Probability distribution of intensity fluctuations of vortex laser beams in the turbulent atmosphere

Numerical simulation is used to analyze statistical characteristics of vortex beams propagating in the atmosphere. The cumulative distribution function and the probability density function of intensity fluctuations are compared for Gaussian beams and vortex beams. It is shown that for propagation conditions in the turbulent atmosphere corresponding to weak fluctuations (Rytov parameter much smaller than unity), intensity fluctuations at the axis of the Gaussian beam have the lognormal distribution, whereas the probability density distribution of the radiation intensity fluctuations at the axis of the vortex beams is well approximated by the exponential distribution characteristic of conditions of saturated fluctuations (Rytov parameter much larger than unity)


INTRODUCTION
To predict the effect of atmospheric turbulence on the functioning of optical systems in the turbulent atmosphere, it is important to know statistical properties of intensity fluctuations of laser beams used in these systems. The cumulative distribution function (CDF) or the probability density function (PDF) characterizes most completely the statistics of these fluctuations (scintillations). It is a good idea to have such a theoretical distribution that simulates intensity fluctuations in the plane of receiving aperture under all possible propagation conditions with the minimal error.
Usually, the Rytov parameter 2 The probability density function PDF is studied quite intensely as can be seen from the literature [1][2][3][4][5][6][7][8]. The currently proposed theoretical models include the lognormal distribution, exponential one, K distribution, lognormal distribution modulated by the exponential distribution, lognormal distribution modulated by the Rician distribution (also known as the Beckman distribution), and gammagamma distribution (gamma distribution modulated by the gamma distribution). Some of these distributions were proposed for different intensity of turbulence: from weak to saturated.
The lognormal and gamma-gamma distributions are now used most widely, according to [4]. These distributions were checked with the results of numerical simulation of propagation of the Gaussian laser beam through statistically homogeneous and isotropic turbulence. The objects under study were the laws of distribution of intensity fluctuations at the beam axis or the laws of distribution of irradiance fluctuations at a receiving aperture of finite size. The proposed distribution models were repeatedly checked experimentally [3,5,[7][8][9]10]. Despite the significant number of theoretical PDF models, they do not work well for a large number of propagation conditions occurring in the atmosphere. Thus, for example, the lognormal distribution significantly understates the frequency of occurrence of giant irradiance spikes observed in the case of strong fluctuations [10]. Some kinds of the distributions include fitting parameters, which are not directly related to values of measured statistical characteristics of irradiance fluctuations, such as the mean intensity and the variance of irradiance fluctuations. These characteristics are quite amendable to calculation based on the existing theories of laser beam propagation in the turbulent medium.
As far as we know, the laws of probability distribution of irradiance fluctuations over the beam cross section remain unstudied.
These studies become urgent in connection with application of beams of special kind and some exotic beams [11]. The emphasis is on vortex beams having the orbital angular momentum [12,13]. In this paper, the numerical simulation is used to compare the probability distribution functions of intensity fluctuations of the vortex Laguerre-Gaussian beam and fundamental Gaussian beam. The probability distribution functions of two beams at different points of their cross section are examined. The numerical results are compared with the analytical models (lognormal, exponential, gamma-gamma distributions) [4] used most widely by specialists in wave propagation through randomly inhomogeneous media. In addition, the probability distribution densities obtained from the numerical simulation of propagation of optical beams in the atmosphere are approximated by the gamma distribution [14][15][16] and the fractional exponential distribution proposed in this paper.
A prerequisite for this study was the result [17], which suggests that the scintillation index of the Laguerre-Gaussian beam corresponds to the case of saturated fluctuations at the axis and periphery of the beam and to the case of weak fluctuations at the maximum of the mean intensity.

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][20][21]. The simulation algorithms were organized in the same way as the algorithms of [13,17]. We used the modified Andrews spectrum of refractive index fluctuations [4], which had the following form 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 0 LG l with the initial field LG of the

RESULTS OF NUMERICAL SIMULATION
The calculated probability density function and the cumulative distribution function of irradiance fluctuations obtained in the numerical experiment (solid curves) are shown in Figs. 1-10. For comparison with the results of numerical simulation, we used the known analytical models of the probability density functions, including the lognormal distribution is a PDF model for the conditions of saturated fluctuations . The gamma-gamma PDF is commonly believed a versatile model of distribution suitable for the entire range of turbulent conditions [4,24] (at least, for a point receiver of radiation [9]) respectively [23]. These parameters are connected to the scintillation index 2 ()  I r through the following equation Advantages of PDF model (9) and results of its use for the plane and spherical waves can be found in [4,24]. It should be noted, however, that the use of analytical estimates of  and  [4] not always leads to the sufficiently close agreement between distribution (9) and the distribution obtained from numerical simulation. Therefore, the additional correction of these parameters is necessary. The gamma model of PDF (m-distribution) [14][15][16] I mI (11) with the two parameters m and I 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 [14]. Then it was used for description of statistical properties of a speckle field [15].
In [16], 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 and without combination with other distributions. For this purpose, using distribution (11) and equations (6) and (7), we calculate the scintillation index and find the value of the parameter  (Fig. 1a). This is in a good agreement with the theoretical and experimental results [4,24,25]. We obtain the analogical result at the ring of the Laguerre-Gaussian beam (Fig. 1b), where its mean intensity is maximal and the condition 2 ( ) 1  I r is fulfilled for the scintillation index. We can also see that the probability density function for the gamma distribution is very close to the lognormal distribution [16] and can be used for approximation of the calculated results.
At the same time, the scintillation index at the axis of the vortex beam satisfies the condition 2 ( ) 1  I r , and PDF is well approximated by the exponential distribution (Fig. 2a)  Thus, we can draw the conclusion that for the considered beams at the cross-sectional points, for which the condition 2 0 ( ) 1    I r is fulfilled, the statistics of intensity fluctuations can be approximated by the gamma distribution.

Moderate turbulence
As was already mentioned, the approximation in the form (11)-(12) becomes inapplicable when the scintillation index 2 ( ) 1  I r , because in this case the probability density becomes infinite for zero values of I(r). Therefore, it is necessary to use a different approximation, which, as in Eq. (11), transforms to model (8)  with the parameter m, which can be found from the following equation:  It can be seen that at ( ) 1.1  I r the probability density distributions obtained in numerical calculations (solid curve) become different from the exponential distribution (dashed curve) and can be well approximated by the fractional exponential distribution (dotted curve). Line designations in Fig. 5a are analogous to Fig. 4. In Fig. 5b, the solid curve is the sample distribution (2). The dashed curve is cumulative distribution function (CDF) (3) corresponding to probability density (13)- (14). The dotted line corresponds to the exponential probability density.
One can see that approximation (13)-(14) provides a satisfactory agreement for the probability density function and the cumulative distribution function. Figures 6-7 show the results of comparison of the numerical calculations with the lognormal distribution, gamma-gamma distribution, and fractional exponential distribution. The calculations were performed for the distribution function and probability density of intensity fluctuations of the Laguerre-Gaussian beam at different distances from the beam center for conditions of moderate turbulence and the atmospheric path length equal to the Rayleigh diffraction length. From here on, the parameters of gamma-gamma distribution (9)  and  were found from fitting to the numerically obtained distribution. It can be seen that CDF of intensity fluctuations is well approximated by the gamma-gamma distribution and the fractional exponential distribution, which give close results in this case. However, the probability density functions for these distributions behave differently, especially, in the zone of low intensity values. Fractional distribution (13)- (14) tends to a finite value, while log-normal (5) and gamma-gamma (9) distributions tend to zero as 0  I . For the lognormal distribution, the difference from the numerical results is large as compared to the other distributions.   Figure 8 shows   Fig. 8, the numerical calculations are compared with the exponential and fractional exponential distributions. In Figs. 9-10, the numerical results are compared with the fractional exponential, lognormal, and gamma-gamma distributions.

Strong turbulence
It can seen that under conditions of strong turbulence the distribution laws of intensity fluctuations do not vary qualitatively over the beam cross section. The parameters  and  of gamma-gamma distribution () GG PI (9) are found here through the fitting to the numerically determined distribution. However, it can be shown that the applicability condition of distribution (9) provides that these parameters should satisfy the conditions