## 3 Results and Discussion  
All of macroscopic electromagnetism, including the propagation of light in a photonic crystal, is governed by the four macroscopic Maxwell's equations with no free charges or currents. One interesting feature of electromagnetism in PCs is that there is no fundamental length scale other than the assumption that the system is macroscopic [1]. Therefore, to study physical phenomena in PCs, one may scale a system from the optical frequency range to the microwave one and vice versa if suitable conditions are fulfilled. For the purposes of demonstration and easier verification by experimentalists, a WR28 (7.11 mm \(\times\) 3.555 mm) rectangular waveguide, usually used for Ka-band millimeter waves, is chosen. The periodic dielectric heterostructure is arranged along the longitudinal \((z)\) direction to form the quasi-1D PCs [34]. In the microwave or millimeter wave frequency ranges, the PC experiments are very popular and more affordable compared to those in the optical frequency ones. Nevertheless, in order to extend the results for general use, the data are normalized for solutionsat all length scale. Consider the quasi-1D PCs with the arrangement shown in Figure 1, 15 double-layer stacks (\(n=30\)) have been included in the waveguide for investigation. The transmittance is calculated using the transfer matrix approach mentioned above. In order to demonstrate the stop band and pass band, the transmittances, expressed as the \(S\)-parameters, \(S_{12}\), in dB, of the lowest TE and TM modes for three quasi-1D PCs have been calculated and plotted in Figure 2. For the three cases, the dielectric constants used are the same as \(\varepsilon_{1}=3.8\) (quartz) and \(\varepsilon_{2}=1.0\) (air), while the thicknesses are varied as (\(t_{1}\), \(t_{2}\)) = (1.00, 3.30), (1.00, 3.60), and (1.00, 3.90) mm, corresponding to the filling ratios \(t_{1}/\Lambda=23.26\%\), 21.74%, and 20.41% for the periods of the stacks \(\Lambda=4.3\), 4.6, and 4.9 mm, respectively. As one can see, the central frequencies of the PBGs for both the \(TE_{10}\) and \(TM_{11}\) modes, as shown in Figure 2a,b respectively, are shifted to lower values as the filling ratio decreases. As mentioned above, the frequency axes in the plots are normalized to the cutoff frequency of \(TE_{10}\) for general use. In order to look for a complete PBG for all possible modes in the quasi-1D PCs, one may want to enlarge a PBG for a specific mode of choice. For the following four cases, the dielectric constants used are (\(\varepsilon_{1}\), \(\varepsilon_{2}\)) = (2.3, 1.0), (3.8, 1.0), (4.9, 1.0), and (11.4, 1.0), while the thicknesses are varied as (\(t_{1}\), \(t_{2}\)) = (2.15, 2.15), (1.00, 3.30), (0.72, 3.58), and (0.27, 4.03) mm, corresponding to the filling ratios \(t_{1}/\Lambda=50.00\%\), 23.26%, 16.74%, and 6.28% for the fixed period of the stacks \(\Lambda=4.3\) mm, respectively. The dielectric constants of 2.3, 3.8, 4.9 and 11.4 used in the chosen microwave frequency range correspond to the dielectric materials, polyethylene, quartz, phenolic resin, and barium sulfate, respectively. As one can see, the calculated transmittances, \(S_{12}\)(dB), of the four quasi-1D PCs with different 15 double-layer stacks for the \(TE_{10}\) and \(TM_{11}\) modes are plotted in Figure 3a,b, respectively. The width of PBGs for both the \(TE_{10}\) and \(TM_{11}\) modes in the quasi-1D PCs is widened to larger sizes as decreasing the filling ratio while increasing the dielectric contrast between \(\varepsilon_{1}\) and \(\varepsilon_{2}\). One may notice that the configurations of these quasi-1D PCs have been specially arranged so that the central frequencies of the PBGs of the \(TE_{10}\) mode are kept the same while tuning the PBG sizes. However, those of the PBGs of the \(TM_{11}\) mode are not centralized. This indicates that a tremendous computational effort is still needed to find a complete PBG in spite of the approach with a closed form developed is very efficient for the quasi-1D PCs.