All our calculations have been performed for ²a = ² = 13.6 appropriate for gallium arsenide (GaAs), and ²b = 1.0 in air. GaAs has been used because this material exhibits facinating optical properties in the infrared region and is representative of many semiconductors. The design of this structure has many degrees of freedom which can be used to optimize the size of the gap, depending on the materials used in the fabrication. Although GaAs is used in this example, they can be replaced by other material with a different index contrast. 1521 plane waves in the Fourier expansion are used to calculate PBGs for the E(H)-polarization. First, the PBG structures and the corresponding density of states (DOS) of an air hole drilled into the center of each square dielectric rod in each unit cell are calculated, as shown in Figure 3.2(a). The parameters in this figure are chosen as a/l=1.63,
β=0.35 (corresponding to filling factor f = 0.34017) and s = 0. The solid (dotted ) curves correspond to the E(H)-polarization. It is shown that there are four PBGs for the E-polarization and two PBGs for the H-polarization. However, overlap of PBG for E- and H-polarization does not exist. Along the left-hand and right-hand margin of this figure the density of photonic states in arbitrary units were plotted.
The eigenfrequencies for 6400 uniformly spaced values of k vectors inside the first Brillouin zone were calulated. In Figure 3.2(b), the calculated result for s = 0.11a and γ = 45◦ is illustrated. The other parameters are the same as those quoted in Figure 3.2(a). A complete PBG with a gapwidth of ∆ω = 0.0415(2πc/a), and a central value, ωg = 0.66335(2πc/a), which is in the region of overlap of E8 and H6 band gaps, is found. Ei and Hi denotes the gaps that appear between the ith and (i + 1)th bands, for the corresponding polarization. In the spectral range of this complete bandgap neither E-polarized nor H-polarized photonic states exist (DOS=0). Notably, modifying the position of circular hole in the square dielectric rod in air seems to lower the frequency of the ninth E-polarization band and the fifth and sixth H-polarization bands at the M (M0) point of the Brillouin zone that depicted in Figure 3.2(a), then the overlap of E8 and H6 band gaps occurs. This result can be understood to be due to the fact that reducing the symmetry of the dielectric distribution in the square rod. Apparently, the band structure in Figure 3.2(b) also exhibits very gently sloped bands near the complete PBG’s edge. Thus, a sharp peak of density of states can be observed due to the flat band. Since the group velocity vg of the modes given by the slope of the dispersion curves, ∂ω/∂k, is expected to be zero or very small and correspondingly, the optical path is expected to be long. Comparison between Figure 3.2(a) and 3.2(b) shows that the zero or small group velocities are observed in a broad region of k-space as the increase of s. The fifth, sixth H-polarization bands and the ninth E-polarization band become
more restricted to a narow spectral region, thus, light waves become more localized as s increased. There are k points between the M–U direction at which the sixth H-polarization and the eighth E-polarization bands are almost flat. That is to say, group velocities of both mode approach to zero. Generally, the zero group velocity appears near photonic band edge only for E- or H-polarization. In this case, the zero group velocity is allowed for both E- and H-polarization simultaneously.
An additional plot in Figure 3.3 provides more information on PCs. The PBG map as the relative shift s of the drilled rod for three different directions of (a) γ = 0◦, (b) γ = 22.5◦ and (c) γ = 45◦. The other parameters are as those in Figure 3.2(a). Only the first ten-bands are involved in this map for both E- and H-polarizations. Notably, for a given γ, the varying region of s is limited, i.e., only from zero to a certain value at which the outermost edge of the internal air circular rod just touches the outermost edge of the square dielectric rod at the lattice. The gap map for E-polarization shown in Figure 3.3(a) exhibits six large gaps. We note that E1 and E3 gaps occur over the range of the shift s within [0, 0.199]a. Moreover, a remarkable gaps H6 occur in the same range for H-polarization. Some other gaps only lie in the intermediate range of s. There are three complete PBG’s in this configuration due to the overlap of E8 with H5; E7 with H5, and E8 with H6 gaps. Comparison among Figure 3.3(a), 3.3(b) and 3.3(c) shows that gap widths strongly depend on the shift of the air hole position. The most important result is the appearance of the overlap of E8 with H6 gap, which occurs for s in the region [0.015, 0.18]a for γ = 0◦, [0.016, 0.215]a for γ = 22.5◦ and [0.014, 0.253]a for γ = 45◦, in turn. One would see this complete PBG to get larger width as the γ is increased.
This complete PBG is always bounded at the top by the upper boundary of the E8 gap. The lower boundary switches from H6 to E8 gap both in Figure 3.3(a) and 3.3(b). Furthermore, its bottom side shown in Figure 3.3(c) is wholly bounded by
the lower boundary of the E8 gap. In fact, since E and H polarized modes are decoupled and are governed by different equations for a right choice of s and γ.
We have also examined the case of an air hole drilled at the center of each square dielectric rod in air. Figure 3.4 shows the PBG map as a function of the parameter β for filling factor f =0.34017. Several gaps in both E- and H-polarization appear and disappear as β is varied. We should note here that one H-polarization and four E-polarization gaps exhibit near β = 0 when air hole is absent. One large complete PBG occurs due to the overlap of H6 and E8 gaps. This complete PBG starts near β = 0 and ends at about 0.34. The gap size ∆ω reaches the maximum value 0.0427(2πc/a) at about β = 0.19 when the same total filling factor f = 0.34017 and a/l = 1.69.