The cause and the evolution of “lens” structure for terahertz signals in hypersonic plasma sheaths

Once a vehicle is moving hypersonically in near space, the air in front of the vehicle is compressed intensively, which results in the formation of a shock surrounding the vehicle, and the air in between is heated. The temperature of the heated air could be up to thousands of Kelvins. Neutral particles would be ionized due to the high temperature. The process gives rise to the plasma sheath, whose electron density could be up to 10²⁰ m⁻³. The communication signal would be shielded by the dense free electrons, which results in the occurrence of the communication blackout.¹ 

THz communication is believed to be a potential solution to the blackout problem.² Many studies have been carried out to investigate the propagation characteristics of THz signals in plasma sheaths. Yuan et al. found that the plasma parameter makes a significant contribution to the transmission rate of THz signals in plasma sheaths.³ The attenuation of THz signals decreases with the signal frequency increasing.⁴ The THz signal attenuation in the plasma sheath mainly attributes to the collisions between electrons and neutral particles. The electron density and the collision frequency determine the THz signal attenuation rate.⁵ On the other hand, once the gradient of the electron density in the plasma sheath is big, the reflection could also make a significant contribution to the THz signal attenuation.⁶ In addition, the ablation particles in plasma sheaths might enhance the attenuation of the THz signals.^7,8 On the other hand, the plasma sheath is significantly inhomogeneous. Thus, the attenuation rates for the signals propagating along different paths in the plasma sheath could be different.⁹ Flight conditions of the vehicle could modulate the THz signal attenuation in the plasma sheath.¹⁰ For example, the THz signal attenuation in the plasma sheath increases with the atmospheric mass density surrounding the vehicle.¹¹ Also, the THz signal attenuation increases with the flight speed.¹² The angle of attack of the vehicle could change the THz signal attenuation.¹³ Besides the signal attenuation, there could be frequency offsets¹⁴ and carrier phase offset^15–18 for the THz signals propagating in plasma sheaths.

On the other hand, previous studies rarely concern the offsets of the propagation direction of THz signals in plasma sheaths. However, some studies on the THz antenna performance in plasma sheaths imply that the propagation direction of THz signals could be changed by plasma sheaths.^19–21 Zhang et al. found that the plasma sheath could act as a compound lens to the THz signals propagating in plasma sheaths, and eventually change the propagation directions for the signals. However, the plasma sheath was considered as a static plasma environment in Zhang’s work, although the plasma sheath always evolves even without any external perturbation.²² 

In this study, the auto-evolution of plasma sheaths is taken into account. The consequent evolution of the lens structure is investigated. The mechanism that dominates the evolution of the lens structure is analyzed. Based on the analysis, the appropriate position to install the onboard antenna is suggested in order to stabilize the transmission direction of THz communication signals.

In this study, the plasma sheath is considered to be the propagation environment for the THz signals. The parameters of the plasma sheath are obtained by solving a hypersonic hydrodynamical model, which has been utilized in many previous studies.^23,24 The shape of the vehicle is identical to that involved in the Radio Attenuation Measurements C-II experiments, which is a blunt cone. The vehicle has a length of 1.29 m and the angle between its wall and main axis is 9°. The model was solved in an unstructured mesh by using the finite volume method. In this study, the angle of attack is assumed to be 0°. Therefore, the plasma sheath is symmetrical with reference to the main axis of the vehicle. In such a case, only half of the plasma sheath is solved.

It should be noted that the plasma sheath is essentially a significantly inhomogeneous flow field. In such a case, the plasma sheath always keeps evolving even when the external perturbation is absent. In this study, the structure evolution of the plasma sheath is concerned, as shown in Fig. 1.

In order to obtain the plasma sheath, a shock traveling hypersonically is initialed at the beginning of the simulation. It enters the simulation box from the top boundary and moves toward the bottom. Figures 1(a)–1(e) and Figs. 1(f)–1(j) illustrate the evolutions of the electron collision frequency and the electron density, respectively. Those two parameters make the most significant contribution to the propagation characteristics of THz waves in plasma sheaths.⁵ The structure evolution of the plasma sheath is obvious according to Fig. 1.

The onboard antenna is assumed to be installed on the wall of the vehicle. In order to reduce the computation cost in solving the THz signal propagation, an area, which is marked with the dashed box in Fig. 1(j), is selected. The electron density distribution of the selected area is shown in Fig. 2.

The sides of the dashed box are not parallel or perpendicular to the axes in Fig. 1. In order to deal with the selected area more conveniently, the area was rotated by 9°, as shown in Fig. 2. Obviously, the x axis in Fig. 2 is the wall of the vehicle. The maximum and the minimum electron densities in the selected area are 10²⁰ and 10¹² m⁻³, respectively. The boundary of the plasma sheath in the selected area is a curve despite the wall being straight. It should be noted that plasma is a dispersive media, and the wavelengths of THz signals are sub-millimeters. Hence, the structure of the plasma sheath in the selected area implies the plasma sheath could act as lenses to THz communication signals that can penetrate the plasma sheath.²⁵ 

The ray tracing method is employed to model the signal propagation in the plasma sheath in this study. The selected area is meshed with adaptive grids, as shown in Fig. 3. In the y direction, the area is divided into N layers. In each layer, the plasma is homogeneous in the x direction. In addition, the plasma parameter in each grid is assumed to be homogeneous. In such a case, the refraction only occurs at the interface of the neighboring grids. In each grid, the signal propagates straightly.

According to Zhang’s study,²⁵ the lens structures are yielded by the inhomogeneity of the plasma sheath. The inhomogeneous electron density and the electron collision frequency make the refraction index within the plasma sheath strongly inhomogeneous. The contour of the refraction index appears as “lens,” which is able to focus or defocus the high-frequency signals propagating through it. Figure 4 illustrates the evolution of the refraction index in the selected area.

Figures 4(a)–4(c) show the contour plot for the refraction index at 0.40, 0.54, and 1.00 ms after the shock enters the simulation box, respectively. In this study, the signal frequency is chosen to be 0.14 THz, which is an atmospheric window frequency in the THz band. According to the figure, the refraction index is significantly inhomogeneous. In addition, it evolves with time. On the other hand, the contours are not flat. Most of the contours are curved, which implies that they might act as lenses. From 0.40 to 0.54 ms, the distribution of the refraction index evolved fast. Hence, the distributions of the refraction index in the area of x > 0.9 m are almost completely different at 0.40 and 0.54 ms. After 0.54 ms, the structure of the refraction index evolves slower. Figure 5 shows more details about the dashed region in Fig. 4(b).

According to Fig. 5, the refraction index near the right bound changes from 0.2 to nearly 1 in several centimeters. The contours are curved. Hence, those areas could act as negative lens to the signals at 0.14 THz. On the other hand, it should be noted that there are some small “bubble-like” structures in the region whose refraction index is around 0.4. The typical size for those plasma bubbles ranges from several millimeters to several centimeters. The wavelength of 0.14 THz is about 2 mm. Therefore, there could be refracted or scattered at those bubbles. Also, it is interesting that diffractions could occur due to those small bubbles although it is not the main concern in this work.

Figure 6 shows the electron density and the refraction indices in the three parallel lines. The lines are illustrated in Fig. 6(a), whose distance to the wall of the vehicle is 0.048, 0.086, and 0.117 m, respectively. Figures 6(b), 6(d), and 6(f) illustrate the electron densities along the three lines. It is obvious that the electron densities along the three lines are different at all three moments. Figures 6(c), 6(e), and 6(g) illustrate the corresponding refraction indices. By comparing the electron densities and the refraction indices at the three moments, the variation trends of the refraction indices are almost opposite to the trends of the variation of the electron densities, including the trends of variations not only in space but also with time. According to Eq. (2), the refraction index is modulated by the electron density and the collision frequency. According to Fig. 6, the electron density mainly determines the refraction index. The contribution made by the collision frequency could be neglected.

On the other hand, Fig. 6 implies that besides the ordinary lens, the lens structure in this work is the so-called “gradient index (GRIN) lens,” whose thickness is constant with the refraction index varies gradually. In addition, it should be noted that it is very difficult to say whether a GRIN lens is positive or negative. It significantly depends on the ratio between the thickness of the GRIN lens and the signal wavelength.

Figure 6 shows that the GRIN lens varies with time and space. At 0.40 ms, the plasma in line 2 acts as a lens only in the area of x < 0.9 m, while the plasma in line 3 acts as a lens only in the area of x > 0.8 m. The refraction index in line 1 varies in a small range in the area of 0.6 m < x < 1.1 m. Hence, it might make an insignificant contribution to the offset of signal transmission direction. Nevertheless, at 0.54 ms, the refraction indices of line 1 and line 3 vary slightly compared with that of line 2. At 1.00 ms, only the plasma in line 1 makes a significant contribution to the offsets of signal transmission direction.

In a given area, there are two sources of free electrons, which are the electrons flow into the given area from neighboring regions, and the local ionization processes within the given area. Figure 7 shows the flow velocities of the plasma along the three parallel lines. By comparing Fig. 7 and Figs. 6(b), 6(d), and 6(f), the plasma convection almost dominates the electron distribution of the plasma sheath. Basically, the convection velocity closer to the top of the vehicle is greater than that near the bottom of the vehicle. In the area near the top of the vehicle, the plasma tends to flow toward the right and the inner plasma sheath, which yields the accumulation of electrons in lines 1 and 2 in the region of x > 0.7 m. As a result, the electron densities in that area increase. In addition, it should be realized that there is also plasma flow into the area covered by the three lines from left, and the plasma within the area flows away toward right. Thus, it finally yields the evolution of refraction indices distribution as shown in Figs. 6(c), 6(e), and 6(g).

Figure 8 illustrates the temporal offsets of signal propagation direction. Positive offset means that the propagation direction was refracted toward the bottom of the vehicle, and negative offset denotes the offset toward the top. The incident angle ranges from −30° to 30°. In Figs. 8(a)–8(c), the propagation direction offsets terminated at vertical dashed lines. Once the incident angle exceeds the angle highlighted by the dashed lines in each panel, total inner reflection occurs. In other words, the gradient of the refraction index is too big to make the refraction angle greater than 90°. In such a case, the propagating signals would be totally reflected and propagated toward the vehicle, i.e., the signals are no longer able to penetrate the plasma sheath even if the signal frequency is higher than the plasma frequency.

At 0.4 ms, the windows of incident angles that could make the signals able to penetrate the plasma sheath are very small compared with those at 0.54 and 1.00 ms. While the onboard antenna is installed at x = 0.8 m, the window ranges from −25° to 15°. Nevertheless, once the antenna is installed at x = 0.9 m or x = 1.0 m, the window ranges from −5° to 5°. On the other hand, at 0.54 ms, once the antenna is at x = 0.8 m, the signals whose incident angle exceeds −25° would be totally reflected. The available window is only 55° in total despite there being no total reflection, while the antenna is at x = 0.9 m or x = 1.0 m. At 1.00 ms, the total inner reflection vanishes.

Moreover, it should be noted that the propagation direction offsets by installing the onboard antenna at x = 1.0 m are always lower than that of x = 0.8 m and x = 0.9 m. The range of direction offsets while the antenna is at x = 1.0 m is −1.0° to 1.0°, which is rather small. Therefore, to install the onboard antenna close to the bottom of the vehicle may help to stabilize the transmission direction of THz communication signals.

In this study, the evolution of the plasma sheath is investigated. Its impact on the offsets of the THz signal propagation direction is analyzed. The analysis shows that the plasma sheath evolves even without any external perturbation. As a result, the refraction index distribution of the plasma sheath also evolves automatically. Electron convection is the main mechanism that dominates the evolution of the refraction index distribution. The inhomogeneous distribution of the refraction index in the plasma sheath acts as GRIN lenses to the propagating THz signals, which consequently change the transmission direction of THz signals in the plasma sheath. The direction of THz signal propagation always varies with time. To install the onboard antenna near the bottom of the vehicle helps stabilize the transmission direction of THz communication signals.

The authors gratefully acknowledged support provided by the National Natural Science Foundation of China (Grant Nos. 41974195 and 41727804), the Natural Science Foundation of Jiangxi for their funding (Grant No. 20224BAB212027), and the Financial Contract of the Science and Technology Project of Jiangxi Province (Grant No. ZBG20230418032). Finally, they appreciated the valuable contributions of the Interdisciplinary Innovation Fund of Natural Science from Nanchang University, specifically Grant Nos. 9166-27060003-YB14 and 9167-28220007-YB2104, in advancing this research.

The authors have no conflicts to disclose.

Kunpeng Peng: Data curation (equal); Software (equal); Writing – original draft (equal). Kai Yuan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). Mingyang Mao: Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Ziyang Zhao: Conceptualization (equal); Formal analysis (equal); Software (equal). Pingsheng Liu: Data curation (equal); Formal analysis (equal); Writing – review & editing (equal). Yuxin Cheng: Conceptualization (equal); Writing – review & editing (equal). Ruiting Mao: Data curation (equal); Software (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.