The optical phase $\phi$ is a key quantity in the physics of light propagating through a turbulent medium. In certain respects, however, the statistics of the phase factor, $\psi = \exp (i\phi)$, are more relevant than the statistics of the phase itself. Here, we present a theoretical analysis of the 2D phase-factor spectrum ${F_\psi}({\boldsymbol \kappa})$ of a random phase screen. We apply the theory to four types of phase screens, each characterized by a power-law phase structure function, ${D_\phi}(r) = (r/{r_c}{)^\gamma}$ (where ${r_c}$ is the phase coherence length defined by ${D_\phi}({r_c}) = 1\;{{\rm rad}^2}$), and a probability density function ${p_\alpha}(\alpha)$ of the phase increments for a given spatial lag. We analyze phase screens with turbulent ($\gamma = 5/3$) and quadratic ($\gamma = 2$) phase structure functions and with normally distributed (i.e., Gaussian) versus Laplacian phase increments. We find that there is a pronounced bump in each of the four phase-factor spectra ${F_\psi}(\kappa)$. The precise location and shape of the bump are different for the four phase-screen types, but in each case it occurs at $\kappa \sim 1/{r_c}$. The bump is unrelated to the well-known “Hill bump” and is not caused by diffraction effects. It is solely a characteristic of the refractive-index statistics represented by the respective phase screen. We show that the second-order $\psi$ statistics (covariance function, structure function, and spectrum) characterize a random phase screen more completely than the second-order $\phi$ counterparts.

中文翻译：

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湍流相位屏的相位因子谱

光学相位$\phi$是光通过湍流介质传播的物理学中的一个关键量。在某些方面，但是，相统计因素，$ \ PSI = \ EXP（I \ PHI）$，比相本身的统计数据更相关。在这里，我们对随机相位屏幕的二维相位因子谱${F_\psi}({\boldsymbol \kappa})$进行了理论分析。我们将该理论应用于四种类型的相位屏幕，每种类型都以幂律相位结构函数为特征，${D_\phi}(r) = (r/{r_c}{)^\gamma}$（其中${r_c }$是由${D_\phi}({r_c}) = 1\;{{\rm rad}^2}$ ) 和概率密度函数定义的相位相干长度给定空间滞后的相位增量${p_\alpha}(\alpha)$。我们使用湍流（$\gamma = 5/3$）和二次（$\gamma = 2$）相位结构函数以及正态分布（即高斯）与拉普拉斯相位增量来分析相位屏幕。我们发现四个相位因子谱${F_\psi}(\kappa)$中的每一个都有明显的凸起。四种相位屏蔽类型的凸块的精确位置和形状不同，但在每种情况下它都发生在$\kappa \sim 1/{r_c}$. 该凸块与众所周知的“Hill 凸块”无关，也不是由衍射效应引起的。它仅是由相应相位屏幕表示的折射率统计的特征。我们表明，二阶$\psi$统计量（协方差函数、结构函数和频谱）比二阶$\phi$对应物更完整地表征随机相位屏幕。