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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on July 14, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(4):453-474; doi:10.1093/qjmam/hbn015
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© The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Pure currents in foliated waveguides

F. Zolla

( Institut Fresnel, UMR 6133, Université Aix-Marseille I, Avenue Escadrille Normandie Niemen, Marseille 13013, France )

G. Bouchitte

( Département de Mathématiques, Université de Toulon et du Var, BP132, 83957 La Garde Cedex, France )

S. Guenneau{dagger}

( Department of Mathematical Sciences, Liverpool University, Liverpool L69 3BX )

{dagger} <guenneau{at}liverpool.ac.uk>

Received 14 November 2006. Revise 27 May 2008.
  Abstract

We give an explicit characterization of obliquely propagating electric modes within a multilayered planar dielectric waveguide of cross-section [0, 1]. This foliated strip, with fixed metallic boundaries at 0 and 1, consists of an integer number N of periodic cells of thickness {eta} = 1/N. We are interested in the high-frequency asymptotic analysis of its spectrum. For this, we rescale the wavelength as {eta} and derive, using a transfer matrix formalism, that for vanishing {eta} the limit spectrum consists of two parts. The first part is a Bloch (or band) spectrum associated with a family of operators acting on Floquet–Bloch eigenfunctions defined on the real line. The second part is a boundary layer spectrum associated with an operator acting on square integrable eigensolutions defined on the positive real line [0, + {infty}) and satisfying a Neumann condition at 0. This latter part is further characterized via a spectral problem on [0, 1] which is supplied with Neumann conditions at both ends. Eventually, we illustrate our discussion by numerical results derived from this auxiliary spectral problem. Moreover, after a suitable rescaling of the field, we prove that the total electromagnetic energy is entirely located on either boundary of the structure. Finally, in the case of transverse propagation, we apply our results to antiplane shear waves propagating within a foliated acoustic waveguide, whose freely vibrating walls at 0 and 1 are shown to support surface waves.


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