Sterilization mechanism of helium/helium–oxygen atmospheric-pressure pulsed dielectric barrier discharge on membrane surface

1.   Introduction

The detection of reentry vehicles is an important issue in aerospace communications, satellite positioning, and tracking. Military means, such as reconnaissance, anti-reconnaissance, midcourse and terminal interception also need to investigate the electromagnetic characteristics of reentry vehicles. When a vehicle reenters the atmosphere, it will be rubbed and compressed sharply by the surrounding atmosphere, and its surface generates infrared radiation and visible light radiation. A plasma sheath and wake are formed separately on the surface and downstream of the vehicle, which significantly change its electromagnetic characteristics [1–6]. At present, the main detection methods of reentry vehicles include ground-, air- and space-based radar, and space-based infrared systems [7–12]. Most of them use the electromagnetic wave scattering and infrared radiation characteristics of the plasma sheath to achieve target recognition. Compared with the detection method of directly targeting the reentry vehicle, the detection of the wake has the advantages of reducing the speed of the tracking target, increasing the stress time, and decreasing the interference of the stealth coating of the vehicle. The methods of detecting wakes mainly include radar echoes, infrared radiation characteristics [13, 14], laser diagnosis [15], etc. However, the literature on excited electromagnetic radiation based on wakes has not yet been reported in depth. Therefore, the characteristics of excited electromagnetic radiation need to be further explored.

In 1983, Thidé et al discovered the characteristics of the echo spectrum while they used a high-intensity and high-frequency transmitter to irradiate ionospheric plasmas, and called it excited electromagnetic radiation [16]. The research on electromagnetic radiation mainly focused on the fields of ionospheric detection and heating, and the equipment such as the European Incoherent Scatter Radar (EISCAT) and High Frequency Active Auroral Research Program (HAARP) were used to heat the ionosphere for observing nonlinear phenomena in plasmas [17–21]. When an incident electromagnetic wave with frequency ${\omega }_0$ illustrates plasmas, it could generate reverse electromagnetic waves of $0.98{\omega }_0–1.04{\omega }_0$ and ion acoustic waves. The incident electromagnetic wave needs to satisfy the dispersion relationship between the two excited waves [22].

Existing research has established that the plasma densities of both the far wake and the near wake are much larger than that in the ionosphere [23, 24]. As a result, the interference of the ionosphere can be ruled out. These conclusions provide a theoretical possibility for detecting electromagnetic radiation of the plasma wake. Therefore, it is worthy of in-depth research to study the electromagnetic radiation characteristics of the plasma wake and explore its application value in the detection of reentry vehicles.

The theory of excited electromagnetic radiation is nonlinear coupling between plasma waves and electromagnetic waves, such as second harmonic generation (SHG), third harmonic generation (THG), stimulated electromagnetic emission (SEE), etc. So far, there is plenty of research on excited electromagnetic radiation. The Particle In Cell (PIC) method starts from the low-frequency decay and particle cyclotron theory and then presents the SEE spectra [25]. The Vlasov and Maxwell equations (VME) could be coupled to observe the burst process of SEE by detecting the electronic phase space vortex [26–28]. The PIC and VME methods are kinetic theories based on the motion of electrons and ions in plasmas. The Zakharov system of equations (ZSE) gives the electric field, as well as the low frequency oscillation, by dividing the three-wave coupling into global and local parts [29, 30]. ZSE is an important component in a THG system, and plays a key role in the research of dynamic collision-free stimulated Raman scattering (SRS) and the attenuation of Langmuir waves in ionized plasmas [31–33]. Both the VME and the PIC methods aim to simulate nonlinear effects in plasmas by using the motion of microscopic particle populations. As a complex plasma environment, the excited electromagnetic radiation of the plasma wakes has not been dealt with. If the detected target transfers from the reentry vehicle to the plasma wake by means of excited electromagnetic radiation, the detected target will become larger and longer, and more response time will be obtained. Therefore, the study of excited electromagnetic radiation is extremely interesting. The calculation of excited electromagnetic radiation by ZSE is based on wave-wave interactions, which is suitable for the study of the plasma wake.

This work investigates the excited electromagnetic radiation characteristics of plasma wakes by ZSE in detail. The rest of the paper is as follows. Section 2 simulates the density distribution of plasma wakes by using the typical sphere-cone model and the Arrhenius chemical reaction model, and analyzes the effects of both flight speed and angle of attack. Section 3 evaluates the excitation conditions of electromagnetic radiation, and discusses the effects of both flight speed and escape angle on the radiation power spectra. Finally, some conclusions are presented in section 4.

2.   Model and methods

2.1   Plasma wake of reentry vehicles model

In this section, the Computational Fluid Dynamics (CFD) numerical method is used to simulate plasma wakes of reentry vehicles in an atmospheric environment. As shown in figure 1, a geometric model of a typical sphere-cone reentry vehicle is established. The radius of the sphere is set to 0.5 m, the axial angle of the sphere is 79.6°, the length of the model is 1.5 m, and its bottom radius is 0.6 m.

Figure  1.  The geometric model of a typical sphere-cone reentry vehicle.

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As the flow field changes dramatically at the head, the grid is densified as shown in figure 2. The quality of 98% grid, which is responsible for the simulation results, is in the range of 0.95–1.

Figure  2.  The grid of simulation area.

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Considering the calculation of both the plasma wake and the flow around the wall, the k-ω equation is selected as the fluid governing equation. The diffusion equation of substances is used to estimate the mass fraction of each substance [34]:

where $R_i$ means the net yield of the chemical reaction, and $S_i$ refers to the production rate of species i caused by the source term during simulation. If there are $N$ species in the model, the $N-1$ conservation equations need to be solved. The rest is the mass fraction of the $N\mathrm{th}$ substance. In this calculation, nitrogen is taken as the last one.

When equation (1) is the mass diffusion equation in advection, $J_i=-\rho D_{i.\mathrm{m}}\mathrm{▽}{Y}_i$ means the diffusion flux of species $i$ and is generated by the concentration gradient, where $D_{i.\mathrm{m}}$ stands for the diffusion coefficient of the species $i$ in the mixture. The mass diffusion equation is expressed as $J_i=-\left(\rho D_{i.\mathrm{m}}+\frac{{\mu }_i}{S{c}_{\mathrm{t}}}\right)\mathrm{▽}{Y}_i$ in viscous flow, and $S{c}_{\mathrm{t}}$ is the Schmidt number in the turbulent model.

The velocity boundary will cause the reflection of a shock wave at the simulation boundary, thus the far-field pressure boundary condition is used at the inlet and the pressure outlet boundary condition is used at the outlet.

The fluid of a reentry vehicle has a series of complex chemical processes, such as vibrational excitation, dissociation, ionization and chemical reaction at the molecular energy level, thus it must be analyzed by thermochemical non-equilibrium theory. The thermochemical non-equilibrium process includes two aspects: first, the temperature is non-equilibrium and changes with time; second, the chemical reaction and the mass fraction of fluid components are non-equilibrium. Therefore, the two-temperature model is chosen to simulate the nonequilibrium in hypersonic flow, which can provide a better flow field prediction than the single temperature model.

In conclusion, the following chemical reaction rate model is selected:

where N stands for the number of chemical reaction substances, $v_{i, r}^\prime$ and $v_{i, r}^{\prime\prime}$ are the stoichiometric coefficient of chemical reactants and the stoichiometric number of products, respectively, ${M_i}$ is substance of $i$. ${k_{{\rm{f}}, r}}$ and ${k_{{\rm{b}}, r}}$ indicates the positive and reverse reaction rate of the reaction $r$. Ultimately, the 7-component chemical reaction model is selected for simulation and is shown in table 1 (N₂, O₂, NO, O, NO⁺, e⁻) [32].

Table  1.  7-component chemical reaction model.

Equation number	Reaction equation
1	${{\rm{N}}_2} + {\rm{M}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\rm{N}} + {\rm{N}} + {\rm{M}}$
2	${{\rm{O}}_2} + {\rm{M}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\rm{O}} + {\rm{O}} + {\rm{M}}$
3	${\rm{NO}} + {\rm{M}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\rm{N}} + {\rm{O}} + {\rm{M}}$
4	${{\rm{N}}_2} + {\rm{O}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\rm{NO}} + {\rm{N}}$
5	${\rm{NO}} + {\rm{O}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\rm{N}} + {{\rm{O}}_2}$
6	${\rm{N}} + {\rm{O}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\rm{N}}{{\rm{O}}^ + } + {{\rm{e}}^ - }$

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Figure 3 describes the distribution (${\mathrm{m}}^{-3}$ ) of electron number density in the plasma wake with the vehicle flying at a height of 48 km, a static pressure of 100 Pa and three Mach numbers (M6, M9, M13).

Figure  3.  Electron number densities of the plasma wake when the Mach numbers are M6 (a), M9 (b), and M13 (c), respectively.

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Figure 3 depicts that the plasma wake forms a narrow 'neck' downstream of the vehicle. As the Mach number increases, the electron number density increases and the 'neck' gradually moves away from the bottom of the vehicle and becomes longer. Considering the great kinetic energy of the hypersonic flow, in the boundary layer, when the viscosity effect slows down the flow rate, part of the lost kinetic energy is converted into the internal energy of the gas. This part of the energy is enough to excite the vibration energy in the molecule, and finally causes the dissociation and ionization of the gas in the boundary layer. For these reasons, chemical reactions produce more ions. The electron number densities of the wake decrease with the increase of the distance away from the vehicle.

Figure 4 illustrates the electron number densities along the central axis of the plasma wake at various velocities. x stands for the range away from the bottom of the reentry vehicle. The reduction rate of the electron number density slows down as x increases. The electron number density first increases and then decreases, where x = 2 m is the demarcation point. There is a negative correlation between the velocity of the vehicle and the reduction rate of the electron number density.

Figure  4.  Electron number densities along the central axis of the plasma wake when Mach numbers are M6 (a), M9 (b), and M13 (c), respectively.

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Figures 5(a) and (b) are the cloud diagrams of the electron number density distribution of the plasma wake when the angles of attack are 10° and 5°, respectively. The wake structures are similar to the case with the zero angle of attack. Figures 3 and 5 show that the plasma wake structures with different angles of attack are all cylindrical. The electron number density along the axial direction decreases with the range away from the vehicle, and the distribution of the electron number densities is basically consistent with the existing research [23, 24].

Figure  5.  Electron number densities of the plasma wake when the angles of attack are (a) 10° and (b) 5°, and the Mach number is M13.

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Thus, a plasma wake model applied to the following section is constructed by utilizing the cylindrical layers, that is, the wake is divided into several cylinders of finite length along the axial direction, and the distribution of plasma parameters in the radial direction is non-uniform. Figure 6 depicts the schematic of an incident wave acting on a plasma wake.

Figure  6.  Schematic diagram of the geometric model of a plasma wake.

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The coordinate system is established with the midpoint of the vehicle tail as the origin, the longitudinal direction of the wake is along the z-axis, and the x-axis is perpendicular to the z-axis. $d$ denotes the radial radius of the non-uniform cylindrical trail, $r$ represents the change in radial dimension and θ indicates the escape angle of the excited electromagnetic radiation. This model takes x as the axis of symmetry. In the plate layer approximation described in figure 6, the plasma wake is divided into 7 layers (the model of 7 layers can ensure the accuracy of the results and greatly reduce the calculation time). The one-dimensional ZSE is applied to each layer. The central layer has the largest electron number density, which then decreases along the direction of the negative z-axis. In section 3, we use this model to calculate the excited electromagnetic radiation spectra in the plasma wakes of the reentry vehicles.

2.2   Excitation conditions of electromagnetic radiation in plasma wakes

Excited electromagnetic radiation originates from the nonlinear interaction between electromagnetic waves and plasmas. In other words, the generation of radiation is due to the vibration of plasmas caused by the incident electromagnetic wave, for example, ion vibration produces ion acoustic wave. The coupling of both the excited radiation and the incident wave makes the particles oscillate in an orderly manner, and then produces electric fields and currents. It is worth noting that the incident frequency of the electromagnetic wave must be higher than the plasma frequency ${\omega _{\rm{p}}} = \sqrt {{n_0}{e^2}/({m_{\rm{e}}}{\varepsilon _0})}$, and the amplitude of the incident wave needs to be higher than the given threshold Considering the plasma wake of a reentry vehicle, there are a large number of electrons and ions generated by chemical reactions and ionization in the plasma wake. As a result, the excited electromagnetic radiation in the plasma wake is feasible.

The ZSE gives the condition that the amplitude of the incident wave must meet by analyzing the THG system, and the threshold value $E_{\mathrm{th}}$ is given by [35, 36] as:

where △Ω stands for the difference between the angular frequency of the incident electromagnetic wave ${\omega }_0$ and the plasma frequency ${\omega }_{\mathrm{p}},$ $k_1$ means the dominant wavenumber of the stimulated emission, $m$ and $M$ are the electron and ion mass respectively, $T_{\mathrm{e}}$ and $T_{\mathrm{i}}$ represent electron temperature and ion temperature, $v_{\mathrm{i}}$ is the ion collision frequency, $v_{\mathrm{e}}$ is the electron collision frequency, $\eta =1+3T_{\mathrm{i}}/T_{\mathrm{e}},$ and $v_{\mathrm{coll}}$ indicates collision damping.

According to equation (3), figure 7 shows the distribution of the electric field threshold of the plasma wake at x = 10 m.

Figure  7.  Distribution of the threshold of excited electromagnetic radiation under different Mach numbers.

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As the Mach number increases, the threshold decreases and the maximum of thresholds exists at all three speeds. It can be seen from equation (3) that $E_{\mathrm{th}}$ is negatively correlated with variable $k_1.$ However, the layered model established in section 2.1 shows that $k_1$ increases with the increase in the value of $z.$

2.3   Excited electromagnetic radiation simulation model of a plasma wake

After the incident electromagnetic wave acts on the plasma wake, the incident wave couples to the plasmas and decays into a low-frequency acoustic wave and a reverse electromagnetic wave. Based on the plasma wake model of the reentry vehicle as shown in figure 6, ZSE describes the parametric instability process in the plasma wake. ZSE is constructed from the two-fluid equation, which describes the coupling process of both the incident wave and the low-frequency sound wave [36]. The ZSE of the plasma wake supports the following assumptions: (1) the high-frequency electromagnetic wave and the low-frequency acoustic wave in the plasma wake can be well separated; (2) the particle motion in the low frequency part is in a quasi-neutral state; (3) the electron velocity in the plasma wake is much smaller than the ion velocity. The ZSE is as follows [37]:

Equations (4) and (5) are the one-dimensional ZSE. indicates the imaginary unit, is the total high frequency electric field, stands for the density perturbation of the ions, represents the total number of ions, is the ionic speed of sound, and means the dielectric constant. $\left\langle {} \right\rangle$ and || stand for the spatial mean and absolute value, respectively.

As the ZSE is dimensionless, the Fourier transform is used to solve it in the frequency domain [38]. The definition of the source spectrum is obtained by the two-scale model [39]:

where means the number of segments in the simulation area, and each segment contains microelements, and thus, constitutes the entire simulation area. There is a nonlinear current source ${S_J}\left( \omega \right) = {\rm{| }}{\left\{ {nE} \right\}_J}{\rm{|}}{{\rm{ }}^2},J = 0 \ldots K - 1$ in each segment of the simulation region. Equation (6) describes the average current source term for the simulation region. It can be seen that the current source term is an analysis of the nonlinear current in a certain simulation area. According to the current source definition, denotes the current source spectrum for the entire simulation area.

The power spectrum reflects the total power on the escape path of the excited electromagnetic radiation in the plasma wake, the source spectrum only describes the intensity of the current source term of the localized plasmas. With reference to the layered structure of figure 6, when the local size of the plasmas meets $L=-0.5z{\omega }_0/\mathrm{\Delta }\mathrm{\Omega },$ the power spectrum of the plasma wake is calculated by [39]:

Here, $Ai$ represents the airy function, $l={\left(c^{2}L/{\omega }_0^2\right)}^{1/3},$ $L$ means the length of the simulation region of the plasma source spectrum, ${\delta }_{\omega }=2\omega {\left({\omega }_{0}L/c\right)}^{2/3}/{\omega }_0,$ and $K=0.5\pi l{\omega }_0/{\epsilon }_0{c}^2.$

Excited electromagnetic radiation in the descending sideband of the wake cannot escape, which can be seen from equation (7) that ${\left(Ai\right)}^2\left(-{\delta }_{\omega }\right)$ will rapidly approach zero with ${\delta }_{\omega }\to -\infty .$

When excited radiation escapes obliquely, the escaping angle θ is no longer zero. As shown in figure 6, these plasma layers have different x-coordinate values. The formula of the power spectrum is the same as that of equation (7), where ${\delta }_{\omega }$ is replaced by:

3.   Numerical analysis of the excited electromagnetic radiation of plasma wakes

3.1   Effects of incident wave parameters on the distribution of excited electromagnetic radiation source spectra

Figure 8 shows the effects of the incident frequencies of electromagnetic waves on the excited electromagnetic radiation source spectra. The range of the plasma wake is 100 m away from the tail of the vehicle. The distribution of electron densities along the z-axis is given in section 2, where the maximum of electron number densities is $n_{\mathrm{e}}=1.45\times {10}^{16}{\mathrm{m}}^{-3},$ the speed is set to M13, and the plasma temperature is $T=6500\mathrm{K}.$

Figure  8.  Source spectra of speed M13 at different incident angular frequencies.

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The three curves in figure 8 are envelope spectra without characteristic peaks at the angular frequency ${\omega }_0=7.9\times {10}^9\mathrm{rad}{\mathrm{s}}^{-1}.$ The spectral line (b) shows distinct secondary peaks, which appear at frequency shifts $\omega =0,\omega =\pm 2k_1$ for the case of the incident angular frequency ${\omega }_0=8.9\times {10}^9\mathrm{rad}{\mathrm{s}}^{-1}.$ The descending part of the source spectrum (c) illustrates the peaks of the cascade structure with the increase of ${\omega }_0.$ All of source spectra in figure 8 have the same feature that the down-shifted sidebands of the source spectra are generally higher than the up-shifted sidebands. If the spectrum is up-shifted, for example, the secondary peak at $+0.93\times {10}^5\mathrm{Hz}$ in the curve (c), denoting this frequency shifted photon as ${\omega }_2$ and the incident wave photon as ${\omega }_0,$ then, the relationship $\hslash {\omega }_2=\hslash {\omega }_0+\hslash {\omega }_1$ may be obtained, which means that the energy of the up-shifted spectrum is greater than that of the incident wave. Otherwise, considering the secondary peak at $-0.93\times {10}^5\mathrm{Hz},$ we have $\hslash {\omega }_2=\hslash {\omega }_0-\hslash {\omega }_1,$ which means that the downshifted wave has less energy than that of the incident wave. From an exothermic point of view, it is easier to generate a down-shifted spectrum than an up-shifted one. Hence, the peaks of the down-shifted spectrum are larger than those of an up-shifted one.

Choosing the locations separately as $x=25\mathrm{m},x=75\mathrm{m},x=150\mathrm{m}$ and setting the incident frequency to ${\omega }_0=1.15\times {10}^9\mathrm{rad}{\mathrm{s}}^{-1},$ figure 9 displays the source spectra with a Mach number of M13. The electron number densities are $n_0=2.75\times {10}^{16}{\mathrm{m}}^{-3},$ $n_0=2.00\times {10}^{16}{\mathrm{m}}^{-3},$ and $n_0=1.75\times {10}^{16}{\mathrm{m}}^{-3},$ respectively.

Figure  9.  Source spectra of speed M13 at different positions of the plasma wake.

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As shown in figure 9, the frequency shifts of the peaks in curves (a), (b) and (c) are consistent. The sharp peaks of the cascade in the source spectra are the embodiments of current sources. The peak structures of curves (b) and (c) are more abundant than that of curve (a) due to the higher electron number density in the wake closer to the bottom of the vehicle. It is interesting to note that there are no more peaks in curve (c) than in curve (b), which indicates that $\mathrm{\Delta }\mathrm{\Omega }={\omega }_0-{\omega }_{\mathrm{p}}$ and peak structures do not constitute a positive correlation. For a fixed incident frequency, we have $\mathrm{\Delta }{\mathrm{\Omega }}_{150\mathrm{m}}>\mathrm{\Delta }{\mathrm{\Omega }}_{75\mathrm{m}}>\mathrm{\Delta }{\mathrm{\Omega }}_{25\mathrm{m}}.$ Meanwhile, the peak intensity is negatively correlated with the range.

3.2   Excited electromagnetic radiation power spectra

In this section, the power spectra are selected to analyze excited electromagnetic radiation in the plasma wake of the reentry vehicle.

According to the model given in figure 6 and the results of section 2, figures 10 (a)–(c) show the power spectra at 25 m, 75 m and 150 m away from the bottom of the vehicle at M6, M9 and M13. The corresponding incident frequencies are selected as ${\omega }_{0,\mathrm{M}6}=6.3\times {10}^8\mathrm{rad}{\mathrm{s}}^{-1},$ ${\omega }_{0,\mathrm{M}9}=3.0\times {10}^9\mathrm{rad}{\mathrm{s}}^{-1}$ and ${\omega }_{0,\mathrm{M}13}=1.2\times {10}^{10}\mathrm{rad}{\mathrm{s}}^{-1}$ for the three speeds, respectively.

Figure  10.  Power spectra at different velocities.

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The distributions of secondary peaks in figure 10 are presented in table 2. In the case of the same incident wave, there is $\mathrm{\Delta }{\mathrm{\Omega }}_{150\mathrm{m}}>\mathrm{\Delta }{\mathrm{\Omega }}_{75\mathrm{m}}>\mathrm{\Delta }{\mathrm{\Omega }}_{25\mathrm{m}},$ that is, $E_{\mathrm{th},150\mathrm{m}}<E_{\mathrm{th},75\mathrm{m}}<E_{\mathrm{th},25\mathrm{m}}.$ The number of shifted peaks at x = 25 m is less than that at x = 75 m and x = 150 m, and these peaks present a cascade structure, where they are satisfied with $f_{\pm 2}\approx 2f_{\pm 1},f_{\pm 3}\approx f_{\pm 2}+f_{\pm 1}.$ It can also be concluded from table 2 that the larger the speed, the greater the frequency shift.

Table  2.  Frequency shifts of figure 10.

	x (m)	$f_{-3}$	$f_{-2}$	$f_{-1}$	$f_1$	$f_2$	$f_3$
M6	25		$-19.4\mathrm{kHz}$	$-7.87\mathrm{kHz}$	$7.08\mathrm{kHz}$	$17.8\mathrm{kHz}$
	75	$-28.6\mathrm{kHz}$	$-14.9\mathrm{kHz}$	$-7.35\mathrm{kHz}$	$7.35\mathrm{kHz}$	$14.7\mathrm{kHz}$	$29.1\mathrm{kHz}$
	150	$-23.8\mathrm{kHz}$	$-17.3\mathrm{kHz}$	$-8.9\mathrm{kHz}$	$7.35\mathrm{kHz}$	$15.2\mathrm{kHz}$	$28.6\mathrm{kHz}$
M9	25	$-1.12\mathrm{MHz}$	$-68.7\mathrm{kHz}$	$-35.0\mathrm{kHz}$	$42.5\mathrm{kHz}$	$83.7\mathrm{kHz}$
	75	$-1.22\mathrm{MHz}$	$-81.2\mathrm{kHz}$	$-41.2\mathrm{kHz}$	$40.0\mathrm{kHz}$	$81.2\mathrm{kHz}$	$1.21\mathrm{MHz}$
	150	$-1.26\mathrm{MHz}$	$-85.0\mathrm{kHz}$	$-42.5\mathrm{kHz}$	$42.5\mathrm{kHz}$	$85.0\mathrm{kHz}$	$1.23\mathrm{MHz}$
M13	25		$-3.70\mathrm{MHz}$	$-1.65\mathrm{MHz}$	$1.65\mathrm{MHz}$	$3.35\mathrm{MHz}$
	75	$-5.60\mathrm{MHz}$	$-3.75\mathrm{MHz}$	$-1.85\mathrm{MHz}$	$1.70\mathrm{MHz}$	$3.35\mathrm{MHz}$	$5.05\mathrm{MHz}$
	150	$-5.70\mathrm{MHz}$	$-3.35\mathrm{MHz}$	$-1.65\mathrm{MHz}$	$1.65\mathrm{MHz}$	$3.35\mathrm{MHz}$	$5.05\mathrm{MHz}$

 | Show Table

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In the power spectra shown in figure 10, compared with the up-shifted sidebands, the down-shifted sidebands still have more cascade peaks, which is consistent with the law of the source spectra as presented in section 3.1. The magnitudes of the power spectra increase with the increase of Mach number. According to the conclusion in section 2.2, incident waves with higher frequency are needed to excite the electromagnetic radiation of the vehicle at a higher speed. As a result, such incident waves generate a higher electromagnetic radiation under the action of three-wave coupling. Therefore, higher speeds produce higher power spectra, and this feature also exists in source spectra at different speeds.

Figure 6 describes the path model, where the plasma parameters along the path of electromagnetic wave propagation are different. It emerges that the distribution of electron densities is the most important factor, horizontal layering is then carried out. Figure 6 shows the model of the escape wave from point A through two points I and E.

According to the path in figure 6, by calculating the source spectra of the ABIH and IDEF regions and reckoning the overall power spectra, figure 11 plots the escaping excited radiation along the 45° direction. The curve (a) selects the plasma wake of the vehicle with M13 and $x=50\mathrm{m}.$ Consequently, the escaping path from the center to leaving the wake is split into two layers. The electron densities in the centers of the first and the second layers are $n_{\mathrm{e}}=2.2\times {10}^{16}{\mathrm{m}}^{-3}$ and $n_{\mathrm{e}}=1.9\times {10}^{16}{\mathrm{m}}^{-3},$ respectively. The plasma temperature and the angular frequency of the incident wave are $T=6800\mathrm{K}$ and ${\omega }_0=1.2\times {10}^9\mathrm{rad}{\mathrm{s}}^{-1},$ respectively.

Figure  11.  Escape power spectra at speed M13 in the direction of 45° at 50 m and 150 m from the bottom of the vehicle.

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In curve (b), the Mach number of the vehicle is M13 and the axial position is $x=150\mathrm{m}.$ The central electron densities on the path are $n_{\mathrm{e}}=1\times {10}^{16}{\mathrm{m}}^{-3}$ and $n_{\mathrm{e}}=9.6\times {10}^{15}{\mathrm{m}}^{-3},$ respectively. The plasma temperature is $T=6000\mathrm{K}.$ The angular frequency of the incident wave is the same as that of curve (a).

The results reveal that both power spectra have the similar peaks and produce some small protrusions next to the secondary peaks. The phenomena can be observed at $-1.6\times {10}^5\mathrm{Hz},$ $-3.3\times {10}^5\mathrm{Hz}$ and $1.7\times {10}^5\mathrm{Hz},$ respectively. As excited radiation escapes outward, it may be coupled with other waves on the path, and continues to produce excited radiation. Therefore, they become small protrusions next to the secondary peaks.

Furthermore, the effects of the escape angles are shown by the curves in figure 12. The parameters of the plasma wake at x = 100 m are selected, where the central electron density of the trail axis is $n_{\mathrm{e}}=1.5\times {10}^{16}{\mathrm{m}}^{-3},$ the Mach number is M13, the plasma temperature is $T_{\mathrm{e}}=6500\mathrm{K},$ and the angular frequency of the incident electromagnetic wave is ${\omega }_0=1\times {10}^{10}\mathrm{rad}{\mathrm{s}}^{-1}.$

Figure  12.  Power spectra $\left\langle {{P_\omega }} \right\rangle$ at speed M13 obtained under different escape angles.

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Figure 12 shows the distribution of power spectra obtained at different escape angles. It indicates that the effects of the escape angle on the magnitudes of the power spectra are slight. However, under the four selected angles, the magnitudes of both the main and secondary peaks at $\theta =\mathrm{\pi }/4$ are generally higher than those of the other three angles. According to the model in figure 6, the escape angle is inversely proportional to the range of the escape path. The long escape path represents a large plasma region. Otherwise, if the range of the escape path is too long, the incident electromagnetic wave cannot meet the excited radiation conditions on the path at the same time. Therefore, the escape angle has a limited effect on excited radiation.

4.   Conclusions

This paper proposes a new method for detecting reentry targets through the excited electromagnetic radiation phenomenon of the plasma wake. The following main conclusions are drawn through the numerical analysis under different conditions.

(1) Due to the conservation of energy $\hslash {\omega }_2=\hslash {\omega }_0\pm \hslash {\omega }_1,$ the peaks of the down-shifted spectrum are larger than those of an up-shifted spectrum.

(2) When the frequency of the incident wave satisfies the excitation conditions, the incident frequency affects both the number of sub-peaks and the shape of the spectral lines, and these peaks present a cascade structure, whose frequency shifts satisfy $f_{\pm 2}\approx 2f_{\pm 1},f_{\pm 3}\approx f_{\pm 2}+f_{\pm 1}.$

(3) The greater the Mach number, the more obvious and stronger the excited electromagnetic radiation spectra are.

(4) While under the incident wave with the same frequency, the spectra on the far wake have 6 peaks, nevertheless, there are only 4 peaks at 25 m.

At present, early warning radar can only search and track long-distance targets. If the research in this area is combined with early warning radar, this work may provide a certain reference for the speed measurement and maneuvering monitoring of the vehicle.