Numerical methods for calculating poles of the scattering matrix with applications in grating theory

TL;DR

This paper introduces a new iterative numerical method for accurately calculating the complex poles of scattering matrices, crucial for understanding waveguide and resonant properties in grating theory, demonstrating high convergence and stability.

Contribution

A novel iterative method for computing scattering matrix poles that accounts for matrix behavior near poles and offers high convergence and stability for large matrices.

Findings

The iterative method converges rapidly and reliably.

It is numerically stable for large scattering matrices.

The method typically finds the nearest pole efficiently.

Abstract

Waveguide and resonant properties of diffractive structures are often explained through the complex poles of their scattering matrices. Numerical methods for calculating poles of the scattering matrix with applications in grating theory are discussed. A new iterative method for computing the matrix poles is proposed. The method takes account of the scattering matrix form in the pole vicinity and relies upon solving matrix equations with use of matrix decompositions. Using the same mathematical approach, we also describe a Cauchy-integral-based method that allows all the poles in a specified domain to be calculated. Calculation of the modes of a metal-dielectric diffraction grating shows that the iterative method proposed has the high rate of convergence and is numerically stable for large-dimension scattering matrices. An important advantage of the proposed method is that it usually converges to the nearest pole.