1. Introduction

Laser beams propagation in atmosphere turbulence has been widely investigated in recent years due to the applications in imaging, remote sensing and optical communication. As beams propagate through atmosphere turbulence, the beam dynamics are largely influenced by turbulence, and several optical effects are discussed including scintillation [1–3], beam wander [4,5], beam spreading [6,7] and angle-of-arrival fluctuations [8]. In addition, the beam quality represented by the propagation factor has also been reported [9–11], and the propagation factor gives an intuitive demonstration about the effects of turbulent environment on the beam quality.

Hypersonic turbulence, due to turbulence in the flow about a high velocity flight vehicle, is different from the atmosphere turbulence and contains complex hypersonic plasma sheaths. Experiments have demonstrated that flight dynamics can cause hypersonic turbulence [12]. When an aircraft reenters the earth’s atmosphere with a high velocity, a plasma sheath surrounding the aircraft’s surface will be constructed due to high temperature from air compression and friction [13]. Hypersonic turbulence can also result in the random wave-front distortion and fluctuation [14], which is similar to atmosphere turbulence. The random effects have a significant influence on communication and imaging due to the fluctuation of optical signal. Comparing with atmosphere turbulence, beams propagation through hypersonic turbulence is relatively less investigated because of the complexity of the environment. However, investigation on beams propagation properties through hypersonic turbulence is useful in aerospace engineering. The effects of hypersonic turbulence on electronic beams propagation are discussed [15], and the emphasis is placed on the polarization properties and the source of noise in electromagnetic signal reception. In order to measure the fractal dimension of hypersonic turbulence, a nano-based planar laser scattering method is proposed to visualize the flow in supersonic mixing layer of the wind tunnel [16]. However, these results mainly focus on the electromagnetic waves and measurements of hypersonic turbulence. More recently, a general theoretical model of hypersonic turbulence is proposed, and an anisotropic power spectrum of the refractive-index fluctuation in hypersonic turbulence is obtained [17]. With this model, the wave structure function and the spatial coherence radius are established, and the bit error rate of free-space optical link is also discussed [18]. To the best of our knowledge, no study has been reported about partially coherent beams propagation through hypersonic turbulence with the anisotropic spectrum, and the evolution of beam properties still remains unknown. And some questions have been promoted: What are the effects of the source and hypersonic turbulence parameters on beams evolution? What are the comparison results between hypersonic turbulent environment and atmosphere turbulent environment? The paper will answer the questions.

In this paper, we have firstly investigated the beam evolution properties through anisotropic hypersonic turbulence based on the extended Huygens-Fresnel formula and second-order moments of the Wigner distribution function (WDF). The analytical expressions for the average intensity, the normalized propagation factor and the mean-squared beam width are derived. The effects of the source and turbulent parameters on the average intensity, beam width and beam quality are detailly discussed, and the comparison between anisotropic hypersonic turbulence and atmosphere turbulence has also been shown. Our results can be helpful in optical communication system in the turbulent environment induced by hypersonic aircraft.

2. Second-order moments of Gaussian Schell-model beams in anisotropic hypersonic turbulence

The cross-spectral density (CSD) of Gaussian Schell-model (GSM) beams in the source plane can be expressed as follows [19]

Here, our aim is to investigate the propagation properties in anisotropic hypersonic turbulence. Within the paraxial approximation, the cross-spectral density at $z$ plane propagating in anisotropic hypersonic turbulence can be obtained with the help of the extended Huygens-Fresnel integral [20–23]

The WDF can be expressed by the following formula [32],

By the virtue of WDF, the moments of order $n_1+n_2+m_1+m_2$ of the WDF for the beam is defined as [33],

3. Statistical properties of Gaussian Schell-model beams in anisotropic hypersonic turbulence

In this section, we study the average intensity distribution, the mean-squared beam width and the beam quality of a GSM beam in anisotropic hypersonic turbulence. Based on Eqs. (12), (24) and (25), we perform the numerical simulations. From the result in [16], the parameter $m=1.4$ is a fixed constant for fully developed turbulence in hypersonic turbulence mixing layer. Note that the maximum thickness of anisotropic hypersonic turbulence is restrict to 0.4m through experiment [17]. The simulation results are given in the following subsections.

3.1 Average intensity and mean-squared beam width

The average intensity distribution under the influence of hypersonic turbulence is presented in Fig. 1. For convenience, we perform the normalized average intensity: $I/I_m$, which always presents the Gaussian shape no matter how the parameter changes, and this fact can also be directly indicated from the Gaussian function of Eq. (12). However, the beam width of GSM beams has changed in hypersonic turbulence. From the results in Figs. 1(a)-1(d), one can find that the beam significantly expands during propagation as shown in Fig. 1(a), while the tendency of beams expansion gets weakened for the larger outer scale and anisotropic parameter or the smaller variance of the refractive-index fluctuation in Figs. 1(b)-1(c). From the $\rho _x$ coordinate of Fig. 1, the large width of Gaussian shape can be seen, indicating that beams propagation is severely affected by anisotropic hypersonic turbulence, for instance, the beams width changes from $w_0=2\textrm {cm}$ to the magnitude of hectometer within a short propagation distance. For intuitively showing the evolution of the beam width, we give a evaluation factor represented as the mean-squared beam width.

 

Fig. 1. The normalized average intensity in hypersonic turbulence with $w_0=0.02\textrm {m}$, $\lambda =3.8\mu\textrm {m}$, $\xi _y=1$, $\delta _0=5\textrm {mm}$ for (a) propagation distance $z$, (b) outer scale $L_0$, (c) anisotropic parameter $\xi _x$ and (d) variance of the refractive-index fluctuation $<n_1^2>$.

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Figure 2 shows the normalized mean-squared beam width versus propagation distance under different source parameters. From Eq. (25), one can clearly see that the normalized mean-squared beam width shows a positive relation with the propagation distance. That is to say, the increased propagation distance leads to the larger beam width in Fig. 2. The evolution of the normalized mean-squared beam width under different source parameters are presented in Figs. 2(a) and 2(b), and the beam expansion becomes slow when the initial beam width increases, which can be indicated from Fig. 2(a), and the normalized mean-squared beam width gradually decreases for a larger initial beam width. However, the beam width almost remains unchanged when changing the initial coherent length from $0.005\textrm {m}$ to $0.3\textrm {m}$ (see the three overlapped lines), for details, the normalized mean-squared beam width versus inital coherent length is presented as the subfigure in Fig. 2(b), where the identical phonomenon can be shown. The result in Fig. 2(b) demonstrates that the normalized mean-squared beam width is not sensitive to the initial coherence in hypersoinc turbulence. The influences of the hypersonic turbulent parameters on the mean-squared beam width of GSM beams are shown in Fig. 3, and it can be seen that the beam width increases almost monotonically with the variance of the refractive-index fluctuation, that is to say, the stronger refractive-index fluctuation of anisotropic hypersonic turbulence leads to the larger beam expansion. In addition, the beam expanding can be decreased for increasing the outer scale and the anisotropic parameter of hypersonic turbulence. However, for a large outer scale or anisotropic parameter, the increment rate of the beam width under the variance of the refractive-index fluctuation becomes slow, for example, the mean-squared beam width just increases from 500 to 2000 for $\xi _x=10$ and generally retains unchanged when the outer scale is equal to $0.4\textrm {m}$. It should be noted that the identical results emerge for changing the anisotropic parameter $\xi _y$.

 

Fig. 2. The normalized mean-squared beam width versus propagation distance under different source parameters with $L_0=0.1m$, $\xi _x=3$, $\xi _y=1$, $<n_1^2>=0.2\times 10^{-24}$ for (a) $\lambda =3.8\mu\textrm{m}$, $z=0.4\textrm {m}$, $\delta _0=5\textrm {mm}$ and (b) $\lambda =3.8\mu\textrm {m}$, $w_0=2\textrm {cm}$, $z=0.4\textrm {m}$.

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Fig. 3. The normalized mean-squared beam width versus the variance of the refractive-index fluctuation under different hypersonic turbulent parameters with $w_0=2\textrm {cm}$, $\delta _0=5\textrm {mm}$, $z=0.4\textrm {m}$ for (a) the outer scale and (b) the anisotrpic parameter.

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3.2 Beam quality evaluation factor

Figure 4 depicts the evolution of the normalized propagation factor in anisotropic hypersonic turbulence versus propagation distance under different source parameters, and the high value of the normalized propagation factor is obvious, and then the beam quality is significantly affected by anisotropic hypersonic turbulence. The increase of the propagation distance leads to the larger normalized propagation factor, which indicates the decline of the beam quality, as shown in Fig. 4. The effect of the initial coherent length on the normalized propagation factor is presented in Fig. 4(b), and the normalized propagation factor firstly increases and then tends to be saturable as the initial coherent length increases, demonstrating that the beam quality evolves from degradation to immutability. The influence of the initial coherent length on the beam quality can also be directly shown in the subfigure of Fig. 4(b). However, the beam quality can be improved for a larger initial beam width in Fig. 4(a), where the normalized propagation factor gradually decreases for increasing the initial beam width. In addition, the decrement trend of the normalized propagation factor becomes slow for further increasing the initial beam width, therefore, the improvement of the beam quality is not sensitive to a much larger initial beam width.

 

Fig. 4. The normalized propagation factor versus propagation distance under different source parameters with $L_0=0.1\textrm {m}$, $\xi _x=3$, $\xi _y=1$, $<n_1^2>=0.2\times 10^{-24}$ for (a) $\lambda =3.8\mu\textrm {m}$, $z=0.4\textrm {m}$, $\delta _0=5\textrm {mm}$ and (b) $\lambda =3.8\mu\textrm {m}$, $w_0=2\textrm {cm}$, $z=0.4\textrm {m}$.

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Next, we will give a clarification of the effects of hypersonic turbulence parameters on the normalized propagation factor as Fig. 5 shows. Figures 5(a) and 5(b) give the normalized propagation factor versus the variance of the refractive-index fluctuation $<n_1^2>$, and one can find that the normalized propagation factor linearly increases with the variance of the refractive-index fluctuation, thus leading to the linear degradation of the beam quality. The influence of the outer scale $L_0$ of anisotropic hypersonic turbulence on the normalized propagation factor is shown in Fig. 5(a), where the value of the outer scale $L_0$ changes from 0 to $0.4\textrm {m}$. It is noted that the maximum thickness of the hypersonic turbulence is 0.4m, thus the outer scale is limited to $0.4\textrm {m}$. In addition, we have plotted the normalized propagation factor versus the outer scale as the subfigure shows in Fig. 5(a). A large outer scale can improve the increment of the normalized propagation factor, and then the beam quality becomes better, while the improvement tends to be saturable when the outer scale is larger than $0.15\textrm {m}$. For the impacts of anisotropic parameters, the details are presented in Fig. 5(b). Similar to the result in Fig. 5(a), for increasing the anisotropic parameter $\xi _x$ from 1 to 10, the normalized propagation factor firstly decreases, and then the rate of improvement becomes slow. It is also noted that increasing $\xi _x$ is equivalent to the increment of $\xi _y$. The influences of the turbulence parameters on the beam quality and beam width can be explained from the evolution of the spatial coherence radius according to Eq. (4). Figure 6 gives an intuitive view on the spatial coherence radius of anisotropic hypersonic turbulence represented by $\rho _{0\xi }$. It is well known that the spatial coherence radius is inversely proportional to the turbulent strength [30]. From Fig. 6, the spatial coherence radius $\rho _{0\xi }$ gradually decreases with the increase of the refractive-index fluctuation $<n_1^2>$ or the decrease of the outer scale $L_0$ and the anisotropic parameter $\xi _x$, and then the turbulent strength becomes stronger, as a result, the turbulence-induced beam distortion gets enhanced, and the degradation of the beam quality and the beam expansion becomes severe. Finally, the more physical explanations are given. As is known, the influence of turbulence depends on the refractive-index, and the larger refractive-index corresponds to the stronger turbulent phase modulation. When the variance of the refractive-index fluctuation increases, the modulation of turbulent phase becomes strong, thus leading to the enhanced wave-front fluctuation of beams. The larger outer scale or anisotropic parameter means the increment of the size or asymmetry of the turbulent eddy, respectively. As the size of the turbulent eddy increases, the modulation area of turbulent environment on the beam spot is larger, and then the wave-front becomes relatively homogeneous. For the eddy with increased asymmetry, the weak modulation of turbulent phase on beams can be indicated because of the decreased modulation thickness induced by the eddy. Therefore, the weak influence of turbulence on beams propagation is easy to be concluded when the outer scale and anisotropic parameter increases. The analysis from Fig. 6 and physical mechanisms matches well with the results in Fig. 1, Fig. 3 and Fig. 5.

 

Fig. 5. The normalized propagation factor under different turbulent parameters with $\lambda =3.8\mu\textrm {m}$, $w_0=2\textrm {cm}$, $\delta _0=5\textrm {mm}$, $z=0.4\textrm {m}$ for (a) $<n_1^2>=0.2\times 10^{-24}$, $\xi _x=3$, $\xi _y=1$ and (c) $<n_1^2>=0.2\times 10^{-24}$, $L_0=0.1\textrm {m}$, $\xi _y=1$.

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Fig. 6. The spatial coherence radius versus the variance of the refractive-index fluctuation under different hypersonic turbulence parameters, for (a) and (b) anisotropic parameter, (c) the outer scale.

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3.3 Comparison between anisotropic hypersonic turbulence and atmosphere turbulence

Finally, we give a comparison between anisotropic hypersonic turbulence and atmosphere turbulence. With the method above, the propagation factor and mean-squared beam width of GSM beams in atmosphere turbulence can be obtained after tedious calculations:

 

Fig. 7. The results comparison between anisotropic hypersonic turbulence ((a1)–(a3)) and atmosphere turbulence ((b1)–(b3)) with $\lambda =3.8\mu\textrm {m}$, $w_0=2\textrm {cm}$, $\delta _0=5\textrm {mm}$.

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4. Conclusion

The analytical expressions for the average intensity, the normalized propagation factor and the normalized mean-squared beam width of Gaussian Schell-model beams in anisotropic hypersonic turbulence have been derived. The evolution of the normalized average intensity proves that beams significantly expand in anisotropic hypersonic turbulence with retaining a Gaussian shape. It is shown that beams with the larger initial beam width or the smaller coherent length lead to the less impact on the beam quality degradation, while the change of the coherent length contributes less to the decrease of beam expanding. The influences of the turbulent parameters on the beam quality and beam width are detailly discussed, and the expansion of beams and the degradation of beam quality can be enhanced with the increase of the variance of the refractive-index fluctuation or the decrease of the outer scale and the anisotropic parameter. The influences of turbulent parameters on beams evolution are indicated from the spatial coherence radius. Moreover, we have made a comparison of the beam properties between anisotropic hypersonic turbulence and atmosphere turbulence, and the result shows the much higher value of the normalized propagation factor and the normalized mean-squared beam width in anisotropic hypersonic turbulence, demonstrating the much stronger influence of anisotropic hypersonic turbulence. Our results intuitively present the beam properties through anisotropic hypersonic turbulence, which can be useful in optical communication field in the hypersonic turbulent environment.