Numerical realization of Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides

The computation of guided modes in photonic crystal wave-guides is a key issue in the process of designing devices in photonic communications. Existing methods, such as the super-cell method, provide an efficient computation of well-confined modes. However, if the modes are not well-confined, the modelling error of the super-cell method becomes prohibitive and advanced methods applying transparent boundary conditions for periodic media are needed. In this work we demonstrate the numerical realization of a recently proposed Dirichlet-to-Neumann approach and compare the results with those of the super-cell method. For the resulting non-linear eigenvalue problem we propose an iterative solution based on Newton's method and a direct solution using Chebyshev interpolation of the non-linear operator. Based on the Dirichlet-to-Neumann approach, we present a formula for the group velocity of guided modes that can serve as an objective function in the optimization of photonic crystal wave-guides.

[1]  Kurt Busch,et al.  Photonic band structure theory: assessment and perspectives , 2002 .

[2]  Christian Wieners,et al.  A computer-assisted proof for photonic band gaps , 2009 .

[3]  Peter Kuchment,et al.  On guided waves in photonic crystal waveguides , 2004 .

[4]  Henri Benisty,et al.  Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal , 2001 .

[5]  Julien Coatléven,et al.  Helmholtz equation in periodic media with a line defect , 2012, J. Comput. Phys..

[6]  T. Krauss.  Why do we need slow light , 2008 .

[7]  A. Sommerfeld,et al.  Elektronentheorie der Metalle , 1967 .

[8]  Mariya Vorobets.  On the Bethe–Sommerfeld conjecture for certain periodic Maxwell operators , 2011 .

[9]  Karl Meerbergen,et al.  The Quadratic Eigenvalue Problem , 2001, SIAM Rev..

[10]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[11]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[12]  E. Cassan,et al.  New approach to describe light refraction at the surface of a photonic crystal , 2010 .

[13]  N. Ahmed,et al.  Discrete Cosine Transform , 1996 .

[14]  Sonia Fliss,et al.  Exact Boundary Conditions for Wave Propagation in Periodic Media Containing a Local Perturbation , 2010 .

[15]  G. Karniadakis,et al.  Spectral/hp Element Methods for Computational Fluid Dynamics , 2005 .

[16]  Y. Lu,et al.  Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices , 2008 .

[17]  A. Figotin,et al.  Localized classical waves created by defects , 1997 .

[18]  P. Kuchment.  Floquet Theory for Partial Differential Equations , 1993 .

[19]  Robert Scheichl,et al.  Convergence Analysis of Planewave Expansion Methods for 2D Schrödinger Operators with Discontinuous Periodic Potentials , 2010, SIAM J. Numer. Anal..

[20]  E. Sargent,et al.  Behavior of light at photonic crystal interfaces , 2005 .

[21]  Steven G. Johnson,et al.  Photonic Crystals: Molding the Flow of Light , 1995 .

[22]  Markus Richter,et al.  On the Spectrum of an Operator Pencil with Applications to Wave Propagation in Periodic and Frequency Dependent Materials , 2008, SIAM J. Appl. Math..

[23]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[24]  Sonia Fliss,et al.  A Dirichlet-to-Neumann Approach for The Exact Computation of Guided Modes in Photonic Crystal Waveguides , 2012, SIAM J. Sci. Comput..

[25]  D. Kressner,et al.  Chebyshev interpolation for nonlinear eigenvalue problems , 2012 .

[26]  T. Krauss,et al.  Systematic design of flat band slow light in photonic crystal waveguides. , 2008, Optics express.

[27]  Sofiane Soussi,et al.  Convergence of the Supercell Method for Defect Modes Calculations in Photonic Crystals , 2005, SIAM J. Numer. Anal..

[28]  P. Kuchment.  The mathematics of photonic crystals , 2001 .

[29]  Dan Givoli,et al.  Recent advances in the DtN FE Method , 1999 .

[30]  P. Kaspar,et al.  Design proposal for a low loss in-plane active photonic crystal waveguide with vertical electrical carrier injection. , 2012, Optics express.

[31]  Kersten Schmidt,et al.  A multiscale hp-FEM for 2D photonic crystal bands , 2011, J. Comput. Phys..

[32]  Kristian Kirsch,et al.  Methods Of Modern Mathematical Physics , 2016 .

[33]  R. Kappeler,et al.  Efficient computation of photonic crystal waveguide modes with dispersive material. , 2010, Optics express.

[34]  Sonia Fliss,et al.  Exact boundary conditions for periodic waveguides containing a local perturbation , 2006 .

[35]  Sonia Fliss,et al.  Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media , 2009 .

[36]  C. Kittel.  Introduction to solid state physics , 1954 .

[37]  Ray T. Chen,et al.  Theory of light refraction at the surface of a photonic crystal , 2005 .

[38]  E. Cassan,et al.  Computation of light refraction at the surface of a photonic crystal using DtN approach , 2010 .

[39]  Stefano Giani,et al.  Adaptive finite element methods for computing band gaps in photonic crystals , 2012, Numerische Mathematik.

[40]  T. Hohage,et al.  Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides , 2010, 1010.1738.

[41]  J. D. Joannopoulos,et al.  Enhancement of nonlinear effects using photonic crystals , 2004, Nature materials.

[42]  Daniele Boffi,et al.  Modified edge finite elements for photonic crystals , 2006, Numerische Mathematik.

[43]  Nicholas Chako,et al.  Wave propagation and group velocity , 1960 .

[44]  Sonia Fliss.  Analyse mathématique et numérique de problèmes de propagation des ondes dans des milieux périodiques infinis localement perturbés , 2009 .