Link between the laws of geometrical optics and the radiative transfer equation in media with a spatially varying refractive index

We proposed in a previous paper [J.-M. Tualle, E. Tinet, Opt. Commun. 228 (2003) 33] a modified radiative transfer equation to describe radiative transfer in a medium with a spatially varying refractive index. The present paper is devoted to the demonstration that this equation perfectly works in the non-absorbing/non-scattering limit, what was contested by Martí-López and coworkers [L. Martí-López, J. Bouza-Domínguez, R.A. Martínez-Celorio, J.C. Hebden, Opt. Commun. 266 (2006) 44]. The assertion that this equation would imply a zero divergence of the rays is also commented.     

Analytical Model of Doppler Spectra of Light Backscattered from Rotating Convex Bodies of Revolution in the Global Cartesian Coordinate System

We present an analytical model of Doppler spectra in backscattering from arbitrary rough convex bodies of revolution rotating around their axes in the global Cartesian coordinate system. This analytical model is applied to analyse Doppler spectra in backseatter from two cones and two cylinders, as well as two ellipsoids of revolution. We numerically analyse the influences of attitude and geometry size of objects on Doppler spectra. The analytical model can give contribution of the surface roughness, attitude and geometry size of convex bodies of revolution to Doppler spectra and may contribute to laser Doppler velocimetry as well as ladar applications.     

A Time Domain Algorithm on the Reconstruction of Rough Surfaces

We consider the inverse scattering problem of a perfectly conducting one-dimensional rough surface in the case that the incident field is unnecessary to be time harmonic. Based on our previous investigation of the frequency domain algorithm, a new time domain algorithm is proposed, in which we approximate the incident pulse by a finite sum of time harmonic fields and then apply the frequency domain algorithm for time harmonic waves. Numerical experiments indicate that the time domain algorithm shows great accuracy of reconstruction of the surface profile and yields significant improvement than the frequency domain algorithm.     

Pairs of opposite decaying evanescent waves in random media and tunneling radiative transfer

A tunneling mechanism of radiative transfer through a dielectric random medium is revealed applying technique of Dyson and Bethe-Salpeter equations for electromagnetic wave multiple scattering by medium inhomogeneities (scatterers) with near fields effects in scattered fields. The mechanism consists in existing inside of a random inhomogeneity a pair of virtually opposite decaying evanescent waves whose interference results in energy flux.     

Local fields close to the surface of nanoparticles and aggregates of nanoparticles

A refined discussion of the near-field scattering of spherical nanoparticles and the electromagnetic fields close to the particle surface is given. New results for the dependence on the distance from the surface and the angular distribution of the scattered light in the near-field are given. It will be shown that the radial component of the electric field leads to striking differences in the phase functions in the near-field and the far-field. Exemplary computations are presented for Ag and Au particles with different size. In a second part the discussion is extended to assemblies of spherical Ag and Au nanoparticles. It will be shown that large near-fields at wavelengths commonly used in SERS experiments are obtained for aggregates. In the near-field scattering intensity “hot spots” mark regions between particles in the aggregate where the near-field is particularly high. Received: 4 May 2001 / Revised version: 20 July 2001 / Published online: 19 September 2001     

Continuum radiative heat transfer modeling in media consisting of optically distinct components in the limit of geometrical optics

Continuum-scale equations of radiative transfer and corresponding boundary conditions are derived for a general case of a multi-component medium consisting of arbitrary-type, non-isothermal and non-uniform components in the limit of geometrical optics. The link between the discrete and continuum scales is established by volume averaging of the discrete-scale equations of radiative transfer by applying the spatial averaging theorem. Precise definitions of the continuum-scale radiative properties are formulated while accounting for the radiative interactions between the components at their interfaces. Possible applications and simplifications of the presented general equations are discussed.     

Conservation laws and angular transverse shifts of the reflected and transmitted light beams

The relation between the angular transverse shifts of the beams reflected and transmitted at a plane interface of two isotropic transparent media is established. The derivation of this relation is based on the conservation law of the transverse component of the total Minkowski linear momentum, which takes place in the processes of the reflection and transmission of wave packets.     

Weak localization effects in the second-harmonic light scattered by random systems of particles

We report rigorous numerical simulations that show the presence of coherent backscattering effects in the second-harmonic generation and scattering of light by random systems of two-dimensional particles. Since the medium composing the particles is assumed to be homogeneous and isotropic, the second-harmonic field is generated mainly by surface effects. For the fundamental frequency, the results present a clear enhanced backscattering peak. In contrast, the second-harmonic scattering patterns present an intensity dip in the backscattering direction.     

Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the mutual coherence function of quasi-monochromatic electromagnetic Gaussian Schell-model (EGSM) beams propagating through turbulent atmosphere is derived analytically. By employing the lateral and the longitudinal coherence length of EGSM beams to characterize the spatial and the temporal coherence of the beams, the behavior of changes in the spatial and the temporal coherence of those beams is studied. The results show that with a fixed set of beam parameters and under particular atmospheric turbulence model, the lateral coherence of an EGSM beam reaches its maximum value as the beam propagates a certain distance in the turbulent atmosphere, then it begins degrading and keeps decreasing along with the further distance. However, the longitudinal coherence length of an EGSM beam keeps unchanging in this propagation. Lastly, a qualitative explanation is given to these results.     

Bounded speed of propagation for solutions to radiative transfer equations

The Radiative Transfer Equation is the nonlinear transport equation

Application of the modified differential approximation for radiative transfer to arbitrary geometry

The first-order spherical harmonics method (or P₁ approximation) has found prolific usage for approximate solution of the radiative transfer equation (RTE) in participating media. However, the accuracy of the P₁ approximation deteriorates as the optical thickness of the medium is decreased. The modified differential approximation (MDA) was originally proposed to remove the shortcomings of the P₁ approximation in optically thin situations. This article presents algorithms to apply the MDA to arbitrary geometry—in particular, geometry with obstructions, and inhomogeneous media. The wall-emitted component of the intensity was computed using a combined view-factor and ray-tracing approach. The Helmholtz equation, arising out of the medium-emitted component, was solved using an unstructured finite-volume procedure. The general procedure was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results. The accuracy of MDA was found to be superior to the P₁ approximation for all optical thicknesses. Its accuracy, when compared with the discrete ordinates method (both S₆ and S₈), was found to be clearly superior in optically thin situations, but problem dependent in optically intermediate and thick situations. For 3D geometries, calculation and storage of the view-factor matrix was found to be a major shortcoming of the MDA. In addition, for inhomogeneous media, calculation of optical distances requires a ray-tracing procedure, which was found to be a bottleneck from a computational efficiency standpoint. Several strategies to reduce both memory and computational time are discussed and demonstrated.     

Infiltration of colloidal crystal with nanoparticles using capillary forces: a simple technique for the fabrication of films with an ordered porous structure

A dipping method has been developed for the infiltration of nanoparticles into an opal template to fabricate high quality inverse opal. Titania and silica inverse opal films, with a uniform color over centimeter dimensions were derived. As there is no need for special substrates or equipment, a widespread application of this method is anticipated. Received: 23 August 2001 / Accepted: 27 August 2001 / Published online: 30 October 2001     

Generalized form of the direct and adjoint radiative transfer equations

The main goal of this paper is to give a rigorous derivation of the generalized form of the direct (also referenced as forward) and adjoint radiative transfer equations. The obtained expressions coincide with expressions derived by Ustinov [Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211]. However, in contrast to [Ustinov EA. Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211] we formulate the generalized form of the direct radiative transfer operator fully independent from its adjoint. To illustrate the application of the derived adjoint radiative transfer operator we consider the angular interpolation problem in the framework of the discrete ordinate method widely used to solve the radiative transfer equation. It is shown that under certain conditions the usage of the solution of the adjoint radiative transfer equation for the angular interpolation of the intensity can be computationally more efficient than the commonly used source function integration technique.     

Magnetic field effects on the optical properties of an azo-dye doped liquid crystal

Homeotropically aligned nematic liquid crystals doped with azo-dye were subjected both to a linear polarized light of a He–Ne laser and to a magnetic field perpendicular to the incident light beam. We found that the emerging light was elliptically polarized when using magnetic field strengths higher than the threshold value for the magnetic Freedericksz transition. The light transmission, the rotatory power (induced by azo-dye) and the ellipticity varied quasiperiodically when increasing magnetic field strength. The number and positions of maxima and minima depend on the cell thicknesses. Changes in the phase difference between the emergent ordinary and extraordinary rays were computed from the experimental data and the magnetic field dependence of the birefringence was determined.     

Generalized solutions of the radiative transfer equations in a singular case

This paper is devoted to the study of the radiative transfer equations: First, we prove a global existence theorem, which allows a blow-up of the opacity _v() when 0. Thus, it extends Mercier's previous result [13]. This proof relies mainly on a nonlinear version of Hille-Yosida theorem: see Crandall-Ligett [9].Then, we prove the uniqueness of the semigroup solving (TR), and some regularity results (in the class of functions with bounded variation).Finally, we prove the convergence of some splitting algorithms associated to (TR).     

Reduction of radiative transfer problems in semi-infinite media to linear Fredholm integral equations

Linear Fredholm integral equations are derived for the Stokes vector of polarized radiation, emergent from a scattering plane parallel semi-infinite medium, by means of the full range orthogonality and completeness properties of Case's eigensolutions. A renormalization concerning the eigenmode with the greatest discrete eigenvalue is applied, which permits us to obtain a new integral equation for the zeroth Fourier component of the radiation field. The kernel of the integral equations is given in terms of Case's eigenfunctions or of the Green's function matrix for an infinite medium. For isotropic scattering, it is shown that the integral equation can be solved by means of a very rapidly convergent Neumann series. Physical arguments lead to the conclusion that the renormalized Fredholm integral equations are well suited also for arbitrary phase matrices.     

An alternative Monte Carlo approach to the thermal radiative transfer problem

The usual Monte Carlo approach to the thermal radiative transfer problem is to view Monte Carlo as a solution technique for the nonlinear thermal radiative transfer equations. The equations contain time derivatives which are approximated by introducing small time steps. An alternative approach avoids time steps by using Monte Carlo to directly sample the time at which the next event occurs. That is, the time is advanced on a natural event-by-event basis rather than by introducing an artificial time step.     

ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton-Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket-bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.     

Comparison of the radiative transfer equations for constant and spatially varying refraction

The radiative transfer equation for scattering media with constant refraction index (RTE) and the radiative transfer equation for scattering media with spatially varying refraction index (RTEvri) are compared by using the principle of conservation of energy. It is shown that the RTEvri, not only accounts for the spatial variations of refraction index, but also contains a term that accounts for the divergence of the rays. The latter term is missing in the RTE. A corrected RTE is proposed.