Fig. 1. Maps of photonic bandgap for TE (electric field perpendicular to the computation plane) and TM polarizations in a two-dimensional hexagonal-lattice structure of air cylinders in GaAs (ε=12.96) as functions of the ratio r/a.

Fig. 2. (a)–(d) One period of the waveguide structure and magnetic field pattern in a PBG waveguide made by putting a dielectric slab of thickness d between two hexagonal lattice PBG mirrors. The radii of air holes in all cases are r=0.38a, with a being the distance between the centers of nearest neighbor air holes. The thickness of the middle slab in (a) and (b) is d=0.66a, and that in (c) and (d) is d=0.52a. The dispersion diagrams for the two dominant guided TM modes of these PBG waveguides calculated by the FDTD technique are shown in (e). The dielectric material is GaAs with relative permittivity ε=12.96. The distance between the centers of the nearest neighbor holes and the radius of each hole in the photonic crystal are a=29 calculation cells and r=0.38a, respectively. The frequency ranges of bandgap (PBG) and single-mode guiding for d=0.52a are shown in the figure.

Fig. 3. (a) One unit cell of the waveguide structure and the magnetic field patterns for different TM modes of: (a), (b) a specially designed waveguide as discussed in the text, and (c) the equivalent corrugated waveguide made by considering only two rows of the photonic crystal adjacent to the middle slab with air above these two rows. (d) Dispersion diagrams of the dominant guided modes corresponding to hexagonal lattice waveguide and corrugated waveguide shown in (a), (b), and (c). The dielectric material in all waveguides is GaAs (ε_s =12.96). The cladding region of the corrugated waveguides is air (ε=1). The period of the hexagonal lattice and the radius of each air column in the PBG waveguide are a=29 calculation cells and r=0.38a, respectively.