Pareto optimization of resonances and minimum-time control

TL;DR

This paper formulates spectral resonance optimization as Pareto and control problems on the Riemann sphere, enabling the design of optimal layered optical cavities with improved resonance properties.

Contribution

It introduces a novel reduction of spectral optimization to control problems, providing explicit methods for computing Pareto optimal resonators and analyzing their properties.

Findings

Explicit Pareto frontiers for resonator optimization

High-accuracy computation of high-Q resonators

Bounds on optimal layer widths

Abstract

The aim of the paper is to reduce one spectral optimization problem, which involves the minimization of the decay rate $|\mathrm{Im} \, k |$ of a resonance $k$, to a collection of optimal control problems on the Riemann sphere $\widehat{\mathbb{C}}$. This reduction allows us to apply methods of extremal synthesis to the structural optimization of layered optical cavities. We start from a dual problem of minimization of the resonator length and give several reformulations of this problem that involve Pareto optimization of the modulus $|k|$ of a resonance, a minimum-time control problem on $\widehat{\mathbb{C}}$, and associated Hamilton-Jacobi-Bellman equations. Various types of controllability properties are studied in connection with the existence of optimizers and with the relationship between the Pareto optimal frontiers of minimal decay and minimal modulus. We give explicit examples of optimal resonances and describe qualitatively properties of the Pareto frontiers near them. A special representation of bang-bang controlled trajectories is combined with the analysis of extremals to obtain various bounds on optimal widths of layers. We propose a new method of computation of optimal symmetric resonators based on minimum-time control and compute with high accuracy several Pareto optimal frontiers and high-Q resonators.