Three-dimensional radiative transfer in an anisotropically scattering, semi-infinite medium: generalized reflection function

Three-dimensional radiative transfer in an anisotropic scattering medium exposed to spatially varying, collimated radiation is studied. The generalized reflection function for a semi-infinite medium with a very general scattering phase function is the focus of this investigation. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The integration is over both the polar and azimuthal angles; hence, the integral equation is said to be in the double-integral form. The double-integral, reflection function formulation can handle a variety of anisotropic phase functions and does not require an expansion of the phase function in a Legendre polynomial series. Complicated kernel transformations of previous single-integral studies are eliminated. Single and double scattering approximations are developed. Numerical results are presented for a Rayleigh phase function to illustrate the computational characteristics of the method and are compared to results obtained with the single-integral method. Agreement between the two approaches is excellent; however, as the transform variable increases beyond five the number of quadrature points required for the double-integral method to produce accurate solutions significantly increases. A new interpolation scheme produces accurate results when the transform variable is large.     

Three-dimensional radiative transfer in an isotropically scattering, plane-parallel medium: generalized X- and Y-functions

The topic of this work is the generalized X- and Y-functions of multidimensional radiative transfer. The physical problem considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, plane-parallel medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is used to derive coupled, integro-differential equations for the source functions at the boundaries of the medium. The resulting equations are said to be in double-integral form because the integration is over both angular variables. Numerical results are presented to illustrate the computational characteristics of the formulation.     

Three-dimensional radiative transfer in an isotropically scattering, semi-infinite medium: generalized reflection function

The focus of this study is the generalized reflection function of multidimensional radiative transfer. The physical situation considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, semi-infinite medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The resulting equation is said to be in double-integral form because the integration is over both angular variables. Computational issues associated with this generalized reflection function formulation are investigated. The source function and reflection function formulations are compared, and the relative merits of the two approaches are discussed.     

Albedo problem for finite plane-parallel medium

The problem of radiation transfer through a scattering and absorbing finite plane-parallel medium is solved using an efficient and accurate method of analysis which utilizes trial functions based on Case's eigenvalues plus a linear combination of exponential integral functions. The proposed trial functions are used on the integral equation reducing it to a system of algebraic equations to be solved for the expansion coefficients which are used to calculate some interesting physical quantities such as the angular radiation intensity and the reflection and the transmission coefficients. Numerical results are obtained for two different external incidence on the left boundary, x=0. The results are compared with the exact results and with those calculated by the Pomraning-Eddington variational method.     

The generalized Ewald-Oseen extinction theorem and the integral equation in electromagnetic scattering

Four different types of response functions are used to derive the generalized Ewald-Oseen extinction theorem and the integral equation for the treatment of scattering of electromagnetic waves from a material medium. No surface terms appear explicitly in the integral equation. The present formulation is specially suited for perturbation calculations since the response functions, which appear, are not necessarily the ones corresponding to free space.     

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

Three-dimensional vector radiative transfer in a semi-infinite medium exposed to spatially varying, polarized radiation is studied. The problem is to determine the generalized reflection matrix for a multiple scattering medium characterized by a 4×4 scattering matrix. A double integral transform is used to convert the three-dimensional vector radiative transfer equation to a one-dimensional form, and a modified Ambarzumian's method is then applied to derive a nonlinear integral equation for the generalized reflection matrix. The spatially varying backscattered radiation for an arbitrarily polarized incident beam can be found from the generalized reflection matrix. For Rayleigh scattering and normal incidence and emergence, the generalized reflection matrix is shown to have five non-zero elements. Benchmark results for these five elements are presented and compared to asymptotic results. When the incident radiation is polarized, the vector approach used in this study correctly predicts three-dimensional behavior, while the scalar approach does not. When the incident radiation is unpolarized, both the vector and scalar approaches predict a two-dimensional distribution of the intensity, but the error in the scalar prediction can be as high as 20%.     

Emergent intensity from a plane-parallel medium exposed to a Gaussian laser beam: A comparison of Rayleigh and isotropic scattering

The focus of this two-dimensional study is the radially varying intensity emergent from a plane-parallel scattering medium exposed to a collimated, Gaussian laser beam directed perpendicular to the upper surface. The method of analysis is the integral transform technique. Specifically, this work uses the generalized reflection and transmission functions from a previous study to construct the emergent intensity with the use of an inverse Hankel transform. Radially varying backscattered and transmitted intensities are calculated for media with isotropic and Rayleigh scattering phase functions and optical thicknesses that range from 0.125 to 8.0. The behavior of the emergent radiation inside and outside the beam is investigated for both narrow and wide beams. A new integration method is implemented to compute the emergent intensity at the beam center. The emergent intensity at the beam center is used to quantify when a one-dimensional model may be used. As expected, for small optical thicknesses and near the beam the phase function has significant influence, while far from the beam multiple scattering reduces the influence of the Rayleigh phase function. Results from this study will be useful in understanding and interpreting more complicated situations, such as those that include polarization.     

A dynamical formulation of one-dimensional scattering theory and its applications in optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrödinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.     

A simple reduction procedure for the three- and N-body scattering equations

A generalized form of the two-body Kowalski-Noyes method is shown to provide a both simple and powerful unitary reduction of the three- and N-body scattering equations. Employing generalized half-off-shell functions that satisfy of-sshell but real and non-singular integral equations, the reduction directly leads to on-shell integral equations for the scattering amplitudes. Physically, it is simple example of how the scattering problem can be split into an internal and an external part.     

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%.     

Radiation transfer with synthetic scattering phase function

Equations connecting the relation between the reflection and transmission functions for a finite slab and those of an infinite one are obtained in terms of an operator which satisfies a semigroup. In trying to calculate the infinite medium reflection function, we use a synthetic scattering function to approximate the Henyey-Greenstein scattering law. Numerical calculations are done and compared with the results obtained from different scattering functions used by other authors. Good results are obtained, especially for |c?1| ?i1 and for τ>1. The deviation in some results is due to the effect of back-scattering.     

Algebraic recurrence relations for the finite-order scattering and transmission functions

In preceding papers (cf. Bellmanet al.(^1,2,3,), integral recurrence for the finite-order scattering and transmission functions have been given in connection with integral recurrence relations for the finite order X- and Y-functions with the aid of an initial-value method. In the present paper, it is shown how to find algebraic recurrence relations for the finite-order scattering and transmission functions in terms of finite-order X- and Y-functions without referring to an initial-value method. These recurrence relations are suitable for the numerical computation of the finite-order scattering and transmission functions by use of a digital computer. Furthermore, in Tables 1–3, a numerical example of the finite-order reflection functions for optical thickness 0·2 with albedo = 1 is listed. Table 4 furnishes an example of the cumulative reflection functions for optical thickness 0·2 with three different albedos.     

Acoustic scattering by arbitrary distributions of disjoint, homogeneous cylinders or spheres

A T-matrix formulation is presented to compute acoustic scattering from arbitrary, disjoint distributions of cylinders or spheres, each with arbitrary, uniform acoustic properties. The generalized approach exploits the similarities in these scattering problems to present a single system of equations that is easily specialized to cylindrical or spherical scatterers. By employing field expansions based on orthogonal harmonic functions, continuity of pressure and normal particle velocity are directly enforced at each scatterer using diagonal, analytic expressions to eliminate the need for integral equations. The effect of a cylinder or sphere that encloses all other scatterers is simulated with an outer iterative procedure that decouples the inner-object solution from the effect of the enclosing object to improve computational efficiency when interactions among the interior objects are significant. Numerical results establish the validity and efficiency of the outer iteration procedure for nested objects. Two- and three-dimensional methods that employ this outer iteration are used to measure and characterize the accuracy of two-dimensional approximations to three-dimensional scattering of elevation-focused beams.     

On Green's function for cylindrically symmetric fields of polarized radiation

Analytic expressions for Green's function describing the process of transfer of polarized radiation in homogeneous isotropic infinite medium in case of cylindrical symmetry and nonconservative scattering are obtained. The solution is based on the set of systems of Abel integral equations of the first kind obtained using the principle of superposition, and the known expression of Green's function for radiation fields with plane-parallel symmetry. Eigenvalue decompositions for the corresponding matrices of generalized spherical functions are found. Using this result the systems of Abel integral equations are diagonalized, and the final solution is obtained.     

Connection between the immersing and transfer-matrix methods in one-dimensional scattering problems

We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.     

Multidimensional radiative transfer in absorbing,emitting, and linearly anisotropic scattering cylindrical medium with space-dependent properties

The integral form of three-dimensional radiative transfer equation for an absorbing, emitting, and linear-anisotropic scattering medium with space-dependent properties is formulated. A product-integration method is subsequently applied to develop a numerical scheme for solving the corresponding integral transfer equations in a two-dimensional, axisymmetric and nonhomogeneous medium subjected to externally incident radiation or bounded by emitting and diffusely-reflecting walls. The numerical solutions for cases of constant, continuous, and stepwise variations of scattering albedo are presented to illustrate its accuracy and flexibility, and validated by comparing with results available in the literature.     

Interior radiances in optically deep absorbing media—I Exact solutions for one-dimensional model

The exact solutions are obtained for a one-dimensional model of a scattering and absorbing medium. The results are given for both the reflected and transmitted radiance for any arbitrary surface albedo as well as for the interior radiance. These same quantities are calculated by the matrix operator method. The relative error of the solutions is obtained by comparison with the exact solutions as well as by an error analysis of the equations. The importance of an accurate starting value for the reflection and transmission operators is shown. A fourth-order Runge-Kutta method can be used to solve the differential equations satisfied by these operators in order to obtain such accurate starting values. Except for extremely large values of the optical thickness of a layer, the reflection and transmission operators calculated from accurate starting values obtained by the Runge-Kutta method are orders of magnitude (a factor of 10¹¹ better in a typical case at unit optical depth) more accurate than those obtained by the use of the single scattering approximation for an optically thin layer. The relative error in the reflection and transmission operators is less than 10^-12 and 10^-8 respectively up to an optical thickness of 32,768 when calculated by this procedure, while the relative error in the interior radiance is less than 10^-8 at all points within a layer of optical thickness 32,768. It is shown that flux conservation is a poor test of the accuracy of a numerical method, since flux is conserved to all orders for a conservative medium when the doubling method is used, no matter how inaccurate the starting values may be.     

Adding and invariant imbedding equations in matrix notation for all the scattering functions

Previous work by the author introduced a radiative formulation, containing a delta interior illumination, that allowed scattering solutions driven by internal sources to be handled in complete analogy to those for the standard problem (external delta illumination scattering through a medium). This analogy was made explicit by defining the three levels of scattering functions, S_s-level, $\text{S}$_s- and S_F-level, and $\text{S}$_F-level, that characterize scattering through, into and out of, and within a finite medium, respectively. For an inhomogeneous medium the invariant imbedding method was employed to solve for these functions. This paper continues the work by showing that: (1) Adding equations can be derived for all the scattering functions using one superposition formula. (2) Adding and invariant imbedding computational methods are closely related and should be used in combination for efficient calculations. (3) A new set of functions can be defined that represent scattering out of a medium driven by thermal sources. (4) All scattering functions can be converted to represent a planetary problem by one adding step. References are given for numerical results using this formulation.     

Many-times formalism and Coulomb interaction

We extend here the many-times formalism, formerly used mainly for particles moving in given classical fields, to interacting particles. In order to minimize the difficulties associated with an equal-time interaction, we limit ourselves to nonrelativistic quantum mechanics and a two-particle interaction, such as that corresponding to the Coulomb force between charged particles. We obtain a set of differential equations which are really not consistent, but they serve as a guide to a formulation in terms of integral equations that has the same perturbation expansion as the usual theory for the scattering of particles. The integral equation for two-particle amplitudes can be modified to give the correct theory for bound states, but this is not the case for more than two particles. We expect that this theory can be generalized to a formulation of relativistic quantum mechanics of interacting particles.     

One-Dimensional Inverse Scattering Problem in Acoustics

We are concerned with the inverse scattering problem (ISP) in acoustics within the Marchenko inversion scheme. The quantum ISP is first discussed and applied in order to exhibit certain characteristics and application prospects of the method which could be useful in extending it to classical systems. We then consider the ISP in acoustics by assuming plane waves propagating in an elastic, isotropic, and linear medium. The wave equation is first transformed into a Schrödinger-like equation which can be brought into the Marchenko integral equation for the associated nonlocal kernel the solution of which provides us the full information of the underlying reflective profile. We apply the method in several model problems where the reflection coefficient of the multi-layer reflective medium is used as input to the ISP and in all cases we obtain excellent reproduction of the original structure of the scatterer. We then applied the inverse scattering scheme to construct profiles with certain predetermined reflection and transmission characteristics.