Math. Model. Nat. Phenom.
Volume 8, Number 1, 2013Harmonic analysis
|Page(s)||143 - 155|
|Published online||28 January 2013|
Nonlinear Eigenvalue Problem for Optimal Resonances in Optical Cavities
Institute of Applied Mathematics and Mechanics of NAS
of Ukraine R. Luxemburg str. 74, Donetsk
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.
Mathematics Subject Classification: 78M50 / 49R05 / 47N50 / 47A55
Key words: photonic crystal / high Q-factor resonator / quasi-normal eigenvalue optimization / nonlinear eigenvalue
© EDP Sciences, 2013
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