Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection

Analytical techniques are used to solve a class of inverse radiative-transfer problems relevant to finite and semi-infinite plane-parallel media. While the assumption of isotropic scattering is made, diffuse reflection is allowed at the surface, for the semi-infinite case, and at both surfaces for the case of a finite layer. For the general case based on a semi-infinite medium, a cubic algebraic equation is used to define the basic result, but for the specific case of a semi-infinite medium illuminated by a constant incident distribution of radiation, very simple exact expressions are developed for the albedo for single scattering ? and the coefficient for diffuse reflection ρ. Analytical results are also developed (again in terms of a cubic algebraic equation) for the case of a finite layer with equal reflection coefficients relevant to the two surfaces. For the general case of a finite layer with unequal reflection coefficients, two specific formulations are given. The first algorithm is based on a system of three quadratic algebraic equations for the two reflection coefficients ρ₁ and ρ₂ and the single-scattering albedo ?. Secondly, an elimination between these three algebraic equations is carried out to yield two coupled algebraic equations for ρ₁ and ρ₂ plus an explicit expression for ? in terms of ρ₁ and ρ₂. In addition, an exact expression for τ₀, the optical thickness of the finite layer, is developed in terms of ?, ρ₁ and ρ₂. As is typical with the considered class of inverse problems in radiative transfer, all surface quantities are either specified or considered available from experimental measurements. All basic results are tested numerically.     

Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method

Application of the lattice Boltzmann method (LBM) recently proposed by Asinari et al. [Asinari P, Mishra SC, Borchiellini R. A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium. Numer Heat Transfer B 2010; 57:126–146] is extended to the analysis of transport of collimated radiation in a planar participating medium. To deal with azimuthally symmetric radiation in planar medium, a new lattice structure for the LBM is used. The transport of the collimated component in the medium is analysed by two different, viz., flux splitting and direct approaches. For different angles of incidence of the collimated radiation, the LBM formulation is tested for the effects of the extinction coefficient, the anisotropy factor, and the boundary emissivities on heat flux and emissive power distributions. Results are compared with the benchmark results obtained using the finite volume method. Both the approaches in LBM provide accurate results.     

Differential approximations for transient radiative transfer in refractive planar media with pulse irradiation

Transient radiative transfer in an anisotropically scattering refractive planar medium with pulse irradiation is solved by various approximation methods, such as P?1, P?1 parabolic, P_1/3 and two-flux. The time-resolved transmittance and reflectance are calculated for various radiative parameters, and are compared with those obtained by the discrete ordinate method (DOM). Among the approximation methods considered, the P_1/3 approximation is the better one, because its results are in overall good agreement with those obtained by the more rigorous DOM, except the transmittance around the peak for neither thin nor very thick slabs. It is found that the curved paths of radiation and the internal reflection of the back scattered radiation enhance the effect of scattering.     

Second-order ellipsometric coefficients

We derive analytic expressions for the reflection amplitudes of s and p polarized electromagnetic radiation incident on a planar interface profile of arbitrary form, to second order in the parameter qa, where q is the component of the wavenumber perpendicular to the interface, and a is a length proportional to the interface thickness. New comparison identities, relating the reflection and transmission amplitudes of the p-wave to those for any reference profile, are derived. The second-order results are obtained by using one of these identities, and an integro-differential form of the p-wave equation.     

Radiative transport in a planar medium exposed to azimuthally unsymmetric incident radiation

An analytical solution is developed for the radiation field engendered by illuminating an absorbing/anisotropically scattering planar medium with an azimuthally unsymmetric incident flux. The boundaries of the medium are allowed to possess both specular and diffuse reflection characteristics. The problem is mathematically reduced to solving a system of azimuthally independent equations which may be analyzed by many existing techniques. A solution scheme is formulated in terms of singular eigenfunctions and solved using a modified F_N method. Computations are performed for the azimuthally symmetric case of irradiation by parallel rays at oblique incidence. Several different phase functions are considered and results are presented for the discrete eigenvalues, the heat flux distributions within the medium, the reflectance and transmittance of the slab and the angular distribution of the intensities at the boundaries.     

Experimental observation of optical diffraction radiation

The optical diffraction radiation of ultrarelativistic electrons at the boundary of a conducting medium is observed experimentally. Backward diffraction radiation, which, like transition radiation, is emitted at the angle of specular reflection from the target, is detected. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 10, 760–764 (25 May 1998)     

Forward light reflection from a moving inhomogeneity

The reflection of monochromatic and quasi-monochromatic pulsed light incident on a moving inhomogeneity in the optical characteristics of a medium having plasma-type dispersion has been analyzed. The velocity V of the inhomogeneity, induced in the medium by an intense laser pulse, has been changed by varying its carrier frequency. It has been shown that the usual back-reflection mode, when the reflected radiation pulse moves in the direction opposite the direction of incident radiation, is implemented only if the velocity V is less than the critical value V _min, which depends on the carrier frequency of the incident radiation pulse. It has been found that reflected radiation moves in the same direction as the incident radiation in a certain range of the velocity V _min < V < V _max (forward reflection). In this case, the reflected radiation pulse begins to lag behind a fast-moving inhomogeneity. When V _max < V < c, where c is the speed of light in vacuum, the group velocity of the incident radiation pulse is less than the speed of inhomogeneity, and there is no reflection. Analytical treatment is supported by numerical simulation.     

Three-dimensional radiative transfer using a fourier-transform matrix-operator method

The three-dimensional equation of transfer for a scattering medium with planar geometry is solved by using a spatial Fourier transform and extending matrix-operator techniques developed previously for the one-dimensional equation. Doubling and adding algorithms were derived by means of an interaction principle for computing the fourier-transformed radiation field. The resulting expressions fully describe the radiative transfer process in a scattering medium, inhomogeneous in the x-, y- and z-directions, illuminated from above by an arbitrarily general intensity field and bounded from below by a surface with completely general reflection properties.     

Equalization of reflectance of parallel-polarized electromagnetic plane waves at normal and oblique incidence of interfaces between transparent media and its use for measurement of the dielectric constant

A new angle of incidence of significance, when considering the reflection of electromagnetic waves at interfaces between transparent media, is defined. At this angle, denoted by φ_e, the reflection coefficient of parallel-polarized radiation is equal in magnitude and opposite in sign to the reflection coefficient at normal incidence. No similar angle exists for the perpendicular polarization. If ε is the relative dielectric constant, i.e., the ratio of the dielectric constant of the medium of refraction to that of the medium of incidence, we find that tan φ_e=(ε²+ε)^1/2. Measurement of φ_e, by equalization of the absolute (intensity) reflectances at normal and oblique incidence, allows ε to be determined using the inverse relation ε=(tan²φ_e+1/4)^1/2−1/2.     

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.     

Simultaneous solution of Kompaneets equation and radiative transfer equation in the photon energy range 1-125 keV

Radiative transfer equation in plane parallel geometry and Kompaneets equation is solved simultaneously to obtain theoretical spectrum of 1-125 keV photon energy range. Diffuse radiation field are calculated using time-independent radiative transfer equation in plane parallel geometry, which is developed using discrete space theory (DST) of radiative transfer in a homogeneous medium for different optical depths. We assumed free-free emission and absorption and emission due to electron gas to be operating in the medium. The three terms n, n² and (∂n/∂x_k) where n is photon phase density and x_k=(hν/kT_e), in Kompaneets equation and those due to free-free emission are utilized to calculate the change in the photon phase density in a hot electron gas. Two types of incident radiation are considered: (1) isotropic radiation with the modified black body radiation I^MB[1] and (2) anisotropic radiation which is angle dependent. The emergent radiation at τ=0 and reflected radiation τ=τ_max are calculated by using the diffuse radiation from the medium. The emergent and reflected radiation contain the free-free emission and emission from the hot electron gas. Kompaneets equation gives the changes in photon phase densities in different types of media. Although the initial spectrum is angle dependent, the Kompaneets equation gives a spectrum which is angle independent after several Compton scattering times.     

Second harmonic generation in a polarized Dirac vacuum

The Dirac vacuum, polarized by a strong electric field E₀, is discussed as nonlinear medium for laser radiation (LR). It is shown that such a medium leads to (a) a LR refractive index appearing, (b) LR polarization plane rotation, (c) LR second harmonic generation. It is proposed to use these effects for E₀ field diagnostics.     

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.     

Time-dependent radiative transfer using Chapman-Enskog maximum entropy method

The time-dependent problems of radiative transfer involve a coupling between radiation and material energy fields and are nonlinear because of proposed temperature dependence of the medium characteristics in semi-infinite medium with Rayleigh anisotropic scattering. By means of the limited flux, Chapman-Enskog and maximum entropy technique the time-dependent radiative transfer equation has been solved explicitly. The maximum entropy method is used to solve the resulting differential equation for radiative energy density. The calculations are carried out for temperature (normalized dimensionless) Θ(x,τ), radiative energy density and net flux with Rayleigh and anisotropic scattering for different space at different times.     

Solid Fabry-Perot etalons for X-rays

Solid Fabry-Perot etalons for X-rays have been constructed using sputter deposition techniques, each etalon consisting of two Layered Synthetic Microstructures (LSM) Bragg diffraction structures separated by a carbon spacer. The individual LS mirrors contain fifteen tungsten layers (t_w = 8.5 Å) separated by carbon layers (t_c = 19.1 Å. The thick carbon spacers act as resonant cavities; for the structures reported on here the spacer thicknesses, t_sp, are 496.6 Å and 981 Å. The structures were characterized at grazing incidence in reflection using Cu Kα (λ = 1.5418 Å) radiation. The measured response of the etalons agrees well with calculation. Observed reflection efficiencies for Cu K_α were approximately 50 percent of that calculated. This discrepancy is believed to be the result of the interfacial roughness (～3.25 Å) between component layers and the sensitivity of the etalon response to the divergence of the incident X-ray beam.     

Neutron optics of strongly absorbing media and interaction of long-wave neutrons with gadolinium films

The modern state of neutron optics of absorbing media is briefly surveyed. In all probability, there are no physics arguments that would constrain, in the case of strong absorption, the applicability of the commonly accepted Fermi-Foldy dispersion law for neutron waves. In accord with previously known results, it is found that the coefficient of reflection of neutrons from the boundary of a strongly absorbing medium tends to unity with decreasing velocity of neutrons incident on this medium. At low neutron energies peculiar to the case of ultracold neutrons, the complex scattering length for neutron-nucleus interaction proves to be constant, whence it follows that the cross section for neutron capture by a free nucleus obeys the 1/v law. The cross section for the analogous process on nuclei within a medium is described by the 1/v′ law, where v′=?k′/m, with k′ being the real part of the neutron wave number in the medium. As the incident-neutron velocity v decreases, the velocity v′ in a medium tends to some limiting value. From the coefficient of reflection of cold neutrons that is measured as a function of the wavelength and the angle of incidence, a refined value is found for the real part of the scattering length for neutron interaction with gadolinium nuclei. An experiment was performed where ultracold neutrons were transmitted through thin samples containing natural gadolinium. In analyzing the results of this experiment, use was made of the value found here for the real part of the neutron-nucleus scattering length. The experiment indicates that the imaginary part of the scattering length is a constant or, what is the same, that, for neutron velocities ranging from 4 to about 120 m/s, the 1/v law for the cross section for neutron capture by a free nucleus is valid to within 6%.     

Long-range correlations upon wave propagation in random media under the conditions of strong internal reflection from their boundaries

The large-scale behavior of the spatial distribution of radiation in a random medium is investigated under the assumption of strong internal reflection from its boundaries. The qualitative variations of the angular coherent backscattering spectrum and long-range spatial intensity correlations in the transmitted and reflected radiation fluxes are predicted. Zh. éksp. Teor. Fiz. 113, 291–312 (January 1998)     

Amplification by reflection from an active medium

The reflectance R of an active (inverted) medium is theoretically shown to be equal to the inverse of the Fresnel reflectance of an absorbing medium when the imaginary parts of the dielectric permeabilities of the active and the absorbing medium differ only in sign. The reflected wave is always amplified (R > 1). The phase shifts upon reflection at the active and the absorbing medium are equal. These results for monochromatic light presuppose a sufficient thickness of the active medium.     

Classical and quantum theory of synergic synchrotron-Čerenkov radiation

The power spectra of ?erenkov, synchrotron, and synchrotron-?erenkov radiation are calculated both classically (by source methods) and quantum mechanically (by mass operator methods). The synergic features of the synchrotron-?erenkov radiation are pointed out. The existence of a transition region near nβ = 1 [n(ω, H) = index of refraction of the medium; $\text{β =}\text{v}\text{c}$, v - velocity of the charged particle] coupled with the intrinsic dispersion of the medium is then applied to the discussion of the suppression of X-ray radiation, the construction of X-ray counters, the detection of the quantum corrections, and the modification of the synchrotron radiation from pulsars.     

Photochemically induced burning of a spectral line in infrared reflection spectra of silicates

The change in IR reflection spectra of natural silicates caused by exposure of the surface of these materials to pulsed radiation of a CO₂ laser (P=10⁷ W/cm²,τ _p =200 ns) was investigated. Burning of a line in the IR reflection spectra in the vicinity of 10.6µm was observed. This burning of a line is attributed to laser-induced selective sublimation of Si-O groups from the silicates. The phenomenon of selective sublimation of silicon dioxide from the silicates was confirmed by x-ray electron-microprobe analysis of laser-irradiated samples.