BIT Numerical Mathematics

, Volume 55, Issue 1, pp 81–115 | Cite as

Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides

  • Sonia Fliss
  • Dirk KlindworthEmail author
  • Kersten Schmidt


The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.


Robin-to-Robin map Photonic crystal wave-guide Surface modes High-order FEM Non-linear eigenvalue problem 

Mathematics Subject Classification

35P30 35Q61 65N30 65Z05 78-04 78M10 


  1. 1.
    Besse, C., Coatléven, J., Fliss, S., Lacroix-Violet I., Ramdani, K.: Transparent boundary conditions for locally perturbed infinite hexagonal periodic media. arXiv:1205.5345v1 (2012)
  2. 2.
    Coatléven, J.: Helmholtz equation in periodic media with a line defect. J. Comput. Phys. 231(4), 1675–1704 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Concepts Development Team. Webpage of numerical C++ library concepts 2. (2013)
  4. 4.
    Fliss, S.: Etude mathématique et numérique de la propagation des ondes dans des milieux périodiques localement perturbés. Ph.D. thesis, École Doctorale de l’École Polytechnique (2009)Google Scholar
  5. 5.
    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
  6. 6.
    Fliss, S., Cassan, E., Bernier, D.: New approach to describe light refraction at the surface of a photonic crystal. JOSA B 27, 1492–1503 (2010)CrossRefGoogle Scholar
  7. 7.
    Fliss, S., Joly, P.: Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media. Appl. Numer. Math. 59(9), 2155–2178 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fliss S., Joly, P., Li J.-R.: Exact boundary conditions for wave propagation in periodic media containing a local perturbation. In: Ehrhardt, M. (ed.) Wave Propagation in Periodic Media—Analysis, Numerical Techniques and practical Applications, Progress in Computational Physics, volume 1, chapter 5, pp. 108–134. Bentham Science Publishers, Sharjah (2010)Google Scholar
  9. 9.
    Frauenfelder, P., Lage, C.: Concepts—an object-oriented software package for partial differential equations. Math. Model. Numer. Anal. 36(5), 937–951 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Joannopoulos, J.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2008)Google Scholar
  11. 11.
    Joly, P., Li, J.-R., Fliss, S.: Exact boundary conditions for periodic waveguides containing a local perturbation. Commun. Comput. Phys. 1(6), 945–973 (2006)zbMATHGoogle Scholar
  12. 12.
    Katō, T.: Perturbation Theory for Linear Operators. Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg (1995)Google Scholar
  13. 13.
    Klindworth, D., Schmidt, K., Fliss, S.: Numerical realization of Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides. Comput. Math. Appl. 67(4), 918–943 (2014)Google Scholar
  14. 14.
    Kuchment, P.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993)CrossRefzbMATHGoogle Scholar
  15. 15.
    Melenk, J.M., Sauter, S.: Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49(3), 1210–1243 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Sauter, S., Schwab, C.: Boundary Element Methods. Springer, Berlin, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    Schmidt, K., Kauf, P.: Computation of the band structure of two-dimensional photonic crystals with hp finite elements. Comput. Methods Appl. Mech. Eng. 198, 1249–1259 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Steinbach, O.: On a generalized \({L}^2\) projection and some related stability estimates in Sobolev spaces. Numer. Math. 90(4), 775–786 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43(2), 235–286 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Wohlmuth, B.I.: A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38(3), 989–1012 (2001)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIAENSTA ParisTechParisFrance
  2. 2.Department of Mathematics, Research Center MATHEONTU BerlinBerlinGermany

Personalised recommendations