Noncommuting Limits in Electromagnetic Scattering: Asymptotic Analysis for an Array of Highly Conducting Inclusions

We consider formulations for the Helmholtz operator for periodic media containing high contrast inclusions in the limit when the wavelength outside the inclusions tends to infinity. Applications are to problems of electromagnetism. The main focus is on the analysis of the effect of noncommuting limits, an effect which indicates that linear boundary value problems of electromagnetism give formally different results for the long wavelength limits in cases where highly conducting inclusions have refractive indices of different orders of magnitude. Specifically, the effective moduli of the homogenized material will depend on the path used to approach the origin in the coordinate space {wave number, (normalized refractive index of the inclusions) -1 }. This mathematical observation gives a physical subtlety which is studied in this paper. The dispersion relation for the lowest frequency (or acoustic mode) is investigated, as are the conditions for existence of an acoustic mode. Cases of both nondispersive and dispersive inclusions are considered.