Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures

Article type: Research Article

Authors: Karabash, Illya M.

Affiliations: Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg str. 74, 83114 Donetsk, Ukraine. Tel.: +38 062 3110830, Fax: +38 062 3110285; E-mails: i.m.karabash@gmail.com, karabashi@mail.ru

Abstract: The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given α∈R a medium that generates a resonance on the line α+iR with a minimal possible modulus of the imaginary part. We consider an admissible family of media that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints 0≤b1≤B(x)≤b2 with certain constants b1,2. The paper gives an accurate definition of optimal structures that ensures their existence. We prove that optimal structures are piecewise constant functions taking only two extreme possible values b1 and b2. This result explains an effect recently observed in numerical experiments. Then we show that intervals of constancy of an optimal structure are tied to the phase of the corresponding resonant mode and write this connection as a nonlinear eigenvalue problem.

Keywords: photonic crystal, resonance perturbations, quasi-normal level

DOI: 10.3233/ASY-2012-1128

Journal: Asymptotic Analysis, vol. 81, no. 3-4, pp. 273-295, 2013