Nonlinear Bang–Bang Eigenproblems and Optimization of Resonances in Layered Cavities

Abstract

We study optimization of quasi-normal-eigenvalues $\omega$ associated with the equation $y^{\prime \prime } = -\omega ^2 B y$ of two-side open optical and mechanical resonators. The coefficient B(x) is assumed to satisfy the side constraints $b_1(x) \le B(x) \le b_2 (x)$ and is subjected to modifications with the aim to move a particular quasi-(normal-)eigenvalue closer to the real line. The existence of various optimizers is rigorously proved including the existence of local minimizers for the decay rate in the case of low contrast. We show that locally extremal quasi-eigenvalues belong to the spectrum $\Sigma ^\mathrm {nl}$ of the bang-bang eigenproblem $y^{\prime \prime } = - \omega ^2 y [ b_1 + (b_2 - b_1) \chi _{_{\scriptstyle \mathbb C_+}}(y^2 ) ]$ (here $\chi _{_{\scriptstyle \mathbb C_+}}(\cdot )$ is the indicator function of the upper complex half-plane $\mathbb C_+$). To achieve this a variational characterization of $\Sigma ^\mathrm {nl}$ is obtained in terms of quasi-eigenvalue perturbations. To address the minimization of the decay rate $| {{\mathrm{Im}}}\omega |$ for a fixed frequency ${{\mathrm{Re}}}\omega$, we develop and rigorously justify a new numerical method based on the above nonlinear equation. This approach excludes an infinite-dimensional unknown $B (\cdot )$ from the optimization process. In a numerical experiment, we compute some of quasi-eigenvalues of minimal decay and compare one of associated structures with recently introduced designs of high quality-factor cavities.