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An Efficient Calculation of Photonic Crystal Band Structures Using Taylor Expansions

Published online by Cambridge University Press:  03 June 2015

Dirk Klindworth  and
Kersten Schmidt
Dirk Klindworth*
Affiliation:
Research Center MATHEON, TU Berlin, Department of Mathematics, Straβe des 17. Juni 136, 10623 Berlin, Germany
Kersten Schmidt*
Affiliation:
Research Center MATHEON, TU Berlin, Department of Mathematics, Straβe des 17. Juni 136, 10623 Berlin, Germany
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

Keywords

35Q6165N3065Z0578-0478M1042.70.Qs71.20.-b02.30.Jr02.70.DhPhotonic crystalsband structure calculationgroup velocityTaylor expansionadaptivity
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 5 , November 2014 , pp. 1355 - 1388
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Babuška, I. and Osborn, J.Eigenvalue problems. Handbook of Numerical Analysis, II:641–787,1991.Google Scholar
[2]Boffi, D., Conforti, M., and Gastaldi, L.Modified edge finite elements for photonic crystals. Numer. Math., 105(2):249–266,2006.Google Scholar
[3]Brandsmeier, H.Standard and generalized hp-finite element discretizations for periodic structures. PhD thesis, ETH Zurich, 2013.Google Scholar
[4]Brandsmeier, H., Schmidt, K., and Schwab, C.A multiscale hp-FEM for 2D photonic crystal bands. J. Comput. Phys., 230(2):349–374,2011.Google Scholar
[5]Brillouin, L.Wave Propagation and Group Velocity. Pure and Applied Physics, Vol. 8. spcademic Press, New York, NY, USA, 1960.Google Scholar
[6]Busch, K.Photonic band structure theory: Assessment and perspectives. C. R. Physique, 3(1):53–66,2002.Google Scholar
[7]Carstensen, C., Gedicke, J., Mehrmann, V., and Miedlar, A.An adaptive homotopy approach for non-selfadjoint eigenvalue problems. Numer. Math., 119(3):557–583,2011.Google Scholar
[8]Concepts Development Team. Webpage of Numerical C++ Library Concepts 2. http://www.concepts.math.ethz.ch, 2013.Google Scholar
[9]de Sterke, C. M. and Sipe, J. E.Envelope-function approach for the electrodynamics of non-linear periodic structures. Phys. Rev. A, 38:5149–5165,1988.Google Scholar
[10]Engström, C. and Richter, M.An the spectrum of an operator pencil with applications to wave propagation in periodic and frequency dependent materials. SIAM J. Appl. Math., 70(1):231–247,2009.CrossRefGoogle Scholar
[11]Fliss, S.A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides. SIAM J. Sci. Comput., 35(2):B438–B461,2013.Google Scholar
[12]Frauenfelder, P. and Lage, C.Concepts - an object-oriented software package for partial differential equations. Math. Model. Numer. Anal., 36(5):937–951,2002.CrossRefGoogle Scholar
[13]Giani, S. and Graham, I. G.Adaptive finite element methods for computing band gaps in photonic crystals. Numer. Math., 121(1):31–64,2012.Google Scholar
[14]Hermann, D., Frank, M., Busch, K., and Wolfe, P.Photonic band structure computations. Opt. Express, 8(3):167–172,2001.Google Scholar
[15]Hoang, V., Plum, M., and Wieners, C.A computer-assisted proof for photonic band gaps. Z. Angew. Math. Phys., 60(6):1035–1052,2009.Google Scholar
[16]Hu, Z. and Lu, Y. Y.Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices. Opt. Quantum Electron., 40(11–12):921–932,2008.Google Scholar
[17]Istrate, E., Green, A. A., and Sargent, E. H.Behavior of light at photonic crystal interfaces. Phys. Rev. B, 71(19):195122,2005.Google Scholar
[18]Jiang, W., Chen, R. T., and Lu, X.Theory of light refraction at the surface of a photonic crystal. Phys. Rev. B, 71:245115,2005.CrossRefGoogle Scholar
[19]Joannopoulos, J. D.Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton, NJ, USA, 2008.Google Scholar
[20]Kaspar, P., Kappeler, R., Erni, D., and Jäckel, H.Average light velocities in periodic media. J. Opt. Soc. Am. B, 30(11):2849–2854,2013.CrossRefGoogle Scholar
[21]Katō, T.Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin & Heidelberg, Germany, 1995.Google Scholar
[22]Klindworth, D., Schmidt, K., and Fliss, S.Numerical realization of Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides. Comput. Math. Appl., 67(4):918–943,2014.CrossRefGoogle Scholar
[23]Kuchment, P.Floquet Theory for Partial Differential Equations. Birkhauser, Basel, Switzerland, 1993.Google Scholar
[24]Kuchment, P.The Mathematics of Photonic Crystals, chapter 7, pages 207–272. SIAM, Philadelphia, PA, USA, 2001.Google Scholar
[25]Lui, S., Keller, H., and Kwok, T.Homotopy method for the large, sparse, real nonsymmetric eigenvalue problem. SIAM J. Matrix Anal. Appl., 18(2):312–333,1997.CrossRefGoogle Scholar
[26]Marletta, M. and Scheichl, R.Eigenvalues in spectral gaps of differential operators. J. Spectr. Theory, 2(3):293–320,2012.Google Scholar
[27]Norton, R. A. and Scheichl, R.Convergence analysis of planewave expansion methods for 2D Schrödinger operators with discontinuous periodic potentials. SIAM J. Numer. Anal., 47(6):4356–4380,2010.Google Scholar
[28]Norton, R. A. and Scheichl, R.Planewave expansion methods for photonic crystal fibres. Appl. Numer. Math., 63(0):88 – 104, 2013.Google Scholar
[29]Odeh, F. and Keller, J. B.Partial differential equations with periodic coefficients and bloch waves in crystals. Journal of Mathematical Physics, 5(11):1499–1504,1964.CrossRefGoogle Scholar
[30]Olivier, S., Benisty, H., Smith, C. J. M., Rattier, M., Weisbuch, C., and Krauss, T. F.Transmission properties of two-dimensional photonic crystal channel waveguides. Opt. Quant. Electron., 34:171–181,2002.Google Scholar
[31]Olivier, S., Rattier, M., Benisty, H., Weisbuch, C., Smith, C. J. M., De La Rue, R. M., Krauss, T. F., Oesterle, U., and Houdre, R.Mini-stopbands of a one-dimensional system: The channel waveguide in a two-dimensional photonic crystal. Phys. Rev. B, 63(11):113311,2001.Google Scholar
[32]Quarteroni, A., Sacco, R., and Saleri, F.Numerical Mathematics, volume 37 of Texts in Applied Mathematics. Springer, Berlin & Heidelberg, Germany, 2nd edition, 2007.Google Scholar
[33]Reed, M. and Simon, B.Methods of Modern Mathematical Physics, volume 4: Analysis of Operators. Academic Press, New York, NY, USA, 1978.Google Scholar
[34]Sauter, S. and Schwab, C.Boundary Element Methods. Springer, Berlin & Heidelberg, Germany, 2011.Google Scholar
[35]Schmidt, K. and Kappeler, R.Efficient computation of photonic crystal waveguide modes with dispersive material. Opt. Express, 18(7):7307–7322,2010.Google Scholar
[36]Schmidt, K. and Kauf, P.Computation of the band structure of two-dimensional photonic crystals with hp finite elements. Comp. Meth. App. Mech. Engr., 198:1249–1259,2009.Google Scholar
[37]Sipe, J. E.Vector k. p approach for photonic band structures. Phys. Rev. E, 62(4):5672–5677, 2000.Google Scholar
[38]Soussi, S.Convergence of the supercell method for defect modes calculations in photonic crystals. SIAM J. Numer. Anal., 43(3):1175–1201,2005.Google Scholar