Abstract

Hypersonic turbulence, causing the environment disturbance, is an important factor influencing optical propagation and optical communication. Based on the extended Huygens-Fresnel principle and second-order moments of the Wigner distribution function, we have derived the analytical expressions of the average intensity, the normalized propagation factor, and the normalized mean-squared beam width of Gaussian Schell-model beams in anisotropic hypersonic turbulence. The evolution properties demonstrate the severe beam expansion and beam quality degradation, and the influences of the source and turbulent parameters are discussed in detail. The comparison between anisotropic hypersonic turbulence and atmosphere turbulence is given, and it is proved that the influence of anisotropic hypersonic turbulence is much stronger. Our results play a guiding role in optical signal transmission in the turbulent environment induced by hypersonic aircraft.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 [13], 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 [911], 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]

$$W\left(\mathbf{r_1},\mathbf{r_2},0\right)=\textrm{exp}\left(-\frac{|\mathbf{r_1}|^2+|\mathbf{r_2}|^2}{4w_0^2}\right)\textrm{exp}\left[-\frac{\left(\mathbf{r_1}-\mathbf{r_2}\right)^2}{2\delta_0^2}\right],$$
where $\mathbf {r_1}=\left (r_{1x}, r_{1y}\right )$ and $\mathbf {r_2}=\left (r_{2x}, r_{2y}\right )$ are two positions vectors at $z=0$. $w_0$ is the transverse beam width at the source plane and $\delta _0$ is the transverse coherent width.

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 [2023]

$$\begin{aligned} W\left(\rho_{\mathbf{1}}, \rho_{\mathbf{2}}, z\right)=&\left(\frac{k}{2\pi z}\right)^2\int\int d^2\mathbf{r_1}\int\int d^2\mathbf{r_1} W(\mathbf{r_1},\mathbf{r_2},0)\\ &\times \textrm{exp}\left(\frac{ik}{2z}\left[(\rho_{\mathbf{1}}-\mathbf{r_1})^2-\left(\rho_{\mathbf{2}}-\mathbf{r_2}\right)^2\right]\right)\\ &\times \left\langle \textrm{exp}\left[\Psi^{*}\left(\rho_{\mathbf{1}},\mathbf{r_1}\right)+\Psi\left(\rho_{\mathbf{2}},\mathbf{r_2}\right)\right]\right\rangle , \end{aligned}$$
where $k=2\pi /\lambda$ is the wave number with $\lambda$ being the wavelength of the source beam. $\rho _{\mathbf {1}}=\left (\rho _{1x}, \rho _{1y}\right )$ and $\rho _{\mathbf {2}}=\left (\rho _{2x}, \rho _{2y}\right )$ are two arbitrary transverse position vectors at the receiver plane, $d^2\mathbf {r_1}\int \int d^2\mathbf {r_2}=dr_{1x}dr_{1y}dr_{2x}dr_{2y}$. $\left\langle \left [\Psi ^{*}\left (\rho _{\mathbf {1}},\mathbf {r_1}\right )+\Psi \left (\rho _{\mathbf {1}},\mathbf {r_1}\right )\right ]\right\rangle $, which dues to the anisotropic hypersonic turbulence, represents the ensemble average and is expressed as [24,25]:
$$\begin{aligned} \left\langle \textrm{exp}\left[\Psi^{*}\left(\rho_{\mathbf{1}},\mathbf{r_1}\right)+\Psi\left(\rho_{\mathbf{2}},\mathbf{r_2}\right)\right]\right\rangle =&\textrm{exp}\left[-\frac{\left(\rho_{\mathbf{1}}-\rho_{\mathbf{2}}\right)\left(\mathbf{r_1}-\mathbf{r_2}\right)+\left(\mathbf{r_1}-\mathbf{r_2}\right)^2}{\rho_{0\xi}^{2}}\right]\\ &\times \textrm{exp}\left[-\frac{\left(\rho_{\mathbf{1}}-\rho_{\mathbf{2}}\right)^2}{\rho_{0\xi}^{2}}\right], \end{aligned}$$
In Eq. (3), $\rho _{0\xi }$ is the spatial coherence radius of a spherical wave in anisotropic hypersonic turbulence [17]. From [2628], the spatial coherence radius $\rho _{0\xi }$ is defined as:
$$\rho_{0\xi}^{-2}=\frac{\pi^2k^2z}{3}\int_{0}^{\infty}\kappa^{'3}\tilde{\psi}_{an}\left(\kappa^{'}\right)\textrm{d}\kappa{'},$$
where $\int _{0}^{\infty }\kappa ^{'3}\tilde {\psi }_{an}\left (\kappa ^{'}\right )\textrm {d}\kappa {'}$ represents the turbulence strength. It should be noted from Eq. (4) that the hypersonic turbulence strength is inversely proportional to the spatial coherence radius $\rho _{0\xi }$. That is to say, the smaller $\rho _{0\xi }$ means the stronger turbulent strength. In Markov approximation, the two-dimensional refractive index power spectrum $\tilde {\psi }_{an}\left (\kappa ^{'}\right )$ for anisotropic hypersonic turbulence can be expresses as follows [17,29,30]:
$$\tilde{\psi}_{an}\left(\kappa^{'}\right)=a\frac{64\pi<n_1^2>L_0^2(m-1)}{\left(1+100\kappa^{'}L_0^2\right)^m}\textrm{exp}\left(-\frac{\kappa^{'}}{\kappa_0}\right),$$
where $\left\langle n_1^2\right\rangle $ is the variance of the refractive-index fluctuation, $L_0$ is the outer scale, $m$ is a constant, $a$ is a fitting parameter, $\kappa ^{'}=\sqrt {\xi _x^2\kappa _x^2+\xi _y^2\kappa _y^2}$, $\xi _x$ and $\xi _y$ are two anisotropic parameters along the $x$ and $y$ directions, respectively, $\kappa _0=\left (2\pi /l_0\right )^{m-0.7}$ and $l_0$ is the inner scale. In hypersonic turbulence, the outer scale $L_0$ and inner scale $l_0$ follow the equation: $L_0/l_0=R_e^{3/4}$ with $R_e=5\times 10^5/\textrm {m}$ [18]. According to the results in [31], when the wavelength $\lambda =3.8\mu\textrm{m}$, the electron-neutral particle collision frequency $v_e=30\textrm {GHz}$ and the mean electron density $n_e=10^{15}/\textrm {m}^{-3}$, the variances of the refractive-index fluctuation is equal to $\left\langle n_1^2\right\rangle =0.2\times 10^{-24}$, and then the mean value of $a$ equals to $\kappa _0^{4.214}$. Substituting Eq. (5) into Eq. (4), the analytical expression of $\rho _{0\xi }$ can be obtained by a long calculation:
$$\begin{aligned} \rho_{0\xi}=&\left[\frac{64a}{10^4}\pi^3k^2z\left\langle n_1^2\right\rangle L_0^2(m-1)\frac{\xi_x^2+\xi_y^2}{6\xi_x^3\xi_y^3}\Gamma(4)\frac{1}{L_0^8}\right]^{-1/2}\\ &\times \left[\left(\frac{\Gamma(m-4)}{\Gamma(m)}+\frac{\Gamma(4-m)}{\Gamma(4)}\left(\frac{1}{100L_0^2\kappa_0}\right)^{m-4}\right)\right]^{-1/2}, \end{aligned}$$
Introducing four variables: $\mathbf {r}=\left (\mathbf {r_1}+\mathbf {r_2}\right )/2$, $\mathbf {r_d}=\mathbf {r_1}-\mathbf {r_2}$, $\mathbf {\rho }=\left (\rho _{\mathbf {1}}+\rho _{\mathbf {2}}\right )/2$, $\rho _{\mathbf {d}}=\rho _{\mathbf {1}}-\rho _{\mathbf {2}}$, we transform the initial cross-spectral density and the ensemble average of anisotropic hypersonic turbulence into the following form:
$$W\left(\mathbf{r_1},\mathbf{r_2},0\right)=W\left(\mathbf{r},\mathbf{r_d},0\right)=W\left(\mathbf{r}+\frac{\mathbf{r_d}}{2},\mathbf{r}-\frac{\mathbf{r_d}}{2},0\right),$$
$$\left\langle \left[\Psi^{*}\left(\rho_{\mathbf{1}},\mathbf{r_1}\right)+\Psi\left(\rho_{\mathbf{1}},\mathbf{r_1}\right)\right]\right\rangle =-\frac{1}{\rho_{0\xi}^2}\left(\mathbf{r_d}^2+\mathbf{r_d}\rho_{\mathbf{d}}+\rho_{\mathbf{d}}^2\right),$$
For the convenience of obtaining the second-order moments, we introduce the coordinate transform: $\mathbf {r=r'}, \mathbf {r_d}=\rho _{\mathbf {d}}+z\mathbf {\kappa _d}/k$, and then Eqs. (7) and (8) become the following form:
$$\begin{aligned} W\left(\mathbf{r_1},\mathbf{r_2},0\right)=&W\left(\mathbf{r^{'}},\mathbf{r_d}+\frac{z}{k}\mathbf{\kappa_d},0\right)\\ =&\textrm{exp}\left[-\frac{\mathbf{r^{'2}}}{2\sigma^2}-\left(\frac{1}{8\sigma^2}+\frac{1}{2\delta^2}\right)\left(\mathbf{r_d}+\frac{z}{k}\kappa_\mathbf{d}\right)^2\right], \end{aligned}$$
$$\left\langle \textrm{exp}\left[\Psi^{*}\left(\rho_{\mathbf{1}},\mathbf{r_1}\right)+\Psi\left(\rho_{\mathbf{2}},\mathbf{r_2}\right)\right]\right\rangle =\textrm{exp}\left[-\frac{\left(3\rho_{\mathbf{d}}^{\mathbf{2}}+\frac{3z}{k}\kappa_{\mathbf{d}}\rho_{\mathbf{d}}+\frac{z^2}{k^2}\kappa_{\mathbf{d}}^{\mathbf{2}}\right)}{\rho_{0\xi}^{2}}\right],$$
By substituting Eqs. (9) and (10) into Eq. (2), the cross-spectral density (CSD) at arbitrary $z$ plane can be expressed as:
$$\begin{aligned} W(\rho_{\mathbf{1}}, \rho_{\mathbf{2}}, z)=W({\rho}, \rho_{\mathbf{d}}, z)=&\left(\frac{1}{2\pi}\right)^2\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d^2\mathbf{r^{'}}d^2 \kappa_{\mathbf{d}}\\ &\times \textrm{exp}\left[-\frac{\mathbf{r^{'2}}}{2w_0^2}-\left(\frac{1}{8w_0^2}+\frac{1}{2\delta_0^2}\right)\left(\mathbf{r_d}+\frac{z}{k} \kappa_{\mathbf{d}}\right)^2\right]\\ &\times \textrm{exp}\left[-\frac{1}{\rho_{0\xi}^2}\left(3\rho_{\mathbf{d}}^{\mathbf{2}}+\frac{3z}{k} \kappa_{\mathbf{d}}\rho_{\mathbf{d}}+\frac{z^2}{k^2}\kappa_{\mathbf{d}}^{\mathbf{2}}\right)\right]\\ &\times \textrm{exp}\left(-i \rho \kappa_{\mathbf{d}}+i\mathbf{r^{'}}\kappa_{\mathbf{d}}\right) , \end{aligned}$$
After tedious calculation and by performing $\rho _{\mathbf {1}}=\rho _{\mathbf {2}}=\rho$, the average intensity of GSM beams in anisotropic hypersoinc turbulence can be obtained as
$$I(\mathbf{\rho}, z)=\frac{w_0^2k^2}{2a_0z^2}\textrm{exp}\left(a_1^2\frac{\mathbf{\rho}^2}{4a_0}\right),$$
where $a_0=1/8w_0^2+1/2\delta _0^2+w_0^2k^2/2z^2+1/\rho _{0\xi }^2$, $a_1=ik/z$.

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

$$h\left(\mathbf{\rho, \theta}, z\right)=\left(\frac{k}{2z}\right)^2\int_{-\infty}^{\infty} W\left(\mathbf{\rho}, \rho_{\mathbf{d}}, z\right)\textrm{exp}\left(-ik\theta\rho_{\mathbf{d}}\right)d^2\rho_{\mathbf{d}},$$
After calculations, the WDF for GSM beams is obtained:
$$\begin{aligned} h\left(\mathbf{\rho, \theta}, z\right)=&\frac{k^2w_0^2}{8\pi^3}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} d^2\rho_{\mathbf{d}}d^2\kappa_{\mathbf{d}}\textrm{exp}\left(-b\rho_{\mathbf{d}}^{\mathbf{2}}-c\kappa_{\mathbf{d}}^{\mathbf{2}}-d\rho_{\mathbf{d}} \kappa_{\mathbf{d}}\right)\\ &\times \textrm{exp}\left(-i\rho\kappa_{\mathbf{d}}-ik\theta\rho_{\mathbf{d}}\right), \end{aligned}$$
In Eq. (14), $a=1/4\sigma ^2+1/\delta _0^2$, $b=a/2+3/\rho _{0\xi }^2$, $c=az^2/2k^2+\sigma ^2/2+z^2/(k^2\rho _{0\xi }^2)$, $d=az/k+3z/(k\rho _{0\xi }^2)$.

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],

$$\left\langle \rho_x^{n_1}\rho_y^{n_2}\theta_x^{m_1}\theta_y^{m_2}\right\rangle =\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\rho_x^{n_1}\rho_y^{n_2}\theta_x^{m_1}\theta_y^{m_2}h\left(\mathbf{\rho, \theta}, z\right)d^2\mathbf{\rho}d^2\mathbf{\theta}}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}h\left(\mathbf{\rho, \theta}, z\right)d^2\mathbf{\rho}d^2\mathbf{\theta}},$$
In fact, $\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }h\left (\mathbf {\rho, \theta , z}\right )d^2\mathbf {\rho }d^2\mathbf {\theta }$ is the total irradiance of the beam. The propagation factor of a partially coherent model beam can be defined in terms of the second-order moments of WDF as follows [911,32]
$$M^2(z)=k\left[\left\langle \mathbf{\rho^2}\right\rangle \left\langle \mathbf{\theta^2}\right\rangle -\left\langle \mathbf{\rho\theta}\right\rangle ^2\right]^{1/2},$$
where
$$\left\langle \mathbf{\rho^2}\right\rangle =\left\langle \rho_x^2\right\rangle +\left\langle \rho_y^2\right\rangle ,$$
$$\left\langle \mathbf{\theta^2}\right\rangle =\left\langle \theta_x^2\right\rangle +\left\langle \theta_y^2\right\rangle ,$$
$$\left\langle \mathbf{\rho \theta}\right\rangle =\left\langle \rho_x \theta_x\right\rangle +\left\langle \rho_y \theta_y\right\rangle ,$$
Substituting Eq. (14) and Eq. (15) into Eqs. (17)–(19), we can get the following expressions for the second-order moments of WDF of GSM beams in anisotropic hypersonic turbulence after tedious calculations by applying the formulas mentioned in [34]
$$\left\langle \mathbf{\rho^2}\right\rangle =2w_0^2+\left(\frac{1}{2w_0^2}+\frac{2}{\delta_0^2}\right)\frac{z^2}{k^2}+\frac{4z^2}{k^2\rho_{0\xi}^2},$$
$$\left\langle \mathbf{\theta^2}\right\rangle =\left(\frac{1}{2w_0^2}+\frac{2}{\delta_0^2}\right)\frac{1}{k^2}+\frac{12}{k^2\rho_{0\xi}^2},$$
$$\left\langle \mathbf{\rho}\mathbf{\theta}\right\rangle =\left(\frac{1}{2w_0^2}+\frac{2}{\delta_0^2}\right)\frac{z}{k^2}+\frac{6z}{k^2\rho_{0\xi}^2},$$
Applying the results in Eqs. (20)–(22) to Eq. (16), we can obtain the analytical expression of the propagation factor of GSM beams in anisotropic hypersonic turbulence
$$M^2(z)=\left(1+\frac{4w_0^2}{\delta_0^2}+\frac{24w_0^2}{\rho_{0\xi}^2}+\frac{2z^2}{k^2\rho_{0\xi}^2w_0^2}+\frac{8z^2}{k^2\rho_{0\xi}^2\delta_0^2}+\frac{12z^2}{k^2\rho_{0\xi}^4}\right)^{1/2},$$
From the result in Eq. (23), when anisotropic hypersonic turbulence is not considered, the propagation factor in free space ($\rho _{0\xi }=\infty$) is: $M^2(z)=1+4w_0^2/\delta _0^2$, which is identical with the result reported in [35]. In numerical simulations, the normalized propagation factor $M_N^2(z)=M^2(z)/M^2(0)$ is usually adopted, and $M_N^2(z)$ can be obtained as:
$$M_N^2(z)=\sqrt{\frac{1+\frac{4w_0^2}{\delta_0^2}+\frac{24w_0^2}{\rho_{0\xi}^2}+\frac{2z^2}{k^2\rho_{0\xi}^2w_0^2}+\frac{8z^2}{k^2\rho_{0\xi}^2\delta_0^2}+\frac{12z^2}{k^2\rho_{0\xi}^4}}{1+\frac{4w_0^2}{\delta_0^2}}},$$
In addition, the mean-squared beam width is another important parameter characterizing light beams in turbulence. According to the definition of the mean-squared beam width of laser beams [36], the analytical expression of the normalized mean-squared beam width can be decribed as:
$$w_N\left(z\right)=\frac{w\left(z\right)}{w\left(0\right)}=\sqrt{\frac{2w_0^2+\left(\frac{1}{2w_0^2}+\frac{2}{\delta_0^2}\right)\frac{z^2}{k^2}+\frac{4z^2}{k^2\rho_{0\xi}^2}}{2w_0^2}}.$$

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.

 figure: Fig. 1.

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 $\xi _y$.

 figure: Fig. 2.

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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 figure: Fig. 3.

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.

 figure: Fig. 4.

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.

 figure: 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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 figure: Fig. 6.

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:

$$M^2(z)=\sqrt{1+\frac{4w_0^2}{\delta_0^2}+8\pi^2k^2Tw_0^2z+\frac{2\pi^2Tz^3}{3w_0^2}+\frac{8\pi^2Tz^3}{3\delta_0^2}+\frac{4\pi^4k^2T^2z^4}{3}},$$
$$w\left(z\right)=2\sqrt{2w_0^2+\left(\frac{1}{2w_0^2}+\frac{2}{\delta_0^2}\right)\frac{z^2}{k^2}+\frac{4z^3\pi^2T}{3}},$$
$$\begin{aligned} T=&\frac{A(\alpha)}{2\alpha-4}\tilde{C}_n^2\Bigg[\kappa_m^{2-\alpha}\left(2\kappa_0^2-2\kappa_m^2+\alpha \kappa_m^2\right)\textrm{exp}\left(\frac{\kappa_0^2}{\kappa_m^2}\right)\\ &\times \Gamma\left(2-\frac{\alpha}{2},\frac{\kappa_0^2}{\kappa_m^2}\right)-2\kappa_0^{4-\alpha}\Bigg], \end{aligned}$$
where $T$ represent the strength of atmosphere turbulence, $\Gamma (\cdot )$ is the Gamma function, $A(\alpha )=\Gamma (\alpha -1)\textrm {cos}(\alpha \pi /2)/4\pi ^2$, $\kappa _m=[\Gamma (5-\alpha /2)A(\alpha )2\pi /3]^{1/(\alpha -5)}/l_0$, $\kappa _0=2\pi /L_0$, $\tilde {C}_n^2$ is the refractive structure constant, $\alpha$ is the spectral index, $L_0$ and $l_0$ are the outer and inner scales of the turbulence, respectively. The spatial coherence radius $\rho _0$ is inversely proportional to the turbulent strength $T$, and the analytical expression is:
$$\rho_0=\left(\frac{3}{\pi^2k^2zT}\right)^{1/2}.$$
The comparison results are presented in Fig. 7. Figures 7(a1)–7(a3) and Figs. 7(b1)–7(b3) are the normalized propagation factor, the normalized mean-squared beam width and the corresponding spatial coherence radius in anisotropic hypersonic turbulence and atmosphere turbulence, respectively. In simulations of Fig. 7, the parameters of atmosphere turbulence are: $\tilde {C}_n^2=10^{-12}\textrm {m}^{3-\alpha }$, $L_0=10\textrm {m}$, $l_0=1\textrm {cm}$, $\alpha =3.11$, and the parameters of anisotropic hypersonic turbulence are: $<n_1^2>=0.2\times 10^{-24}$, $L_0=0.4\textrm {m}$, $\xi _x=10$, $\xi _y=1$. According to the foregoing researches, one can find that the parameters of atmosphere turbulence indicate the strong turbulent strength [37,38]. Figures 7(a3) and 7(b3) gives the spatial coherence radius in two kinds of turbulence, and the spatial coherence radius in hypersonic turbulence is much smaller comparing with the atmosphere turbulence, which indicates the much stronger turbulent strength. The optical effects can be directly shown from the normalized propagation factor in Figs. 7(a1) and 7(b1) and the beam width in Figs. 7(a2) and 7(b2), and the normalized propagation factor and mean-squared beam width under hypersonic turbulence is much larger than that under atmosphere turbulence with the same propagation distance. For instance, when the propagation length is equal to $0.4\textrm {m}$, the value of the normalized propagation factor can reach $10^6$ order of magnitude in hypersonic turbulence and is only 1.0004 in atmosphere turbulence, and the beam width remains unchanged in atmosphere turbulence. From the results in Fig. 7, we can conclude that the influence of anisotropic hypersonic turbulence on beams propagation is much stronger than atmosphere turbulence. The results shown in this paper evaluate the effects of the hypersonic turbulent channel on beams evolution properties, which might be useful in free-space optical communication of hypersonic aircraft-to-ground link.

 figure: Fig. 7.

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.

Funding

National Natural Science Foundation of China (61571183, 61971184); Natural Science Foundation of Hunan Province (2017JJ1014).

Disclosures

The authors declare no conflicts of interest.

References

1. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011). [CrossRef]  

2. X. Yi, Z. Liu, and P. Yue, “Inner-and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012). [CrossRef]  

3. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009). [CrossRef]  

4. W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018). [CrossRef]  

5. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015). [CrossRef]  

6. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef]  

7. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef]  

8. L. Cui, “Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015). [CrossRef]  

9. X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015). [CrossRef]  

10. Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016). [CrossRef]  

11. H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014). [CrossRef]  

12. T. C. Lin and L. K. Sproul, “Influence of reentry turbulent plasma fluctuation on EM wave propagation,” Comput. Fluids 35(7), 703–711 (2006). [CrossRef]  

13. M. Kim, M. Keidar, and I. D. Boyd, “Electrostatic manipulation of a hypersonic plasma layer: Images of the two-dimensional sheath,” IEEE Trans. Plasma Sci. 36(4), 1198–1199 (2008). [CrossRef]  

14. J. T. Li and L. X. Guo, “Research on electromagnetic scattering characteristics of reentry vehicles and blackout forecast model,” J. Electromagnet. Wave. 26(13), 1767–1778 (2012). [CrossRef]  

15. B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014). [CrossRef]  

16. Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008). [CrossRef]  

17. J. Li, S. Yang, L. Guo, and M. Cheng, “Anisotropic power spectrum of refractive-index fluctuation in hypersonic turbulence,” Appl. Opt. 55(32), 9137–9144 (2016). [CrossRef]  

18. J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017). [CrossRef]  

19. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef]  

20. E. Wolf and G. Gbur, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef]  

21. J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015). [CrossRef]  

22. X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017). [CrossRef]  

23. M. Tang and D. Zhao, “Regions of spreading of Gaussian array beams propagating through oceanic turbulence,” Appl. Opt. 54(11), 3407–3411 (2015). [CrossRef]  

24. J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016). [CrossRef]  

25. Y. Wu, Y. Zhang, and Y. Zhu, “Average intensity and directionality of partially coherent model beams propagating in turbulent ocean,” J. Opt. Soc. Am. A 33(8), 1451–1458 (2016). [CrossRef]  

26. X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018). [CrossRef]  

27. M. Tang, D. Zhao, X. Li, and J. Wang, “Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Commun. 407, 392–397 (2018). [CrossRef]  

28. M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015). [CrossRef]  

29. L. Cui, B. Xue, and F. Zhou, “Generalized anisotropic turbulence spectra and applications in the optical waves’ propagation through anisotropic turbulence,” Opt. Express 23(23), 30088–30103 (2015). [CrossRef]  

30. M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015). [CrossRef]  

31. J. Li, T. Gong, L. Guo, and S. Yang, “Wave structure function of electromagnetic waves propagating through anisotropic hypersonic turbulence,” In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (2017), pp. 1843–1844.

32. S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013). [CrossRef]  

33. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef]  

34. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. (Academic, 2007).

35. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]  

36. X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011). [CrossRef]  

37. Z. Song, Z. Liu, K. Zhou, Q. Sun, and S. Liu, “Propagation factors of multi-sinc Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 24(2), 1804–1813 (2016). [CrossRef]  

38. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016). [CrossRef]  

References

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  • |

  1. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
    [Crossref]
  2. X. Yi, Z. Liu, and P. Yue, “Inner-and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012).
    [Crossref]
  3. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
    [Crossref]
  4. W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
    [Crossref]
  5. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
    [Crossref]
  6. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
    [Crossref]
  7. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
    [Crossref]
  8. L. Cui, “Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015).
    [Crossref]
  9. X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015).
    [Crossref]
  10. Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016).
    [Crossref]
  11. H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
    [Crossref]
  12. T. C. Lin and L. K. Sproul, “Influence of reentry turbulent plasma fluctuation on EM wave propagation,” Comput. Fluids 35(7), 703–711 (2006).
    [Crossref]
  13. M. Kim, M. Keidar, and I. D. Boyd, “Electrostatic manipulation of a hypersonic plasma layer: Images of the two-dimensional sheath,” IEEE Trans. Plasma Sci. 36(4), 1198–1199 (2008).
    [Crossref]
  14. J. T. Li and L. X. Guo, “Research on electromagnetic scattering characteristics of reentry vehicles and blackout forecast model,” J. Electromagnet. Wave. 26(13), 1767–1778 (2012).
    [Crossref]
  15. B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
    [Crossref]
  16. Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
    [Crossref]
  17. J. Li, S. Yang, L. Guo, and M. Cheng, “Anisotropic power spectrum of refractive-index fluctuation in hypersonic turbulence,” Appl. Opt. 55(32), 9137–9144 (2016).
    [Crossref]
  18. J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017).
    [Crossref]
  19. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref]
  20. E. Wolf and G. Gbur, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
    [Crossref]
  21. J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
    [Crossref]
  22. X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
    [Crossref]
  23. M. Tang and D. Zhao, “Regions of spreading of Gaussian array beams propagating through oceanic turbulence,” Appl. Opt. 54(11), 3407–3411 (2015).
    [Crossref]
  24. J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
    [Crossref]
  25. Y. Wu, Y. Zhang, and Y. Zhu, “Average intensity and directionality of partially coherent model beams propagating in turbulent ocean,” J. Opt. Soc. Am. A 33(8), 1451–1458 (2016).
    [Crossref]
  26. X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
    [Crossref]
  27. M. Tang, D. Zhao, X. Li, and J. Wang, “Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Commun. 407, 392–397 (2018).
    [Crossref]
  28. M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
    [Crossref]
  29. L. Cui, B. Xue, and F. Zhou, “Generalized anisotropic turbulence spectra and applications in the optical waves’ propagation through anisotropic turbulence,” Opt. Express 23(23), 30088–30103 (2015).
    [Crossref]
  30. M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
    [Crossref]
  31. J. Li, T. Gong, L. Guo, and S. Yang, “Wave structure function of electromagnetic waves propagating through anisotropic hypersonic turbulence,” In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (2017), pp. 1843–1844.
  32. S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
    [Crossref]
  33. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
    [Crossref]
  34. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. (Academic, 2007).
  35. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999).
    [Crossref]
  36. X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
    [Crossref]
  37. Z. Song, Z. Liu, K. Zhou, Q. Sun, and S. Liu, “Propagation factors of multi-sinc Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 24(2), 1804–1813 (2016).
    [Crossref]
  38. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
    [Crossref]

2018 (3)

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

M. Tang, D. Zhao, X. Li, and J. Wang, “Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Commun. 407, 392–397 (2018).
[Crossref]

2017 (2)

X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
[Crossref]

J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017).
[Crossref]

2016 (6)

2015 (8)

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref]

M. Tang and D. Zhao, “Regions of spreading of Gaussian array beams propagating through oceanic turbulence,” Appl. Opt. 54(11), 3407–3411 (2015).
[Crossref]

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref]

L. Cui, B. Xue, and F. Zhou, “Generalized anisotropic turbulence spectra and applications in the optical waves’ propagation through anisotropic turbulence,” Opt. Express 23(23), 30088–30103 (2015).
[Crossref]

M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
[Crossref]

Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
[Crossref]

L. Cui, “Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015).
[Crossref]

X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015).
[Crossref]

2014 (2)

H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
[Crossref]

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

2013 (2)

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

2012 (2)

J. T. Li and L. X. Guo, “Research on electromagnetic scattering characteristics of reentry vehicles and blackout forecast model,” J. Electromagnet. Wave. 26(13), 1767–1778 (2012).
[Crossref]

X. Yi, Z. Liu, and P. Yue, “Inner-and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012).
[Crossref]

2011 (2)

X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
[Crossref]

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

2010 (1)

2009 (2)

2008 (2)

M. Kim, M. Keidar, and I. D. Boyd, “Electrostatic manipulation of a hypersonic plasma layer: Images of the two-dimensional sheath,” IEEE Trans. Plasma Sci. 36(4), 1198–1199 (2008).
[Crossref]

Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
[Crossref]

2006 (1)

T. C. Lin and L. K. Sproul, “Influence of reentry turbulent plasma fluctuation on EM wave propagation,” Comput. Fluids 35(7), 703–711 (2006).
[Crossref]

2002 (2)

1999 (1)

Bai, B.

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

Bai, Y.

Baykal, Y.

Borghi, R.

Boyd, I. D.

M. Kim, M. Keidar, and I. D. Boyd, “Electrostatic manipulation of a hypersonic plasma layer: Images of the two-dimensional sheath,” IEEE Trans. Plasma Sci. 36(4), 1198–1199 (2008).
[Crossref]

Cai, Y.

Chen, Y.

Cheng, M.

Cheng, M. J.

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
[Crossref]

Cheng, Z.

Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
[Crossref]

Chu, X.

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Cincotti, G.

Cui, L.

Deng, Z.

Du, S.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Eyyuboglu, H. T.

Fu, S.

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

Fu, X.

Gao, C.

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

Gao, J.

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
[Crossref]

Gao, Z.

Gbur, G.

Gong, T.

J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017).
[Crossref]

J. Li, T. Gong, L. Guo, and S. Yang, “Wave structure function of electromagnetic waves propagating through anisotropic hypersonic turbulence,” In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (2017), pp. 1843–1844.

Gori, F.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. (Academic, 2007).

Gu, Y.

Guo, L.

J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017).
[Crossref]

J. Li, S. Yang, L. Guo, and M. Cheng, “Anisotropic power spectrum of refractive-index fluctuation in hypersonic turbulence,” Appl. Opt. 55(32), 9137–9144 (2016).
[Crossref]

M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
[Crossref]

J. Li, T. Gong, L. Guo, and S. Yang, “Wave structure function of electromagnetic waves propagating through anisotropic hypersonic turbulence,” In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (2017), pp. 1843–1844.

Guo, L. X.

J. T. Li and L. X. Guo, “Research on electromagnetic scattering characteristics of reentry vehicles and blackout forecast model,” J. Electromagnet. Wave. 26(13), 1767–1778 (2012).
[Crossref]

He, L.

Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
[Crossref]

Hu, M.

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Hu, Z. D.

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
[Crossref]

Huang, W.

Huang, X.

Huang, Y.

Ji, G.

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Ji, X.

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Jin, Y.

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Keidar, M.

M. Kim, M. Keidar, and I. D. Boyd, “Electrostatic manipulation of a hypersonic plasma layer: Images of the two-dimensional sheath,” IEEE Trans. Plasma Sci. 36(4), 1198–1199 (2008).
[Crossref]

Kim, M.

M. Kim, M. Keidar, and I. D. Boyd, “Electrostatic manipulation of a hypersonic plasma layer: Images of the two-dimensional sheath,” IEEE Trans. Plasma Sci. 36(4), 1198–1199 (2008).
[Crossref]

Korotkova, O.

Li, J.

J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017).
[Crossref]

J. Li, S. Yang, L. Guo, and M. Cheng, “Anisotropic power spectrum of refractive-index fluctuation in hypersonic turbulence,” Appl. Opt. 55(32), 9137–9144 (2016).
[Crossref]

J. Li, T. Gong, L. Guo, and S. Yang, “Wave structure function of electromagnetic waves propagating through anisotropic hypersonic turbulence,” In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (2017), pp. 1843–1844.

Li, J. T.

J. T. Li and L. X. Guo, “Research on electromagnetic scattering characteristics of reentry vehicles and blackout forecast model,” J. Electromagnet. Wave. 26(13), 1767–1778 (2012).
[Crossref]

Li, X.

M. Tang, D. Zhao, X. Li, and J. Wang, “Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Commun. 407, 392–397 (2018).
[Crossref]

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Liang, C.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Lin, T. C.

T. C. Lin and L. K. Sproul, “Influence of reentry turbulent plasma fluctuation on EM wave propagation,” Comput. Fluids 35(7), 703–711 (2006).
[Crossref]

Liu, L.

Liu, S.

Liu, X.

Liu, Y.

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

Liu, Z.

Luo, M.

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Ma, Y.

Mei, Z.

Peng, X.

Pu, J.

Qiu, Z.

Qu, J.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. (Academic, 2007).

Santarsiero, M.

Shi, L.

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

Shi, X.

Song, Z.

Sproul, L. K.

T. C. Lin and L. K. Sproul, “Influence of reentry turbulent plasma fluctuation on EM wave propagation,” Comput. Fluids 35(7), 703–711 (2006).
[Crossref]

Sun, Q.

Tang, M.

Tian, L.

Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
[Crossref]

Vahimaa, P.

Wang, D. L.

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
[Crossref]

Wang, J.

M. Tang, D. Zhao, X. Li, and J. Wang, “Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Commun. 407, 392–397 (2018).
[Crossref]

Wang, X.

Wen, W.

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Wolf, E.

Wu, Y.

Xie, K.

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

Xu, H. F.

Xu, J.

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

Xue, B.

Yang, S.

J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017).
[Crossref]

J. Li, S. Yang, L. Guo, and M. Cheng, “Anisotropic power spectrum of refractive-index fluctuation in hypersonic turbulence,” Appl. Opt. 55(32), 9137–9144 (2016).
[Crossref]

J. Li, T. Gong, L. Guo, and S. Yang, “Wave structure function of electromagnetic waves propagating through anisotropic hypersonic turbulence,” In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (2017), pp. 1843–1844.

Yao, M.

Yi, S.

Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
[Crossref]

Yi, X.

Yu, J.

Yuan, Y.

Yue, P.

Zeng, A.

Zhang, B.

Zhang, Y.

Zhang, Y. X.

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
[Crossref]

Zhang, Z.

Zhao, D.

Zhao, H.

Zhao, Y.

Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
[Crossref]

Zhou, F.

Zhou, K.

Zhou, L.

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Zhou, P.

Zhou, Y.

Zhu, Y.

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
[Crossref]

Y. Wu, Y. Zhang, and Y. Zhu, “Average intensity and directionality of partially coherent model beams propagating in turbulent ocean,” J. Opt. Soc. Am. A 33(8), 1451–1458 (2016).
[Crossref]

Zou, C.

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Appl. Opt. (2)

Comput. Fluids (1)

T. C. Lin and L. K. Sproul, “Influence of reentry turbulent plasma fluctuation on EM wave propagation,” Comput. Fluids 35(7), 703–711 (2006).
[Crossref]

IEEE Trans. Plasma Sci. (2)

M. Kim, M. Keidar, and I. D. Boyd, “Electrostatic manipulation of a hypersonic plasma layer: Images of the two-dimensional sheath,” IEEE Trans. Plasma Sci. 36(4), 1198–1199 (2008).
[Crossref]

B. Bai, X. Li, Y. Liu, J. Xu, L. Shi, and K. Xie, “Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves,” IEEE Trans. Plasma Sci. 42(10), 3365–3372 (2014).
[Crossref]

J. Electromagnet. Wave. (1)

J. T. Li and L. X. Guo, “Research on electromagnetic scattering characteristics of reentry vehicles and blackout forecast model,” J. Electromagnet. Wave. 26(13), 1767–1778 (2012).
[Crossref]

J. Opt. Soc. Am. A (5)

New J. Phys. (1)

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Opt. Commun. (3)

M. Tang, D. Zhao, X. Li, and J. Wang, “Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Commun. 407, 392–397 (2018).
[Crossref]

J. Li, S. Yang, L. Guo, M. Cheng, and T. Gong, “Bit error rate performance of free-space optical link under effect of plasma sheath turbulence,” Opt. Commun. 396, 1–7 (2017).
[Crossref]

W. Wen, Y. Jin, M. Hu, X. Liu, Y. Cai, C. Zou, M. Luo, L. Zhou, and X. Chu, “Beam wander of coherent and partially coherent Airy beam arrays in a turbulent atmosphere,” Opt. Commun. 415, 48–55 (2018).
[Crossref]

Opt. Express (13)

X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
[Crossref]

X. Yi, Z. Liu, and P. Yue, “Inner-and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012).
[Crossref]

L. Cui, “Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015).
[Crossref]

X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015).
[Crossref]

Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016).
[Crossref]

H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
[Crossref]

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref]

L. Cui, B. Xue, and F. Zhou, “Generalized anisotropic turbulence spectra and applications in the optical waves’ propagation through anisotropic turbulence,” Opt. Express 23(23), 30088–30103 (2015).
[Crossref]

M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
[Crossref]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref]

Z. Song, Z. Liu, K. Zhou, Q. Sun, and S. Liu, “Propagation factors of multi-sinc Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 24(2), 1804–1813 (2016).
[Crossref]

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[Crossref]

Opt. Laser Technol. (1)

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Opt. Lett. (4)

Photonics Res. (2)

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

J. Gao, Y. Zhu, D. L. Wang, Y. X. Zhang, Z. D. Hu, and M. J. Cheng, “Bessel-Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30–34 (2016).
[Crossref]

Sci. China, Ser. G: Phys., Mech. Astron. (1)

Y. Zhao, S. Yi, L. Tian, L. He, and Z. Cheng, “The fractal measurement of experimental images of supersonic turbulent mixing layer,” Sci. China, Ser. G: Phys., Mech. Astron. 51(8), 1134–1143 (2008).
[Crossref]

Other (2)

J. Li, T. Gong, L. Guo, and S. Yang, “Wave structure function of electromagnetic waves propagating through anisotropic hypersonic turbulence,” In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (2017), pp. 1843–1844.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. (Academic, 2007).

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Figures (7)

Fig. 1.
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 $<n_1^2>$ .
Fig. 2.
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) $\lambda =3.8\mu\textrm {m}$ , $w_0=2\textrm {cm}$ , $z=0.4\textrm {m}$ .
Fig. 3.
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.
Fig. 4.
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) $\lambda =3.8\mu\textrm {m}$ , $w_0=2\textrm {cm}$ , $z=0.4\textrm {m}$ .
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) $<n_1^2>=0.2\times 10^{-24}$ , $L_0=0.1\textrm {m}$ , $\xi _y=1$ .
Fig. 6.
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.
Fig. 7.
Fig. 7. The results comparison between anisotropic hypersonic turbulence ((a1)–(a3)) and atmosphere turbulence ((b1)–(b3)) with $\lambda =3.8\mu\textrm {m}$ , $w_0=2\textrm {cm}$ , $\delta _0=5\textrm {mm}$ .

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

W ( r 1 , r 2 , 0 ) = exp ( | r 1 | 2 + | r 2 | 2 4 w 0 2 ) exp [ ( r 1 r 2 ) 2 2 δ 0 2 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 d 2 r 1 d 2 r 1 W ( r 1 , r 2 , 0 ) × exp ( i k 2 z [ ( ρ 1 r 1 ) 2 ( ρ 2 r 2 ) 2 ] ) × exp [ Ψ ( ρ 1 , r 1 ) + Ψ ( ρ 2 , r 2 ) ] ,
exp [ Ψ ( ρ 1 , r 1 ) + Ψ ( ρ 2 , r 2 ) ] = exp [ ( ρ 1 ρ 2 ) ( r 1 r 2 ) + ( r 1 r 2 ) 2 ρ 0 ξ 2 ] × exp [ ( ρ 1 ρ 2 ) 2 ρ 0 ξ 2 ] ,
ρ 0 ξ 2 = π 2 k 2 z 3 0 κ 3 ψ ~ a n ( κ ) d κ ,
ψ ~ a n ( κ ) = a 64 π < n 1 2 > L 0 2 ( m 1 ) ( 1 + 100 κ L 0 2 ) m exp ( κ κ 0 ) ,
ρ 0 ξ = [ 64 a 10 4 π 3 k 2 z n 1 2 L 0 2 ( m 1 ) ξ x 2 + ξ y 2 6 ξ x 3 ξ y 3 Γ ( 4 ) 1 L 0 8 ] 1 / 2 × [ ( Γ ( m 4 ) Γ ( m ) + Γ ( 4 m ) Γ ( 4 ) ( 1 100 L 0 2 κ 0 ) m 4 ) ] 1 / 2 ,
W ( r 1 , r 2 , 0 ) = W ( r , r d , 0 ) = W ( r + r d 2 , r r d 2 , 0 ) ,
[ Ψ ( ρ 1 , r 1 ) + Ψ ( ρ 1 , r 1 ) ] = 1 ρ 0 ξ 2 ( r d 2 + r d ρ d + ρ d 2 ) ,
W ( r 1 , r 2 , 0 ) = W ( r , r d + z k κ d , 0 ) = exp [ r 2 2 σ 2 ( 1 8 σ 2 + 1 2 δ 2 ) ( r d + z k κ d ) 2 ] ,
exp [ Ψ ( ρ 1 , r 1 ) + Ψ ( ρ 2 , r 2 ) ] = exp [ ( 3 ρ d 2 + 3 z k κ d ρ d + z 2 k 2 κ d 2 ) ρ 0 ξ 2 ] ,
W ( ρ 1 , ρ 2 , z ) = W ( ρ , ρ d , z ) = ( 1 2 π ) 2 d 2 r d 2 κ d × exp [ r 2 2 w 0 2 ( 1 8 w 0 2 + 1 2 δ 0 2 ) ( r d + z k κ d ) 2 ] × exp [ 1 ρ 0 ξ 2 ( 3 ρ d 2 + 3 z k κ d ρ d + z 2 k 2 κ d 2 ) ] × exp ( i ρ κ d + i r κ d ) ,
I ( ρ , z ) = w 0 2 k 2 2 a 0 z 2 exp ( a 1 2 ρ 2 4 a 0 ) ,
h ( ρ , θ , z ) = ( k 2 z ) 2 W ( ρ , ρ d , z ) exp ( i k θ ρ d ) d 2 ρ d ,
h ( ρ , θ , z ) = k 2 w 0 2 8 π 3 d 2 ρ d d 2 κ d exp ( b ρ d 2 c κ d 2 d ρ d κ d ) × exp ( i ρ κ d i k θ ρ d ) ,
ρ x n 1 ρ y n 2 θ x m 1 θ y m 2 = ρ x n 1 ρ y n 2 θ x m 1 θ y m 2 h ( ρ , θ , z ) d 2 ρ d 2 θ h ( ρ , θ , z ) d 2 ρ d 2 θ ,
M 2 ( z ) = k [ ρ 2 θ 2 ρ θ 2 ] 1 / 2 ,
ρ 2 = ρ x 2 + ρ y 2 ,
θ 2 = θ x 2 + θ y 2 ,
ρ θ = ρ x θ x + ρ y θ y ,
ρ 2 = 2 w 0 2 + ( 1 2 w 0 2 + 2 δ 0 2 ) z 2 k 2 + 4 z 2 k 2 ρ 0 ξ 2 ,
θ 2 = ( 1 2 w 0 2 + 2 δ 0 2 ) 1 k 2 + 12 k 2 ρ 0 ξ 2 ,
ρ θ = ( 1 2 w 0 2 + 2 δ 0 2 ) z k 2 + 6 z k 2 ρ 0 ξ 2 ,
M 2 ( z ) = ( 1 + 4 w 0 2 δ 0 2 + 24 w 0 2 ρ 0 ξ 2 + 2 z 2 k 2 ρ 0 ξ 2 w 0 2 + 8 z 2 k 2 ρ 0 ξ 2 δ 0 2 + 12 z 2 k 2 ρ 0 ξ 4 ) 1 / 2 ,
M N 2 ( z ) = 1 + 4 w 0 2 δ 0 2 + 24 w 0 2 ρ 0 ξ 2 + 2 z 2 k 2 ρ 0 ξ 2 w 0 2 + 8 z 2 k 2 ρ 0 ξ 2 δ 0 2 + 12 z 2 k 2 ρ 0 ξ 4 1 + 4 w 0 2 δ 0 2 ,
w N ( z ) = w ( z ) w ( 0 ) = 2 w 0 2 + ( 1 2 w 0 2 + 2 δ 0 2 ) z 2 k 2 + 4 z 2 k 2 ρ 0 ξ 2 2 w 0 2 .
M 2 ( z ) = 1 + 4 w 0 2 δ 0 2 + 8 π 2 k 2 T w 0 2 z + 2 π 2 T z 3 3 w 0 2 + 8 π 2 T z 3 3 δ 0 2 + 4 π 4 k 2 T 2 z 4 3 ,
w ( z ) = 2 2 w 0 2 + ( 1 2 w 0 2 + 2 δ 0 2 ) z 2 k 2 + 4 z 3 π 2 T 3 ,
T = A ( α ) 2 α 4 C ~ n 2 [ κ m 2 α ( 2 κ 0 2 2 κ m 2 + α κ m 2 ) exp ( κ 0 2 κ m 2 ) × Γ ( 2 α 2 , κ 0 2 κ m 2 ) 2 κ 0 4 α ] ,
ρ 0 = ( 3 π 2 k 2 z T ) 1 / 2 .

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