# Structural Tunable Plasma Photonic Crystals in Dielectric Barrier Discharge
## 3 Results and Discussion
The real part of the relative dielectric constant \(\varepsilon_{\text{P}}\) given in Equation (2) is used to calculate the dispersion relations of the plasma photonic crystals, while the imaginary part is responsible for dissipation and absorption of the EM waves, which is not taken into account here. \[\text{Re}\Big{(}\varepsilon_{\text{P}}\Big{)}=1-\frac{\omega_{\text{pe}}^{2}} {\omega^{2}+\
u_{\text{m}}^{2}} \tag{2}\]  
Figure 2: Plasma photonic crystals with different symmetries when increasing of the applied voltage. The left column: snapshots of the triangular plasma photonic crystal (**a**) and square plasma photonic crystal (**d**). The middle column: the corresponding Fourier spectra of triangular lattice (**b**) and square lattice (**e**). Right column: the voltage-current waveforms associated with the triangular lattice (**c**) and square lattice (**f**). The discharge parameters are: (**a**) the applied voltage \(U=5.3\) kV, discharge power \(p=32\) W (**d**) \(U=6.0\) kV, \(p=46\) W. The frequency of the applied voltage \(f=55\) kHz. The working gas is 100% air. The distance of the gas gap \(d=1.1\) mm. Gas pressure \(p=300\) torr and the exposure time \(t=40\) ms.  
Simulations of the plasma photonic crystals with different configurations were completed by use of COMSOL Multiphysics software based on the finite element method. The Lorentz-Drude model was employed. The mesh division is selected as a free triangle mesh, and the largest mesh element is set as \(8.33\times 10^{-5}\) m, the curvature factor is 0.25. The areas enclosed by the black and rhombus in Figure 3a,c indicate the basic units of different plasma photonic crystals. Floquet periodic boundary conditions are applied at two sets of opposite edges of the square and rhombus. According to [PERSON]'s theorem, the eigenvalue frequencies corresponding to different wave vectors k along the irreducible Brillouin zone boundaries M-\(\Gamma\)-X-M and M-\(\Gamma\)-K-M are calculated, respectively. By scanning the coefficient k through the parameterized sweep built in COMSOL, the dispersion relations of EM waves for different plasma structures can be obtained. Figure 4 illustrates the band diagrams of the triangular and the square plasma photonic crystals corresponding to the plasma structures shown in Figure 2. Due to the facts that there may have fluctuations of electron densities in the ensemble of filaments forming the crystal lattice, the band diagrams under different electron densities are studied. For similar conditions to the present experiments, \(10^{13}\leq\mathrm{n_{e}}\leq 10^{16}\) cm\({}^{-3}\)[36; 38; 39; 40]. Obviously, one can see that an increase of the electron density leads to formation of more band gaps for both of the square and triangular plasma photonic crystals. Moreover, the positions of the band gaps move to the higher frequency and the sizes of the band gaps are increased remarkably. Here we take the band diagram at the electron density \(\mathrm{n_{e}}=10^{15}\) cm\({}^{-3}\) for example, which order of magnitude most commonly occurs in the discharges [39; 40]. We can see that three unidirectional band gaps form for the triangular lattice as shown in Figure 4c. They locate at the position of 0.58-0.76\(\omega\)a/2\(\pi\)rc in M-\(\Gamma\) direction, and at 0.74-0.99\(\omega\)a/2\(\pi\)rc and 1.08-1.26\(\omega\)a/2\(\pi\)rc in K-M direction, respectively. When the plasma photonic crystal is tuned to the square lattice by increasing the applied voltage as shown in Figure 4g, a remarkably different band structure forms. An absolute band gap is produced at the position of 0.72-0.77\(\omega\)a/2\(\pi\)rc, and the EM waves within this frequency range cannot pass through the plasma crystals. Consequently, with the symmetrical  
Figure 3: Schematics of a square lattice (**a**) and a triangular lattice (**c**) and the corresponding reduced Brillouin zone boundaries (**b–d**).