Implementation and application of PyNE sub-voxel R2S for shutdown dose rate analysis

1.   Introduction

As a novel type of photonic crystals (PCs), plasma photonic crystals (PPCs) have attracted widespread attention [1–17]. The PPC is a periodic array composed of discharge plasma and dielectric material, which was proposed by Hojo in 2004 and demonstrated experimentally by Sakai in 2005 [5, 6]. Compared with the traditional PC, the significant advantage of the PPC is the tunability in controlling the propagation of electromagnetic waves [7–12]. In 2014, Zhang et al studied the transmission characteristics of microwaves through 1D PPCs and found that there are band gaps in 5–13 GHz. The central frequency of the band gap was changed by about 0.5 GHz by varying the electron density from 10¹¹ to 10¹² cm⁻³, and changed by about 2 GHz by varying the lattice constant [13]. In 2015, Wang et al showed that the frequency shift of the band gap is about 0.3 GHz (between 13 GHz and 15 GHz) by changing the plasma current to change the electron density from 10¹¹ to 4 × 10¹¹ cm⁻³ [11]. In 2019, Sun et al introduced a class of 3D plasma/metal/dielectric photonic crystals with a band gap in the 120–170 GHz spectral range. The shift in the central frequency of band gaps is below 1 GHz when the layers of the micro-plasma columns introduced to the scaffold are changed [14]. Since 2010, self-organized patterns in dielectric barrier discharge (DBD) have been proposed to generate tunable PPCs. In 2010, Dong et al obtained tunable plasma photonic crystals with variable lattice constants of a hexagonal structure through the self-organization of filaments in DBD [15]. Fan et al obtained a series of 2D plasma photonic crystals, evolving from square to hexagon, by altering the voltage through filament self-organization, from which the band edge frequencies of band gaps can be changed from 27 to 124 GHz [16]. In 2014, Wang et al reported on PPCs with the structures of square, square superlattice, and hexagonal lattice coupled to square lattice by using a square meshed electrode in uniform gap DBD, which realized the combination of self-organization and artificial modulation of the discharge filaments. The simulations of the band gaps of a square superlattice PPC showed that the band edge frequency can be tuned from 20 to 100 GHz [17]. This is in addition to the effect of the external magnetic fields on the band gap [18, 19]. In 2013, Guo et al investigated the properties of the photonic band structures (PBSs) of 1D plasma PPCs tuned accordingly after being exposed to an external magnetic field [18]. In a word, there are several methods of generating PPCs with tunable band gaps. The first one is to change the electron density and diameter of discharge plasma in fixed array electrodes, whose tunable range in the band gap is relatively narrow because the symmetry and lattice constants of PPCs cannot be easily changed. The second one is to change the symmetry, lattice constant, and electron density of discharge plasma columns through self-organized discharge patterns. The third one is to add an external magnetic field. In order to further promote the application of PPCs in the mm-wave region, a simple device to obtain PPCs with controllable symmetry, tunable parameters and wider band gaps is highly desired. In addition, in recent years, the feasibility of using 1D micro plasmonic photonic crystals (MPPCs) for terahertz bandgap regulation has been demonstrated at both theoretical and simulation levels [20, 21].

Pattern formation is a typical nonlinear self-organized phenomenon. It exists widely in nature such as cloud formations and animal coat markings. It can also be studied in experimental systems including reaction-diffusion systems, the Faraday system, the nonlinear optical system, and the dielectric barrier discharge (DBD) system [22–25]. As a newly developed pattern system in recent years, DBD has demonstrated many unique applications in PPCs besides its many advantages in basic pattern research, such as abundant patterns and convenient experimental setups. DBD is an AC gas discharge, whose device is generally composed of two electrodes, at least one of which is covered with a dielectric layer [26–31]. As the applied voltage increases to the gas breakdown threshold, some filaments appear in the gas gap, and the discharge filaments can self-organize into diverse regular patterns under the appropriate conditions. Up to now, some different types of square patterns have been obtained in DBD. In 2003, Dong et al observed a square pattern in DBD, which is an interleaving of two square sublattices A and B with temporal inversion ABBA in one cycle of voltage [32]. In 2006, Dong et al reported a square superlattice pattern, which is an interleaving of two different transient square sublattices s and l, and the discharge sequence in one cycle of voltage is s-l-l-s-l-l [33]. In 2016, Wei et al obtained a white-eye square superlattice pattern and found that the pattern consists of four different transient sublattices, by using an intensified charge-coupled device (ICCD) camera and photomultiplier tubes [34]. It is worth pointing out that these square patterns result from the self-organization of filaments in a uniform discharge gap in DBD. As is well known, pattern formation is dependent upon the nonlinear resonance of wave vectors. If a periodic modulation is inserted into the uniform discharge gap, which is equivalent to adding a wave vector, the self-organization of the filaments will become complex and some novel patterns should emerge. Here, we report five patterns with square symmetry including three new spatiotemporal patterns, besides that of the square pattern and the square superlattice pattern mentioned above, by introducing a square periodic modulation into the uniform discharge gap in DBD. These patterns will promote not only the dynamics development of the DBD pattern, but also the application of PPC in the mm-wave region because of their controllable and tunable properties.

In this work, a specially designed grid PC is placed between two plate electrodes in DBD, which forms a grid modulation in the uniform discharge gap in DBD. Five patterns with square symmetry including three new patterns, which are either new in appearance or in spatiotemporal dynamics, are obtained as the applied voltage increases. From the viewpoint of PC application, this device realizes the conversion from PC with only unidirectional band gaps to five types of PPCs with omnidirectional band gaps. The tunable and controllable properties in controlling the propagation of electromagnetic waves in the mm-wave region are greatly advanced.

2.   Experimental setup and simulation

A schematic diagram of the experimental setup is shown in figure 1. A special grid PC is designed and placed between two quartz plates on two water containers in a DBD system. The grid PC is composed of 16 × 16 square arrays of gas columns (with a diameter of 1 mm and a lattice constant of 2 mm) and glass in which gas columns were drilled. The dielectric constants of glass and gas are 2 and 1 respectively. The water electrodes consist of two cylindrical containers sealed with 2.5 mm thick quartz plates and filled with water. The copper ring in the water electrode is connected to the power supply. The discharge cell is placed into a chamber filled with air/argon mixed gas. A high-voltage probe (Tektronix P6015A) and a current probe (Tektronix TCP 0030A) are used to detect the current and voltage, and a digital phosphor oscilloscope (Tektronix DPO 4054B) is used to record the current and voltage waveforms. Images of patterns are taken by a digital camera (Canon EOS 6D). ICCD (pro. 120PH0047) and PMT (PMT, RCA 7265) are used to measure the spatio-temporal structure of the patterns. The instantaneous images in successive discharge current pulses can be captured by ICCD through the superposition of multiple discharge cycles. The optical signals of different discharge filaments (plasma columns) in each applied voltage cycle can be detected by photomultiplier tube systems due to its high sensitivity. The emission spectrum of an individual plasma column is measured by a spectrograph (Acton Advanced SP 2750 A, CCD: 1340 × 400 pixels) to estimate electron density. Simulations of PC and PPCs with different configurations are completed by using COMSOL Multiphysics software.

Figure  1.  Schematic diagram of the experimental setup. A grid PC is placed between two quartz plates on two water containers.

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3.   Results

3.1   Spatio-temporal dynamics and formation mechanism of patterns

The parameters in our experiments are selected to obtain more types of PPCs to realize the wide adjustment range of the band gap. After a lot of experiments, five PPCs can be obtained by only altering the applied voltage under experimental conditions. Figure 2 shows a grid PC and five kinds of discharge patterns with square symmetry with the applied voltage increasing. From the viewpoint of PC application, it realizes the conversion from PC to five types of PPCs, which is the combination of the plasma pattern and PC. Figure 2(a) shows the front view picture of the grid PC. When the applied voltage is 3.72 kV, the discharge filaments (plasma columns) with the radius of 0.2 mm appear in half of all holes and form a simple square pattern (named 'square Ⅰ pattern') with the lattice constant of $2 \sqrt{2}$ mm as shown in figure 2(b). With the voltage increasing to 5.21 kV, the bright discharge filaments with the radius of 0.2 mm occur in all of the holes and form another square pattern (named 'square Ⅱ pattern') with the lattice constant of 2 mm, as shown in figure 2(c). When the voltage reaches 6.32 kV, the superlattice Ⅰ pattern with the lattice constant of 4 mm appears as shown in figure 2(d). It is composed of two types of filaments, one is thin filaments with the radius of 0.2 mm, and the other is bright thick filaments with a radius of 0.3 mm. As the voltage increases to 7.28 kV, the discharge filaments with a radius of 0.3 mm increase and become brighter and form a square superlattice Ⅱ pattern with the lattice constant of $2 \sqrt{2}$ mm as shown in figure 2(e). It can be seen obviously that the unit cell in figure 2(d) is different from that in figure 2(e). When the applied voltage reaches 7.84 kV, bright thick filaments occupy all of the holes and form a square pattern (named 'square Ⅲ pattern') with the lattice constant of 2 mm, as shown in figure 2(f). These well-defined plasma structures have high spatial symmetry, which resemble the 'atom' crystals in condensed matter physics.

Figure  2.  Grid PC and five kinds of square patterns with different applied voltages. (a) Picture of front view of the grid PC device. (b) Square Ⅰ pattern, U = 3.72 kV. (c) Square Ⅱ pattern, U = 5.21 kV. (d) Square superlattice Ⅰ pattern, U = 6.32 kV. (e) Square superlattice Ⅱ pattern, U = 7.28 kV. (f) Square Ⅲ pattern, U = 7.84 kV. Other experimental parameters: argon concentration Φ = 95% in mixed gas of argon and air, gas pressure p = 50.5 kPa, driven frequency f = 53 kHz, gas gap d = 2 mm, and exposure time of the pictures t = 25 ms.

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As we all know, the spatio-temporal structure of a pattern plays a vital role, not only in explaining the formation mechanism of the pattern, but also in the calculation of the dispersion relation of PPCs. For example, the dispersion relation of PPCs is greatly dependent upon the lattice constant of each sublattice if the pattern consists of two or more spatio-temporal sublattices [16]. Therefore, the spatio-temporal dynamics of five patterns is studied by photomultiplier tube (PMT) and ICCD.

The spatio-temporal structures of three kinds of square patterns (Ⅰ–Ⅲ) are studied through measurements of time correlation between lights from two adjacent discharge filaments by two photomultiplier tubes. Figure 3 shows that the square Ⅰ pattern has one current pulse in a half voltage cycle and all filaments discharge simultaneously, which was never observed in uniform gap dielectric barrier discharge. In the square Ⅱ pattern, two adjacent filaments, A and B, discharge alternately in two current pulses in a half voltage cycle and the discharge order is A-B-B-A in each voltage cycle, as shown in figure 4(a). Therefore, the square Ⅱ pattern is an interleaving of sublattice A and sublattice B with temporal inversion, as shown schematically in figure 4(b). The structure of each sublattice is the same as the square Ⅰ pattern with a lattice constant of $2 \sqrt{2}$ mm. Figure 5 shows the spatio-temporal structure measurements of the square Ⅲ pattern. Figure 5(a) shows that each filament discharges twice, one at the rising edge of the applied voltage $\left(\frac{\left|\mathrm{d} U_{\mathrm{app}}\right|}{\mathrm{d} t}>0\right)$ which is denoted by A₁ or B₁, and the other at the falling edge of the applied voltage $\left(\frac{\left|\mathrm{d} U_{\mathrm{app}}\right|}{\mathrm{d} t}<0\right)$ which is denoted by A₂ or B₂. It is the same as the discharge of big spots in the square superlattice pattern in [33]. It is worth pointing out that two adjacent discharge filaments, A and B, do not discharge simultaneously, at the rising edge nor at the falling edge. There is a time difference of several hundred nanoseconds within a current pulse. If the discharge sequence is A₁-B₁-B₂-A₂ in a half voltage cycle, the sequence will be B₁-A₁-A₂-B₂ in the next half voltage cycle. From the viewpoint of the temporal inversion of ABBA, it has double temporal inversions, i.e. A₁-B₁-B₁-A₁ at the rising edge while B₂-A₂-A₂-B₂ at the falling edge. To the best of our knowledge, it is the most complex temporal inversion pattern observed in DBD. It results from the influence of wall charges. As is well known, the discharge will occur at the falling edge of the applied voltage if the wall charges deposited at the rising edge are more than enough. The wall charges deposited during B₁ are slightly more than those during A₁, if B₁ discharges after A₁ at the rising edge, resulting in B₂ discharging hundreds of nanoseconds before A₂ at the falling edge, the discharge sequence is A₁-B₁-B₂-A₂. In the next half cycle, B₁ discharges before A₁ because the residual wall charges (after B₂ at the last falling edge) on the B₁ position are more than those on A₁ and then A₂ discharges before B₂, forming the discharge sequence of B₁-A₁-A₂-B₂. Thus, the square Ⅲ pattern consists of four sublattices A₁, A₂, B₁, and B₂ with a complex temporal sequence of A₁-B₁-B₂-A₂-B₁-A₁-A₂-B_2. In the following calculation of PPC, considering the discharge time difference between A₁ and B₁, as well as A₂ and B₂, is only several hundred nanoseconds within a current pulse, it can be roughly regarded as that all filaments discharge simultaneously at the rising edge of the applied voltage as well as at the falling edge. It is found that the radius of the filament is 0.3 mm at the rising edge of the applied voltage, while 0.2 mm at the falling edge, from the picture taken by the ICCD. Therefore, the temporal evolution of the square Ⅲ pattern is a square lattice composed of bright thick filaments with a lattice constant of 2 mm at the rising edge, and a square lattice composed of thin filaments with a lattice constant of 2 mm at the falling edge.

Figure  3.  Time correlation between lights from two adjacent spots A and B in the square Ⅰ pattern. U stands for the waveform of the applied voltage, I stands for the waveform of the current.

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Figure  4.  (a) Time correlation between lights from two adjacent spots A and B in the square Ⅱ pattern. U stands for the waveform of applied voltage, I stands for the waveform of current. (b) Schematic diagram of discharge sequence of the pattern in one cycle.

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Figure  5.  Time correlation between lights from two adjacent spots A and B in the square Ⅲ pattern. U stands for the waveform of applied voltage, I stands for the waveform of current. (b) Schematic diagram of discharge sequence of the PPC in one cycle. A₁, B₁ represent the discharge filaments at the rising edge of the applied voltage; A₂, B₂ represent the discharge filaments at the falling edge of the applied voltage.

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The spatial-temporal structures of two kinds of square superlattice patterns (Ⅰ, Ⅱ) are studied by ICCD and PMT. Figure 6 shows a series of time-resolved photos of the square superlattice Ⅰ pattern. In figure 6(a), the bright thick spot is denoted by L, and the thin spot between two bright thick spots along a line is denoted by D(S), and the thin spot between two bright thick spots along a diagonal is denoted by D(C). Five holes of the grid PC device are circled to show the sizes of the discharge filaments in figure 6(a). Figures 6(c)–(e) show the instantaneous images correlated to the current pulse phases Δt₁ = 500 ns at the falling edge of voltage, Δt₂ = 900 ns, and Δt₃ = 1000 ns at the rising edge of voltage respectively, which are denoted in the waveforms of the voltage and current in figure 6(b). They are integrated over 10 voltage cycles to obtain sufficient light signals. Figures 6(c) and (e) show that the bright thick spots (L) discharge in Δt₁ with a radius of 0.2 mm (denoted as L₁) and in Δt₃ with a radius of 0.3 mm (denoted as L₂). Obviously, the spots L₁ at the falling edge of the voltage are smaller than L₂ at the rising edge, which is consistent with the experimental results in previous work [35]. Figure 6(d) shows that thin spots (D(C) and D(S)) with a radius of 0.2 mm discharge in Δt₂. Figure 6(f) is the superposition of figures 6(c)–(e).

Figure  6.  Instantaneous images of the square superlattice Ⅰ pattern. (a) Picture of the square superlattice Ⅰ pattern, L stands for bright thick spots, D(S) stands for thin spots between two bright thick spots along a line, D(C) stands for thin spots between two bright thick spots along a diagonal. Five holes of the grid PC device are circled to show the sizes of the discharge filaments. (b) Waveforms of the applied voltage and current of the pattern. (c)–(e) Images exposed corresponding to the current pulse phases denoted by Δt₁, Δt₂, and Δt₃ in (b), respectively, and the images are integrated for 10 voltage cycles. (f) The superposition of (c)–(e).

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By observing figures 6(d)–(f), it is found that the distance between spot D(C) and spot L is different from that between D(S) and L although D(C) and D(S) both discharge during Δt₂. According to the wall charge theory, the discharge time of the two thin spots should be different. In order to measure the difference between the discharge moments of the two thin spots, i.e., the fine structure, two PMTs are used to measure the time correlation between lights of spots D(C) and D(S) as PMT is more sensitive than ICCD. Figure 7 shows that their discharge moments are slightly different, by about several hundred nanoseconds. D(C) spots will discharge after D(S) spots in the next half voltage cycle if the D(C) spots discharge before the D(S) spots in this half voltage cycle. Thus, the sequence of discharge will be D(C)-D(S)-D(S)-D(C) in one voltage cycle, which presents a temporal inversion.

Figure  7.  Time correlation between lights from two different thin spots D(C) and D(S) denoted in figure 6(a).

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It can be concluded from figures 6 and 7 that the square superlattice Ⅰ pattern consists of four sublattices. Figure 8 gives the diagram of discharge sequence in one voltage cycle, in which C and S represent spots D(C) and D(S) respectively. Obviously, the discharge sequence in one voltage cycle of the square superlattice Ⅰ pattern is L₁-D(C)-D(S)-L₂-L₁-D(S)-D(C)-L₂.

Figure  8.  Schematic diagram of the discharge sequence of four sublattices of the square superlattice Ⅰ pattern. L₁ and L₂ represent the discharge filaments at the falling and rising edges of the applied voltage, respectively. C and S represent spots D(C) and D(S) respectively.

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As demonstrated in [25, 36], the formation of every sublattice is dependent upon the electric field produced by wall charges deposited by all discharged sublattices. The discharge will occur firstly at the positions with the largest wall charges field because of positive effect on discharge. Considering that the wall charges deposited at L spots during the discharge at the rising edge of applied voltage are almost consumed during the discharge at the falling edge, D(C) or D(S) spots should discharge before L spots. If the D(C) sublattice is ignited firstly, the next sublattice will be D(S) rather than L. When the applied electric field becomes larger, L spots are ignited. Therefore, the discharge sequence should be D(C)-D(S)-L₂-L₁ in this half voltage. The D(S) spots accumulate more wall charges than the D(C) spots because they discharge after the D(C) spots in this half voltage cycle, causing the D(S) spots discharge before the D(C) spots in next half voltage cycle, and vice versa. Therefore, the discharge sequence is D(C)-D(S)-L₂-L₁-D(S)-D(C)-L₂-L₁ in one voltage cycle.

In addition, it is found that the radii of D(C), D(S), and L₁ are 0.2 mm, while that of L₂, is 0.3 mm, from the picture taken by ICCD. Considering that the difference between discharge moment of D(C) and D(S) is only several hundred nanoseconds in one current pulse, two sublattices can be roughly regarded as discharging simultaneously during Δt₂ for simplicity. So, the temporal evolution of the square superlattice Ⅰ pattern in the following calculation of PPC will be square lattice (D(C)+D(S)) with lattice constant of 4 mm, square lattice L₂ with lattice constant of 4 mm, and a square lattice L₁ with lattice constant of 4 mm.

Figure 9 shows a series of time-resolved photos of the square superlattice Ⅱ pattern. In figure 9(a), the bright thick spot is denoted by L, and the thin spot is denoted by D. Figures 9(c)–(e) give the instantaneous pictures correlated to the current pulse phases Δt₁ = 900 ns at the falling edge of voltage, Δt₂ = 800 ns and Δt₃ = 1300 ns at the rising edge of voltage respectively, which are denoted in figure 9(b). They are integrated over 20 voltage cycles to obtain sufficient light signals. Figures 9(c) and (e) show that the bright thick spots (L) discharge twice, one with radius of 0.2 mm in Δt₁ (denoted as L₁), and the other with radius of 0.3 mm in Δt₃ (denoted as L₂). Figure 9(d) shows that thin spots D with radius of 0.2 mm discharge in Δt₂. Figure 9(f) is the superposition of figures 9(c)–(e). The discharge sequence of the pattern is D-L₂-L₁, which is similar to the square superlattice pattern in [33]. The temporal evolution of the square superlattice Ⅱ pattern in the following calculation of PPC will be square lattice D, square lattice L₁, and square lattice L₂, whose lattice constants are all $2 \sqrt{2}$ mm.

Figure  9.  Instantaneous images of the square superlattice Ⅱ pattern. (a) Picture of the square superlattice Ⅱ pattern, L stands for bright thick spots, D stands for thin spots. (b) Waveforms of the applied voltage and current of the pattern. (c)–(e) Images exposed corresponding to the current pulse phases denoted by Δt₁, Δt₂, Δt₃ in (b), respectively. The images are integrated for 20 voltage cycles. (f) The superposition of (c)–(e).

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As is well known, pattern formation in DBD is dependent upon the wall charges deposited during the discharges in uniform DBD. In this work, the grid PC is inserted into the uniform discharge gap, which is equivalent to adding a wave vector prior to the filaments self-organization. It results in some new physical phenomena demonstrated in some novel spatio-temporal patterns including square Ⅰ, square Ⅲ, and square superlattice Ⅰ, which have not been observed in uniform gap DBD. Figures 10(a)–(f) are the Fast Fourier transform (FFT) corresponding to figures 2(a)–(f) respectively. It can be seen that the basic vector of the FFT is $1 / \sqrt{2}$, 1, 1/4, $1 / \sqrt{2}$, and 1 times of K_grid (the basic wave vector of the grid) in figures 10(b)–(f) respectively, indicating many types of resonances of wave vectors. In figure 10(b). the resonance results from the discharge can only occur at adjacent holes along a diagonal rather than along a line due to the dielectric character of the grid device. As there is almost no interaction between adjacent filaments in the square Ⅰ pattern, they discharge at the same time and a new spatio-temporal structure of the square pattern occurs. Although the nonlinear resonance relationship of the wave vectors of square Ⅱ and square Ⅲ patterns is similar to that of grid PC, the number of wall charges in both patterns is different. In the square Ⅱ pattern with a voltage of 5.21 kV, two adjacent filaments discharge alternately due to the effect of wall charges. While the filaments in the square Ⅲ pattern with a higher voltage of 7.84 kV accumulate more wall charges, which cause all filaments to discharge again at the falling edge of voltage and two adjacent filaments discharge alternately, either at the rising edge or the falling edge. It is a novel spatio-temporal structure of the square superlattice pattern and has not been observed in uniform gap DBD. Comparing with the square superlattice Ⅱ pattern, the square superlattice Ⅰ pattern has one more sublattice D(S) and its basic wave vector is $1 / 2 \sqrt{2} K_{\text {grid }}$, which should be related to the dielectric character of the grid device. In a word, the above experiments not only achieve the control of the symmetry of the pattern by additional wave vector, but also enrich the pattern types in DBD and promote the development of pattern dynamics.

Figure  10.  Fourier transform corresponding to figures 2(a)–(f).

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In addition, the experiment provides a method for obtaining PPCs with different symmetries and parameters.

3.2   Electron density measurements by optical emission spectra

From the viewpoint of PC application, this device realizes the conversion from PC to five kinds of PPC. For the calculation of the dispersion relations of PPCs, the electron density of the plasma column is essential. In our previous works, the electron density in filaments in different DBD patterns in argon gas mixture is measured by Stark broadening and a shift of argon spectral line 696.54 nm. It is found that the electron density in filaments is in the order of 10¹⁶ cm⁻³ in atmospheric pressure DBD [37, 38], and the electron densities increase from 8.4 × 10¹⁴ to 1.5 × 10¹⁵ cm⁻³ with gas pressure increasing from 1 × 10⁴ to 5 × 10⁴ Pa in sub-atmospheric pressure DBD [39]. Recently, a method for estimating the average electron density of a plasma column by measuring line-ratio I₁/I₂ through optical emission spectroscopy was suggested, in which I₁ and I₂ are the intensities of emission of lines 738.4 nm and 763.5 nm respectively [40]. For simplicity, the line-ratio method is used to determine the electron density in this work. Figure 11 shows the emission spectra of plasma columns in each PPC in the range of 730–770 nm. Figure 11(a) shows the emission spectra of square Ⅰ, Ⅱ and Ⅲ PPCs. Emission spectra of square superlattice Ⅰ PPC are shown in figure 11(b), where L, D(S) and D(C) are denoted in figure 6(a). Emission spectra of square superlattice Ⅱ PPC are shown in figure 11(c), where L and D stand for bright thick spots and thin spots respectively. It is worth pointing out that the electron densities of L₁ (discharging at the rising edge of the applied voltage) and L₂ (discharging at the falling edge) are averaged in measurements because their emission spectra cannot be measured separately. The results of the line intensities and the average electron density in above PPCs are listed in table 1. It reveals that the electron density of different plasma column changes from 1.6 × 10¹⁵ to 3.00 × 10¹⁵ cm⁻³, which is consistent with the electron measurement results in DBD by using Stark broadening and a shift of argon line 696.54 nm [37–39].

Figure  11.  Emission spectra of single plasma column of PPCs in the range of 730–770 nm. The intensities of lines 738.4 nm and 763.5 nm are used to estimate the average electron density of different kinds of plasma columns. (a) Emission spectra of square Ⅰ, square Ⅱ and square Ⅲ PPCs. (b) Emission spectra of square superlattice Ⅰ PPC, L, D(S) and D(C) are denoted in figure 6(a). (c) Emission spectra of square superlattice Ⅱ PPC, L and D are denoted in figure 9(a).

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Table  1.  Electron density for different plasma columns in PPCs.

		I738.4nm	I763.5nm	I738.4/I763.5nm	Electron density (1015 cm−3)
Sq Ⅰ		400	1739	0.230	1.86
Sq Ⅱ		1178	5013	0.2349	2.01
Sq-super Ⅰ	L	2969	13364	0.2222	1.60
	D(C)	1143	4779	0.2392	2.10
	D(S)	1522	6678	0.2279	1.70
Sq-super Ⅱ	L	3805	16506	0.2305	1.90
	D	2879	12258	0.2348	2.00
Sq Ⅲ		2361	9468	0.2493	3.00

 | Show Table

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3.3   Simulation of the band diagrams of PPCs

The schematic views of the square lattice PC and three types of square lattice PPCs with their basic units, and their corresponding irreducible Brillouin zones are shown in figures 12 and 13 respectively. The PC is a 2D periodic structure of alternating gas and glass. The PPC is composed of plasma columns immersed in gas with ε_gas = 1, and glass with ε_glass = 4. For the plasma, its dielectric constant is varied with frequency and written as:

Figure  12.  Schematics of a square lattice PC (a) and its corresponding irreducible Brillouin zone (shaded triangle) (b). Here r is the radius of the gas columns, a₁, a₂ denote the base vectors of the lattice; ε_glass and ε_gas represent the dielectric constant of glass and gas, respectively. The areas enclosed by the black square in the middle of (a) indicate the basic unit of the PC. In (b), the triangular region surrounded by Γ-X-M-Γ is the irreducible Brillouin zone which is utilized for calculating of the dispersion relations.

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Figure  13.  Schematics of basic units of different square lattice PPCs and their corresponding irreducible Brillouin zone. (a) Basic unit of the first type of square lattice PPC. (b) Basic unit of the second type of square lattice PPC. In (c), the area enclosed by the black square at the middle is the basic unit of the third type of square lattice PPC. In the following calculation of PPCs, the corresponding basic unit will be specified. In (d), the triangular region surrounded by M-Γ-X-M is the irreducible brillouin zone. Here r_p is the radius of the plasma column, ε_plasma, ε_glass and ε_gas represent the dielectric constant of plasma, glass, and gas respectively.

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where ν_m, ω_pe, and ω are the electron collision frequency, the plasma frequency, and the frequency of the incident electromagnetic wave, respectively. $\omega_{\mathrm{pe}}=\left(e^2 n_{\mathrm{e}} / \varepsilon_0 m\right)^{1 / 2}$, where n_e of the only variable is the electron density, which is estimated by measuring line-ratio I₁/I₂ through optical emission spectroscopy, which is consistent with that by using Stark broadening and the shift of argon spectral line, as shown in table 1. As stated above, the electron density can be estimated by optical spectrum methods including the Stark broadening method and line-ratio method [37–40]. The electron collision frequency ν_m = 29 GHz, which is calculated by the empirical formula [1].

The five types of PPCs are 2D structures alternating of three materials with dielectric constants ε_gas, ε_glass and ε_plasma. As shown above, their basic unit structure, lattice constant, electron density, and radius of plasma column change with the applied voltage. In other words, the PPCs in this work can be tunable by the applied voltage through self-organization of filaments although a square lattice PC device is inserted into the uniform gap.

PPCs with different structures are simulated by COMSOL Multiphysics software which is based on finite element method. According to Bloch's theorem, the eigenvalue frequencies corresponding to different wave vectors k are calculated, which are along the boundaries of irreducible Brillouin zone Γ-X-M-Γ of PC and M-Γ-X-M of PPCs. Through COMSOL's built-in parametric sweep scan the coefficient k, a series of eigenvalue frequencies corresponding to k can be obtained, and the band diagrams of EM waves of different plasma structures can be obtained through data processing [3, 41].

Figure 14 shows the band diagrams of the transverse-magnetic (TM) mode for the PC device. One unidirectional bandgap locates at the position of 37.94–42.53 GHz in the Γ-X direction. Obviously, there is no omnidirectional band gap.

Figure  14.  Band diagram of the TM mode for square PC, the lattice constant a = 2 mm, the radius r = 0.5 mm, whose basic unit diagram is shown in figure 12(a). The shadow strips indicate the positions of the band gaps.

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Figures 15(a)–(d) illustrate the band diagrams of the TM mode for the square Ⅰ, Ⅱ, and Ⅲ PPCs. Figure 15(a) shows the dispersion relation of square Ⅰ PPC. It is found that three unidirectional band gaps locate at the positions of 41.09–44.31 GHz and 61.67–64.11 GHz in the X-M direction, 59.87–62.12 GHz in the M-Γ direction, and an omnidirectional band gap appears from 29.49 to 34.38 GHz. In square Ⅱ PPC, the parameters of the two sublattices A and B are the same (in figure 4) although they discharge in two current pulses. So only the dispersion relation of one sublattice PPC is calculated and shown in figure 15(b). It is found that three unidirectional band gaps locate at the position of 59.91–62.15 GHz in the M-Γ direction, 41.12–44.34 GHz, and 61.69–64.12 GHz in the X-M direction, respectively, and an omnidirectional band gap occurs from 29.50 to 34.43 GHz, whose central frequency and gap width are slightly higher (less than 0.1%) than those of square Ⅰ PPC (from 29.49 to 34.38 GHz) respectively. It indicates that the band gaps of two lattice PPCs with the same parameters, except the electron density, are almost the same as if their electron densities are of the same order. In addition, although the electron density of the plasma column is dynamic in a short time, the effect on the energy band is negligible. Figures 15(c) and (d) show the band diagrams of the TM mode for the square Ⅲ PPC. According to figure 5, all plasma columns in the square Ⅲ PPC discharge twice, one of which at the rising edge of the applied voltage and the other at the falling edge. Figure 15(c) shows the dispersion relation of the sublattice in the square Ⅲ PPC discharging at the rising edge of the applied voltage. One can see that an omnidirectional band gap is produced from 53.51 to 57.46 GHz and two unidirectional band gaps locate at the positions of 76.75–82.30 GHz and 87.19–95.20 GHz in the Γ-X direction. Figure 15(d) shows the dispersion relation of the sublattice in the square Ⅲ PPC discharging at the falling edge of the applied voltage. One can see that one unidirectional band gap locates at the position of 52.36–56.34 GHz in the M-Γ direction, and three unidirectional band gaps locate at the positions of 39.15–52.52 GHz, 76.02–80.13 GHz and 82.23–93.00 GHz in the Γ-X direction. With the radius of plasma column r decreasing from 0.3 mm in figure 15(c) to 0.2 mm in figure 15(d), the central frequency and the width of the two unidirectional band gaps in the Γ-X direction decrease, meanwhile the omnidirectional band gap disappears and two unidirectional band gaps in the M-Γ and Γ-X directions arise instead.

Figure  15.  Band diagrams of the TM mode for square Ⅰ, Ⅱ, Ⅲ PPCs. (a) Band diagram of the square Ⅰ PPC, whose basic unit diagram is shown in figure 13(a), the electron density n_e = 1.86 × 10¹⁵ cm⁻³, the lattice constant a = $2 \sqrt{2}$ mm, the radius of the plasma column r = 0.2 mm. (b) Band diagram of the sublattice A or B in the square Ⅱ PPC in figure 4, whose basic unit diagram is shown in figure 13(a), n_e = 2.01 × 10¹⁵ cm⁻³, a = $2 \sqrt{2}$ mm, r = 0.2 mm. (c) The band diagram of the sublattice L₂ in the square Ⅲ PPC, whose basic unit diagram is shown in figure 13(c), n_e = 3.00 × 10¹⁵ cm⁻³, a = 2 mm, r = 0.3 mm. (d) Band diagram of the sublattice L₁ in the square Ⅲ PPC, whose basic unit diagram is shown in figure 13(c), a = 2 mm, r = 0.2 mm.

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The experimental calculations are consistent with Sakai's discovery that the transmitted microwaves at 70–75 GHz show a change of energy flow direction [6].

Figure 16 shows the band diagrams of the TM mode for the square superlattice Ⅰ PPC which is composed of sublattice L₁, sublattice D(C)+D(S), and sublattice L₂. Figure 16(a) shows the dispersion relation of the sublattice L₂. One can see that two unidirectional band gaps locate at the position of 24.76–27.02 GHz in M-Γ direction, and 18.93–24.29 GHz in the Γ-X direction. Figure 16(b) shows the dispersion relation of the sublattice D(C)+D(S). It can be seen that two unidirectional band gaps locate at the positions of 47.02–48.77 GHz in the M-Γ direction, and 43.38–48.62 GHz in the X-M direction. An omnidirectional band gap locates at the position of 26.76–59.44 GHz. Figure 16(c) shows the dispersion relation of the sublattice L₁. It can be seen that one unidirectional band gap is produced at the position of 24.58–26.78 GHz in the M-Γ, and another unidirectional band gap in the Γ-X direction 18.89–22.97 GHz.

Figure  16.  Band diagrams of the TM mode for square superlattice Ⅰ PPC. (a) Band diagram of the sublattice L₂, whose basic unit diagram is shown in figure 13(c), n_e = 1.60 × 10¹⁵ cm⁻³, a = 4 mm, r = 0.3 mm. (b) Band diagram of the sublattice D(C)+D(S), whose basic unit diagram is shown in figure 13(b), the n_e = 1.90 × 10¹⁵ cm⁻³ is the average of sublattice D(C) and D(S), a = 4 mm, r = 0.2 mm. (C) Band diagram of the sublattice L₁, n_e = 1.60 × 10¹⁵ cm⁻³, a = 4 mm, r = 0.2 mm.

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Figure 17 shows the band diagrams of the TM mode for the square superlattice Ⅱ PPC which is composed of sublattice L₁, sublattice D, and sublattice L₂. Figure 17(a) presents the band diagram of the sublattice L₂, one can find an omnidirectional band gap from 30.32 to 36.67 GHz and five unidirectional band gaps, which locate at the positions of 53.22–55.34 GHz and 60.30–64.00 GHz in the M-Γ direction, 41.79–45.14 GHz, 57.29–59.01 GHz and 62.91–65.04 GHz in the X-M direction respectively. Figure 17(b) presents the band diagram of the sublattice D. There are three unidirectional band gaps, which locate at the positions of 59.91–62.14 GHz in M-Γ direction and 41.12–44.33 GHz and 61.69–64.12 GHz in X-M direction, and an omnidirectional band gap is produced from 29.44 to 34.46 GHz. Figure 17(c) presents the band diagram of the sublattice L₁, one can see that a unidirectional band gap locates at the position of 59.91–62.13 GHz in the M-Γ direction, and two unidirectional band gaps locate at the positions of 41.10–44.32 GHz and 61.68–64.11 GHz in the X-M direction, and an omnidirectional band gap is formed from 29.41 to 34.40 GHz.

Figure  17.  Band diagrams of the TM mode for square superlattice Ⅱ PPC, U = 7.28 kV, the schematic diagram of the basic unit is shown in figure 13(a). (a) Band diagram of the sublattice L₂, n_e = 1.90 × 10¹⁵ cm⁻³, a = $2 \sqrt{2}$ mm, r = 0.3 mm. (b) Band diagram of the sublattice D, n_e = 2.00 × 10¹⁵ cm⁻³, a = $2 \sqrt{2}$ mm, r = 0.2 mm. (c) Band diagram of the sublattice L₁, n_e = 1.90 × 10¹⁵ cm⁻³, a = $2 \sqrt{2}$ mm, r = 0.2 mm.

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Figure 18 shows the variations in the width and position of bandgaps of PC and five types of PPCs (square Ⅰ, square Ⅱ, square superlattice Ⅰ, square superlattice Ⅱ, and square Ⅲ PPC) with the applied voltage in the directions of M-Γ, Γ-X, and X-M respectively. Figure 18(a) shows the variations in bandgap-width and position with the applied voltage in the direction of M-Γ. There is no band gap of PC, however, the band gaps of PPCs in this direction cover from 24 to 64 GHz excluding two narrow bands in the ranges of 36.67–47.02 GHz and 48.77–52.36 GHz. Therefore, the band gap in this direction can be tuned from 24 to 64 GHz excluding two narrow bands by changing the applied voltage. It should have many potential applications in controlling the propagation of electromagnetic waves. For example, the band filter for the electromagnetic wave in 52–58 GHz in this direction can be realized by adjusting the applied voltage to 7.84 kV for forming square Ⅲ PPC, i.e. the electromagnetic wave in the 52–58 GHz in this direction cannot pass through when the applied voltage is 7.84 kV. Another potential application is large frequency range filtering. Figure 18(a) illustrates that the electromagnetic wave with a frequency in three bands denoted by yellow shadow zones will disappear and only that in two narrow bands in the ranges of 36–47 GHz and 49–52 GHz can pass through if the applied voltage is changed from 0 to 8 kV back and forth. Figure 18(b) gives the variations of bandgaps-width and position with the applied voltage in direction of Γ-X. It is found that there is a band gap of PC located at 37.94–42.53 GHz. Five PPCs produce Eight band gaps, resulting in the tunable range of band gaps is increased to 18–96 GHz, except the range of 57.46–76.20 GHz in direction of Γ-X. Figure 18(c) shows the variations of bandgaps-width and position with the applied voltage in direction of X-M. There is no band gap of PC in this direction. Due to three band gaps of square Ⅰ PPC and square Ⅱ PPC, two band gaps of square superlattice Ⅰ PPC, four band gaps of square superlattice Ⅱ PPC, and one band gap of square Ⅲ PPC, the band gaps of PPCs can be tuned from 26 to 65 GHz in direction of X-M, excluding the ranges of 36.67–41.09 GHz and 48.62–53.51 GHz. In summary, the tunable ranges of the band gaps of PC and five PPCs in directions of M-Γ, Γ-X, and X-M are 24–64 GHz (except 36.67–47.02 GHz and 48.77–52.36 GHz), 18–96 GHz (except 57.46–76.02 GHz), and 26–65 GHz (except 36.67–41.09 GHz and 48.62–53.51 GHz) respectively.

Figure  18.  Variations in the position and sizes of bandgaps with the applied voltage in the directions of M-Γ (a), Γ-X (b) and in the direction of X-M (c). (PC without plasma, U = 0 kV). The red words Sq Ⅰ, Sq Ⅱ, Sq-super Ⅰ, Sq-super Ⅱ, Sq Ⅲ represent square Ⅰ, square Ⅱ, square superlattice Ⅰ, square superlattice Ⅱ, square Ⅲ PPC, respectively. Yellow shadow zones indicate the range of bandgaps.

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Figure 19 shows the variations in the size and position of omnidirectional bandgaps of five types of PPCs. It can be clearly seen that there is no omnidirectional band gap in PC, while it arises after plasma columns filling into PC. The size and the position of the bandgap can be tuned by the applied voltage because the spatio-temporal structure of plasma columns formed by the self-organization process is dependent upon the applied voltage. Comparing with the omnidirectional bandgap of square Ⅰ and Ⅱ PPC, the center frequency of the bandgap of the square superlattice Ⅰ PPC shifts downward and the width decreases, while the omnidirectional bandgap of square superlattice Ⅱ PPC changes to the higher frequency and wider width. The bandgaps of these four types of PPCs are in the ranges of 29.49–34.38 GHz, 29.50–34.43 GHz, 26.76–29.44 GHz, and 29.41–36.67 GHz respectively. With the applied voltage increasing, the center frequency of the bandgap of square Ⅲ PPC jumps to 55.49 GHz and the range is 53.51–57.46 GHz. Similar to the description of figure 18(a), these tunable omnidirectional band gaps should have many potential applications such as band filter, large frequency range filtering and so on. It is believed that these tunable omnidirectional band gaps in the range from 26 to 57 GHz except the range of 36.67–53.51 GHz greatly advance the tunable and controllable properties in controlling the propagation of electromagnetic waves in the mm-wave region.

Figure  19.  Variations of the position and sizes of omnidirectional bandgaps with the applied voltage.

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In summary, five types of PPCs with different structures and parameters are obtained through the combination of photonic crystal and dielectric barrier discharge patterns. Since there is no measurement device in our lab at present, direct comparisons between the experimental and simulation results cannot be done. The simulation result is credible as the order of the bandgap is the same as that of a similar plasma lattice which was verified by Sakai et al [6]. The experimental results have achieved a controllable and tunable band gap in the range of 18–96 GHz, which can be widely used to control the propagation of electromagnetic waves in the mm-wave region. As is well known and shown above, the band gap is immutable once the PC is fabricated. Its tunable frequency range is narrow and there is no omnidirectional bandgap. However, the PPCs can produce many unidirectional bandgaps and omnidirectional band gaps in different frequency bands only by changing the applied voltage. These PPCs are tunable mm-wave resonators in essence. They can be used as a filter device in different frequency ranges of electromagnetic waves. As metamaterials, they have potential applications in multichannel communications, millimeter-wave spectroscopy, fundamental studies of multiple, and coupled resonators. The method suggested in this experiment can generate tunable and controllable PPCs, overcoming not only the limits of the fixed structure and invariable lattice constant of PPC produced by the fixed array electrodes method but also the uncontrollable symmetry transitions of PPC formed by the plasma columns self-organization method.

4.   Conclusion

In this work, we report five types of patterns with square symmetry, including three novel types obtained by inserting a specially designed grid PC device into a DBD system. They are studied using optical equipment including an ICCD camera, photomultiplier tubes, and a spectrograph. The three novel types of patterns are a square pattern with one structure, a square superlattice Ⅰ pattern with four sublattices and a (1/4) K_grid (K_grid is the basic wave vector of the grid), and another square pattern with a complex inversion discharge sequence, respectively. Some new physical phenomena occur in these patterns. From the FFT of five patterns, the basic wave vector is $1 / \sqrt{2}$, 1, 1/4, $1 / \sqrt{2}$, and 1 times the K_grid, indicating many types of resonances of wave vectors. Different from the uniform gap DBD, the new phenomena such as square Ⅰ with a single structure, and the square Ⅲ pattern with the complex inversion discharge sequence of A₁-B₁-B₂-A₂-B₁-A₁-A₂-B₂, arise from the nonlinear resonance of wave vectors (one of which is a grid basic wave vector) and the wall charge field distribution under a grid modulation. In a word, these results enrich the pattern types as well as controlling pattern symmetry by adding a wave vector in DBD, which can greatly promote the development of pattern dynamics.

From the viewpoint of PC application, this device realizes the conversion from PC to five types of PPCs with symmetry controllability and bandgap tunability. The average electron density of plasma columns in different PPCs is estimated by measuring line ratio I₁/I₂ through optical emission spectroscopy. It is found that the electron density in filaments is approximately in the order of 10¹⁵ cm⁻³. The electron density, the lattice constant, the basic unit of different PPCs, and the radius of the plasma column can be tuned by altering the applied voltage on purpose, which cannot be easily changed in the PC once it is fabricated. The band diagrams of the PC and PPCs under a transverse-magnetic wave have been simulated. The results show that the large number of band gaps, including omnidirectional band gaps, and a wide tunable range, can be obtained by changing the applied voltage. The positions of unidirectional band gaps vary from 18 to 96 GHz, and those of omnidirectional band gaps change from 26 to 57 GHz. Comparing with the original PC with only one unidirectional band gap, the five types of PPCs have tunable and omnidirectional band gaps, which greatly advance the tunable and controllable properties and expand the range in controlling the propagation of electromagnetic waves in the mm-wave region. They can be used as a filter device in different frequency ranges of electromagnetic waves. As metamaterials, they have potential applications in multichannel communications, millimeter-wave spectroscopy, and fundamental studies of multiple, and coupled, resonators. The method suggested in this experiment can generate tunable and controllable PPCs, overcoming not only the limits of fixed structure and the invariable lattice constant of PPC produced by the fixed array electrodes method, but also the uncontrollable symmetry transitions of PPC formed by the plasma columns self-organization method.

The experiment results not only enrich the pattern types in the DBD system and promote the development of pattern dynamics, but also propose a flexible method for the fabrication of tunable PPCs with different structures and parameters. There are many potential applications, such as band filters, large frequency range filtering and so on, in controlling the propagation of electromagnetic waves in the mm-wave region.