1. Introduction

Active implementation of free-space optical communications (FSOs) and the development of related technologies have caused a growing interest in mitigation of the turbulence-induced degradation of the efficiency of such systems. Knowledge of the dependence of the statistics of laser irradiance fluctuation (scintillation) on the turbulence strength and the parameters of an optical system, such as the receiver aperture size, is necessary for development of communication equipment.

The probability density function (PDF) [1] is one of the most widely used tools for analyzing scintillations at the receiver aperture and is one of the key characteristics for analysis of the atmospheric effects on FSO systems. The PDF was studied starting from the 1980s both on a receiver [2,3] after propagation through turbulent atmosphere and after scattering off a surface [4]. Since then, laser power fluctuations have been studied in various fluctuation regimes and many PDFs have been proposed for describing various situations. Attempts to find a distribution suitable for describing experimental data for any observed atmospheric conditions have not been successful. Models describing a wider range of turbulence conditions required an increasing of number of fitting parameters in the resulting distributions. At the same time, different one-parametric models are convincingly fitting the experimental results under scintillation regime, meaning there are different types of PDF behavior depending on the turbulence strength, characteristics of a receiving optical system, and received laser radiation. By now, there are a large number of both one-parameter and two-parameter distributions, which were introduced for various experimental data.

An example of the one-parameter distribution is the lognormally modulated exponential PDF, which was proposed for describing the strong scintillation regime based on an experiment at a 1 km long path with 0.1 cm and 2.5 cm apertures with values of the Rytov index for a plane wave $\beta _0^2 = 1.23C_n^2{k^{7/6}}{L^{11/6}} = [{4.8 - 17.2} ]$ where $C_n^2$ is refractive index structure parameter, $k = {{2\pi } / \lambda }$ is a wave number, $\lambda $ is a wavelength, L is a propagation length [3]. In addition, Churnside and Hill [3] mentioned both the aperture averaging effect and a good agreement between the lognormally modulated exponential distribution and the experiment for large apertures. The results of the analysis for a few typical cases were reported for strong, moderate, and weak scintillations at 0.05-2.4 km paths with the submillimeter apertures [5]. It was demonstrated that the observed statistics depended on the path length and the turbulence strength [5]. In addition, Churnside and Frehlich [5] concluded that at least a two-parameter distribution is needed to describe experimental results under any atmospheric conditions. Afterwards numerical simulations were performed for propagation of both plane and spherical waves through a turbulent atmosphere [6,7] for a range of turbulence conditions characterized by $\beta _0^2 = [{0.024 - 15.8} ]$ (after conversion to the Rytov index for plane waves). However, the considered heuristic PDF models failed to demonstrate the agreement with the simulations in some of the key cases [7]. Results of experiments along 1.5 km and 12 km paths with receiver apertures 0.1 cm, 12 cm, and 15 cm in diameter were compared in detail with ten theoretical models. It was noted that many models are characterized by a limited application range (fitted experiment for a small range of scintillation index values), and some of them, including the gamma distribution, demonstrated bad results in all the cases considered [8]. In [9], the statistics of laser scintillations were simulated for long paths of 50, 75, and 100 km. In this case, the gamma-gamma and lognormal distributions worked well for the considered apertures in the weak scintillation regime, while the gamma-gamma distribution was a better fit for the regime of moderate scintillations [9]. For the strong scintillation regime, two ranges were separated, namely, for apertures larger and smaller than the coherence scale. These ranges were better approximated by the lognormal and gamma-gamma distributions, respectively [9]. Analogously, a good agreement with the experiment was shown for the gamma-gamma and lognormal distributions in the moderate-to-strong regimes of laser radiation scintillations for apertures much smaller than, or equal to (or larger than) the coherence radius, respectively [10,11]. The effect of the coherence scale on the changing of statistics was exemplified in [11] with the experimental data for short near-surface paths 100 and 125 m long for six aperture diameters from 0.016 to 1.3 cm. It was shown in [12,13] that the scintillation statistics can vary within the beam cross section depending on the scintillation index value at a given point. The averaging effect of the aperture was found to decrease the scintillation index and to lead to the changing of the power scintillation statistics [14–15]. Most of the mentioned papers presented the results for only a few sizes of the receiver aperture and a few values of the turbulence strength along the propagation path. The continuous dependence of the PDF distribution on these parameters was not considered.

The existing publications demonstrated that PDFs are different for a point aperture or a very large aperture, as well as for strong, moderate, or weak scintillations. To the best of our knowledge, there are no publications about the PDF behavior at the transitions between different scintillation regimes and the specific values of the characteristic parameters at which these transitions occur. Knowledge of such critical values of the characteristic parameters, if any, would help to provide accurate recommendations for the choice of specific PDF distributions for a range of conditions a communication system operates in. This information has a significant practical value and can help with the a priori choice of the probability distribution corresponding to a received signal. In this paper, we consider the PDF evolution for a wide range of turbulent conditions and aperture sizes varying in the range d = [0.055-14.4] cm with a step size of about 0.11 cm. Variations of the turbulence conditions corresponded to variations of the coherence radius in the range ${r_0} = 1.67{({C_n^2{k^2}L} )^{ - {3 / 5}}} = [{1.4 - 18.6} ]$ cm and variation of the Rytov parameter in the range $\beta _0^2 = [{0.35 - 25.4} ]$.

The use of probability distributions with two (or more) parameters can be justified from the viewpoint of their derivation from theory. However, it can be inconvenient because of possible uncertainties in the parameter values, which may demand independent measurements during the operation of an FSO system. These measurements are not always possible or can significantly increase the cost of installation and maintenance of an optical communication line. For this reason, we consider only analytical distributions depending on one parameter, namely, the scintillation index. The scintillation index is selected because it is measured directly at the aperture. This information is always available at the receiver and can be used immediately to determine the statistics of observed scintillations. The distributions analyzed in this paper were chosen in a way that at least one of them can successfully fit experimental data under any considered conditions according to the selected criterion, namely, the Pearson criterion [16]. The selected models are the lognormal distribution (LN), the gamma distribution (GD), the fractional exponential distribution (FE) [12], and the fractional gamma distribution (FG) proposed in this paper for description of the aperture averaging effect. These four theoretical models are compared with the results of the experiment at the 7 km long path.

Section 2 of this paper describes the experimental setup and atmospheric conditions during the experiment. The theoretical models used for approximation of the experiment and the technique of their comparison with the experimental data are discussed in Section 3. Section 4 describes the results, and provides examples of PDFs for different scintillation regimes. We also discuss there the dependence of the PDF of the measured power on turbulence strength and aperture size as well as consideration of the characteristic parameters at which the type of PDF distribution changes. The experimental results are compared with the analytical distributions for all 27 considered measurement trials.

2. Experimental setup

The measurements were carried out at an atmospheric propagation path between a window of the laboratory in the Fitz Hall on the campus of the University of Dayton, USA and the roof of the VA Medical Center. To cover the entire considered range of the refractive index structure parameter ($C_n^2$) values, the measurements were conducted on two days with a total measurement time of more than 6 hours. The turbulence data were obtained with a scintillometer operating along nearly the same optical path. The schematic of the experiment is shown in Fig. 1.

 

Fig. 1. Schematic illustration of the experimental setup.

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A collimated Gaussian laser beam with the diameter d = 1.2 cm (at the e⁻² level) was formed by a single-mode fiber and a lens system installed on the roof of the VA Medical Center. The laser system was mounted on a gimbal to allow for fine adjustment of the direction of the laser beam. The emitted beam propagated along the 7 km long nearly horizontal path having only a small vertical tilt of approximately 1.15°. The height of the receiver end of the optical path at UD Fitz Hall side was approximately h_UD= 20 m, while the laser beam was sent from the roof of VA Medical Center with height h_VAMC= 40 m. The urban type of the ground might result in inhomogeneity of the turbulence along the path.

We used a single-mode fiber-coupled laser module (Photop model FGSM-532-050-10-10-00-A with a wavelength of 532 nm) as a source with a transmitted power of 2 mW. We were able to use a shear plate to collimate the laser transmitter. Considering the optical path difference between beams reflected by the front and back sides and well expressed interference patterns on the shear plate, we can conclude that the coherence length was at least 3.5 cm The source laser’s spectral width Δw can be estimated as $\Delta w \le {{{\lambda ^2}} / {{\rho _0}}}$ where ${\rho _0}$ is the coherence length, λ is the wavelength. In laboratory test measurements we found that the source's output power fluctuations were smaller than 1% rms. Additional tests with the laser light passing through an interference filter with full width at half maximum of 5 nm did not result in higher power fluctuations. This means that potential laser frequency shifts are much smaller than the bandwidth of the used interference filter and did not impact the scintillation measurements. A transmitter being a part of the commercial Scintec BLS 2000 scintillometer [17] was also installed on the roof at a distance of about 1 m from the laser aperture. The receivers for the laser beam and the scintillometer were set in the laboratory on the fifth floor of the Fitz Hall of the University of Dayton, about 1 m apart. The radiation incident on the 14.4 cm receiver aperture of a telescope was re-imaged onto the sensor of a 12-bit CCD camera, Imperx Bobcat BO620, with a resolution of 320 × 256 pixels operating with a time resolution of 2 fps and integration time 10⁻⁴ s.

For the purpose of calibration and correction of the background, we took dark images while the laser source was switched off and found that the background noise of each pixel was approximately 0.15% of the dynamical range. An interference filter (center wavelength: 532 nm, full width half max: 5 nm) and a diaphragm with an aperture diameter of 0.05 cm were located in the focal plane of the telescope. They reduced the impact of environmental light to a level that had no noticeable impact on the recorded images. Therefore, no extra background correction was necessary to eliminate influence of environmental light. The lens system and the beam reducer both reduced the beam size by the factor of 36. As one can see in Fig. 2(b), the image of the telescope pupil plane is larger than the camera sensor. Due to this fact, a single pixel of the camera corresponded to a size of 0.0554 cm in the entrance aperture plane. To minimize the influence of background radiation, a narrow-band interference filter was installed in front of the second lens of the receiving optical system. The start of each measurement trial was synchronized with the start of scintillometer measurements which provided one $C_n^2$ value per minute.

 

Fig. 2. Turbulence strength and laser beam intensity scintillation patterns: (a) values of refractive index structure parameter $C_n^2$ during each of the 27 measurement trials, (b) example of pupil plane images corresponding to different refractive index structure parameter values.

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Before the start of each measurement the direction of the emitted laser beam was fine tuned to make sure the receiving aperture is in the center of the laser beam footprint which has, although depending on turbulence strength, a typical diameter of about 50 cm at the receiver plane. The goal of the experiment was to study statistical properties of scintillations for apertures of different sizes at different turbulence strengths. We analyzed the apertures starting from the smallest one, a single pixel, which was approximately considered as a circle with a diameter of 0.0554 cm. The next considered aperture consisted of a square of four pixels. Then we increased the aperture radius by one pixel until it reached 14.4 cm. The apertures with radii of at least 4-5 pixels have shapes which are close to a circle. Scintillations at each of the 132 considered apertures were analyzed separately.

It is clear that with this data processing technique, the error in calculating statistical characteristics, such as the scintillation index, increases for small apertures having a size of few pixels due to the noise of the CCD array and possible flares or spurious interference on optical elements of the system. We estimate this error as 5-7% for the worst case of measurement at a one-pixel aperture by comparing neighbor pixels.

The measurements were conducted during two days, and they covered the entire range of $C_n^2$ values from 1.25E-16 to 1.59E-14 m^-2/3, which corresponds to the range of the coherence radius of ${r_0} = [{1.4 - 18.6} ]$ cm. Please note that while instantaneous $C_n^2$ values vary in this range, the values of $C_n^2$ averaged over the trial changes in the smaller range as it shown in Table 1. The values of $C_n^2$ for every minute of every trial are given in Fig. 2(a). Examples of pupil images at the aperture of the 14.4 cm telescope at the path end for different values of the turbulence strength are presented in Fig. 2(b). The date, starting time, and duration of every trial, as well as the corresponding values of $C_n^2$ averaged over the trial and ${r_0}$ are given in Table 1.

Table 1. Experimental trials: date, starting time, and duration.

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As a side note, we want to point out that while each trial was characterized by the average value of $C_n^2$, future studies may actually use the dates and times of the trials presented in the Table 1 for numerical simulation of details of the turbulence distribution along the path using numerical weather predictive methods. There are many trials characterized by close values of integrated turbulence strength along the path. Regardless possible differences in the turbulence distribution, trials with close ${r_0}$ results in similar histograms with similar dependencies on aperture size (see Section 4 for details).

Before proceeding to the analysis of the measurements we present the PDFs used for comparison with the experiment.

3. Statistical models and method of their comparison

3.1 Experimental and analytical PDF distributions

In the case of an experiment, where the value of power at the receiver aperture was measured N times, the power distribution function can be written as follows:

Since the set of experimental data is discrete and finite, we need to approximate PDF by constructing histograms. The number of bins for constructing histograms was chosen depending on the number of analyzed experimental values of measured power in the bucket (PIB), where the bucket is the considered aperture size. Each interval (bin size) should contain a sufficient number of measured values of PIBs to reduce the estimation error. On the other hand, this number should be small enough to avoid significant smoothing which can distort the form of the distribution.

There are many analytical models used to approximate the probability density of scintillation at the receiver aperture. By the reasons mentioned above, we consider four distributions in this paper. One of the most widely used distributions is the lognormal distribution [18].

Lognormal PDF is described by an equation:

Along with the lognormal distribution, the gamma distribution demonstrated good results for approximation of the PDFs of Gaussian and vortex beams at values of scintillation index from 0 to 1 [12,13]. The gamma model of PDF (m-distribution) [4,19] is

For the case $\sigma _{\textrm{PIB}}^2 > 1$, the fractional exponential distribution was proposed recently for describing the PDF at a point [12,13]. It also demonstrated good fitting for small (point) apertures under conditions of strong scintillations [15]. This distribution is described by the equation

This distribution for the normalized intensity/power works well when P_FE(PIB = 0) > 1. It should be noted that if the scintillation index is $\sigma _{\textrm{PIB}}^2 = 1$, then Eq. (6) and Eq. (8) transform into the exponential distribution, which has a value equal to unity for the occurrence of zero values.

In addition to these three distributions, we consider here the fractional gamma distribution, which is introduced to describe the effect of aperture averaging under conditions of moderate and strong scintillation (scintillation index larger than unity). The averaging effect of the aperture at strong scintillation manifests itself as follows. At an aperture with the size reaching some critical value, points with nonzero intensity values necessarily appear along with points with zero intensity. The integration of the received power across the aperture results in finite (non-zero) value. Consequently, the probability of zero power values tends to zero with increasing aperture size. As a result, the PDF at zero changes significantly from a value exceeding unity to zero, while the PDF for larger PIB values changes only slightly. This transformation of the fractional exponential distribution is described by the proposed fractional gamma distribution

3.2 Comparison of experimental and analytical results

When constructing experimental histograms, the entire range of PIB values (from zero to the maximal value) was divided into N bins. In the case of strong scintillation, the probability to observe zero and near-zero power values increases sharply. This effect is described by the fractional exponential function [13,15]. The successful description of this effect requires the bin to be small enough to detail the PDF behavior around zero. In this study, we chose N = 200. This allowed us to divide a histogram into small enough bins, whose size was sufficient to track abrupt changes in the PDF near zero. To compare the experimental histograms, which show the number of measured PIB values in each bin n_i, with the theoretical distributions, it is necessary to construct the corresponding histograms for theoretical data too. The theoretical histograms were constructed as

The correspondence of the theoretical distribution to the experimental results was checked by calculating the chi-square criterion according to the classical equation

The corresponding experimental and theoretical values for the overflow bin were calculated as

Thus, Eq. (14) converted to the form

Below we present the results for Pearson criterion normalized by the critical value (as it is defined in Eq. (17)). The normalized criterion smaller than unity corresponds to the condition for the null hypothesis not rejected, i.e. the condition that the theoretical distribution corresponds to the experimental data.

In the case of a finite number of experimental values, the Pearson criterion depends, to some extent, on the number of PIB values. It is practically impossible to choose the optimal bin length for all PDF parts. If a relatively small bin is chosen, the PDF form in zones of abrupt changes can be described better, but the accuracy of estimation can degrade in the zones where the experimental values are insufficient. The division of a histogram into the overflow and underflow bins partially lifts these restrictions, because the distributions of tail bins are combined into far larger ones, which allows us to have a sufficiently small bin for the rest part of the distribution. It should be emphasized that all distributions in this study were compared for the case of constructing them at N = 200. If a different, in particular, much smaller number of bins is chosen, this is critical only for a small range near zero in the strong scintillation regime, where it is necessary to describe the effect of the increasing probability of very low power values observed at the aperture. In other cases, a change in the number of bins can only slightly change the values of $\chi _{ns}^2$ without affecting the conclusions drawn in the paper. It should be noted that the values of the overflow and underflow bins are omitted in the figures.

Formally, the value of the relative Pearson criterion smaller than or equal to unity means there is a good agreement between the theoretical and experimental results. We ignored small (5-10%) fluctuations of this criterion, if they occurred simultaneously for the values of $\chi _{ns}^2$ of all the considered distributions, because in this case they are explained by abrupt changes in the experimental PDF due to experimental noise (noise of the CCD array, possible interference on the optical elements of the system, and so on) or lack of experimental statistics rather than by the better of worse agreement of different analytical functions with the experiment. In three of 27 cases considered, the criterion $\chi _{ns}^2$ fluctuated near 1.2 and 1.5 rather than around unity. Possible reasons for this behavior are discussed in section 4.5, but the behavior of the statistics in these cases qualitatively corresponds to those observed in other trials.

We would like to mention that applying another well-known criterion, the Anderson-Darling test as it was performed in Ref. [15], leads to the same conclusions as was obtained by using the Pearson criterion (see Appendix).

4. Analysis of experimental results

4.1 Strong scintillation regime

Consider several examples of the evolution of $\chi _{ns}^2$ and the scintillation index as functions of the aperture radius in the cases of strong scintillation regime (Fig. 3). It should be noted that, first, with the smallest aperture size (point aperture), the fractional exponential (FE) distribution is the best choice in all presented trials (Fig. 3(a),(b),(e),(f)). The criterion $\chi _{ns}^2$ for the FE distribution takes values smaller than unity and, at least half as much as for fractional gamma (FG) and gamma distribution (GD). As the aperture size increases, FG distribution becomes the best approximation (that is, having the smallest value of $\chi _{ns}^2$). In the following, the most appropriate fit changes from FG to GD (Fig. 3(a),(b),(e),(f)). The values of $\chi _{ns}^2$ for the LN distribution are almost always higher than 3 and, for this reason, are not shown in the plots. Although the aperture sizes at which FE, FG, and GD change to each other, are different for each of the cases at hand, we can find the parameters having fixed values at this change.

 

Fig. 3. Сhi-square metric of (a) trial 1, (b) trial 5, (e) trial 7, (f) trial 27; and the scintillation index of (c) trial 1, (d) trial 5, (g) trial 7, (h) trial 27 as functions of the aperture size for the case of strong in-pixel scintillations.

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The most appropriate fit changes from FG to FE at d/a ∼ 1, where a ≈ 0.314r_0, which follows from the relationship for the isoplanatic angle, path length, and coherence radius ${\theta _0} = 0.314{{{r_0}} / L}$ [20,21]. The changing of FG and GD occurs when the scintillation index achieves some critical value $\sigma _{\textrm{PIB,crit}}^2 \approx 0.75$ (Fig. 3(c),(d),(g),(h)), which is illustrated by vertical lines connecting the $\chi _{ns}^2$ plots and the corresponding plots of the scintillation index as a function of the aperture size.

To better understand the reasons for the change of the distributions that describe the experimental data in the best way, consider the PDF distributions in Fig. 4. For the aperture with the smallest considered size d = 0.0554 cm and the scintillation index $\sigma _{PIB}^2 > 2$, the experimental PDF has a characteristic feature, namely, the sharp increase in the probability of observing low-intensity values up to the level higher than two, which can be clearly seen in Fig. 4(a). The FE distribution qualitatively copies this behavior, whereas the LN, to the contrary, has a sharp decline near zero.

 

Fig. 4. Experimental and analytical PDFs for strong in-pixel scintillations (trial 7) at apertures with d equal to (a) 0.0554, (b) 0.665, (c) 2.88, (d) 5.98, (e) 12.0, and (f) 14.4 cm.

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At the same time, the GD overestimates the probability of low-intensity values and underestimates the probability of normalized power values in the range from 0.3 to 0.8 as can be seen from Fig. 4(a).

This is because the GD takes on an infinite value for a point aperture and zero intensity. The best distribution for a small aperture under conditions of strong scintillations is the FE (Fig. 3(a),(b),(e),(f)). This result is quite typical and was observed in all the considered trials corresponding to the strong scintillation regime ($\sigma _{PIB}^2 > 1$). It is in a good agreement with the results of numerical simulation reported in Refs. [12] and [15].

The high probabilities of observing low-power values are also found for larger apertures, up to the change from FE to FG. After the change, the ${\chi ^2}$ for FE becomes larger than unity, while the FG criterion is smaller than unity. This change occurs, for example, at d = 0.4 cm for trial 1 (Fig. 3(a)) and d = 0.8 cm for trial 5 (Fig. 3(b)), which correspond to approximately the same characteristic size, namely, the size of the isoplanatic region d/a ∼ 1. The accuracy in determination of this size related to the aperture size (two pixels) normalized to the corresponding size of the isoplanatic region is shown in Fig. 3(a),(b),(e),(f). Section 4.4 discusses the value of the characteristic scale of transition from FE to FG and the error of its estimation in detail. The mentioned transition is associated with the qualitative change of the experimental PDF, namely, with disappearance of high probabilities for small values of the measured power, which is explained by the averaging effect of the aperture.

Scintillation of the electric field at a point and its immediate vicinity is determined by changes in turbulence along the path that the wave is traversing. For the points in the aperture at distances not exceeding the size of the isoplanatic region, we can believe that the wave has passed through the same, or at least highly correlated, inhomogeneities of the refractive index. Therefore scintillations of all points of the isoplanatic region should be highly correlated as well. The well-known methods of compensation for distortions in adaptive optics are based on this fact. In the current context, it means that the aperture averaging does not change significantly the scintillations of received power at the scales smaller than isoplanatic size a, because they are correlated or even coincide.

At distances larger than a, other inhomogeneities of the refractive index contribute significantly. Consequently, at the time when a low-intensity value is observed at the aperture of size a, the field intensity beyond the aperture can be high, because it is determined by the different distribution of the refractive index along the path of the other portion of the optical wave. Therefore, apertures larger than a average the observed intensity more efficiently, thus reducing the probability of observing very low or very high relative power values.

Due to this isoplanatism we will sometimes refer to apertures, whose size is smaller than a, as point apertures. With a further increase in the aperture size, the FG remains the most appropriate of the considered distributions until the scintillation index falls below some critical value $\sigma _{\textrm{PIB}}^2 = \sigma _{\textrm{PIB,crit}}^2$, which was $\sigma _{\textrm{PIB,crit}}^2 \approx 0.75$ for our calculations.

After this value, the FG changes to the GD. A significant change of the choice of the bin size can result in a small change of the value of $\sigma _{\textrm{PIB,crit}}^2$, but such change is not significant. We estimated the critical value of the scintillation index here as $\sigma _{\textrm{PIB,crit}}^2 = [0.7 - 0.8]$.

The PDF behavior in the transition states between the different scintillation regimes is shown in Fig. 4(b),(d). One can see that both FG and FE (Fig. 4(b)) are close to the experimental histograms but demonstrate different qualitative behavior. The GD and FG distributions (Fig. 4(d)) are very close to each other and both approximate the experimental histograms well. With a further increase in the aperture, the FG distribution is less and less suitable (Fig. 4е), while the LN does not improve its result (Fig. 4(e),(f)).

4.2 Moderate scintillation regime

Consider the experimental measurement trials in moderate scintillation ($\sigma _{PIB}^2 \approx 1$) observed at the smallest aperture, namely trial 10 and trial 11. The evolution of $\chi _{ns}^2$ and the scintillation index as functions of the aperture radius are shown in Fig. 5. The scintillation index in the point aperture (∼0.0554 cm) for these trials is 0.98 and 0.9, respectively. Since the condition $\sigma _{PIB}^2 \le 1$ is fulfilled, the FE distribution is no longer suitable for describing scintillations at a small aperture, and the best approximation is the FG (Fig. 5(a),(b)). It should be noted that the GD, while not being the best approximation, also satisfies the Pearson criterion (Fig. 5(a)).

 

Fig. 5. Chi-square metric of (a) trial 10, (b) trial 11; and the scintillation index of (c) trial 10, (d) trial 11, as functions of the aperture size for the case of moderate in-pixel scintillations.

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The observed behavior of the chi-squared criterion is explained by the change of the type of the experimental PDF, which now tends to zero for small power values (Fig. 6(a),(b)), whereas the FE distribution was proposed to describe the case $\sigma _{PIB}^2 > 1$, in which this probability increases near zero. The transition from FG to GD occurs at the same critical value of the scintillation index $\sigma _{\textrm{PIB,crit}}^2 \approx 0.75$ as in the strong scintillation regime. It should be noted that the LN distribution turns out to be the best approximation of the experiment for trial 11 approximately at the time of transition from FG to GD (Fig. 5(b)).

 

Fig. 6. Chi-square metric of (a) trial 10, (b) trial 11; and the scintillation index of (c) trial 10, (d) trial 11, as functions of the aperture size for the case of moderate in-pixel scintillations.

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Since there is no drastic increase of the probability near zero PIB values in the considered scintillation regime, there is no need to decrease the bin size to describe the part of the PDF function near zero. For convenient presentation, we plotted the PDF with N = 50 for trial 10 (Fig. 6). With this choice of N, the histogram step becomes so large that the tendency of GD to zero is smoothed for low intensity values (Fig. 6(a)).

FG is the best approximation for the point aperture (Fig. 6а). At the time of transition ($\sigma _{\textrm{PIB,crit}}^2 \approx 0.75$) the GD and FG distributions demonstrate the almost complete coincidence (Fig. 6(b)), as in the other considered cases (Fig. 4(d)). With a further increase of the aperture, the LN and GD become closer to each other (Fig. 4(c),(d)).

4.3 Weak scintillation regime

Consider two examples of PDF comparison for the case that the scintillation index at the point aperture is $\sigma _{\textrm{PIB}}^2 \ll 1$. The results shown in Fig. 7 demonstrate that the FE and FG proposed for the cases of strong and moderate scintillations are far from the experimental results in the case of weak scintillations, while the GD well describes the experiment. We need to mention that LN is also very close to the experiment data.

 

Fig. 7. Chi-square metric of (a) MN12, (b) trial 14, and experimental and analytical PDFs of trial 14 for the aperture diameter d of (c) 0.0554 and (d) 5.98 cm for the case of weak in-pixel fluctuations.

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A few PDFs plotted with N = 50 illustrate the similarity between the LN and GD (Fig. 7(c),(d)). The other two PDFs distributions are omitted because they differ significantly from the experiment.

4.4 PDF type transformation: aperture and scintillation index characteristic values

The critical spatial scale, at which the transition between the FE and FG occurs, was already discussed in Section 4.1. Here we give the estimates of this scale for all the measurement series, in which the strong scintillation regime was observed at the point aperture (Fig. 8(a)). One can see that the estimation of the critical size ranges from 0.6 to 1.6 of the size of isoplanatic region. We assume that the scatter of values is more due to the estimation error than to physical differences in the observed series. The error of estimation is a sum of the error in determining the aperture, at which the transition occurs, (doubled pixel size) and the error in determining the isoplanatic region, which directly depends on the error of the scintillometer data. The first contribution can be roughly estimated as 20% of the measured value with allowance for the fact that the radius of the considered apertures is usually about 5-7 pixels (this value is given in Fig. 3(a),(b),(e),(f) for trials 1, 5, 7, 27).

 

Fig. 8. Values of (a) d/a, (b) the scintillation index which correspond to the aperture size of the transition between FE to FG (see Fig. 4(b)) and FG to GD (see Fig. 4(d)) as the best approximations, respectively.

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The contribution of the error of determining $C_n^2$ is a sum of the scintillometer error, which can be estimated as roughly 10% by analysis of short term scintillometer fluctuations, and the inhomogeneity of $C_n^2$ values during the measurement. Due to the latter, the accuracy of estimates worsens as the observation time increases. On the other hand, the longer observation time allows us to increase the number of frames and accumulate the needed statistics. Consequently, an increase in the measurement time can either improve or worsen the estimate of d/a. Thus, the scatter of the values shown in Fig. 8(a) is expected.

We also found that, with increasing aperture size, the transition from FG to GD distribution always happens at the same critical value of the scintillation index, namely $\sigma _{\textrm{PIB,crit}}^2 \approx 0.75$, which is observed at different apertures for different turbulence strengths. The estimations performed for the trials containing this transition are given in Fig. 8(b).

Despite the mentioned estimation errors, we believe that the obtained rough evaluations reflect the observed patterns, namely, the relation between the qualitative change in the experimental histograms (corresponding to the transition between the analytical distributions) and the characteristic values of the aperture size and scintillation index.

4.5 Results overview

The transition between the different PDFs as the best approximation to experiments does not mean that a distribution, which is not the best in some range, is necessarily poorly suited to describe experimental results. The region of good approximation of the experiment by an analytical distribution is often much larger than the region of the best approximation. Table 2 gives the values of the normalized diameters of the receiver apertures and the corresponding scintillation indexes for all the considered cases in the ranges, where the considered analytical distributions provide good approximation ($\chi _{ns}^2 \le 1$) for the experimental data with three exceptions discussed below. In Table 2, the boundary values corresponding to the minimal or maximal considered aperture are given in red. Three numbers of the trials, for which we had to increase the threshold value of $\chi _{ns}^2$, are given in italic (trial 6, trial 7, trial 11).

Table 2. Experimental results

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It should be noted that in three trials of the 27 considered, we obtained approximation results, for which the criterion $\chi _{ns}^2$ was slightly larger than unity, as can be seen from Fig. 3(e) for trial 7. Mostly, this is the result of scintillation index value around unity. Nevertheless, we had to change the threshold value of $\chi _{ns}^2$ up to 1.2 for trial 7 and trial 11 and up to 1.5 for trial 6. All the other results given in Table 2 were obtained for $\chi _{ns}^2 \le 1$.

A surprising result was obtained for trial 20, for which three distributions (FE, FG, and GD) turned out to be a good approximation. We believe that this is due to the insufficient sample duration (4 minutes, 480 frames, as can be seen from Table 1 and Fig. 2). Correspondingly, with the chosen 200 bins for analysis, which was dictated by the need to describe the low-intensity part in the strong scintillation regime, the number of PIB values falling into the bins were insufficient.

In the trials which have sufficient numbers of PIB values per bin, the FE distribution demonstrates a good fit for small apertures (d ≤ a) and the scintillation index $\sigma _{PIB}^2 > 1$. For larger apertures or smaller values of the scintillation index, the FG begins to provide a good approximation starting from 0.2a-0.5a. Thus, even under the condition that the turbulence strength is not known exactly beforehand, its rough estimate can allow us to successfully apply both distributions, passing to the FG somewhat earlier than the optimal transition (as the data in Fig. 8 suggest). However, we should keep in mind that for the scintillation index $\sigma _{PIB}^2 > 1$ scintillation is accompanied by the high probability of appearance of dislocation points [22] (points with zero intensity values). The FG distribution provides zero value for the probability of dislocations. Consequently, its use for estimating the probability of signal fading in communication systems will lead to completely incorrect results. To the contrary, the use of the FE distribution for the conditions of $\sigma _{PIB}^2 < 1$ will lead to the significant overestimation of the probability of low PIB values. Therefore, it should be kept in mind that the close values of the criterion $\chi _{ns}^2$ for these two distributions under the condition $\sigma _{PIB}^2 \approx 1$ are associated with the choice of histogram bins nearby zero intensity values. Therefore, when using them for practical purposes, it is essential to understand the physical processes that cause scintillation and the role of the averaging effect of the aperture on power scintillation.

It should be noted that the FG performed well up to scintillation index values of 0.5-0.7, while the GD demonstrated a good result starting from 0.8 and to the minimal considered values of about 0.05. The GD distribution has never been a good approximation for $\sigma _{\textrm{PIB}}^2 > 1$ (except for the case of trial 20 discussed above). The LN distribution performed well mainly at $\sigma _{\textrm{PIB}}^2 \le 0.3$, with the exception of trial 11 ($\sigma _{\textrm{PIB}}^2 \le 0.8$), whose $\chi _{ns}^2$ is shown in Fig. 5(b). The data presented in Table 2 summarize the dependence of the acceptable approximation of experimental data by analytical distributions on the scintillation regime.

5. Conclusions

Consideration of the PDF for the aperture size ranging within d = [0.0554-14.4] cm and, correspondingly, d/a = [0.009-32.8], and the turbulence characterized by the coherence radius ${r_0} = [{1.4 - 18.6} ]$ cm has allowed us to formulate some regularities being of both basic and practical interest.

First, it was shown that the experimental histograms can be well approximated by the analytical one-parameter distributions depending only on the scintillation index measured at the receiver for the considered scintillation regimes characterized by the Rytov index for plane waves $\beta _0^2(plane) = [{0.35 - 25.4} ]$ and considered range of the receiver aperture sizes from $d \ll a$ to $d \gg a$.

Second, it was shown that at the aperture having the size comparable with the isoplanatic region and smaller, scintillations can be described by the fractional exponential distribution in the regime of strong scintillations ($\sigma _{\textrm{PIB}}^2 > 1$) and by the gamma distribution in other cases ($\sigma _{\textrm{PIB}}^2 \le 1$), as well as by the lognormal distribution at $\sigma _{\textrm{PIB}}^2 \ll 1$.

Third, the fractional gamma (FG) distribution was proposed. It is a one-parameter distribution providing a good approximation for the non-point aperture (d ≥ a) under conditions of strong and moderate scintillations.

Fourth, it was shown that for the regime of strong scintillations ($\sigma _{\textrm{PIB}}^2 > 1$) the effect of aperture averaging begins to manifest itself at the aperture size equal to the size of the isoplanatic region а. As a result of the aperture averaging, the transition from the fractional exponential distribution to the fractional gamma distribution occurs at d/a > 1. FG distribution better approximates experimental histograms at the large apertures (d/a > 1), because it takes into account the fact that the probability of zero power values becomes equal to zero due to the averaging effect of the aperture.

Fifth, it was shown that the transition from the FG to the GD distribution for non-point apertures is determined by a critical value of the scintillation index $\sigma _{\textrm{PIB,crit}}^2 \approx 0.75$ measured at the aperture. Thus, at $\sigma _{\textrm{PIB}}^2 > \sigma _{\textrm{PIB,crit}}^2$ the fractional gamma distribution provides the best approximation for the PDF, whereas at $\sigma _{\textrm{PIB}}^2 < \sigma _{\textrm{PIB,crit}}^2$ it is changed to the gamma distribution.

Note that the considered set of the analytical one-parameter distribution functions FE, FG, and GD provide approximately the same good level of approximation for all aperture sizes. This set turned out to be sufficient to approximate the PDF for all experimental series of random power scintillations observed in the series of experiments described above. In the range of the transition from one distribution to another, these distributions have a very similar form, which ensures the smoothness of these transitions.

Appendix: Comparison of result analysis based on of Pearson criterion and Anderson Darling test

In this paper, the Pearson criterion was used for analysis of PDF fitting. This criterion made it possible to carry out a quantitative analysis of the PDF at the arbitrary intervals of intensity values. Such detailed analysis allowed the authors to propose a new distribution (fractional exponential distribution). At the same, there is another powerful method, Anderson Darling (AD) test, which was used for analysis of PDF fitting [15]. We would like to compare here analysis of the same data set (trial 7) performed by using different methods, namely Pearson criterion and AD test. This should help readers to compare results presented in this paper with the results of other publications which are based on using AD test for the data analysis.

Here we provide only short description of AD test while more details on this method and its implementation for PDF analysis can be found in [15]. The value of A, characterized the AD test statistic, is calculated as

(18)$$A ={-} n - \sum\nolimits_{k = 1}^n {\frac{{2k - 1}}{n}[{\ln ({1 - F({y_{n - k + 1}}) + \ln (F({y_k}))} )} ]} . $$

If A is smaller than some critical value α_CRIT (for example α_CRIT= 2.49 for the significance level 0.05) it means that, according to test statistic, there is no sufficient evidence to reject the PDF. Following [15] we also limit the number of points used in calculating and used only 672 points (one fifth of the points).

The A is plotted versus the aperture diameter in Fig. 9(a). For the comparison, the corresponding dependence of chi-squared metric on the aperture diameter is plotted in Fig. 9(b). Please note, that Fig. 9(b) is a copy of Fig. 3(e). Even the chi-squared metric was useful for quantitative analysis of PDF at the arbitrary intervals of intensity values, one can see that dependencies of A, the AD characteristic, presented in Fig. 9(a) are more smooth and easy to interpret than χ² dependencies presented in Fig. 9(b).

 

Fig. 9. Dependencies of (a) α, and (b) χ² on aperture diameter for considered PDFs.

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One can see that the change of FE to FG as the best approximation of the experimental histogram happens at the aperture diameter reach size of d = 1.1a, i.e. the isoplanatic size in the error limits. The change of FG to GD as the best approximation also happens at approximately the same aperture diameter, d = 5.7 cm for AD test and d = 6 cm for Pearson criterion, which approximately correspond to $\sigma _{PIB}^2 = 0.75$ according to Fig. 3(g). Thus, the results of the AD test support the conclusions made in this paper by using Pearson criterion.

Funding

US Office of Naval Research (N00014-17-1-2535); Russian Foundation for Basic Research (18-29-20115\18).

Acknowledgments

The authors are thankful to Dr. M.A. Vorontsov and Dr. T. Weyrauch for useful discussions, constructive remarks, and interest to this study.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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