Overdamped modes and optimization of resonances in layered cavities

Abstract

We study the problem of optimizing the imaginary parts $\mathrm{Im}\, \omega$ of quasi-normal-eigenvalues $\omega$ associated with the equation $y'' = -\omega^2 B y $. It is assumed that the coefficient $B(x)$, which describes the structure of an optical or mechanical resonator, is constrained by the inequalities $0 \le b_1 \le B(x) \le b_2 $. Extremal quasi-normal-eigenvalues belonging to the imaginary line ${\mathrm{i}} {\mathbb R}$ are studied in detail. As an application, we provide examples of $\omega$ with locally minimal $|\mathrm{Im}\, \omega|$ (without additional restrictions on $\mathrm{Re}\, \omega$) and show that a structure generating an optimal quasi-normal-eigenvalue on ${\mathrm{i}} {\mathbb R}$ is not necessarily unique.