Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media

We consider the solution of the Helmholtz equation with absorption -@Du(x)-n(x)^2(@w^2+i@e)u(x)=f(x), x=(x,y), in a 2D periodic medium @W=R^2. We assume that f(x) is supported in a bounded domain @W^i and that n(x) is periodic in the two directions in @W^e=@W@?@W^i. We show how to obtain exact boundary conditions on the boundary of @W^i, @S"S that will enable us to find the solution on @W^i. Then the solution can be extended in @W in a straightforward manner from the values on @S"S. The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems.

[1]  Steven G. Johnson,et al.  Photonic Crystals: The Road from Theory to Practice , 2001 .

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

[3]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[4]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[5]  H. Dym,et al.  Operator theory: Advances and applications , 1991 .

[6]  F. Brezzi,et al.  On the coupling of boundary integral and finite element methods , 1979 .

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

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

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

[10]  Alexander Figotin,et al.  Localization of light in lossless inhomogeneous dielectrics , 1998 .

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

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

[13]  Frank Schmidt,et al.  Solving Time-Harmonic Scattering Problems Based on the Pole Condition I: Theory , 2003, SIAM J. Math. Anal..

[14]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[15]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[16]  R. Kleinman,et al.  Second International Conference on Mathematical and Numerical Aspects of Wave Propagation , 1993 .

[17]  George Herrmann,et al.  Floquet waves in anisotropic periodically layered composites , 1992 .

[18]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[19]  Alexander Figotin,et al.  Midgap Defect Modes in Dielectric and Acoustic Media , 1998, SIAM J. Appl. Math..

[20]  Jerome K. Butler,et al.  Floquet Multipliers of Periodic Waveguides via Dirichlet-to-Neumann Maps , 2000 .

[21]  李幼升,et al.  Ph , 1989 .

[22]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[23]  Frank Schmidt,et al.  Solving Time-Harmonic Scattering Problems Based on the Pole Condition II: Convergence of the PML Method , 2003, SIAM J. Math. Anal..

[24]  Christophe Hazard,et al.  On the solution of time-harmonic scattering problems for Maxwell's equations , 1996 .