## Abstract

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.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The modern physics of optical wave propagation through turbulent media, pioneered by Tatarskii and coworkers in the 1950s and 1960s [1,2], relies on Maxwell’s electromagnetic field theory, the Obukhov–Corrsin theory [3,4] of fully developed scalar turbulence, and the mathematics of stochastic processes and random fields [5]. The starting point of Tatarskii’s propagation theory is the scalar Helmholtz equation. By means of the geometrical-optics method and Rytov’s “method of smooth perturbations,” Tatarskii derived analytical and empirically testable predictions for a wide range of optical observables, such as the spectra and variances of irradiance fluctuations and angle-of-arrival fluctuations. It was clear that the geometrical-optics method cannot account for diffraction effects, and, initially, it was hoped that the Rytov method described the interplay between refraction and diffraction accurately for most cases of practical relevance. After the discovery of the “saturation of scintillation” [6,7], however, the majority of researchers in the field have become aware that the Rytov method is reliable only if the irradiance fluctuations are small compared to the mean irradiance. This regime is usually referred to as the “weak-scattering regime” or “weak-fluctuation regime.” The theoretical and observational progress in the field since the 1970s has been documented in various standard texts and collections (e.g., [8–18]).

An alternative approach to Tatarskii’s methods is to represent turbulent layers as “phase screens” and to solve the problem of optical propagation through random phase screens by means of the Huygens–Fresnel principle. This method was pioneered by Booker, Ratcliffe, and other radio scientists in the late 1940s and 1950s [19,20]. In the 1960s, Goodman [21] showed that the Huygens–Fresnel principle can be approximated as a 2D input–output relationship for an optical wave passing through a linear, shift-invariant optical system, such that the optical output field is the 2D spatial convolution of the optical input field and the optical system’s 2D spatial impulse response function. Goodman’s optical systems approach is the 2D equivalent to the standard theory of linear, time-invariant systems, or “LTI systems” (e.g., [22,23]). Goodman applied this concept first to deterministic optical systems [21,24] and later to random optical systems, such as a slab of turbulent air through which light propagates [25,26]. In 1985, Booker, Ferguson, and Vats [27] analyzed, for the weak-scattering regime, the scintillation index $\sigma _I^2$ of a plane wave propagating through classical, fully developed turbulence. They showed that the values of $\sigma _I^2$ predicted by the Huygens–Fresnel principle and by the Rytov theory are equal to each other if the phase screen representing the distributed 3D turbulence is placed half-way between the source (input) plane and the observation (output) plane.

In recent decades, the system approach pioneered by Goodman has been widely used in imaging physics [28,29] and computational optics [30–32].

All these methods to analyze and predict optical wave propagation through
turbulent media have in common is that it is usually not the phase $\phi$ that matters but the *phase factor* $\psi$, where $\psi$ is the complex exponential of the phase, $\psi = \exp
(i\phi)$. Nevertheless, most theoretical and
observational research has been done on the basis of phase statistics
rather than *phase-factor* statistics.

The purpose of this paper is to present and discuss phase-factor statistics for 2D random phase screens. The three main statistics are the phase-factor covariance function, ${B_\psi}(\textbf{r})$, the phase-factor structure function, ${D_\psi}(\textbf{r})$, and, most importantly, the 2D phase-factor spectrum, ${F_\psi}({\boldsymbol \kappa})$. We will show that, in contrast to the phase counterparts ${B_\phi}(\textbf{r})$, ${D_\phi}(\textbf{r})$, and ${F_\phi}({\boldsymbol \kappa})$, the three phase-factor statistics ${B_\psi}(\textbf{r})$, ${D_\psi}(\textbf{r})$, and ${F_\psi}({\boldsymbol \kappa})$ are “well-behaved” at all spatial lags $\textbf{r}$ and all wave vectors ${\boldsymbol \kappa}$.

The paper is organized as follows. Section 2 develops the three main statistics of the phase factor of a random phase screen: the covariance function, the structure function, and the spatial spectrum. In Section 3, we analyze four specific phase screens, each of which is characterized by a power-law phase structure function and by a probability density function of the phase increments. We consider phase structure functions with power-law exponents $\gamma = 5/3$ or $\gamma = 2$ and phase increments with either Gaussian or Laplacian probability densities. The results are discussed in Section 4. Summary and conclusions are given in Section 5.

## 2. GENERAL THEORY

#### A. Applying the Huygens–Fresnel Principle to a Random Phase Screen

Goodman ([21], p. 60) applied the Huygens–Fresnel principle to optical propagation through a phase screen and showed that the scalar, complex, optical field $U(\textbf{x})$ in a certain observation plane (the “output field”) can be approximated (assuming the Fresnel approximation is valid) as the 2D convolution of an input field ${U_0}(\textbf{x})$ and a 2D spatial impulse response function $h(\textbf{x})$:

An important special case is an initially unperturbed plane wave propagating through a thin turbulent layer and thereby undergoing phase distortions $\phi (\textbf{x})$, such that the input field (the optical field immediately after propagating through the turbulent layer) is

Here, $A$ is the complex optical field of the unperturbed wave, and is the phase factor, where $\phi (\textbf{x})$ is the random phase screen representing the phase distortions due to the turbulent layer. Therefore, $U(\textbf{x})$ is the convolution of the phase factor and of the impulse response function:If $U(\textbf{x})$ and $\psi (\textbf{x})$ are statistically homogeneous (that is, if all statistics of $U$ and of $\psi$ are independent of $\textbf{x}$), then we can write $U(\textbf{x})$ and $\psi (\textbf{x})$ as stochastic Fourier–Stieltjes integrals,

It can be easily shown that

where $H({\boldsymbol \kappa})$ is the 2D transfer function, that is, the 2D Fourier transform of the 2D impulse response function $h(\textbf{x})$:*phase-factor*spectrum ${F_\psi}({\boldsymbol \kappa})$ and not the phase spectrum ${F_\phi}({\boldsymbol \kappa})$ that determines the spectral characteristics of $U$ in the general case. In the following, we express ${F_\psi}({\boldsymbol \kappa})$ in terms of the phase structure function ${D_\phi}(\textbf{r})$ and of the probability density function ${p_\alpha}(\alpha)$ of the spatial phase increments $\alpha$.

Before we do so, however, we briefly consider the simplest case of an optical system, namely, a slab of non-turbulent air. Let $z$ be the distance between the input plane (which we assume to coincide with the phase-screen plane) and the observation (output) plane. Then, according to the Fresnel approximation,

#### B. Phase-Factor Covariance Function

Consider the covariance function of $\psi (\textbf{x})$:

Now, let ${p_\alpha}(\alpha)$ be the probability density of $\alpha (\textbf{r})$ for a given spatial lag $\textbf{r}$. Then,

We see that ${B_\psi}(\textbf{r})$ is affected by higher-order statistical moments of the phase increments $\alpha$, while ${D_\phi}(\textbf{r})$ is a purely second-order statistic (the variance) of $\alpha$. This has profound implications for the usefulness of ${D_\phi}(\textbf{r})$ for the analysis and prediction of higher-order statistics of the complex optical field $U$, such as the scintillation index or other statistics of irradiance fluctuations.#### C. Phase Screens with Gaussian versus Laplacian Phase Increments

It is a standard assumption (e.g., [1,34]) that the turbulent phase increment for a fixed $\textbf{r}$ is a zero-mean Gaussian random variable, such that

While the assumption of Gaussian phase increments is widely accepted, its physical justification relies on the central limit theorem, which is not universally applicable. Therefore, the possibility of phase screens with non-Gaussian phase increments should be taken into account; see, e.g., [37]. Here, we consider, as a counter-example, the two-sided exponential, or Laplacian, probability density function:

#### D. Phase-Factor Structure Function

The second-order structure function of the phase factor is defined as

If the separation $|\textbf{r}|$ is sufficiently small, the first term ($n = 1$) in (35) is the leading term, which gives

That is, for small $|\textbf{r}|$, the second-order phase-factor structure function is approximately equal to the second-order phase structure function. This is to be expected because, for sufficiently small $|\textbf{r}|$, the phase increments are small compared to 1, such that $\psi = \exp i\phi \approx 1 + i\phi$, and we obtain ${D_\psi}(\textbf{r}) \approx {D_\phi}(\textbf{r})$ directly. In general, however, all even moments of the phase increments, that is, all even-order phase structure functions $D_\phi ^{(2n)}(\textbf{r})$, contribute to the second-order phase-factor structure function ${D_\psi}(\textbf{r})$.An important, very general relationship between structure function and covariance function of a random function is

(e.g., [1], p. 10, Eq. 1-14).Equation (38) leads to specific expressions for ${D_\psi}(\textbf{r})$ in terms of ${D_\phi}(\textbf{r})$, depending on the respective probability density function of the phase increments. For Gaussian phase increments, inserting (26) into (38) gives

For Laplacian phase increments, inserting (29) into (38) gives For both Gaussian and Laplacian phase increments, we obtain ${D_\psi}(\textbf{r}) \approx {D_\phi}(\textbf{r})$ for very small $|\textbf{r}|$ and ${D_\psi}(\textbf{r}) \approx 2$ for very large $|\textbf{r}|$. For intermediate $|\textbf{r}|$, however, ${D_\psi}(\textbf{r})$ depends on the probability density of the phase increments.#### E. Phase-Factor Spectrum

According to the Wiener–Khintchine theorem, the 2D phase-factor spectrum ${F_\psi}({\boldsymbol \kappa})$ is the Fourier transform of ${(2\pi)^{- 2}}{B_\psi}(\textbf{r})$:

Now, we know that the variance of $\psi$ is

On the other hand, according to Parseval’s theorem, $\langle |\psi {|^2}\rangle$ is equal to the integral of ${F_\psi}({\kappa _x},{\kappa _y})$ over the entire ${\kappa _x}$-${\kappa _y}$ plane. Therefore, In the case of isotropy in the $x$-$y$ space, changing to polar coordinates, ${\rm d}{\boldsymbol \kappa} = {\rm d}{\kappa _x}{\rm d}{\kappa _y} = 2\pi \kappa {\rm d}\kappa$, gives#### F. Dimensionless Phase-Factor Spectrum

It is useful to present ${F_\psi}(\kappa)$ in a dimensionless form. First, we normalize the spatial lag $r$ with the phase coherence length ${r_c}$, which we define by the relationship

That is, we define ${r_c}$ as the spatial lag for which the standard deviation of the phase increments is equal to 1 rad. Equation (49) defines ${r_c}$ unambiguously if ${D_\phi}({r_c})$ increases strictly with $r$, which is the case, for example, for power-law functions with exponents $\gamma$ larger than zero.Introducing the dimensionless spatial lag

and substituting $x$ for $r$ in (45) gives where is the dimensionless phase-factor spectrum, and where we have introduced the dimensionless wavenumber Substituting $x^\prime = Kx$ and renaming the new integration variable $x^\prime $ back to $x$ leads to an alternative equation for ${f_\psi}(K)$:## 3. ANALYSIS OF FOUR KINDS OF RANDOM PHASE SCREENS

We have shown that, in general, the phase-factor spectrum ${F_\psi}({\boldsymbol \kappa})$ of a random phase screen depends on both ${D_\phi}(\textbf{r})$ and ${p_\alpha}(\alpha)$. In the following, we will analyze four specific random phase screens, each of which is characterized by a power-law phase structure function (with exponents $\gamma = 5/3$ or $\gamma = 2$) and a phase-increment distribution (either Gaussian or Laplacian).

#### A. Power-Law Phase Structure Functions

A power-law phase structure function has the form

where the exponent $\gamma$ is a dimensionless constant. Note that (55) is consistent with (49), the defining equation for ${r_c}$ for any strictly increasing ${D_\phi}(r)$.Turbulent phase screens are characterized by $\gamma = 5/3$, as shown by Tatarskii [1] on the grounds of the Obukhov–Corrsin theory [3,4] of fully developed, scalar turbulence. For mathematical expediency, turbulent phase structure functions are sometimes (e.g., [34]) approximated by quadratic phase structure functions ($\gamma = 2$). It has to be kept in mind, however, that a quadratic ${D_\phi}(r)$ represents plane, randomly tilted wavefronts, while wavefronts characterizing optical turbulence are not only randomly tilted but also randomly curved. Therefore, certain aspects of optical turbulence cannot be captured by means of quadratic phase structure functions; see Charnotskii’s ([39], pp. 714ff.) critical discussion.

#### B. Phase-Factor Covariance Functions

Figure 1 shows the covariance functions ${B_\psi}(r/{r_c})$ of the phase factor $\psi$ for four different phase-screen models: phase screens with an ${r^{5/3}}$ and ${r^2}$ power-law phase structure function, each with either Gaussian- or Laplacian-distributed phase increments. The explicit functional forms of these four ${B_\psi}(r/{r_c})$ models are

All four ${B_\psi}(r/{r_c})$ have in common that ${B_\psi}(r/{r_c}) \to 1$ for $r/{r_c} \to 0$ and ${B_\psi}(r/{r_c}) \to 0$ for $r/{r_c} \to \infty$. The two ${B_\psi}(r/{r_c})$ for Gaussian phase increments drop more rapidly to zero than the two ${B_\psi}(r/{r_c})$ for Laplacian phase increments.

The nature of the decrease of ${B_\psi}(r/{r_c})$ with increasing $r/{r_c}$ can be more clearly seen in Fig. 2, where the four ${B_\psi}(r/{r_c})$ are shown in double-logarithmic scaling. The two ${B_\psi}(r/{r_c})$ with Laplacian increments show power-law asymptotes with exponents ${-}5/3$ and ${-}2$, respectively, consistent with (57). The two ${B_\psi}(r/{r_c})$ with Gaussian increments, however, decrease exponentially, consistent with (56). That is, whether ${B_\psi}(r/{r_c})$ decreases exponentially or follows a power law for $r \gg {r_c}$ is determined by ${p_\alpha}(\alpha)$, not by the value of $\gamma$.

#### C. Phase-Factor Structure Functions

Figure 3 shows the phase-factor structure functions ${D_\psi}(r/{r_c})$ for the four phase-screen models. As expected, ${D_\psi}(r/{r_c}) \approx {D_\phi}(r/{r_c})$ for $r/{r_c} \ll 1$. That is, for small $r$, ${D_\psi}(r/{r_c})$ follows the same ${r^\gamma}$ power law, regardless of whether the phase increments are Gaussian or Laplacian. In all four cases, ${D_\psi}(r/{r_c}) \to 2$ for $r/{r_c} \to \infty$, in contrast to ${D_\phi}(r/{r_c})$, which increases without bound for $r/{r_c} \to \infty$.

In the intermediate range, where $r$ is comparable to ${r_c}$, the four ${D_\psi}(r/{r_c})$ differ significantly from each other.

#### D. Phase-Factor Spectra

In this section, we evaluate numerically the dimensionless phase-factor spectrum ${f_\psi}(K)$ introduced in Section 2.F for the four phase-screen categories that we have specified, and we compare the four spectra with the high-wavenumber asymptote obtained from Tatarskii’s classical theory [1].

Inserting (56) and (57) into (54) gives, respectively,

We have numerically evaluated ${f_\psi}(K)$ given in (58) and (59) for $\gamma = 5/3$ and for $\gamma = 2$ by means of the composite Simpson rule ([40], p. 96, Eq. 3.6.4), and the results are shown in Fig. 4.

Additionally, we show in Fig. 4 (blue line) the high-wavenumber ${K^{- 11/3}}$ asymptote that we have retrieved independently from Tatarskii’s classical theory [1] as follows. Starting with our definition of the dimensionless phase spectrum,

we use Tatarskii’s classical models for the 2D phase spectrum ([1], p. 101, Eq. 6.34; p. 25, Eq. 1.58), and, for the phase structure function ([1], p. 155, Eq. 7.101), where $k = 2\pi /\lambda$ is the optical wavenumber, $L$ is the thickness of the layer assumed to be filled with fully developed, homogeneous and isotropic turbulence, and $C_n^2$ is the refractive-index structure parameter. We equate (49) and (62) and obtain an expression for the phase coherence length, and (60) gives That is, the parameter ${k^2}LC_n^2$ cancels out when we make the transition from ${F_\phi}(\kappa)$ to its dimensionless counterpart, ${f_\phi}(K)$, and $\gamma$ is the only parameter that determines ${f_\phi}(K)$.In the discussion that follows, we distinguish between three wavenumber regimes: a low-wavenumber regime, where $K \ll 1$, a high-wavenumber regime, where $K \gg 1$, and a transition regime, where $K \sim 1$.

### 1. Low-Wavenumber Regime

At low wavenumbers, $K \ll 1$, the spectra for Gaussian phase increments are flat, while the behavior of the spectra for Laplacian phase increments is less clear.

Now, we evaluate $\mathop {\lim}\nolimits_{K \to 0} {f_\psi}(K)$ analytically for Gaussian phase increments. Because ${B_\psi}(x)$ decreases exponentially for $x \gt 1$ and drops below ${10^{- 4}}$ for $x \gt 6$ (see Fig. 2) and because $|{J_0}(Kx)| \le 1$ for all $x$, we know that only $x$ values small compared to 6 can significantly contribute to the integral in (52). This means that we can safely approximate ${J_0}(Kx) = 1$ if $K$ is small compared to 6. Then, (52) gives

Note that this reasoning is valid only for the case of Gaussian phase increments. For Laplacian increments, ${B_\psi}(x) \approx 2{x^{- \gamma}}$ for $x \gg 1$, such that ${B_\psi}(x)x \approx 2{x^{1 - \gamma}}$ for $x \gg 1$, and the integral in (65), which relies on the approximation ${J_0}(Kx) \approx 1$, diverges unless $\gamma \gt 2$. That is, the approximation of (65) is not applicable for Laplacian phase screens with $\gamma = 5/3$ or $\gamma = 2$.For Gaussian phase increments, inserting (56) into (65) gives

### 2. High-Wavenumber Regime

Figure 4 shows clearly that for $K \gg 1$ our numerical ${f_\psi}(K)$ results for $\gamma = 5/3$ converge to the ${K^{- 11/3}}$ power-law phase spectrum retrieved from Tatarskii’s classical theory and given in (64), regardless of whether the phase increments are Gaussian or Laplacian distributed. This is consistent with the expectation that the functional form of ${p_\alpha}(\alpha)$ is irrelevant for ${f_\psi}(K)$ at high wavenumbers and that ${f_\psi}(K) \approx {f_\phi}(K)$ for $K \gg 1$.

For phase screens with quadratic phase structure functions ($\gamma = 2$), the phase-factor spectra drop exponentially at a steeper rate for Gaussian than for Laplacian phase increments.

### 3. Transition Regime

A remarkable feature of the four ${f_\psi}(K)$ graphs shown in Fig. 4 is that all of them exceed the Tatarskii asymptote in the transition regime where $K \sim 1$. In order to examine these spectral “bumps” more closely, in Fig. 5, we show the compensated dimensionless phase-factor spectra,

for turbulent phase screens ($\gamma = 5/3$) with Gaussian and Laplacian phase increments. (We do not show the compensated spectra for the Laplacian phase screens because they do not show ${K^{- 11/3}}$ power-law “tails”.) The constant Tatarskii asymptote, ${g_\psi}(K)$ = 0.45, is shown as well.The bumps in ${f_\psi}(K)$ shown in Fig. 4 stand out as pronounced maxima of ${g_\psi}(K)$ in Fig. 5. For Gaussian phase increments, ${g_\psi}(K)$ reaches a maximum value of 1.26 at $K= 1.8$, and, for Laplacian phase increments, the ${g_\psi}(K)$ maximum occurs at $K = 2.8$, and its value is 0.98. That is, the Gaussian ${f_\psi}(K)$ and the Laplacian ${f_\psi}(K)$ exceed the spectral densities predicted by the Tatarskii asymptote $0.45{K^{- 11/3}}$ by factors up to 2.8 and 2.2, respectively.

## 4. DISCUSSION

#### A. Phase-Factor Statistics versus Phase Statistics

Before we discuss the phase-factor spectrum ${F_\psi}(\kappa)$ in more detail, a few general remarks on phase-factor statistics versus phase statistics are in order.

The analytical treatment of optical turbulence is complicated by the fact that a power-law phase structure function ${D_\phi}(r)$ increases without bound for an increasing spatial lag $r$. Therefore, the phase covariance function ${B_\phi}(r)$ does not exist in this case, and the Wiener–Khintchine relationship, which states that the phase spectrum ${F_\phi}(\kappa)$ is the Fourier transform of ${B_\phi}(r)$, is not directly applicable. While it is possible to calculate ${F_\phi}(\kappa)$ from the structure function, it remains a problem that ${F_\phi}(\kappa) \to \infty$ for $\kappa \to 0$.

In contrast to the phase statistics ${D_\phi}(r)$, ${B_\phi}(r)$, and ${F_\phi}(\kappa)$, the *phase-factor* counterparts ${D_\psi}(r)$, ${B_\psi}(r)$, and ${F_\psi}(\kappa)$ are all “well-behaved” for all values
of $r$ and $\kappa$, respectively. In particular, ${D_\psi}(r)$ and ${B_\psi}(r)$ are finite for $r = 0$ and $r \to \infty$, and ${F_\psi}(\kappa)$ is finite for $\kappa = 0$ and $\kappa \to
\infty$.

#### B. Bump in the Phase-Factor Spectrum

Perhaps the most important result of our analysis is that there is a pronounced bump in the phase-factor spectrum at wavenumbers comparable to the phase-coherence wavenumber

It is to be expected, and we have confirmed numerically, that ${F_\psi}(\kappa)$ for turbulent phase screens ($\gamma = 5/3$) varies like ${\kappa ^{- 11/3}}$ for $\kappa \gg {\kappa _c}$ and approaches a constant spectral density for $\kappa \ll {\kappa _c}$, regardless of whether the phase increments are Gaussian or Laplacian. This means that the compensated dimensionless spectrum, ${g_\psi}(K) = {f_\psi}(K){K^{11/3}}$, increases like ${K^{11/3}}$ for $K \ll 1$ and approaches a constant value for $K \gg 1$. Our numerical results show, for both Gaussian and Laplacian phase increments, that $\mathop {\lim}\nolimits_{K \to \infty} {g_\psi}(K) = 0.45$, the universal constant that we retrieved analytically from Tatarskii’s classical predictions.It is somewhat surprising, however, that ${g_\psi}(K)$ has a pronounced maximum in the transition region, where $K$ is of order 1 (and $\kappa$ is of order $1/{r_c}$).

It is not immediately clear what causes this bump in ${f_\psi}(K)$ and the maximum in ${g_\psi}(K)$. The bump is certainly not related to the well-known “Hill bump” in the compensated refractive-index spectrum ${\Phi _n}(\kappa){\kappa ^{11/3}}$ [41,42] because throughout this paper we have assumed that the phase structure function ${D_\phi}(r)$ follows a power law, which implies that ${\Phi _n}(\kappa)$ also follows a power law.

Diffraction effects can also be excluded as a possible cause of the
bump in ${F_\psi}(\kappa)$, simply because all the phase-screen
statistics, including ${F_\psi}(\kappa)$, were derived from relationships in
the geometrical-optics limit, i.e., $r \gg \sqrt
{L\lambda}$, where $L$ is the thickness of the “slab” of
turbulence that the phase screen represents. Moreover, a phase screen
is *a priori* solely determined by
path-length fluctuations associated with refractive-index
fluctuations. In other words, diffraction effects play no role in the ${f_\psi}(K)$ bumps (Fig. 4), the ${g_\psi}(K)$ maxima (Fig. 5), or in any of the other phase-screen statistics
that we have discussed in this paper.

#### C. Phase Coherence Length and Fried Length

We have shown that the phase coherence length for turbulent ($\gamma = 5/3$) phase screens defined in (55) is of the form

if we apply (55) to Tatarskii’s phase structure function model, (62). Our ${r_c}$ is physically equivalent to the Fried length ${r_0}$, except that Fried ([43], p. 1430, Eq. 6.3) defined his ${r_0}$ as where, in Fried’s words ([43], p. 1430), “the apparently arbitrary constant 6.88 was chosen on the basis of the analysis of the performance of an optical heterodyne detection system.” In (55), we used the coefficient 1 instead of Fried’s coefficient 6.88. By equating Fried’s $6.88(r/{r_0}{)^{5/3}}$ with our ${(r/{r_c})^{5/3}}$, we find that Fried’s length ${r_0}$ is ${6.88^{3/5}} = 3.18$ times larger than our ${r_c}$. In a different paper ([44], p. 1376f.), Fried writes “The significance of the factor 6.88, which is more precisely given by $2\{(24/5)\Gamma {(6/5)\} ^{5/6}}$, is contained in the fact that it makes the knee of curve A … occur at $D = {r_0}$.” Here, “curve A” is a graph of a normalized telescope resolution as function of the normalized aperture diameter $D/{r_0}$.It is important to note, as we have emphasized in the previous subsection, that ${r_c}$ and ${r_0}$ are pure geometrical-optics characteristics, which are unaffected by diffraction effects. This is contrary to Fried’s remark ([44], p. 1376, after Eq. 5.1): “The theory, it should be noted, is based on the Rytov approximation to solve the wave equation …,” which could be misread as suggesting that diffraction effects do play a role in any of the relationships that lead to the equations for ${r_c}$ or ${r_0}$ as stated above.

#### D. Gaussian versus Laplacian Phase Screens

It is well-established that spatial increments of turbulent temperature
fluctuations have “exponential tails” (e.g., [45]) consistent with the Laplace distribution.
However, the central limit theorem is usually invoked (e.g., [34]) to argue that, in spite of the
Laplacian temperature increments, the phase increments tend to be
normally (Gaussian) distributed. This argument relies on the
assumption that a large number of statistically independent (or
quasi-independent) refractive-index fluctuations of comparable
magnitude contribute to the phase fluctuations, and therefore also to
the spatial phase *increments*. This
assumption is questionable if the “slab” of turbulent air represented
by the phase screen has a thickness that is comparable to, or smaller
than, the outer scale of turbulence.

In this paper, we present no empirical evidence that real-world phase screens are non-Gaussian. Rather, we have used the Laplacian probability density as a counter-example and have evaluated the resulting phase-screen statistics for Gausian versus Laplacian phase screens. We have shown that only in the high-wavenumber regime ($\kappa \gg r_c^{- 1}$) are the phase-factor statistics (covariances, structure functions, and spectra) the same for Gaussian versus Laplacian phase increments.

## 5. SUMMARY AND CONCLUSIONS

We have analyzed the covariance function ${B_\psi}(r)$, the structure function ${D_\psi}(r)$, and the 2D spectrum ${F_\psi}(\kappa)$ of the phase factor $\psi = \exp (i\phi)$, where $\phi$ is the phase for four different types of random, 2D phase screens $\phi (\textbf{x})$. Each of these phase screens is characterized by a power-law phase structure function (with exponents $\gamma = 5/3$ or $\gamma = 2$) and a model for the probability density function of the spatial phase increments (either Gaussian or Laplacian). Our main results are as follows:

- 1. The phase-factor statistics ${B_\psi}(r)$, ${D_\psi}(r)$, and ${F_\psi}(\kappa)$ are all “well-behaved.” In particular, in contrast to their phase counterparts ${B_\phi}(r)$, ${D_\phi}(r)$, and ${F_\phi}(\kappa)$, they show no singularities at $r = 0$, $r \to \infty$, $\kappa = 0$, and $\kappa \to \infty$ if the underlying phase structure function ${D_\phi}(r)$ follows the power laws ${r^{5/3}}$ or ${r^2}$.
- 2. We have defined a phase coherence length, ${r_c}$, by requiring ${D_\phi}({r_c}) = 1\;{{\rm rad}^2}$. For $\gamma = 5/3$, our ${r_c}$ is physically equivalent to Fried’s coherence length ${r_0}$, except for the numerical factor ${r_0}/{r_c}= 6.88^{3/5} = 3.18$. We show that both ${r_c}$ and ${r_0}$ are pure phase-screen statistics based on optical path-length (“eikonal”) fluctuations. Both ${r_c}$ and ${r_0}$ are unaffected by any assumptions or approximations regarding the propagation after passing through the phase screen.
- 3. We have formulated the second-order phase-factor structure function ${D_\psi}(r)$ as an infinite sum of second- and higher-order (of even order) phase structure functions, $D_\phi ^{(2n)}(r)$. For $r \ll {r_c}$, we find ${D_\psi}(r) \approx {D_\phi}(r)$. For $r \gg {r_c}$, however, ${D_\psi}(r)$ saturates to the universal value of 2, while ${D_\phi}(r)$ increases without bound.
- 4. For turbulent ($\gamma = 5/3$) phase screens, at high wavenumbers ($\kappa \gg r_c^{- 1}$), the dimensionless phase-factor spectra ${f_\psi}(K)$ (where $K = \kappa {r_c}$ is a dimensionless wavenumber) for Gaussian and Laplacian phase increments converge to the same power law, ${f_\psi}(K) = 0.45 {K^{- 11/3}}$. The only parameter that affects this result is $\gamma = 5/3$.
- 5. Independently, we have retrieved the $K \gg 1$ asymptote of ${f_\psi}(K)$ from Tatarskii’s [1] relationships ${F_\psi}(\kappa) = 2\pi {k^2}L \times 0.033C_n^2{\kappa ^{- 11/3}}$ and ${D_\phi}(r) = 2.91{k^2}LC_n^2{r^{5/3}}$, and we find the same power law and the same coefficient, ${f_\psi}(K) = 0.45 {K^{- 11/3}}$.
- 6. For phase screens with quadratic ($\gamma = 2$) phase structure functions, ${f_\psi}(K)$ drops exponentially for $K \gg 1$, more rapidly for Gaussian than for Laplacian phase increments.
- 7. At low wavenumbers ($\kappa \ll r_c^{- 1},K \ll 1$), all four $\psi$ spectra become flat, which is consistent with the fact that $\psi$ becomes uncorrelated for $r \gg {r_c}$.
- 8. At intermediate wavenumbers ($\kappa \sim r_c^{- 1},K \sim 1$), each of the four ${f_\psi}(K)$ shows a “bump,” in the case of $\gamma = 5/3$ exceeding the high-wavenumber asymptote, ${f_\psi}(K) = 0.45 {K^{- 11/3}}$, by a factor of up to 2.8 for Gaussian phase increments and a factor of up to 2.2 for Laplacian phase increments.

As we have seen, the second-order phase-factor ($\psi$) statistics (covariance function, structure function, and 2D spectrum) exhibit a rich variety of phase-screen characteristics that are not as directly, or not at all, captured by means of second-order phase ($\phi$) statistics characterizing the same phase screen. There are two obvious advantages of characterizing phase screens by means of $\psi$ statistics rather than $\phi$ statistics. First, the (second-order) $\psi$ statistics account for the probability density ${p_\alpha}(\alpha)$ of the phase increments $\alpha$, while the (second-order) $\phi$ statistics are invariant with respect to the functional form of ${p_\alpha}(\alpha)$; second, the second-order $\psi$ statistics account explicitly for effects resulting from a finite phase coherence length ${r_c}$, while the second-order $\phi$ statistics are simply the second-order path-length (eikonal) statistics re-scaled with the constant factor ${k^2}$.

## Funding

Air Force Office of Scientific Research (FA9550-17-C-0021, FA9550-18-1-0009, FA9550-18-C-0011); National Science Foundation (AGS-1547476).

## Acknowledgment

The author thanks the two anonymous reviewers for their constructive comments and David Voelz for fruitful discussions.

## Disclosures

The author declares that there are no conflicts of interest related to this paper.

## Data Availability

No data were generated or analyzed in the presented research.

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