The hexagonal super-lattice pattern composed of the light spot and the dim spot is firstly observed and investigated in the discharge of gas mixture of air and argon by using the dielectric barrier discharge device with double water electrodes. It is found that the dim spot is located at the center of its surrounding three light spots by observing the discharge image. Obviously, the brightness of the light spot and the dim spot are different, which indicates that the plasma states of the light spot and the dim spot may be different. The optical emission spectrum method is used to further study the several plasma parameters of the light spot and the dim spot in different argon content. The emission spectra of the N2 second positive band（C3Πu→B3Πg）are measured, from which the molecule vibration temperatures of the light spot and the dim spot are calculated. Based on the relative intensity ratio of the line at 391.4 nm and the N2 line at 394.1 nm, the average electron energies of the light spot and the dim spot are investigated. The broadening of spectral line 696.57 nm（2P2→1S5） is used to study the electron densities of the light spot and the dim spot. The experiment shows that the molecule vibration temperature, average electron energy and the electron density of the dim spot are higher than those of the light spot in the same argon content. The molecule vibration temperature and electron density of the light spot and dim spot increase with the argon content increasing from 70% to 95%, while average electron energies of the light spot and dim spot decrease gradually. The short-exposure image recorded by a high speed video camera shows that the dim spot results from the surface discharges(SDs). The surface discharge induced by the volume discharge (VD) has the decisive effect on the formation of the dim spot. The experiment above plays an important role in studying the formation mechanism of the hexagonal super-lattice pattern with light spot and dim spot. In addition, the studies exert influences on the application of surface discharge and volume discharge in different fields.