Characterization of Electromagnetic Cavity Resonators by Integral Identities

Abstract

This paper is intended to derive a pair of integral identities for the Dirichlet and Neumann Laplacian eigenvalue problems. These integral identities show the relationship between the $L^2$ integral of the eigenfunction in the domain and that of the eigenfunction on the boundary of the domain and exhibit a property linking the Dirichlet and Neumann Laplacian problems. Based on these identities we obtain the upper and lower bounds for the quality (Q) factors of the Dirichlet and Neumann electromagnetic cavity resonators. One of our objectives is to show that the conjecture that the Q-factors of the Dirichlet electromagnetic resonators are proportional to the volume of the cavity and inversely proportional to the surface of the cavity holds if the (two dimensional) cavity is a disk but that it does not generally hold.