Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T01:57:21.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear Eigenvalue Problem for Optimal Resonances in OpticalCavities

Published online by Cambridge University Press:  28 January 2013

I. M. Karabash
Get access

Abstract

The paper is devoted to optimization of resonances in a 1-D open optical cavity. Thecavity’s structure is represented by its dielectric permittivity functionε(s). It is assumed thatε(s) takes values in the range1 ≤ ε1 ≤ ε(s) ≤ ε2.The problem is to design, for a given (real) frequency α, a cavity havinga resonance with the minimal possible decay rate. Restricting ourselves to resonances of agiven frequency α, we define cavities and resonant modes with locallyextremal decay rate, and then study their properties. We show that such locally extremalcavities are 1-D photonic crystals consisting of alternating layers of two materials withextreme allowed dielectric permittivities ε1 andε2. To find thicknesses of these layers, a nonlineareigenvalue problem for locally extremal resonant modes is derived. It occurs thatcoordinates of interface planes between the layers can be expressed via arg-function ofcorresponding modes. As a result, the question of minimization of the decay rate isreduced to a four-dimensional problem of finding the zeroes of a function of twovariables.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Akahane, Y., Asano, T., Song, B., Noda, S.. High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature, 425 (2003), 944–947. CrossRefGoogle Scholar
Albeverio, S., Hryniv, R., Mykytyuk, Ya.. Inverse spectral problems for coupled oscillating systems : reconstruction from three spectra. Methods Funct. Anal. Topology, 13 (2007), No. 1, 110–123. Google Scholar
S. Burger, J. Pomplun, F. Schmidt, L. Zschiedrich. Finite-element method simulations of high-Q nanocavities with 1D photonic bandgap. Proc. SPIE Vol. 7933 (2011), 79330T (Physics and Simulation of Optoelectronic Devices XIX).
Cox, S., Zuazua, E.. The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J., 44 (1995), No. 2, 545–573. CrossRefGoogle Scholar
Heider, P., Berebichez, D., Kohn, R.V., Weinstein, M.I.. Optimization of scattering resonances. Struct. Multidisc. Optim., 36 (2008), 443–456. CrossRefGoogle Scholar
Gubreev, G.M., Pivovarchik, V.N.. Spectral analysis of the Regge problem with parameters. Funktsional. Anal. i Prilozhen., 31 (1997), No. 1, 70–74 (Russian); Engl. transl. : Funct. Anal. Appl., 31 (1997), No. 1, 54–57. CrossRefGoogle Scholar
I.S. Kac, M.G. Krein. On the spectral functions of the string. Supplement II in Atkinson, F. Discrete and continuous boundary problems. Mir, Moscow, 1968. Engl. transl. : Amer. Math. Soc. Transl., Ser. 2, 103 (1974), 19–102.
Kao, C.-Y., Santosa, F.. Maximization of the quality factor of an optical resonator. Wave Motion, 45 (2008), 412–427. CrossRefGoogle Scholar
I.M. Karabash. Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures. To appear in Asymptotic Analysis (see also the preprint of the paper arXiv :1103.4117v5 [math.SP]).
I.M. Karabash. Optimization of quasi-normal eigenvalues for Krein-Nudelman strings. Integral Equations and Operator Theory, DOI : 10.1007/s00020-012-2014-4
T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin-Heidelberg-New York, 1980.
Keldysh, M.V.. On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations. Doklady Akad. Nauk SSSR, 77 (1951), 11–14 (Russian). Google Scholar
Krein, M.G.. On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability. Prikl. Mat. Meh., 15 (1951), 323–348 (Russian); English transl. : Amer. Math. Soc. Transl.(2), 1 (1955), 163–187. Google Scholar
Krein, M.G., Nudelman, A.A.. On direct and inverse problems for the boundary dissipation frequencies of a nonuniform string. Dokl. Akad. Nauk SSSR, 247 (1979), No. 5, 1046–1049 (Russian). Engl. transl. : Soviet Math. Dokl., 20 (1979), No. 4, 838–841. Google Scholar
L.D. Landau, E.M. Lifshitz. Electrodynamics of continuous media. Pergamon, 1984.
Leung, P.T., Liu, S.Y., Tong, S.S., Young, K.. Time-independent perturbation theory for quasinormal modes in leaky optical cavities. Phys. Rev. A, 49 (1994), 3068–3073. CrossRefGoogle ScholarPubMed
Moro, J., Burke, J.V., Overton, M.L.. On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure. SIAM J. Matrix Anal. Appl., 18 (1997), No. 4, 793–817. CrossRefGoogle Scholar
Notomi, M., Kuramochi, E., Taniyama, H.. Ultrahigh-Q nanocavity with 1d photonic gap. Opt. Express, 16 (2008), 11095. CrossRefGoogle ScholarPubMed
Pivovarchik, V.N., Inverse problem for a smooth string with damping at one end. J. Operator Theory, 38 (1997), No. 2, 243–263. Google Scholar
Pivovarchik, V., van der Mee, C.. The inverse generalized Regge problem. Inverse Problems, 17 (2001), No. 6, 1831–1845. CrossRefGoogle Scholar
M. Reed, B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978.
Rempe, G.. One atom in an optical cavity : spatial resolution beyond the standard diffraction limit. Appl. Phys. B, 60 (1995), 233–237. CrossRefGoogle Scholar
Shubov, M.A.. Spectral operators generated by damped hyperbolic equations. Integral Equations Operator Theory, 28 (1997), No. 3, 358–372. CrossRefGoogle Scholar
Ujihara, K.. Quantum theory of a one-dimensional optical cavity with output coupling. Field quantization. Phys. Rev. A, 12 (1975), 148–158. CrossRefGoogle Scholar
Vahala, K.J.. Optical microcavities. Nature, 424 (2003), 839–846. CrossRefGoogle ScholarPubMed
Y. Yamamoto, F. Tassone, H. Cao. Semiconductor cavity quantum electrodynamics. Springer, New York, 2000.