## Abstract

Statistical characteristics of radiation intensity in the cross section of laser beams propagating in the turbulent atmosphere are analyzed by using numerical simulations. It is shown that for arbitrary values of the propagation distance, Rytov parameter, beam type, and position of the analyzed point in the beam cross section, the probability density function (PDF) of radiation intensity is fully determined by the scintillation index and the average intensity. In the case of moderate and weak intensity fluctuations characterized by the scintillation index smaller than unity, the probability density function is determined by the gamma distribution. For the case of strong fluctuations (with the scintillation index larger than unity), a new analytical expression for PDF is proposed, which well approximates PDFs obtained in numerical experiments under different conditions of propagation of different-type beams.

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

## 1. Introduction

To predict the influence of turbulence on operation of optical systems of different application in the atmosphere, it is important to know statistical properties of intensity fluctuations of beams used in these laser systems. The probability density function (PDF) most thoroughly characterizes the statistics of these fluctuations. A lot of theoretical models of PDF [1–8] such as, for example, the lognormal distribution, exponential distribution, K-distribution, lognormal distribution modulated by the exponential one, lognormal distribution modulated by the Rician one (also known as the Beckmann distribution), and the gamma-gamma distribution, is proposed now. The lognormal distribution and the gamma-gamma distribution are used most widely according to [4]. These distributions were tested with the results of numerical simulation of propagation of the Gaussian laser beam through statistically homogeneous and isotropic turbulence. The mentioned models were repeatedly checked in experiments [3,5,7–9,10].

Despite the large number of theoretical models of PDF, they fail for many propagation conditions occurring in the atmosphere. Many PDF models are not versatile and can be used only for certain turbulence intensity. Thus, for example, the lognormal distribution far understates the frequency of occurrence of giant intensity spikes observed in the case of strong fluctuations [10]. Some types of PDFs such as, for example, the gamma-gamma distribution, include the parameters not related directly to values of measured statistical characteristics of intensity fluctuations. This casts some doubts on the possibility of using these models for prognostic purposes.

It should also be noted that the statistical analysis of laser beam propagation in the turbulent atmosphere is usually restricted to determination of the distribution laws of intensity fluctuations at the axis of the Gaussian beam or the distribution laws of fluctuations of optical flux power received by a finite-size aperture. As far as we know, the probability distribution laws of intensity fluctuations over the beam cross section remain unstudied yet. These studies become urgent in connection with the promise of using the so-called exotic beams [11], which differ markedly from the Gaussian beams. Vortex beams having orbital angular momentum are the most common examples of such beams [12].

In this paper, we use numerical simulation to compare the probability distribution functions of intensity fluctuations of the vortex Laguerre—Gaussian beam and the fundamental Gaussian beam [13]. This study was initiated by the results of recent paper [14], which considered intensity fluctuations in the cross section of the Laguerre—Gaussian beam. It was shown that at the end of an atmospheric path, for which the Rytov parameter is equal to 0.1 corresponding to the case of weak turbulence, the scintillation index at the axis of the beam and at the periphery takes values exceeding unity. At the same time, at the ring (where the average intensity is maximal), the scintillation index is approximately equal to 0.1. That is, in the cross section of the Laguerre—Gaussian beam, fluctuations corresponding to the cases of weak, moderate, and strong turbulence occurred simultaneously [4].

To construct the versatile analytical model, in this paper we study PDFs of the vortex and Gaussian beams at different points of their cross sections. Numerical results are compared with the most common analytical models (lognormal, exponential, gamma, and gamma-gamma distributions) [4,15–17]. The particular attention is paid to the case of fluctuations, for which the scintillation index exceeds unity. For this case, our numerical results are in the closest agreement with the experimental data and the lognormally modulated exponential model of PDF presented in [5]. This model differs principally from other models by the fact that for the case of strong fluctuations, PDF for the intensity normalized to its average value takes a finite value exceeding unity at zero.

## 2. Numerical model

The propagation of laser beams was simulated through solution of the parabolic wave equation [18]. Atmospheric turbulence was represented by a set of phase screens [19–22]. The simulation algorithms were organized in the same way as the algorithms of [13,14]. We used the modified Andrews spectrum of refractive index fluctuations [4], which had the following form

_{0}= 2π/

*M*

_{0}, κ

*= 3.3*

_{a}*m*

_{0},

*m*

_{0}and

*M*

_{0}are the inner and outer scales of atmospheric turbulence, $C_n^2$ is the structure characteristic of the refractive index. In the calculations

*M*

_{0}= 20

*a*and

*m*

_{0}= 0.08

*a*(

*a*is the beam radius)

*.*The turbulence was considered isotropic and homogeneous. The turbulent conditions of propagation at the path were specified with the Rytov parameter $\beta _0^2 = 1.23C_n^2{k_0}^{{7 \mathord{\left/ {\vphantom {7 6}} \right.} 6}}{z^{{{11} \mathord{\left/ {\vphantom {{11} 6}} \right.} 6}}}$, which depends on $C_n^2$, the path length

*z*, and the wave number of laser radiation ${k_0} = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right.} \lambda }$ [4]. From the simulated array of random realizations (

*N*= 5 000) of the wave field ${E_i}({\textbf r},z)$, a sample (empirical) distribution function of random intensity fluctuations was constructed by the standard equation [23] where $\theta (I)$ is the Heaviside function, ${I_i}({\textbf r},z) = {|{{E_i}({\textbf r},z)} |^2}$ are random values of intensity. ${F_N}(I)$ is an approximation of the cumulative distribution function $F(I)$. At the known probability density function of intensity fluctuations $P(I)$, the cumulative distribution function can be calculated as [4,23] The probability density of intensity fluctuations was approximated through the construction of a histogram with the use of a smoothing procedure.

We have studied the statistical characteristics of intensity fluctuations of the Laguerre–Gaussian beam ${\textrm{LG}}_0^l$ with the initial field distribution

*l*is the topological charge. Equation (4) describes the Gaussian beam (G) if

*l*= 0 and the circulation mode LG

_{0}

^{1}of the Laguerre–Gaussian beam if

*l*= 1.

## 3. Results and discussion

#### 3.1 Analyzed analytical models of PDF

For comparison with the results of numerical simulation, we used the known analytical models of the probability density functions.

Lognormal PDF

Exponential PDF

The gamma-gamma PDF is commonly believed a versatile distribution model suitable for the entire range of turbulent conditions [4,24] (at least, for a point receiver of radiation [9])

The gamma model of PDF [15–17]

*k*and $\langle{I({\textbf r})} \rangle$ serves as a basis for construction of the gamma-gamma model. The gamma distribution was initially developed for approximation of the probability density of the amplitude of a wave field scattered by a rough surface [15]. Then it was used for description of statistical properties of a speckle field [16]. In [17], in combination with the exponential distribution, it was also used for approximation of PDF of intensity of a speckle field passed through atmospheric turbulence. As was shown by our calculations (presented below), model (11) can be used for description of statistical properties of the intensity of radiation passed through the turbulent atmosphere without combination with other distributions. For this purpose, using Eq. (7), we calculate the scintillation index and find the value of the parameter

*k*It should be noted that if $\sigma _I^2({\textbf r}) = 1$, then distribution (11) transforms into distribution (8). For distribution (9) to transform into the negative exponential distribution, the fulfillment of $\sigma _I^2({\textbf r}) = 1$ is not sufficient, it is also necessary for $\alpha \to \infty$. This corresponds to the condition $\beta \to 1$[24].

For strong turbulent fluctuations ($\sigma _I^2({\textbf r}) > 1$), the following lognormally modulated exponential model of PDF was proposed by Churnside [5]:

In [25] for the case $\sigma _I^2({\textbf r}) > 1$, we have proposed the following PDF model, which can be called a fractional exponential distribution:

*m*can be found from the following equation: The distribution for the normalized intensity, as well as Eq. (13), is characterized by a higher-than-unity finite value of

*P*(

_{F}*I*= 0). If the scintillation index $\sigma _I^2({\textbf r}) = 1$, then Eqs. (13) and (15) transform into exponential distribution (8). It should be noted that for the normalized representation of this PDF at zero, we obtain the value identically equal to unity. Since the random field $I({\textbf r})$ is statistically isotropic [4], we re-designate $\sigma _I^2({\textbf r}) = \sigma _I^2(r)$ and $\langle{I({\textbf r})} \rangle = \langle I \rangle$.

#### 3.2 Results of numerical simulation

The most widely used measure of turbulent distortions of a wave beam is the Rytov parameter $\beta _0^2$ [4]. Depending on the value of this parameter, propagation conditions can be conditionally divided into the cases of weak ($\beta _0^2 < 1$) and strong ($\beta _0^2 > 1$) turbulence. However, when we analyze statistical characteristics of the field in the cross section of limited beams, for $\beta _0^2 < 1$ the scintillation index in some zones of the beam may exceed unity. Analogously, for $\beta _0^2 > 1$ the scintillation index may be smaller than unity. It can be seen from the results of simulation of propagation of the Gaussian and vortex beams for different propagation conditions corresponding to $\beta _0^2 = 0.1$ and 1.0 (see Fig. 1) that the value of the scintillation index vary considerably in the beam cross section. That is, the Rytov parameter cannot serve an indicator for determination of the PDF type in the cross section of limited beams.

The analysis of numerical results has shown that the form of PDF of normalized intensity is determined by the value of the scintillation index. At the same time, the PDF form differs qualitatively for the situations with $\sigma _I^2(r) < 1$ (weak scintillations) and $\sigma _I^2(r) > 1$ (strong scintillations). That is why we consider these situations separately, regardless of values of the Rytov parameter.

Below we present the results of comparison of numerical PDFs with different analytical models of PDF for the Gaussian beams and beams with the topological charge *l* = 1.

### 3.2.1 Weak intensity fluctuations ($\sigma _I^2(r) < 1$)

The results of calculation of the probability density function of intensity fluctuations obtained in the numerical experiment for the conditions of weak turbulence at a path with *β*_{0}^{2} = 0.1 are depicted in Figs. 2–4. These distributions were obtained at the distance *z* = 0.1*z _{d}*

_{.}. These results correspond to the points in the beam cross section, where the scintillation index is smaller than unity or approximately equal to unity. It follows from Eqs. (11) – (12) that the gamma distribution increases infinitely as the intensity tends to zero, when $\sigma _I^2(r) > 1$ (

*k*<1). That is why for the cases $\sigma _I^2(r) \ge 1$ (Figs. 2(a) and (b)) the parameter

*k*was taken

*k*= 1 in Eq. (11). In this case, the gamma distribution transforms into exponential distribution (8), for which $\sigma _I^2(r) = 1$.

It can be seen from Fig. 2 that for the weak turbulence ($\beta _0^2 \ll 1$), intensity fluctuations at the axis of the Gaussian beam satisfy the condition $\sigma _I^2(r) \ll 1$ and have the lognormal distribution (Fig. 2(a)). This is in a good agreement with the theoretical and experimental results [4,24–27]. We obtain the analogous result at the ring of the Laguerre–Gaussian beam (Fig. 2(b)), where its mean intensity is maximal and the scintillation index $\sigma _I^2(r) \ll 1$ as well. We can also see that the probability density function for the gamma distribution is very close to the lognormal distribution [17] and can be used for approximation of the calculated results.

At the same time, for the weak turbulence ($\beta _0^2 < 1$) the scintillation index at the axis of the vortex beam satisfies the condition $\sigma _I^2(r) \approx 1$, and PDF is well approximated by the exponential distribution (Fig. 3(a)) that is usually used for the case of strong fluctuations $(\beta _0^2 \gg 1)$. The analogous conditions ($\sigma _I^2(r) \approx 1$) are also fulfilled at the beam periphery and at other beam points, where the curve of the dependence $\sigma _I^2(r)$ intersects the level of 1 (see Fig. 1). At all these points, we obtain the exponential distribution for PDF, in particular, for the beam with *l* = 2.

It should be noted that the lognormal distribution in this case differs significantly from the exponential one, and the gamma distribution transforms into the exponential one at $\sigma _I^2(r) = 1$. It should be noted that, as was assumed in [24], distribution (8) can take place only in the case of saturated intensity fluctuations.

The analysis of numerical results has shown that at $\sigma _I^2(r) > 0.4$ the lognormal distribution begins to differ markedly from numerical PDF, which can be well approximated by the gamma distribution (Fig. 4) with the parameter *k* inversely proportional to the scintillation index.

Thus, we can conclude that for the considered beams at the cross-sectional points, for which the condition $0 < \sigma _I^2(r) \le 1$ is fulfilled, the statistics of intensity fluctuations can be approximated by the gamma distribution.

### 3.2.2 Strong intensity fluctuations ($\sigma _I^2(r) > 1$)

The principal difference of the results presented below for $\sigma _I^2(r) > 1$ from the case of the scintillation index smaller than unity is that PDF of normalized intensity at zero takes a finite value exceeding unity.

The distributions shown in Fig. 5 were obtained at the distance *z* = 0.1*z _{d}* (

*z*=

_{d }*ka*

^{2}/2), for the conditions close to moderate turbulence ($\beta _0^2$ = 1). The circles are drawn from the results of numerical simulation, the red dashed curves correspond to exponential probability density (8), the black dashed curves correspond to lognormally modulated exponential (Churnside) probability density (13), and the blue curves are drawn by Eqs. (15)–(16).

It is seen that at ${\sigma _I}(r) > 1$ the numerically obtained probability density distributions become different from the exponential distribution and are well approximated by the fractional exponential distribution and the lognormally modulated exponential distribution (Churnside model). The value of PDF at zero for the numerical results exceeds the corresponding value for the exponential model.

Figure 6 shows the distributions at the points with the highest values of the scintillation index (σ* _{I}*(

*r*) = 1.50 and σ

*(*

_{I}*r*) = 1.65). The calculations were performed for the conditions of strong turbulence ($\beta _0^2$ = 10) for different path lengths (

*z*= 0.1

*z*and

_{d}*z*= 2.0

*z*) and at different points of the cross section. One can see that for these values of the scintillation index the numerical results differ widely from the exponential model. In addition to the exponential PDF, Fig. 6 shows also other distributions, namely, the lognormal PDF, log-normally modulated exponential (Churnside) PDF, and fractional exponential PDF.

_{d}Parameters of analytical models (5), (13), and (15) were determined by setting the average intensity <*I*> and the scintillation index $\sigma _I^2(r)$ equal to the corresponding values obtained in the numerical experiment. These two parameters are sufficient to construct distributions (5), (13), and (15). For construction of gamma-gamma PDF *P _{GG}*(

*I*) described by Eq. (9), it is needed to find the parameters α and β. The procedure for calculation of these parameters is described in [4]. It requires the use of data not related directly to the measured statistical characteristics of intensity fluctuations (average value and variance). For their determination, it is necessary to measure or to calculate additional parameters (inner and outer scales, Rytov parameter, beam parameters).

The comparison of analytical distributions with the results of numerical simulation demonstrates that as the intensity approaches zero, the lognormal distribution differs markedly from numerical results. This is connected with the fact that for this distribution, as the intensity tends to zero, PDF tends to zero too, whereas the numerical calculations yield a finite value of PDF.

The similar remark is also true for the gamma-gamma distribution, for which the value at zero tends to zero or infinity. This can be demonstrated in the following way. We simplify the representation of *P _{GG}*(

*I*) in the area of low intensity values. Using Eq. (9) and equation 8.485 from [28]

*P*(0) tends to the value coinciding with the value of the of probability density of exponential distribution at zero. For other values of the parameters α and β, the gamma-gamma distribution either tends to zero or increases unlimitedly as the intensity tends to zero. Thus, the unlimited increase of

_{GG}*P*(

_{GG}*I*) at $\alpha < 1$ or $\;\beta < 1$ follows from Eq. (19)

It should be noted that Eq. (17) is valid only for noninteger values of the index ν (ν = α - β). If we assume the existence of integer or noninteger values of this index, then with equations 8.446 and 8.447.1 from [28] we can find the estimate of *P _{GG}*(

*I*) at zero intensity values. This estimate will also give zero and infinity

*P*(0) values.

_{GG}As was already mentioned, the approximation by gamma distribution (11)–(12) becomes inapplicable when the scintillation index $\sigma _I^2(r) > 1$, because in this case the probability density becomes infinite for zero values of *I*. Therefore, it is necessary to use a different approximation, which as in Eq. (11), transforms to model (8) if $\sigma _I^2(r) = 1$.

The finite value of PDF at zero was obtained not only in the calculations presented in this paper, but also in our previous papers [25,29]. In addition, this condition was fulfilled in experiments [5]. The analytical proof and detailed discussion of this result can be found in [29].

Among the considered models, the log-normally modulated exponential PDF and the fractional exponential PDF satisfy this condition. For these models the values of PDF in zero exceeds the value of the exponential PDF, and this excess is larger for higher values of $\sigma _I^2.$

If we compare the results in the zone of average intensity, then we can notice that the fractional exponential PDF is closest to the results of numerical simulation, while the log-normally modulated exponential PDF differs widely, and even the larger difference is observed for the lognormal PDF.

For comparison of the distributions in the area of high intensity values several times exceeding the average value, Fig. 6 is not informative, because all the distributions overlap in the linear scale. For this purpose, Fig. 7 shows the results in the log scale. The calculations were performed for the scintillation index σ* _{I}*(

*r*) = 1.84 and σ

*(*

_{I}*r*) = 2.1. The Rytov parameter in the calculations corresponded to the conditions of strong turbulence ($\beta _0^2$ = 10), and the distance length was

*z*=

*z*. Here we can also see the good agreement of proposed model (15) with the numerical results for small (near zero) and moderate values of intensity. In addition, to be noted is some difference of the analytical models for large intensity values exceeding more than fivefold the average intensity (so-called giant spikes).

_{d}The probability of occurrence of giant spikes was analyzed in [10]. The results of numerical simulation were compared with the lognormal PDF. It was shown that the probability of spike appearance in numerical calculations is much higher than predicted by the analytical model. The comparison of the analytical models at high intensity region (see Fig. 7) demonstrates that the values of the log-normally modulated exponential PDF and fractional exponential PDF far exceed the values of the lognormal PDF. The fractional exponential PDF gives the highest PDF values. Thus, this model predicts the highest probability of appearance of giant spikes.

Figure 8 depicts the probability density distributions of the intensity normalized to its average value for the Gaussian and Laguerre-Gaussian beams. For the Gaussian beam, the parameter σ* _{I}*(

*r*) is equal to 1.84, and for the Laguerre-Gaussian beam this parameter has the close value σ

*(*

_{I}*r*) = 1.82. The result for the Gaussian beam in the non-normalized coordinates is also shown in Fig. 7(a). These results are obtained for different beams at different points. One can see that at very close values of the parameter σ

*(*

_{I}*r*) for these cases, the numerically obtained probability density distributions of the intensity normalized to its average values nearly coincide and are well approximated by the fractional exponential PDF.

## 4. Conclusions

The results presented in this paper cover a wide range of turbulent propagation conditions for laser beams of different types. The Rytov parameter ranged within 0.1 ≤ $\beta _0^2$ ≤ 10.0, and the propagation distance was 0.1*z _{d}* ≤

*z*≤ 10

*z*. The analysis of intensity statistics of optical beams propagating in the turbulent medium suggests that in all the cases, regardless of the beam type, turbulent conditions, and position of the observation point, the probability density of intensity fluctuations is unambiguously determined by the values of the average intensity and the scintillation index. For intensity fluctuations with the scintillation index $\sigma _I^2(r) < 1$, the probability density is well approximated by gamma distribution (11), for which the PDF value at zero tends to zero.

_{d}The situation alternates qualitatively for intensity fluctuations with the scintillation index $\sigma _I^2(r) > 1$. In this case, PDF at zero tends to a finite value exceeding the corresponding value for the exponential distribution. This behavior of PDF can be described by neither the lognormal distribution nor the gamma-gamma distribution, since PDF at zero for them takes zero or infinite values. This PDF behavior at zero is well approximated by the log-normally modulated exponential PDF (13) and fractional exponential PDF (15). It should be noted that as the scintillation index tends to unity, all three distributions (11), (13), and (15) tend to the exponential distribution.

Thus, we can conclude that we have proposed a versatile approach based on the scintillation index as a measure of intensity fluctuations, which determines the PDF distribution laws, in the analysis of statistical characteristics of radiation intensity fluctuations for limited beams of arbitrary type. The value of the scintillation index unambiguously determines the probability density distribution of the intensity normalized to its average value at a given point. In the case that direct measurements of the scintillation index and the average intensity are impossible, these parameters can be found from numerical simulation. This approach significantly (three to four orders of magnitude) decreases the volume of computations needed for construction of PDF in numerical simulation. The knowledge of this distribution allows prior or on-line estimation of the efficiency of operation of different optical systems in the turbulent atmosphere. In particular, it allows one to estimate the probability of signal fading. According to the results obtained, for these estimates it is sufficient to know the measured (calculated) values of the scintillation index and the average intensity.

## Funding

Russian Science Foundation (18-19-00437); Russian Foundation for Basic Research (18-29-20115\18); Ministry of Education and Science of the Russian Federation (8.1039.2017).

## References

**1. **S. M. Flatté, C. Bracher, and G. Wang, “Probability density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A **11**(7), 2080–2092 (1994). [CrossRef]

**2. **R. J. Hill and R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A **14**(7), 1530–1540 (1997). [CrossRef]

**3. **J. H. Churnside and R. G. Frehlich, “Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere,” J. Opt. Soc. Am. A **6**(11), 1760–1766 (1989). [CrossRef]

**4. **L. C. Andrews and R. L. Phillips, * Laser Beam Propagation through Random Media* (SPIE, 2005). [CrossRef]

**5. **J. H. Churnside and R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A **4**(4), 727–733 (1987). [CrossRef]

**6. **S. D. Lyke, D. G. Voelz, and M. C. Roggemann, “Probability density of aperture-averaged irradiance fluctuations for long range free space optical communication links,” Appl. Opt. **48**(33), 6511–6527 (2009). [CrossRef]

**7. **J. R. W. Mclaren, J. C. Thomas, J. L. Mackintosh, K. A. Mudge, K. J. Grant, B. A. Clare, and W. G. Cowley, “Comparison of probability density functions for analyzing irradiance statistics due to atmospheric turbulence,” Appl. Opt. **51**(25), 5996–6002 (2012). [CrossRef]

**8. **R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express **20**(12), 13055–13064 (2012). [CrossRef]

**9. **F. S. Vetelino, C. Young, and L. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. **46**(18), 3780–3790 (2007). [CrossRef]

**10. **S. L. Lachinova and M. A. Vorontsov, “Giant irradiance spikes in laser beam propagation in volume turbulence: analysis and impact,” J. Opt. **18**(2), 025608 (2016). [CrossRef]

**11. **D. L. Andrews, * Structured Light and its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces* (Academic, 2008).

**12. **A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics **3**(2), 161–204 (2011). [CrossRef]

**13. **V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Properties of vortex beams formed by an array of fibre lasers and their propagation in a turbulent atmosphere,” Quantum Electron. **46**(8), 726–732 (2016). [CrossRef]

**14. **V. P. Aksenov and V. V. Kolosov, “Scintillations of optical vortex in randomly inhomogeneous medium,” Photonics Res. **3**(2), 44–47 (2015). [CrossRef]

**15. **M. Nakagami, “The m distribution—a general formula of intensity distribution of rapid fading,” in * Statistical Methods in Radio Wave Propagation*, W. C. Hoffman, ed. (Pergamon, 1960).

**16. **J. W. Goodman, “Statistical properties of laser speckle patterns,” *Chap. 2 in Laser Speckle and Related Phenomena*, J. C. Dainty, Ed., Springer-Verlag, New York (1975).

**17. **V. S. R. Gudimetla and J. F. Holmes, “Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere,” J. Opt. Soc. Am. **72**(9), 1213–1218 (1982). [CrossRef]

**18. **S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, * Principles of Statistical Radiophysics. Vol. 4 Wave Propagation through Random Media* (Springer, 1988).

**19. **J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. **10**(2), 129–160 (1976). [CrossRef]

**20. **P. A. Konyaev and V. P. Lukin, “Thermal distortions of focused laser beams in the atmosphere,” Appl. Opt. **24**(3), 415–421 (1985). [CrossRef]

**21. **V. V. Dudorov, G. A. Filimonov, and V. V. Kolosov, “Algorithm for formation of an infinite random turbulent screen,” Proc. SPIE **6160**, 61600R (2005). [CrossRef]

**22. **J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. **27**(11), 2111–2126 (1988). [CrossRef]

**23. **A. W. van der Vaart, * Asymptotic Statistics* (Cambridge University Press, 1998). [CrossRef]

**24. **M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. **40**(8), 1554–1562 (2001). [CrossRef]

**25. **V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Probability distribution of intensity fluctuations of vortex laser beams in the turbulent atmosphere,” arXiv:1802.03172 (2018).

**26. **A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE **63**(5), 790–811 (1975). [CrossRef]

**27. **M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity Relations and their Experimental Verification for Strong Intensity Fluctuations of Laser Radiation,” in * Laser Beam Propagation in the Atmosphere*, J.W. Strohbehn, ed. (Springer-Verlag, 1978).

**28. **I. S. Gradshteyn and I. M. Ryzhik, * Tables of Integrals, Series and Products* (Academic Press, 2000).

**29. **V. P. Aksenov and V. V. Kolosov, “Probability density of field and intensity fluctuations of structured light in a turbulent atmosphere,” J. Opt. **21**(3), 035605 (2019). [CrossRef]