## 3 Results and Discussion
Figure 2: Calculated transmittance, \(S_{12}\)(dB), of the quasi-1D PCs with 15 double-layer stacks enclosed in the rectangular waveguide for (**a**) \(TE_{10}\) and (**b**) \(TM_{11}\) modes of light propagation. The dielectric constants used are \(\varepsilon_{1}=3.8\) and \(\varepsilon_{2}=1.0\), and the corresponding filling ratios are \(t_{1}/\Lambda=23.26\%\), 21.74%, and 20.41% in the three cases. The frequency axes are normalized to the cutoff frequency of \(TE_{10}\). A better way to identify PBGs more accurately and efficiently is to perform the BS calculation and further compute the PDOS [15; 16]. Figure 4 shows the band structures of (a) TE\({}_{10}\) and (b) TM\({}_{11}\) modes for the quasi-1D PC with the dielectric constants \(\varepsilon_{1}=3.8\), \(\varepsilon_{2}=1\) and the thicknesses \(t_{1}=1.00\) mm, \(t_{2}=3.30\) mm, corresponding to the case presented in Figures 2 and 3 in a black solid line with a filling ratio of 23.26%. The PBGs of the TE\({}_{10}\) mode show no overlap with those of TM\({}_{11}\) mode, all marked in gray stripes in the figure. However, from Figure 4, one may notice that there are no photon states in the frequency ranges below the cutoff frequency of the \(TE_{10}\) mode nor within the first PBG of \(TE_{10}\) which is under the cutoff frequency of the \(TM_{11}\) mode. Note that only the lowest TE amd TM modes are plotted in Figure 4. As there might be other modes involved within the frequency range of the PBG of interest, it is easier to identify a complete PBG from the PDOS plots compared to the BS ones. Figure 5 shows the PDOS of TE\({}_{10}\), TE\({}_{01}\), TE\({}_{11}\), and TM\({}_{11}\) modes for the same quasi-1D PC. As one can see, the PDOS of the TE\({}_{10}\) and TM\({}_{11}\) modes are consistent with the BS calculations shown in Figure 4. The PDOS contributed from the TE\({}_{01}\) and TE\({}_{11}\) modes tend to fill up the first PBG of the TE\({}_{10}\) mode. Other higher order modes are not considered as their cutoff frequencies are too high to contribute any PDOS in the frequency range of the PBG. Finally, the combined PDOS of TE\({}_{10}\), TE\({}_{01}\), TE\({}_{11}\), and TM\({}_{11}\) modes shows no photon states in some frequency ranges. The first one is below the cutoff frequency of the \(TE_{10}\) mode, (0-0.77)\(\ell_{c}\), the second one is within the first PBG of \(TE_{10}\) but under the cutoff frequency of the \(TE_{01}\) mode, (1.26-1.48)\(\ell_{c}\), and the third one is the overlap of the PBGs of TE\({}_{10}\), TE\({}_{01}\), and TE\({}_{11}\) modes, (1.79-1.87)\(\ell_{c}\). Therefore, a \"complete PBG\" can be obtained for some frequency ranges and categorized into three types: (1) below the cutoff frequency of the fundamental TE mode, (2) within the PBG of the fundamental TE mode but below the cutoff frequency of the next higher order mode, and (3) within an overlap of the PBGs of either TE modes, TM modes, or both. Figure 3: Calculated transmittance, \(S_{12}\)(dB), of the quasi-1D PCs with 15 double-layer stacks enclosed in the rectangular waveguide for (**a**) \(TE_{10}\) and (**b**) \(TM_{11}\) modes of light propagation. The dielectric constants used are \(\varepsilon_{1}=2.3\), \(3.8\), \(4.9\), and \(11.4\) and \(\varepsilon_{2}=1\), and the corresponding filling ratios are \(t_{1}/\Lambda=50.00\%\), \(23.26\%\), \(16.74\%\), and \(6.28\%\) in the four cases. The frequency axes are normalized to the cutoff frequency of \(TE_{10}\).