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.     

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.     

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.     

Radiative transfer in a two-dimensional rectangular medium exposed to diffuse radiation

The integral equation for radiative transfer in a two-dimensional rectangular scattering medium exposed to diffuse radiation is solved numerically by removing the singularity. This method yielded accurate results except at very large optical thicknesses. Graphical and tabular results for the source function, flux, and intensity are presented. The source function is also calculated using the first term of a Taylor series expansion. The Taylor series is fairly accurate for small optical thicknesses and columnar geometries. A method is presented for extending these results to the problem of a strongly anisotropic scattering phase function which is made up of a spike in the forward direction superimposed on an isotropic phase function.     

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.     

Relation between multidimensional radiative transfer in cylindrical and rectangular coordinates with anisotropic scattering

An exact integral equation is derived for the source function in a three-dimensional rectangular medium which scatters anisotropically. The upper boundary of the finite medium is exposed to collimated radiation, while the lower boundary has no radiation incident on it. The problem is multidimensional because the incident radiation varies spatially. The scattering phase function is represented by a series of Legendre polynomials. A double Fourier transform reduces the problem to a one-dimensional integral equation for the source function. The transformed equation is compared with the integral equation for a two-dimensional cylindrical medium which scatters anisotropically and is exposed to Bessel-varying collimated radiation. A simple relation is found between the two source functions which will greatly reduce the number of computations required for the three-dimensional case. The relation also illustrates the wide utility of the generalized one-dimensional source function. Simplification of the two-dimensional rectangular case to the generalized source function is also presented. The results are extended to problems with a strong anisotropic phase function which has a diffraction spike in the forward direction.     

Exact expressions for radiative transfer in a three-dimensional rectangular geometry

The equations for the source function, flux, and scattered intensity normal to the surface are formulated in cartesian coordinates for a 3-D rectangular absorbing, emitting, isotropically scattering medium exposed to both diffuse and collimated radiation. Simplifications of these equations for certain important geometries and uniform loading are presented. Also, superposition of these equations and radiative equilibrium are discussed. For pure scattering, the source function at the center of the square and cubic geometries is analytically determined for the diffuse boundary condition. The generalized 3-D equations are shown to reduce to the familiar 1-D results. Also, the equations for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function are expressed in terms of the isotropic expressions.     

Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering

Exact integral equations are derived describing the source function and radiative flux in a two-dimensional, radially infinite cylindrical medium which scatters anisotropically. The problem is two-dimensional and cylindrical because of axisymmetric loading. Radially varying collimated radiation is incident normal to the upper surface while the lower boundary has no radiation incident upon it. The scattering phase function is represented by a spike in the forward direction plus a series of Legendre polynomials. The two-dimensional integral equations are reduced to a one-dimensional form by separating variables for the case when the radial variation of the incident radiation is a Bessel function. The one-dimensional form consists of a system of linear, singular Fredholm integral equations of second kind. Other more complex boundary conditions are shown to be solvable by a superposition of this basic Bessel function case. Diffusely incident radiation is also considered.     

Variational principles for scattering in the generator coordinate method

The generator coordinate method (GCM) wave function is used as a trial function in a Kohn type variational principle for scattering phase shifts. It is shown that a GCM trial function is a solution of the variational equations if the Hill-Wheeler integral equation is satisfied subject to an appropriate boundary condition. A new method for introducing the scattering boundary condition is presented. There is a uniqueness theorem for the phase shift.     

The influence of anisotropic scattering on the radiative intensity in a gray, plane-parallel medium calculated by the DRESOR method

Even though there have been many ways to treat complex anisotropic scattering problems, in most of the cases only the radiation flux or its dimensionless data were provided, and radiative intensity with high directional resolution could merely be seen. In this paper, a comprehensive formulation for the DRESOR method was proposed to deal with the anisotropic scattering, emitting, absorbing, plane-parallel media with different boundary conditions. The method was validated by the data from literature and the integral formulation of RTE. The DRESOR value plays an important role in the DRESOR method, and how it is determined by the anisotropic scattering was demonstrated by some typical results. The intensities with high directional resolution at any point can be given by the present method. It was found that the scattering phase function has little effect on the intensity for thin optical thickness, for example, 0.1. And there is the largest boundary intensity for the medium with the largest forward scattering capability, and the smallest one with the largest backward scattering capability. An attractive phenomenon was observed that the scattering of the medium makes the intensity at boundary can not reach the blackbody emission capability with the same temperature, even if the optical thickness tends to very large. It was also revealed that the scattering of the medium does not mean it cannot alter the magnitude of the energy; actually, stronger scattering causes the energy to have more chance to be absorbed by the medium, and indirectly changes the energy magnitude in the medium. Finally, it is easy to deduce all the associated quantities such as the radiation flux, the incident radiation and the heat source from the intensity, just as done in literature.     

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.     

On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media

The ability of the finite volume method (FVM) and the discrete ordinates method (DOM) to model radiative heat transfer in acute forward anisotropic scattering media has been investigated. The test case involves a purely scattering medium in a cubic enclosure, irradiated by one boundary with diffuse emission. Four phase functions have been considered: three of the Henyey-Greenstein type with respective asymmetry factors of 0.2, 0.8 and 0.93, and a Mie phase function with a strong forward scattering peak (computed for a size parameter of 245 and corresponding to an asymmetry factor of 0.93). Results obtained with the FVM are in good agreement with Monte Carlo reference solutions, whatever the level of acute anisotropic scattering (for asymmetry factors up to 0.93). The DOM combined with the renormalization procedures of the phase function proposed by Kim and Lee (Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures. Int J Heat Mass Transfer 1988;31:1711-21. [1]) and Wiscombe (On initialization error and flux conservation in the doubling method. JQSRT 1976;18:637-58. [2]) provides accurate results only for the smallest asymmetry factor. As the asymmetry factor increases, the renormalization procedures induce strong modifications in the values of the discretized phase function resulting in an underestimation of the effective attenuation by scattering. This error has been found to increase with optical thickness. In fact, when using the DOM, results would be more accurate combining this method with a Delta-Eddington approximation of the phase function, instead of using the actual phase function which is altered too much by renormalization.     

The Iterative-DRESOR method to solve radiative transfer in a plane-parallel, anisotropic scattering medium with specular-diffuse boundaries

Using the intensity with high directional resolution obtained by the Basic-DRESOR method as an initial guess, which is substituted into the integrated radiative transfer equation (IRTE), an iterative algorithm is proposed, called the Iterative-DRESOR method. This method can reduce the error levels of the intensity from several percent using the Basic-DRESOR method to a level of less than 1.0×10⁻⁶ with acceptable computation costs. The method is also validated against the exact heat flux in literature in some cases. It further clarifies some uncertain results for the reflectance in a pure, linearly anisotropic scattering medium with specular-diffuse boundaries. The directional distributions of intensity are obviously influenced by the reflecting modes of the boundary, especially in the zone near the boundary. The reflecting mode of an emitting boundary has little effect on the transmittance or reflectance. The reflecting mode of a non-emitting boundary also has little effect on the transmittance, but it obviously influences the reflectance. The difference between the reflectance for specular and diffuse boundaries increases at first, and then decreases, as the optical thickness of the medium increases. The difference will decrease as the scattering albedo of the medium increases, and it is negligible when the medium is pure scattering. The effect of the scattering phase function of the medium on the difference can also not be ignored. The Iterative-DRESOR method is expected to strengthen the capability of the Monte Carlo method to produce accurate results and to validate the results of other methods to solve RTE.     

Normalization of ultrasonic scattering measurements to obtain average differential scattering cross sections for tissues

An integral expression is derived for the normalization of ultrasonic scattering data to obtain an average differential scattering cross section per unit volume for tissue which is modeled as a random, fluidlike medium. The expression assumes narrowband signals and involves the incident beam, receiver beam pattern, and time gates. The beams and gates combine to form a window which limits the scattering volume. The derivation of the expression requires that the dimensions of the window be large compared to the correlation length of the scattering medium. Numerical values of the normalizing integral are given for cylindrical and beamlimited scattering volumes as a function of incident frequency and scattering angle under the assumptions of Gaussian beams and rectangular time gates. A set of curves is presented to relate the percent difference between the result for backscatter from a cylindrical scattering volume and from a beamlimited scattering volume which does not include the truncation effect of the cylinder boundary. Although similar in form to normalizations used by others, the integral in this paper is obtained from a derivation which treats physical parameters rigorously and provides a precise statement of conditions which are sufficient to obtain system-independent scattering data.     

Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm

A novel continuation method is presented for solving the inverse medium scattering problem of the Helmholtz equation, which is to reconstruct the shape of the inhomogeneous medium from boundary measurements of the scattered field. The boundary data is assumed to be available at multiple frequencies. Initial guesses are chosen from a direct imaging algorithm, multiple signal classification (MUSIC), along with a level set representation at a certain wavenumber, where the Born approximation may not be valid. Each update via recursive linearization on the wavenumbers is obtained by solving one forward and one adjoint problem of the Helmholtz equation.     

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.     

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.     

A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer

The Sensitivity Function-based Conjugate Gradient Method (SFCGM) is described. This method is used to solve the inverse problems of function estimation, such as the local maps of absorption and scattering coefficients, as applied to optical tomography for biomedical imaging. A highly scattering, absorbing, non-reflecting, non-emitting medium is considered here and simultaneous reconstructions of absorption and scattering coefficients inside the test medium are achieved with the proposed optimization technique, by using the exit intensity measured at boundary surfaces. The forward problem is solved with a discrete-ordinates finite-difference method on the framework of the frequency-domain full equation of radiative transfer. The modulation frequency is set to 600 MHz and the frequency data, obtained with the source modulation, is used as the input data. The inversion results demonstrate that the SFCGM can retrieve simultaneously the spatial distributions of optical properties inside the medium within a reasonable accuracy, by significantly reducing a cross-talk between inter-parameters. It is also observed that the closer-to-detector objects are better retrieved.     

多层球对高斯波束散射的德拜级数研究

基于广义洛伦兹-米氏理论,利用多层球粒子散射系数的德拜级数展开公式,提出了一种新的研究多层球粒子对高斯波束散射的方法。计算结果与已有的广义洛伦兹-米氏理论算法的计算结果吻合得很好。利用该方法有效分离了折射率分布满足指数变化规律的多层球粒子对高斯波束散射的远区散射场中多阶彩虹的干涉强度分布。数值模拟了双层球的归一化双一阶彩虹强度分布以及各层的一阶彩虹艾里结构。最后分析讨论了高斯波束的入射位置和束腰半径对多层球单阶彩虹强度分布的影响。     

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.