Figure 1. The dispersion relation for E polarization (left-hand side) and the threshold of laser oscillation (right-hand side) of a two-dimensional photonic crystal composed of a regular square array of circular air cylinders formed in a dielectric material with a dielectric constant of 2.1. The ordinate is the normalized frequency where ω, a, and c denote the angular frequency of the radiation field, the lattice constant of the two-dimensional crystal, and the light velocity in vacuum, respectively. The radius of the air cylinders was assumed to be 0.28 times the lattice constant. The number of the lattice layers was assumed to be eight. The dispersion relation was presented from Γ to X points in the two-dimensional Brillouin zone. The solid lines represent the dispersion relation for symmetric modes that can be excited by an incident plane wave, whereas the dashed line represents that of an antisymmetric (uncoupled) mode that does not contribute to the light propagation. The threshold of laser oscillation is given by the imaginary part of the dielectric constant of the host material. Note that the third lowest symmetric mode has a small group velocity over its entire spectral range (group-velocity anomaly) and the lasing threshold for this mode is smaller than that for the lowest and the second lowest modes by about two orders of magnitude. Also note the decrease of the threshold at the upper band edges of the latter.

Figure 2. The sum of the transmittance and the reflectance for E polarization in a logarithmic scale for the third symmetric band of the two-dimensional photonic crystal as a function of the normalized frequency, ω a/2π c, and the imaginary part of the dielectric constant, ∊″. The same parameters as Fig. 1 were used for numerical calculation and the incident light was propagated in the Γ-X direction. We assumed that the front and the rear surfaces of the crystal were perpendicular to the propagation direction and that the distance between each surface and the center of the first air cylinder was half a lattice constant. Note that the sum is divergent for a certain combination of ωa/2πc and ∊″.

Figure 3. The sum of the transmittance and the reflectance for E polarization in a logarithmic scale for the second band of the two-dimensional photonic crystal. The same parameters as Fig. 2 were used for numerical calculation.

Figure 4. The sum of the transmittance and the reflectance for E polarization in a logarithmic scale for the first band of the two-dimensional photonic crystal. The same parameters as Fig. 2 were used for numerical calculation.

Figure 5. The dispersion relation for H polarization (left-hand side) and the threshold of laser oscillation (right-hand side) of the two-dimensional photonic crystal. The same parameters as Fig. 1 were used for numerical calculation.