Sebastien Guenneau, Chris Poulton and Alexander Movchan

University of Liverpool

Department of Mathematical Sciences

M & O Building, Peach Street

Liverpool, L69 3BX

England

This talk presents analysis of electromagnetic and elastodynamic waves
conically propagating through a doubly periodic array of cylindrical
fibres. A new method, based on a multiple scattering approach, has
been proposed to reduce these spectral problems for partial
differential equations to certain algebraic problems of the Rayleigh
type: its matrix elements decay exponentially away from the main
diagonal, giving rise to higher-order multipole coefficients that
decay similarly quickly.
We obtain a formulation in terms of an eigenvalue problem
that enables us to construct the high-order dispersion
curves and to study both photonic and phononic bang gap
structures in oblique Incidence [1].
We also address the question of a singular perturbation
induced by the conical incidence parameter for both
electromagnetic and elastic modes. We finally discuss some
effective properties for ferro-magnetic photonic crystal
fibres in the long wavelength limit [2].

References:

[1] Guenneau, S., Poulton C. G. and Movchan, A. B. Oblique
propagation of electromagnetic and elastic waves for an
array of cylindrical fibres, Proc. Roy. Soc. (submitted)

[2]
Poulton, C. G., Botten, L. C., McPhedran, R. C.,
Nicorovici, N. A., Movchan, A. B. Non-commuting limits in
electromagnetic scattering: asymptotic analysis for an array
of highly conducting inclusions, SIAM J. Appl. Math., 61
(2001) 1706-1730