# A Survey of Structure of Atmospheric Turbulence in Atmosphere and Related Turbulent Effects
## 3 Theoretical Studies of Turbulent Effect
### Theoretical Studies of Kolmogorov Turbulent Effect  
The classical theory for propagation through random media derives from the solution of stochastic Helmholtz equations, which is a governing partial differential equation for the scalar field. The Born and Rytov perturbation methods for solving the stochastic Helmholtz equations are introduced first. Both approximation methods are limited to weak fluctuations condition. The parabolic equation method is one of the most general methods that is theoretically applicable under all atmospheric conditions [1]. The extended Huygens-Fresnel principle was developed by [PERSON] and [PERSON] [28] in the United States and [PERSON] and [PERSON] in the former Soviet Union [29], and the corresponding optical field has the following form:  
\[U(\vec{r},L)=-\frac{ik}{2\pi L}\exp(ikL)\int\int_{-\infty}^{\infty}d^{2}sU_{0} (\vec{s},0)\exp\left[\frac{ik|\vec{s}-\vec{r}|^{2}}{2L}+\Psi(\vec{r},\vec{s}) \right], \tag{6}\]  
where \(\vec{r}\) and \(\vec{s}\) are the distance vectors transverse to the propagation axis, \(k\) represents the optical wave number, \(U_{0}(\vec{s},0)\) denotes the unperturbed field in the absence of turbulence at the point \((\vec{s},0)\), and \(\Psi(\vec{r},\vec{s})\) is the random part of the complex phase of a spherical wave propagating in the turbulent medium from point \((\vec{s},0)\) to point \((\vec{r},L)\). The extended Huygens-Fresnel principle is applicable for first-order and second-order field moments under weak or strong fluctuation conditions of atmospheric turbulence. Another approximation method was introduced based on a modification of the Rytov theory that permits extending many weak-fluctuation results into the moderate-to-strong fluctuations regimes. Moreover, the corresponding optical field has the following form:  
\[U(\vec{r},L)=U_{0}(\vec{r},L)\exp[\Psi_{X}(\vec{r},L)]\exp[\Psi_{Y}(\vec{r},L)], \tag{7}\]  
where \(\Psi_{X}(\vec{r},L)\) and \(\Psi_{Y}(\vec{r},L)\) are statistically independent complex phase perturbations due only to the large-scale and small-scale effects of the effective refractive index, respectively, on propagating wave. By using the Rytov method, the first and second moments of the optical field were examined from which several important statistical quantities were deduced, mainly mutual coherence function, spatial coherence radius for plane and spherical wave and a Gaussian-beam wave, angel-of-arrival fluctuations and image jitter for a plane wave, beam wander of a Gaussian-beam wave, and angular and temporal frequency spectra for a plane wave [1]. Based on the parabolic equation method and the extended Huygens-Fresnel principle, second-order statistics were studied in strong fluctuation conditions. Using the parabolic equation method, the mean irradiance and mutual coherence function for a Gaussian-beam wave were deduced in the strong fluctuation regime. Based on the extended Huygens-Fresnel principle, the second-order moments of the complex phase perturbation were investigated, and mean irradiance, beam spread, and mutual coherence function for a Gaussian-beam wave were deduced in the strong fluctuation regime. In addition, applying the method of effective beam parameters, the spatial coherence radius and beam wander for a Gaussian-beam wave were deduced in the strong fluctuations condition [1]. The various fourth-order statistical quantities such as the scintillation index and the irradiance covariance function were also examined. By using the Rytov method, the scintillation index, covariance function of irradiance, temporal spectrum of irradiance for a plane wave, a spherical wave, and a Gaussian-beam wave were deduced, and beam wander for a Gaussian-beam wave were developed in weak irradiance fluctuation regimes. Furthermore, phase variance, phase structure functions, covariance function, and temporal power spectrum for a plane wave and a Gaussian-beam wave were also developed in weak fluctuations regimes [1].