# Structural Tunable Plasma Photonic Crystals in Dielectric Barrier Discharge
## 3 Results and Discussion
Here a is the lattice constant; **r** is the radius of the plasma column; \(\mathrm{a_{1}}\), \(\mathrm{a_{2}}\) denote the base vectors of the lattices; \(\mathrm{\epsilon_{a}}\) and \(\mathrm{\epsilon_{p}}\) indicate the dielectric constants of the gas and plasmas, respectively. The areas enclosed by the black and rhombus at the top left of (**a,c**) indicate the basic unit of the plasma photonic crystals. In figures (**b,d**), the triangular regions surrounded by M-\(\Gamma\)-X-M and M-\(\Gamma\)-K-M are the irreducible Brillouin regions which are utilized for calculating of the dispersion relations. deformation from the triangular lattice to square lattice, the band structures change significantly, which can be utilized to modulate the propagation of EM waves of different frequencies. By changing the applied voltage, the lattice constant of the plasma structures can be dynamically controlled. As shown in Figure 5, the triangular lattices with different lattice constants are obtained in our experiment. More filaments are ignited and the lattice constants decrease by increasing the applied voltage. All these plasma structures are very stable, which can last for tens of minutes unless the discharge parameters are changed. To quantitatively characterize the spatial order of the plasma structures, the pair correlation functions (PCF) are calculated from the discharge density distributions. The PCFs have been used to characterize complex plasma crystals to show the probability density (\(\rho_{\rm r}\)) to find a filament at distance r from a given filament [41]. One can clearly see that the pair correlation function displays a series of isolated peaks, in which the first peak is the strongest, giving the value of the lattice constant. The lattice constant \(a=2.68\) mm, \(2.59\) mm, \(2.48\) mm, respectively, which decreases  
Figure 4: The dispersion relations under different electron densities for the triangular lattice (**a–d**) and the square lattice (**e–h**) shown in Figure 2a,b. The shadow strips indicate the positions of the band gaps. The plasma collision frequency \(\upsilon_{\rm m}=160\) GHz. with an increase of the applied voltage. The positions of \(\rho_{\rm r}\) peaks for the ideal triangular lattices are given by the red short lines at the bottom of Figure 5d-f. Obviously, they matched well with the \(\rho_{\rm r}\) peaks of the plasma structures obtained in our experiment, indicating that these plasma lattices are very regular and close to the ideal triangular lattices. The voltage-current waveforms for different plasma photonic crystals are illustrated in Figure 5g-i. One can see that more current spikes appear with an increase of the applied voltage, owing to the formation of more discharge filaments. As expected, the band diagrams change correspondingly when the lattice constant is reduced with increasing of the applied voltage. One can find in Figure 5j that a complete band gap is produced when \(a=2.68\) mm, which locates at the position of 1.18-1.22\(\omega\)a/2\(\pi\)c. Besides, two unidirectional band gaps form in the M-\(\Gamma\) direction at the positions of 0.74-0.90\(\omega\)a/2\(\pi\)rc and 1.47-1.63\(\omega\)a/2\(\pi\)c, respectively, and two unidirectional band gaps are produced in the K-M direction at the positions of 1.23-1.47\(\omega\)a/2\(\pi\)c and 1.68-1.76\(\omega\)a/2\(\pi\)rc. When the lattice constant is reduced from 2.68 mm to 2.48 mm, as shown in Figure 5l, the width of the complete band gap becomes larger and its position is changed to 1.19-1.27\(\omega\)a/2\(\pi\)rc. In addition, the positions of the other four unidirectional band gaps are also changed, which locate at 0.78-0.94\(\omega\)a/2\(\pi\)c and 1.51-1.71\(\omega\)a/2\(\pi\)rc in the M-\(\Gamma\) direction and at 1.28-1.52\(\omega\)a/2\(\pi\)c and 1.71-1.82\(\omega\)a/2\(\pi\)rc, in the K-M direction, respectively. Obviously, the photonic bands move towards higher frequencies with a decrease of the lattice constant and the size of each band gap is increased.