Nonlinear Eigenvalue Problem for Optimal Resonances in Optical Cavities

Institute of Applied Mathematics and Mechanics of NAS of Ukraine R. Luxemburg str. 74, Donetsk 83114, Ukraine

^∗ i.m.karabash@gmail.com, karabashi@mail.ru

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 ε(s). It is assumed that ε(s) takes values in the range 1 ≤ ε₁ ≤ ε(s) ≤ ε₂. The problem is to design, for a given (real) frequency α, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency α, 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 ε₁ and ε₂. 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.