Optical extinction of an assembly of spherical particles in an absorbing medium: Application to silver clusters in absorbing organic materials

The theory presented by Gerardy and Ausloos for the calculation of the linear optical response of aggregates of spherical particles is analytically continued for absorbing embedding media. The method is based on the calculation of the extinction rate by a single particle embedded in an absorbing matrix. Explicit expressions for the extinction and scattering cross-sections are given. The method is applied to calculate the energy losses in several organic matrices with embedded silver clusters. Comparison with experimental data shows a very good agreement. Received: 21 December 1998     

Study on the algorithm of computational ghost imaging based on discrete fourier transform measurement matrix

We have developed a model and realized an algorithm for the calculation of the coefficient of coherent (direct) transmission of light through a layer of liquid crystal (LC) droplets in a polymer matrix. The model is based on the Hulst anomalous diffraction approximation for describing the scattering by an individual particle and the Foldy-Twersky approximation for a coherent field. It allows one to investigate polymer dispersed LC (PDLC) materials with homogeneous and inhomogeneous interphase surface anchoring on the droplet surface. In order to calculate the configuration of the field of the local director in the droplet, the relaxation method of solving the problem of minimization of the free energy volume density has been used. We have verified the model by comparison with experiment under the inverse regime of the ionic modification of the LC-polymer interphase boundary. The model makes it possible to solve problems of optimization of the optical response of PDLC films in relation to their thickness and optical characteristics of the polymer matrix, sizes, polydispersity, concentration, and anisometry parameters of droplets. Based on this model, we have proposed a technique for estimating the size of LC droplets from the data on the dependence of the transmission coefficient on the applied voltage.     

A new solution to the problem of scattering of a plane wave by a multilayer confocal spheroid

We have constructed a solution to the problem of scattering by a nonconfocal multilayer particle. The main difficulty was to join expansions constructed in two spheroidal systems on either side of each boundary. As a result of a detailed consideration of relations between scalar wave spheroidal and spherical functions, we have succeeded in finding a representation of the former in terms of the latter and vice versa. In the final form, the joining of solutions is described by only one matrix, which depends on coefficients of representations of angle spheroidal functions in terms of associated Legendre functions of the first kind. Since the problem has been solved using an approach that involves the method of extended boundary conditions, the dimension of the system for numerical determining unknown coefficients is equal to the number of terms that are taken into account in field expansions and does not depend on the number of particle layers. Previously performed numerical calculations for confocal particles have shown a very high efficiency of the algorithm not only for particles that are close to spheres in shape, but also for strongly prolate and strongly oblate spheroids. In addition, the algorithm makes it possible to calculate optical properties of particles that have dozens of layers.     

Selection of response measurement locations to improve inverse force determination

The forces obtained by inverse methods are prone to errors. These arise due to a combination of errors in the measurements and high condition numbers in the matrix of transfer functions to be inverted. Ill-conditioning of the frequency response function matrix causes measurement errors to be magnified significantly. When the condition numbers are small, the measurement errors simply propagate without much amplification. Due to modal behaviour of the structure, the condition numbers can vary significantly over the frequency range and with the spatial location of the response measurements. The spatial variation can be quite considerable across the structure. The potential for using this characteristic to improve force determination is explored in this paper as an alternative to matrix regularization methods. The aim is to reduce error magnification in inverse methods by an ‘optimal’ spatial distribution of response locations. A method is proposed which is based on the minimization of the average condition number across the frequency range. If many possible locations are available, however, this can involve excessive calculation. An approximate method is therefore proposed which results in consistently good location selection for use in inverse force determination but involves much less computational effort. The error reduction in reconstructed forces is found to be significant in numerical simulations on a simply supported plate and in validation experiments.     

Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis

Matrix-based Gaussian paraxial ray-tracing is commonly used for first estimates in early design stages of optical systems. However, the usual techniques are suitable only for rays in systems with a common straight-line optical axis, and cannot be used in systems containing mirrors and/or prisms. Attard extended these methods to matrix-based skew ray-tracing for systems containing cylindrical lenses with orthogonal cylinder axes and spherical lenses with combinations of cylindrical lenses, but had problems with non-straight and non-coplanar optical relations. This paper develops a novel general matrix method for paraxial skew ray-tracing in systems with non-coplanar optical axes containing spherical and flat boundary surfaces. First-order Taylor series expansion is used to approximate skew ray-tracing equations in simple repetitive linear matrix form. Sufficiently good accuracy is obtained if the proposed method is restricted to skew rays in the immediate neighborhood of the optical axis, as demonstrated by numerical examples. This study extends earlier Matrix-based paraxial ray-tracing design technique to include non-coplanar systems containing mirrors and/or reflecting prisms such as projectors, with special application potential for first estimates in early design stages of 3D optical systems.     

Application of spheroidal functions in theory of light scattering by nonspherical particles: Separation of variables method

The separation of variables method (SVM), which uses a spheroidal basis, is proposed. According to this method, fields are presented in the form of expansion in terms of spheroidal functions. The previously conducted analysis of various methods using a spherical basis showed that the SVM is applicable in a broader area for numerical calculations, while the proposed approach using a spheroidal basis yields reliable results in the case of spheroids with a high degree of asphericity where other methods and approaches cannot be used. Importantly, the method includes an SVM that uses a spherical basis as the limiting case. Thus, the proposed method has all chances of being highly efficient for calculation of optical characteristics of various nonspherical particles in a wide range of parameters of the formulated problem.     

Construction of the collision integral kernel for the nonlinear Boltzmann equation from its matrix elements

The methods for constructing the kernels of collision integrals emerging in the expansion of the non-linear Boltzmann kinetic equation in spherical harmonics are investigated. The techniques developed for calculating the kernel from the known matrix elements of the collision integral using the averaging over a number of computational parameters make it possible to substantially improve the accuracy of the proposed algorithm. In kernel calculations, Maxwell molecules were simulated using the asymptotic technique. This makes it possible to approximate the analytic expression of the kernel known in this case to a high degree of accuracy.     

Inverse problem for particle size distributions of atmospheric aerosols using stochastic particle swarm optimization

As a part of resolving optical properties in atmosphere radiative transfer calculations, this paper focuses on obtaining aerosol optical thicknesses (AOTs) in the visible and near infrared wave band through indirect method by gleaning the values of aerosol particle size distribution parameters. Although various inverse techniques have been applied to obtain values for these parameters, we choose a stochastic particle swarm optimization (SPSO) algorithm to perform an inverse calculation. Computational performances of different inverse methods are investigated and the influence of swarm size on the inverse problem of computation particles is examined. Next, computational efficiencies of various particle size distributions and the influences of the measured errors on computational accuracy are compared. Finally, we recover particle size distributions for atmospheric aerosols over Beijing using the measured AOT data (at wavelengths λ=0.400, 0.690, 0.870, and 1.020 μm) obtained from AERONET at different times and then calculate other AOT values for this band based on the inverse results. With calculations agreeing with measured data, the SPSO algorithm shows good practicability.     

Estimation of interfacial forces in time domain for linear systems

A structural path rank ordering process under transient excitations requires a good knowledge of the interfacial path forces, which are difficult to directly measure. Four time domain methods to estimate the interfacial forces are proposed and comparatively evaluated with application to linear time-invariant, proportionally damped discrete systems. First, the transient response is derived by modal analysis and a direct time domain technique to calculate the interfacial forces is outlined. Next, the frequency domain estimation methods, based on the sub-system concept are reviewed, and an inverse Fourier transform scheme is introduced. An indirect method of estimating interfacial force in transient state is then developed through an inverse procedure of modal analysis. The sub-system approach is employed to obtain the interfacial forces based on the forced vibration response of the original system and modal data of the sub-system. Finally, an approximate time domain scheme is suggested that could be used only if the system properties are known or precisely estimated. Although the proposed indirect methods are designed for eventual experimental applications, this article provides numerical feasibility studies via a simple source-path-receiver system (with parallel vibration paths) that has five translational degrees of freedom. The proposed methods are compared under ideal impulse force excitation input and a periodic sawtooth load (without and with Gaussian noise) to observe the starting transients as well as subsequent motions and interfacial forces. Preliminary comparisons with a laboratory experiment are very promising.     

New recursive solution of the problem of scattering of electromagnetic radiation by multilayer spheroidal particles

A new recursive algorithm for the solution of the problem of scattering of light (of an arbitrarily polarized plane electromagnetic wave) by multilayer confocal spheroidal particles is constructed. This approach preserves the advantages of the two approaches proposed earlier by us for single-layer and two-layer spheroids (special choice of scalar potentials and utilization of the basis of wave spheroidal harmonics) and for homogeneous axially symmetric particles (formulation of the problem in terms of surface integral equations, calculation of the potentials inside the particle from the potentials of the incident radiation, and calculation of the potentials of the scattered radiation from the potentials inside the particle). In the case of multilayer particles, the potential inside each shell is a sum of two terms. The first has the properties of the incident radiation (no singularities inside the volume enclosed by the external boundary of the shell), whereas the second term has the properties of the scattered radiation (satisfies the radiation conditions at infinity). Therefore, as the calculation progresses from one layer to the next (from the core to the outer shell), the dimensionality of the reduced linear matrix equations for the unknown expansion coefficients of the scattered field potentials does not increase with respect to the case of a homogeneous spheroid. The algorithm is particularly simple and lucid (as far as possible for such a complex problem). In the case of spherical multilayer particles, the solution can be found explicitly.     

Propagation of polarized light in media with large discrete inhomogeneities

An approximate analytical method of solution of the vector radiative-transfer equation for an optically isotropic medium with large-scale inhomogeneities is proposed. The method is based on an assumption about the distinct anisotropy of single scattering. The method was used to calculate the polarization characteristics of light multiply scattered in a monodisperse medium with large spherical particles.     

On the probabilistic approach for Gaussian Berezin integrals

We present a novel approach to Gaussian Berezin correlation functions. A formula well-known in the literature expresses these quantities in terms of submatrices of the inverse matrix appearing in the Gaussian action. By using a recently proposed method to calculate Berezin integrals as an expectation of suitable functionals of Poisson processes, we obtain an alternative formula which allows one to skip the calculation of the inverse of the matrix. This formula, previously derived using different approaches (in particular by means of the Jacobi identity for the compound matrices), has computational advantages which grow rapidly with the dimension of the Grassmann algebra and the order of correlation. By using this alternative formula, we establish a mapping between two fermionic systems, not necessarily Gaussian, with short and long range interaction, respectively.     

半空间球面波函数叠加法重构声源直接辐射声场

提出了基于半空间球面波函数叠加的声场重构方法,以重构含有限声阻抗边界半空间中声源直接辐射的声场.在半空间中多极子声源声压场的解析解的基础上,构造出以边界声阻抗为参量的半空间球面波函数的正交基;通过求逆获得半空间总声压解的基函数系数,同时也获得声源直接辐射声场即自由空间中的基函数系数,进而重构出声源直接辐射的声场.在边界...     

Dielectric function and phonon spectrum of grey tin

The dielectric function of α-Sn has been evaluated from a pseudopotential calculation of the valence electron wavefunctions and energy bands. The result is used, together with an approximate method of calculating the inverse of the dielectric function matrix, to calculate the phonon dispersion curves. Fairly good overall agreement with the experimental dispersion curves has been obtained.     

Calculation of the collision integral linear kernel for Maxwellian molecules in the isotropic case

Application of the method of nonlinear moments to solve the Boltzmann equation generates the need to sum a series that is the expansion of the distribution function in basis functions. This series converged only if the Grad test is fulfilled. Such a limitation can be removed if the expansion of the distribution function is summed over the index related to only the expansion in velocity magnitude. In this case, the distribution function and the collision integral become expanded in only spherical harmonics and the expansion coefficients satisfy integro-differential equations. The kernels of these equations are the sums of the Sonine polynomials in the velocities of colliding and outgoing particles multiplied by matrix elements of the collision integral. For a number of arguments, the direct calculation of the kernels requires that a very large number of terms in the sum be taken into consideration. In this respect, an approach seems to be promising in which the asymptotics of the matrix elements and Sonine polynomials at large indices are used and summation over index is replaced by integration. In this paper, we apply this approach to calculate the linear kernel in the isotropic case, assuming that interaction between particles is described by a pseudopower law. With this approach, the collision integral kernel can be calculated with a high accuracy using as little as a few tens of series terms and the asymptotic estimate of the residue.     

Light scattering by multilayer axially symmetric particles

An exact solution to the problem of light scattering by multilayer axially symmetric particles is derived and some aspects of its computer-aided implementation are discussed. The main specific features of the solution are (i) separation of the incident, scattered, and internal fields into two parts and special selection of the scalar potentials for each of them; (ii) expansion of the potentials in terms of spherical wave functions; (iii) formulation of the problem in the form of surface integral equations; and (iv) solution of the reduced systems of the linear algebraic equations for the coefficients of the potential expansions. Mathematical justification of the solution is discussed, which is formulated in the recursive and nonrecursive form (for the T-matrix). The developed computer program has shown that the proposed approach makes it possible to consider axially symmetric particles with essentially different internal structures (i.e., with a spherical core, oblate spheroidal shell, or prolate spheroidal intermediate layer). The results of calculations of the optical properties of the multilayer nonspherical particles are presented and discussed.     

Specific features of the dielectric response of heterogeneous systems with a polar matrix containing electrically active inclusions

A theoretical model has been proposed which makes it possible to introduce the correction into the complex permittivity of electrically active condensed systems that contain a polar liquid matrix and low-dimensional solid particles with a developed electrically active surface. It has been established that there is an interfacial electrical interaction between surface charges of the solid component and polar molecules of the liquid matrix. The processes occurring during this interaction lead to the appearance of an intrinsic internal electric field in the system under investigation. The contribution from surface effects to the formation of the polarized state of the polar liquid medium has been investigated in terms of the proposed model. The possibility of controlling the processes of local change in the structure of the polar liquid component under the effect of the internal electric field by varying the electrically active specific surface area of the solid phase and the intrinsic dipole moment of molecules of the liquid matrix has been analyzed in the studied systems. The conditions providing for the appearance of a controlled potential gradient of the internal electric field have been determined for the dispersed systems under consideration. The parameter that makes it possible to evaluate the contribution from the interfacial electrical contact interaction to the dielectric response of the liquid component and the entire system has been introduced for the first time.     

Calculation of electromechanical coupling coefficient of Lamb waves in multilayered plates

Two methods have been always used to calculate the electromechanical coupling coefficient of a Lamb wave in a multilayered plate: one is an approximate method using the acoustic velocity difference under different electric boundary conditions and the other is the Green's function method. The Green's function method is more accurate but more complicated, because an 8N-order matrix is used for calculating the electromechanical coupling coefficient of the Lamb wave in an N-layered plate, which induces great computation loads and some calculation deviations. In this paper, a transfer matrix method is used for calculating the electromechanical coupling coefficient of Lamb waves in a multilayered plate, in which only an 8-order matrix is needed regardless of the number of layers of the plate. The results show that the transfer matrix method can obtain the same accuracy as those by the Green's function method, but the computation load and deviation are greatly decreased by avoiding the use of a high order matrix used in the Green's function method.     

Method for retrieving the refractive index of ordered particles from data on the photonic band gap

A method for retrieving the refractive index of spherical particles arranged into ordered structures is proposed. It is based on the solution of the inverse problem using data on the photonic band gap. The solution has been obtained within the quasi-crystalline approximation of the multiple wave scattering theory and the transfer-matrix method. Quantitative results are presented for systems of silicon oxide particles. The effective refractive indices of synthetic opal particles have been found from the available experimental data on the spectral position of the photonic band gap. The described technique is applicable for retrieving not only the refractive index of particles but also other characteristics of ordered particulate structures from the coherent transmittance spectra.     

梯度界面层对复合材料有效光学性质的影响

 研究了球形颗粒被无规地浸没在基质中,具有梯度分界面和光滑分界面的球形颗粒复合物界面层对介电响应和光学性质的影响;利用第一原理近似,计算了在稀释极限下具有梯度分布界面层复合物的有效介电常数,讨论了其实部和吸收光谱随外加场频率的变化关系。对结果进行比较,发现梯度分布的界面层使复合材料在等离子体共振频率处的有效介电常数的涨落明显减小,吸收谱线明显宽化并伴有“蓝移”现象,梯度界面层能很好地控制等离子体共振和减小有效介电响应的涨落。