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

Abstract

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.

Higher order asymptotic homogenization and wave propagation in periodic composite materials

“…High-frequency homogenization using two-scale asymptotic methods in a combination with the Floquet-Bloch approach was proposed by Allaire & Conca (1998) and Cherednichenko & Guenneau (2007). Localization of electromagnetic modes in periodic lattices with defects was explored by Movchan et al (2007).…”

Origami with negative refractive index to generate super-lenses

“…Such boundary conditions can be easily derived from the divergence theorem applied to the flux through the interface between each thin bridge η and the large region to which they are connected. In the case of constant curvature a, that is when h − = h + = 1 + a 2 h 2 /2, this leads the following resonant frequency (see equation (4.12) in [21]):…”

Homogenisation for hexagonal lattices and honeycomb structures

“…Much of the behaviour is then interpreted using dispersion curves relating phase shift across the cell to frequency and the resulting iso-frequency contours or Bloch dispersion curves are vital interpretive tools and originate from Brillouin's seminal work (9). Complementary to the study of perfect lattice systems are those containing defects (26) or Green's function excitations (6,29) with exact Green's solutions available for discrete hexagonal, honeycomb (22) or square (16,28) lattice systems. None the less these exact solutions, given typically as integrals or in elliptic functions, often resist simple interpretation (28) and are complicated by transitions from propagating to stopband regimes: It is attractive to alternatively replace a discrete lattice system, or other basically periodic medium, with an effective continuum to avoid the detailed interactions between lattice elements.…”