Method of analytical regularization in computational photonics

Abstract

We discuss the advantages of the conversion of electromagnetic field problems to the Fredholm second‐kind integral equations (analytical regularization) and Fredholm second‐kind infinite‐matrix equations (analytical preconditioning). Special attention is paid to specific features of the characterization of metals and dielectrics in the optical range and their effect on the problem formulation and on the methods applicable to the mentioned conversion.

Fast Converging Cfie-Mom Analysis of Electromagnetic Scattering From Pec Polygonal Cross-Section Closed Cylinders

“…Well-posed integral equations in the functional space to which the solution belongs can be obtained by means of analytical regularization method, i.e., by recasting the equation at hand as a second-kind Fredholm integral equation [19]. Indeed, Fredholm's theory [27] allows to state that the solution of the discretized and truncated counterpart of a second-kind Fredholm integral equation converges to the exact solution of the problem if unique.…”

“…The desired uniqueness property of such a kind of formulation resides in the orthogonality of the null spaces of EFIE and MFIE formulations. Unfortunately, the discretization and truncation of CFIE when its EFIE component contains a hypersingular term leads to "approximate solutions" which do not necessarily converge to the exact solution of the problem, and in any case, the sequence of condition numbers of truncated systems is divergent due to the unboundedness of the involved operator [19]. On the other hand, it has been widely noted that classical discretization schemes (such as, Rao-Wilton-Glisson discretization) applied to MFIE produced worse results than EFIE, and more sophisticated approaches have to be employed in order to achieve more accurate solutions [20][21][22][23][24][25][26].…”

Gaussian Beam Electromagnetic Scattering From Pec Polygonal Cross-Section Cylinders

“…Hence, for such kind of applications, a shaped beam (Gaussian beam) represents a more realistic incident field. It is worth noting that, despite a Gaussian beam can be represented as a continuous spectrum of plane waves, the analysis of the scattering deserves attention since the convergence of a numerical scheme is strictly related to the functional space to which the free term belongs [6]. On the other hand, the field pattern obtained for a Gaussian beam does not exactly follow the geometrical optics rules of reflection and transmission satisfied by a plane wave [7].…”

“…Anyway, due the unboundedness of the involved operator or of its inverse, the sequence of condition numbers diverges. In order to overcome these problems, the Method of analytical preconditioning is applied in this paper [6]. After defining the functional spaces to which the unknown and free term belong, it consists in individuating a suitable expansion basis for the surface current density in a Galerkin scheme such that the most singular part of the integral operator is invertible with a continuous two-side inverse.…”

Plane Wave Diffraction by a Finite Parallel-Plate Waveguide With Sinusoidal Wall Corrugation

“…Various analytical and numerical methods have been developed thus far and diffraction phenomena have been investigated for a number of periodic structures [1]. It is well known that the Riemann-Hilbert problem technique [2][3][4], the analytical regularization methods [4][5][6][7], the Yasuura method [8][9][10], the integral and differential method [11], the point matching method [12], and the Fourier series expansion method [13,14] are efficient for the analysis of diffraction problems involving periodic structures. The Wiener-Hopf technique [15][16][17][18] is known as a powerful approach for analyzing electromagnetic wave problems associated with canonical geometries rigorously, and can be applied efficiently to the problems of diffraction by specific periodic structures such as gratings.…”