Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T16:37:07.189Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics for Helmholtz and Maxwell Solutions in 3-D Open Waveguides

Published online by Cambridge University Press:  20 August 2015

Carlos Jerez-Hanckes  and
Jean-Claude Nédélec
Carlos Jerez-Hanckes*
Affiliation:
Seminar für Angewandte Mathematik, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland Escuela de Ingeniería, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile
Jean-Claude Nédélec*
Affiliation:
Escuela de Ingeniería, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile
*
Corresponding author.Email:[email protected]
Email address:[email protected]
Get access

Abstract

We extend classic Sommerfeld and Silver-Müller radiation conditions for bounded scatterers to acoustic and electromagnetic fields propagating over three isotropic homogeneous layers in three dimensions. If X= (x1,x2,x3)ϵℝ3, with x3 denoting the direction orthogonal to the layers, standard conditions only hold for the outer layers in the region ∣x3∣ > ∣∣x∣γ, for γϵ(1/4,1/2) and x large. For ∣x3∣ < ∣∣x∣∣γ and inside the slab, asymptotic behavior depends on the presence of surface or guided modes given by the discrete spectrum of the associated operator.

Keywords

35C1535Q6078A45Open waveguideradiation conditionHelmholtz equation
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 629 - 646
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ablowitz, M. and Fokas, A.. Complex Variables: Introduction and Applications. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, 2nd edition, 2003.Google Scholar
[2]Bleistein, N.. Mathematical Methods for Wave Phenomena. Computer Science and Applied Mathematics. Academic Press, Orlando, USA, 1984.Google Scholar
[3]Bonnet-Ben, A.-S. Dhia, Dakhia, G., Hazard, C., and Chorfi, L.. Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition. SIAM J. Appl. Math., 70(3):677–693, 2009.Google Scholar
[4]Ciraolo, G. and Magnanini, R.. A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides. Math. Methods Appl. Sci., 32(10):1183–1206, 2009.Google Scholar
[5]Durán, M., Muga, I., and Nédélec, J.-C.. The Helmholtz equation in a locally perturbed half-space with non-absorbing boundary. Archive for Rational Mechanics and Analysis, 191(1):143–172, 2009.CrossRefGoogle Scholar
[6]Harris, J. G.. Linear Elastic Waves. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, second edition, 2001.Google Scholar
[7]Jerez-Hanckes, C.. Modeling Elastic and Electromagnetic Surface Waves in Piezoelectric Tranducers and Optical Waveguides. PhD thesis, Ecole Polytechnique, Palaiseau, France, 2008.Google Scholar
[8]Jerez-Hanckes, C. and Nédélec, J. C.. Asymptotics for Helmholtz and Maxwell solutions in 3-D open waveguides. Technical report, Seminar for Applied Mathematics, ETH Zurich, 2010.Google Scholar
[9]Leis, R.. Initial Boundary Value Problems in Mathematical Physics. Teubner, B.G., Stuttgart, 1986.CrossRefGoogle Scholar
[10]Murray, J.. Asymptotic Analysis. Number 48 in Applied Mathematical Sciences. Springer-Verlag, Inc., New York, USA, 1984.Google Scholar
[11]Olyslager, F.. Discretization of continuous spectra based on perfectly matched layers. SIAM J. Appl. Math., 64(4):1408–1433, 2004.Google Scholar
[12]Weder, R.. Spectral and scattering theory for wave propagationin perturbed stratified media. Number 87 in Applied Mathematical Sciences. Springer-Verlag, Inc., New York, USA, 1981.Google Scholar