Defect Modes in Quasi-One-Dimensional Photonic Waveguides — Application to the Resonant Tunneling between Two Continua

Abstract

For the last few years, we were studying quasi-one-dimensional photonic materials [1, 2]. These materials are composed of an infinite one-dimensional waveguide (the backbone) along which stars of N’ finite monomode side branches (or stubs) are grafted at N equidistant sites, N and N’ being integers (see Fig. 1). This star waveguide is described by two structural and two compositional parameters, namely the periodicity d₁, the length d₂ of each stub, and the relative dielectric permittivity ε₁ of each medium with i=1 for the backbone and i=2 for the side branches. We developed a theoretical model in order to compute the dispersion relation for an infinite number of sites (N→∞) and the transmission coefficient through the waveguide when the side branches are grafted at a finite number of nodes. In this model, electromagnetic waves only propagate in the interior of the waveguides. Two boundary conditions at the free end of the side branches were considered, namely the vanishing of either the electric field (E=0) or the magnetic field (H=0). With these conditions, the impedance at the extremity of the side branches becomes, respectively, Z=0 and Z=∞. We have shown that the band structure of star waveguides may exhibit very narrow pass bands separated by large forbidden bands. These gaps originate both from the periodicity of the system and the resonance states of the grafted branches which play the role of resonators. Wide gaps/narrow bands can be obtained by an appropriate choice of the parameters, in particular the ratio between the two characteristic lengths d₁ and d₂. Increasing the number N’ of side branches grafted on each node, results in larger forbidden bands. The choice of the boundary condition at the free end of the resonators plays also an important role and the condition E=0 is more favourable to the opening of large gaps. Unlike in the usual photonic crystals where the contrast in dielectric properties between the constituent materials is a critical parameter in determining the existence of gaps, relatively wide gaps exist for homogeneous star waveguides where the branches and the backbone are constituted of the same material. One would also emphasize that the width of the forbidden bands of star waveguides is, in general, much larger than the one observed in usual superlattices made of a quarter-wave stack of alternating indices of refraction [3, 4].