Infinite Systems Of Linear Algebraic Equations Research Articles

Electromagnetic field of the circular magnetic source in biconical section

The problem of axially-symmetric electromagnetic wave diffraction from the perfectly conducting biconical scatterer formed by the finite cone placed in the semi-infinite conical region is solved rigorously using the mode-matching and analytical regularization techniques. The problem is reduced to the infinite systems of linear algebraic equations (ISLAE) of the second kind. The obtained equations admit the reduction procedure and can be solved with a given accuracy for any geometrical parameters and frequency. The numerical examples of the solution are presented. The analysis of the source location influences on the far-field pattern for different geometrical parameters of the bicone is carried out.

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.

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 limacons 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%.

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 aef and bef 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%.

Analysis of the extended boundary condition method: an electrostatic problem for Chebyshev particles

The extended boundary condition method (EBCM) in solving the electrostatic problem for axisymmetric Chebyshev particles the surface of which is described by the equation r(θ) = a(1 + ɛcosnθ) is studied. The main attention is paid to the case n = 1. The problem is reduced to solving infinite systems of linear algebraic equations (ISLAEs) for expansion coefficients of internal and “scattered” fields in terms of a spherical basis where matrix elements of the fields are integrals of products of Legendre functions and power functions. Radii of convergence R2 and R1 of these expansions, respectively, have been found analytically. For the considered particles, depending on perturbation parameter ɛ, conditions of the applicability of the EBCM have been obtained, i.e., conditions of correct construction of the T matrix (R1 maxr(θ)) Rayleigh hypotheses under which expansions of the “scattered” and internal fields in terms of the spherical basis converge up to the surface of the particle beyond and inside it, respectively. In the particular case n = 1, numerical calculations have been performed; in this process, integral ISLAE elements have been represented as finite sums the summands of which depend on gamma functions. Calculations of matrix elements by explicit formulas have made it possible to considerably increase the dimension of the solved reduced ISLAE with preservation of the necessary accuracy. Analysis of results of numerical calculations verified their agreement with theoretical conclusions.

The Rayleigh hypothesis and the region of applicability of the extended boundary condition method in electrostatic problems for nonspherical particles

The electrostatic problem of excitation of a homogeneous axisymmetric particle in a constant electric field is considered. The approach is based on the surface integral equations arising in the extended boundary condition method (EBCM). The electrostatic fields are related to the scalar potentials, which are represented as expansions in the eigenfunctions of the Laplace operator in the spherical coordinate system. Unknown expansion coefficients are determined from infinite systems of linear algebraic equations (ISLAE). Analytic analysis of the solvability of the ISLAE is performed, and the convergence radii of expansions are obtained. It is shown that the EBCM can be applied in the far zone of a particle, i.e., the T matrix, can be constructed even if the Rayleigh hypothesis (the expansions converge up to the boundary of the particle) is not satisfied. However, a weaker restriction appears, which can be reduced to the requirement that there exists a spherical shell inside which the expansions of excited and internal fields simultaneously converge. The case of spheroids, as well as pseudospheroids that arise from spheroidal particles by inversion, is studied in detail. It is shown that the EBCM is applicable to spheroids for any ratio a/b of semiaxes and to pseudospheroids for a/b < 1 + \(\sqrt 2 \). The external Rayleigh hypothesis, i.e., the convergence of the expansion of a “scattered” field up to the surface of a particle is valid for a spheroid if a/b < \(\sqrt 2 \). The internal hypothesis, i.e., the convergence of an internal field, is always valid because the field is uniform inside the spheroid. For a pseudospheroid, the hypotheses are valid for a/b < \(\sqrt 2 \) and a/b < \(\sqrt 2 \), respectively. The relation and similarity between the results obtained for the wave and electrostatic problems are discussed.