Study on the transmission characteristics and band structure of 2D and 3D plasma photonic crystals

Yichao Liang; Yichao Liang; Zhuqing Liang; Zhen Liu; Peng Jun; Peng Jun; Dianqing Qiu

doi:10.1364/OE.460396

1. Introduction

Plasma Photonic Crystal (PPC) has been widely concerned by scholars from all over the world since it was proposed in 2004 [1]. Compared with the traditional dielectric photonic crystal (PhC), the photonic bandgap (PBG) of PPC is tunable because the plasma is a dispersive medium and its dielectric constant can be adjusted by electron density. Moreover, the plasma electron density can be adjusted by an external electrical signal, so the adjustment of the PPC bandgap is also easy to achieve. In addition, the electromagnetic properties of the PPC can be changed by the magnetic field because the plasma produces magneto-optical effects under the action of the external magnetic field. At present, the research on PPC is flourishing in various countries. In the future, PPC is expected to be applied in the fields of communication, microwave device manufacturing, and military target stealth.

Sakai [2–4] et al. obtained 2D PPC by using atmospheric pressure gas discharge and conducted in-depth research on its transmission characteristics and band structure. The experimental results show that there is a stable PBG in the PPC, and the PBG can be adjusted by changing the discharge voltage. Wang, Dong [5,6], et al. constructed spatiotemporally tunable PPCs using grid-like DBD discharges, studied the transient sublattices of PPCs under continuous discharge pulses, and gave their temporal evolutions. Sun et al [7]. fabricated a 3D PPC, which was based on PDMS material through stereolithography, and etched micro-channels in the material as discharge channels to generate plasma. The PPC operates in the 120-170 GHz range, and the dynamic properties of the PPC are achieved through the control of the plasma. PPC has been extensively studied by Wang, Cappeli, et al. They constructed 2D [8,9] and 3D PPCs [10] using glow discharge tubes and experimented with their transmission characteristics. The results show that PPC can achieve stable and tunable PBG in the S to X band. In Ref. [11], they constructed a double negative dielectric device composed of double split ring resonators array and plasma discharge tubes, and in Ref. [12], they constructed a reconfigurable plasma-dielectric hybrid photonic crystal using plasma discharge tubes and silicon nitride spheres.

Since rare gas-mercury glow discharge plasma is easy to obtain and control, its properties are relatively stable. Therefore, most of the PPCs in the published literature use rare gas-mercury glow discharge tubes as the basic PPC unit [13–15].

The study of rare gas-mercury glow discharge started at the beginning of the 20th century. Due to the limitation of experimental data and computational power, a large number of empirical formulas were used in the early literature for calculations. Since the 1970s, scholars use the collisional radiative model (CRM) [16–20] to describe the low-pressure rare gas-mercury glow discharge process. Although CRM is a relatively simple model, the computational results of CRM are in good agreement with experimental results and have achieved great success in engineering. On the experimental side, the diagnostic methods of low-temperature plasma parameters have also been improved, and methods such as the Langmuir probe method, Optical Emission Spectrometry (OES), and microwave diagnosis have been widely used. Each of these methods has its advantages, but microwave diagnosis is suitable for rapid diagnosis of low-temperature plasma electron density because of its low interference with the plasma and the simplicity of the experimental method.

The bandgap of PPC may have different generation mechanisms. It is not possible to accurately analyze the type of bandgap simply by observing the transmission spectra of PPC. It is necessary to accurately calculate the band structure of PPC [21,22]. Zhang [23–25], Qi [26] et al. conducted exhaustive studies of the Plane Wave Expansion (PWE) method. Chaudhari et al. [27] calculated the band structure of PPC using the Finite Differential Time Domain (FDTD) method. Fietz [28] calculated the band structure of 3D PPC using the finite element method (FEM). Zhang [29] studied the influence of the nonlinear effect of plasma on the transmission spectra of one-dimensional PPC. These calculation methods have their advantages and disadvantages, so it is necessary to combine the calculation results of multiple methods.

In this paper, 2D and 3D PPC are constructed based on the low-pressure argon-mercury glow discharge plasma tubes. To analyze the change of plasma average electron density with discharge current, the discharge process was calculated using the CRM model, and the average electron density in the tube was diagnosed by microwave diagnosis. Based on the experimental and computational results of average electron density, a layered model of the discharge tube is established. The transmission spectra of PPC are studied by simulation and measurement. To analyze the source of bandgap in the transmission spectra, the band structure of PPC is calculated using the Symplectic Finite Differential Time Domain (SFDTD) and PWE method.

2. Mathematical model and experiment diagnosis of positive column of discharge tube

2.1 Mathematical model of positive column region

In this paper, the 8 energy levels CRM of the Hg atom is used to model the positive column. The model has the following assumptions:

(1) Due to the high excitation and ionization energy of Ar atoms, only Hg atoms are excited;
(2) The positive column is long enough, and the plasma is evenly distributed along the axial direction;
(3) The plasma satisfies the quasi-neutral condition and is isotropic;
(4) It is assumed that the electron energy distribution function (EEDF) is Maxwellian distribution.
(5) According to Schottky's theory [30], the radial electron density distribution conforms to the first kind of zero-order Bessel function;

A simplified Hg atomic energy level diagram is shown in Fig. 1. The arrow lines indicate the process of Hg atoms located at higher energy levels transiting to lower energy levels and emitting photons.

Fig. 1. Simplified Hg atomic energy level diagram (level heights not drawn to scale)

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Rate equations for electron density n_e and Hg atomic density n_i (i = g, p, q, r, s, t, u) for each state, as well as electron temperature T_e and gas temperature T_g [16–19,31]:

(1)$$\begin{aligned} \frac{d}{{dt}}{n_e} &= \sum\limits_i {{S_i}{n_i}{n_e}} + {K_{rr}}{n_r}{n_r} + {K_{pq}}{n_p}{n_q} + {K_{ps}}{n_q}{n_s} + {K_{qs}}{n_q}{n_s} + {K_{qr}}{n_q}{n_r}\\ &\textrm{ } + {K_{ss}}{n_s}{n_s} + {K_{rs}}{n_r}{n_s} - {D_a}{\nabla ^2}{n_e} \end{aligned}$$

(2)$$\frac{d}{{dt}}{n_p} = \sum\limits_i {{C_{ip}}{n_i}{n_e}} - \sum\limits_i {{C_{pi}}{n_p}{n_e}} - {S_p}{n_p}{n_e} - {K_{pq}}{n_p}{n_q} - {K_{ps}}{n_p}{n_s} + \sum\limits_{n > p} {{A_{np}}{n_n}} - \sum\limits_{n < p} {{A_{pn}}{n_p}}$$

(3)$$\begin{aligned} \frac{d}{{dt}}{n_q} &= \sum\limits_i {{C_{iq}}{n_i}{n_e}} - \sum\limits_i {{C_{qi}}{n_q}{n_e}} - {S_q}{n_q}{n_e} - {K_{pq}}{n_p}{n_q} - {K_{qs}}{n_q}{n_s} - {K_{qr}}{n_q}{n_r}\\ &\textrm{ } + \sum\limits_{n > q} {{A_{nq}}{n_n}} - \sum\limits_{n < q} {{A_{qn}}{n_q}} \end{aligned}$$

(4)$$\frac{d}{{dt}}{n_r} = \sum\limits_i {{C_{ir}}{n_i}{n_e}} - \sum\limits_i {{C_{ri}}{n_p}{n_e}} - {S_r}{n_r}{n_e} - {K_{rr}}{n_r}{n_r} + \sum\limits_{n > r} {{A_{nr}}{n_n}} - \sum\limits_{n < r} {{A_{rn}}{n_r}}$$

(5)$$\begin{aligned} \frac{d}{{dt}}{n_s} &= \sum\limits_i {{C_{is}}{n_i}{n_e}} - \sum\limits_i {{C_{si}}{n_s}{n_e}} - {S_s}{n_s}{n_e} - {K_{ss}}{n_s}{n_s} - {K_{ps}}{n_p}{n_s} - {K_{rs}}{n_r}{n_s}\\ &\textrm{ } + \sum\limits_{n > s} {{A_{ns}}{n_n}} - \sum\limits_{n < s} {{A_{sn}}{n_s}} \end{aligned}$$

(6)$$\frac{d}{{dt}}{n_t} = \sum\limits_i {{C_{it}}{n_i}{n_e}} - \sum\limits_i {{C_{ti}}{n_t}{n_e}} - {S_t}{n_t}{n_e} + \sum\limits_{n > t} {{A_{nt}}{n_n}} - \sum\limits_{n < t} {{A_{tn}}{n_t}}$$

(7)$$\frac{d}{{dt}}{n_u} = \sum\limits_i {{C_{iu}}{n_i}{n_e}} - \sum\limits_i {{C_{ui}}{n_u}{n_e}} - {S_u}{n_u}{n_e} + \sum\limits_{n > u} {{A_{nu}}{n_n}} - \sum\limits_{n < u} {{A_{un}}{n_u}}$$

(8)$$\frac{3}{2}k\frac{d}{{dt}}{T_e} = e{\mu _e}{E^2} - v\frac{{3{m_e}}}{{{M_a}}}k({T_e} - {T_g}) - \sum\limits_i {(\frac{5}{2}k{T_e} + e{U_{ie}}){S_i}{n_i}} - \sum\limits_{i \ne j} {e{U_{ij}}{C_{ij}}{n_i}}$$

(9)$$\frac{3}{2}k{n_{Ar}}\frac{d}{{dt}}{T_g} = v\frac{{3{m_e}}}{{{M_a}}}k({T_e} - {T_g}){n_e} - \frac{4}{{{R^2}}}({T_g} - {T_{wall}})$$

In Eq. (8), the electric field is calculated by Ohm's law:

(10)$$J = \frac{I}{{0.43\pi {\textrm{R}^2}}} = e{n_e}{\mu _e}E$$

According to Schottky's theory [30], the last term on the right-hand side of Eq. (1) is:

(11)$${D_a}{\nabla ^2}{n_e} = \frac{{ - 5.784{D_a}{n_e}}}{{{R^2}}}$$

The collision frequency v can be estimated by referring to the empirical formula in Ref. [16]:

(12)$$v = 1.5 \times {10^9}{P_{Ar}}{\left( {\frac{{k{T_e}}}{e}} \right)^{1.65}}$$

In Eq. (1) to Eq. (12), k is the Boltzmann constant, e is the electron charge, m_e is the electron mass, P_Ar is the argon pressure in torr. M_a is the average atomic mass of the argon-mercury mixed gas. U_ij is the excitation or ionization energy (j = ion) between the i state and the j state. C_ij represents the excitation/de-excitation rate coefficient between states, and S_i represents the ionization coefficient of each state. The calculation of excitation/de-excitation rate coefficients and ionization coefficients can be found in [19,32]. T_g is the gas temperature. D_a is the bipolar diffusion coefficient, μ_e is the electron mobility, and the calculation method can be found in Ref. [19]. K_ij (i,j = p,q,r,s) is the rate coefficients for heavy-particle collisions [33,34]. A_ij is the transition probability. When the transition occurs between atoms in the excited state, the transition probability is taken from the Ref. [20]; and when the transition occurs between the excited state and the ground state, that is, resonance radiation, the transition probability A_qg and A_sg should be the effective transition probabilities, A_qg* and A_sg*, are calculated in Ref. [18,31]. The cross sections of mercury atoms are derived from the Ref. [20,33,35]. R is the radius of the discharge tube, R = 7.5 mm.

Since the partial pressure of mercury vapor is very small, the pressure in the tube can be considered as the partial pressure of argon, and the partial pressure of argon, P_Ar, is about 1 torr, and the partial pressure of mercury vapor can be calculated by the formula given in Ref. [19]. The number of argon atoms, n_Ar, is calculated from the ideal gas equation of state. The wall temperature of the discharge tube T_wall is 300 K. For comparison, the experimental and CRM results of the average electron density are given in Section 2.2.

CRM is essentially a set of rigid ordinary differential equations (ODEs), which are solved in this paper using Matlab.

2.2 Diagnostic methods and results of electron density in the positive column

In this paper, the plasma electron density is measured by microwave diagnosis, and the experimental principle is shown in Fig. 2.

Fig. 2. Schematic diagram of the principle of microwave diagnosis

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For a uniform plasma with weak collision, low density, and isotropy, assuming that the thickness of the plasma is L₀ when the microwave passes through the plasma, its phase changes compared with the air of the same thickness [36]. The average electron density is calculated from the phase change:

(13)$${n_e} = \frac{{\int_0^{{L_0}} {{n_e}(x)dx} }}{{{L_0}}} = \frac{{2{\varepsilon _0}{m_e}c\omega }}{{{e^2}{L_0}}} \approx 118.6\frac{{f\varDelta \varphi }}{{{L_0}}}$$

In Eq. (12), ε₀ is the vacuum permittivity, f is the frequency of the incident wave, the unit is Hz, Δφ is the phase change before and after the plasma is turned on, the unit is rad, and L₀ is the diameter of the discharge tube, the unit is cm. The unit of n_e is cm^-3. The result of Eq. (12) reflects the average electron density of the plasma.

During the experiment, two antennas were placed opposite each other, two metal plates were used to construct a slit, and the plasma was placed behind the slit. The S21 parameter was measured using the continuous wave (CW) mode of the vector network analyzer (VNA). The phase change of the S21 parameter before and after the plasma was turned on was recorded. The diagnostic result of the average electron density can be obtained by Eq. (12).

The VNA used in the experiment is Agilent N5224A, the antenna is a pair of standard gain horn antennas, and the frequency range is 26.5-40GHz. The discharge current was measured using a Tektronix TCP0150 current probe. Figure 3 shows the comparison between the results of CRM and microwave diagnosis.

Fig. 3. Comparison of CRM and experimental results of the average electron density

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Both experimental and CRM results show that the average electron density is on the order of 10¹⁷/m³ within a given current variation range. And as the discharge current increases, the average electron density in the tube increases continuously. The reason is that the discharge current increases and the electrode temperature increases. According to the principle of thermionic emission, the number of electrons escaping from the electrode increases accordingly, and under the action of the panning effect, the average electron density in the discharge tube also increases. In addition, the increased current may also cause ionization of argon gas, thereby increasing the average electron density.

Figure 3 shows that the experimental and CRM results are in good agreement, proving the effectiveness of CRM. However, CRM is a 0D model and does not take into account the distribution of plasma in space. In fact, the distribution of plasma in the positive column is not completely uniform, which may lead to discrepancies between the experimental and calculated results. In addition, in the case of normal glow discharge, the plasma in the tube may not reach the local thermodynamic equilibrium (LTE) state, and its EEDF will deviate from the Maxwellian distribution in the high-energy region, which will also cause errors in the calculation.

3. Transmission characteristics and band structure of PPC

3.1 Modeling of PPC unit

Since most of the area in the discharge tube is in the positive column, and the cathode area and the anode area occupy a small area, it is assumed that the discharge tube is uniformly distributed along the axial direction during modeling.

It can be seen from Fig. 3 that when the discharge current I is 36.9mA, the CRM and experimental results of average electron density are between 3.6 × 10¹⁶/m³ and 9.6 × 10¹⁶/m³, and when the discharge current I is 158.3 mA, the CRM and experimental results of average electron density range from 3.04 × 10¹⁷/m³ to 4.24 × 10¹⁷/m³. Therefore, the average electron density of the established model should be within the above range.

Schottky's theory [30] points out that the distribution of electron density along the radial direction conforms to the zeroth-order Bessel function of the first kind. Thus, a discharge tube model with multi-layer plasma can be built. The higher the number of layers, the closer the plasma distribution is to the theoretical model. However, too many layers will lead to an increase in the amount of calculation. To simplify the calculation, the model established in this paper is shown in Fig. 4.

Fig. 4. Layered model of discharge tube (not drawn according to actual size)

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In practice, the electron density is the largest at the center of the discharge tube, decreases in the radial direction, and is zero at the wall. To simplify the simulation, an equivalent model of the discharge tube is established. In this section, models are established for two working conditions with discharge currents of 36.9mA and 158.3mA, respectively, as shown in Fig. 4.

The dielectric constant of the plasma uses the Drude model:

(14)$$ \varepsilon_p=1-\frac{\omega_p^2}{\omega(\omega+j v)} $$

Where

(15)$${\omega _p} = \sqrt {\frac{{{e^2}{n_e}}}{{{\varepsilon _0}{m_e}}}}$$

In Eq. (13), ω_p is the plasma angular frequency, v is the collision frequency, ω is the incident wave angular frequency, and j is the imaginary unit.

The geometric dimensions of the model in Fig. 4: R1 = 8 mm, R2 = 7.5 mm, R3 = 5.625 mm, R4 = 3.5 mm, R5 = 1.875 mm.

When I = 36.9 mA, set the electron density of Plasma1 n_e1= 2.92 × 10¹⁷/m³, the electron density of Plasma2 n_e2= 2.14 × 10¹⁷/m³, the electron density of Plasma3 n_e3= 1.08 × 10¹⁷/m³, the average electron density is 0.92 × 10¹⁶/m³. When I = 158.3 mA, set the electron density of Plasma1 n_e1= 1.09 × 10¹⁸/m³, the electron density of Plasma2 n_e2= 8.04 × 10¹⁷/m³, the electron density of Plasma3 n_e3= 4.06 × 10¹⁷/m³, the average electron density is 3.46× 10¹⁷/m³.

Results of CRM indicate that the collision frequency v is about 4 GHz. Equation (12) mainly considers the collision between electrons and Ar atoms, but in practice, electrons also collide with excited argon atoms and Hg atoms, so the result of Eq. (12) may be slightly conservative. If the dielectric constant of the plasma is the Drude model and the electron density is determined, the collision frequency only affects the loss of electromagnetic waves in the plasma. Therefore, according to the experimental results, the collision frequency was adjusted to 5 GHz.

The relative permittivity of quartz glass is 3.75.

The experimental and calculated transmission spectra are obtained as:

(16)$$ \text { Transmission }=\mathrm{S}_{21on}-\mathrm{S}_{21off} $$

In Eq. (14), “Transmission” represents the transmission spectra, “S21_on”represents the S21 parameter when the plasma is on, and “S21_off” represents the S21 parameter when the plasma is off, and the unit of S21 is dB.

3.2 Transmission characteristics and band structure of 2D PPC

In this section, a 2D PPC was constructed using 6 × 6 discharge tubes, and the transmission spectra of the 2D PPC were measured and calculated when the discharge currents were 36.9mA and 158.3mA. The experiments were conducted in a 6m×6m×6m microwave anechoic chamber, and the PPC was placed on low-scattering support. The antenna is a pair of double-ridged broadband antennas, and the frequency ranges from 1GHz to 18GHz. Electromagnetic waves propagate in the negative direction of the Z-axis.

The schematic diagram and photo of the experiment are shown in Fig. 5, where the lattice constant of the photonic crystal a is 25mm. Transmission spectra were calculated using CST studio software. The experimental and simulated results are shown in Fig. 6. In the legend of Fig. 6, “TE” represents TE polarization, “TM” represents TM polarization, “exp” represents experimental results, and “sim” represents calculation results.

Fig. 5. 2D PPC

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Fig. 6. Experimental and simulated transmission spectra of 2D PPC

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Figure 6(a) shows that under TE polarization, a wide and deep bandgap appears around 2.5 GHz, which may be caused by the localized surface plasmon (LSP) [37], LSP has two significant effects: first, electromagnetic waves are strongly absorbed and scattered; second, there is a locally enhanced electric field on the surface of plasma. LSP does not appear under TM polarization, but plasma has a cutoff effect on electromagnetic waves, and electromagnetic waves below the cut-off frequency cannot transmit. The cutoff frequency is about 1 GHz.

Figure 6(b) shows that in the experimental results, multiple bandgaps appear under TE polarization, located at 4.2 GHz, 6.3 GHz, 12.2 GHz, and 14.8 GHz, respectively. It can be seen that as the electron density increases, the plasma distribution in the radial direction is more uneven, and the number of bandgaps increases. Under the TM polarization, the cutoff frequency is about 3 GHz, and a clear bandgap appears around 11.95 GHz and 14.55 GHz. However, the generation of the bandgap cannot be accurately analyzed based on the transmission spectra alone and needs to be analyzed in combination with electric field distribution and band structure.

In Fig. 6, the experimental and simulation results of the transmission spectra are in good agreement. This shows that the established PPC model is valid. The simulation result of the transmission spectra has positive values, and the reason is shown in Eq. (14). At certain frequencies, when the plasma is turned off, the quartz tube will hinder the transmission of electromagnetic waves, resulting in the attenuation of the transmission spectra, while the electromagnetic waves can pass when the plasma is turned on. Positive values appear when the transmission spectra in both cases are subtracted.

The error sources of the experimental and simulation results may include: the plasma tube in the experiment has a limited length, the plasma distribution may be uneven along the axial direction, and the plasma distribution near the electrode is significantly different from the positive column; In the simulation, the plasma tube is considered to be infinitely long, and the plasma is considered to be uniformly distributed along the axial direction. In practice, according to the measured results of Verweij [38], the distribution of electron density along the radial direction will deviate from Schottky’s theoretical solution [30], which may also be one of the sources of differences between experiments and simulations.

In addition, due to the limited number of plasma tubes, the PPC cannot completely cover the antenna aperture under TM polarization, and the low-frequency electromagnetic waves may be diffracted and enter the receiving antenna. Therefore, in Fig. 6, the experimental and simulation results under TM polarization have relatively large errors within 1-3 GHz.

Figure 7 shows the distribution of the electric field amplitude in the PPC at different frequencies with I = 36.9 mA. Figure 7 (a) shows that under TE polarization, when Freq = 1.00 GHz, the electromagnetic wave can propagate in the PPC, and the energy is concentrated in the medium other than the plasma. This indicates that the Fano mode is generated in PPC [39], and in the band structure of PPC, the Fano mode is located below the flat band. When Freq = 2.20 GHz, a locally enhanced electric field appears at the interface between plasma and air, and the electromagnetic wave is strongly attenuated during the propagation process, which is consistent with the phenomenon generated by LSP. When Freq = 11.35 GHz, the electromagnetic wave will have some energy loss during the propagation, which may be related to the scattering of the electromagnetic wave by the periodically distributed quartz medium. 12.48 GHz is in the passband, and the electric field distribution is different from the Fano mode.

Figure 7(b) shows that under TM polarization, when Freq = 1.00 GHz, the electromagnetic wave is cut off by plasma, and the electromagnetic wave cannot propagate in the PPC. When Freq = 11.80 GHz, the propagation of electromagnetic waves is also hindered. Other frequencies belong to the passband, and electromagnetic waves can propagate normally.

Fig. 7. I = 36.9 mA, electric field distribution of 2D PPC

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Figure 8 shows the distribution of the electric field amplitude in the PPC at different frequencies with I = 158.3 mA. Figure 8 (a) shows that the electric field distribution at Freq = 1.00 GHz is similar to the result at the same frequency in Fig. 7(a) for TE polarization, indicating that the Fano mode exists in the passband with a frequency lower than the LSP bandgap. When Freq = 5.26 GHz, a locally enhanced electric field appears at the interface of plasma2 and plasma3, and when Freq = 6.26 GHz, a locally enhanced electric field appears at the interface of plasma1 and plasma2. This indicates that both bandgaps are LSP bandgaps. When Freq = 11.35 GHz, the electric field distribution is similar to that at the same frequency in Fig. 7(a), which indicates that the attenuation of electromagnetic waves at 11.35 GHz should be due to the scattering of electromagnetic waves by the periodically distributed quartz medium.

Fig. 8. I = 158.3 mA, electric field distribution of 2D PPC

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Figure 8(b) shows that with TM polarization, compared with Fig. 7(b), the cutoff effect of plasma on electromagnetic waves is stronger when Freq = 1.00 GHz, which indicates that under TM polarization, the incident frequency is lower than the cutoff frequency, the greater attenuation of the electromagnetic wave. When Freq = 5.26 GHz, it is located in the passband, and the electromagnetic wave transmission will not be disturbed. When Freq = 11.95 GHz and 14.55 GHz, the transmission of electromagnetic waves is hindered. According to the Drude model, the relative permittivity of plasma is close to air at higher frequencies, so electromagnetic waves may be scattered by periodically distributed quartz at this frequency. Moreover, under different polarization modes, the influence of quartz on the propagation of electromagnetic waves is different.

In order to analyze the causes of bandgaps in the transmission spectra, the band structure of 2D PPC was calculated using SFDTD and PWE, respectively, and the results are shown in Fig. 9 to Fig. 10. In Fig. 9 and Fig. 10, Simulation represents the simulation result of the transmission spectra, and Experiment represents the experimental result of the transmission spectra.

Fig. 9. I = 36.9 mA, transmission spectra and band structure of 2D PPC

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Fig. 10. I = 158.3 mA, transmission spectra and band structure of 2D PPC

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Figure 9(a) shows that under TE polarization, the band structure of PPC has a flat band around 2.5 GHz, and the cause for the flat band is LSP, which is also the common feature of metal photonic crystals and PPC under TE polarization. The flat band indicates that the group velocity of the electromagnetic wave propagating in the PPC is 0, and plasma is a lossy medium, so the electromagnetic wave energy will be dissipated by the plasma. On bounded surfaces, LSP can be generated by direct irradiation of electromagnetic waves, which is essentially a resonance phenomenon. The incident electric field exerts a force on the free electrons in the plasma, thus causing the free electrons to be deflected relative to the positive ions, and the Coulomb force between the positive ions and the electrons acts as an effective restoring force to pull the driven deflected electrons toward its equilibrium position, thus causing resonance. Therefore, it can be judged that the bandgap of 2.5 GHz is the LSP bandgap.

Figure 9(b) shows that under TM polarization, electromagnetic waves with frequencies less than 1 GHz are cut off by the plasma. The transmission spectra in Fig. 9 have attenuation around 11.95 GHz and 14.55 GHz, but no obvious bandgap is seen in the band structure. The reason is that at high frequencies, the relative permittivity of the plasma is close to that of air, and only the quartz tube affects the propagation of electromagnetic waves. But the quartz tube is thin, so the attenuation peak seen in the transmission spectra is very sharp and cannot be resolved in the band structure.

Compared with Fig. 9, the band structure in Fig. 10 is more complicated. Multiple LSP band gaps appear in Fig. 10(a). According to the band structure, it can be judged that the band gaps at 4.2 GHz, and 6.3 GHz are LSP band gaps. It can be seen in Fig. 10(b) that with the increase of the average electron density, the cutoff frequency under TM polarization increases. Figure 10 shows that the increase in the average electron density mainly leads to changes in the number and position of the LSP bandgap.

3.3 Transmission characteristics and band structures of 3D PPC

In this section, a 3D PPC was constructed using 6 × 7 discharge tubes, and the transmission spectra of the 3D PPC were measured and calculated. The schematic diagram of the experiment and the experimental photo is shown in Fig. 11, where a = 25mm, the electromagnetic wave propagates along the negative direction of the Z-axis, and the polarization direction is the Y-axis. The experimental and simulation results are shown in Fig. 12. In the legend of Fig. 12, “exp” represents the experimental result, and “sim” represents the simulation result.

Fig. 11. 3D PPC

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Fig. 12. Experimental and calculation results of 3D PPC transmission spectrum

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Figure 12(a) shows that when I = 36.9mA, a deep bandgap appears around 2.4GHz. Since there are some discharge tubes whose axis is perpendicular to the polarization direction of the electromagnetic wave in 3D PPC, the electromagnetic wave will generate LSP on these discharge tubes, resulting in absorption of the electromagnetic wave. Multiple bandgaps appear in Fig. 12(b), which are located around 1 GHz, 4 GHz, 7 GHz, and 13 GHz, respectively. The bandgap around 1 GHz is different from other band gaps, and its characteristics are similar to the cut-off bandgap in Fig. 6(b), indicating that the electromagnetic wave is cut off by the plasma, and the reason is that there are discharge tubes whose axis are parallel to the polarization direction in the 3D PPC. The rest of the bandgaps have similar characteristics to the LSP bandgap, and the exact cause of the bandgap needs to be judged based on the electric field distribution and the band structure.

Figure 13 shows the electric field distribution at different discharge currents and different frequencies. Figure 13(a) shows that at Freq = 1.00 GHz, the electromagnetic wave produces a Fano mode on the discharge tube with axis parallel to the X-axis, while the electric field distribution is similar to the phenomenon under 2D TM polarization on the discharge tube with the axis parallel to the Y-axis. At this time, the electromagnetic wave propagation is affected, but not completely cut off by the plasma. This indicates that in the 3D PPC designed in this paper, the cutoff effect of the plasma on the electromagnetic waves is weakened and the cutoff frequency is shifted to lower frequencies due to the presence of the Fano mode. When Freq = 2.09 GHz, in the discharge tube with the axis parallel to the X-axis, the electromagnetic wave produces a locally enhanced electric field on the plasma, indicating that the bandgap here is the LSP bandgap. When Freq = 8.60 GHz, the electromagnetic wave is located in the passband and can propagate normally. When Freq = 12.93 GHz, the main influence on the electromagnetic wave is the quartz medium, and the periodic distribution of The quartz medium will affect the electromagnetic wave transmission.

Fig. 13. Electric field distribution of 3D PPC

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Figure 13(b) shows that when Freq = 1.00 GHz, the electromagnetic wave is completely cut off by the plasma in the discharge tube whose axis is parallel to the Y-axis, indicating that this frequency lies within the cut-off bandgap. When Freq is 4.19 GHz and 6.83 GHz, LSP appears in plasma2 and plasma3 respectively, resulting in the attenuation of electromagnetic wave energy. When Freq = 12.63 GHz, the main influence on electromagnetic wave propagation is quartz.

To analyze the causes of bandgaps in the transmission spectra, the band structure of the 3D PPC was calculated using the SFDTD and the PWE, respectively, and the results are shown in Fig. 14.

Fig. 14. Transmission spectra and band structure of 3D PPC

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Figure 14(a) shows that when I = 36.9 mA, the cut-off bandgap is below 0.7 GHz. A flat band appears around 2.4 GHz, indicating that the bandgap here is the LSP bandgap. Figure 14(b) shows that when I = 158.3 mA, the cut-off bandgap is below 1.5 GHz, and flat bands appear around 4 GHz and 7 GHz, indicating that with the increase of the discharge current, the electron density in the tube increases, and its radial-gradient increases, and multiple LSP bandgaps appear in the transmission spectra. Under the same discharge current, the cutoff frequency of the 3D PPC is lower than that of the 2D TM polarization. The reason may be that there are discharge tubes perpendicular to and parallel to the polarization direction of the electromagnetic wave in the 3D PPC. Fano modes may weaken the plasma's cutoff effect on electromagnetic waves.

Figure 10 and Fig. 14 show that both SFDTD and PWE can effectively calculate the band structure of PPC, and results of the two methods are in good agreement. The SFDTD method is an improvement of the FDTD method. It is essentially a high-order format FDTD. Different from ordinary high-order FDTD, it uses the symplectic format for time discrete format, which can reduce the dissipation in the calculation process. While the PWE has a clear physical concept and is easy to implement, when the number of media changes, the calculation formula needs to be re-derived, and the calculation formula becomes extremely complicated with the increase of media types.

The band structure calculated by SFDTD may “miss” or “misreport” the eigenfrequency because the calculated result of SFDTD depends on the number and position of the excitation source and monitor. If the photonic crystal is composed of a lossy medium such as a metal or a plasma, the energy recorded by the monitor will weaken over time, resulting in the flat band being indistinguishable; PWE will calculate some pseudo eigenfrequencies when calculating band structure under TE polarization because the presence of LSP in PPC makes it difficult to converge when solving for the complex eigenvalues. Therefore, it is necessary to compare the calculation results of various methods to comprehensively analyze the band structure of PPC.

4. Conclusion

1) Although CRM is a mathematical model that does not consider the spatial distribution of plasma, its results are consistent with the trend and order of magnitude of the diagnostic results of microwave interferometry. It shows that CRM is an effective engineering model that can be used to quickly evaluate the parameters of rare gas-mercury glow discharge plasma.
2) The simulation and experimental results of the PPC transmission spectra are in good agreement, indicating that the layered model established in this paper for the plasma tube is reasonable and effective.
3) There are different types of band gaps in PPC. In the 2D PPC, the LSP band gap generates under the TE polarization, while the cutoff bandgap generates under the TM polarization. With the increase of the average electron density, the number of LSP bandgap will increase under TE polarization, the cutoff frequency will be higher under TM polarization. Both cutoff bandgap and LSP bandgap exist in the 3D PPC and the position of the cutoff bandgap shifts to low frequencies due to the appearance of the Fano mode.
4) The PWE is easy to implement but has poor versatility. If the photonic crystal contains a variety of media, the calculation formula is extremely complicated, and the calculation amount will also be greatly increased. When the PWE calculates the band structure of the photonic crystal with dispersive media, a large number of pseudo eigenfrequencies appear in the results. The SFDTD has good versatility but requires a large amount of calculation. It needs to test the number, location, and calculation time of excitation sources and monitors many times to obtain correct results.

Acknowledgments

Thanks to Associate Professor Liu Yang, Department of Light Sources and Illuminating Engineering, Fudan University, and Gu Heng, CTO of Shanghai Forbay Electronic Technology Co., Ltd, for their enthusiastic help. They gave massive guidance to the work of the CRM in this paper. The author would like to express his sincere gratitude to them for their noble spirit of sharing their knowledge selflessly. Mr. Gu passed away in March 2022, the author hereby expresses our deep condolences to Mr. Gu, and may his soul rest in peace.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Study on the transmission characteristics and band structure of 2D and 3D plasma photonic crystals

Abstract

1. Introduction

2. Mathematical model and experiment diagnosis of positive column of discharge tube

2.1 Mathematical model of positive column region

2.2 Diagnostic methods and results of electron density in the positive column

3. Transmission characteristics and band structure of PPC

3.1 Modeling of PPC unit

3.2 Transmission characteristics and band structure of 2D PPC

3.3 Transmission characteristics and band structures of 3D PPC

4. Conclusion

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (14)

Equations (16)

Optics Express