Photonic bandgap is a key parameter of photonic crystal structure to envisage mirror application. In this case, plane wave expansion (PWE) method is used to make simulation for photonic bandgap of 2D dimensional photonic crystal structures [13I.A. Sukhoivanov IV, Guryev, ‘Physics and Practical Modeling: Photonic Crystals., Springer: Heidelberg, 2009.
[http://dx.doi.org/10.1007/978-3-642-02646-1] ]. Here, simulation is made for finite area of proposed photonic crystal structure which contains six columns of air holes on GaAs substrate material. Photonic bandgap depends on structure parameters such as lattice spacing, radius of air holes, number of air holes including structure and configuration of crystal. In this case, we have chosen different combination of radius and lattice spacing, which is mentioned in the previous section. Using data from Table 1, (different combination of radius and lattice spacing), simulation is done for photonic bandgap of photonic crystal structure using plane wave expansion method. The simulation results for (100 nm, 200 nm) and (100 nm, 300 nm) of (r,a) are shown in Figs. (2a and 2b) respectively.

Fig. (2a) Simulation graph for photonic bandgap of 2D photonic crystal structure having radius of air hole, 100 nm and lattice spacing of 200 nm.

Fig. (2b) Simulation graph for photonic bandgap of 2D PCs having radius of air hole, 100 nm and lattice spacing of 300 nm.

Simulation for other combination, such as (16 nm,100 nm), (17nm,100 nm), (18 nm,100 nm), (20 nm,100 nm), (25 nm,100 nm), (30 nm,100 nm), (35 nm,100 nm), (40 nm,100 nm), (45 nm,100 nm), (48 nm,100 nm), (49 nm,100 nm), (50 nm,100 nm), (100 nm, 250 nm), (100 nm, 350 nm), (100 nm, 400 nm), (100 nm, 450 nm), (100 nm, 500 nm), (100 nm, 550 nm), (100 nm, 600 nm) and (100 nm, 650 nm) of radius and lattice spacing are done but not shown here. Fig. (2a) represents the dispersion diagram (photonic bandgap) of 2D photonic crystal structure with gallium arsenide as background material having lattice constant of 200 nm and radius of air holes, 100 nm. Similarly, Fig. (2b) represents the dispersion diagram (photonic bandgap) of 2D photonic crystal structure with gallium arsenide as background materials having lattice constant of 300 nm and radius of air holes, 100 nm. From these figures, it is seen that normalized frequency (a/λ) is taken along vertical axis with respect to wave vector (K in m^-1) along horizontal axis. Here, the difference between higher and lower band of normalized frequency is shown in these diagrams. This difference indicates that the range of normalized frequencies cannot pass through the same structure or these ranges of electromagnetic waves are completely reflected from such structures. Then, the wavelength range corresponding to each forbidden normalized frequency is computed. These wavelength ranges of electromagnetic wave will be reflected from such photonic crystal structure. Since reflected beam or light is nothing but the amount of photonic bandgap of photonic crystal structure, photonic band gap corresponding to each dispersion curve have been computed for the same. We know that photonic band gap is determined by taking difference between bottom of the higher band to the top of the lower band. Using above concept, it is seen from Fig. (2a) that higher and lower bands of normalized frequency are overlapped. This indicates that there is no band gap exists between them (upper and lower band). It is understood that no photonic band gap is available in the same graph. And also it is inferred that no light will be reflected, if gallium arsenide photonic crystal structure will have 200 nm of lattice constant and 100 nm of radius of air holes. So it is concluded that such dimensions of the same structure is not suitable for mirror application. However, bandgap is clearly found in Fig. (2b). In this figure lower and higher, a/λ are shown by arrow mark. Apart from this the wavelength corresponding to each normalized frequency (a/λ). is found. From this figure, it is observed that lower wavelength λ2 is 853 nm corresponding to higher a/λ and higher wavelength λ1 is 1197 nm corresponding to lower a/λ. It is also found that wavelength range of 853 nm - 1197 nm cannot propagate through it. It indicates that such wavelength range is completely reflected from such structure. It is also realized that the photonic crystal structure (2D triangular) with gallium arsenide as background material having lattice constant of 300 nm and radius of air hole 100 nm reflects wavelength from 853 nm -1197 nm. And the wavelength band width corresponding to same range (Δλ= 344 nm) is calculated. In other word, one can say that 2 D gallium arsenide triangular photonic crystals structure having lattice spacing of 300 nm and radius of air holes of 100 nm is suitable for an optical mirror for the range of wavelength, 853 nm-1197 nm with wavelength band width of 344 nm. Similarly simulation is done for other lattice spacing and radius ((16 nm,100 nm), (17nm,100 nm), (18 nm,100 nm), (20 nm,100 nm), (25 nm,100 nm), (30 nm,100 nm), (35 nm,100 nm), (40 nm,100 nm), (45 nm,100 nm), (48 nm,100 nm), (49 nm,100 nm), (50 nm,100 nm), (100 nm, 250 nm), (100 nm, 350 nm), (100 nm, 400 nm), (100 nm, 450 nm), (100 nm, 500 nm), (100 nm, 550 nm), (100 nm, 600 nm) and (100 nm, 650 nm). We also computed the lower wavelength, higher wavelength and wavelength width corresponding to each combination. Using above values, graphs are plotted between lower wavelength, higher wavelength and wavelength width along y-axis and radius and lattice spacing along x-axis. The above variations for first and second set of data are shown in Figs. (3a and 3b) respectively.

Fig. (3a) Variation of higher wavelength, lower wavelength and wavelength width/band with radius of air holes, when lattice spacing of the structure is taken of 100 nm.

Fig. (3a) represents the variation of higher wavelength (λ1), lower wavelength (λ2) and wavelength width/band (Δλ) with respect to radius of air hole, which varies from 16 nm to 50 nm, when lattice spacing is 100 nm. Similarly, Fig. (3b) represents the variation of higher wavelength, lower wavelength and wavelength width/band with respect to lattice spacing of structure, which varies from 200 nm to 650 nm, when radius of air holes is 100 nm. From Fig. (3a), it is seen that higher wavelength and lower wavelength are taken along primary y-axis, where wavelength width is taken along secondary y-axis with respect to radius of air hole along primary x-axis. Similarly from Fig. (3b), it is seen that higher wavelength and lower wavelength are taken along primary y-axis, where wavelength width is taken along secondary y-axis with respect to radius of air hole along primary x-axis. From Fig. (3a), it is also observed that both higher and lower wavelengths corresponding to radius of air holes gradually decrease for example; both higher and lower wavelength decrease from 446 nm to 180 nm with the increasing of radius of air hole which varies from 16 nm to 50 nm, however an interesting variation of wavelength width (λ1-λ2) is found with respect to the same radius, for example; λ1-λ2 increases from 0 to 130 nm with radius from 16 nm to 35 nm and further decreases to 0. It is also seen that though the wavelength values of higher and lower band are same at radius of 16 nm and 50 nm, the track of variation is different. The same is clearly mentioned in Fig. (3a). Similarly from Fig. (3b), it is also observed that higher and lower wavelengths corresponding to lattice spacing gradually increases for example; both higher and lower wavelength increases from 365 nm to 2910 nm with the increasing of lattice spacing which varies from 200 nm to 650 nm, however an interesting variation of wavelength width (λ1-λ2) is found with respect to same lattice spacing for example ‘λ1-λ2’ increases from 0 to 344 nm with lattice spacing from 200 nm to 300 nm and further decreases to 0. It is also seen that though the wavelength values of higher and lower band are same at lattice spacing of 200 nm and 650 nm, the track of variation is found to be different. The same is clearly mentioned in Fig. (3b).