Comparison of the light scattering methods using the spherical basis

A comparative analysis of the widely known methods for solving the problem of light scattering by nonspherical particles—of the method of separation of variables (MSV), of the extended boundary condition method (EBCM), and the point-matching method (PMM), which use the spherical wave functions as a basis for the expansions of the fields—is carried out. In the scientific literature, these methods have been analyzed independently of one another in spite of their evident similarity: The same expansion coefficients are determined from similar set of equations and all optical characteristics are calculated with the same formulas. The ranges of applicability of the methods for dielectric spheroids and Chebyshev particles are studied in the same manner. It was found that, when considering the far-field zone, theoretical conditions of mathematical correctness of the EBCM and the MSV, apparently, differ fundamentally, although, as was shown, the methods themselves are extremely closely related. The performed numerical calculations suggest that the EBCM is preferable for spheroids, the MSV is preferable for Chebyshev particles, and the PMM, which is the most time-consuming method, gives satisfactory results in many cases when two other methods are inapplicable. Since the methods supplement one another well and their programs differ only in several tens of operators, we propose combining these methods within the framework of one universal program.     

An Ellipsoidal Model for Small Multilayer Particles

This paper presents an ellipsoidal model that is constructed for small layered nonspherical particles and methods for constructing “effective” multilayer ellipsoids, the light-scattering properties of which would be close to the corresponding properties of original particles. In the case of axisymmetric particles, prolate or oblate spheroids (ellipsoids of revolution) are implied. Numerical calculations of the polarizability and scattering cross sections of small layered nonspherical particles, including nonconfocal (similar) spheroids, Chebyshev particles, and pseudospheroids, are performed by different approximate and rigorous methods. Approximate approaches involve the use of an ellipsoidal model, in which the polarizability of a layered particle is determined in two ways. In the first case, the polarizability is calculated in the approximation of confocal spheroids, while, in the second case, it is sought as a linear combination of the polarizabilities of embedded spheroids proportionally to the volumes of layers. Among rigorous methods, the extended boundary conditions method and the generalized separation of variables method are applied. On the basis of a comparison of the results obtained with rigorous and approximate approaches, their drawbacks and advantages are discussed.     

Extended boundary condition method in combination with field expansions in terms of spheroidal functions

The problem of light scattering by nonspherical particles, which arises in many applications, is nowadays most frequently solved by the method of extended boundary conditions in combination with the expansion of the fields in terms of spherical wave functions. However, such an approach encounters difficulties if the shape of particles is far from spherically symmetric, even in the simplest case of spheroids with the semiaxis ratio a/b > 5?10. A new approach to solving this problem is proposed, which also applies the extended boundary condition method but involves the expansion of the fields in terms of spheroidal functions. In this case, to obtain effective solutions for strongly prolate and oblate particles, the fields are divided in two parts with known properties and specific scalar potentials are used for each part. The basic relations of the approach are presented and some results of calculations of the optical properties of spheroids and spheroidal Chebyshev particles that are performed using computer codes realizing this approach are given. The convergence of the results for different cases and the domain of applicability of the method are discussed.     

Electrostatic solution and rayleigh approximation for small nonspherical particles in a spheroidal basis

The electrostatic problem for the case of axially symmetric particles is analyzed in a spheroidal basis. In this case, the wavenumber is zero and Maxwell’s equations are reduced to the Laplace equation for scalar potentials. An alternative approach involves solving integral equations that are similar to those obtained within the framework of the extended boundary conditions method. The scalar potentials are represented as expansions in terms of eigenfunctions of the Laplace equation in a spheroidal frame of reference, and unknown expansion coefficients are determined from an infinite set of linear algebraic equations (the separation of variables method). These two approaches yield exact solutions of the problem in the case of axially symmetric particles, which coincide with known solutions in particular cases. Investigation of infinite systems allowed finding the boundaries where these algorithms are valid. Numerical calculations showed that, for spheroidal Chebyshev particles (i.e., perturbed spheroids), the Rayleigh approximation based on the electrostatic solution is applicable in a wide range of the problem parameters and is in fair agreement with the results obtained using the discrete dipole approximation.     

Light scattering by multilayered axisymmetric particles: Solution of the problem by the separation of variables method

A new solution to the problem of light scattering by multilayered particles possessing axial symmetry is obtained. Two methods are applied for this purpose. One is the separation of variables method with expansion of fields in terms of spherical wave functions, and the other is a novel approach based on the separation of fields into axisymmetric and nonaxisymmetric parts and on the choice of specific scalar potentials for each of them. A specific feature of the new solution is that the dimension of truncated linear algebraic systems used for determining unknown expansion coefficients of fields does not increase with an increasing number of layers. Using double-and three-layer spheroidal and Chebyshev particles of different shape and size as examples, the domain of applicability of the solution presented is compared with that of the solution previously obtained by the extended boundary conditions method. Except for nearly spherical particles, the solution presented is shown to be more favorable than the previously obtained solution.     

On the impact of non-sphericity and small-scale surface roughness on the optical properties of hematite aerosols

We perform a comparative modelling study to investigate how different morphological features influence the optical properties of hematite aerosols. We consider high-order Chebyshev particles as a proxy for aerosol with a small-scale surface roughness, and spheroids as a model for nonspherical aerosols with a smooth boundary surface. The modelling results are compared to those obtained for homogeneous spherical particles. It is found that for hematite particles with an absorption efficiency of order unity the difference in optical properties between spheres and spheroids disappears. For optically softer particles, such as ice particles at far-infrared wavelengths, this effect can be observed for absorption efficiencies lower than unity. The convergence of the optical properties of spheres and spheroids is caused by absorption and quenching of internal resonances inside the particles, which depend both on the imaginary part of the refractive index and on the size parameter, and to some extent on the real part of the refractive index. By contrast, small-scale surface roughness becomes the dominant morphological feature for large particles. This effect is likely to depend on the amplitude of the surface roughness, the relative significance of internal resonances, and possibly on the real part of the refractive index. The extinction cross section is rather insensitive to surface roughness, while the single-scattering albedo, asymmetry parameter, and the Mueller matrix are strongly influenced. Small-scale surface roughness reduces the backscattering cross section by up to a factor of 2-3 as compared to size-equivalent particles with a smooth boundary surface. This can have important implications for the interpretation of lidar backscattering observations.     

Near- and far-field light scattering by nonspherical particles: Applicability of methods that involve a spherical basis

The separation of variables method for coordinate system, the extended boundary condition method, and the point-matching method that are used to solve the problem of light scattering by nonspherical particles are considered from a unified viewpoint. It is shown that, if the mathematical correctness condition (the Rayleigh hypothesis) holds, these methods are interrelated and are equivalent. The applicability ranges of the methods in the near- and far-field zones are analyzed, discussed, and compared on both analytical (based on analytical investigations) and practical (based on numerical calculations) grounds.     

On optical properties of nonspherical inhomogeneous particles

To solve the problem of light scattering by multilayer scatterers of an arbitrary axisymmetric shape, a separation of variables method that involves special scalar potentials and their expansions in spherical functions is developed. The approach is shown to yield highly exact results even for particles that have 100 layers or more. A graphic library that illustrates the optical properties of layered and homogeneous (with an effective refractive index) spheroids, spheres, and Chebyshev particles of various shapes and sizes (about 650 figures) is created and is put on the Internet. It is noted that the linear polarization of radiation transmitted forward through a polydisperse medium containing partially oriented nonspherical porous particles strongly depends on the structure of scatterers. It is shown that the difference between the degrees of polarization of layered and corresponding homogeneous scatterers can exceed 200–300%.     

On the applicability of a spherical basis for spheroidal layered scatterers

The applicability of an analog of the extended boundary condition method, which is popular in light-scattering theory, is studied in combination with the standard spherical basis for the solution of an electrostatic problem appearing for spheroidal layered scatterers the sizes of which are small as compared to the incident radiation wavelength. In the case of two or more layers, polarizability and other optical characteristics of particles in the far zone are shown to be undeterminable if the condition under which the appearing systems of linear equations for expansion coefficients of unknown fields are Fredholm systems solvable by the reduction method is broken. For two-layer spheroids with confocal boundaries, this condition is transformed into a simple restriction on the ratio of particle semiaxes a/b< $\sqrt 2 $ + 1. In the case of homogeneous particles, the solvability condition is that the radius of convergence of the internal-field expansion must exceed that of the expansion of an analog of the scattering field. Since homogeneous spheroids (ellipsoids) are unique particles inside which the electrostatic field is homogeneous, it is shown that the solution can be always found in this case. The obtained results make it possible to match in principle the results of theoretical and numerical determinations of the domain of applicability for the extended boundary condition method with a spherical basis for spheroidal scatterers.     

Analysis of the method of generalized separation of variables in the problem of light scattering by small axisymmetric particles

In the problem of light scattering by small axisymmetric particles, we have constructed the Rayleigh approximation in which the polarizability of particles is determined by the generalized separation of variables method (GSVM). In this case, electric-field strengths are gradients of scalar potentials, which are represented as expansions in term of eigenfunctions of the Laplace operator in the spherical coordinate system. By virtue of the fact that the separation of variables in the boundary conditions is incomplete, the initial problem is reduced to infinite systems of linear algebraic equations (ISLAEs) with respect to unknown expansion coefficients. We have examined the asymptotic behavior of ISLAE elements at large values of indices. It has been shown that the necessary condition of the solvability of the ISLAE coincides with the condition of correct application of the extended boundary conditions method (ЕВСМ). We have performed numerical calculations for Chebyshev particles with one maximum (also known as Pascal’s snails or limaçons of Pascal). The obtained numerical results for the asymptotics of ISLAE elements and for the matrix support theoretical inferences. We have shown that the scattering and absorption cross sections of examined particles can be calculated in a wide range of variation of parameters with an error of about 1–2% using the spheroidal model. This model is also applicable in the case in which the solvability condition of the ISLAE for nonconvex particles is violated; in this case, the SVM should be considered as an approximate method, which frequently ensures obtaining results with an error less than 0.1–0.5%.     

Surface-integral formulation for electromagnetic scattering in spheroidal coordinates

We derive surface-integral expressions for the Q matrices in spheroidal coordinates that allow us to compute the T matrix in spheroidal coordinates. This approach combines the advantages of the null-field method (also referred to as the extended boundary condition method) with those of the separation of variables method. For spheroidal particles we obtain explicit Q matrix expressions that display the expected symmetry properties and yield correct results in the spherical limit. Compared to surface-integral expressions for spheroids in spherical coordinates, our results are considerably simpler because the integrands do not contain radial functions.     

Light Scattering by Small Multilayer Particles: A Generalized Separation of Variables Method

A Rayleigh approximation is constructed for light scattering by small multilayer axisymmetric particles, in which their polarizability is determined by the generalized separation of variables method (SVM). In this method, scalar potentials, the gradients of which yield the electric-field strengths, are represented as expansions in spherical harmonics of the Laplace equation. Unknown coefficients of expansions are determined from the boundary conditions, which are reduced to infinite systems of linear algebraic equations (ISLAEs), since the separation of variables is incomplete. The T matrix of the electrostatic problem, principal element T₁₁ of which is proportional to the particle polarizability, is determined. The necessary condition for the ISLAEs solvability for the SVM coincides with the condition of the correct application of the extended boundary conditions method (EBCM). However, numerical calculations in which finite-dimensional (i.e., reduced) systems are solved, yield different results in ranges of variation of parameters that are close to the boundary of the range of applicability. An analysis of the numerical calculations of the scattering and absorption cross sections for two-layer confocal spheroids, an exact solution for which can be obtained using spheroidal harmonics, shows that the SVM is preferable to the EBCM. It turned out that the proposed method yields workable results in a wider range of variation of parameters. Even outside the range of applicability, in which it should be regarded as a certain approximate solution, its use in a number of cases is quite acceptable. Additional calculations for three-layer nonconfocal spheroids, as well as for three-layer similar pseudospheroids and Pascal’s snails, which can be obtained from spheroids as a result of the inversion with respect to the coordinate origin and one of the foci, respectively, confirm these inferences. We note that, for certain values of the parameters, the shapes of the latter particles are nonconvex.     

On scattering of light by small axially symmetric particles

Light scattering by small dielectric particles of an arbitrary axially symmetric shape is analyzed. A simple approximate expression that governs the polarizability of the particle is found under the assumption of field homogeneity inside of these particles. The expression includes four relatively simple one-dimensional integrals that can be calculated analytically for some types of particles (except for spheroids). A comparison with the numerical data obtained for various Chebyshev particles and finite cylinders showed that the obtained approximation yields acceptable results, even when the shape of scatterers is significantly different from spheroidal. For spheroids, our approximation coincides with the Rayleigh one.     

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.     

Analysis of the modified point-matching method in the electrostatic problem for axisymmetric particles

An integral modification of the generalized point-matching method (GPMMi) in the electrostatic problem for axisymmetric particles is developed. Scalar potentials that determine electric fields are represented as expansions in terms of eigenfunctions of the Laplace operator in the spherical coordinate system. Unknown expansion coefficients are determined from infinite systems of linear algebraic equations (ISLAEs), which are obtained from the requirement of a minimum of the integrated residual in the boundary conditions on the particle surface. Matrix elements of ISLAEs and expansion coefficients of the “scattered” field at large index values are analyzed analytically and numerically. It is shown analytically that the applicability condition of the GPMMi coincides with that for the extended boundary conditions method (ЕВСМ). As model particles, oblate pseudospheroids \(r\left( \theta \right) = a\sqrt {1 - {^2}{{\cos }^2}\theta } ,\;{^2} = 1 - {\raise0.7ex\hbox{${{b^2}}$} \!\mathord{\left/ {\vphantom {{{b^2}} {{a_2}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{a_2}}$}} \geqslant 0\) with semiaxes a = 1 and b ≤ 1 are considered, which are obtained as a result of the inversion of prolate spheroids with the same semiaxes with respect to the coordinate origin. For pseudospheroids, the range of applicability of the considered methods is determined by the condition \({\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$b$}} < \sqrt 2 + 1\). Numerical calculations show that, as a rule, the ЕВСМ yields considerably more accurate results in this range, with the time consumption being substantially shorter. Beyond the ЕВСМ range of applicability, the GPMMi approach can yield reasonable results for the calculation of the polarizability, which should be considered as approximate and which should be verified with other approaches. For oblate nonconvex pseudospheroids (i.e., at \({\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$b$}} \geqslant \sqrt 2 \)), it is shown that the spheroidal model works well if pseudospheroids are replaced with ordinary “effective” oblate spheroids. Semiaxes a_ef and b_ef of the effective spheroids are determined from the requirement of the particle volumes, as well as from the equality of the maximal longitudinal and transverse dimensions of particles or their lengths. As a result, the polarizability of pseudospheroids can be calculated by simple explicit formulas with an error of about 0.5–2%.     

局域共振型声学超材料薄板带隙特性的能量解法

研究局域共振型声学超材料的带隙特性时,面临的首要问题是寻求一种带隙特性的高效解法。为此,本文以局域共振型声学超材料薄板为研究对象,提出了一种基于能量泛函变分原理及正交多项式级数展开的带隙计算方法。该方法通过合并基体板及共振子的动、势能及外力功建立元胞的能量泛函,经过变分运算得到元胞的振动控制方程,进而引入第一类Chebyshev正交多项式作为基函数将控制方程进行离散。为克服无法直接对离散控制方程中的广义坐标施加周期边界条件的困难,该方法将连续边界条件弱化到若干选定的配点上以离散方式实现,并将对应的一组线性约束条件通过拉格朗日乘子法施加到系统中。仿真结果表明,该方法具有较高的计算精度和效率,并可应用于"局域共振子-夹层板"等复合结构的带隙特性研究。本文方法为局域共振型薄壁超材料的带隙求解提供了新的思路,丰富了声学超材料的振-声学理论,并为薄壁超材料的工程设计和优化提供了技术支撑。      

Light Scattering Simulation and Measurement of Monodisperse Spheroids using a Phase Doppler Anemometer

Mathematical tools are provided for the computation of the scattered field produced by non-spherical particles moving through the measurement volume of a phase Doppler anemometer. The phase distribution of a spheroid with random orientation is computed by using the rigorous extended boundary condition method and the ray theory. In a phase Doppler experiment the spheroid parameters are obtained by fitting the measured phase distribution with the simulated phase distribution. The numerical simulations are supported by experimental results on monodisperse spheroids.     

Attenuation and scattering of light by a system of chaotically oriented axially symmetric particles: Analytical averaging in the modified <Emphasis Type="Italic">T</Emphasis>-matrix method

A procedure for analytical averaging of the attenuation and scattering cross sections for systems of chaotically oriented axially symmetric particles was developed for the first time within the framework of the modified T-matrix method and the method of separation of variables for spheroids. These approaches essentially complement each other: one is applicable to axially symmetric scatterers of different shape but is inefficient if the ratio of the maximum to the minimum size of the particles exceeds 3–5; the other is applicable only to spheroids, but the ratio of the major semiaxis to the minor one can be considerable, for example, 100 and larger.     

The effect of the size,shape, and structure of metal nanoparticles on the dependence of their optical properties on the refractive index of a disperse medium

The effect of the size, shape, and structure of gold and silver nanoparticles on the dependence of their extinction and integral scattering spectra on the dielectric environment has been investigated. Calculations were performed using the Mie theory for spheres and nanoshells and the T-matrix method for chaotically oriented bispheres, spheroids, and s cylinders with hemispherical ends. The sensitivity of plasmon resonances to variations in the refractive index of the environment in the range 1.3–1.7 for particles of different equivolume size, as well as to variations in the thickness of the metal layer of nanoshells, was studied. For nanoparticles with an equivolume diameter of 15 nm, the maximal shifts of plasmon resonances due to variation in the refractive index of the environment are observed for bispheres and the shifts decrease in the series nanoshells, s cylinders or spheroids, and spheres. For particles 60 nm in diameter, the largest shifts of plasmon resonances occur for nanoshells and the shifts decrease in the series bispheres, s cylinders or spheroids, and spheres. All other conditions being the same, silver nanoparticles are more sensitive to the resonance tuning due to a change in the dielectric environment.     

Light scattering by dielectric particles with axial symmetry: II

A new method is developed for calculating optical characteristics of axially symmetric particles. Electromagnetic fields are separated into two (axisymmetric and non-axisymmetric) parts. The light scattering problem is formed in the integral form and solved independently for each of the parts by using specially chosen scalar potentials. The potentials are expanded into series in spherical wave functions, and the expansion coefficients are calculated from solving the infinite systems of linear algebraic equations. The applicability of the proposed method for solving the problem of light scattering by Chebyshev particles, spheroids, and finite circular cylinders is briefly discussed, and some results of calculations performed for these particles are presented.