effects. The effective radiative properties such as the absorption and scattering coefficients, and phase function are retrieved from the calculation of mean-free paths of scattering and absorption, and the angular scattering probability of radiation propagating in the dispersed medium. The model accounts for the two geometric effects called here as non-point scattering and ray transportation effects. The successful comparison of the current model with data of radiative properties and transmittances of particle beds in a non-absorbing medium reported in the literature confirm its suitability. It is shown that: (i) for opaque or absorbing particles (not systematically opaque), the non-point scattering is the dominant geometric effects whereas both non-point scattering and ray transportation effects occur for weakly absorbing and transparent particles. In the later cases, these two geometric effects oppose and may cancel out. This may explain why the Independent scattering theory works well for packed of quasi-transparent particles; (ii) the non-point scattering and ray transportation effects can be captured through the scattering and absorption coefficients while using the classical form of phase function. This enables using the standard radiative transfer equation (RTE); (iii) the surrounding medium absorption can be accounted for without any homogenization rule. It contributes to increasing the effective absorption coefficient of the composite medium as expected but, at the same time, it reduces the particle extinction; and (iv) the current transfer calculation predicts remarkably the results of direct Monte Carlo (MC) simulation. This study tends therefore to confirm that the RTE can be applied to densely packed media by using effective radiative properties.     

随机介质中的矢量辐射传输理论

本文讨论了极化电磁波在随机介质中多次散射,传输和热辐射的斯托克斯矢量的辐射传输理论。其中包括随机分布离散的球形和非球形粒子的矢量辐射传输方程,离散坐标-特征分析法,付利叶变换,迭代法等数值解。讨论了非球形粒子的穆勒矩阵。并研究了密集分布的散射粒子介质的辐射传输理论,考虑了密集粒子散射的相干性,计算了有效传播常数。理论及数值结果与实验作了很好的比较。     

非均匀散射层矢量辐射传输(VRT)方程高阶散射解的迭代法

将散射介质层在z轴方向划分成薄层，用薄层的一阶散射强度、Fourier变换和迭代方法求解散射介质整层的矢量辐射传输(VRT)方程的高阶散射解.该方法将一阶散射与高阶散射迭代结合起来，计算公式简明，可计算高阶迭代解，计算时间少.计算结果与一层均匀散射介质的VRT方程一阶Mueller矩阵解、半空间均匀散射介质二阶Mueller矩阵解、以及离散坐标-特征值特征矢量法的VRT热辐射的数值解作了全面的比较.提出并讨论了非均匀散射层主动与被动VRT方程的高阶解.本计算程序可以通用于非球形粒子多层结构及非均匀介质的散射和热辐射计算. 关键词： VRT方程 分层 迭代解     

Polarization imaging of multiply-scattered radiation based on integral-vector Monte Carlo method

A new integral-vector Monte Carlo method (IVMCM) is developed to analyze the transfer of polarized radiation in 3D multiple scattering particle-laden media. The method is based on a “successive order of scattering series” expression of the integral formulation of the vector radiative transfer equation (VRTE) for application of efficient statistical tools to improve convergence of Monte Carlo calculations of integrals. After validation against reference results in plane-parallel layer backscattering configurations, the model is applied to a cubic container filled with uniformly distributed monodispersed particles and irradiated by a monochromatic narrow collimated beam. 2D lateral images of effective Mueller matrix elements are calculated in the case of spherical and fractal aggregate particles. Detailed analysis of multiple scattering regimes, which are very similar for unpolarized radiation transfer, allows identifying the sensitivity of polarization imaging to size and morphology.     

Radiative properties of scattering and absorbing dense media: theory and experimental study

We investigate the validity of the radiative transfer equation to model transmission of light through an absorbing and scattering medium. Assuming that radiative transfer equation is valid, the inverse scattering problem for non-polarized radiative transfer in one-dimensional absorbing and scattering media is solved using a parameter identification method. We discuss how to identify the albedo, phase function and extinction coefficient of the medium. We present experimental data that confirm that this approach is robust and can be used to make reliable predictions of the behavior of scattering absorbing systems.     

Coherent backscattering of light by complex random media of spherical scatterers: numerical solution

Novel Monte Carlo techniques are described for the computation of reflection coefficient matrices for multiple scattering of light in plane-parallel random media of spherical scatterers. The present multiple scattering theory is composed of coherent backscattering and radiative transfer. In the radiative transfer part, the Stokes parameters of light escaping from the medium are updated at each scattering process in predefined angles of emergence. The scattering directions at each process are randomized using probability densities for the polar and azimuthal scattering angles: the former angle is generated using the single-scattering phase function, whereafter the latter follows from Kepler's equation. For spherical scatterers in the Rayleigh regime, randomization proceeds semi-analytically whereas, beyond that regime, cubic spline presentation of the scattering matrix is used for numerical computations. In the coherent backscattering part, the reciprocity of electromagnetic waves in the backscattering direction allows the renormalization of the reversely propagating waves, whereafter the scattering characteristics are computed in other directions. High orders of scattering (~10 000) can be treated because of the peculiar polarization characteristics of the reverse wave: after a number of scatterings, the polarization state of the reverse wave becomes independent of that of the incident wave, that is, it becomes fully dictated by the scatterings at the end of the reverse path. The coherent backscattering part depends on the single-scattering albedo in a non-monotonous way, the most pronounced signatures showing up for absorbing scatterers. The numerical results compare favourably to the literature results for nonabsorbing spherical scatterers both in and beyond the Rayleigh regime.     

非均匀随机散射介质层的多次散射和热辐射的分层迭代解

为纳入矢量辐射传输方程的多次散射，将非均匀平行分层分布的随机散射介质划分成多个薄 层，利用各薄层的低阶Mueller矩阵解推导出整层的高阶散射迭代解.用这一方法计算了具有 占空比、粒子大小和温度廓线非均匀分布的平行分层随机密集球形粒子积雪层的极化热辐射 .数值结果与一层均匀随机散射层的离散坐标-特征值方法作了比较，讨论了多次散射和热辐 射与散射层各特征参数的关系，与积雪层微波遥感数据进行了对比. 关键词： VRT方程 平行分层 高阶散射解 迭代解     

Depolarization of multiply scattered light in transmission through a turbid medium with large particles

An approximate analytical method for solving the vector radiative transfer equation is proposed. The method is based on the assumption that single scattering of light by large-scale inhomogeneities occurs predominantly through small angles. The method is applied to calculate the polarization state of multiply scattered light. The results obtained are discussed for various turbid media.     

Pomraning-Eddington approximation for radiative transfer in a spherical turbid medium

The Pomraning-Eddington approximation is used to solve the radiative transfer problem for anisotropic scattering in a spherical homogeneous turbid medium with diffuse and specular reflecting boundaries. This approximation replaces the radiative transfer integro-differential equation by a second-order differential equation which has an analytical solution in terms of the modified Bessel function. Here, we calculate the partial heat flux at the boundary of anisotropic scattering on a homogeneous solid sphere. The calculations are carried out for spherical media of radii 0.1, 1.0 and 10 mfp and for scattering albedos between 0.1 and 1.0. In addition, the calculations are given for media with transparent, diffuse reflecting and diffuse and specular reflecting boundaries. Two different weight functions are used to verify the boundary conditions. Our results are compared with those given by the Galerkin technique and show greater accuracy for thick and highly scattering media.     

Pomraning-Eddington approximation for radiative transfer in a spherical turbid medium

Abstract

The Pomraning-Eddington approximation is used to solve the radiative transfer problem for anisotropic scattering in a spherical homogeneous turbid medium with diffuse and specular reflecting boundaries. This approximation replaces the radiative transfer integro-differential equation by a second-order differential equation which has an analytical solution in terms of the modified Bessel function. Here, we calculate the partial heat flux at the boundary of anisotropic scattering on a homogeneous solid sphere. The calculations are carried out for spherical media of radii 0.1, 1.0 and 10 mfp and for scattering albedos between 0.1 and 1.0. In addition, the calculations are given for media with transparent, diffuse reflecting and diffuse and specular reflecting boundaries. Two different weight functions are used to verify the boundary conditions. Our results are compared with those given by the Galerkin technique and show greater accuracy for thick and highly scattering media.     

Random walk of polarized light in turbid media

We study the propagation of polarized light in turbid media as a random walk of vector photons. Both propagation and polarization directions of light are found to isotropize, following a power law of the number of scattering events. The characteristic length scale governing light isotropization and linear depolarization, the isotropization length , is derived using the exact Mie scattering for spherical particles. A simple relation is obtained for Rayleigh-Gans scatterers where is the transport mean free path and is the mean cosine of scattering angles.     

Heat transfer in a spherical turbid medium with conduction and radiation

The heat transfer through a spherical media with conduction and radiation is considered. The medium is considered to be turbid and anisotropically scattering with diffusely reflecting boundaries of constant temperatures. The radiative transfer problem is solved using the Galerkin method. An iterative method is used to solve the nonlinear relation between the radiative transfer equation and the conductive energy equation. Calculations are carried out and compared for a homogeneous, isotropically scattering medium with isothermal, transparent boundaries. The results show good agreement with previous work. Calculations are carried out for inhomogeneous media with isotropic, and forward and backward anisotropic scattering. The boundaries of the media are considered to be isothermal and may be transparent or diffusely reflecting boundaries. The calculations are used to study the effects of the single scattering albedo, the anisotropic scattering parameter, the conduction-radiation parameter and the heat source.     

Iterative approach of high-order scattering solution for vector radiative transfer of inhomogeneous random media

By stratifying a random scatter media into multiple thin layers in the vertical z direction, the first-order scattering solution of each thin layer is employed to derive high-order scattering solution of whole random media. Using the Fourier transform and Mueller matrices in discrete ordinates, an iterative approach to solve high-order scattering solution of vector radiative transfer (VRT) equation is newly developed. Numerical results are well compared with the Mueller matrix solutions of the first order for a single layer medium, second order for a half-space, and the results of the discrete ordinate and eigen analysis method. It demonstrates our approach as feasible, effective and especially applicable to high-order solution of VRT for both bistatic scattering and thermal emission of inhomogeneous non-spherical scatter media.     

低太阳高度角条件下的天空偏振模式模拟及大气折射影响研究

为模拟低太阳高度角条件下的天空偏振模式,自主开发了考虑大气球形几何及大气折射效应的辐射传输模式VSPART,并将其运用于漫射光偏振特性仿真.在模式中,基于射线追踪法实现了光线传播轨迹的追踪和入射光偏振态及透过率的计算,基于矩阵算法实现了辐射传输方程的求解.将VSPART模拟结果与基准值、SPDISORT模拟值进行了比较,验证了模型的准确性.在瑞利散射大气和含气溶胶大气条件下,模拟并分析了漫射光偏振度及偏振方向的分布特征,讨论了大气球形几何及折射效应对天空偏振度的影响.结果表明,低太阳高度角条件下,随着波长增加,瑞利散射大气对应的偏振度整体随之增强,中性点向大天顶角方向移动;气溶胶的存在并不改变天空偏振度分布特征,但对偏振方向影响显著,随着光学厚度的增加,天空偏振度值迅速降低;中性点的偏移可能与低阶散射过程紧密相关;大气球形几何和折射效应的主要影响区域为地平线区域、两中性点附近及天顶区域;瑞利散射大气条件下,随着波长增加,大气球形几何及折射效应的影响逐步减弱,特别在中性点附近及天顶区域,其影响逐步消失;随着气溶胶光学厚度的增加,其影响随之增强.     

Radiative transfer in stochastically inhomogeneous media

New radiative transfer theory is developed for stochastically inhomogeneous scattering media. The three-dimensional shapes and large scale (compared to the mean free path) structures of the media are modeled by stochastic interfaces separating regions of different scattering properties. The small scale fluctuations are characterized by a pair-correlation function. The radiative transfer equation is extended to include individual scattering and propagation probabilities of a ray for each subregion as well as the probability for a ray to cross the interface between two subregions. The propagation probability is found to depend on the entire preceding path of the ray; the present formulation accounts for the two previous scatterings. A new adding/doubling algorithm is developed to solve this problem numerically. Transmission through a cloud layer and backward scattering seem to be particularly sensitive to inhomogeneities.     

Theoretical study of combined radiative conductive and convective heat transfer in semi-transparent porous medium in a spherical enclosure

A new method for the solution of the radiative transfer equation in spherical media based on a modified discrete ordinates method is extended to study radiative, conductive and convective heat transfer in a semi-transparent scattering porous medium. The set of differential equations is solved using the fourth-order Runge-Kutta method. Various results are obtained for the case of combined radiative and conductive heat transfer, as well as for the interaction of those modes with convection. The effects of some radiative properties of the medium on the heat transfer rate are examined.