Guided and defect modes in periodic dielectric waveguides

Periodic dielectric materials offer a great deal of control over the propagation of light.1 Usually the basic idea is to design periodic dielectric structures that have a band gap for a particular frequency range. In a recent study2 Meade et al. showed that defects in a three-dimensional (3-D) periodic dielectric material (photonic crystal) can admit localized states within a band gap. It is the purpose of this paper to investigate the characteristics and properties of modes in periodic dielectric waveguides. In particular, we are interested in understanding the nature of guided modes and whether defects can lead to highly localized resonances. Before we present our model systems it is useful to consider guided modes in a uniform dielectric waveguide. A schematic rectangular waveguide is shown in Fig. 1(a). Because of the translational symmetry along the slab the mode can be characterized by wave vector kk. The dispersion relation vskkd is also sketched in Fig. 1(a). Modes above the light line sv ­ kkd are radiating modes, with a continuous spectrum. Below the light line are guided modes, which have a discrete band structure. The guided modes that make up a given band have the same symmetry and similar polarization properties. We can consider a small periodic index contrast along the waveguide to be a weak perturbation of a uniform slab. (A typical example is the corrugated grating on the surface of a thin-film waveguide in a distributed-feedback laser.) For such a system a unit cell of length a is repeated along the waveguide, and modes can be characterized by a wave vector kk that is confined to the first Brillouin zone. The perturbation-induced coupling between the modes at kk ­ pya and kk ­ 2pya opens up small gaps at the zone edge [see Fig. 1(b)]. Light cannot propagate along the waveguide with a frequency within the gap since there are no permitted propagating states. This phenomenon can also be readily understood by Bragg’s condition—multiple reflections from the different unit cells conspire to interfere destructively within the structure. In distributed-feedback laser systems the translational symmetry is broken by placement of a phase shift into the structure. The phase shift can be designed (for example, by use of a ly4 phase shift) so that a weakly localized mode is created inside the gap.3 This mode would