# Euler-Lagrange equations for full topology optimization of the Q-factor in leaky cavities

## TL;DR

This paper develops a mathematical framework using Euler-Lagrange equations for optimizing the Q-factor in lossy optical cavities, enabling better design of cavities with minimal decay rates.

## Contribution

It introduces a novel set of nonlinear Maxwell equations derived from topology optimization principles, incorporating Pareto optimality and multi-parameter perturbation theory.

## Key findings

- Derivation of Euler-Lagrange equations for cavity decay rate optimization
- Formulation of nonlinear Maxwell systems with switching functions
- Discussion of parallels with optimal control theory

## Abstract

We derive Euler-Lagrange equations for the topology optimization of decay rate in 3-d lossy optical cavities. This leads to a new class of time-harmonic differential or integro-differential equations, which can be written as nonlinear Maxwell systems with switching functions of special types. Our approach is based on the notion of Pareto optimal frontier and on the multi-parameter perturbation theory for eigenfrequencies. Parallels with optimal control theory are discussed.