Fresnel Boundary Conditions Research Articles

Boundary-layer structures arising in linear transport theory.

We consider boundary-layer structures that arise in connection with the transport of neutral particles (e.g., photons or neutrons) through a participating medium. Such boundary-layer structures were previously identified by the authors in certain particular cases [Phys. Rev. E 104, L032801 (2021)2470-004510.1103/PhysRevE.104.L032801]. Extending the previous work to anisotropic scattering and general Fresnel boundary conditions, this contribution presents computational algorithms which (1) resolve the aforementioned layers as well as previously unreported boundary layers associated with Fresnel boundary transmission and reflection, and (2) yield accurate simulations at fixed computational cost for transport under phase functions with arbitrarily strong anisotropy. The present paper additionally includes (3) Mathematical proofs which justify the numerical methods proposed for resolution of boundary-layer structures. The impact of the new theory on algorithmic performance is demonstrated through a series of 1D computational benchmarks that emulate typical photon- and neutron-transport applications such as, e.g., optical tomography, and nuclear reactor analysis and design. Experimental results for transmission of photons through turbid media are presented, exhibiting close agreement between simulated and experimental data. As illustrated by means of a variety of numerical results, the proposed boundary-layer-based approach tackles transport problems with unprecedented accuracy and efficiency.

Parallel inverse-problem solver for time-domain optical tomography with perfect parallel scaling

This paper presents an efficient parallel radiative transfer-based inverse-problem solver for time-domain optical tomography. The radiative transfer equation provides a physically accurate model for the transport of photons in biological tissue, but the high computational cost associated with its solution has hindered its use in time-domain optical-tomography and other areas. In this paper this problem is tackled by means of a number of computational and modeling innovations, including (1) A spatial parallel-decomposition strategy with perfect parallel scaling for the forward and inverse problems of optical tomography on parallel computer systems; and, (2) A Multiple Staggered Source method (MSS) that solves the inverse transport problem at a computational cost that is independent of the number of sources employed, and which significantly accelerates the reconstruction of the optical parameters: a six-fold MSS acceleration factor is demonstrated in this paper. Finally, this contribution presents (3) An intuitive derivation of the adjoint-based formulation for evaluation of functional gradients, including the highly-relevant general Fresnel boundary conditions—thus, in particular, generalizing results previously available for vacuum boundary conditions. Solutions of large and realistic 2D inverse problems are presented in this paper, which were produced on a 256-core computer system. The combined parallel/MSS acceleration approach reduced the required computing times by several orders of magnitude, from months to a few hours.

Method for Calculating the Optical Properties of Composites Based on a Transparent Matrix with Residual Porosity and Metal Nanoparticles

A method is proposed for calculating the optical characteristics of composites based on a transparent matrix with residual porosity and metal nanoparticles by solving the transport equation for monochromatic radiation employing a spherical harmonic method and Fresnel boundary conditions. The method is tested by modeling the transport of monochromatic radiation at four wavelengths of practical importance in cyclotrimethylenetrinitramine–aluminum nanoparticle composites with a Rayleigh distribution of pores with respect to radius. It is shown that in the case of small-radius pores the reflectivity increases as the mass fraction of nanoparticles is increased. In the case of pores with large radii, a kink corresponding to complete filling of the pores appears on the dependence of the transmission on the nanoparticle mass fraction. The application of these results to inverse problems in the spectroscopy of lightscattering system is discussed.

A study into light scattering and absorption by aluminum nanoparticles in PETN

The paper is devoted to experimental and theoretical research into nanopartides' optic properties in pentaerythritol tetranitrate (petn) matrix. A photometric sphere was applied for the transmittance and sum of transmittance and reflectance measurement of petn pressed pellets containing aluminum nanoparticles at the light wavelength 643 nm. The theory of light propagation in terms of spherical harmonics solution of radiative transfer equation in the slab geometry with Fresnel boundary conditions was developed. The properties of aluminum nanoparticles were evaluated in terms of Mie theory. The absorbed energy distribution inside the sample was calculated. It was shown that the Beer's type law is applicable approximately. The apparent light absorption cross section determined, which takes into account both scattering and absorption, is bigger than the geometrical one. The aluminum refractive index value, estimated during comparison of theory with the experimental data, agrees well with the handbook's data.

Radiance distribution simulation in a transparent medium with fresnel boundaries containing aluminum nanoparticles

The radiative transfer equation in a slab with Fresnel boundaries was solved by the spherical harmonics method in the case of a transparent medium containing aluminum nanoparticles. A computer program was developed, the angular radiance distribution on the slab boundaries and in the slab center were calculated. It was concluded that Fresnel boundary conditions make the angular distribution on the slab boundaries asymmetric. The use of angular radiance distribution for solving the reverse problems of scattering spectroscopy was discussed.

An application of the K-integrals for solving the radiative transfer equation with Fresnel boundary conditions

We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials.

A note on radiative transfer in a finite layer

An ADO (analytical discrete ordinates) solution is used to establish a concise and accurate result for a basic radiative transfer problem in a finite layer described by the grey equation of transfer with general anisotropic scattering. As a specific application, the solution is evaluated for the case of Fresnel boundary conditions to yield numerical results (of a high standard) for several specific cases.

The constant source problem with Fresnel reflection

The classic constant source problem for a half-space is generalized to include the effect of refraction at the boundary by inclusion of the Fresnel boundary conditions. The problem is solved using the Wiener–Hopf technique with both specular and diffuse reflection. The non-singular Fredholm integral equations that arise for the surface angular distribution are solved numerically and the solutions are illustrated by a number of results in graphical and tabular forms. The significant effect of refraction on the surface flux and current and the associated angular distributions is highlighted.

Radiation transport and internal reflection in a two region, turbid sphere

We construct an integral equation for the flux intensity in a scattering and absorbing two-region turbid spherical medium using the integro-differential form of the radiative transfer equation. The sphere is uniformly irradiated by an external source of arbitrary angular distribution and contains a distributed volume source. Anisotropic scattering is accounted for by the transport approximation. The Fresnel boundary conditions, which incorporate reflection and refraction, are used at the outer surface and at the interface between the two regions. In this respect, some new interfacial boundary conditions are introduced. For the special case of a non-scattering medium, we obtain exact solutions for specular reflection. Some numerical examples are given which show qualitative agreement with some recent work of other authors. Of particular interest are the emergent angular distribution and the albedo of the surface as a function of the refractive index and the radii of the two regions. We also draw attention to the fact that the boundary conditions at the interface differ according to the relative values of the refractive indices in the two regions. The interfacial boundary conditions for use in diffusion theory are derived and compared with those of Aronson [Boundary conditions for diffusion of light. J opt Soc Am 1995;12:2532]. In appendix B, we show how diffusion theory may be used to include scattering into the problem in a simple way.

Radiation transport and internal reflection in a sphere

We construct an integral equation for the flux intensity in a scattering and absorbing medium using the integro-differential form of the radiative transfer equation in a sphere. The sphere is uniformly irradiated by an external source of arbitrary angular distribution. The Fresnel boundary conditions, which incorporate reflection and refraction, are used. For the special cases of a non-scattering medium, and in the limit of an optically transparent medium, we obtain exact solutions for specular and diffuse refection. Some numerical examples are given which give qualitative agreement with some recent work of Tian and Chiu (JQSRT, 2005).

The albedo problem with Fresnel reflection

The classic albedo problem for a half-space is generalised to include the effect of refraction at the boundary by inclusion of the Fresnel boundary conditions. The problem is solved using the Wiener–Hopf technique with both specular and diffuse reflection. The non-singular Fredholm integral equations that arise for the surface angular distribution are solved numerically and the solutions are illustrated by a number of results in graphical and tabular form. The significant effect of refraction on the albedo and the associated angular distributions is highlighted.

The optical bistability of polarisation in B5NH4 crystal caused by the optical Kerr effect

We present experimental results concerning the polarisation optical bistability in the B 5 NH 4 crystal together with an attempt of its theoretical explanation based on third-order optical nonlinearity and Fresnel's boundary conditions. In our experiments a strong laser beam is transmitted through a crystal plate of B 5 NH 4 . The polarisation state of the transmitted light depends in a bistable way on the intensity of the incident light.

Observation of the effect of refraction on x rays diffracted in a grazing-incidence asymmetric Bragg geometry.

The positions of x-ray diffraction peaks from a polycrystalline iron oxide thin film have been measured in a grazing-incidence, asymmetric Bragg geometry for incidence angles between 0.09\ifmmode^\circ\else\textdegree\fi{} and 1.04\ifmmode^\circ\else\textdegree\fi{}. The peak positions are shifted from the Bragg angles 2${\ensuremath{\theta}}_{B}$ due to refraction of the incident x rays as they penetrate the air-solid interface. The shifts are quantitatively consistent with those obtained by applying Fresnel boundary conditions for reflection and refraction of a plane electromagnetic wave at an interface between air and a uniform, absorbing dielectric and they provide confidence in the validity of the Fresnel boundary conditions in the x-ray regime.

The Ewald–Oseen theorem in the x-ray frequency region: A microscopic analysis

We address the problem of the proper microscopic description of the propagation of x rays in condensed matter. In the optical region of the spectrum, Fresnel boundary conditions may be used; these boundary conditions can be regarded as a consequence of the Ewald–Oseen extinction theorem. This theorem is, however, generally derived in the dipole limit, where the wavelength of the light is large compared with the dimensions of the scatterers. Since x-ray wavelengths are comparable to atomic dimensions, it is necessary to re-examine the validity of the Ewald–Oseen theorem and the consequent boundary conditions for x-ray optics. In this paper, we demonstrate that the Ewald–Oseen extinction theorem may also be derived in the limit where the radiation frequency is high compared with atomic transition frequencies, a condition which holds for x-ray radiation. This result therefore justifies the use of macroscopic dielectric theory with Fresnel boundary conditions to describe the x-ray reflectivity of liquid surfaces.