A MATHEMATICAL THEORY FOR FANO RESONANCE IN A PERIODIC ARRAY OF NARROW SLITS

This work concerns resonant scattering by a perfectly conducting slab with periodically arranged subwavelength slits, with two slits per period. There are two classes of resonances, corresponding to poles of a scattering problem. A sequence of resonances has an imaginary part that is nonzero and on the order of the width epsilon of the slits; these are associated with Fabry-Perot resonance, with field enhancement of order 1/epsilon in the slits. The focus of this study is another class of resonances which become real valued at normal incidence, when the Bloch wavenumber kappa is zero. These are embedded eigenvalues of the scattering operator restricted to a period cell, and the associated eigenfunctions extend to surface waves of the slab that lie within the radiation continuum. When 0 < vertical bar kappa vertical bar << 1, the real embedded eigenvalues will be perturbed as complex-valued resonances, which induce the Fano resonance phenomenon. We derive the asymptotic expansions of embedded eigenvalues and their perturbations as resonances when the Bloch wavenumber becomes nonzero. Based on the quantitative analysis of the diffracted field, we prove that the Fano-type anomalies occur for the transmission of energy through the slab, and we show that the field enhancement is of order 1/(kappa epsilon), which is stronger than Fabry-Perot resonance.