Introduction

Over the past few decades, significant advancements have been made in the study of beam transmission through atmospheric turbulence1,2,3,4, resulting in the development of numerous models of refractive index power spectrum and physical models of beam transmission in such conditions5,6,7,8,9. Atmospheric turbulence primarily degrades the performance of optical waves as they traverse turbulence vortices. The scintillation effect caused by turbulence can cause deep random fading of the optical signal at the detector, increase the bit error rate (BER) and potential signal interruptions. Scintillation index (SI) is a crucial metric for quantifying the scintillation phenomenon on beams due to turbulence and is widely employed to assess the quality of beam transmission10,11,12.

The advancement of aerospace technology has spurred extensive research into space target detection and high-speed aircraft13,14,15,16. When an aircraft flies at high speed (Ma > 1), the air surrounding an aircraft is compressed into shock waves, creating a boundary layer along aircraft’s surface. As the distance from the head increases, the boundary layer gradually becomes turbulent, and the medium around the aircraft becomes complex17. The turbulence near the surface of an aircraft differs significantly from atmospheric turbulence. Compressible turbulence generally satisfies the assumption that the air velocity is greater than the speed of sound and often occurs near the aircraft, which leads to compressible turbulence with a small range and strong fluctuation. The essential difference between compressible and atmosphere turbulence is that the velocity field divergence of compressible turbulence is not 0. The research of Xu, Guo, Mackey, Gao, et al. has shown the turbulence near the aircraft’s surface significantly impacts the quality of beam transmission18,19,20,21. Optical distortions caused by air disturbances are referred to as aero-optical effects. Compressible turbulence is distinguished by its short-range effects and large fluctuations in the refractive index. However, research on the transmission model of beam in compressible turbulence is relatively limited, and the transmission behavior of beams in compressible turbulence remains unclear. This gap in research is partly due to the compressibility in such turbulence, which arises from high fluid velocities, contrasting with classical atmospheric turbulence models that assume incompressible flow22. Therefore, before a rigorous physical model of beam transmission in compressible turbulence is established, it is essential to identify suitable methods to obtain transmission results that can help elucidate the beam’s evolution characteristics in such environments.

The research of Rabinovich, Garnier, Wu, Li, et al. indicates various turbulent media exhibit anisotropy23,24,25,26. Turbulence anisotropy reflects the degree of vortex distortion in different directions. Regardless of the type of anisotropic turbulence, beams will exhibit phenomena such as scintillation23, attenuation27, drift28, and expansion29 during transmission. If the existing formulas and results of anisotropic atmospheric turbulence can be used to correspond to the solutions of beam transmission in more complex turbulence, equivalent solutions can be obtained in advance before deriving the strict formulas for beam transmission in complex turbulence. Consequently, Yahya Baykal and Yalcin Ata have linked different turbulence parameters, using other turbulence parameters to represent atmospheric refractive index structure constant \(C_{n}^{2}\)30,31, with the goal of using \(C_{n}^{2}\) to equate other turbulence parameters, achieving the use of extensive solutions of beam transmission in atmospheric turbulence to find solutions for beam in other turbulence. The advantage of this approach lies in its ability to insert expressions for atmospheric structure parameters of anisotropic turbulence into existing physical solutions, thereby enabling the easy derivation of corresponding solutions for similar physical entities in other anisotropic turbulent environments. Thus, the equivalent method can also be applied to compressible turbulence. However, in the original method30, the transmission distances of the beam in both types of turbulence were considered consistent. In reality, though, compressible turbulence typically involves transmission distances on the meter scale, while atmospheric turbulence operates on the kilometer scale. The difference necessitates improving the equivalent method to adapt to turbulence parameters under different transmission distances.

In this paper, we use the scintillation index as an indicator of beam transmission results to establish the relationship between compressible turbulence and atmospheric turbulence. The equivalence indicator is not limited to the scintillation index; it can also include beam expansion, drift variance, and other parameters. This paper focuses on equivalence from a single perspective. By equating the effects caused by parameter changes in compressible turbulence and atmospheric turbulence on the beam, we use the extensive solutions of beam transmission in atmospheric turbulence to correspond to solutions in compressible turbulence, promoting the research of beam transmission behavior in compressible turbulence.

The first section provides an introduction to the background and implementation of the proposed method. The second section derives the scintillation index of light in compressible turbulence and details the theoretical framework of the equivalent method, which links the scintillation index of beam transmitted through atmospheric turbulence with that through compressible turbulence using the equivalent structure function. The third section presents a numerical analysis of the equivalent method to confirm its applicability across different transmission distances. The fourth section summarizes the findings of the paper.

Method

In this section, the expressions for the scintillation index of a Gaussian beam transmitted in anisotropic compressible turbulence and classical atmospheric turbulence under weak fluctuation conditions are given. The atmospheric optical turbulence parameters are used to equate the optical parameters of compressible turbulence, and the equivalent factor is obtained.

Under weak fluctuation conditions, the SI at the center of the Gaussian beam can be obtained from the variance of the logarithmic amplitude fluctuations, that is \(\sigma _{I}^{2}=4\sigma _{\chi }^{2}\)32. Under the Rytov approximation, the axial center SI of a Gaussian beam transmitted in anisotropic compressible turbulence can be expressed as32:

$$\begin{gathered} \sigma _{{I,\left( {\text{C}\text{T}} \right)}}^{2}=8{\uppi ^2}{k^2}{L_{\left( {\text{C}\text{T}} \right)}}\int_{0}^{1} {\int_{0}^{\infty } {\kappa {\Phi _{n\left( {\text{C}\text{T}} \right)}}\left( {\kappa ^{\prime}} \right)} } \exp \left( { - \frac{{{\Lambda _{\left( {\text{C}\text{T}} \right)}}{L_{\left( {\text{C}\text{T}} \right)}}{\kappa ^2}{\zeta ^2}}}{k}} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \left\{ {1 - \cos \left[ {\frac{{{L_{\left( {\text{C}\text{T}} \right)}}{\kappa ^2}}}{k}\zeta \left( {1 - {{\bar {\Theta }}_{\left( {\text{C}\text{T}} \right)}}\zeta } \right)} \right]} \right\}\text{d}\kappa \text{d}\zeta \hfill \\ \end{gathered}$$
(1)

where \(\kappa\) is spatial wavenumber, \(\kappa ^{\prime}=\sqrt {\xi _{{x\left( {\text{C}\text{T}} \right)}}^{2}\kappa _{x}^{2}+\xi _{{y\left( {\text{C}\text{T}} \right)}}^{2}\kappa _{y}^{2}+\kappa _{z}^{2}}\), \({\kappa ^{\prime}_x},{\kappa ^{\prime}_y},{\kappa ^{\prime}_z}\) are the spatial wavenumbers in x, y, and z directions, respectively. ξx and ξy are anisotropic factors (According to the Markov approximation, the anisotropy perpendicular to the propagation axis is considered). If ξx = ξy = 1, turbulence is isotropic. \({\Phi _{n\left( {\text{C}\text{T}} \right)}}\) is the refractive index power spectral function of anisotropic compressible turbulence. \(\zeta =z/L\) is normalized distance, z is the distance on the propagation axis. L(CT) refers to the propagation distance of beam in compressible turbulence. k is optical wavenumber, \(k=2\uppi /\lambda\), \(\lambda\) is wavelength. Other parameters are expressed as

$$\begin{gathered} {\Lambda _{\left( {\text{C}\text{T}} \right)}}=\frac{{{\Lambda _{0\left( {\text{C}\text{T}} \right)}}}}{{\Theta _{{0\left( {\text{C}\text{T}} \right)}}^{2}+\Lambda _{{0\left( {\text{C}\text{T}} \right)}}^{2}}},{\Theta _{\left( {\text{C}\text{T}} \right)}}=\frac{{{\Theta _{0\left( {\text{C}\text{T}} \right)}}}}{{\Theta _{{0\left( {\text{C}\text{T}} \right)}}^{2}+\Lambda _{{0\left( {\text{C}\text{T}} \right)}}^{2}}}, \hfill \\ {{\bar {\Theta }}_{\left( {\text{C}\text{T}} \right)}}=1 - {\Theta _{\left( {\text{C}\text{T}} \right)}},{\Theta _{0\left( {\text{C}\text{T}} \right)}}=1 - \frac{{{L_{\left( {\text{C}\text{T}} \right)}}}}{{{F_0}}},{\Lambda _0}=\frac{{2{L_{\left( {\text{C}\text{T}} \right)}}}}{{k{W_0}}} \hfill \\ \end{gathered}$$
(2)

where F0 is the phase front radius of curvature and \({W_0}\) is the beam radius width. \({\Phi _{n\left( {\text{C}\text{T}} \right)}}\) is expressed as22,33

$$\begin{gathered} {\Phi _{n\left( {\text{C}\text{T}} \right)}}\left( {\kappa ^{\prime}} \right)=\frac{{{{\left( {{K_{GD}}{{\bar {\rho }}^2}\gamma R\bar {T}} \right)}^2}}}{{4\uppi {{\bar {a}}^4}{{\bar {P}}^2}}}\left\{ {{B_P}{\varepsilon ^{4/3}}{y^{ - 4/3}}{l^{13/3}}{{\left[ {1+\left( {\xi _{{x\left( {\text{C}\text{T}} \right)}}^{2}\kappa _{x}^{2}+\xi _{{y\left( {\text{C}\text{T}} \right)}}^{2}\kappa _{y}^{2}+\kappa _{z}^{2}} \right){l^2}} \right]}^{ - 13/6}}} \right. \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {{\kern 1pt} {\kern 1pt} {\text{+}}{R^2}\beta \chi {\varepsilon ^{ - 1/3}}{l^{11/3}}{{\left[ {1+\left( {\xi _{{x\left( {\text{C}\text{T}} \right)}}^{2}\kappa _{x}^{2}+\xi _{{y\left( {\text{C}\text{T}} \right)}}^{2}\kappa _{y}^{2}+\kappa _{z}^{2}} \right){l^2}} \right]}^{ - 11/6}}} \right\} \hfill \\ \end{gathered}$$
(3)

where \(\gamma\) is the ratio of specific heats for air, \({K_{GD}}\) is the Gladstone-Dale constant, \(\bar {\rho }\),\(\bar {T}\),\(\bar {P}\) is the average air density, temperature, and pressure, respectively. y is the distance of the turbulence from the aircraft, \(\bar {a}\)is the average value of sound velocity, \({B_P}\) is a dimensionless constant, \(\varepsilon\) is the turbulence dissipation rate, l is the characteristic length scale (l = L0/2π, L0 is outer scale of turbulence), R is the gas constant, \(\beta\)is a constant, and \(\chi\) is the average temperature dissipation rate.

For convenience in calculations, the constants of the refractive index spectrum of compressible turbulence can be set to

$${C_1}=\frac{{{{\left( {{K_{GD}}{{\bar {\rho }}^2}\gamma R\bar {T}} \right)}^2}}}{{4\uppi {{\bar {a}}^4}{{\bar {P}}^2}}}{B_P}{\varepsilon ^{4/3}}{y^{ - 4/3}},{C_2}=\frac{{{{\left( {{K_{GD}}{{\bar {\rho }}^2}\gamma R\bar {T}} \right)}^2}}}{{4\uppi {{\bar {a}}^4}{{\bar {P}}^2}}}{R^2}\beta \chi {\varepsilon ^{ - 1/3}}$$
(4)

Meanwhile, simplify calculations using the following Eq. 

$${\kappa _x}=\frac{{{q_x}}}{{{\xi _{x\left( {\text{C}\text{T}} \right)}}}},{\kappa _x}=\frac{{{q_y}}}{{{\xi _{y\left( {\text{C}\text{T}} \right)}}}},q=\sqrt {q_{x}^{2}+q_{y}^{2}}$$
(5)
$$\text{d}{\kappa _x}\text{d}{\kappa _y}=\frac{{q\text{d}q\text{d}\theta }}{{{\xi _{x\left( {\text{C}\text{T}} \right)}}{\xi _{y\left( {\text{C}\text{T}} \right)}}}}$$
(6)

Therefore, substituting Eqs. (3), (4), (5) and (6) into Eq. (1) yields

$$\begin{gathered} \sigma _{{I\left( {CT} \right)}}^{2}=4\uppi {k^2}{L_{\left( {\text{C}\text{T}} \right)}}\int_{0}^{{2\uppi }} {\int_{0}^{1} {\int_{0}^{\infty } {\frac{q}{{{\xi _{x\left( {\text{C}\text{T}} \right)}}{\xi _{y\left( {\text{C}\text{T}} \right)}}}}\left[ {{C_1}{l^{13/3}}{{\left( {1+{q^2}{l^2}} \right)}^{ - 13/6}}+{C_2}{l^{11/3}}{{\left( {1+{q^2}{l^2}} \right)}^{ - 11/6}}} \right]} } } \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \exp \left[ { - \frac{{{\Lambda _{\left( {\text{C}\text{T}} \right)}}{L_{\left( {\text{C}\text{T}} \right)}}{q^2}{\zeta ^2}}}{k}\left( {\frac{{{{\cos }^2}\theta }}{{\xi _{{x\left( {\text{C}\text{T}} \right)}}^{2}}}+\frac{{{{\sin }^2}\theta }}{{\xi _{{y\left( {\text{C}\text{T}} \right)}}^{2}}}} \right)} \right] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \left\{ {1 - \cos \left[ {\frac{{{L_{\left( {\text{C}\text{T}} \right)}}{q^2}}}{k}\left( {\frac{{{{\cos }^2}\theta }}{{\xi _{{x\left( {\text{C}\text{T}} \right)}}^{2}}}+\frac{{{{\sin }^2}\theta }}{{\xi _{{y\left( {\text{C}\text{T}} \right)}}^{2}}}} \right)\zeta \left( {1 - {{\bar {\Theta }}_{\left( {\text{C}\text{T}} \right)}}\zeta } \right)} \right]} \right\}\text{d}q\text{d}\zeta \text{d}\theta \hfill \\ \end{gathered}$$
(7)

Integrating Eq. (7) with respect to q, we obtain

$$\begin{gathered} \sigma _{{I\left( {CT} \right)}}^{2} = \frac{{\pi k^{2} L_{{\left( {CT} \right)}} }}{{\xi _{{x\left( {CT} \right)}} \xi _{{y\left( {CT} \right)}} }}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left\{ {2C_{1} \exp \left[ {\frac{{\Lambda _{{\left( {CT} \right)}} L_{{\left( {CT} \right)}} \zeta ^{2} }}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]} \right.} } \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \left[ {\frac{{\Lambda _{{\left( {CT} \right)}} L_{{\left( {CT} \right)}} \zeta ^{2} }}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]^{{7/6}} \Gamma \left[ { - \frac{7}{6},\frac{{\Lambda _{{\left( {CT} \right)}} L_{{\left( {CT} \right)}} \zeta ^{2} }}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right] \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - C_{1} \exp \left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {i + \Lambda _{{\left( {CT} \right)}} \zeta - i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]\left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {i + \Lambda _{{\left( {CT} \right)}} \zeta - i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}} \right. \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \left. {\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]^{{7/6}} \Gamma \left[ { - \frac{7}{6},\frac{{L_{{\left( {CT} \right)}} \zeta \left( {i + \Lambda _{{\left( {CT} \right)}} \zeta - i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right] \hfill \\ {\kern 1pt} {\kern 1pt} - C_{1} \exp \left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {\Lambda _{{\left( {CT} \right)}} \zeta - i + i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]\left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {\Lambda _{{\left( {CT} \right)}} \zeta - i + i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]^{{7/6}} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \Gamma \left[ { - \frac{7}{6},\frac{{L_{{\left( {CT} \right)}} \zeta \left( {\Lambda _{{\left( {CT} \right)}} \zeta - i + i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right] + 2C_{2} \exp \left[ {\frac{{\Lambda _{{\left( {CT} \right)}} L_{{\left( {CT} \right)}} \zeta ^{2} }}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right] \hfill \\ {\kern 1pt} \times \left[ {\frac{{\Lambda _{{\left( {CT} \right)}} L_{{\left( {CT} \right)}} \zeta ^{2} }}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]^{{5/6}} \Gamma \left[ { - \frac{5}{6},\frac{{\Lambda _{{\left( {CT} \right)}} L_{{\left( {CT} \right)}} \zeta ^{2} }}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right] \hfill \\ {\kern 1pt} {\kern 1pt} - C_{2} \exp \left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {i + \Lambda _{{\left( {CT} \right)}} \zeta - i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]\left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {i + \Lambda _{{\left( {CT} \right)}} \zeta - i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]^{{5/6}} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \Gamma \left[ { - \frac{5}{6},\frac{{L_{{\left( {CT} \right)}} \zeta \left( {i + \Lambda _{{\left( {CT} \right)}} \zeta - i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right] \hfill \\ - C_{2} \exp \left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {\Lambda _{{\left( {CT} \right)}} \zeta - i + i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]\left[ {\frac{{L_{{\left( {CT} \right)}} \zeta \left( {\Lambda _{{\left( {CT} \right)}} \zeta - i + i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]^{{5/6}} \hfill \\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \Gamma \left[ { - \frac{5}{6},\frac{{L_{{\left( {CT} \right)}} \zeta \left( {\Lambda _{{\left( {CT} \right)}} \zeta - i + i\bar{\Theta }_{{\left( {CT} \right)}} \zeta } \right)}}{{kl^{2} }}\left( {\frac{{\cos ^{2} \theta }}{{\xi _{{x\left( {CT} \right)}}^{2} }} + \frac{{\sin ^{2} \theta }}{{\xi _{{y\left( {CT} \right)}}^{2} }}} \right)} \right]} \right\}d\zeta d\theta \hfill \\ \end{gathered}$$
(8)

where \(\Gamma \left( \cdot \right)\) is Gamma function. Since Eq. (8) is rather complex, obtaining its analytical solution is difficult. Therefore, subsequent discussions will rely on its numerical solution. This also underscores the complexity of the formula for beam propagation in compressible turbulence and the necessity of using equivalent atmospheric turbulence parameters to explain the transmission results in compressible turbulence. The equivalent method will aid in comprehending the propagation behavior of beams in compressible turbulence.

Similarly, using Rytov theory, the SI of a Gaussian beam on axial center position during propagation in atmospheric turbulence can be expressed as

$$\begin{gathered} \sigma _{{I\left( {\text{A}\text{T}} \right)}}^{2}=8{\uppi ^2}{k^2}{L_{\left( {\text{A}\text{T}} \right)}}\int_{0}^{1} {\int_{0}^{\infty } {\kappa {\Phi _{n\left( {\text{A}\text{T}} \right)}}\left( {\kappa ^{\prime}} \right)} } \exp \left( { - \frac{{{\Lambda _{\left( {\text{A}\text{T}} \right)}}{L_{\left( {\text{A}\text{T}} \right)}}{\kappa ^2}{\zeta ^2}}}{k}} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \left\{ {1 - \cos \left[ {\frac{{{L_{\left( {\text{A}\text{T}} \right)}}{\kappa ^2}}}{k}\zeta \left( {1 - {{\bar {\Theta }}_{\left( {\text{A}\text{T}} \right)}}\zeta } \right)} \right]} \right\}\text{d}\kappa \text{d}\zeta \hfill \\ \end{gathered}$$
(9)

where L(AT) is the propagation distance of light in compressible turbulence, \({\Phi _{n\left( {\text{A}\text{T}} \right)}}\) is the atmospheric turbulence refractive index power spectral density function. Here, the basic form used is the anisotropic non-Kolmogorov spectrum, as described as32

$${\Phi _{n\left( {\text{A}\text{T}} \right)}}\left( {\kappa ^{\prime}} \right){\text{=}}=\frac{{A\left( \alpha \right)C_{n}^{2}{\xi _{x\left( {\text{A}\text{T}} \right)}}{\xi _{y\left( {\text{A}\text{T}} \right)}}}}{{{{\left( {\xi _{{x\left( {\text{A}\text{T}} \right)}}^{2}\kappa _{x}^{2}+\xi _{{y\left( {\text{A}\text{T}} \right)}}^{2}\kappa _{y}^{2}} \right)}^{\alpha /2}}}}$$
(10)

where \(C_{n}^{2}\) represents the generalized atmospheric refractive index structure constant, which is a physical quantity measuring turbulence intensity. \(\alpha\) denotes the spectral power law. The refractive index spectrum assumes zero inner scale and infinite outer scale for turbulence. \(A\left( \alpha \right)\) is expressed as

$$A\left( \alpha \right)=\frac{1}{{4\uppi }}\Gamma \left( {\alpha - 1} \right)\cos \left( {\frac{{\uppi \alpha }}{2}} \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 3<\alpha <4$$
(11)

Substituting Eq. (10) into Eq. (9), the expression for \(\sigma _{{I,\left( {\text{A}\text{T}} \right)}}^{2}\) as follows:

$$\sigma _{{I\left( {\text{A}\text{T}} \right)}}^{2}=\frac{{\sigma _{R}^{2}}}{{\sin \left( {\alpha \uppi /4} \right)}}\operatorname{Re} \left[ {{i^{\frac{\alpha }{2} - 1}}{}_{2}{F_1}\left( {1 - \frac{\alpha }{2},\frac{\alpha }{2};1+\frac{\alpha }{2};{{\bar {\Theta }}_{\left( {\text{A}\text{T}} \right)}}+i{\Lambda _{\left( {\text{A}\text{T}} \right)}}} \right) - \frac{{\alpha \Lambda _{{\left( {\text{A}\text{T}} \right)}}^{{\frac{\alpha }{2} - 1}}}}{{2\left( {\alpha - 1} \right)}}} \right]$$
(12)

where 2F1 is the hypergeometric function, \(\sigma _{R}^{2}\) is Rytov variance, which is expressed as34

$$\begin{gathered} \sigma _{R}^{2}\left[ {\alpha ,{\xi _{x\left( {\text{A}\text{T}} \right)}},{\xi _{y\left( {\text{A}\text{T}} \right)}}} \right]= - 2\frac{{\Gamma \left( {\alpha - 1} \right)}}{\alpha }\Gamma \left( {1 - \frac{\alpha }{2}} \right)\sin \left( {\frac{{\uppi \alpha }}{4}} \right)\cos \left( {\frac{{\uppi \alpha }}{2}} \right)C_{n}^{2}{k^{3 - \frac{\alpha }{2}}}L_{{\text{A}\text{T}}}^{{\frac{\alpha }{2}}} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \frac{1}{{2\uppi }}{\int_{0}^{{2\uppi }} {\left( {\frac{{{{\cos }^2}\theta }}{{\xi _{{x\left( {\text{A}\text{T}} \right)}}^{2}}}+\frac{{{{\sin }^2}\theta }}{{\xi _{{y\left( {\text{A}\text{T}} \right)}}^{2}}}} \right)} ^{\frac{\alpha }{2} - 1}}{\text{d}}\theta \hfill \\ \end{gathered}$$
(13)

The Rytov variance fundamentally corresponds to the SI of a plane wave propagating in weak fluctuating turbulence. To find the equivalent factor, we need to equate Eq. (8) with Eq. (12), that is \(\sigma _{{I\left( {\text{C}\text{T}} \right)}}^{2}=\sigma _{{I\left( {\text{A}\text{T}} \right)}}^{2}\). Therefore, according to Eqs. (12) and (13), \(\sigma _{{I\left( {\text{A}\text{T}} \right)}}^{2}\) can be decomposed into the product of \(C_{n}^{2}\) and an equivalent factor D(\(\sigma _{{I\left( {\text{A}\text{T}} \right)}}^{2}=C_{n}^{2}D\)), yielding:

$$C_{n}^{2}=\frac{{\sigma _{{I\left( {\text{C}\text{T}} \right)}}^{2}\left( {{C_1},{C_2},{L_{\text{C}\text{T}}}} \right)}}{{D\left( {{L_{\text{A}\text{T}}}} \right)}}$$
(14)
Table 1 Comparison between the proposed method and the original method.
Full size table

The equivalent method uses parameters of compressible turbulence to characterize the effect on SI analogous to atmospheric turbulence parameters through an equivalent factor. Additionally, compared to the original equivalent method, a new dimension of propagation distance has been introduced. This is because beams propagate shorter distances in compressible turbulence and longer distances in atmospheric turbulence. The comparison between the proposed method and the original method is shown in Table 1. Therefore, improving the equivalent method is necessary to adapt to turbulent environments with different transmission distances. However, this improved method also has limitations. It cannot equivalently account for SI variations beyond the beam center, primarily due to the different diffraction effects experienced by beam at various transmission distances.

Results

The parameters to analyze the characteristics of wave propagation are given as \(\lambda =1.06{\kern 1pt} {\kern 1pt} \upmu \text{m}\), the characteristic scale l is related to the external scale, l = L0/2π, we let l = 0.4 m. The thickness of compressible turbulence near the aircraft is generally less than 1.0 m, so we choose the optical wave transmission distance L(CT) within 1.0 m. Parameter \(\Theta\) in Gaussian beam satisfies \(0<\Theta <1\), so \(\bar {\Theta }\) satisfies \(0<\bar {\Theta }<1\), we make\(\bar {\Theta }\) = 0.5. \(R{\text{=}}8.31\text{J} \cdot {\left( {\text{m}\text{o}\text{l} \cdot \text{K}} \right)^{-1}}\) is the thermodynamic constant.\({B_p}\), \(\varepsilon\), \(\beta\), \(\chi\) are all from Refs35,36. , \({B_p} \approx 8\),\(\varepsilon {\text{=0}}{\text{0.5}}{\kern 1pt} {\kern 1pt} {{\text{m}}^2} \cdot {{\text{s}}^{{\text{-}}3}}\) ,\(\beta {\text{=}}0.8\), and\(\chi \approx 1{\kern 1pt} {\kern 1pt} {\text{K}^2} \cdot {{\text{s}}^{ - 1}}\). y is the distance of the turbulence from the wall, we do not consider the influence of the turbulent dissipation region, so y is approximately the transmission distance L(CT), \(y \approx {L_{\left( {\text{C}\text{T}} \right)}}\).

Fig. 1
figure 1

Variation of atmospheric turbulence \(C_{n}^{2}\) with L(AT) while keeping the optical parameters of anisotropic compressible turbulence unchanged.

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Figure 1 shows the effect of L(AT) (atmospheric transmission distance) on the atmospheric refractive index structure constant within the equivalent factor. It means that under the condition of a fixed transmission distance and turbulence parameters in anisotropic compressible turbulence, any point on the curve in Fig. 1 can be used to equate the optical parameters of compressible turbulence (LCT = 1.0 m, C2 = 10–10 m-2/3) with the optical parameters of atmospheric turbulence. From Fig. 1, it can be seen that increasing the anisotropy of the turbulence can effectively reduce the required \(C_{n}^{2}\) and the transmission distance in the equivalent factor. This indirectly indicates that increasing the turbulence anisotropy can effectively reduce the scintillation effect caused by turbulence on the beam. Moreover, with the required equivalent compressible turbulence parameters remaining constant, the atmospheric parameter \(C_{n}^{2}\) inversely changes with L(AT), meaning \(C_{n}^{2}\) and L(AT) can be interconverted.

Fig. 2
figure 2

Variation of atmospheric turbulence \(C_{n}^{2}\) with L(CT) while keeping L(AT) and C2.

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Figure 2 shows the change in \(C_{n}^{2}\) caused by variations in L(CT) for a fixed atmospheric transmission distance. Compared to the original equivalent method, the improved method allows the transmission distance in the equivalent turbulence parameters to vary, meaning the transmission distances in the two types of turbulence can be different. Therefore, under the condition of a constant equivalent factor, increasing the transmission distance L(CT) of the equivalent turbulence parameters will increase the atmospheric refractive index structure constant \(C_{n}^{2}\). L(CT) and \(C_{n}^{2}\) are positively correlated, which physically means that increasing the transmission distance of the beam in compressible turbulence will enhance the scintillation effect of the turbulence on the beam, thereby increasing the equivalent parameters and the scintillation index of the beam transmitted in atmospheric turbulence. Moreover, increasing the anisotropy factor of the turbulence can still effectively reduce the required \(C_{n}^{2}\) in atmospheric turbulence.

Fig. 3
figure 3

Variation of atmospheric turbulence \(C_{n}^{2}\) with C2 while keeping transmission distance in turbulence.

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Figure 3 shows the variation of the equivalent atmospheric turbulence parameter \(C_{n}^{2}\) with the compressible turbulence parameter C2 for a fixed transmission distance in both types of turbulence. When the transmission distances in the two types of turbulence are not equal, the improved equivalent method still retains the functionality of the original method. That is, with the equivalent factor remaining constant, the turbulence parameters of the two types of turbulence intensities (C2 and \(C_{n}^{2}\)) can still be mutually measured. It means that the impact caused by changes in turbulence intensity C2 during beam transmission in compressible turbulence can be mapped to changes in \(C_{n}^{2}\) during beam transmission in atmospheric turbulence.

From Figs. 1, 2 and 3, it can be observed that increasing the anisotropic factors of the turbulence can effectively reduce the scintillation effect, leading to a decrease in the equivalent parameters. We attribute the emergence of particular phenomenon to two reasons: (a) When considering the turbulent vortex as a lens, an increase in the anisotropic factor leads to an increase in the lens’s radius of curvature. This reduces the lens’s impact on the beam’s transmission path. (b) The SI in anisotropic compressible turbulence is related to the vortex structure. Increasing the anisotropic factor enlarges the vortex structure, creating regions of low air density within the vortex. Low density results in a lower refractive index. Numerous vortices with higher anisotropic factors tend to produce continuous, low-density turbulent regions along the propagation path. These continuous low-density turbulence regions improve the SI performance of the beam.

Fig. 4
figure 4

SI Variation with L(AT) and \(C_{n}^{2}\), (a) 3D color graph of SI Variation with L(AT) and \(C_{n}^{2}\), (b) Distribution of L(AT) -\(C_{n}^{2}\) at the cross-section SI = 0.4.

Full size image
Fig. 5
figure 5

SI Variation with L(CT) and C2, (a) 3D color graph of SI Variation with L(CT) and C2, (b) Distribution of L(CT) - C2 at the cross-section SI = 0.4.

Full size image

To further illustrate the applicability of using atmospheric parameters to equivalently represent compressible turbulence parameters, we can discuss the results of SI under different turbulence parameters and then use the corresponding turbulence environment with the same SI as the equivalent parameters. Figure 4 shows the variation of SI with L(AT) and \(C_{n}^{2}\), while Fig. 5 shows the variation of SI with L(CT) and C2. We use a constant SI (SI = 0.4) to illustrate the situation. The intersection of the plane \(\sigma _{{I,\left( {\text{C}\text{T}} \right)}}^{2}=\sigma _{{I,\left( {\text{A}\text{T}} \right)}}^{2}=\)0.4 [Fig. 4(a) and Fig. 5(a)]is shown in Fig. 4(b) and Fig. 5(b). The black line in Fig. 4(a) represents the intersection of the atmospheric turbulence parameters L(AT) -\(C_{n}^{2}\) on the plane \(\sigma _{{I,\left( {\text{A}\text{T}} \right)}}^{2}\)= 0.4, and the red line in Fig. 5(a) represents the intersection of the compressible turbulence parameters L(CT) - C2 on plane \(\sigma _{{I,\left( {\text{C}\text{T}} \right)}}^{2}\)= 0.4. Thus, any point on the red line can be equivalently represented by any point on the black line, with the corresponding SI being constant. It means that any combination of L(CT) and C2 on the red line can be equivalently represented by any combination of L(AT) and \(C_{n}^{2}\) on the black line. This equivalence is not unique, a combination of compressible optical turbulence parameters can be equivalently represented by numerous combinations of atmospheric turbulence parameters, and vice versa. The mutual equivalence relationship means that for a measured scintillation index in compressible turbulence, one can directly use the intersection plane SI = constant to obtain the required equivalent atmospheric turbulence parameters. Therefore, the improved equivalent turbulence parameter method not only incorporates the dimension of transmission distance but also significantly enhances the applicability of the equivalent method.

Conclusion

In this paper, we developed an improved equivalent turbulence method to represent anisotropic atmospheric turbulence structure parameters in anisotropic compressible turbulence. The method equates the optical parameters in compressible turbulence to those in atmospheric turbulence using the SI as an indicator. The improved equivalent method liberates beam from the constraint of equal transmission distances between two types of turbulence, adapting to the characteristics of compressible turbulence with a small range and significant turbulence disturbance. By extending the dimension of transmission distance, the equivalent method simplifies complex derivations and calculations needed to obtain beam transmission results in anisotropic compressible turbulence, aiding in finding solutions for light in compressible turbulence. The results indicate that increasing the distance over which a beam travels through the atmosphere turbulence increases the equivalent structure constant, thereby reducing the equivalent \(C_{n}^{2}\). Increasing the distance and turbulence intensity of beam propagation in compressible turbulence significantly enhances the equivalent \(C_{n}^{2}\). Increasing the anisotropy of turbulence effectively reduces the equivalent \(C_{n}^{2}\) required. In summary, we use the optical solution of atmospheric turbulence to equate the optical solution of compressible turbulence, and obtains the transmission results of beam in compressible turbulence in a simpler way, which helps to clear the transmission behavior of beam in compressible turbulence.