Application of the iterative approach to modal methods for the solution of Maxwell's equations

TL;DR

This paper introduces an iterative approach to modal methods for Maxwell's equations that significantly reduces computational complexity and memory usage, enabling large-scale eigenmode calculations on standard hardware.

Contribution

It develops an iterative modal method that decreases computational complexity from cubic to quadratic and memory demand from quadratic to linear in the number of eigenmodes.

Findings

Reduces computational complexity from third to second power of eigenmodes

Decreases memory demand from second to first power of eigenmodes

Enables calculations with hundreds of thousands of eigenmodes without supercomputers

Abstract

In this work we discuss the possibility to reduce the computational complexity of modal methods, i.e. methods based on eigenmodes expansion, from the third power to the second power of the number of eigenmodes. The proposed approach is based on the calculation of the eigenmodes part by part by using shift-and-invert iterative technique and by applying the iterative approach to solve linear equations to compute eigenmodes expansion coefficients. As practical implementation, the iterative modal methods based on polynomials and trigonometric functions as well as on finite-difference scheme are developed. Alternatives to the scattering matrix (S-matrix) technique which are based on pure iterative or mixed direct-iteractive approaches allowing to markedly reduce the number of required numerical operations are discussed. Additionally, the possibility of diminishing the memory demand of the whole algorithm from second to first power of the number of modes by implementing the iterative approach is demonstrated. This allows to carry out calculations up to hundreds of thousands eigenmodes without using a supercomputer.