Maxwell eigenmodes in product domains

TL;DR

This paper analyzes Maxwell eigenmodes in three-dimensional product domains, deriving explicit eigenpairs for various geometries and revealing common polarization structures and special TEM modes.

Contribution

It provides a unified framework for understanding Maxwell eigenmodes in product domains, linking eigenvalues to scalar Laplace operators and explicitly solving for key geometries.

Findings

Eigenvalues in Cartesian domains are sums of scalar Laplace eigenvalues.

Eigenvectors have a tensor product structure.

Explicit eigenpairs are found for cuboids and cylinders.

Abstract

This paper is devoted to Maxwell modes in three-dimensional bounded electromagnetic cavities that have the form of a product of lower dimensional domains in some system of coordinates. The boundary conditions are those of the perfectly conducting or perfectly insulating body. The main case of interest is products in Cartesian variables. Cylindrical and spherical variables are also addressed. We exhibit common structures of polarization type for eigenmodes. In the Cartesian case, the cavity eigenvalues can be obtained as sums of Dirich-let or Neumann eigenvalues of positive Laplace operators and the corresponding eigenvectors have a tensor product form. We compare these descriptions with the spherical wave function Ansatz for a ball and show why the cavity eigenvalue of the ball are also Dirichlet or Neumann eigenvalues of some scalar operators. As application of our general formulas, we find explicit eigenpairs in a cuboid, in a circular cylinder, and in a cylinder with a coaxial circular hole. This latter example exhibit interesting '' TEM '' eigenmodes that have a one-dimensional vibrating string structure, and contribute to the least energy modes if the cylinder is long enough.