Plane-parallel Medium Research Articles

Diffuse and coherent backscattering of polarized light: Polarization ratios for a discrete random medium composed of nonspherical particles

For a sparse, disordered, plane-parallel particulate medium, we analyze quantitatively the effects of particle microphysical properties on the values of the linear and circular backscattering polarization ratios. Using numerically exact T-matrix and vector radiative-transfer codes, we performed computations for the following models: (1) a semi-infinite homogeneous layer composed of randomly oriented, polydisperse oblate spheroids with the real part of the refractive index equal to 1.2, 1.4 and 1.6 and the imaginary part of the refractive index equal to 0 and 0.01; (2) a semi-infinite homogeneous layer composed of randomly oriented, polydisperse, oblate circular cylinders with the refractive index 1.4; (3) finite homogeneous layers of various optical thickness composed of randomly oriented, polydisperse, oblate spheroids with the refractive index 1.55. Our computations demonstrate that the values of the polarization ratios depend substantially on particle shape, real and imaginary parts of the particle refractive index, particle size relative to the wavelength, illumination geometry and optical thickness.

Radiative transfer in a spherical, emitting, absorbing and anisotropically scattering medium

The atmospheres of planets (including Earth) and the outer layers of stars have often been treated in radiative transfer as plane-parallel media, instead of spherical shells, which can lead to inaccuracy, e.g. limb darkening. We give an exact solution of the radiative transfer specific intensity at all points and directions in a finite spherical medium having arbitrary radial spectral distribution of: source (temperature), absorption, emission and anisotropic scattering. The power and efficiency of the method stems from the spherical numerical gridding used to discretize the transfer equations prior to matrix solution: the wanted ray and the rays which scatter into it both have the same physico-geometric structure. Very good agreement is found with an isotropic astrophysical benchmark [Avrett EH, Loeser R. Methods in radiative transfer. In: Kalkofen W, editor. Cambridge: Cambridge University Press; 1984. pp. 341–79]. We introduce a specimen arbitrary forward– side–back phase scattering function for future comparisons. Our method directly and exactly addresses spherical symmetry with anisotropic scattering, and could be used to study the Earth's climate, nuclear power (neutron diffusion) and the astrophysics of stars and planets.

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.

Coupled conductive–radiative heat transfer problem for two-layer slab

The coupled conductive radiative transfer problem in two homogeneous layers slab of anisotropic scattering with specularly reflecting boundaries has been considered. A Galerkin-iterative technique is used to solve the coupled conductive radiative heat equations in integral forms for the two layers. Numerical results are obtained for the temperature, the conductive, radiative and the total heat fluxes for the two homogeneous layers with isotropic and anisotropic scattering. The calculations are also carried out for homogeneous plane parallel medium with anisotropic scattering which show good agreement with the published calculations.

Non-gray radiative and conductive heat transfer in single and double glazing solar collector glass covers

A rigorous model for the radiative heat transfer combined with the conduction and the convection has been applied for a solar collector glazing. The glass cover is analysed as a non-gray plane-parallel medium subjected to solar and thermal irradiation in one-dimensional case, using the radiation element method by ray emission model. The model allows the calculation of the steady-state heat flux and the temperature distribution within the glass cover. The spectral dependence of the relevant radiation properties of glass (i.e. specular reflectivity, refraction angle and absorption coefficient) is taken into account. Both collimated and diffuse incident irradiations are applied at the boundary surfaces using the spectral solar model proposed by Bird and Riordan. Single and double glasses commonly used for the glass cover are considered. The optical constant of a commercial clear glass material have been used. These optical constants of real and imaginary parts of the complex refractive index of the glass, determined by the author from 0.19 to 5 μm combined by those reported by Rubin from 6 to 300 μm have been used. The calculation has been performed for both single and double glazing cover at low and high temperature of the absorber. The effect of the thickness of the single glass cover has been also discussed. The result shows that increasing the thickness of the single glass cover, the steady heat flux decreases at both low and high temperature of the absorber. It has been also shown that the double-glazing cover assembly is more suitable than the single one at high temperature of the absorber.

Theoretical approach of a flat-plate solar collector taking into account the absorption and emission within glass cover layer

A rigorous theoretical approach of a flat-plate solar collector with a black absorber considering the glass cover as an absorbing–emitting media is presented. The glass material is analyzed as a non-gray plane-parallel medium subjected to solar and thermal irradiations in one-dimensional case using the Radiation Element Method by Ray Emission Model (REM 2). The optical constants of a clear glass window proposed by Rubin have been used. These optical constants, 160 values of real part n and imaginary part k of the complex refractive index of a clear glass, cover the range of interest for calculating the solar and thermal radiative transfer through the glass cover. The computational time for predicting the thermal behavior of solar collector was found to be prohibitively long for the non-gray calculation using 160 values of n and k. Therefore a suitable semi-gray model is proposed for rapid calculation. The profile of the efficiency curve obtained in the present study was found to be not linear in shape. Indeed, the heat loss from the collector is a combination of convection and radiation and highly non linear. The effect of the outside convective heat transfer on the efficiency curve is also studied. In fact, when the convection is the dominant heat transfer mode compared with the radiation one, the profile of the efficiency curve is more or less straight line. Consequently, the heat loss coefficient could be calculated using Klein model. It has been also shown that the effect of the wind speed on the glass cover mean temperature is very important. This effect increases with the increase of the mean absorber temperature.

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.

Numerical Method for the Solution of the Equation of Radiation Transfer. Reflection and Transmission Coefficients for an Optically Thin Plane-Parallel Layer

We study the problem of the choice of initial approximation for the reflection and transmission coefficients in numerical methods based on the principle of “interaction.” The disadvantages of the approximation of single scattering are demonstrated and the regularities of propagation of light in media with strongly anisotropic scattering are analyzed. Semianalytic expressions proposed for the evaluation of the initial approximation enable one to determine the characteristics of the light field in plane-parallel media with relative errors of about 10−5 within the framework of the algorithm of “adding” of layers.

Theoretical approach of a flat plate solar collector with clear and low-iron glass covers taking into account the spectral absorption and emission within glass covers layer

Accurate modeling of solar collector system using a rigorous radiative model is applied for the glass cover which represents the most important component of the system and greatly affects the thermal performance. The glass material is analyzed as a non-gray plane-parallel medium subjected to solar and thermal irradiations in one dimensional case using the radiation element method by ray emission model (REM 2). The optical constants of a clear and low-iron glass materials proposed by Rubin have been used. These optical constants, 160 values of real part n and imaginary part k of the complex refractive index of such materials, cover the range of interest for calculating the solar and thermal radiative transfer through the glass cover. The computational times for predicting the thermal behavior of solar collector were found to be prohibitively long for the non-gray calculation using 160 values of n and k for both glasses. Therefore, suitable semi-gray models have been proposed for rapid calculation. The temperature distribution within the glass cover shows a good agreement with that obtained with iterative method in case of clear glass. It has been shown that the effect of the non-linearity of the radiative heat exchange between the black plate absorber and the surroundings on the shape of the efficiency curve is important. Indeed, the thermal loss coefficient is not constant but is a function of temperature, due primarily to the radiative transfer effects. Therefore, when the heat exchange by radiation is dominant compared with the convective mode, the profile of the efficiency curve is not linear. It has been also shown that the instantaneous efficiency of the solar collector is higher in case of low-iron glass cover.

Time-dependent neutron transport in finite media using Pomraning–Eddington approximation

The time-dependent monoenergetic neutron transport equation in a finite plane-parallel medium with linear anisotropic scattering and with specular-reflecting boundaries is considered. The problem is solved using Pomraning–Eddington approximation and by using weight functions to force the boundary conditions to be fulfilled. Numerical calculations for the reflectivity and transmissivity of the proposed medium are calculated at different times and different thicknesses.

Albedo problem in anisotropic semi-infinite medium

Radiation transfer problem in an absorbing and anisotropic scattering homogeneous semi-infinite plane parallel medium subjected to externally incident radiation is considered. Trial functions based on Case's eigen modes and exponential integral function are used. Our numerical results for medium albedo, forward and backward intensities, total intensity and current density are reported and compared with the available published results.

Solution of the radiative transfer equation in the separable approximation

The radiative transfer equation in plane-parallel media with coherent isotropic scattering and finite optical depth is solved analytically, utilizing the formalism of meromorphic functions. The solution obtained is well suited for applications to accretion disks due to the imposed property of the slab, such as the presence of the plane of symmetry. The practical performance of the method is discussed and its correctness is checked by comparing the results obtained with those of other methods.

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.

Analytic inverse radiative transfer equations for atmospheric and hydrologic optics.

Two new sets of analytical equations are derived with which the albedo of single scattering and the coefficients of a Legendre polynomial expansion of the scattering phase function can be determined for a source-free, homogeneous plane-parallel medium uniformly illuminated over the surfaces. The equations, essentially linear in the unknowns, require measurements of the radiance in the interior of the medium, but no iterative forward-problem calculations are needed. Sets of equations for both unpolarized and polarized radiation applications are given, as well as a side-by-side comparison with previously known sets of analytic inversion equations. Applications of the equations are suggested.

Combined non-gray radiative and conductive heat transfer in solar collector glass cover

A rigorous approach for the radiative heat transfer analysis in solar collector glazing is developed. The model allows a more accurate prediction of thermal performance of a solar collector system. The glass material is analysed as a non-gray plane-parallel medium subjected to solar and thermal irradiations in the one-dimensional case using the Radiation Element Method by Ray Emission Model (REM by REM). This method is used to analyse the combined non-gray convective, conductive and radiative heat transfer in glass medium. The boundary surfaces of the glass are specular. The spectral dependence of the relevant radiation properties of glass (i.e. specular reflectivity, refraction angle and absorption coefficient) are taken into consideration. Both collimated and diffuse incident irradiation are applied at the boundary surfaces using the spectral solar model proposed by Bird and Riordan. The optical constants of a commercial ordinary clear glass material have been used. These optical constants (100 values) of real and imaginary parts of the complex refractive index of the glass material cover the range of interest for calculating the solar and thermal radiative heat transfer through the solar collector glass cover. The model allows the calculation of the steady-state heat flux and temperature distribution within the glass layer. The effect of both conduction and radiation in the heat transfer process is examined. It has been shown that the real and imaginary parts of the complex refractive index have a substantial effect on the layer temperature distribution. The computational time for predicting the combined heat transfer in such a system is very long for the non-gray case with 100 values of n and k. Therefore, a simplified non-gray model with 10 values of n and k and two semi-gray models have been proposed for rapid computations. A comparison of the proposed models with the reference non-gray case is presented. The result shows that 10 bandwidths could be used for rapid computation with a very high level of accuracy.

Radiative transfer computations for optical beams

In this paper, we present a method for computing direct numerical simulations of narrow optical beam waves propagating and scattering in a plane-parallel medium. For these computations, we use Fourier and Chebyshev spectral methods for three-dimensional radiative transfer that also includes polar and azimuthal angle dependences. We treat anisotropic scattering with peaked forward scattering by using a Clenshaw–Curtis quadrature rule for the polar angle and an extended trapezoid rule for the azimuthal angle. To verify our results, we compare this spectral method to Monte Carlo simulations.

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Linearly Anisotropically Scattering Plane-Parallel Medium: Part 2

The SKN (Synthetic Kernel) approximation is proposed for solving radiative transfer problems in linearly anisotropically scattering homogeneous and inhomogeneous participating plane-parallel medium. The radiative integral equations for the incident energy and the radiative heat flux using synthetic kernels are reduced to a set of coupled second-order differential equations for which proper boundary conditions are established. Performance of the three quadrature sets proposed for isotropic scattering medium are further tested for linearly anisotropically scattering medium. The method and its convergence with respect to the proposed quadrature sets are explored by comparing the results of benchmark problems using the exact, P11, and S128 solutions. The SKN method yields excellent results even for low orders using appropriate quadrature set.

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Isotropically Scattering Plane-Parallel Medium: Part 1

A high order approximation, the SKN method—a mnemonic for synthetic kernel—is proposed for solving radiative transfer problems in participating medium. The method relies on approximating the integral transfer kernel by a sum of exponential kernels. The radiative integral equation is then reducible to a set of coupled second-order differential equations. The method is tested for one-dimensional plane-parallel participating medium. Three quadrature sets are proposed for the method, and the convergence of the method with the proposed sets is explored. The SKN solutions are compared with the exact, PN, and SN solutions. The SK1 and SK2 approximations using quadrature Set-2 possess the capability of solving radiative transfer problems in optically thin systems.

Photon density wave for imaging through random media

The passage of a photon density wave through random media has been investigated extensively for medical imaging based on the diffusion approximation. In this paper, the photon density wave is studied based on the exact time-dependent vector radiative transfer theory. Both continuous and pulse photon density waves are analysed in a plane parallel medium using Mie scattering and the discrete ordinates method. The photon density wave shows superior properties over regular waves in several aspects. It has a narrower angular spectrum and maintains the original pulse shape. It also preserves the degree of polarization and increases the cross-polarization discrimination. These properties of a photon density wave suggest its potential for improving imaging. Thus, we apply the photon density wave to an imaging problem and show that it improves the quality of the images compared to other conventional imaging techniques.

Discrete-ordinates solution for radiative-transfer problems

The discrete-ordinates method is used to solve radiative-transfer problems, in plane-parallel media. We present a generalized analytical discrete-ordinates formulation for solving single- and multi-region problems. Included in our model are internal sources, reflecting and emitting boundaries, incident distribution of radiation on each surface and a beam incident on one surface. Numerical results for three test problems are reported.