In this study, a square quartz grid dielectric barrier discharge device is designed, in which a square dot frame lattice pattern and a square cross dot halo lattice pattern are observed for the first time. These plasma patterns, combined with the quartz grid, constitute a plasma photonic crystal with a square lattice arrangement. Solving the Laplace equation reveals that the applied electric field in the gas gap exhibits a square lattice distribution. The spatiotemporal evolution of the above two patterns is characterized using two photomultiplier tubes and an intensified charge coupled device. Results indicate that the square dot frame lattice pattern consists of two substructures: corner dots and square frame. The corner dots discharge before the square frame. The emission intensities of the corner dots and square frame differ significantly, with a brightness ratio of approximately 8:5. The square cross dot halo lattice pattern consists of three substructures: central dots, corner dots, and rhombic frame. The discharge sequence is central dots→corner dots→rhombic frame. The brightness ratio among the three substructures is approximately 17:16:4. This variation indicates that the intensity of electron avalanches differs across distinct substructure locations. Notably, the pattern substructures discharge synchronously across all holes of the quartz grid. The emission spectra of the square cross dot halo lattice pattern are measured using a spectrograph. The vibrational temperatures at the central dot, corner dot, and rhombic frame are 2527±55 K, 2559±57 K, and 2611±59 K, respectively. It indicates a non-uniform vibrational temperature distribution within each grid hole. Specifically, the relative populations of nitrogen molecules across different vibrational energy levels vary among distinct substructures. It reveals that the substructures of the pattern are in different plasma states. Theoretically, the electric field distributions of the square cross dot halo lattice pattern at different times are simulated by solving the Poisson equation. It effectively elucidates the formation mechanism of the pattern in the square quartz grid dielectric barrier discharge device. This work not only advances the study of pattern dynamics but also provides a new approach for the design of plasma photonic crystals.