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Transmission characteristics of terahertz waves propagation in magnetized plasma using the WKB method
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Transmission characteristics of terahertz waves propagation in magnetized plasma using the WKB method
T
⁎
Huan Yua, Guanjun Xua,b,c,d, , Zhengqi Zhenga,b,c a
School of Information Science and Technology, East China Normal University, Shanghai 200241, China Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, Shanghai 200241, China Engineering Center of SHMEC for Space Information and GNSS, East China Normal University, Shanghai 200241, China d Peng Cheng Laboratory, Shenzhen 518052, China b c
A R T IC LE I N F O
ABS TRA CT
Keywords: Terahertz waves propagation Magnetized plasma Transmittance Attenuation WKB
The communication between a ground station and an aircraft is seriously affected by the plasma sheath when the aircraft re-enters the Earth. The transmission characteristics of terahertz waves in a plasma sheath with magnetized plasma are studied in this paper. The well-developed Wentzel–Kramers–Brillouin (WKB) method is used to calculate the transmittance and attenuation of electromagnetic waves in plasma. Numerical simulation results demonstrate that the plasma density, plasma slab thickness, and collision frequency significantly influence the attenuation and transmittance of terahertz waves. Furthermore, for terahertz waves through the plasma, the peak value of the attenuation and transmittance occur at different external magnetic field strengths. Moreover, the effects of the plasma density models on the transmission characteristics are compared. These results are of great significance for improving the communication quality between the ground station and the aircraft, and will provide a potential theoretical basis for solving the ‘blackout’ problem.
1. Introduction When a high-speed aircraft returns to the Earth, the speed of flight can reach tens of times the speed of sound. The front end of the vehicle forms strong shock waves, and a lot of kinetic energy is converted into thermal energy. This high temperature ionizes the materials of the aircraft and produces a plasma sheath around the aircraft [1]. The plasma sheath will attenuate, absorb, and reflect the electromagnetic (EM) waves during the re-entry process. Therefore, the communication performance is seriously degraded by the plasma sheath, which leads to a communication failure between the ground station and the aircraft [2,3]. During this period, the aircraft will be in a state of blind monitoring, with interruption communications. This phenomenon is referred to as the ‘blackout’ problem [4,5]. Therefore, it is of great importance to study the characteristics of EM waves propagation in plasma. The research into EM waves propagation in plasma slabs in recent years has shown that a high-frequency EM wave is more able to penetrate the plasma. Therefore, plenty of researchers have focused on the terahertz frequencies [6,7]. Yuan et al. studied the absorption properties of an EM wave in the range of 0.1–0.2 THz in a magnetized homogeneous plasma [8]. Zheng first investigated the transmission characteristics, such as transmittance, reflectance, and absorbance, in [9], for a terahertz wave passing through a non-magnetized homogeneous plasma. After that, the oscillation period of the reflectivity curve was obtained and the transmission properties were also verified in a shock tube. In [10], the effects of different parameters, such as the plasma density, collision frequency, slab
⁎
Corresponding author. E-mail address: [email protected] (G. Xu).
https://doi.org/10.1016/j.ijleo.2019.05.061 Received 14 March 2019; Accepted 20 May 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
Fig. 1. The physics model of terahertz wave propagation through plasma.
thickness, and the frequency of the wave, on the absorption rate of terahertz waves in a magnetized and non-uniform plasma were also discussed. In addition to the scattering matrix method (SMM) used in the above studies, another two methods, Wentzel–Kramers–Brillouin (WKB) method and finite-difference time-domain (FDTD) method have been applied to investigate the characteristics of EM wave propagation in a magnetic and non-uniform plasma [11–13]. Tian et al. studied the attenuation characteristics of terahertz waves in plasmas using the WKB method, assuming that the plasma density model is double exponential [14]. Note that the plasma density model is usually simplified to be a parabolic distribution or a Gaussian distribution [15,16]. In the above research, only the right-hand polarized wave passing through the plasma is considered. However, both the right-hand polarized wave and lefthand polarized wave should be taken into account simultaneously. In addition, the effects of the plasma under different distribution models should be further analyzed. In the present paper, we mainly investigate the attenuation properties of the terahertz waves passing through a magnetic nonuniform plasma. The well-developed WKB method is applied. In addition, the influence of various plasma parameters on the absorbance and transmittance is analyzed. The rest of this paper is organized as follows: Section 2 introduces the physical model and the use of the WKB method. Numerical simulations and analyses of the relevant results are presented in Section 3. The final conclusions of this paper are summarized in Section 4. 2. Physical model The model of the physics of the plasma slab is shown in Fig. 1. Assume that the physics model is divided into free space, a plasma layer, and free space from left to right. We also assume that the plasma in the range of the spatial region 0 ≤ z ≤ d is magnetized and the direction of the magnetic field is parallel to the z-axis. Suppose that the incident wave is a plane wave and enters the non-uniform magnetized plasma vertically along the xOz-plane, which means the angle of incident is zero, i.e., θ1 = θ2 = θ3 = 0. In fact, the relative permittivity of the plasma plays a crucial role in analyzing the transmission characteristic of a terahertz wave through a plasma slab [17,18]. According to the Appleton's formula, the expression for the relative permittivity of the magnetized plasma can be expressed as [20]:
εr = 1 −
ωp2 / ω2 1−j
ven ω
±
ωce ω
. (1)
e 2/ ε
where ωp = Ne 0 me is the electron plasma frequency and ven represents the collision frequency. Ne is defined to the electron density. ωce = eB/me and ω = 2πf denote as the electron cyclotron frequency and the wave angular frequency, respectively. e denotes the charge of electron and the mass of electron is me. ϵ0 is the vacuum permittivity, and B is the symbol of the magnetic strength. The symbols of ‘+’ and ‘−’ mean left-hand and right-hand polarized, respectively. In contrast to the former mentioned research [8,9], both the left-hand polarized waves and right-hand polarized waves propagate in the plasma are considered in this paper. According to Maxwell's equations, the wave equation in the plasma slab can be written as
→ → ∇2 E + (kp2 + k 0 2) E = 0,
(2)
Here, kp = ω/ c εr is the wave number in the plasma and k0 = ω/c is the wave number in free space. The electric field equation of an EM wave passing through the plasma is derived by the WKB method as follows [19].
Ey (z ) = Ey (0) exp ⎡j ⎣
∫
kp2 (z ) − k 0 2 (z ) dz⎤, ⎦
(3)
where Ey (0) is the electric field intensity at the incident boundary of the plasma slab, and Ey (z ) represents the electric field intensity when the EM waves have passed the plasma. z is the position of the plasma slab. The coefficient of j in this expression can be denoted by Im(·). According to Eq. (3), the transmittance coefficient, T, can be obtained when the EM wave reaches the right boundary of the plasma region as follows. 245
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
Fig. 2. Different plasma density distribution models versus the plasma thickness.
T=
Ey (d ) Ey (0)
{
∫0
= exp −2Im ⎡ ⎣
z
kp2 (z ) − k 0 2 (z ) dz⎤ ⎦
}
(4)
Note that Eq. (4) denotes the total attenuation after the EM wave has propagated through the whole of the plasma slab. Therefore, the attenuation (Att) can also be obtained as following
{
∫0
Att = −10 log T = 8.686 Im ⎡ ⎣
z
}
kp2 (z ) − k 0 2 (z ) dz⎤ . ⎦
(5)
The above derivation yields the attenuation and transmittance of an EM wave passing through the plasma. In the following section, the influence of various parameters on the attenuation and transmittance of the EM waves propagating in the plasma slab will be analyzed based on Eqs. (5) and (4), respectively.
3. Numerical simulations and discussion Since the mass of the electron is much larger than that of the ions and other particles in the plasma, the plasma density can be approximately expressed by the electron density [15,20]. Both research and experiments show that the plasma flow generated by the friction with the atmosphere has a smaller density of plasma outside and a larger density in the middle when the aircraft re-enters the atmosphere at high speed [19]. In this paper, four widely used density distribution models, namely, the uniform distribution, parabolic distribution, Gaussian distribution, and double exponential distribution, are used as presented in Fig. 2. The corresponding density functions are listed in Table 1. Note that the plasma density varies with the thickness d of the plasma in the z-axis direction. The maximum density is ne, and z0 is the midpoint of the plasma slab. In the following subsection, the influence of different plasma parameters, such as the maximum electron density ne, the plasma thickness d, the magnetic field strength B, and the collision frequency ven , on the attenuation and transmission of an terahertz wave in a magnetized plasma are first investigated with the parabolic distribution density model. After that, the attenuation and transmission characteristics of terahertz waves in plasma are further analyzed with different density models. The main parameters used in the simulations are given in Table 2.
Table 1 The plasma density models used in the simulations. Number
Distribution model
Plasma density function
Case 1 Case 2
Uniform Parabolic
Ne = ne
Case 3
Gaussian
e
Case 4
Double exponential
(
2
) + 1⎤⎥⎦ N = n exp ⎡−2.3 ( ) ⎤ ⎢ ⎥ ⎣ ⎦ ⎧ n exp ⎡0.9 − 2) − 0.9⎤, (z ≤ z ≤ d) ⎪ ⎢ ( ⎥ ⎣ ⎦ N = ⎨ ⎤ ⎡ − n exp 0.9 0.9 , ⎪ ( ) ⎥⎦ (0 ≤ z ≤ z ) ⎢ ⎣ ⎩ 1
Ne = ne ⎡− ⎢ ⎣ 2
z z0
−1
z z0
e
2
2
z z0
e
0
e
e
246
z z0
2
0
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
Table 2 Simulation parameters of the plasma slab. Parameters
Symbol (unit)
Values
Peak electron density Plasma thickness Collision frequency Magnetic field strength Wave frequency
ne (m3) d (cm) ven (THz) B (T) f (THz)
1 × 1016–1 × 1020 5–20 1 × 10−3–1 × 101 0.01–10 0.1–1
3.1. Effects of different magnetic field strengths on terahertz waves Fig. 3 shows the effects of different values of B on the attenuation and transmittance. The plasma model with parabolic distribution in Case 2 is adopted here. Note that the electron density function of the parabolic distribution is centrally symmetric. We also assume that the terahertz wave is vertically incident on the plasma. The plasma thickness is d = 10 cm, the collision frequency ven = 1 × 10−2 THz, and the peak electron density ne = 1 ×1018/m3. Fig. 3(a) and (b) depicted the case for a right-hand polarized wave throughout the plasma and Fig. 3(c) and (d) consider the lefthand polarized wave. The following results can be obtained by comparing Fig. 3(a) and (b). The attenuation first increases to its maximum value and then decreases in a certain frequency range when B ≥ 2 T. After that, the attenuation decreases with an increase in the frequency of the wave. On the contrary, the transmittance will be a minimum in a certain frequency range as shown in Fig. 3(b). As the magnetic field strength increases, the attenuation peak and transmittance valley move to a higher frequency. Note that the maximum point and minimum point appear later with an increase of magnetic field strength. In Fig. 3(c) and (d), the attenuation decreases with the increase of the frequency of the terahertz wave, while the transmittance increases during this process. In addition, with the amplitude of the magnetic field B increasing, the attenuation decreases while the transmittance increases. According to the above simulation results, the effect of absorption by the plasma on re-entry communication can be effectively mitigated by a rational use of the magnetic field strength and wave frequency.
3.2. Effects of different plasma parameters on terahertz waves Fig. 4 shows the effects of different peak electron densities on the attenuation and transmittance. The amplitude of the magnetic field is B = 1 T, which is constant throughout the whole plasma. Only the right-hand polarized wave is considered in the following
Fig. 3. Effects of different magnetic field strengths on attenuation and transmittance. 247
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
Fig. 4. Effects of different peak plasma densities on attenuation and transmittance.
simulations for simplicity. The attenuation of the plasma is depicted in Fig. 4(a) and the transmittance in Fig. 4(b) versus the maximum plasma density ven . The plasma thickness is d = 10 cm, the collision frequency ven = 1 × 10−2 THz. Intuitively, the attenuation of the terahertz wave throughout the non-uniform magnetized plasma decreases with the increase of plasma peak electron density. In addition, the signal at a higher frequency is more easily propagated through the plasma. From Fig. 4(b), we can see that the higher the electron density, the smaller the transmittance of the terahertz wave throughout the non-uniform magnetized plasma at the same frequency. For instance, the transmittance is so small that almost all the terahertz waves are absorbed when ne = 5 × 1018/m3. On the contrary, when ne = 1 × 1017/m3, the transmittance is nearly 1. This can be explained by the fact that the higher the plasma density, the more wave energy will be absorbed by the electrons and then transferred to neutral particles by collisions. Therefore, the attenuation of the terahertz wave increases. The effect of the plasma thickness on the propagation of terahertz wave in a plasma is illustrated in Fig. 5(a) and (b), respectively. The collision frequency ven = 1 × 10−2 THz, and the peak electron density ne = 1 ×1018/m3. As shown in Fig. 5(a), the attenuation decreases with an increase of the frequency of the wave, whereas the transmittance increases. In addition, a larger plasma thickness induces a higher attenuation and a lower transmittance as shown in Fig. 5. It is easy to understand that the terahertz wave has more difficulty passing through the plasma when the plasma thickness is large. The attenuation and transmittance of an terahertz wave passing through the plasma versus different collision frequencies ven are demonstrated in Fig. 6. The plasma thickness is d = 10 cm, and the peak electron density ne = 1 ×1018/m3. From Fig. 6(a), the attenuation changes little with an increase of the frequency once ven ≥ 0.5 THz. However, the attenuation decreases with increasing frequency when ven ≤ 0.1 THz. In addition, decreasing the collision frequency is an alternative way to decrease the attenuation and
Fig. 5. Effects of different plasma thicknesses on the attenuation and transmittance. 248
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
Fig. 6. Effects of different collision frequencies on attenuation and transmittance.
increase the transmittance, as shown in Fig. 6. This can be explained by the fact that less of the energy of the terahertz wave is absorbed when the collision frequency is very small.
3.3. Effects of different electron density models on terahertz waves In this subsection, the effects of different plasma density models (different cases in Table 1) on terahertz waves attenuation and transmittance are taken into account. The other plasma parameters are set as B = 1 T, d = 10 cm, ven = 0.1 THz. Note that the peak electron density in Fig. 7(a) and (b) are ne = 1 × 1018/m3, and it increases to ne = 5 × 1018/m3 in Fig. 7(c) and (d).
Fig. 7. Effects of different plasma densities on attenuation and transmittance. 249
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
According to Fig. 7(a), the attenuation of the propagation of an terahertz wave in plasma with a double exponential distribution (Case 4) is the highest, and the attenuation under the parabola distribution (Case 2) is close to that under the uniform distribution (Case 1), while the attenuation under a Gaussian distribution (Case 3) is dozens of times smaller than that under the uniform distribution at the same frequency of the incident wave. Form Fig. 7(b), we can see that some valleys appear when f ≤ 0.1 THz. In this region, the plasma with Gaussian distribution exhibits larger transmittance than the other distribution models. However, the transmittance of the plasma with these four distribution models all tend to 1 when f ≥ 0.8 THz. It can be explained that the plasma density with double-exponential distribution (Case 4) fluctuates sharply while the density with Gaussian distribution (Case 1) decreases along the propagation direction as shown in Fig. 2. In addition, we can conclude that the terahertz waves with higher frequency are more beneficial to propagate through plasma. The attenuation and transmittance in Fig. 7(c) and (d) have almost the same variation tendency with Fig. 7(a) and (b). However, their value are smaller than that in Fig. 7(a) and (b). In addition, the attenuation peaks due to the absorption resonance shift to the larger frequency in Fig. 7(c). This is understandable due to the larger peak plasma density in Fig. 7(c) and (d). From the above simulations and discussion, the attenuation and transmittance of the terahertz wave propagating in the plasma are obviously affected by the plasma density. Since the plasma density distribution model is directly related to the external structure of the flight, the flight speed, and the pressure around the flight, etc., therefore the influence of these factors on the attenuation and transmittance are very important and will be further investigated in our future studies. 4. Conclusion In this paper, the transmission characteristics, including the attenuation and transmittance, of terahertz waves propagation in a plasma slab with different plasma density distribution models have been analyzed. Simulation results demonstrate that the plasma density, plasma thickness, and collision frequency play a crucial role in the attenuation and transmittance of the terahertz wave. At a high collision frequency, the attenuation decreases significantly with an increase in the frequency of the incident wave. Both the attenuation peak and transmittance valley appear with an increase of magnetic field strength. In addition, the attenuation and transmission of terahertz waves propagation in a plasma with a parabolic distribution are similar to those under a uniform distribution, whereas the attenuation in a plasma with a parabolic distribution is far smaller than that of a double exponential distribution. According to the above analyses, the interruption of communication caused by the ‘blackout’ can be mitigated by adding an appropriate magnetic field and changing the plasma density distribution. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (under Grant No. 61801181, 61771197), the Open Research Fund of Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University. References [1] T.K. Statom, Plasma parameters from reentry signal attenuation, IEEE Trans. Plasma Sci. 46 (2018) 494–502. [2] Y.Y. Zhang, G.J. Xu, Z.Q. Zheng, Propagation of terahertz waves in a magnetized, collisional, and inhomogeneous plasma with the scattering matrix method, Optik 182 (2019) 618–624. [3] M. Kundrapu, J. Loverich, K. Beckwith, P. Stoltz, A. Shashurin, M. Keidar, Modeling radio communication blackout and blackout mitigation in hypersonic vehicles, J. Spacecr. Rockets. 52 (2015) 853–862. [4] L. Shi, B.W. Bai, Y.M. Liu, X.P. Li, Navigation antenna performance degradation caused by plasma sheath, J. Electromagn. Waves Appl. 27 (2013) 518–528. [5] J.T. Li, L.X. Guo, Research on electromagnetic scattering characteristics of reentry vehicles and blackout forecast model, J. Electromagn. Waves Appl. 26 (2012) 1767–1778. [6] A.V. Kulizhsky, Uniform integral representation for the fluctuating field propagating through the multi-scale media, Waves Random Complex Media 27 (2017) 289–307. [7] J.H. Booske, Plasma physics and related challenges of millimeter-wave-to-terahertz and high power microwave generation, Phys. Plasmas 15 (2008) 055502. [8] C.X. Yuan, Z.X. Zhou, H.G. Sun, et al., Propagation of broadband terahertz pulses through a dense-magnetized-collisional bounded plasma layer, Phys. Plasmas 17 (2010) 113304. [9] L. Zheng, L.Q. Zhao, S. Zhang, et al., Studies of terahertz wave propagation in non-magnetized plasma, Acta Phys. Sin. 61 (2012) 245202. [10] Y. Wang, C. Yuan, Y. Liang, J. Yao, Z. Zhou, Ponderomotive force induced nonlinear interaction between powerful terahertz waves and plasmas, Optik 175 (2018) 250–255. [11] B.J. Hu, G. Wei, L.L. Sheng, SMM analysis of reflection absorption and transmission from non-uniform magnetized plasma slab, IEEE Trans. Plasma Sci. 27 (1999) 1131–1136. [12] D.K. Kalluri, Frequency transformation of a whistler wave by a collapsing plasma medium in a cavity: FDTD solution, IEEE Trans. Antennas Propag. 57 (2009) 1921–1930. [13] J.F. Liu, H. Lv, Y.C. Zhao, Y. Fang, X.L. Xi, Stochastic PLRC-FDTD method for modeling wave propagation in unmagnetized plasma, IEEE Antennas Wireless Propag. Lett. 17 (2018) 1024–1028. [14] Y.X. Tian, W.Z. Yan, X.L. Gu, et al., Study on the attenuation characteristic of terahertz wave through a non-uniform magnetized plasma, J. Electromagn. Waves Appl. 32 (2018) 1249–1260. [15] J. Zhao, L.L. Zhang, T. Wu, et al., Wavelength scaling of terahertz wave absorption via preformed air plasma, IEEE Trans. Terahertz Sci. Technol. 6 (2016) 846–850. [16] J. Ji, K.F. Tong, H. Xue, P. Huang, Quadratic recursive convolution (QRC) in dispersive media simulation of finite-difference time-domain (FDTD), Optik 138 (2017) 542–549. [17] G.J. Xu, Z.H. Song, Interaction of terahertz waves propagation in a homogeneous, magnetized, and collisional plasma slab, Waves Random Complex Media (2018). [18] S. Kumar, R.K. Singh, R.P. Sharma, Strong terahertz generation by beating of two super Gaussian laser beams in axially magnetized plasma, IEEE Trans. Terahertz Sci. Technol. 6 (2016) 491–496. [19] L.J. Guo, L.X. Guo, J.T. Li, Propagation of electromagnetic waves on a relativistically moving nonuniform plasma, IEEE Antennas Wireless Propag. Lett. 16 (2017) 137–140. [20] P.K. Shukla, B. Eliasson, Nonlinear interactions between electromagnetic waves and electron plasma oscillations in quantum plasmas, Phys. Rev. Lett. 99 (2007) 096401.
250
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Transmission characteristics of terahertz waves propagation in magnetized plasma using the WKB method
T
⁎
Huan Yua, Guanjun Xua,b,c,d, , Zhengqi Zhenga,b,c a
School of Information Science and Technology, East China Normal University, Shanghai 200241, China Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, Shanghai 200241, China Engineering Center of SHMEC for Space Information and GNSS, East China Normal University, Shanghai 200241, China d Peng Cheng Laboratory, Shenzhen 518052, China b c
A R T IC LE I N F O
ABS TRA CT
Keywords: Terahertz waves propagation Magnetized plasma Transmittance Attenuation WKB
The communication between a ground station and an aircraft is seriously affected by the plasma sheath when the aircraft re-enters the Earth. The transmission characteristics of terahertz waves in a plasma sheath with magnetized plasma are studied in this paper. The well-developed Wentzel–Kramers–Brillouin (WKB) method is used to calculate the transmittance and attenuation of electromagnetic waves in plasma. Numerical simulation results demonstrate that the plasma density, plasma slab thickness, and collision frequency significantly influence the attenuation and transmittance of terahertz waves. Furthermore, for terahertz waves through the plasma, the peak value of the attenuation and transmittance occur at different external magnetic field strengths. Moreover, the effects of the plasma density models on the transmission characteristics are compared. These results are of great significance for improving the communication quality between the ground station and the aircraft, and will provide a potential theoretical basis for solving the ‘blackout’ problem.
1. Introduction When a high-speed aircraft returns to the Earth, the speed of flight can reach tens of times the speed of sound. The front end of the vehicle forms strong shock waves, and a lot of kinetic energy is converted into thermal energy. This high temperature ionizes the materials of the aircraft and produces a plasma sheath around the aircraft [1]. The plasma sheath will attenuate, absorb, and reflect the electromagnetic (EM) waves during the re-entry process. Therefore, the communication performance is seriously degraded by the plasma sheath, which leads to a communication failure between the ground station and the aircraft [2,3]. During this period, the aircraft will be in a state of blind monitoring, with interruption communications. This phenomenon is referred to as the ‘blackout’ problem [4,5]. Therefore, it is of great importance to study the characteristics of EM waves propagation in plasma. The research into EM waves propagation in plasma slabs in recent years has shown that a high-frequency EM wave is more able to penetrate the plasma. Therefore, plenty of researchers have focused on the terahertz frequencies [6,7]. Yuan et al. studied the absorption properties of an EM wave in the range of 0.1–0.2 THz in a magnetized homogeneous plasma [8]. Zheng first investigated the transmission characteristics, such as transmittance, reflectance, and absorbance, in [9], for a terahertz wave passing through a non-magnetized homogeneous plasma. After that, the oscillation period of the reflectivity curve was obtained and the transmission properties were also verified in a shock tube. In [10], the effects of different parameters, such as the plasma density, collision frequency, slab
⁎
Corresponding author. E-mail address: [email protected] (G. Xu).
https://doi.org/10.1016/j.ijleo.2019.05.061 Received 14 March 2019; Accepted 20 May 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
Fig. 1. The physics model of terahertz wave propagation through plasma.
thickness, and the frequency of the wave, on the absorption rate of terahertz waves in a magnetized and non-uniform plasma were also discussed. In addition to the scattering matrix method (SMM) used in the above studies, another two methods, Wentzel–Kramers–Brillouin (WKB) method and finite-difference time-domain (FDTD) method have been applied to investigate the characteristics of EM wave propagation in a magnetic and non-uniform plasma [11–13]. Tian et al. studied the attenuation characteristics of terahertz waves in plasmas using the WKB method, assuming that the plasma density model is double exponential [14]. Note that the plasma density model is usually simplified to be a parabolic distribution or a Gaussian distribution [15,16]. In the above research, only the right-hand polarized wave passing through the plasma is considered. However, both the right-hand polarized wave and lefthand polarized wave should be taken into account simultaneously. In addition, the effects of the plasma under different distribution models should be further analyzed. In the present paper, we mainly investigate the attenuation properties of the terahertz waves passing through a magnetic nonuniform plasma. The well-developed WKB method is applied. In addition, the influence of various plasma parameters on the absorbance and transmittance is analyzed. The rest of this paper is organized as follows: Section 2 introduces the physical model and the use of the WKB method. Numerical simulations and analyses of the relevant results are presented in Section 3. The final conclusions of this paper are summarized in Section 4. 2. Physical model The model of the physics of the plasma slab is shown in Fig. 1. Assume that the physics model is divided into free space, a plasma layer, and free space from left to right. We also assume that the plasma in the range of the spatial region 0 ≤ z ≤ d is magnetized and the direction of the magnetic field is parallel to the z-axis. Suppose that the incident wave is a plane wave and enters the non-uniform magnetized plasma vertically along the xOz-plane, which means the angle of incident is zero, i.e., θ1 = θ2 = θ3 = 0. In fact, the relative permittivity of the plasma plays a crucial role in analyzing the transmission characteristic of a terahertz wave through a plasma slab [17,18]. According to the Appleton's formula, the expression for the relative permittivity of the magnetized plasma can be expressed as [20]:
εr = 1 −
ωp2 / ω2 1−j
ven ω
±
ωce ω
. (1)
e 2/ ε
where ωp = Ne 0 me is the electron plasma frequency and ven represents the collision frequency. Ne is defined to the electron density. ωce = eB/me and ω = 2πf denote as the electron cyclotron frequency and the wave angular frequency, respectively. e denotes the charge of electron and the mass of electron is me. ϵ0 is the vacuum permittivity, and B is the symbol of the magnetic strength. The symbols of ‘+’ and ‘−’ mean left-hand and right-hand polarized, respectively. In contrast to the former mentioned research [8,9], both the left-hand polarized waves and right-hand polarized waves propagate in the plasma are considered in this paper. According to Maxwell's equations, the wave equation in the plasma slab can be written as
→ → ∇2 E + (kp2 + k 0 2) E = 0,
(2)
Here, kp = ω/ c εr is the wave number in the plasma and k0 = ω/c is the wave number in free space. The electric field equation of an EM wave passing through the plasma is derived by the WKB method as follows [19].
Ey (z ) = Ey (0) exp ⎡j ⎣
∫
kp2 (z ) − k 0 2 (z ) dz⎤, ⎦
(3)
where Ey (0) is the electric field intensity at the incident boundary of the plasma slab, and Ey (z ) represents the electric field intensity when the EM waves have passed the plasma. z is the position of the plasma slab. The coefficient of j in this expression can be denoted by Im(·). According to Eq. (3), the transmittance coefficient, T, can be obtained when the EM wave reaches the right boundary of the plasma region as follows. 245
Optik - International Journal for Light and Electron Optics 188 (2019) 244–250
H. Yu, et al.
Fig. 2. Different plasma density distribution models versus the plasma thickness.
T=
Ey (d ) Ey (0)
{
∫0
= exp −2Im ⎡ ⎣
z
kp2 (z ) − k 0 2 (z ) dz⎤ ⎦
}
(4)
Note that Eq. (4) denotes the total attenuation after the EM wave has propagated through the whole of the plasma slab. Therefore, the attenuation (Att) can also be obtained as following
{
∫0
Att = −10 log T = 8.686 Im ⎡ ⎣
z
}
kp2 (z ) − k 0 2 (z ) dz⎤ . ⎦
(5)
The above derivation yields the attenuation and transmittance of an EM wave passing through the plasma. In the following section, the influence of various parameters on the attenuation and transmittance of the EM waves propagating in the plasma slab will be analyzed based on Eqs. (5) and (4), respectively.
3. Numerical simulations and discussion Since the mass of the electron is much larger than that of the ions and other particles in the plasma, the plasma density can be approximately expressed by the electron density [15,20]. Both research and experiments show that the plasma flow generated by the friction with the atmosphere has a smaller density of plasma outside and a larger density in the middle when the aircraft re-enters the atmosphere at high speed [19]. In this paper, four widely used density distribution models, namely, the uniform distribution, parabolic distribution, Gaussian distribution, and double exponential distribution, are used as presented in Fig. 2. The corresponding density functions are listed in Table 1. Note that the plasma density varies with the thickness d of the plasma in the z-axis direction. The maximum density is ne, and z0 is the midpoint of the plasma slab. In the following subsection, the influence of different plasma parameters, such as the maximum electron density ne, the plasma thickness d, the magnetic field strength B, and the collision frequency ven , on the attenuation and transmission of an terahertz wave in a magnetized plasma are first investigated with the parabolic distribution density model. After that, the attenuation and transmission characteristics of terahertz waves in plasma are further analyzed with different density models. The main parameters used in the simulations are given in Table 2.
Table 1 The plasma density models used in the simulations. Number
Distribution model
Plasma density function
Case 1 Case 2
Uniform Parabolic
Ne = ne
Case 3
Gaussian
e
Case 4
Double exponential
(
2
) + 1⎤⎥⎦ N = n exp ⎡−2.3 ( ) ⎤ ⎢ ⎥ ⎣ ⎦ ⎧ n exp ⎡0.9 − 2) − 0.9⎤, (z ≤ z ≤ d) ⎪ ⎢ ( ⎥ ⎣ ⎦ N = ⎨ ⎤ ⎡ − n exp 0.9 0.9 , ⎪ ( ) ⎥⎦ (0 ≤ z ≤ z ) ⎢ ⎣ ⎩ 1
Ne = ne ⎡− ⎢ ⎣ 2
z z0
−1
z z0
e
2
2
z z0
e
0
e
e
246
z z0
2
0
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Table 2 Simulation parameters of the plasma slab. Parameters
Symbol (unit)
Values
Peak electron density Plasma thickness Collision frequency Magnetic field strength Wave frequency
ne (m3) d (cm) ven (THz) B (T) f (THz)
1 × 1016–1 × 1020 5–20 1 × 10−3–1 × 101 0.01–10 0.1–1
3.1. Effects of different magnetic field strengths on terahertz waves Fig. 3 shows the effects of different values of B on the attenuation and transmittance. The plasma model with parabolic distribution in Case 2 is adopted here. Note that the electron density function of the parabolic distribution is centrally symmetric. We also assume that the terahertz wave is vertically incident on the plasma. The plasma thickness is d = 10 cm, the collision frequency ven = 1 × 10−2 THz, and the peak electron density ne = 1 ×1018/m3. Fig. 3(a) and (b) depicted the case for a right-hand polarized wave throughout the plasma and Fig. 3(c) and (d) consider the lefthand polarized wave. The following results can be obtained by comparing Fig. 3(a) and (b). The attenuation first increases to its maximum value and then decreases in a certain frequency range when B ≥ 2 T. After that, the attenuation decreases with an increase in the frequency of the wave. On the contrary, the transmittance will be a minimum in a certain frequency range as shown in Fig. 3(b). As the magnetic field strength increases, the attenuation peak and transmittance valley move to a higher frequency. Note that the maximum point and minimum point appear later with an increase of magnetic field strength. In Fig. 3(c) and (d), the attenuation decreases with the increase of the frequency of the terahertz wave, while the transmittance increases during this process. In addition, with the amplitude of the magnetic field B increasing, the attenuation decreases while the transmittance increases. According to the above simulation results, the effect of absorption by the plasma on re-entry communication can be effectively mitigated by a rational use of the magnetic field strength and wave frequency.
3.2. Effects of different plasma parameters on terahertz waves Fig. 4 shows the effects of different peak electron densities on the attenuation and transmittance. The amplitude of the magnetic field is B = 1 T, which is constant throughout the whole plasma. Only the right-hand polarized wave is considered in the following
Fig. 3. Effects of different magnetic field strengths on attenuation and transmittance. 247
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Fig. 4. Effects of different peak plasma densities on attenuation and transmittance.
simulations for simplicity. The attenuation of the plasma is depicted in Fig. 4(a) and the transmittance in Fig. 4(b) versus the maximum plasma density ven . The plasma thickness is d = 10 cm, the collision frequency ven = 1 × 10−2 THz. Intuitively, the attenuation of the terahertz wave throughout the non-uniform magnetized plasma decreases with the increase of plasma peak electron density. In addition, the signal at a higher frequency is more easily propagated through the plasma. From Fig. 4(b), we can see that the higher the electron density, the smaller the transmittance of the terahertz wave throughout the non-uniform magnetized plasma at the same frequency. For instance, the transmittance is so small that almost all the terahertz waves are absorbed when ne = 5 × 1018/m3. On the contrary, when ne = 1 × 1017/m3, the transmittance is nearly 1. This can be explained by the fact that the higher the plasma density, the more wave energy will be absorbed by the electrons and then transferred to neutral particles by collisions. Therefore, the attenuation of the terahertz wave increases. The effect of the plasma thickness on the propagation of terahertz wave in a plasma is illustrated in Fig. 5(a) and (b), respectively. The collision frequency ven = 1 × 10−2 THz, and the peak electron density ne = 1 ×1018/m3. As shown in Fig. 5(a), the attenuation decreases with an increase of the frequency of the wave, whereas the transmittance increases. In addition, a larger plasma thickness induces a higher attenuation and a lower transmittance as shown in Fig. 5. It is easy to understand that the terahertz wave has more difficulty passing through the plasma when the plasma thickness is large. The attenuation and transmittance of an terahertz wave passing through the plasma versus different collision frequencies ven are demonstrated in Fig. 6. The plasma thickness is d = 10 cm, and the peak electron density ne = 1 ×1018/m3. From Fig. 6(a), the attenuation changes little with an increase of the frequency once ven ≥ 0.5 THz. However, the attenuation decreases with increasing frequency when ven ≤ 0.1 THz. In addition, decreasing the collision frequency is an alternative way to decrease the attenuation and
Fig. 5. Effects of different plasma thicknesses on the attenuation and transmittance. 248
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Fig. 6. Effects of different collision frequencies on attenuation and transmittance.
increase the transmittance, as shown in Fig. 6. This can be explained by the fact that less of the energy of the terahertz wave is absorbed when the collision frequency is very small.
3.3. Effects of different electron density models on terahertz waves In this subsection, the effects of different plasma density models (different cases in Table 1) on terahertz waves attenuation and transmittance are taken into account. The other plasma parameters are set as B = 1 T, d = 10 cm, ven = 0.1 THz. Note that the peak electron density in Fig. 7(a) and (b) are ne = 1 × 1018/m3, and it increases to ne = 5 × 1018/m3 in Fig. 7(c) and (d).
Fig. 7. Effects of different plasma densities on attenuation and transmittance. 249
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According to Fig. 7(a), the attenuation of the propagation of an terahertz wave in plasma with a double exponential distribution (Case 4) is the highest, and the attenuation under the parabola distribution (Case 2) is close to that under the uniform distribution (Case 1), while the attenuation under a Gaussian distribution (Case 3) is dozens of times smaller than that under the uniform distribution at the same frequency of the incident wave. Form Fig. 7(b), we can see that some valleys appear when f ≤ 0.1 THz. In this region, the plasma with Gaussian distribution exhibits larger transmittance than the other distribution models. However, the transmittance of the plasma with these four distribution models all tend to 1 when f ≥ 0.8 THz. It can be explained that the plasma density with double-exponential distribution (Case 4) fluctuates sharply while the density with Gaussian distribution (Case 1) decreases along the propagation direction as shown in Fig. 2. In addition, we can conclude that the terahertz waves with higher frequency are more beneficial to propagate through plasma. The attenuation and transmittance in Fig. 7(c) and (d) have almost the same variation tendency with Fig. 7(a) and (b). However, their value are smaller than that in Fig. 7(a) and (b). In addition, the attenuation peaks due to the absorption resonance shift to the larger frequency in Fig. 7(c). This is understandable due to the larger peak plasma density in Fig. 7(c) and (d). From the above simulations and discussion, the attenuation and transmittance of the terahertz wave propagating in the plasma are obviously affected by the plasma density. Since the plasma density distribution model is directly related to the external structure of the flight, the flight speed, and the pressure around the flight, etc., therefore the influence of these factors on the attenuation and transmittance are very important and will be further investigated in our future studies. 4. Conclusion In this paper, the transmission characteristics, including the attenuation and transmittance, of terahertz waves propagation in a plasma slab with different plasma density distribution models have been analyzed. Simulation results demonstrate that the plasma density, plasma thickness, and collision frequency play a crucial role in the attenuation and transmittance of the terahertz wave. At a high collision frequency, the attenuation decreases significantly with an increase in the frequency of the incident wave. Both the attenuation peak and transmittance valley appear with an increase of magnetic field strength. In addition, the attenuation and transmission of terahertz waves propagation in a plasma with a parabolic distribution are similar to those under a uniform distribution, whereas the attenuation in a plasma with a parabolic distribution is far smaller than that of a double exponential distribution. 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