Two-dimensional isotropic scattering in a semi-infinite cylindrical medium

Beginning with the integral equation for the source function, the solutions for the source function, flux and intensity at the boundary of a two-dimensional, isotropically scattering cylindrical medium are found. The incident radiation is collimated and normal to the surface of the medium and depends only on the radial coordinate. For a Bessel function boundary condition, separation of variables is used to reduce the source function integral equation to a one-dimensional equation. The resulting integral equation is shown to be the same as that for the two-dimensional planar case. Solutions for other boundary conditions are then shown to be superpositions of the Bessel function solution. Numerical results are presented for a Gaussian distribution of incident radiation which closely models a laser beam. These multiple scattering results are compared to the single scattering approximation. Also, the solution for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function is expressed in terms of the isotropic results.     

Unconventional application of the two-flux approximation for the calculation of the Ambartsumyan-Chandrasekhar function and the angular spectrum of the backward-scattered radiation for a semi-infinite isotropically scattering medium

It is commonly accepted that the Schwarzschild-Schuster two-flux approximation (1905, 1914) can be employed only for the calculation of the energy characteristics of the radiation field (energy density and energy flux density) and cannot be used to characterize the angular distribution of radiation field. However, such an inference is not valid. In several cases, one can calculate the radiation intensity inside matter and the reflected radiation with the aid of this simplest approximation in the transport theory. In this work, we use the results of the simplest one-parameter variant of the two-flux approximation to calculate the angular distribution (reflection function) of the radiation reflected by a semi-infinite isotropically scattering dissipative medium when a relatively broad beam is incident on the medium at an arbitrary angle relative to the surface. We do not employ the invariance principle and demonstrate that the reflection function exhibits the multiplicative property. It can be represented as a product of three functions: the reflection function corresponding to the single scattering and two identical h functions, which have the same physical meaning as the Ambartsumyan-Chandrasekhar function (H) has. This circumstance allows a relatively easy derivation of simple analytical expressions for the H function, total reflectance, and reflection function. We can easily determine the relative contribution of the true single scattering in the photon backscattering at an arbitrary probability of photon survival Λ. We compare all of the parameters of the backscattered radiation with the data resulting from the calculations using the exact theory of Ambartsumyan, Chandrasekhar, et al., which was developed decades after the two-flux approximation. Thus, we avoid the application of fine mathematical methods (the Wiener-Hopf method, the Case method of singular functions, etc.) and obtain simple analytical expressions for the parameters of the scattered radiation. Note that the simplicity of the expressions is supplemented with unexpectedly high accuracy. The results demonstrate the unknown possibilities offered by the two-flux approximation, which is the simplest approximate method to solve the equations of transport theory. We assume that the method can be employed in the calculations of the angular characteristics of the reflected radiation for media whose single scattering is described using complicated (in comparison with isotropic) laws.     

Pure-triplet scattering for radiative transfer in binary discrete random finite media with reflective boundaries and internal source of radiation

The stochastic solution of the monoenergetic radiative transfer equation in a finite slab random medium with pure-triplet anisotropic scattering is considered. The random medium is assumed to consist of two randomly mixed immiscible fluids labelled by 1 and 2. The extinction function, the scattering kernel, and the internal source of radiation are treated as discrete random variables, which obey the same statistics. The theoretical model used here for stochastic media transport assumes Markovian processes and exponential chord length statistics. The boundaries of the medium under consideration are considered to have specular and diffuse reflectivities with an internal source of radiation inside the medium. The ensemble-average partial heat fluxes are obtained in terms of the average albedos of the corresponding source-free problem, whose solution is obtained by using the Pomraning-Eddington approximation. Numerical results are calculated for the average forward and backward partial heat fluxes for different values of the single scattering albedo with variation of the parameters that characterize the random medium. Compared to the results obtained by Adams et al. in the case of isotropic scattering based on the Monte Carlo technique, it can be demonstrated that we have good comparable data.     

Pure-triplet scattering for radiative transfer in binary discrete random finite media with reflective boundaries and internal source of radiation

The stochastic solution of the monoenergetic radiative transfer equation in a finite slab random medium with pure-triplet anisotropic scattering is considered. The random medium is assumed to consist of two randomly mixed immiscible fluids labelled by 1 and 2. The extinction function, the scattering kernel, and the internal source of radiation are treated as discrete random variables, which obey the same statistics. The theoretical model used here for stochastic media transport assumes Markovian processes and exponential chord length statistics. The boundaries of the medium under consideration are considered to have specular and diffuse reflectivities with an internal source of radiation inside the medium. The ensemble-average partial heat fluxes are obtained in terms of the average albedos of the corresponding source-free problem, whose solution is obtained by using the Pomraning–Eddington approximation. Numerical results are calculated for the average forward and backward partial heat fluxes for different values of the single scattering albedo with variation of the parameters that characterize the random medium. Compared to the results obtained by Adams et al. in the case of isotropic scattering based on the Monte Carlo technique, it can be demonstrated that we have good comparable data.     

Application of the singularity-subtraction technique to isotropic scattering in a planar layer

A numerical procedure for solving a singular Fredholm integral equation, which describes radiative transfer in an absorbing and isotropically scattering layer exposed to collimated radiation, is studied. The integral equation is solved by subtracting out the singularity and then approximating the integral term by a Gaussian quadrature. Bidirectional and hemispherical properties are found from the source function. Any arbitrary directional distribution of incident radiation can be handled by superimposing the collimated radiation case. In particular, the case of isotropic incident radiation is presented. The method gives accurate results, even for the extreme conditions of large optical thickness and large angle of incidence.     

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.     

Source function expansion method for radiative transfer in a two-layer slab

Radiative heat transfer in an isotropically scattering, absorbing and emitting two-layer slab with specularly reflecting boundaries is solved by expanding the source function by Legendre polynomials in the space variable in the integral form of the equation of radiative transfer. The reflectivity and the transmissivity of the slab for an externally incident isotropic radiation are determined. The S-1 solution yields results which are sufficiently accurate for most engineering applications.     

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.     

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.     

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.     

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.     

Two-dimensional rayleigh scattering in a semi-infinite cylindrical medium exposed to a laser beam

A modification of Ambarzumian's method is used to develop the integro-differential equations for the source function, flux, and intensity at the boundary of a two-dimensional, semi-infinite cylindrical medium with second order Legendre phase function scattering. The incident radiation is collimated, normal to the top surface, and is dependent only on the radial coordinate. Boundary conditions which vary as a Bessel function and as a Gaussian distribution are investigated. The Gaussian distribution approximates a laser beam. Numerical results are presented in graphical and tabular forms for a Rayleigh scattering medium. The results are compared with those of isotropic scattering.     

Single and double scattering approximations for a two-dimensional cylindrical medium

Large spatial frequency expansions for the source function, radiative flux, and intensity are obtained for an isotropically scattering finite two-dimensional medium exposed to collimated radiation. With these expansions, the single and double scattering results are obtained which are valid at small optical distances away from the incident radiation. Results are presented for a circular disk, exponential distribution and a Gaussian distribution of incident radiation.     

Using a plane wave approximation in simulation of radiation distribution in an X-ray Talbot interferometer

This study is devoted to the analysis of the plane-wave approximation applicability to X-ray radiation incident on an object. Based on simple calculations, it is shown that an X-ray tube focal spot 0.4 × 0.8 mm in size at distances of ~1 m can be considered as a point source; however, the plane-wave approximation for such X-ray radiation with energy E = 22keV is valid for radiation source–object distances much longer than 10 m.     

Two-dimensional isotropic scattering in a finite thick cylindrical medium exposed to a laser beam

Graphical and tabular results are presented for the back-scattered intensity from a finite two-dimensional cylindrical medium exposed to a Gaussian beam of radiation. Also, results for the source function and flux at the boundaries are presented. The influence of optical thickness and albedo are most pronounced at large optical radii. The semi-infinite results can be used to approximate the finite case for small optical radii. Ranges for single, double, and multiple scattering are discussed. For locations far from the incident beam, the results can be expressed in terms of universal functions independent of beam size. A method is presented for extending the isotropic results to the anisotropic case where the phase function is made up of a spike superimposed on an otherwise isotropic phase function.     

Optical equivalence of isotropic ensembles of ellipsoidal particles in the Rayleigh-Gans-Debye and anomalous diffraction approximations and its consequences

Light scattering by isotropic ensembles of ellipsoidal particles is considered in the Rayleigh-Gans-Debye approximation. It is proved that randomly oriented ellipsoidal particles are optically equivalent to polydisperse randomly oriented spheroidal particles and polydisperse spherical particles. Density functions of the shape and size distributions for equivalent ensembles of spheroidal and spherical particles are presented. In the anomalous diffraction approximation, equivalent ensembles of particles are shown to also have equal extinction, scattering, and absorption coefficients. Consequences of optical equivalence are considered. The results are illustrated by numerical calculations of the angular dependence of the scattering phase function using the T-matrix method and the Mie theory.     

Light scattering by homogeneous and multilayer ellipsoids in the quasi-static approximation

The quasi-static approximation for homogeneous and multilayer ellipsoidal particles is considered in detail. Expressions for the elements of the amplitude scattering matrix, as well as for the absorption and scattering cross sections, are presented for an arbitrary orientation of the particles relative to the incident radiation. The accuracy and the domain of applicability of the approximation are investigated for the elements of the scattering matrix of absorbing particles. Comparison of some approximate and exact methods showed that the quasi-static approximation gives good results (it is substantially more preferable than the Rayleigh and Rayleigh-Gans approximations), if the ratio of the largest size of a particle (or the boundaries of the main layer) to the smallest one does not exceed ～3. A new rule of the effective medium theory for multilayer ellipsoidal particles is proposed, which adequately takes into account their structure.     

用蒙特卡罗方法计算光通过薄层随机分布粒子的散射

本文利用蒙特卡罗方法计算准直光束通过薄层随机分布粒子散射的透射和反射光强,并和输运理论的扩散近似结果做了比较。当散射接近各向同性时两者符合良好。当散射明显地成为各向异性时,蒙特卡罗方法的结果是合理的,而输运理论的扩散近似失效。     

Two-dimensional linearly anisotropic scattering in a finite-thick cylindrical medium exposed to a laser beam

Integro-differential equations are developed for the source function, flux and intensity at the boundaries of a two-dimensional finite-thick medium which scatters in a linear fashion. The incident radiation is collimated, normal to the upper surface of the medium and dependent only on the radial coordinate. Two radial distributions are investigated: (1) a Bessel function and (2) a Gaussian laser beam. The solution for the Gaussian beam is constructed from the Bessel solution. Numerical results are presented in graphical and tabular forms for both boundary conditions. Comparisons are made between forward and isotropic scattering and between the finite and semi-infinite cases.     

Analysis of combined conduction and radiation heat transfer in presence of participating medium by the development of hybrid method

The current study addresses the mathematical modeling aspects of coupled conductive and radiative heat transfer in the presence of absorbing, emitting and isotropic scattering gray medium within two-dimensional square enclosure. A blended method where the concepts of modified differential approximation employed by combining discrete ordinate method and spherical harmonics method, has been developed for modeling the radiative transport equation. The gray participating medium is bounded by isothermal walls of two-dimensional enclosure which are considered to be opaque, diffuse and gray. The effect of various influencing parameters i.e., radiation-conduction parameter, surface emissivity, single scattering albedo and optical thickness has been illustrated. The adaptability of the present method has also been addressed.