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A perturbation approach for near bound-state resonances of photonic crystal with defect

Published online by Cambridge University Press:  09 July 2015

J. LIN
J. LIN*
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA email: jzl0097@auburn.edu
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Abstract

This paper is concerned with scattering resonances of a 1D photonic crystal of finite extent. We propose a general perturbation approach to study the resonances that are close to the bound-state frequency of the infinite structure when some defect is embedded in the interior. It is shown that near bound-state resonances exist on the complex plane and the distance between the resonance and the associated bound-state frequency decays exponentially as a function of the number of periodic cells. A numerical approach based upon the perturbation theory is also proposed to calculate the near bound-state resonances accurately.

Keywords

photonic crystalscattering resonancesHelmholtz equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 1 , February 2016 , pp. 66 - 86
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Akahane, Y., Asano, T., Song, B. & Noda, S. (2003) High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature 425 (no. 6961), 944–947.Google Scholar
[2] Ahlfors, L. (1953) Complex Analysis, 2nd ed., McGraw-Hill, New York.Google Scholar
[3] Dobson, D., Santosa, F., Shipman, S. & Weinstein, M. (2013) Resonances of a potential well with a thick barrier. SIAM J. Appl. Math. 73, 1489–1512.CrossRefGoogle Scholar
[4] Figotin, A. & Gorentsvieig, V. (1998) Localized electromagnetic waves in a layered periodic dielectric medium with a defect. Phy. Rev. B 58, 180–188.Google Scholar
[5] Figotin, A. & Klein, A. (1997) Localized classical waves created by defects. J. Stat. Phys. 86, 165–177.Google Scholar
[6] Figotin, A. & Klein, A. (1998) Midgap defect modes in dielectric and acoustic media. SIAM J. Appl. Math. 58, 1748–1773.Google Scholar
[7] Figotin, A. & Kuchment, P. (1996) Band-gap structure of spectra of periodic dielectric and acoustic media. I. scalar model. SIAM J. Appl. Math. 56, 68–88.Google Scholar
[8] Heider, P., Berebichez, D., Kohn, R. V. & Weinstein, M. I. (2008) Optimization of scattering resonances. Struct. Multidiscip. Optim. 36, 443–456.CrossRefGoogle Scholar
[9] Kao, C. Y. & Santosa, F. (2008) Maximization of the quality factor of an optical resonator. Wave Motion 45, 412–427.Google Scholar
[10] Kato, T. (1995) Perturbation Theory for Linear Operators, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[11] Khitrova, G., Gibbs, H. M., Jahnke, F., Kira, M. and Koch, S. W. (1999) Nonlinear optics of normal-mode-coupling semiconductor microcavities. Rev. Mod. Phys. 71, 1591–1639.Google Scholar
[12] Knopp, K. (1996) Theory of Function, Parts I and Parts II, Dover Publications, New York.Google Scholar
[13] Krantz, S. (2001) Function Theory of Several Complex Variables, 2nd ed., American Mathematical Society.CrossRefGoogle Scholar
[14] Iantchenko, A. (2006) Resonance spectrum for one-dimensional layered media. Appl. Anal. 85 (11), 1383–1410.Google Scholar
[15] Lin, J. & Santosa, F. (2013) Resonances of a finite one-dimensional photonic crystal with a defect. SIAM J. Appl. Math. 73, 1002–1019.CrossRefGoogle Scholar
[16] Lin, J. & Santosa, F. (2015) Scattering resonances for a two-dimensional potential well with thick barrier. SIAM J. Math. Anal. 47, 1458–1488.Google Scholar
[17] Michler, P. et al. (2000) A quantum dot single-photon turnstile device. Science 290, 2282–2285.Google Scholar
[18] Ramdani, K. & Shipman, S. (2008) Transmission through a thick periodic slab. Math. Models Methods Appl. Sci. 18, 543–572.Google Scholar
[19] Reed, M. & Simon, B. (1978) Methods of Modern Mathematical Physics, Vol. 4, Analysis of operators. Academic Press, New York.Google Scholar
[20] Soussi, S. (2005) Convergence of the supercell method for defect modes calculations in photonic crystals. SIAM J. Numer. Anal. 43, 1175–1201.Google Scholar
[21] Spillane, S. M., Kippenberg, T. J. & Vahala, K. J. (2002) Ultralow-threshold Raman laser using a spherical dielectric microcavity. Nature 415, 621–623.Google Scholar
[22] Tang, S. & Zworski, M. (2000) Resonance expansions of scattered waves. Commun. Pure. Appl. Math. 53, 1305–1334.Google Scholar
[23] Zworski, M. (1999) Resonances in physics and geometry, Notices of the AMS, 46, 319–328.Google Scholar
[24] Vahala, K. J. (2004) Optical Microcavities, World Science Publishing, Singapore.Google Scholar