# Photonic Stopband Filters Based on Graphene-Pair Arrays
## 3 Photonic Bandgaps
Meanwhile, for a non-zero transmittance at the right rising edge of stopband, the transmittance decreases with the increase in the chemical potential for a fixed incident wavelength. Figure 3b gives the relationship between the stopband width of light and the chemical potential of graphene. The gap width is equal to the deviation between the upper and lower boundaries of stopband. One can see that the lower boundary of bandgap is unchanged as the graphene chemical potential increases, while the upper boundary increases with the increase in the chemical potential. Hence, the gap width can be modulated by changing the value of the graphene chemical potential. Figure 3: (**a**) Transmittance around the first bandgap for different values of graphene chemical potential. (**b**) The first stopband gapwidth varying with the chemical potential. (**c**) Transmittance around the second bandgap for different values of chemical potential. (**d**) Gapwidth of the second stopband varying with the chemical potential. Compared with the first stopband, the gap width of the second stopband is narrower for the same value of graphene chemical potential as shown in Figure 3c. Similarly, the right transmittance from the graphene-pair arrays is zero for the light wave that locates in the bandgap, which forms a stopband of light waves. On both sides of the stopband, the transmittance is relatively lower for a larger value of chemical potential. The left boundary (trailing edge) of the bandgap stays the same position for different values of the chemical potential, while on the right boundary (rising edge) of the bandgap, the curve gets steeper for a lower value in chemical potential. Otherwise, in comparison with the first bandgap, the extinction ratio at the rising edge of the second bandgap is better. Similar to the first bandgap, the inflection point on the right boundary of the second bandgap moves rightwards as the chemical potential increases. Specifically, the width of the second bandgap is proportional to the chemical potential as shown in Figure 3d. Therefore, one can modulate the second stopband gap width by changing the chemical potential, but the starting position of the second bandgap always remains unchanged. Figure 4a gives the real part of graphene surface conductivity \(\mathrm{Re}(\sigma)\) for different values of chemical potential. For three given values of \(\mu\) = 0.3, 0.4 and 0.5 eV, there are three curves, correspondingly, of \(\mathrm{Re}(\sigma)\) as the light frequency changes. The real part of graphene surface conductivity decreases with the frequency of light increasing for a fixed value of graphene chemical potential. On the other hand, a larger value of \(\mathrm{Re}(\sigma)\) can be obtained by tuning the chemical potential up to a bigger value, as the incident wavelength is determined. The parameter \(\mathrm{Re}(\sigma)\) is the counterpart of the imaginary part of the graphene equivalent dielectric constant \(\mathrm{Im}(\varepsilon_{g})\), which represents the optical loss in graphene. Therefore, for a larger \(\mathrm{Re}(\sigma)\) of graphene, it means that the corresponding optical loss is greater in systems. With the increase in light frequency, the loss gradually decreases as well. This approves the phenomenon of each passband transmittance in sequence increasing with the frequency of incident light as shown in Figure 2a. Figure 4b demonstrates \(\mathrm{Re}(\sigma)\) variation in the parameter space composed of the chemical potential and light frequency. One can see that, except for the corner of the parameter space on the left, the value of \(\mathrm{Re}(\sigma)\) increases by raising the graphene chemical potential or turning down the light frequency. Since \(\mathrm{Re}(\sigma)\) governs the loss in graphene  
Figure 4: **(a,c)** Real part and imaginary part of graphene surface conductivity varying with the light frequency for different values of chemical potential, respectively.