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Nonlinear Eigenvalue Problem for Optimal Resonances in Optical Cavities

Published online by Cambridge University Press:  28 January 2013

I. M. Karabash
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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.

Keywords

photonic crystalhigh Q-factor resonatorquasi-normal eigenvalue optimizationnonlinear eigenvalue
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 143 - 155
Copyright
© EDP Sciences, 2013

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