Nonlinear eigenvalue problem for optimal resonances in optical cavities

TL;DR

This paper investigates the design of 1-D optical cavities with minimal decay resonances at a given frequency, revealing that extremal structures are layered photonic crystals with specific dielectric properties.

Contribution

It introduces a nonlinear eigenvalue problem characterizing extremal resonances and reduces the cavity design problem to a four-dimensional zero-finding task.

Findings

Extremal cavities are layered photonic crystals with specific permittivity bounds.

Coordinates of layer interfaces relate to the arg-function of resonant modes.

Decay rate minimization reduces to solving a four-dimensional nonlinear problem.

Abstract

The paper is devoted to optimization of resonances in a 1-D open optical cavity. The cavity's structure is represented by its dielectric permittivity function e(s). It is assumed that e(s) takes values in the range 1 <= e_1 <= e(s) <= e_2. The problem is to design, for a given (real) frequency, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency, we define cavities and resonant modes with locally extremal decay rate, and then study their properties. We show that such locally extremal cavities are 1-D photonic crystals consisting of alternating layers of two materials with extreme allowed dielectric permittivities e_1 and e_2. To find thicknesses of these layers, a nonlinear eigenvalue problem for locally extremal resonant modes is derived. It occurs that coordinates of interface planes between the layers can be expressed via arg-function of corresponding modes. As a result, the question of minimization of the decay rate is reduced to a four-dimensional problem of finding the zeroes of a function of two variables.