A. Experimental setup

Figure 1 presents the schematic diagram of our DBD device. The left electrode is a water electrode made by filling tap water into a cylindrical tube with an inner diameter of 75 mm. It is sealed with glass plates with a thickness of 2.0 mm. The water also acts as a coolant and transparent media to observe and realize stable discharges. A metallic ring is immersed in water and connected to the ground. The right electrode is a mesh electrode connected to a sinusoidal voltage. The stainless-steel mesh is composed of 4 × 4 square unit cells and covered with a 2.0 mm-thick quartz glass layer. The side length of each square unit cell L = 6 mm, which is fixed unless otherwise stated. The frequency of the supply voltage can be varied between 10 and 100 kHz, and the amplitude is adjusted between 0 and 10 kV. A square quartz glass spacer with a thickness of 1.5 mm is placed between two electrodes as the boundary of the discharge area. The mesh-water electrodes are in a vacuum chamber, in which the gas composition and pressure can be controlled. An intensified charge-coupled device (ICCD) camera (Andor DH334T) is utilized to record discharges from the end of the water electrode. The discharge voltage waveform and the current waveform are detected via a high-voltage probe (Tektronix P6015A 1000×) and a current probe (Tektronix TCP0030A), which are input into a digital phosphor oscilloscope (Tektronix TDS3054B) for display and recording.

Optical diagnostics is utilized to detect the photonic band gaps of the plasma lattices. The microwaves are excited at a millimeter wave source (Agilent vector network analyzer, ZNB40, 10.0 MHz–40.0 GHz) and launched from a microwave broad-band horn antenna (18.0–26.5 GHz, XB-GH42-20K, and 26.5–40 GHz, XB-GH28-20K). It transmits through the aperture with an opening of 4 cm × 8 mm adjacent to PPC. A pyramidal horn antenna receiver (XB-GH42-20K, XB-GH28-20K) is placed on the other hand. The microwave with the transverse magnetic (TM) mode is used. The distance between the transmitting and receiving horn antennas is about 30 cm. The positions of band gaps can be detected via the transmission spectra S21.

B. Experimental phenomenon

Figure 2 illustrates the reconfiguration of the Lieb lattice with increasing supply voltage. A simple square lattice forms when the gas discharge is just ignited at U = 2.6 kV. The filaments have a large size and locate at four corners of the square unit cell. When the supply voltage is raised to 3.1 kV, a Lieb lattice appears, in which four large filaments form at the corner sites, while four small filaments emerge at the edge-center sites of each square unit cell [Fig. 2(b)]. Further increasing of the supply voltage gives rise to a Lieb superlattice with the presence of localized quasi-uniform discharges in the middle of each square unit [Fig. 2(c)]. Here, the size and brightness of the filaments at corner sites decrease, which are almost comparable with the filaments at edge-center sites. When the supply voltage is high at U = 4.0 kV, a complex Lieb superlattice is realized, which can be considered as a Lieb lattice immersed in a plasma background. It is noticed that dark regions appear in the vicinity of each filament, which are the inhibition zones due to the spreading of surface charges along the dielectric surface.1515. J. P. Boeuf, B. Bernecker, T. Callegari, S. Blanco, and R. Fournier, “Generation, annihilation, dynamics and self-organized patterns of filaments in dielectric barrier discharge plasmas,” Appl. Phys. Lett. 100, 244108 (2012). https://doi.org/10.1063/1.4729767 The reconfiguration between different plasma lattices takes place rapidly only in a few seconds. Rapid response is essential for designing fast-modulated and multifunctional devices. When the supply voltage decreases, reverse reconfiguration from complex Lieb superlattice to square lattice will take place. As far as we know, we present the first experimental demonstration of tunable plasma Lieb lattices in DBD. New Lieb superlattices have also been realized. It can be anticipated that the photonic band diagrams will change correspondingly with the reconfiguration of various lattices.Figure 3 shows the phase diagram of various plasma lattices as a function of air concentration, applied voltage, and gas pressure. One can see that the same reconfiguration sequence from square to Lieb lattice to Lieb superlattice is achieved with increasing applied voltage, which is insensitive to the air concentration. Lieb lattice can be realized in a wide range of gas pressure and air concentration [Fig. 3(b)]. It is robust and easily produced with no need for rigorous experimental conditions. Remarkably, all these plasma lattices can even be obtained in 100% ambient air. The radii of plasma filaments can also be adjusted by changing the discharge parameters. Such features are beneficial for low-cost production and wide applications.In order to reveal the discharge dynamics of each plasma lattice, a high-speed camera is applied for spatial–temporal resolved measurements. As shown in Fig. 4, for a square lattice, only one single current pulse appears in each half cycle of the supply voltage. This suggests that all of the filaments volley within a time window of about 420 ns. They repeat in the same manner in each half cycle of the supply voltage with the jitter of current pulses less than 30 ns. The good temporal periodicity enables one to control the formation of a square lattice by adjusting the voltage frequency.Figure 5 displays the spatiotemporal dynamics of the Lieb lattice, in which two distinct discharge stages have been obtained in each half-cycle of the supply voltage. The transient Lieb lattice appears during the first current pulse [Fig. 5(b)], while the square sublattice comprised filaments located at edge-center sites (named as square edge-center sublattice) forms during the second current pulse [Fig. 5(c)]. Given the fact that these snapshots are integrated for ten voltage cycles to get sufficient brightness for observation, it may pose a question of whether all of the filaments in the square sublattice initiate twice in each half-cycle of the supply voltage, corresponding to two consecutive current pulses, or whether, in fact, the square sublattice results from the integration of the filaments that appear only once, corresponding to either the first current pulse or the second pulse. To answer this question, the temporal correlations between different filaments have been studied by the use of photomultipliers as illustrated in Fig. 6. It can be seen that the filament at the corner sites (S1) is ignited during the first current pulse and exhibits good temporal order. Notably, the edge-center filament (S2) also appears only once in each half-cycle of the supply voltage. However, it is ignited randomly with time, corresponding to either the first current pulse or the second pulse. The temporal disorder always accompanies the spatial disorder, which implies that a part of the edge-center filaments in the square sublattice appear during the first current pulse, while the left edge-center filaments initiate during the second pulse. Hence, the Lieb lattice results from the spatiotemporal integration of corner filaments and edge-center filaments over multiple voltage cycles.Figures 7 and 8 show the temporally resolved measurements of the Lieb superlattice lattice and complex Lieb superlattice, respectively. Three distinct current pulses appear in each half-cycle of the supply voltage as illustrated in Fig. 7(a). Similar to the Lieb lattice, the filaments at corner sites are ignited during the first current pulse. The filaments at the edge-center sites occur randomly in time, either during the first current pulse or during the second pulse. Interestingly, during the third current pulse, localized quasi-uniform discharges emerge in the middle of each unit cell and arrange in a square lattice as indicated in Fig. 7(d). We name this square sublattice as the square middle sublattice, which is harmonic with the driving voltage and exhibits good temporal periodicity. It is noted that defects may emerge in these instantaneous discharges, such as the defects in the fourth cell in Fig. 7(b) and the third cell in Fig. 7(c) in the top line. The presence of defects results from the random fluctuations produced in discharge processes including the essentially stochastic particle motion, collision ionization, photoionization, etc. For the complex Lieb superlattice as illustrated in Fig. 8, the former two discharge stages are similar to that of the Lieb superlattice, while a quasi-uniform plasma background emerges during the third current pulse. The empty “holes” in these plasma backgrounds give the ranges of breakdown-inhibition zones of isolated filaments, due to the spread of surface charges along the dielectric surface as described above. Thus, both Lieb superlattice and complex Lieb superlattice form on the base of the primary Lieb lattice, with the presence of additional quasi-uniform discharges occurring at the third current pulse.Three distinct discharge stages can be identified based on the above dynamical behaviors as demonstrated in Fig. 9. In stage-I, the filaments located at the corner sites of square unit cells are ignited, which corresponds to the first current pulse in each half cycle of the supply voltage [Fig. 9(a)]. It is a temporally stable discharge state that is harmonic with the driving period and exhibits good temporal periodicity. In stage-II, the filaments at the edge-center sites initiate, which emerge randomly either during the first current pulse or the second pulse [Fig. 9(b)]. It is a temporally accidental discharge state that displays a remarkable temporal disorder. In stage-III, weak quasi-uniform discharges emerge, which occur during the third current pulse in the regions beyond the isolated filaments [Fig. 8(c)]. It is also a temporally stable discharge state, which is harmonic with the driving period and has good temporal periodicity. With reconfiguration from square lattice–Lieb lattice–Lieb superlattice–complex Lieb superlattice as the supply voltage increases, the discharge stages change accordantly. Only stable stage-I occurs in a square lattice, stable stage-I and dynamical stage-II occur in a Lieb lattice, while three discharge stages, i.e., stable stage-I, dynamical stage-II, and stable stage-III take place in two types of Lieb superlattices. We stress that the dynamical stage-II emerges in all of these Lieb-like lattices, giving rise to complexity and unique spatial–temporal properties.A joint action of surface charges, space charges, and two-dimensional Laplacian fields induced by mesh electrodes is responsible for the formation of three discharge stages. As shown in Fig. 10, the strength of the Laplacian field is largest at the corner sites of each square unit, leading to the prior stage-I discharge at these positions. The Laplacian field at the edge sites Ee is slightly lower but comparable with that of the corner sites Ec with Ee/Ec = 0.95. This will lead to two cases for edge-center filaments, depending on the densities of surface charges and space charges produced in the previous half cycle. It is known that the surface charges are not only able to inhibit the discharges but also establish an electric field to aid the next discharge when the polarity of the supply voltage has been reversed. On the other hand, a number of space charges survive in the gas gap after the extinguishment of the discharges, and they serve as the seed charges to ignite the next discharge. Therefore, if the densities of surface charges and space charges are sufficiently large, the edge-center filaments will emerge during the first current pulse with the help of the Laplacian field. Alternatively, if the densities of surface charges and space charges are relatively low, the edge-center filaments will be ignited during the second current pulse, where the momentary AC voltage has been further increased. Owing to the inevitable fluctuations in the accumulation of surface charges and space charges, the filaments at edge-center sites show remarkable temporal disorder, occurring either during the first current pulse or the second pulse. Stage-III discharges are ignited in the middle of each square unit cell or in the large-scale background. Laplacian field strengths in these regions are the lowest so the discharges occur during the third current pulse, where a higher momentary voltage has been reached to meet the breakdown threshold. The formation of three discharge stages in Lieb-like lattices results from a combined effect of surface charges, space charges, and mesh electrodes.