Nonlinear bang-bang eigenproblems and optimization of resonances in layered cavities

Quasi-normal-eigenvalue optimization is studied under constraints $b_1(x) \le B(x) \le b_2 (x)$onstructurefunctions$B$ of 2-side open optical or mechanical resonators. We prove existence of various optimizers and provide an example when different structures generate the same optimal quasi-(normal-)eigenvalue. To show that quasi-eigenvalues locally optimal in various senses are in the spectrum ${\mathrm{\Sigma }}^{nl}$ of the bang-bang eigenproblem $y\mathrm{"}=-{\omega }^{2}y[b_1+(b_2-b_1){\chi }_{{\mathbb{C}}_{+}}(y^2)]$, where ${\chi }_{{\mathbb{C}}_{+}}(\cdot )$ is the indicator function of the upper complex half-plane ${\mathbb{C}}_{+}$, we obtain a variational characterization of the nonlinear spectrum ${\mathrm{\Sigma }}^{nl}$ in terms of quasi-eigenvalue perturbations. To address the minimization of the decay rate $\mathrm{\mid }\mathrm{I}\mathrm{m}\omega \mathrm{\mid }$, we study the bang-bang equation and explain how it excludes an unknown optimal $B$ from the optimization process. Computing one of minimal decay structures for 1-side open settings, we show that it resembles gradually size-modulated 1-D stack cavities introduced recently in Optical Engineering. In 2-side open symmetric settings, our example has an additional centered defect. Nonexistence of global decay rate minimizers is discussed.