Reciprocity and orthogonality relations for ring resonators

A general and rigorous derivation of the reciprocity and orthogonality relations for ring resonators is given without resorting to matrix representations. The general form of the integral equations appropriate to the study of ring resonators containing at least one hard aperture is discussed, and a reciprocity relation for a very general class of ring resonator is established using a theorem concerning so-called Hilbert-Schmidt kernels. It is shown that under very general conditions linear ring resonators are reciprocal and that the transverse eigenmodes for propagation in any direction around the resonator are biorthogonal to those for propagation in the opposite direction.