Coherent transmission and reflection of a two-dimensional planar photonic crystal

A method for modeling the radial distribution function for particles of a two-dimensional planar photonic crystal in the form of a monolayer of spatially ordered monodisperse spherical particles is proposed. The coherent transmission and reflection coefficients for layers under normal illumination are calculated in the quasi-crystalline approximation of the multiple wave scattering theory. The dependence of the coherent transmission and reflection of the layer on the degree of ordering of the spherical particles is investigated. The influence of the long-range order on the coherent transmission and reflection coefficients for layers with triangular, square, and hexagonal lattices is estimated. Monolayers of weakly absorbing dielectric and strongly absorbing metallic particles are considered.     

Small-slope approximation for electromagnetic wave scattering at a rough interface of two dielectric half-spaces

Abstract

The small-slope approximation (SSA) for wave scattering at the rough interface of two homogeneous half-spaces is developed. This method bridges the gap between two classical approaches to the problem: the method of small perturbations and the Kirchhoff (or quasi-classical) approximation. In contrast to these theories, the SSA is applicable irrespective of the wavelength of radiation, provided that the slopes of roughness are small compared with the angles of incidence and scattering.

The resulting expressions for the SSA are given for the entries of an S-matrix that represents the scattering amplitudes of plane waves of different polarizations interacting with the rough boundary. These formulae are quite general and are valid, in fact, for waves of different origins. Apart from the shape of the boundary, some functions in these formulae are coefficients of the expansion of the S-matrix into a power series in terms of elevations. These roughness independent functions are determined by a specific scattering problem. In this paper they are calculated for the case of electromagnetic scattering at the interface of two dielectric half-spaces. In contrast to an earlier paper by the author, where only the formulae for the reflected field were presented, in this paper both reflected and transmitted fields are considered in detail.

The a priori symmetry relations that this scattering problem should obey (reciprocity and energy conservation) are formulated in terms of the S-matrix.

The statistical moments of scattering amplitudes are directly related to the mean-reflection coefficient and scattering cross sections, which are usually determined experimentally. The corresponding formulae are given here for the case of Gaussian space-homogeneous statistics of roughness.     

Method for Describing the Angular Distribution of Optical Radiation Scattered by a Monolayer of Ordered Spherical Particles (Normal Illumination)

We have developed a method for describing the angular distribution of intensity of radiation scattered by a monolayer of homogeneous spatially ordered monodisperse spherical particles normally illuminated by a plane circularly polarized electromagnetic wave. The method is based on the quasicrystalline approximation (QCA) of the theory of multiple scattering of waves (TMSW) using the multipole expansion of fields and the tensor Green function in vectorial spherical wavefunctions. The method is applied for analyzing the characteristics of radiation scattered by a partially ordered monolayer and a monolayer with a nonideal lattice. The results of calculations are compared with the available experimental data on the position of the first-order diffraction peak on the angular and spectral dependences of the intensity of radiation scattered by a closely packed monolayer with a nonideal triangular lattice of SiO₂ particles. Good conformity of the results has been established.     

非常密集随机分布球粒子相关散射的数值模拟

讨论非常密集(占空比>0.4)随机分布球粒子的散射.用Monte Carlo方法和随机搅拌产生非常密集随机分布粒子,求解电场体积分方程的解,获得散射、吸收和消光系数及有效介电常数.与现有的准晶体近似、具有相干势的准晶体近似和Maxwell-Garnett混合方程等几种方法作了比较.非常密集随机分布粒子表现出群聚性产生的大粒子散射效应.     

Light Reflection and Transmission by layers with oriented spheroidal particles

On the basis of the numerical solution of the radiative transfer equation (RTE), the influence of non-sphericity of oriented spheroidal particles on radiation intensity formation in a layer with a multiple scattering regime is considered. For solving the RTE with characteristics depending on the propagation direction of the radiation beam, an algorithm is developed based on the method of layer doubling. Calculations are presented of angle correlations for intensities of radiation in a layer with Fresnel reflection on the boundaries, and reflection and transmission coefficients.     

Optical properties of structures composed of periodic,quasi-periodic,and aperiodic sequences of particulate monolayers

The spectra of the coherent transmission and reflection coefficients of multilayers consisting of the periodic, Fibonacci (quasi-periodic), and Thue–Morse (aperiodic) sequences of plane-parallel monolayers of monodisperse spherical alumina and silica particles are investigated using the quasi-crystalline approximation (QCA) and the transfer matrix method (TMM). The additional opportunities for the transmission and reflection spectra manipulation in comparison with the periodic sequence of monolayers are demonstrated. Photonic band gaps in the spectra of the particulate structures are shifted to the short-wavelength range in comparison with those for systems of homogeneous layers. The shift is larger for the Thue–Morse sequence. The widths of the photonic band gaps for particulate systems are narrower than the ones for multilayers consisting of homogeneous layers of an equivalent volume of matter. The results can be used to create optical, optoelectronics, and photonics devices—for example, multispectral filters, light emitting diodes, solar cells, displays.     

Scattering and Transmission by a Monolayer of Spheres: A Study on the Monolayer Structure

The angular distribution of scattered light and the transmission of radiation through a monolayer of monodisperse spherical particles at variable particle concentration are studied. The scattering of light by a single particle is calculated with the classical Lorentz‐Mie theory. For a monolayer of mono‐dispersed spherical particles, if the monolayer density is less than 0.5 and the particle size parameter is larger than 5, effects from multiple scattering and dependent scattering can be excluded so that only steric interactions are considered. It is found that the scattering pattern, especially in the forward and backward directions, and the transmission are strongly dependent on the monolayer density.     

Multiple scattering of acoustic waves by a half-space of distributed discrete scatterers with modified T-matrix approach

In studying the multiple scattering of acoustic waves by a half-space of distributed discrete scatterers, the quasicrystalline approximation(QCA) approach together with the hole correction (HC) or the pair distribution functions (PDF) have been used extensively, in which a system of simultaneous equations must be solved to determine the effective propagation constant and the expansion coefficients of the coherent exciting field. In this paper, we analyse the same problem under Foldy's approximation (EFA) by using the so-called modified T-matrix approach (MTMA) which was first proposed by Twersky. Two equations in a considerably simple and clear form are obtained for determining the effective propagation constant and the amplitude of the coherent transmitted field, as the scatterers are identical spheres. The numerical results in the low-frequency limit are also discussed in brief.     

The layer multiple-scattering method for calculating transmission coefficients of 2D phononic crystals

We extend the layer multiple-scattering theory (LMST) to elastic waves propagating in two-dimensional (2D) periodical composites. The formalism to calculate the reflection and transmission coefficients for elastic waves through finite slabs is presented. In this spirit, the crystal is viewed as a sequence of identical monolayer which has one-dimensional (1D) periodicity along a given direction. The reflection and transmission coefficients for a multilayer slab can be obtained by a double-layer scheme through the calculation of the scattering matrix of a monolayer. To demonstrate the application of this formalism, we calculate transmission coefficients for systems consisting of pure solid components or mixing (solid and fluid) components. The validity of this method is checked by both band structure calculations and transmission measurement of ultrasonic experiment.     

On the theory of X-ray coherent reflection from a rough surface

A solution of the problem of X-ray specular reflection from a statistically rough surface is presented. It is based on using the Green function formalism. The Kirchhoff formula is used to describe the transmission of the wave field through a rough interface. Generally, microscopically exact expressions for the coefficients of transmission through a rough surface and reflection from it are obtained by taking multiple scattering effects into account. Averaging of the obtained expressions over possible realizations of random roughness of the interface between media allows to obtain rigorous expressions for specular reflection and transmission coefficients. The behavior of exact solutions in the limiting case of infinite correlation lengths is studied. It is shown that, in this case, the obtained solution corresponds to the Debye-Waller normalization. Expressions for the reflection and transmission coefficients making it possible to carry out numerical calculations are obtained in the Bourret approximation of multiple-scattering theory.     

Albedo problem for pure-triplet scattering in finite planar media

Pure triplet scattering in neutron transport through a finite plane-parallel medium is considered. Galerkin approximation is introduced to calculate some functionals arising in radiative transfer problems such as reflection and transmission coefficients. Some of numerical results are compared with the results calculated by Pomraning-Eddington Variational method.     

Coherent transmission and angular structure of light scattering by monolayer films of polymer dispersed liquid crystals with inhomogeneous boundary conditions

We consider monolayer polymer films with oriented droplets of a nematic liquid crystal (LC). Relations for the coherent transmission coefficients of a layer of oriented ellipsoidal droplets and for the intensity of light scattered by monolayers of spherical and spheroidal droplets have been obtained. The amplitude-phase screen model and the interference approximation of the theory of multiple wave scattering have been used. To describe light scattering by an individual ellipsoidal droplet with inhomogeneous surface binding, we have developed an anomalous diffraction approximation. For monolayers of spherical LC droplets, the coherent scattering coefficients and the angular scattering structure have been analyzed. The internal structure of nematic droplets have been calculated by the relaxation method based on the solution of the minimization problem of the free energy volume density. We have studied basic regular features of light scattering by a monolayer with homogeneous and inhomogeneous boundary conditions at the LC-polymer interface. We show that, for films that contain droplets with inhomogeneous boundary conditions of the tangentially normal type, the angular structure of the scattered light is asymmetric with respect to the polar scattering angle.     

弱电离尘埃等离子体层反射与透射的SO-FDTD方法分析

将弱电离尘埃等离子体复介电常数表示以jω 为自变量的分式多项式形式, 利用移位算子时域有限差分(SO-FDTD)方法, 给出了弱电离尘埃等离子体频域本构关系的时域形式. 推导了弱电离尘埃等离子体电磁特性的时域递推关系式. 用SO-FDTD方法计算了弱电离尘埃等离子体层的反射和透射系数, 分析了尘埃粒子浓度和尘埃粒子半径的变化对反射系数和透射系数的影响. 结果表明: 有尘埃粒子时的反射系数比没有尘埃粒子时反射系数小, 而透射系数要比没有尘埃粒子时的透射系数大; 当尘埃粒子浓度或尘埃粒子半径增大时, 其反射系数均减小.     

Light absorption by ordered planar structure of crystalline silicon particles: Simulation as applied to enhancement of the solar cell performance

Light absorption by a single spherical crystalline silicon (c-Si) particle and by monolayers of submicron c-Si particles arranged in triangular lattice is examined within the wavelength range from 0.28 to 1.12 μm. Data for spectral and integrated (over the terrestrial solar spectral irradiance “Global tilt” ASTM G173-03) absorption coefficients of monolayer demonstrating the possibility to enhance light absorption by a particulate active layer of crystalline silicon are presented. They are calculated in the quasicrystalline approximation of the theory of multiple scattering of waves. It is shown that in the narrow wavelength intervals (up to 10 nm) the spectral absorption coefficient of monolayer can be more than 100 times larger than the one of the plane-parallel plate of equal volume of material. Some results on the transmittance and reflectance of partially ordered monolayer are presented as well. The potentialities of light absorption enhancement in a single monolayer and three-monolayer system are considered.     

Energy conservation of the scattering from one-dimensional random rough surfaces in the high-frequency limit

Energy conservation of the scattering from one-dimensional strongly rough dielectric surfaces is investigated using the Kirchhoff approximation with single reflection and by taking the shadowing phenomenon into account, both in reflection and transmission. In addition, because no shadowing function in transmission exists in the literature, this function is presented here in detail. The model is reduced to the high-frequency limit (or geometric optics). The energy conservation criterion is investigated versus the incidence angle, the permittivity of the lower medium, and the surface rms slope.     

On the high-frequency limit of the second-order small-slope approximation

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

A practical cross-section for scattering from rough surfaces at very low grazing angles in both the forward and backward directions

In this paper we develop an extension of the small slope approximation (SSA) for scattering from randomly rough Dirichlet surfaces, which includes some multiple scattering. This extension is designated by SSA+. We focus on scattering at very low grazing angles where multiple scattering of both the incident and scattered fields is of importance. Numerical results for the SSA+ bistatic scattering cross-section for very low forward grazing angles are presented using the Gaussian roughness spectrum and for both very low forward and very low backward grazing angles using the Pierson–Moskowitz and modified power law spectra. The results are restricted to an angle of incidence of 80°. It is shown that when the lowest-order SSA gives reasonably accurate results, the SSA+ increases the accuracy up to at least the final 0.2° of grazing in the forward direction. In the backward direction, the SSA+ gives good results for the Pierson–Moskowitz spectrum, but the results are less dramatic.     

On the high-frequency limit of the second-order small-slope approximation

Abstract

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

Reprint of: Two-dimensional planar photonic crystals: Calculation of coherent transmittance and reflectance at normal illumination under the quasicrystalline approximation

The technique to calculate coherent transmission and reflection coefficients of two-dimensional (2D) planar photonic crystal (PC) at the normal to layer plane illumination is proposed. It is based on the quasicrystalline approximation of the theory of multiple scattering of waves. At this approximation spatial particle correlations are characterized by the radial distribution function. We propose the method of the 2D planar PC radial distribution function simulation. It consists in the calculation of coordination circles of ideal crystal lattice and following blurring them into the “rings” with fuzzy edges to describe the crystal lattice of an actual crystal. The width of the “rings” depends on the distance from coordinate origin. The blurring technique is proposed and discussed. The method allows simulation of the PCs lattices with various ordering degrees and spatial particle correlations. The results of numerical calculations of coherent transmittance and reflectance of monolayers with different orderings and refractive indices of spherical monodisperse particles are displayed.     

Two-dimensional planar photonic crystals: Calculation of coherent transmittance and reflectance at normal illumination under the quasicrystalline approximation

The technique to calculate coherent transmission and reflection coefficients of two-dimensional (2D) planar photonic crystal (PC) at the normal to layer plane illumination is proposed. It is based on the quasicrystalline approximation of the theory of multiple scattering of waves. At this approximation spatial particle correlations are characterized by the radial distribution function. We propose the method of the 2D planar PC radial distribution function simulation. It consists in the calculation of coordination circles of ideal crystal lattice and following blurring them into the “rings” with fuzzy edges to describe the crystal lattice of an actual crystal. The width of the “rings” depends on the distance from coordinate origin. The blurring technique is proposed and discussed. The method allows simulation of the PCs lattices with various ordering degrees and spatial particle correlations. The results of numerical calculations of coherent transmittance and reflectance of monolayers with different orderings and refractive indices of spherical monodisperse particles are displayed.