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

Article

Abstract

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.

Keywords

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 

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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

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