Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets

Abstract

Laser cavities are open systems, in that energy can leak to the outside via output coupling. The ‘‘normal modes’’ are therefore quasinormal modes, with eigenvalues that are complex and eigenfunctions that extend outside the cavity, such that any normalization integral is dominated by the region outside; in short, such systems are non-Hermitian. This paper addresses the question: How is the complex eigenvalue (i.e., the mode frequency) changed when the cavity is perturbed by a small change of dielectric constant? The usual time-independent perturbation theory fails because of non-Hermiticity. By generalizing the work of Zeldovich [Sov. Phys.—JETP 12, 542 (1961)] for scalar fields in one dimension, we express the change of frequency in terms of matrix elements involving the unperturbed eigenfunctions, so that the problem is reduced to quadrature. We then apply the formalism to shape perturbations of a dielectric microdroplet, and give analytic formulas for the frequency shifts of the morphology-dependent resonances. These results are, surprisingly, independent of the radial wave function, so that all integrals can be performed and explicit algebraic expressions are given for axially symmetric perturbations.