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Propagation characteristics of terahertz wave in hypersonic plasma sheath considering high temperature air chemical reactions
Optik - International Journal for Light and Electron Optics 208 (2020) 164090
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Original research article
Propagation characteristics of terahertz wave in hypersonic plasma sheath considering high temperature air chemical reactions
T
Kai Chena,b, Degang Xua,b, Jining Lia,b,*, Xingning Genga,b, Kai Zhonga,b, Jianquan Yaoa,b a b
School of Precision Instrument and Optoelectronics Engineering, Institute of Laser and Optoelectronics, Tianjin University, Tianjin 300072, China Key Laboratory of Optoelectronics Information Technology (Ministry of Education), Tianjin University, Tianjin 300072, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Terahertz wave Propagation characteristics Plasma sheath Scattering matrix method
In this work, the propagation characteristics of terahertz wave in the hypersonic plasma sheath is demonstrated and analyzed based on the realistic vehicle model and the high temperature air thermal chemical reactions. Based on the simulated flow field distribution around the hypersonic vehicle, the electron densities of the plasma sheath at different Mach numbers are calculated by adopting the equilibrium constant method of the 7-species reacting air model. And then the propagation model of terahertz wave in plasma sheath is established using the scattering matrix method. The calculated and simulated results show that the plasma electron density is strongly influenced by Mach number, and the higher frequency and smaller incident angle of terahertz wave are advantageous for a better transmittance and lower loss. Furthermore, the effects of an external magnetic field on the propagation characteristics of left-hand and right-hand polarized terahertz wave in plasma sheath are studied. The external magnetic field indeed reduces the transmission loss of the left-handed wave. This work promises that terahertz waves have good potential applications in the detection and the communication for the hypersonic vehicles in plasma sheath.
1. Introduction When the vehicles are flying at a hypersonic speed, the high-temperature and high-pressure environment is formed outside the vehicles surface due to the fierce friction of the atmosphere, which ionizes the surrounding air and produces a plasma sheath surrounding the hypersonic vehicles. The presence of the sheath impedes or even interrupts the propagation of the communication signals, which causes the famous “blackout” problem [1,2]. In addition, the plasma sheath invalidates the most electromagnetic wave radar. The rapid development of the space technology, especially the exploitation of the near space makes the plasma sheath a pressing problem to be solved. The NASA's RAM project in 1970s explores the attenuation of the microwave electromagnetic waves in plasma medium, and put forward several theories and methods to reduce the blackout problem [3,4]. Since then, many attempts have been made to reduce the impact of the plasma sheath on communication [5–14]. And many effective methods have been developed to study the propagation characteristics, including the finite difference time domain (FDTD) method [15], the analytic method [16], the impedance matching (IM) method [17], and so on [18–20]. Guo et al. investigated the propagation characteristics of terahertz (THz) wave in magnetized ⁎ Corresponding author at: School of Precision Instrument and Optoelectronics Engineering, Institute of Laser and Optoelectronics, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.ijleo.2019.164090 Received 15 November 2019; Accepted 17 December 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 208 (2020) 164090
K. Chen, et al.
plasma with inhomogeneous electron density and collision frequency using scattering matrix method [21]. Zhang et al. investigated the transmission characteristics of THz wave in linear distribution plasma using Runge-Kutta exponential time differencing finitedifference time-domain (RKETD-FDTD) method [22]. The scattering of electromagnetic wave from bell-like and Epstein profiles plasma is studied by Soltanmoradi et al. using a Green function method [23]. The recent research shows that using the electromagnetic wave with higher frequencies than the plasma resonance frequency can reduce the shielding effect of plasma medium effectively. The frequency of THz wave with a range of 0.1–10 THz, which is corresponding to the plasma with the densities of 1020–1024 m−3, is much higher than the resonance frequency of the plasma sheath. And THz wave is more insensitive to temperature than the infrared rays, and has higher spatial resolution than the microwave. Besides, THz wave communication has the characteristics of large capacity, good directivity, confidentiality and strong anti-jamming performance. Therefore, THz technology provides an effective method to solve the blackout problem [24]. Most of the previous works focus on the relative ideal distribution of the plasma electron density and collision frequency and ignore the influence of atmospheric conditions and Mach number of hypersonic vehicles. Thus, in this work, the flow field distribution is from the RAM C-III model at the height of 30 km with different Mach number. And the plasma electron density is calculated through the equilibrium constant method considering 7-species high temperature chemical reactions in near space atmospheric environment. Scattering matrix method is used to establish the interaction model between THz wave and plasma sheath. The propagation characteristics of THz wave with different incident angles is studied. Furthermore, in order to further improve the transmission of THz wave, the propagation characteristics of left-hand and right-hand polarized THz wave in plasma sheath with different external magnetic field intensity are also analyzed. 2. Hypersonic vehicle flow field simulation As is well-known, the plasma electron density depends on the flow field distribution of pressure, temperature and air density around a hypersonic vehicle. In this work, the RAM C-III vehicle model [25] is selected as the hypersonic vehicle to simulate the flow field of the plasma sheath. The schematic of this model is shown in Fig. 1. The model can be seen as a blunt cone with a 159.5 mmradius spherical nose with a half-cone angle of 9∘, and a cone length of 1295.4 mm. When calculating the flow field around the vehicle, the boundary conditions are set according to the pressure and temperature of the atmosphere at the height of 30 km, where the background pressure and temperature are 11.80 mbar and 232.73 K respectively. The flow fields at Mach numbers of 9, 11, 13 and 15 conditions are simulated by the commercial software ANSYS Fluent using k-ε viscous model. And the results of the pressure (left column), temperature (middle column) and air density (right column) distributions are shown in Fig. 2. As shown in Fig. 2, the maximum values of three parameters all increases with the Mach number increasing. The pressure increases from 1.25 × 105 Pa to 5.62 × 105 Pa, the temperature increases from 4310 K to 5350 K, and the air density increases from 1.10 × 10−1 kg/m3 to 3.92 × 10−1 kg/m3. This is because that the increasing of Mach number leads to the enhanced friction between the vehicle and the air surrounding, which results in the enhancement of the chemical reactions. 3. Mathematical model 3.1. Equilibrium constant method The equilibrium chemistry constant method [26] is based on the basic principles of the chemical reactions, thermodynamics, and balances of the elemental and species. It is an effectively way to calculate the content of each particle in the high temperature flow field. Assuming the air in near space is only consists of nitrogen and oxygen, and the volume ratio is 79:21. In the 7-species gas model, when the air around the hypersonic vehicles is ionized, the independent chemical reactions include four reactions as
O2 ⇌ O + O,
N2 ⇌ N + N,
Fig. 1. Schematic of the hypersonic vehicle. 2
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Fig. 2. Flow field distribution of pressure, temperature and air density at different Mach numbers around the vehicle.
NO ⇌ N + O, O + N ⇌ NO+ + e−. The partial pressure of each component (O2, N2, O, N, NO, NO+, e−) is denoted as P1, P2, …, P7. According to Dalton law of partial pressures, the total pressure P satisfies (1)
P = P1 + P2 + P3 + P4 + P5 + P6 + P7. The charge balance equation gives as
(2)
P6 = P7. And the element balance equation gives as
(3)
2P1 + P3 + P5 + P 6 = α (2P2 + P4 + P5 + P6),
where α is the molar fraction ratio of elements nitrogen and oxygen, which equals to the volume ratio. Considering the four reactions, the following relations exist
K eq1 =
K eq2 =
K eq1 =
K eq1 =
P3 2 P P1 P
(4)
P4 2 P P2 P
(5)
( ),
( ), P3 P4 P P P5 P
,
P6 P 7 P P P3 P4 P P
,
(6)
(7)
where Keq1, Keq2, Keq3, Keq4 are the equilibrium constants in the appropriate units of the above reactions respectively. They can be expressed as
K eq, r = exp(B1r + B2r ln Z + B3r Z + B4r Z 2 + B5r Z 3),
(8) 3
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K. Chen, et al.
Fig. 3. Schematic diagram of THz wave propagation model in plasma.
where Z = 1000/T, T is the flow field temperature in the one-temperature model, and the constants Bir are presented in Ref. [27]. Solving all the equations above, the mole fraction and mass fraction of each component can be obtained under the conditions of T and P, then the electron number density of plasma sheath can be obtained by
ne = NA
ρY7 , M7
(9)
where NA is the Avogadro constant, ρ is the air density, Y7 is the mass fraction of electron, and M7 is the molar mass of electron [28]. 3.2. Scattering matrix method The scattering matrix method is an analytical method to solve electromagnetic waves propagation in the medium [29]. As shown in Fig. 3, it divides the inhomogeneous plasma medium into plenty of layers, and each layer is considered as homogeneous medium. So the reflection coefficient and transmission coefficient of the entire plasma medium can be obtained by solving the field distribution and the boundary conditions in each layer [30]. Many applications have shown that this method is an effective way to deal with the electromagnetic wave propagation in plasma sheath. 2 + ωpi2 , where ωpe and ωpi are the electron and ion oscillation frequencies, The plasma frequency is represented as ωp = ωpe respectively, and ωpe ≫ ωpi. So the plasma frequency of mth layer can be expressed as
ωp, m ≈ ωpe, m =
ne, m e 2 , ε 0 me
(10) −12
−27
where ne is the electron density, ε0 = 8.85 × 10 F/m is the vacuum permittivity, me = 1.67 × 10 kg is the electron mass, mi is the ion mass, e = 1.6 × 10−19 C is the electron charge. The permittivity of plasma sheath outside the vehicles is ε = ε0εr, where εr is the relative permittivity of the plasma medium. In the mth layer it can be expressed as
εr , m = 1 −
2 ωp2, m (ω ± ωce) ω [(ω ± ωce)2 + νen, m]
− ωp2, m νen, m , j 2 ω (ω ± ωce)2 + νen, m
(11)
where ω is the angular frequency of the incident wave, and νen is the plasma collision frequency, which is calculated by the empirical formula [31]
νen = 5.8 × 1012T −1/2P ,
(12)
where T is the flow field temperature in degrees Kelvin, P is the flow field pressure in degrees atm. ωce is the electron cyclotron frequency expressed as ωce = eB/me, where B is the external magnetic induction intensity. The ‘ ± ’ sign indicates the polarization direction: ‘+’ means the left-hand polarization and ‘−’ means the right-hand polarization. The plasma wave number is expressed as km = k 0 εr , m , where k0 = ω/c is the wave number in vacuum, c = 3 ×108 m/s is the speed of light in vacuum. For the electromagnetic waves in the oblique incidence condition, km = kθ,m cosθm, where θm is the angle between the electromagnetic wave in the mth layer and the normal of the incident plane. And there is
sin θm =
εr , m − 1 sin θm − 1. εr , m
(13)
Considering that the system state is stable, a THz wave propagates along the xOz plane and from air into the plasma medium. And the electric field of the THz wave is parallel to the plane, the magnetic field is perpendicular to the plane. The schematic diagram of the THz wave propagation model in multi-layer plasma sheath is shown in Fig. 3. The inhomogeneous plasma sheath is divided into n layers. 4
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The total electric field in the region (0) can be expressed as
Ez,0 = E0 (e−jk 0 z + Ae jk 0 z ).
(14)
And the total electric field in the mth layer can be expressed as
Ez, m = E0 (Bm e−jkm z + Cm e jkm z ).
(15)
Only the transmitted wave exists in region (n + 1), so the electric field is
Ez, n + 1 = E0 De−jkn + 1 z ,
(16)
where A is the total reflection coefficient, and D is the total transmission coefficient. Bm is the transmission coefficient and Cm is the reflection coefficient of the mth layer. According to the continuity of electric field at the interface of different media
⎛ Bm ⎞ = Sm ⎛ Bm − 1⎞. ⎝ Cm ⎠ ⎝ Cm − 1 ⎠ ⎜
⎟
⎜
⎟
(17)
The scattering matrix of the mth layer Sm can be expressed by −1
e−jkm dm e jkm dm ⎞ Sm = ⎜⎛ ⎟ −jkm dm − k e jkm dm k e m ⎝ m ⎠ e−jkm − 1 dm − 1 e jkm − 1 dm − 1 ⎞ ⎛ ×⎜ ⎟. −jk dm − 1 − k jk dm − 1 m−1 e m−1 ⎝ km − 1 e m − 1 ⎠
(18)
At the incident plane
⎛ B1⎞ = S1 A . 1 ⎝ C1 ⎠ ⎜
()
⎟
(19)
And at the exit plane
⎛ Bn ⎞ = Vp·D , ⎝ Cn ⎠ ⎜
⎟
(20)
where Vp is expressed by
Vp =
1 ⎛ kn + kn + 1 e j (kn− kn + 1) dp ⎞ ⎜ ⎟. 2kn ⎝ kn − kn + 1 e j (kn+ kn + 1) dp ⎠
(21)
The global scattering matrix Sg is given by 2
⎞ ⎛ Sg = ⎜ ∏ Sm ⎟ S1. ⎝m=n ⎠
(22)
So the entire propagation can be expressed as
()
Sg A = Vp·D . 1
(23)
Sg can also be expressed as Sg = (Sg1, Sg2), so Eq. (23) can be transformed as
( DA) = −(S
g1
− Vp)−1·Sg 2,
(24)
Then the total reflection and transmission coefficients A and D can be obtained. Consequently, the reflectance, transmittance and absorbance are 2 ⎧ R = |A| T = |D|2 ⎨ ⎩Q = 1 − R − T
(25)
4. Result and discussion The pressure, temperature and air density values of the flow field is extracted in the direction perpendicular to the vehicle surface at the position of x = 350 mm, as the red dash line shown in Fig. 1. The distribution of the plasma electron density and the collision frequency are shown in Fig. 4. According to Fig. 4, with the increasing of the distance to the vehicle surface, the electron density decreases firstly, then increases to an extreme value, and decreases again at last. The extreme value increases with Mach number. This is because that with the 5
Optik - International Journal for Light and Electron Optics 208 (2020) 164090
K. Chen, et al.
Fig. 4. Distribution of plasma electron density and collision frequency at different Mach number. (a) Plasma electron density and (b) plasma collision frequency.
compression of the shock wave and the viscous action of air intensify and the rate of the chemical reactions in air accelerate increasing, more and more gas molecules are ionized. The blunt nose diffuses the compressed and heated air from the vehicle head to a certain distance away from the vehicle, and the distance increases as Mach number increasing. The maximum value of the electron density in the vehicle surface at Mach numbers of 9, 11, 13 and 15 conditions are 3.30 × 1011 cm−3, 6.90 × 1012 cm−3, 1.008 × 1013 cm−3 and 1.78 × 1013 cm−3, respectively. As shown in Fig. 4(b), the maximum collision frequency also increases with the Mach number. This is due to the fact that the collision rate between particles accelerates. With the distance to the vehicle surface increasing, the collision frequency has a peak value, which attributes to the combined action of the temperature and pressure according to Eq. (12). 4.1. Effects of Mach number on propagation characteristics The transmittance, reflectance and absorbance of THz wave in plasma sheath with different Mach numbers are shown in Fig. 5. The reflectance is scaled in decibels to a better show about the oscillation characteristics. In Fig. 5, the transmittance decreases with Mach number and increases with the THz frequency of the incident wave. The reflectance and absorbance increase with Mach number and decrease with the THz frequency. The decibel scale of reflectance has a periodic oscillation, which is due to the multiple reflections at the plasma boundaries. The xth order resonance occurs when k = xπ/d [23], and the oscillation frequency in the present conditions is 0.03 THz. For the increasing of Mach number results in the increasing of plasma electron density, which leads to the increasing of electron quantity. Therefore, the absorption of electrons in plasma makes the transmittance decreasing and the reflectance and absorbance increasing. Due to the frequency increasing of the incident THz wave, the rate of the periodic change of the electromagnetic field is accelerated. However, the electron oscillation frequency cannot catch up with the THz wave, which results in less absorption and the transmittance increasing. At the frequency of 0.83 THz, the transmittance at Mach number of 15 condition reaches 95%. Therefore, the communication system operating at THz band can reduce the impact of blackout effectively. 4.2. Effects of incident angle on propagation characteristics The effect of the incident angle on the propagation characteristics of THz wave in plasma sheath are calculated in Mach number of 15 condition. The results of the transmittance, reflectance and absorbance are shown in Fig. 6. As shown in Fig. 6, the transmittance decreases while the reflectance and absorbance increase with the incident angle increasing. When the THz wave incident obliquely, the horizontal component of the electromagnetic wave must be continuous under the constraint of Snell law. A part of the electromagnetic wave will be reflected without entering the plasma, thus the reflectance increases. Meanwhile, when increasing the incident angle, the length of THz wave interacting with the plasma increases. So more
Fig. 5. Propagation characteristics of THz wave in plasma sheath with different Mach numbers. (a) Transmittance, (b) reflectance, (c) absorbance. 6
Optik - International Journal for Light and Electron Optics 208 (2020) 164090
K. Chen, et al.
Fig. 6. Propagation characteristics of THz wave in plasma sheath with different incident angles. (a) Transmittance, (b) reflectance, (c) absorbance.
energy is consumed in the plasma sheath, which causes the transmittance decreasing and absorbance increasing. As shown in Fig. 6(b), the oscillation frequencies at the incident angles of 30∘, 60∘, 80∘ are 0.040 THz, 0.065 THz and 0.175 THz, which is also because of the interacting length change. At below 0.28 THz range of 80∘ incident angle condition, the absorbance increases with THz frequency. This is because that the transmission is neglectable and the reflection plays a major role in low frequency range, and the absorbance increases with the decreasing of the reflectance. For the incident angle of 0∘ condition, the transmittance reaches 90% at 0.58 THz. However, for 80∘ incident angle condition, the frequency is 1.40 THz for the same transmittance. Therefore, a smaller incident angle induces a smaller communication loss at the same frequency. 4.3. Effects of external magnetic field on propagation characteristics The calculation results of the propagation characteristics of the left-hand polarized and right-hand polarized THz wave in plasma sheath are presented in Figs. 7 and 8, respectively, with different external magnetic field intensities in the Mach 15 and 0∘ incident angle condition. The direction of the external magnetic field is parallel to the incident wave. As shown in Fig. 7, for the left-hand polarized THz wave, the transmittance increases while the reflectance and absorbance decrease with the magnetic field intensity increasing. Because the polarization direction of the left-hand wave is opposite to the electron cyclotron direction in plasma and the increasing of the magnetic field intensity rises the electron cyclotron frequency, less of the THz wave energy is absorbed by the electrons, which reduces the transmission loss. The transmittance in unmagnetized plasma sheath is 43.9% at 0.2 THz, while the transmittance rises as high as 91.2% with 15 T magnetic field intensity. Therefore, the application of an external magnetic field can greatly improve the transmission of the left-hand polarized THz wave in plasma sheath. As shown in Fig. 8, for the right-hand polarized THz wave, the transmittance decreases while the reflectance and absorbance increase with the magnetic field intensity increasing. It is obvious that a resonant peak appears in the propagation characteristics. This is because that the polarization direction of the right-hand wave is the same as the electron cyclotron direction. The frequency of the peak has a blue shift with the increasing of the magnetic field intensity, which is equal to the electron cyclotron resonance frequency. In our condition, the resonance frequencies are 0.14 THz, 0.28 THz and 0.42 THz at B = 5 T, 10 T and 15 T, respectively, and the range is about 0.09 THz. This can be used to modulate the communication band in plasma sheath by adjusting the external magnetic field. 5. Conclusion In this work, the flow field distributions of the RAM C-III hypersonic vehicle at different Mach numbers are simulated. Then the electron density of the plasma sheath is calculated by applying 7-species reacting equilibrium constant method. On this basis, the propagation characteristics of THz wave in plasma sheath are analyzed. In addition, the effects of external magnetic field on the lefthand and right-hand polarized THz wave in plasma sheath are also discussed. The results show that the plasma electron density increases with Mach number. The transmittance of THz wave in plasma sheath increases with the frequency and decreases with the incident angle, while the reflectance and absorbance change on the contrary. For the left-hand polarized THz wave, the propagation capacity significantly improves with the external magnetic field. And for the right-hand polarized THz wave, a resonance in
Fig. 7. Propagation characteristics of the left-hand polarized THz wave in plasma sheath with different magnetic field intensities. (a) Transmittance, (b) reflectance, (c) absorbance. 7
Optik - International Journal for Light and Electron Optics 208 (2020) 164090
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Fig. 8. Propagation characteristics of the right-hand polarized THz wave in plasma sheath with different magnetic field intensities. (a) Transmittance, (b) reflectance, (c) absorbance.
transmittance, reflectance and absorbance occurs at the electron cyclotron frequency, which can be modulated by the magnetic field. These results provide an effective way to study the hypersonic vehicle plasma sheath and the propagation characteristics of THz wave. And THz wave has a great potential application in solving the problem of blackout in hypersonic vehicle communication and detection. Conflict of interest The authors declare that there is no conflict of interest. Acknowledgements This work is supported by the Equipment Pre-Research Foundation (No. 614041 5010202) and the National Natural Science Foundation of China (No. 61705162). References [1] M. Keidar, M. Kim, I.D. Boyd, Electromagnetic reduction of plasma density during atmospheric reentry and hypersonic flights, J. Spacecr. Rockets 45 (3) (2008) 445–453. [2] Y. Takahashi, K. Yamada, T. Abe, Prediction performance of blackout and plasma attenuation in atmospheric reentry demonstrator mission, J. Spacecr. Rockets 51 (6) (2014) 1954–1964. [3] C.J. Schexnayder Jr., J.S. Evans, P.W. Huber, Comparison of theoretical and experimental electron density for RAM C flights, NASA Spec. Publ. 252 (1971) 277. [4] P. Huber, N. Akey, W. Croswell, C. Swift, The entry plasma sheath and its effects on space vehicle electromagnetic systems, NASA Tech. Note 252 (1971) 1–630. [5] I. Belov, V.Y. Borovoy, V. Gorelov, A. Kireev, A. Korolev, E. Stepanov, Investigation of remote antenna assembly for radio communication with reentry vehicle, J. Spacecr. Rockets 38 (2) (2001) 249–256. [6] C. Thoma, D. Rose, C. Miller, R. Clark, T. Hughes, Electromagnetic wave propagation through an overdense magnetized collisional plasma layer, J. Appl. Phys. 106 (4) (2009) 043301. [7] M. Kim, M. Keidar, I.D. Boyd, Analysis of an electromagnetic mitigation scheme for reentry telemetry through plasma, J. Spacecr. Rockets 45 (6) (2008) 1223–1229. [8] K. Yuan, J. Chen, L. Shen, X. Deng, M. Yao, L. Hong, Impact of reentry speed on the transmission of obliquely incident THz waves in realistic plasma sheaths, IEEE Trans. Plasma Sci. 46 (2) (2018) 373–378. [9] J. Chen, K. Yuan, L. Shen, X. Deng, L. Hong, M. Yao, Studies of terahertz wave propagation in realistic reentry plasma sheath, Prog. Electromagn. Res. 157 (2016) 21–29. [10] S. Lei, Z. Lei, Y. Bo, L. Xiaoping, Telemetry channel capacity assessment for reentry vehicles in plasma sheath environment, Plasma Sci. Technol. 17 (12) (2015) 1006. [11] L. Min, X. Haojun, W. Xiaolong, L. Hua, S. Huimin, S. Quan, Z. Yanhua, Numerical and experimental investigation on the attenuation of electromagnetic waves in unmagnetized plasmas using inductively coupled plasma actuator, Plasma Sci. Technol. 17 (10) (2015) 847. [12] H. Zhou, X. Li, K. Xie, Y. Liu, B. Yao, W. Ai, Characteristics of electromagnetic wave propagation in time-varying magnetized plasma in magnetic window region of reentry blackout mitigation, AIP Adv. 7 (2) (2017) 025114. [13] C. Yuan, Z. Zhou, X. Xiang, H. Sun, H. Wang, M. Xing, Z. Luo, Propagation properties of broadband terahertz pulses through a bounded magnetized thermal plasma, Nucl. Instrum. Methods Phys. Res. B 269 (1) (2011) 23–29. [14] Y. Tian, Y. Han, Y. Ling, X. Ai, Propagation of terahertz electromagnetic wave in plasma with inhomogeneous collision frequency, Phys. Plasmas 21 (2) (2014) 023301. [15] J.-X. Liu, L. Ju, P. Du, Y.-J. Liu, H.-W. Yang, An improved cascaded SO-FDTD method for high temperature magnetized plasma, Comput. Phys. Commun. 235 (2019) 153–158. [16] L. Zheng, Q. Zhao, S. Liu, X. Xing, Y. Chen, Theoretical and experimental studies of terahertz wave propagation in unmagnetized plasma, J. Infrared Millim. Terahetrz Waves 35 (2) (2014) 187–197. [17] C. Yuan, Z. Zhou, X. Xiang, H. Sun, S. Pu, Propagation of broadband terahertz pulses through a dense-magnetized-collisional-bounded plasma layer, Phys. Plasmas 17 (11) (2010) 113304. [18] G. He, Y. Zhan, N. Ge, Adaptive transmission method for alleviating the radio blackout problem, Prog. Electromagn. Res. 152 (2015) 127–136. [19] M. Wang, H. Li, Y. Dong, G. Li, B. Jiang, Q. Zhao, J. Xu, Propagation matrix method study on THz waves propagation in a dusty plasma sheath, IEEE Trans. Antennas Propag. 64 (1) (2015) 286–290. [20] L. Zhao, W.M. Bao, C.Y. Gong, An overview of the research of plasma sheath, Adv. Mater. Res. 1049–1050 (2014) 1518–1521. [21] L.J. Guo, L. Guo, J. Li, Propagation of terahertz electromagnetic waves in a magnetized plasma with inhomogeneous electron density and collision frequency, Phys. Plasmas 24 (2) (2017) 022108. [22] Y. Zhang, H.-Y. Xu, Z.-K. Yang, C.-K. Zhu, Q.-K. Xi, H.-W. Yang, Analysis of the transmission properties of terahertz wave in the plasma medium, Plasmonics 12 (3) (2017) 921–927. [23] E. Soltanmoradi, B. Shokri, V. Siahpoush, Study of electromagnetic wave scattering from an inhomogeneous plasma layer using Green's function volume integral equation method, Phys. Plasmas 23 (3) (2016) 033304.
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Propagation characteristics of terahertz wave in hypersonic plasma sheath considering high temperature air chemical reactions
T
Kai Chena,b, Degang Xua,b, Jining Lia,b,*, Xingning Genga,b, Kai Zhonga,b, Jianquan Yaoa,b a b
School of Precision Instrument and Optoelectronics Engineering, Institute of Laser and Optoelectronics, Tianjin University, Tianjin 300072, China Key Laboratory of Optoelectronics Information Technology (Ministry of Education), Tianjin University, Tianjin 300072, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Terahertz wave Propagation characteristics Plasma sheath Scattering matrix method
In this work, the propagation characteristics of terahertz wave in the hypersonic plasma sheath is demonstrated and analyzed based on the realistic vehicle model and the high temperature air thermal chemical reactions. Based on the simulated flow field distribution around the hypersonic vehicle, the electron densities of the plasma sheath at different Mach numbers are calculated by adopting the equilibrium constant method of the 7-species reacting air model. And then the propagation model of terahertz wave in plasma sheath is established using the scattering matrix method. The calculated and simulated results show that the plasma electron density is strongly influenced by Mach number, and the higher frequency and smaller incident angle of terahertz wave are advantageous for a better transmittance and lower loss. Furthermore, the effects of an external magnetic field on the propagation characteristics of left-hand and right-hand polarized terahertz wave in plasma sheath are studied. The external magnetic field indeed reduces the transmission loss of the left-handed wave. This work promises that terahertz waves have good potential applications in the detection and the communication for the hypersonic vehicles in plasma sheath.
1. Introduction When the vehicles are flying at a hypersonic speed, the high-temperature and high-pressure environment is formed outside the vehicles surface due to the fierce friction of the atmosphere, which ionizes the surrounding air and produces a plasma sheath surrounding the hypersonic vehicles. The presence of the sheath impedes or even interrupts the propagation of the communication signals, which causes the famous “blackout” problem [1,2]. In addition, the plasma sheath invalidates the most electromagnetic wave radar. The rapid development of the space technology, especially the exploitation of the near space makes the plasma sheath a pressing problem to be solved. The NASA's RAM project in 1970s explores the attenuation of the microwave electromagnetic waves in plasma medium, and put forward several theories and methods to reduce the blackout problem [3,4]. Since then, many attempts have been made to reduce the impact of the plasma sheath on communication [5–14]. And many effective methods have been developed to study the propagation characteristics, including the finite difference time domain (FDTD) method [15], the analytic method [16], the impedance matching (IM) method [17], and so on [18–20]. Guo et al. investigated the propagation characteristics of terahertz (THz) wave in magnetized ⁎ Corresponding author at: School of Precision Instrument and Optoelectronics Engineering, Institute of Laser and Optoelectronics, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.ijleo.2019.164090 Received 15 November 2019; Accepted 17 December 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
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plasma with inhomogeneous electron density and collision frequency using scattering matrix method [21]. Zhang et al. investigated the transmission characteristics of THz wave in linear distribution plasma using Runge-Kutta exponential time differencing finitedifference time-domain (RKETD-FDTD) method [22]. The scattering of electromagnetic wave from bell-like and Epstein profiles plasma is studied by Soltanmoradi et al. using a Green function method [23]. The recent research shows that using the electromagnetic wave with higher frequencies than the plasma resonance frequency can reduce the shielding effect of plasma medium effectively. The frequency of THz wave with a range of 0.1–10 THz, which is corresponding to the plasma with the densities of 1020–1024 m−3, is much higher than the resonance frequency of the plasma sheath. And THz wave is more insensitive to temperature than the infrared rays, and has higher spatial resolution than the microwave. Besides, THz wave communication has the characteristics of large capacity, good directivity, confidentiality and strong anti-jamming performance. Therefore, THz technology provides an effective method to solve the blackout problem [24]. Most of the previous works focus on the relative ideal distribution of the plasma electron density and collision frequency and ignore the influence of atmospheric conditions and Mach number of hypersonic vehicles. Thus, in this work, the flow field distribution is from the RAM C-III model at the height of 30 km with different Mach number. And the plasma electron density is calculated through the equilibrium constant method considering 7-species high temperature chemical reactions in near space atmospheric environment. Scattering matrix method is used to establish the interaction model between THz wave and plasma sheath. The propagation characteristics of THz wave with different incident angles is studied. Furthermore, in order to further improve the transmission of THz wave, the propagation characteristics of left-hand and right-hand polarized THz wave in plasma sheath with different external magnetic field intensity are also analyzed. 2. Hypersonic vehicle flow field simulation As is well-known, the plasma electron density depends on the flow field distribution of pressure, temperature and air density around a hypersonic vehicle. In this work, the RAM C-III vehicle model [25] is selected as the hypersonic vehicle to simulate the flow field of the plasma sheath. The schematic of this model is shown in Fig. 1. The model can be seen as a blunt cone with a 159.5 mmradius spherical nose with a half-cone angle of 9∘, and a cone length of 1295.4 mm. When calculating the flow field around the vehicle, the boundary conditions are set according to the pressure and temperature of the atmosphere at the height of 30 km, where the background pressure and temperature are 11.80 mbar and 232.73 K respectively. The flow fields at Mach numbers of 9, 11, 13 and 15 conditions are simulated by the commercial software ANSYS Fluent using k-ε viscous model. And the results of the pressure (left column), temperature (middle column) and air density (right column) distributions are shown in Fig. 2. As shown in Fig. 2, the maximum values of three parameters all increases with the Mach number increasing. The pressure increases from 1.25 × 105 Pa to 5.62 × 105 Pa, the temperature increases from 4310 K to 5350 K, and the air density increases from 1.10 × 10−1 kg/m3 to 3.92 × 10−1 kg/m3. This is because that the increasing of Mach number leads to the enhanced friction between the vehicle and the air surrounding, which results in the enhancement of the chemical reactions. 3. Mathematical model 3.1. Equilibrium constant method The equilibrium chemistry constant method [26] is based on the basic principles of the chemical reactions, thermodynamics, and balances of the elemental and species. It is an effectively way to calculate the content of each particle in the high temperature flow field. Assuming the air in near space is only consists of nitrogen and oxygen, and the volume ratio is 79:21. In the 7-species gas model, when the air around the hypersonic vehicles is ionized, the independent chemical reactions include four reactions as
O2 ⇌ O + O,
N2 ⇌ N + N,
Fig. 1. Schematic of the hypersonic vehicle. 2
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Fig. 2. Flow field distribution of pressure, temperature and air density at different Mach numbers around the vehicle.
NO ⇌ N + O, O + N ⇌ NO+ + e−. The partial pressure of each component (O2, N2, O, N, NO, NO+, e−) is denoted as P1, P2, …, P7. According to Dalton law of partial pressures, the total pressure P satisfies (1)
P = P1 + P2 + P3 + P4 + P5 + P6 + P7. The charge balance equation gives as
(2)
P6 = P7. And the element balance equation gives as
(3)
2P1 + P3 + P5 + P 6 = α (2P2 + P4 + P5 + P6),
where α is the molar fraction ratio of elements nitrogen and oxygen, which equals to the volume ratio. Considering the four reactions, the following relations exist
K eq1 =
K eq2 =
K eq1 =
K eq1 =
P3 2 P P1 P
(4)
P4 2 P P2 P
(5)
( ),
( ), P3 P4 P P P5 P
,
P6 P 7 P P P3 P4 P P
,
(6)
(7)
where Keq1, Keq2, Keq3, Keq4 are the equilibrium constants in the appropriate units of the above reactions respectively. They can be expressed as
K eq, r = exp(B1r + B2r ln Z + B3r Z + B4r Z 2 + B5r Z 3),
(8) 3
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Fig. 3. Schematic diagram of THz wave propagation model in plasma.
where Z = 1000/T, T is the flow field temperature in the one-temperature model, and the constants Bir are presented in Ref. [27]. Solving all the equations above, the mole fraction and mass fraction of each component can be obtained under the conditions of T and P, then the electron number density of plasma sheath can be obtained by
ne = NA
ρY7 , M7
(9)
where NA is the Avogadro constant, ρ is the air density, Y7 is the mass fraction of electron, and M7 is the molar mass of electron [28]. 3.2. Scattering matrix method The scattering matrix method is an analytical method to solve electromagnetic waves propagation in the medium [29]. As shown in Fig. 3, it divides the inhomogeneous plasma medium into plenty of layers, and each layer is considered as homogeneous medium. So the reflection coefficient and transmission coefficient of the entire plasma medium can be obtained by solving the field distribution and the boundary conditions in each layer [30]. Many applications have shown that this method is an effective way to deal with the electromagnetic wave propagation in plasma sheath. 2 + ωpi2 , where ωpe and ωpi are the electron and ion oscillation frequencies, The plasma frequency is represented as ωp = ωpe respectively, and ωpe ≫ ωpi. So the plasma frequency of mth layer can be expressed as
ωp, m ≈ ωpe, m =
ne, m e 2 , ε 0 me
(10) −12
−27
where ne is the electron density, ε0 = 8.85 × 10 F/m is the vacuum permittivity, me = 1.67 × 10 kg is the electron mass, mi is the ion mass, e = 1.6 × 10−19 C is the electron charge. The permittivity of plasma sheath outside the vehicles is ε = ε0εr, where εr is the relative permittivity of the plasma medium. In the mth layer it can be expressed as
εr , m = 1 −
2 ωp2, m (ω ± ωce) ω [(ω ± ωce)2 + νen, m]
− ωp2, m νen, m , j 2 ω (ω ± ωce)2 + νen, m
(11)
where ω is the angular frequency of the incident wave, and νen is the plasma collision frequency, which is calculated by the empirical formula [31]
νen = 5.8 × 1012T −1/2P ,
(12)
where T is the flow field temperature in degrees Kelvin, P is the flow field pressure in degrees atm. ωce is the electron cyclotron frequency expressed as ωce = eB/me, where B is the external magnetic induction intensity. The ‘ ± ’ sign indicates the polarization direction: ‘+’ means the left-hand polarization and ‘−’ means the right-hand polarization. The plasma wave number is expressed as km = k 0 εr , m , where k0 = ω/c is the wave number in vacuum, c = 3 ×108 m/s is the speed of light in vacuum. For the electromagnetic waves in the oblique incidence condition, km = kθ,m cosθm, where θm is the angle between the electromagnetic wave in the mth layer and the normal of the incident plane. And there is
sin θm =
εr , m − 1 sin θm − 1. εr , m
(13)
Considering that the system state is stable, a THz wave propagates along the xOz plane and from air into the plasma medium. And the electric field of the THz wave is parallel to the plane, the magnetic field is perpendicular to the plane. The schematic diagram of the THz wave propagation model in multi-layer plasma sheath is shown in Fig. 3. The inhomogeneous plasma sheath is divided into n layers. 4
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The total electric field in the region (0) can be expressed as
Ez,0 = E0 (e−jk 0 z + Ae jk 0 z ).
(14)
And the total electric field in the mth layer can be expressed as
Ez, m = E0 (Bm e−jkm z + Cm e jkm z ).
(15)
Only the transmitted wave exists in region (n + 1), so the electric field is
Ez, n + 1 = E0 De−jkn + 1 z ,
(16)
where A is the total reflection coefficient, and D is the total transmission coefficient. Bm is the transmission coefficient and Cm is the reflection coefficient of the mth layer. According to the continuity of electric field at the interface of different media
⎛ Bm ⎞ = Sm ⎛ Bm − 1⎞. ⎝ Cm ⎠ ⎝ Cm − 1 ⎠ ⎜
⎟
⎜
⎟
(17)
The scattering matrix of the mth layer Sm can be expressed by −1
e−jkm dm e jkm dm ⎞ Sm = ⎜⎛ ⎟ −jkm dm − k e jkm dm k e m ⎝ m ⎠ e−jkm − 1 dm − 1 e jkm − 1 dm − 1 ⎞ ⎛ ×⎜ ⎟. −jk dm − 1 − k jk dm − 1 m−1 e m−1 ⎝ km − 1 e m − 1 ⎠
(18)
At the incident plane
⎛ B1⎞ = S1 A . 1 ⎝ C1 ⎠ ⎜
()
⎟
(19)
And at the exit plane
⎛ Bn ⎞ = Vp·D , ⎝ Cn ⎠ ⎜
⎟
(20)
where Vp is expressed by
Vp =
1 ⎛ kn + kn + 1 e j (kn− kn + 1) dp ⎞ ⎜ ⎟. 2kn ⎝ kn − kn + 1 e j (kn+ kn + 1) dp ⎠
(21)
The global scattering matrix Sg is given by 2
⎞ ⎛ Sg = ⎜ ∏ Sm ⎟ S1. ⎝m=n ⎠
(22)
So the entire propagation can be expressed as
()
Sg A = Vp·D . 1
(23)
Sg can also be expressed as Sg = (Sg1, Sg2), so Eq. (23) can be transformed as
( DA) = −(S
g1
− Vp)−1·Sg 2,
(24)
Then the total reflection and transmission coefficients A and D can be obtained. Consequently, the reflectance, transmittance and absorbance are 2 ⎧ R = |A| T = |D|2 ⎨ ⎩Q = 1 − R − T
(25)
4. Result and discussion The pressure, temperature and air density values of the flow field is extracted in the direction perpendicular to the vehicle surface at the position of x = 350 mm, as the red dash line shown in Fig. 1. The distribution of the plasma electron density and the collision frequency are shown in Fig. 4. According to Fig. 4, with the increasing of the distance to the vehicle surface, the electron density decreases firstly, then increases to an extreme value, and decreases again at last. The extreme value increases with Mach number. This is because that with the 5
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Fig. 4. Distribution of plasma electron density and collision frequency at different Mach number. (a) Plasma electron density and (b) plasma collision frequency.
compression of the shock wave and the viscous action of air intensify and the rate of the chemical reactions in air accelerate increasing, more and more gas molecules are ionized. The blunt nose diffuses the compressed and heated air from the vehicle head to a certain distance away from the vehicle, and the distance increases as Mach number increasing. The maximum value of the electron density in the vehicle surface at Mach numbers of 9, 11, 13 and 15 conditions are 3.30 × 1011 cm−3, 6.90 × 1012 cm−3, 1.008 × 1013 cm−3 and 1.78 × 1013 cm−3, respectively. As shown in Fig. 4(b), the maximum collision frequency also increases with the Mach number. This is due to the fact that the collision rate between particles accelerates. With the distance to the vehicle surface increasing, the collision frequency has a peak value, which attributes to the combined action of the temperature and pressure according to Eq. (12). 4.1. Effects of Mach number on propagation characteristics The transmittance, reflectance and absorbance of THz wave in plasma sheath with different Mach numbers are shown in Fig. 5. The reflectance is scaled in decibels to a better show about the oscillation characteristics. In Fig. 5, the transmittance decreases with Mach number and increases with the THz frequency of the incident wave. The reflectance and absorbance increase with Mach number and decrease with the THz frequency. The decibel scale of reflectance has a periodic oscillation, which is due to the multiple reflections at the plasma boundaries. The xth order resonance occurs when k = xπ/d [23], and the oscillation frequency in the present conditions is 0.03 THz. For the increasing of Mach number results in the increasing of plasma electron density, which leads to the increasing of electron quantity. Therefore, the absorption of electrons in plasma makes the transmittance decreasing and the reflectance and absorbance increasing. Due to the frequency increasing of the incident THz wave, the rate of the periodic change of the electromagnetic field is accelerated. However, the electron oscillation frequency cannot catch up with the THz wave, which results in less absorption and the transmittance increasing. At the frequency of 0.83 THz, the transmittance at Mach number of 15 condition reaches 95%. Therefore, the communication system operating at THz band can reduce the impact of blackout effectively. 4.2. Effects of incident angle on propagation characteristics The effect of the incident angle on the propagation characteristics of THz wave in plasma sheath are calculated in Mach number of 15 condition. The results of the transmittance, reflectance and absorbance are shown in Fig. 6. As shown in Fig. 6, the transmittance decreases while the reflectance and absorbance increase with the incident angle increasing. When the THz wave incident obliquely, the horizontal component of the electromagnetic wave must be continuous under the constraint of Snell law. A part of the electromagnetic wave will be reflected without entering the plasma, thus the reflectance increases. Meanwhile, when increasing the incident angle, the length of THz wave interacting with the plasma increases. So more
Fig. 5. Propagation characteristics of THz wave in plasma sheath with different Mach numbers. (a) Transmittance, (b) reflectance, (c) absorbance. 6
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Fig. 6. Propagation characteristics of THz wave in plasma sheath with different incident angles. (a) Transmittance, (b) reflectance, (c) absorbance.
energy is consumed in the plasma sheath, which causes the transmittance decreasing and absorbance increasing. As shown in Fig. 6(b), the oscillation frequencies at the incident angles of 30∘, 60∘, 80∘ are 0.040 THz, 0.065 THz and 0.175 THz, which is also because of the interacting length change. At below 0.28 THz range of 80∘ incident angle condition, the absorbance increases with THz frequency. This is because that the transmission is neglectable and the reflection plays a major role in low frequency range, and the absorbance increases with the decreasing of the reflectance. For the incident angle of 0∘ condition, the transmittance reaches 90% at 0.58 THz. However, for 80∘ incident angle condition, the frequency is 1.40 THz for the same transmittance. Therefore, a smaller incident angle induces a smaller communication loss at the same frequency. 4.3. Effects of external magnetic field on propagation characteristics The calculation results of the propagation characteristics of the left-hand polarized and right-hand polarized THz wave in plasma sheath are presented in Figs. 7 and 8, respectively, with different external magnetic field intensities in the Mach 15 and 0∘ incident angle condition. The direction of the external magnetic field is parallel to the incident wave. As shown in Fig. 7, for the left-hand polarized THz wave, the transmittance increases while the reflectance and absorbance decrease with the magnetic field intensity increasing. Because the polarization direction of the left-hand wave is opposite to the electron cyclotron direction in plasma and the increasing of the magnetic field intensity rises the electron cyclotron frequency, less of the THz wave energy is absorbed by the electrons, which reduces the transmission loss. The transmittance in unmagnetized plasma sheath is 43.9% at 0.2 THz, while the transmittance rises as high as 91.2% with 15 T magnetic field intensity. Therefore, the application of an external magnetic field can greatly improve the transmission of the left-hand polarized THz wave in plasma sheath. As shown in Fig. 8, for the right-hand polarized THz wave, the transmittance decreases while the reflectance and absorbance increase with the magnetic field intensity increasing. It is obvious that a resonant peak appears in the propagation characteristics. This is because that the polarization direction of the right-hand wave is the same as the electron cyclotron direction. The frequency of the peak has a blue shift with the increasing of the magnetic field intensity, which is equal to the electron cyclotron resonance frequency. In our condition, the resonance frequencies are 0.14 THz, 0.28 THz and 0.42 THz at B = 5 T, 10 T and 15 T, respectively, and the range is about 0.09 THz. This can be used to modulate the communication band in plasma sheath by adjusting the external magnetic field. 5. Conclusion In this work, the flow field distributions of the RAM C-III hypersonic vehicle at different Mach numbers are simulated. Then the electron density of the plasma sheath is calculated by applying 7-species reacting equilibrium constant method. On this basis, the propagation characteristics of THz wave in plasma sheath are analyzed. In addition, the effects of external magnetic field on the lefthand and right-hand polarized THz wave in plasma sheath are also discussed. The results show that the plasma electron density increases with Mach number. The transmittance of THz wave in plasma sheath increases with the frequency and decreases with the incident angle, while the reflectance and absorbance change on the contrary. For the left-hand polarized THz wave, the propagation capacity significantly improves with the external magnetic field. And for the right-hand polarized THz wave, a resonance in
Fig. 7. Propagation characteristics of the left-hand polarized THz wave in plasma sheath with different magnetic field intensities. (a) Transmittance, (b) reflectance, (c) absorbance. 7
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Fig. 8. Propagation characteristics of the right-hand polarized THz wave in plasma sheath with different magnetic field intensities. (a) Transmittance, (b) reflectance, (c) absorbance.
transmittance, reflectance and absorbance occurs at the electron cyclotron frequency, which can be modulated by the magnetic field. These results provide an effective way to study the hypersonic vehicle plasma sheath and the propagation characteristics of THz wave. And THz wave has a great potential application in solving the problem of blackout in hypersonic vehicle communication and detection. Conflict of interest The authors declare that there is no conflict of interest. Acknowledgements This work is supported by the Equipment Pre-Research Foundation (No. 614041 5010202) and the National Natural Science Foundation of China (No. 61705162). References [1] M. Keidar, M. Kim, I.D. Boyd, Electromagnetic reduction of plasma density during atmospheric reentry and hypersonic flights, J. Spacecr. Rockets 45 (3) (2008) 445–453. [2] Y. Takahashi, K. Yamada, T. Abe, Prediction performance of blackout and plasma attenuation in atmospheric reentry demonstrator mission, J. Spacecr. Rockets 51 (6) (2014) 1954–1964. [3] C.J. Schexnayder Jr., J.S. Evans, P.W. Huber, Comparison of theoretical and experimental electron density for RAM C flights, NASA Spec. Publ. 252 (1971) 277. [4] P. Huber, N. Akey, W. Croswell, C. Swift, The entry plasma sheath and its effects on space vehicle electromagnetic systems, NASA Tech. Note 252 (1971) 1–630. [5] I. Belov, V.Y. Borovoy, V. Gorelov, A. Kireev, A. Korolev, E. Stepanov, Investigation of remote antenna assembly for radio communication with reentry vehicle, J. Spacecr. Rockets 38 (2) (2001) 249–256. [6] C. Thoma, D. Rose, C. Miller, R. Clark, T. Hughes, Electromagnetic wave propagation through an overdense magnetized collisional plasma layer, J. Appl. Phys. 106 (4) (2009) 043301. [7] M. Kim, M. Keidar, I.D. Boyd, Analysis of an electromagnetic mitigation scheme for reentry telemetry through plasma, J. Spacecr. Rockets 45 (6) (2008) 1223–1229. [8] K. Yuan, J. Chen, L. Shen, X. Deng, M. Yao, L. Hong, Impact of reentry speed on the transmission of obliquely incident THz waves in realistic plasma sheaths, IEEE Trans. Plasma Sci. 46 (2) (2018) 373–378. [9] J. Chen, K. Yuan, L. Shen, X. Deng, L. Hong, M. Yao, Studies of terahertz wave propagation in realistic reentry plasma sheath, Prog. Electromagn. Res. 157 (2016) 21–29. [10] S. Lei, Z. Lei, Y. Bo, L. Xiaoping, Telemetry channel capacity assessment for reentry vehicles in plasma sheath environment, Plasma Sci. Technol. 17 (12) (2015) 1006. [11] L. Min, X. Haojun, W. Xiaolong, L. Hua, S. Huimin, S. Quan, Z. Yanhua, Numerical and experimental investigation on the attenuation of electromagnetic waves in unmagnetized plasmas using inductively coupled plasma actuator, Plasma Sci. Technol. 17 (10) (2015) 847. [12] H. Zhou, X. Li, K. Xie, Y. Liu, B. Yao, W. Ai, Characteristics of electromagnetic wave propagation in time-varying magnetized plasma in magnetic window region of reentry blackout mitigation, AIP Adv. 7 (2) (2017) 025114. [13] C. Yuan, Z. Zhou, X. Xiang, H. Sun, H. Wang, M. Xing, Z. Luo, Propagation properties of broadband terahertz pulses through a bounded magnetized thermal plasma, Nucl. Instrum. Methods Phys. Res. B 269 (1) (2011) 23–29. [14] Y. Tian, Y. Han, Y. Ling, X. Ai, Propagation of terahertz electromagnetic wave in plasma with inhomogeneous collision frequency, Phys. Plasmas 21 (2) (2014) 023301. [15] J.-X. Liu, L. Ju, P. Du, Y.-J. Liu, H.-W. Yang, An improved cascaded SO-FDTD method for high temperature magnetized plasma, Comput. Phys. Commun. 235 (2019) 153–158. [16] L. Zheng, Q. Zhao, S. Liu, X. Xing, Y. Chen, Theoretical and experimental studies of terahertz wave propagation in unmagnetized plasma, J. Infrared Millim. Terahetrz Waves 35 (2) (2014) 187–197. [17] C. Yuan, Z. Zhou, X. Xiang, H. Sun, S. Pu, Propagation of broadband terahertz pulses through a dense-magnetized-collisional-bounded plasma layer, Phys. Plasmas 17 (11) (2010) 113304. [18] G. He, Y. Zhan, N. Ge, Adaptive transmission method for alleviating the radio blackout problem, Prog. Electromagn. Res. 152 (2015) 127–136. [19] M. Wang, H. Li, Y. Dong, G. Li, B. Jiang, Q. Zhao, J. Xu, Propagation matrix method study on THz waves propagation in a dusty plasma sheath, IEEE Trans. Antennas Propag. 64 (1) (2015) 286–290. [20] L. Zhao, W.M. Bao, C.Y. Gong, An overview of the research of plasma sheath, Adv. Mater. Res. 1049–1050 (2014) 1518–1521. [21] L.J. Guo, L. Guo, J. Li, Propagation of terahertz electromagnetic waves in a magnetized plasma with inhomogeneous electron density and collision frequency, Phys. Plasmas 24 (2) (2017) 022108. [22] Y. Zhang, H.-Y. Xu, Z.-K. Yang, C.-K. Zhu, Q.-K. Xi, H.-W. Yang, Analysis of the transmission properties of terahertz wave in the plasma medium, Plasmonics 12 (3) (2017) 921–927. [23] E. Soltanmoradi, B. Shokri, V. Siahpoush, Study of electromagnetic wave scattering from an inhomogeneous plasma layer using Green's function volume integral equation method, Phys. Plasmas 23 (3) (2016) 033304.
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