When vehicles fly at hypersonic velocities within the atmosphere, they become enveloped in a plasma sheath that prevents radio communication between the aerospace and the ground control center, which is the well-known blackout problem.^[1–4] In order to mitigate or eliminate the reentry blackout problem, a number of techniques have been proposed, such as high power, high frequencies, low frequencies, lasers, magnetic fields, and so on.^[3,5–7] The terahertz wave (THz), commonly referred to as T-rays, sub-millimeter, or far infrared, refers to the electromagnetic radiation at the frequency from 0.1 THz to 10 THz, has attracted much attention recently partially because it is one of the potential candidates in the high frequencies communication methods. The techniques for the generation of THz wave were developed rapidly in recent years^[8–11] and promising applications of THz waves can be found in various fields.^[12] As the basis of communication between the hypersonic vehicles and the ground control center, the interaction between plasma and THz wave (absorption, reflection, and transmission) is a key issue to be analyzed. Over the past few years, the propagation characteristics of the THz wave in plasma have been investigated by different methodologies, such as the analytical method,^[13–17] experimental method,^[18] scattering matrix method (SMM),^[19,20] propagation matrix method (PMM),^[21] finite-difference time-domain (FDTD) method,^[22–28] Wentzel–Kramer–Brillouin (WKB) method,^[29,30] and so on.

The FDTD method has been widely used to solve the interaction problems between the THz wave and plasma because of its capability in simulating the propagation characteristics of a wideband THz wave through inhomogeneous dispersive plasma. Wang et al.^[22,27,28] investigated the propagation properties of the THz wave in a time-varying dusty plasma slab using the auxiliary differential equation finite-difference time-domain (ADE-FDTD) method. Zheng et al.^[23] studied the propagation characteristics of the THz wave in unmagnetized plasma by the Z transform finite-difference time-domain (ZT-FDTD) method. Theoretical results were compared with experimental measurements in Ref. [20], where good agreements were achieved. The broadband propagation characteristics of the THz wave in an anisotropic magnetized plasma were investigated by Tian et al.^[24] using the JE convolution FDTD (JEC-FDTD) method. Gao et al.^[25] investigated the characteristics of reflection, transmission, and absorption of THz waves in a sandwich type microplasma structure by the ZT-FDTD method.

Although studies of propagation characteristics of the THz wave in plasma media can be found in several works, most of the previous research focused on the normally incident wave cases, i.e., the incident plane wave propagates at an angle of zero with respect to the normal of the media interfaces, limited attention was paid to the oblique incidence cases. In practice, the angle of incidence changes over a wide range during flight, which indicates that the oblique incidence wave cases require more consideration. Recently, using the impedance matching method, Rahmani et al.^[31] investigated the transmission, reflection, and absorption of s-polarized wave obliquely incident on the plasma slab which has a bell-like electron density distribution. The SMM was adopted by Yuan et al.^[32] to analyze the transmission of obliquely incident THz waves in the plasma sheaths covering a blunt coned vehicle, in which the plasma sheaths were generated based on a hypersonic fluid model. Cao et al.^[33] studied the propagation characteristics of obliquely incident THz waves though dusty plasma using the PMM, where the electron density distribution of the dusty plasma was assumed to be parabolic. Based on the twice reflection model, Tian et al.^[34] investigated the obliquely incident wave through a nonuniform magnetized plasma slab, in which both the electron density and the collision frequency across the plasma were assumed to be in a Gaussian profile. To solve the problem under study, the non-uniform plasma was divided into a series of layers with uniform electron density.^[34] Nevertheless, this dividing method fails when one encounters plasma with a high gradient electron density distribution. To overcome this difficulty, the FDTD is a good choice.

The THz wave obliquely incident on a plasma sheath, which is non-uniform in half space, is a two-dimensional problem for the FDTD method. Thus it will cost a lot of memory and computation time in the simulation. To save computational memory and time, in this paper, the FDTD method combined with the phase matching condition is utilized to solve the propagation problem of obliquely incident THz waves through an inhomogeneous plasma. With the help of phase matching condition, the problem is simplified to a pseudo one-dimensional problem, which can greatly reduce the memory required and improve the efficiency. To describe the distribution of the electron density in an inhomogeneous plasma, the Gaussian profile is introduced which corresponds to the flow field around the hypersonic aircraft at a high altitude of space.^[35]

The rest of this paper is structured as follows. In Section 2, the physical model of the propagation problem is given, with a brief description of the method used in solving the problem under study. Numerical results and discussions concerning the transmission features of the THz wave through an inhomogeneous plasma are presented in Section 3. Section 4 concludes this paper.

A geometric sketch of the problem under study is shown in Fig. 1, where a plane THz wave propagates through a plasma sheath around the hypersonic vehicle at an arbitrary incident angle. The plasma sheath is non-uniform in half space, which is inhomogeneous in y direction and uniform in x and z directions. The thickness of the plasma sheath is d in y direction and infinite in the other two directions. The incident THz wave is assumed to propagate through the plasma sheath in the plane of xoy at an incident angle θ with respect to the normal of the plasma sheath interfaces.

Fig. 1.

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	Fig. 1. Schematic diagram of a THz wave propagating through nonuniform plasma layer.

As shown in Fig. 1, when the THz wave propagates through the plasma slab in the xoy plane, the electric field E and magnetic field H satisfy ∂E/∂z = 0 and ∂H/∂z = 0, in which ∂/∂z means derivative to z. The fields in the plasma sheath can be divided to two modes: (i) the transverse magnetic (TM_z) mode, which is composed of E_z, H_x, H_y, and (ii) the transverse electric (TE_z) mode, which is composed of H_z, E_x, E_y. In the following, we will only consider the TE_z mode case since the TM_z case can be obtained in the same manner.

The fields in plasma can be described by Maxwell’s equations in the frequency domain

For the TE_z mode, the propagation equation in the xoy plane can be expressed as

For simplicity, two new auxiliary variables and are defined as

The permittivity of unmagnetized plasma is given as^[36]

Substituting Eq. (13) into Eq. (9) yields

With the transition relationship from the frequency domain to time domain, jω → ∂/∂t, we can obtain the expressive forms of Eqs. (15) and (16) in the time domain

If we assume that the function in the time domain has the form

The expression of can be obtained in the same way as

For the TM case, the transmission coefficient and absorption coefficient can be calculated in the same way by substituting electric field E_z for magnetic field H_z.

Numerical simulations are implemented based on the theoretical treatments presented in Section 2. The incident THz wave of cosine-modulated Gaussian pulse in the SO-FDTD is given by

3.1. Validity of the proposed method

Before we start to analyze the propagation characteristics of the THz wave through an inhomogeneous plasma, the validity of the method adopted in this paper is verified by comparing the results obtained by our method with those obtained by an analytical method.^[35] The propagation characteristics of THz waves with different incident angles in a homogeneous plasma slab are taken into consideration.

The FDTD problem space consists of 474 cells, and the homogeneous plasma occupies 71–404 cells. The cell size is Δy = 1.5 × 10⁻⁵ m, and the thickness of the plasma layer is 5~mm. Both ends of the cell space are treated by uniaxial perfectly matched layer absorbing boundary condition to eliminate unwanted reflections. The plasma parameters are

Comparisons of the transmission coefficients of the THz wave in the plasma slab calculated by our method and those obtained using the analytical solution are displayed in Fig. 2. As shown in Fig. 2, very good agreements are achieved for both TE and TM waves, which partially indicates the correctness of our method.

Fig. 2.

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	Fig. 2. Comparisons of the transmission coefficients of a plasma slab versus the incident angles between analytical solutions and SO-FDTD method: (a) TE wave, (b) TM wave.

The plasma density stemmed from the reentry hypersonic flight in the stagnation region could reach 10²¹ m⁻³ and the plasma frequency corresponding to this electron density can reach 0.28 THz.^[24] In the following simulations, the Gaussian electron density profile is chosen to stand for the variation of electron density, which corresponds to the non-uniform plasma flow field around the hypersonic aircraft at a high altitude of space^[35]

where d = y₂ − y₁ = 8 cm is the thickness of the plasma, y₁ = −d/5, y₂ = 4d/5, , , in which α₁ and α₂ represent the distribution form of electron density. The maximum electron density is assumed to be n_emax = 10²¹ m⁻³. The distribution of Gaussian profile is plotted in Fig. 3, in which the collision frequency is υ_c = 1.5 × 10¹⁰ Hz, and the plasma frequency ω_p is calculated as

Fig. 3.

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	Fig. 3. Plasma electron density distribution in Gaussian profile.

3.2. Influences of incident angle

In order to investigate the influence of the incident angle on the propagation characteristics of a THz wave in plasma, the incident angle of the THz wave is varied in steps of 15° from 0° to 90° in the simulations. The transmission and absorption coefficients for TE wave are calculated and shown in Fig. 4 for θ = 0°, 15°, 30°, and 45°. As shown in Fig. 4(a), the transmission coefficient increases as the incident angle decreases or the frequency of the THz wave increases. When the incident wave frequency approaches 1 THz, the transmission coefficient tends to 0 dB, which indicates that the THz wave can be used to release the blackout problem. For the normal incidence case, when the frequency of the incident wave is f = 0.3 THz, the transmission coefficient T ≈ −20 dB. As the incident wave frequency increases, the transmission coefficient increases and tends to 0 dB. However, for θ > 15° cases, when f = 0.3 THz, the transmission coefficient is far less than −20 dB, which indicates that the THz wave cannot penetrate the plasma. As shown in Fig. 4(b), the absorption coefficient increases at first and then decreases as the incident frequency increases. For each incident angle case, there is a maximum absorption coefficient, and the corresponding frequency moves to a larger value as the incident angle increases. It indicates that when a THz wave is used in communication through plasma, the frequency corresponds to the maximum absorption coefficient should be avoided. Furthermore, it is interesting to find that the larger the incident angle is, the smoother the absorption coefficient becomes.

Fig. 4.

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	Fig. 4. (a) Transmission and (b) absorption coefficients versus incident angles.

3.3. Influences of thickness

Comparisons of transmission and absorption coefficients for plasma slab of different thickness are displayed in Fig. 5. As shown in Fig. 5, as the thickness of the plasma increases, the transmission coefficient decreases and the absorption coefficient increases. As displayed in Fig. 5(a), in the relatively low frequency range (approximately f < 0.35 THz), the plasma thickness has little effect on the transmission coefficient, and the incident EM wave with frequency less than 0.3 THz cannot penetrate the plasma. Similar to the cases shown in Fig. 4(b), there is an absorption peak for each thickness, as shown in Fig. 5(b). Nevertheless, the frequencies corresponding to the absorption peaks are the same for different thickness of plasma slab. It is interesting to find that the increase of the plasma slab thickness broadens the absorption peak.

Fig. 5.

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	Fig. 5. (a) Transmission and (b) absorption coefficients versus thickness of plasma slab.

3.4. Influences of polarization

Figure 6 illustrates the effects of polarization on the transmission and absorption coefficients. As shown in Fig. 6(a), the polarization of the incident wave has little influence on the transmission coefficient for the cases under study. The transmission coefficient increases as the incident angle decreases or the frequency increases, which is the same as we obtained in Fig. 4(a). For the cases of incident angles 30° and 45°, the transmission coefficients for TM wave and TE wave almost overlapped with each other. Figure 6(b) shows that there is a maximum absorption coefficient for each incident angle and the corresponding frequency is the same for different polarizations. The absorption coefficient increases at first and then decreases as the frequency increases. In addition, as shown in Fig. 6(b), the curve of absorption coefficient for TE wave is much smoother than that for the TM wave case.

Fig. 6.

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	Fig. 6. (a) Transmission and (b) absorption coefficients versus polarization.

The variations of power transmission and absorption coefficients for THz wave obliquely incident on a non-uniform plasma slab are analyzed by the SO-FDTD method combined with the phase matching condition. The factors that may affect the propagation characteristics of THz wave through plasma, including the incident angle of the incident wave, the thickness of the plasma, and the polarization of the wave, are studied. The results reveal that the incident angle can affect the propagation properties greatly, while the polarization of the incident wave has little influence on the propagation properties for the cases under study. When the THz wave is employed to the communication in the ionization blackout situation, the incident angle should be considered, and the frequency corresponding to the maximum absorption should be avoided. This study will be useful to provide a reference for the exploration of potential techniques to overcome ionization blackout problem as well as to provide insights to the interactions between THz wave and plasma.