Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator

Kirill Cherednichenko (1) , Sébastien Guenneau (2, 3)

1 School of Mathematics [Cardiff]
2 FRESNEL - Institut FRESNEL
3 Department of Maths Sciences

This paper is devoted to the asymptotic behavior of the spectrum of the three-dimensional Maxwell operator in a bounded periodic heterogeneous dielectric medium T = [-T,T]3, T > 0, as the structure period , such that -1 T is a positive integer, tends to 0. The domain T is extended periodically to the whole of 3, so that the original operator is understood as acting in a space of T-periodic functions. We use the so-called Bloch-wave homogenization technique which, unlike the classical homogenization method, is capable of characterizing a renormalized limit of the spectrum (called the Bloch spectrum) [6]. The related procedure is concerned with sequences of eigenvalues of the resolvent of the order of the square of the medium period, which correspond to the oscillations of high-frequencies of order -1. The Bloch-wave description is obtained via the notion of two-scale convergence for bounded self-adjoint operators, and a proof of the 'completeness' of the limiting spectrum is provided. The results obtained theoretically are illustrated by finite element computations.

Domaines: Optique [physics.optics]

Kirill Cherednichenko, Sébastien Guenneau. Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator. Journal Waves Random Complex Media, 2007, 17 (4), pp.627-651. ⟨10.1080/17455030701551930⟩. ⟨hal-00186688⟩