Figure 1. The dispersion relation for E polarization in a two-dimensional photonic crystal composed of a regular square array of circular air-rods 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 abscissa represents the wave vector in the two-dimensional Brillouin zone where Γ, X, and M stand for (0, 0), (π/a, 0), and (π/a, π/a). The radius of the air-rods was assumed to be 0.28 times the lattice constant. Symmetric and antisymmetric modes are denoted by S and A, respectively. Note that the third lowest S mode and the lowest A mode in the Γ-X direction has a small group velocity over their whole frequency ranges.

Figure 2. The comparison between the dispersion relation (the left-hand side) and the transmission spectrum (the right-hand side) for E polarization in the Γ-X direction. The same parameters as Fig. 1 were assumed for numerical calculation. The number of the lattice layers was assumed to be 16 for the calculation of the transmission spectrum. The dashed line in the left-hand side represents an A (antisymmetric) mode.

Figure 3. The sum of the transmittance and the reflectance for E polarization as a function of the normalized frequency calculated for a crystal with (a) 16, (b) 8, (c) 4, and (d) 2 layers of air-rods formed in the dielectric material with a dielectric constant of 2.1-0.01i (solid line with filled circles) and a uniform plate of the same thickness with a spatially averaged dielectric constant (dashed line). The negative imaginary part of the dielectric constant stands for the inverted population of the impurity atoms. The incident light was assumed to be propagated in the Γ-X direction. Note that the sum can be greater than unity because of the stimulated emission.

Figure 4. The dispersion relation for H polarization in the two-dimensional photonic crystal composed of a regular square array of circular air-rods formed in a dielectric material. The same parameters as Fig. 1 were assumed for numerical calculation. Symmetric and antisymmetric modes are denoted by S and A as before. Note that the third lowest S mode and the lowest A mode in the Γ-X direction show the group-velocity anomaly as E polarization.

Figure 5. The comparison between the dispersion relation (the left-hand side) and the transmission spectrum (the right-hand side) for H polarization in the Γ-X direction. The same parameters as Fig. 2 were assumed for numerical calculation. The dashed line in the left-hand side represents an A mode.

Figure 6. The sum of the transmittance and the reflectance for E polarization as a function of the normalized frequency calculated for a crystal with (a) 16, (b) 8, (c) 4, and (d) 2 layers of air-rods formed in the dielectric material with a dielectric constant of 2.1- 0.01i (solid line with filled circles) and a uniform plate of the same thickness with a spatially averaged dielectric constant (dashed line). The incident light was assumed to be propagated in the Γ-X direction.