Asymptotic estimates for localized electromagnetic modes in doubly periodic structures with defects

The paper presents analytical and numerical models describing localized electromagnetic defect modes in a doubly periodic structure involving closely located inclusions of elliptical and circular shapes. Two types of localized modes are considered: (i) an axi-symmetric mode for the case of transverse electric polarization with an array of metallic inclusions; (ii) a dipole type localized mode that occurs in problems of waveguide modes confined in a defect region of an array of cylindrical fibres, and propagating perpendicular to the plane of the array. A thin bridge asymptotic analysis is used for case (i) to establish double-sided bounds for the frequencies of localized modes in macro-cells with thin bridges. For the case (ii), the electric and magnetic fields independently satisfy Helmholtz equations, but are coupled through the boundary conditions. We show that the model problem associated with localized vibration modes is the Dirichlet problem for the Helmholtz operator. We characterize defect modes by introducing a parameter called the 'effective diameter'. We show that for circular inclusions in silica matrix, the effective diameter is accurately represented by a linear function of the inclusion radius.