Stochastic radiative transfer in a finite plane-parallel medium with general boundary conditions

The time-independent radiative transfer problem in a scattering and absorbing planar random medium with general boundary conditions and internal energy source is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The problem is solved in terms of the solution of the corresponding free-source problem with simple boundary conditions which is solved using Pomraning-Eddington approximation in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. The average partial heat fluxes are calculated in terms of the albedoes of the source-free problem. Results are obtained for isotropic and anisotropic scattering for specular and diffused reflecting boundaries.     

Radiative transfer in a multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely incident parallel rays

The ADO (analytical discrete ordinates) method, a pre-processing procedure, and the break-point analysis developed for azimuthally symmetric problems in a previous work are generalized and used to solve a radiative-transfer problem defined by a finite, plane–parallel, multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination in the form of obliquely incident parallel rays. Illumination is modeled by Dirac distributions in each of the two angles (polar and azimuthal) that define the direction of propagation of the incident rays. Accurate numerical results are tabulated for two sets of test problems.     

An application of the K-integrals for solving the radiative transfer equation with Fresnel boundary conditions

We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials.     

Finite element approach for radiative transfer in multi-layer graded index cylindrical medium with Fresnel surfaces

Because the optical plane defined by the incidence and reflection direction at a cylindrical surface has a complicated relation with the local azimuthal angle and zenith angle in the traditional cylindrical coordinate system, it is difficult to deal with the specular reflective boundary condition in the solution of the traditional radiative transfer equation for cylindrical system. In this paper, a new radiative transfer equation for graded index medium in cylindrical system (RTEGCN) is derived based on a newly defined cylindrical coordinate system. In this new cylindrical coordinate system, the optical plane defined by the incidence and reflection direction is just the isometric plane of the local azimuthal angle, which facilitates the RTEGCN in dealing with cylindrical specular reflective boundaries. A least squares finite element method (LSFEM) is developed for solving radiative transfer in single and multi-layer cylindrical medium based on the discrete ordinates form of the RTEGCN. For multi-layer cylindrical medium, a radial basis function interpolation method is proposed to couple the radiative intensity at the interface between two adjacent layers. Various radiative transfer problems in both single and multi-layer cylindrical medium are tested. The results show that the present finite element approach has good accuracy to predict the radiative heat transfer in multi-layer cylindrical medium with Fresnel surfaces.     

On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media

The ADO (analytical discrete ordinates) method is used to establish a concise and accurate solution for a multi-layer radiative-transfer problem with Fresnel boundary and interface conditions. A finite plane-parallel medium composed of a number (K) of sub-strata with different material properties is considered to be illuminated by isotropically incident radiation. While a general result is obtained, emphasis in the numerical work is given to computing accurately the currents and the intensities that exit each of the two exterior surfaces. Monochromatic forms (with anisotropic scattering) of the radiative-transfer equation are used, and numerical results are given for several specific cases. The complications introduced by the Fresnel boundary and interface conditions are well resolved, so that the numerical results obtained are thought to define a very high standard.     

Reciprocity principle for radiative transfer models that use periodic boundary conditions

Many numerical models use periodic boundary conditions in solving the radiative transfer through heterogeneous media specified over a fixed domain. A reciprocity principle applicable to solutions from these models is derived for the common situation of a scattering and absorbing heterogeneous medium that is illuminated over the entire domain from a single direction. The derived reciprocity principle states that the domain-averaged bidirectional reflectance distribution function remains invariant when incoming and outgoing directions are interchanged, regardless of the heterogeneity of the medium and the size of the domain. This reciprocity principle provides a simple and useful benchmark test for radiative transfer models that use periodic boundary conditions.     

Many polarized radiative transfer models

This note is an introduction to the reprint of the 1991 JQSRT article “A new polarized atmospheric radiative transfer model” by K.F. Evans and G.L. Stephens. We discuss the significance of the article, how our two plane-parallel polarized radiative transfer codes came about, how our codes have been used, and more recent developments in polarized radiative transfer modeling.     

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.     

Observation of radiative transfer of polarized light

The results of an experiment sensitive to polarization and angle-dependent effects of resonance line radiation transport in a collisional environment are presented. Having measured both the excited atomic state population and alignment as a function of distance from the initial excitation, considerable anisotropy in the alignment is found. Included in the discussion are the results of an angle-average model calculation, which allows us to point out, in a qualitatively way, some of the important physics of the problem.     

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

On the use of a nascent delta function in radiative-transfer calculations for multi-layer media subject to Fresnel boundary and interface conditions

The “pre-processing” procedure and the “break-point” analysis developed in a previous work based on the ADO (analytical discrete ordinates) method are used, along with a nascent delta function to describe the polar-angle dependence of an incident beam, to solve the classical albedo problem for radiative transfer in a plane-parallel, multi-layer medium subject to Fresnel boundary and interface conditions. As a result of the use of a nascent delta function, rather than the Dirac distribution, to model the polar-angle dependence of the incident beam, the computational work is significantly simplified (since a particular solution is not required) in comparison to an approach where both the polar-angle and the azimuthal-angle dependence of the incident beam are formulated in terms of Dirac delta distributions. The numerical results from this approach are (when a sufficiently small “narrowness” parameter is used to define the nascent delta) found to be in complete agreement with already reported (high-quality) results for a set of challenging multi-layer problems.     

Nongray radiative transfer in a semi-infinite medium: H-function

Numerical results are presented for the dimensionless emissive power at the boundary of a nongray semi-infinite medium in radiative equilibrium. The adsorption coefficient of an array of equal intensity, nonoverlapping bands or lines. Specifically, the rectangular, triangular, exponential, Doppler and Lorentz profiles are considered.     

Two-dimensional radiative transfer in a finite scattering planar medium

The source function, radiative flux, and intensity at the boundaries are calculated for a two-dimensional, scattering, finite medium subjected to collimated radiation. The scattering phase function is composed of a spike in the forward direction super-imposed on an isotropic background. Exact radiative transfer theory is used to formulate the problem and Ambarzumian's method is used to obtain results. Using the principle of superposition, the results for any step variation in incident radiation are expressed in terms of universal functions for the semi-infinite step case. Two-dimensional effects are most pronounced at large optical thicknesses and albedos.     

Corrections to the classical continuity boundary conditions at the interface of a composite medium

Using the multiple-scales homogenization method, we derive generalized sheet transition conditions (GSTCs) for electromagnetic fields at the interface between two media, one of which is free-space and the other a certain type of composite material. The parameters in these new boundary conditions are interpreted as effective electric and magnetic surface susceptibilities, which themselves are related to the geometry of the scatterers that constitute the composite. We show that the effective tangential E and H fields are not continuous across the interface except in the limit when the lattice constant (the spacing between the scatterers—atoms, molecules or inclusions in the case of a composite material) of the composite medium is very small compared to a wavelength. We derive first-order corrections to the classical continuity conditions. For naturally occurring materials whose lattice constants are on an atomic scale, these effects are shown to be negligible for waves at optical frequencies or lower. However, once the lattice constant becomes a significant fraction of a wavelength (which is the case for many artificial dielectrics and metamaterials), the corrections can be important. In previous work we have alluded to the fact that such a GSTC is needed to correctly account for the surface effects when extracting the effective material properties of a metamaterial. The results of this current paper justify the assumptions made in that previous work. In general, these GSTCs will result in corrections to the classical Fresnel reflection and transmission coefficients (which are themselves merely zeroth-order approximations to the actual reflection and transmission coefficients), and in a separate publication we will use these GSTCs to address this issue.     

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.     

The radiative transfer for polarized radiation at second order in cosmological perturbations

This article investigates the full Boltzmann equation up to second order in the cosmological perturbations. Describing the distribution of polarized radiation by using a tensor valued distribution function, the second order Boltzmann equation, including polarization, is derived without relying on the Stokes parameters.     

Layer-edge conditions for multislab atmospheric radiative transfer problems

We derive nonstandard layer-edge conditions for efficient solution of multislab atmospheric radiative transfer problems. We begin by defining a local radiative transfer problem on the lowermost layer of a multislab model atmosphere and we consider a standard discrete ordinates version of this local problem. We then make use of a recently developed computational method in order to derive layer-edge conditions involving incident, reflected and transmitted radiation. These layer-edge conditions for the lowermost layer are given in terms of inherent optical properties of the layer, the solar zenith angle and the quadrature set used in the discrete ordinates approach. They can be used to increase the efficiency of our computational method in solving practical problems in atmospheric radiative transfer. Moreover, they are amenable to incorporation into other discrete ordinates methods. To illustrate, we report numerical results for two atmospheric model problems.     

The use of the itFN method for radiative transfer problems with reflective boundary conditions

The F_N method is used to compute the net radiative heat flux relevant to radiative transfer in an anisotropically scattering, plane-parallel medium with specularly and diffusely reflecting boundaries.     

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized laser beam: spatially varying reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized, Gaussian laser beam directed perpendicular to the surface is studied. The focus of this investigation is the 4×4, spatially varying reflection matrix that can be used to determine the normally backscattered radiation when the polarization of the incident radiation is specified. An inverse integral transform is used to construct the spatially varying reflection matrix from the generalized reflection matrix found in a previous study. The elements of this matrix depend on location specified by optical radius and azimuthal angle. The azimuthal variation is found by performing part of the inverse transform analytically, while the radial variation is described by five functions that are calculated numerically via an inverse Hankel transform. Benchmark numerical results for these five functions are presented, and the effects of beam radius and particle concentration are discussed. Expressions that describe the behavior of the reflection functions at small and large optical radii are developed, and comparisons are made to the one-dimensional and scalar situations. The scalar approximation fails to predict the three-dimensional effects produced by the polarized beam, and even when the incident radiation is unpolarized, the error in the scalar reflection function can be as high as 20%.