Maxwell eigenmodes in product domains
Martin Costabel,
Monique Dauge
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 Dirichlet 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
In Maxwell's Equations Analysis and Numerics,
(De Gruyter, Radon Series on Computational and Applied Mathematics) (2019).
