Open Access

# Overdamped modes and optimization of resonances in layered cavities

### Abstract

We study the problem of optimizing the imaginary parts $\mathrm{Im}\, \omega$ of quasi-normal-eigenvalues $\omega$ associated with the equation $y'' = -\omega^2 B y$. It is assumed that the coefficient $B(x)$, which describes the structure of an optical or mechanical resonator, is constrained by the inequalities $0 \le b_1 \le B(x) \le b_2$. Extremal quasi-normal-eigenvalues belonging to the imaginary line ${\mathrm{i}} {\mathbb R}$ are studied in detail. As an application, we provide examples of $\omega$ with locally minimal $|\mathrm{Im}\, \omega|$ (without additional restrictions on $\mathrm{Re}\, \omega$) and show that a structure generating an optimal quasi-normal-eigenvalue on ${\mathrm{i}} {\mathbb R}$ is not necessarily unique.

Key words: Dissipation frequencies, damped string, layered optical cavity, optimal design, spectral optimization.

### Article Information

 Title Overdamped modes and optimization of resonances in layered cavities Source Methods Funct. Anal. Topology, Vol. 23 (2017), no. 3, 252-260 MathSciNet MR3707519 Milestones Received 12/05/2017; Revised 31/05/2017 Copyright The Author(s) 2017 (CC BY-SA)

### Authors Information

Illya M. Karabash
Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 1 Dobrovolskogo, Slovyansk, 84100, Ukraine

Olga M. Logachova
Siberian State University of Geosystems and Technologies, Department of Higher Mathematics, Institute of Geodesy and Management, 10 Plakhotnogo, Novosibirsk, 630108, Russia
Novosibirsk State University of Economics and Management, Department of Mathematics and Natural Sciences, 56 Lomonosov str., Novosibirsk, 630099, Russia

Ievgen V. Verbytskyi
National Technical University of Ukraine ”Igor Sikorsky Kyiv Polytechnic Institute”, Department of Industrial Electronics, Faculty of Electronics, 16 Politekhnichna, block 12, Kyiv, 03056, Ukraine

### Citation Example

Illya M. Karabash, Olga M. Logachova, and Ievgen V. Verbytskyi, Overdamped modes and optimization of resonances in layered cavities, Methods Funct. Anal. Topology 23 (2017), no. 3, 252-260.

### BibTex

@article {MFAT986,
AUTHOR = {Karabash, Illya M. and Logachova, Olga M. and Verbytskyi, Ievgen V.},
TITLE = {Overdamped modes and optimization of resonances in
layered cavities},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {23},
YEAR = {2017},
NUMBER = {3},
PAGES = {252-260},
ISSN = {1029-3531},
MRNUMBER = {MR3707519},
URL = {http://mfat.imath.kiev.ua/article/?id=986},
}

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