Nonlinear Eigenvalue Problem for Optimal Resonances in Optical Cavities

I. M. Karabash¹

¹ Institute of Applied Mathematics and Mechanics of NAS of Ukraine R. Luxemburg str. 74, Donetsk 83114, Ukraine

Mathematical modelling of natural phenomena, Tome 8 (2013) no. 1, pp. 143-155

Résumé

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.

Détail

DOI : 10.1051/mmnp/20138110

Bibliographie

[1] Y. Akahane, T. Asano, B. Song, S. Noda Nature 2003 944 947

[2] S. Albeverio, R. Hryniv, Ya. Mykytyuk Methods Funct. Anal. Topology 2007 110 123

[3] 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).

[4] S. Cox, E. Zuazua Indiana Univ. Math. J. 1995 545 573

[5] P. Heider, D. Berebichez, R.V. Kohn, M.I. Weinstein Struct. Multidisc. Optim. 2008 443 456

[6] G.M. Gubreev, V.N. Pivovarchik Funktsional. Anal. i Prilozhen. 1997 70 74

[7] 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.

[8] C.-Y. Kao, F. Santosa Wave Motion 2008 412 427

[9]

[10] I.M. Karabash. Optimization of quasi-normal eigenvalues for Krein-Nudelman strings. Integral Equations and Operator Theory, DOI : 10.1007/s00020-012-2014-4

[11] T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin-Heidelberg-New York, 1980.

[12] M.V. Keldysh Doklady Akad. Nauk SSSR 1951 11 14

[13] M.G. Krein Prikl. Mat. Meh. 1951 323 348

[14] M.G. Krein, A.A. Nudelman Dokl. Akad. Nauk SSSR 1979 1046 1049

[15] L.D. Landau, E.M. Lifshitz. Electrodynamics of continuous media. Pergamon, 1984.

[16] P.T. Leung, S.Y. Liu, S.S. Tong, K. Young Phys. Rev. A 1994 3068 3073

[17] J. Moro, J.V. Burke, M.L. Overton SIAM J. Matrix Anal. Appl. 1997 793 817

[18] M. Notomi, E. Kuramochi, H. Taniyama Opt. Express 2008 11095

[19] V.N. Pivovarchik J. Operator Theory 1997 243 263

[20] V. Pivovarchik, C. Van Der Mee Inverse Problems 2001 1831 1845

[21] M. Reed, B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978.

[22] G. Rempe Appl. Phys. B 1995 233 237

[23] M.A. Shubov Integral Equations Operator Theory 1997 358 372

[24] K. Ujihara Phys. Rev. A 1975 148 158

[25] K.J. Vahala Nature 2003 839 846

[26] Y. Yamamoto, F. Tassone, H. Cao. Semiconductor cavity quantum electrodynamics. Springer, New York, 2000.