Photonic Band Gap Calculations: Inward and Outward Integral Equations and the KKR Method

Abstract

Our main goals have been to calculate photonic bands on various periodic dielectric lattices, to clarify some controversy regarding existence of a photonic band gap for fcc lattice of dielectric spheres [1], and finally treat impurities in a photonic crystal [2]. Motivated by the search for a photonic band gap [3] we have tried to adapt one of the standard methods of electron band theory — namely the scalar bulk Kohn-Korringa-Rostocker (KKR) method [4, 5] to the form appropriate for photons [2, 6, 7]. We recall that in the case of photons the role of a (periodic) potential plays υ(r) = ε(r) − ε₀. Here ε(r) is the dielectricity of a medium and ε₀ is its host value which is assumed to be uniform and homogeneous [3, 6]. A dielectric “atom” V _s is called a connected region where υ(r) ≠ 0. The reason of our choice was that the KKR method proceeds analytically as far as possible and enables to go beyond nearly-free photon and plane wave approximations. The latter were cast into doubts for discontinuous potentials [1], and by a rigorous proof on the existence of finite number of gaps in two and higher dimensions for the periodic Schrodinger operator [8]. The KKR starts with an integral equation which is after expansion in a suitable basis transformed into an algebraic one. A band structure then follows from the conditions of solvability of the algebraic equation which is vanishing of a determinant of a matrix (see below) which determines dispersion relation and eventually photonic bands. The electronic KKR method is known to lead to a very compact scheme if the perturbing periodic potential υ(r) is spherically symmetric within inscribed spheres and zero (constant) elsewhere [4, 5]. In the case of electrons already p-wave approximation gives agreement within 2% with experiment.