Parasitic modulation effect caused by dynamic plasma in low frequency

When a hypersonic vehicle reentry into the atmosphere, the high-density plasma sheath generated by strong aerodynamic heating may shield the aerospace telemetry, tracing, and control (TT&C) signal in the microwave frequency band. The high-density plasma sheath may cause the interruption of the TT&C between ground station and vehicle for several to ten minutes.^1–4 This phenomenon is known as “radio blackout.”

To allow the uninterrupted monitoring of spacecraft reentry, several methods have been proposed to mitigate radio blackout.^4–11 The existing mitigation methods mainly fall into two categories: changing the propagation environment or the electromagnetic (EM) wave propagation mode. Changing the propagation environment involves interventions that chemically or physically control the plasma spatial distribution and reduce the electron density at the antenna window. Changing the EM wave propagation mode includes changing the frequency of the EM waves, improving the EM wave transmission capacity, establishing the plasma propagation channel model of the corresponding frequency band, and improving the existing communication methods to reduce the information error rate. These methods are still being continuously explored, deriving more optimized and reasonable feasible solutions to promote the method of mitigate radio blackout.

In recent years, expanding the possibilities of new propagation windows has been proposed as a potential solution.¹² In accordance with the theory of EM wave propagation,¹³ when the EM wave frequency is far below than the plasma frequency (namely, $ω c ≪ ω p$⁠), the efficiency of energy transfers from the EM wave to plasma decreases. Hence, the attenuation of low-frequency (LF) EM waves (Note: in this paper, the term “low frequency” differs from the standard radio frequency spectrum definition. Here, the term low frequency refers to frequencies below the microwave range.), which means the EM wave frequency is significantly lower than the plasma frequency, will be significantly lower than that of high-frequency EM waves. In this context, Liu et al. discovered in their research that LF EM waves exhibit unique magnetic field attenuation characteristics when propagating through plasmas.¹⁴ Unlike microwave EM waves, the frequency of LF EM waves is significantly lower than the plasma frequency, resulting in significantly lower attenuation of the magnetic component of LF EM waves in dynamic plasmas compared to the attenuation of the electric component.¹⁵ Consequently, using the magnetic field component of LF EM waves to maintain the minimum essential communications may be considered a feasible solution for mitigating radio blackout.¹⁶ 

When a LF magnetic field is incident on a plasma, induced currents are formed on the plasma surface. The path of these induced currents is constrained by the geometric shape of the plasma.¹⁷ On the other hand, microwave EM waves, with wavelengths significantly smaller than the size of the plasma sheath, are not affected by the plasma model. For example, in the case of a cylindrical plasma model, the induction current forms a path similar to a multi-turn coil along the plasma surface, generating an induced magnetic field that weakens the incident LF magnetic field. Furthermore, the magnitude of the induction current is closely related to the conductivity of the transmission medium and the plasma sheath can be considered as a dielectric layer with a complex permittivity and dynamic and time-varying characteristics. Moreover, Xie et al. experimentally verified the attenuation mechanism of the LF magnetic component through ground-based trials.^17–19 Thus, the unique attenuation mechanism of the magnetic component of LF EM waves and the dynamic conductivity of the transmission medium contribute to the distinct propagation mechanism of LF EM waves in plasma.

Meanwhile, the dynamic time-varying characteristics of the plasma sheath can cause amplitude and phase parasitic modulation on the propagation of communication signals.^20,21 For the microwave band, the parasitic modulation effect has been confirmed as the main causes of communication deterioration.²² Yang et al. pointed out that the dynamic plasma sheath can cause phase fluctuations in microwave signals and lead to a specific rotation of MPSK constellation.²⁰ For the LF band, recent studies have touched on the propagation characteristics of EM waves in dynamic plasma sheaths,¹⁶ but there has not been much detailed mechanism analysis on the parasitic modulation effects. Therefore, it is necessary to analyze the mechanism of the LF parasitic modulation effect to find feasible solutions to eliminate or mitigate the impact of the parasitic modulation effects.

Thus, based on the propagation equivalent model of the LF magnetic field in a cylindrical time-varying plasma,¹⁶ this paper has analyzed the mechanism of the LF parasitic modulation effect. We simulated the amplitude attenuation and phase shift of LF EM wave propagation in the plasma sheath and derived the relevant formulas for amplitude and phase variations. The analysis reveals that there is a sinusoidal correlation between the amplitude attenuation and phase shift, and within the range of phase-shift values, the amplitude and phase consistently maintain a positive correlation. To analyze the specific clockwise rotation of constellation maps, we also derived the coordinate formula of QPSK signal constellation points propagating in plasma. Numerical simulations are conducted to evaluate the influence of the parasitic modulation effect on the propagation of phase modulation signal under different parameters. The increase in carrier frequency results in a reduction in the penetration capability of LF signals through the plasma sheath, while an increase in the plasma collision frequency can mitigate the distortion of the signals. Meanwhile, we verified the parasitic modulation effect of LF EM wave in the dynamic plasma sheath by the experiment.

The remainder of this article is organized as follows. Section II illustrates the propagation mechanism of LF EM waves in the dynamic plasma sheath and derives the specific correlation between signal amplitude attenuation and phase shift. Section III deduces the constellation point function of the QPSK signal and simulates the parasitic modulation effect of LF EM waves in dynamic time-varying plasma. Section IV experimentally verifies the parasitic modulation effect of low-frequency magnetic wave. A detailed discussion, conclusion, and possible extensions are given in Sec. V.

According to Eqs. (7) and (8), the curve of attenuation coefficient $α k ( t )$ and phase-shift coefficient $β k ( t )$ varying with average plasma density $N e ¯$ are simulated by the parameter values as follows.

The numerical simulations adopt an average plasma density $N e ¯$ varying from $1 × 10 11 / cm 3$ to $1 × 10 13 / cm 3$⁠, a collision frequency v_e of $1.5 × 10 9 / s$⁠, a radius of the cylindrical time-varying plasma r of 50 mm, a thickness d of 40 mm, a length l of 1500 mm, and an EM wave frequency f_c of 10 MHz. The electron density fluctuation is set to $σ Δ = 0.3$⁠. The simulation results are shown in Fig. 2.

In accordance with the simulation results, as the average plasma density increases, the attenuation coefficient and phase-shift coefficient of the signal monotonically decrease. It can be inferred that LF EM waves will still be affected by the dynamic plasma sheath, resulting in parasitic modulation effects on the signal. Both α_k and β_k are functions of $N e ¯$⁠, and larger time-varying plasma frequencies can cause more severe amplitude attenuation and phase shift of the signal simultaneously. Therefore, we not only need to study the independent characteristics of attenuation coefficient and phase-shift coefficient, but also need to jointly consider and discuss the correlation between them. With the exponential variation of the time-varying plasma frequency, the phase-shift coefficient of the signal remains negative and within the range of $( − π 2 , 0 )$⁠. So, compared to the previously discovered intermediate frequency parasitic modulation effect, the LF parasitic modulation effect will cause smaller phase rotation on the signal, and the rotation angle is within the range of $( − π 2 , 0 )$⁠.

In this section, the correlation between amplitude attenuation and phase attenuation of the transmission coefficient of LF EM waves in the cylindrical plasma sheath special effect model is derived and simulated.

In this section, we analyzed the constellation point function of the QPSK signal and simulated the propagation of LF magnetic field waves in the plasma sheath to verify the special rotation of the QPSK signal.

Based on Eq. (13), the propagation of QPSK LF wave signals in dynamic plasma has been simulated. The numerical simulations adopt an average plasma density $N e ¯$ of $5 × 10 11 cm − 3$⁠, a collision frequency v_e of $2 × 10 9 / s$⁠, a radius of the cylindrical time-varying plasma r of 50 mm, a thickness d of 40 mm, a length l of 1500 mm, and an EM wave frequency f_c of 10 MHz. The electron density fluctuation is set to $σ Δ = 0.3$⁠.

As shown in Fig. 4, with the channel response of time-varying plasma added to the simulation channel, the constellation points of the signal at the receiver are no longer concentrated in the ideal mapping position, and the constellation diagram has an obvious clockwise rotation phenomenon. The fundamental reason for the constellation rotation of the phase modulation signal is the strong correlation between the amplitude and phase of the signal in plasma.³⁰ Moreover, the simulation results show that parasitic modulation will still occur when LF EM waves pass through the time-varying plasma. Therefore, for LF EM waves, time-varying plasma will still cause the parasitic modulation effect and produce multiplicative interference on QPSK, thus threatening the reliability of communication signals.

Next, taking the constellation point in the first quadrant as an example, we simulated the correlation characteristics of the horizontal and vertical coordinates of the constellation point, namely, I and Q values, when the LF QPSK signal is affected by the parasitic modulation effect caused by the plasma sheath.

We adopt an average plasma density $N e ¯$ varying from $1 × 10 11 / cm 3$ to $1 × 10 13 / cm 3$⁠. The channel parameters are consistent with the simulation above. The simulation results are shown in Fig. 5. As the average plasma density increases, the abscissa I of the constellation point shows a trend of first increasing and then decreasing, with the inflection point near the average plasma density of $2 × 10 17$⁠. At the same time, the vertical coordinate of the constellation shows a trend of decreasing first and then increasing, with the inflection point at the average plasma density greater than $1 × 10 18$⁠. Therefore, as the average plasma density increases, the constellation will exhibit a clockwise rotation phenomenon, as shown in Fig. 6. This is consistent with the constellation point rotation phenomenon in the simulation diagram in Fig. 4.

In this section, we simulate the effects of different carrier frequency and plasma collision frequency on the QPSK channel with dynamic plasma channel response and Gaussian noise. In order to evaluate the effect of parasitic modulation caused by time-varying plasma on demodulation at different carrier frequencies and collision frequencies, the error vector amplitude (EVM) of QPSK signals in plasma is measured in this simulation.

The numerical simulations adopt an EM wave frequency f_c varying from 1 to 30 MHz, an average plasma density $N e ¯$ of $8 × 10 11 cm − 3$⁠, a collision frequency v_e of $2 × 10 9 / s$⁠, a radius of the cylindrical time-varying plasma r of 50 mm, a thickness d of 40 mm, and a length l of 1500 mm. The electron density fluctuation is set to $σ Δ = 0.3$⁠. The simulation results are shown in Fig. 7. It is seen that with the increase in the carrier frequency, the phase rotation of the constellation is significantly intensified, and the distance between any two constellation points will rapidly decrease.

The numerical simulations adopt a collision frequency v_e varying from $1 × 10 9 / s$ to $5 × 10 9 / s$⁠, an average plasma density $N e ¯$ of $8 × 10 11 cm − 3$⁠, a radius of the cylindrical time-varying plasma r of 50 mm, a thickness d of 40 mm, a length l of 1500 mm, and an EM wave frequency f_c of 10 MHz. The electron density fluctuation is set to $σ Δ = 0.3$⁠. The simulation results are shown in the Fig. 9. With the increase in collision frequency of time-varying plasma, all constellation points show similar regular rotation, and the phase rotation degree of constellation gradually decreases, and LF EM wave is affected by dynamic plasma. As shown in the Fig. 10, with the growth of the collision frequency, the EVM gets smaller. When the collision frequency approaches to $2 × 10 9 / s$⁠, the EVM reduces to about 10%. This indicates that an increase in plasma collision frequency can reduce distortion in LF signals to some extent.

The experimental device includes a software radio module based on the graphic programming software LabVIEW for signal source, sink, signal modulation, and demodulation, as well as hardware peripherals such as USRP transmitter and receiver, power amplifier, ring antenna, iso-scale magnetic antenna, vacuum cavity, and time-varying plasma generator. The block diagram of the experimental device is shown in Fig. 11(a). The phase modulation signals (QPSK) were produced by LabVIEW with a carrier frequency of 10 MHz. The bit rate is 25 kbps. Then, the QPSK signal is converted into a radio frequency signal through USRPX310. After the radio frequency signal passes through the power amplifier, the ring transmitting antenna realizes the LF EM wave radiation of the scaled warhead position in the plasma environment. The distance between the warhead and the plasma spout is 200 mm. The receiving antenna is placed inside the scaled warhead, and a ceramic window is set on the surface of the warhead where the receiving antenna is placed for signal transmission. The rest of the warhead is made of stainless-steel material. During the experiment, the warhead was cooled down using a water-cooling system. The receiving antenna is connected to the USRP receiver outside the instrument cavity via a coaxial cable, and the signal is transmitted to the software radio receiver to observe the penetration signal status and record experimental data. Figure 11(b) shows the photograph of the experimental setup. Figure 11(c) shows the installation details of the receiving antenna. The setting of the magnetic field antenna can be referred to Ref. 18.

The experiment utilized inductively coupled plasma (ICP) to generate the plasma. The power supply was an alternating current source with an oscillation frequency of 440 kV. The power supplied varied between 50 and 500 kW, and different frequencies were used to produce plasma in various states, thereby achieving a wide range of electron densities.³¹ As shown in Fig. 11(b), the plasma nozzle had a diameter of 220 mm, suggesting that the ejected plasma could be approximated as a cylindrical shape with a diameter of about 160 mm. The cone diameter of the warhead was 110 mm, and the average thickness of the plasma traversed by the EM waves was approximately 35 mm. In the experiment, plasma density waveforms were obtained using a combination of microwave and probe diagnostics. The microwave diagnosis system mainly includes three parts: a vector network analyzer (VNA), a high-temperature resistant focusing antenna, and a low loss stable phase cable.³²  Figure 12 shows the measured plasma density values over a duration of 10 ms. The average plasma densities corresponding to different experimental conditions, as diagnosed, are $1.76 × 10 11$ and $5.02 × 10 11 cm − 3$⁠, with collision frequencies of $2.75 × 1 0 9 / s$ and $2.35 × 1 0 9 / s$⁠, respectively. Figures 12(a) and 12(b) indicate that the fluctuation levels of plasma density for experiment status 1 and status 2 are in the range of 0.2–0.25. The plasma density and collision frequency corresponding to different experimental states, as obtained from diagnostics, are presented in Table II. The specific process and principle of plasma generation can be found in Refs. 31–33.

Figures 13(a), 13(b), 13(d), and 13(e) show the phase changes of the constellation graph of QPSK signals before and after the LF EM wave passes through the dynamic plasma sheath with an average electron density of $1.76 × 10 11 cm − 3$ and $5.02 × 10 11 cm − 3$⁠. The experimental results were obtained by a software radio receiver program written by USRPX310 and LabVIEW. As shown as in Fig. 13(b), as the LF EM wave passes through the plasma, the constellation is no longer focused on the ideal position, and the constellation appears obvious rotation. Meanwhile, as shown as in Fig. 14, the eye map trajectory becomes unclear and the opening angle becomes smaller. Thus, it confirming that the LF signal is still affected by the parasitic modulation effect in the plasma environment. In Figs. 13(e) and 14(e), with the increase in plasma density, the rotation of constellation is significantly intensified, and the production distance between any two points is obviously decreased. The eye trace gradually becomes chaotic and the opening and closing degree significantly decreases, which further deteriorates the performance of the system. Therefore, as the average plasma density increases, the signal exhibits stronger parasitic modulation effects, which imposes more severe interference on the signal.

To demonstrate the consistency between the experimental results and the theoretical analysis, we simulated the experimental results based on the propagation equivalent model of the LF magnetic field in cylindrical time-varying plasma. The simulation parameters (including electron density and plasma thickness) should be the same as the experimental conditions. As shown in Figs. 13(c), 13(f), 14(c), and 14(f), the simulation results are basically consistent with the experimental results in terms of constellation attenuation and phase rotation changes, which proves the reliability of theoretical derivation.

This paper focus on the effect of parasitic modulation caused by time-varying plasma on the phase modulation of LF EM wave, which is been theoretically analyzed. Experimental and simulation results confirm that LF EM waves are affected by parasitic modulation when passing through time-varying plasma. However, compared to microwave EM waves, LF EM waves exhibit unique parasitic modulation mechanisms. They experience significantly lower levels of parasitic modulation, and under the influence of parasitic modulation, the constellation diagram of LF QPSK signals only undergoes limited clockwise rotation. Additionally, the comparative simulations between microwave and LF signals under the same plasma parameters demonstrate that LF EM waves have stronger penetration capabilities in time-varying plasma. Therefore, using LF waves in maintaining the minimum essential communication may be a more reliable choice than microwave EM wave transmission. Furthermore, numerical simulation results indicate that reducing the carrier frequency or increasing the plasma collision frequency can enhance the penetration capability of LF signals through the plasma sheath. In the future, communication systems suitable for LF ionospheric communication can be developed based on the influence of plasma parameters on LF waves.

This work was supported in part by the National Natural Science Foundation of China under Grant No. 62071355.

The authors have no conflicts to disclose.

Yuxuan Gao: Data curation (supporting); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). Xiaoping Li: Formal analysis (equal). Min Yang: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Kai XIE: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Longjie Qiao: Data curation (equal); Investigation (equal); Methodology (equal); Software (lead). Haoyan Liu: Writing – review & editing (equal). Chengguang Li: Formal analysis (equal). Donglin Liu: Resources (equal). Lei Quan: Resources (equal). Mingxing Wu: Formal analysis (equal).

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