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Abstract
Statistical probability distributions characterizing received optical power fluctuations, or scintillation, enable performance predictions of space-to-ground optical communication systems. In this paper, we present measurements of stellar scintillation over a wide range of elevation angles and turbulence conditions collected simultaneously with a 5 cm and 40 cm telescope aperture, which allows a comparison between minimal and significant aperture averaging conditions. The measured data is compared to a reasonable set of candidate probability distribution functions (PDFs), including lognormal, which is most often cited in the literature for weak to moderate scintillation. For scintillation indices (SIs) less than about 0.2, the Nakagami-m distribution provides the best representation of the collected data for both apertures and imposes a greater lasercom link penalty than a lognormal distribution, which has been inaccurately implemented as the default probability distribution in the literature. For larger values of the SI, the scintillation is best characterized by a Gamma-Gamma distribution. Additionally, the measured temporal covariance for weak to moderate scintillation conditions is found to be in reasonably good agreement with theoretical predictions.
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1. Introduction
During the last decade the international technical community has expressed interest in deploying small satellites equipped with optical terminals for communication links with ground stations in rural and urban locations [1–3]. However, in the downlink from space-to-ground, such systems can be significantly degraded by atmospheric turbulence-induced received power fluctuations. Turbulence-induced fading can degrade the bit error rate (BER) performance, requiring extra transmitted power to overcome the scintillation power penalty. As is well known, the temporal fluctuation period of the scintillation is sufficiently long so that the signal strength is constant over a single bit period for data rates of interest. Thus, in the presence of a fluctuating signal, the required BER is considered a conditional probability that must be integrated over the probability density function (PDF) of the fluctuating signal to obtain the corresponding unconditional BER, from which the scintillation power penalty can be readily obtained.
With the growing interest in deploying optical ground stations for communicating with satellites, there is a corresponding need to characterize the atmospheric environment for the intended ground site locations. This includes identifying the scintillation probability density functions (PDF) that most suitably predict received power statistics under a variety of atmospheric and operational conditions. While the use of satellite assets might be considered the most appropriate for measuring scintillation (especially for temporal considerations), they may not be readily available or may impart systematic errors or other challenges. Thus, using readily available and “willing” stellar sources is a viable approach to start with. Astonishingly, however, empirical measurements of stellar scintillation statistics are limited to only one publication in the literature [4]. In Ref. [4], a lognormal PDF fit was reported to be in good agreement with the observed 550 nm data obtained with a small 5 cm telescope aperture diameter under weak scintillation conditions (SI = 0.058). Note that no other probability distributions were considered for comparison to the data nor were any empirical results presented for larger apertures where aperture averaging is known to produce significant smoothing of the collected power fluctuations.
When a 5 cm aperture is used there is very limited aperture averaging and the telescope behaves essentially as a “point aperture receiver”. In practice, however, ground-based optical terminals will implement larger aperture telescopes (e.g., 30–80 cm), which mitigate scintillation by aperture averaging. Except for Ref. [4], the literature is void of empirical measurements of aperture averaged stellar scintillation statistics, and it is generally assumed that the scintillation statistics obey a log-normal distribution in the weak to moderate turbulence regime. In this paper we present scintillation statistics obtained from simultaneous stellar measurements using two different size apertures. The chosen 5 and 40 cm diameter telescopes provide test cases for minimal and significant aperture averaging, respectively. Measurements for this investigation were taken over a wide range of elevation angles and atmospheric turbulence conditions.
Here, as well as elsewhere in the literature, we characterize the magnitude of the stellar signal fluctuations by the scintillation index (SI), which is a dimensionless quantity defined as the variance of collected power divided by the square of the mean power:
In addition to measuring the SI, we compare the PDFs of our measured data to reasonable candidate PDF distributions, including lognormal and gamma-gamma. As is demonstrated below, the preferred distribution, which most accurately describes the scintillation data for the 5 and 40 cm apertures, are different, as well as unanticipated. It is interesting to note that although the lognormal distribution was expected for weak scintillation (i.e., SI <<1), it never provided the best model description for our measured data. Further, we do not claim that the present study is definitive in solving the long-standing problem of obtaining the correct stellar scintillation PDF. Rather, our intent is to provide the system analyst with an “engineering model” that is sufficiently accurate in describing received power fluctuations over a broad range of elevation angles and turbulence conditions.
In the following discussion, the terms “signal” and “signal power” are used interchangeably. The experimental setup and considerations are provided in Sec. 2. SI results from the 5 cm and 40 cm data are presented in Sec. 3. In Sec. 4 we present representative results for the temporal autocorrelation function obtained with simultaneous collects using the two apertures. Section 5 covers the candidate probability distribution functions (PDFs). In Sec. 6, the PDFs of the 5 and 40 cm aperture data sets are compared to the candidate distributions. In Sec. 7, we give an illustrative example of the scintillation power penalty for the OOK modulation format for both apertures and the candidate PDFs. In Sec. 4, the measured temporal covariance for weak to moderate scintillation conditions was compared to theoretical predictions, where reasonably good agreement was obtained. As such, this paper should be of interest to both the free-space optical communications and the remote sensing communities.
2. Experimental setup, data acquisition, and noise considerations
Scintillation data was collected at The Aerospace Corporation optical ground station located in El Segundo, CA which is approximately at sea level. As shown in Fig. 1, the ground station incorporates a single two-axis gimble system that supports a 40 cm diameter main telescope and an auxiliary 5 cm diameter telescope. The focal lengths of the telescopes are 3.1 m and 20 cm, respectively. The two telescopes are separated by approximately 40 cm and are each outfitted with an InGaAs camera (Xenics Cheetah 640) containing a 640 × 512 array of 20-µm pixels positioned at the focal plane. Data was captured over a 12 – 50° range of elevation angles. For the 5 cm aperture, data was typically recorded at rates between 390 and 1000 frames per second (fps), whereas for the 40 cm aperture, data was collected between 500 and 850 fps. Integration times were set to be commensurate with the frame duration and data capture overhead and were generally between 50 and 70% of the collection window. All frame rates implemented were sufficient to capture the dynamics of the scintillation.
Fig. 1. Telescope pair used for scintillation measurements. The 5-cm diameter telescope resides approximately 40 cm below the larger, 40-cm diameter telescope.
Using the windowing capability of each camera, a desired a subset of pixels could be designated to form the active collection area of each array, or region of interest (ROI). The size of the ROI was chosen to include illuminated pixels and a small border of non-illuminated pixels to accommodate spot movement due to platform jitter and atmospherics, and intensity profile changes due atmospheric induced wave-front distortion, which could lead to erroneous scintillation measurements. Minimizing the ROI, while avoiding clipping, was critical for maximizing the signal-to-noise ratio and enabling faster processing and higher frame rates. Temporal scintillation waveforms were generated by summing and averaging the intensity collected by all the pixels within the ROI and subtracting a background level, determined by off-pointing the telescope so the stellar image was not contained within the ROI. In this paper we are concerned with scintillation statistics (PDF) and thus normalize the time varying stellar signal (camera pixel readings) by the mean value over the interval of the collect. The normalized values are dimensionless and are equal to what would be derived from the absolute energy or power. Likewise, the SI is defined as a normalized dimensionless quantity that can be derived directly from the camera readings.
Scintillation data using the 5 cm aperture scope was collected spanning a period from December 2018 through February 2019, and simultaneously using the 5 and 40 cm aperture scopes during May 2020. As described above, the later data was taken with a pair of Xenics Cheetah cameras, whereas earlier 5 cm data was collected with either a Xenics Cheetah or Bobcat camera, which have similar characteristics. For the dual-aperture experiments, data sets nominally contained 60,000, and occasionally 120,000 points, thus being collected over periods of ∼2 to 4 minutes for each elevation angle. Background collects, usually taken between successive scintillation scans, contained the same number of points to enable accurate determination of the baseline signal level. Successive collects within a series were recorded within a few minutes of each other as the elevation angle (angle from the horizon) progressed slowly with time. Experiments were run during clear nights to avoid intensity drops due to time-varying cloud coverage, which can be problematic in coastal areas. Our collection band is centered around 1064 nm to be consistent with our cubesat downlinks (as first reported in Ref. [2]). The received power is filtered on both apertures with a 50 nm spectral bandpass filter. Selected stars with sufficient IR emission included Betelgeuse, Arcturus, and Sirius. Analysis of the scintillation PDFs is presented in Sec. 6 below.
The noise background for the experiment was derived from collects taken while pointing the telescope away from the star of interest and toward an empty nearby region of space. These measurements were necessary to characterize the noise contributions and intrinsic DC offset, or bias, of the collection system, which is critical for determining the average value of the scintillation intensity used to calculate the SI. For our system the dominant noise source is the detector array readout noise, which was normally distributed. Two stressing situations arise when (1) the collected signal is weak and the scintillation causes intensity fluctuations to drop into the noise floor of the detector as shown in Fig. 2, and (2) the scintillation induced fluctuations are on the order of the detector noise fluctuations. The former condition is most likely to occur for low elevation collects (larger SIs) with the 5 cm aperture. Under certain strong scintillation conditions with this aperture, subtraction of the average background level, sometimes yielded a small quantity (< 0.7%) of negative values. These points were removed from the data set and so not included in the analysis. Calculations with and without the negative values varied the SI value at most by a few percent for SIs near unity and up to ∼7% for SIs > 1. This condition was never observed for the data collected with the 40 cm aperture as exemplified in Fig. 3. The latter condition was a concern for very weak turbulence conditions which could render the signal statistics susceptible to coloration by the detector noise. For our experiments, the variance of the collected signal (scintillation plus detector noise), was generally observed to be an order of magnitude or greater than the variance of the detector noise, and thus signal fluctuations were representative of the SI. (The smallest observed ratio between the scintillation-plus-noise and noise-only variances was approximately seven.)
Fig. 2. Received optical power under moderately strong scintillation conditions, SI = 0.50 with 5 cm aperture collected at 395 fps for Arcturus. Lefthand plot displays the raw signal and background (detector noise) with their respective variances. Righthand plot shows the normalized signal power post background subtraction on a log scale (dB).
Fig. 3. Simultaneous collect with a 40 cm aperture at 850 fps for Arcturus. Lefthand plot displays the raw signal and background (detector noise) with their respective variances. Righthand plot shows the normalized signal power post background subtraction on a log scale (dB).
3. Scintillation index results: single and simultaneous aperture collects
Figure 4 and Fig. 5 displays that range of SIs we observed for our ground station in El Segundo using the 5 and 40 cm apertures, respectively. As others have noted [4], the turbulence strength can vary over the course of a few minutes. Small changes in the turbulence strength on the order of 10% over this time scale were common, although larger changes were also observed. The figures include plots of the theoretically predicted SIs vs elevation angle using the HV-5/7 atmospheric model. This calculation requires the selection of a PDF (discussed in Sec. 4) to generate an SI from the Rytov variance that is derived from the turbulence profile integral over the optical path length.
Fig. 4. SIs observed for the 5 cm aperture telescope collects between 2018 and 2020. HV-5/7 theoretical curve includes an aperture averaging factor derived from analysis in Ref. [5]. Aperture averaging for the 5 cm aperture is essentially negligible.
Fig. 5. SIs observed for the 40 cm aperture telescope collects during 2020. HV-5/7 theoretical curve includes an aperture averaging factor derived from analysis in Ref. [5].
The SI behavior follows the general trend of the HV-5/7 model for both apertures, but there is a large variation of SI values over the course of time. The SIs appear skewed toward larger than expected values at the lower elevation angles in the 5 cm data, where they approach and exceed unity. While most of the collects were taken in a nearly due west direction over the Pacific Ocean, we also recorded data at other azimuthal angles. Overall, we did not observe a correlation between the azimuthal angle and the strength of the turbulence.
One main objective of this effort was to collect stellar scintillation using two apertures, 5 and 40 cm, simultaneously to get an unambiguous correspondence between the underlying condition/physics of the atmospheric channel and the mitigation of scintillation using an aperture size reasonable for a ground station. The separation between the two apertures was approximately 40 cm so they are correlated within the 50 nm collection bandwidth. Representative data showing the SI reduction, presented in Tables 1–4 below for weak, moderate, and strong turbulence conditions, is a factor of 2-3 smaller than would be derived from the scintillation reduction reported in Ref. [4]. For most cases we observed about an order of magnitude decrease in the SI between the two apertures. The aperture averaging factor we observed for large SIs does not appear consistent with values that might be derived from Ref. [5].
4. Temporal autocorrelation: ratio of the scintillation autocorrelation times for simultaneous data collects using two different aperture sizes
In this section we compare the temporal autocorrelation functions derived from data collected simultaneously from the 5 and 40 cm apertures. Figure 6 shows the temporal autocorrelation functions plotted for scintillation data collected on a single evening for Arcturus over elevation angles ranging from 29 to 37 degrees (see Table 2) and azimuthal angles near -90 deg (0 and -90 deg azimuth angles are due North and West, respectively). Table 5 summarizes the experimental and theoretical autocorrelation time ratios for the two apertures (see Appendix A) which are in good agreement. The last entry in Table 5 displays the results for Betelgeuse collected under different conditions (see Table 3, az = -96 deg) that show nonconformance to the theory, which is not applicable when the scintillation is strong. While it was beyond the scope of this work to make systematic measurements of the correlation times with azimuthal direction, we did observe correlation times in the range of ∼5-10 ms for the 5 cm aperture. Correlation times for the 40 cm aperture were about 3 times larger, consistent with weak scintillation theory in Ref. [5] (see Appendix A). Physically, it is expected that the temporal autocorrelation time increases for increasing aperture size. This is because spatial scales less than the order of the dimensions of the aperture are averaged out, and only spatial scales larger than the aperture contribute, which take a longer time to pass over the aperture.
Fig. 6. Temporal autocorrelation function example. The lefthand plot compares data to weak scintillation theory [6] for the 5 cm aperture at an elevation angle of 29°. The righthand plot shows the data for the simultaneous collects with the two apertures. The ratio of the correlation times for the two apertures for this collect was measured to be 3.7 vs. a theoretical value of 3.3 derived from the analysis in Ref. [5].
Table 1. Data collected for Arcturus on 05/22/2020 (weak scintillation)
Table 2. Data collected for Arcturus on 05/15/2020 (moderate scintillation)
Table 3. Data collected for Betelgeuse on 05/03/2020 (strong scintillation)
Table 4. Data collected for Betelgeuse on 05/01/2020 (strong scintillation)
Table 5. Temporal autocorrelation times for simultaneous aperture collects
5. Stellar scintillation statistics: candidate PDF models
Of the several proposed models in the literature describing scintillation effects under various conditions [7–10], we choose four candidate probability distributions for comparison to our scintillation measurements: lognormal distribution (LN), Gamma-Gamma distribution (GG), Inverse Gaussian Gamma distribution (IGG), and Nakagami-m distribution (NK). Historically weak to moderate scintillation conditions have been assumed to follow a LN distribution [6]. As alluded to in the Introduction, Ref. [4] is the only paper in the literature we are aware of that reports empirical measurements of stellar scintillation statistics and shows that a lognormal fit was in good agreement with observed 550 nm data under weak scintillation conditions obtained with a 5 cm telescope aperture. However, no comparison to other distributions was performed. The GG distribution has been proposed to model intensity fluctuations from weak to strong scintillation conditions [11]. The recently introduced IGG distribution has also been proposed to model intensity fluctuations from weak to strong scintillation conditions [12]. The NK model was chosen as a candidate here because it has become the preferred model that describes radio frequency scintillation effects due to atmospheric turbulence in the ionosphere [13]. It is important to note that the Nakagami-m is a purely phenomenological description. It was developed to fit experimental data; no one has yet succeeded in deriving it from first principles, as has been done for the log-normal distribution that was derived for optical scintillation for horizontal propagation in the weak scattering regime. In as much as optical downlink scintillation is primarily due to turbulence in the (high altitude) tropopause we feel that the NK distribution is a legitimate model to consider. An exponential Weibull (EW) distribution has been proposed that is claimed to offer an excellent fit to experimental data (and simulations) under all aperture averaging conditions, and under weak and moderate turbulence conditions [14]. However, we exclude this distribution from consideration because, in contrast to the PDFs given above, the parameters in the EW distribution are not linked to the scintillation index or any physical parameter.
The candidate PDFs are defined in terms of collected signal power ($x$) normalized to its mean. For a given measured scintillation index the PDFs for $x \ge 0$ are given by Eqs. (2–5).
In Eqs. (4) and (5) ${K_n}(\cdot )$ is the modified Bessel function of the second kind of order n. For a measured SI and assuming either the GG or the IGG is a valid distribution we perform a least squares fit for $\alpha $ and $\beta $ while satisfying the relationship SI = 1/α + 1/β + 1/αβ.
Thus, the parameters for each of the candidate PDFs given above are such that these test distributions have a mean-normalized signal power of unity, and a corresponding normalized variance equal to the measured SI. Additionally, to ascertain which of the candidate distributions best describes received power fluctuations corresponding rms errors weighted by the PDF of the data are presented. Weighting the error estimate by the PDF of the data tends to emphasize the part of the PDF near the peak and deemphasize the tails, as data near the peak is more likely to highlight differences in the PDFs. In this way, low probability fades approaching the noise floor or stressing the dynamic range of the detection system would have less influence than the more frequently occurring moderate fluctuations.
6. Results: comparison of measured data with candidate distributions
Figure 7 shows representative experimental PDFs obtained simultaneously with the 5 and 40 cm aperture telescopes. The impact of aperture averaging leads to a narrower collected signal power distribution for the larger aperture and reduction in the SI as discussed above.
Fig. 7. Comparison of experimental probability distributions of scintillation data captured with 5 cm and 40 cm aperture telescopes.
The observed normalized scintillation distribution derived from the 5 cm aperture data is compared with the four model PDFs in Fig. 8. The SIs for all four examples in the figure exceed 0.2. (For SIs below 0.2 we observe a shift in the PDF as discussed below). Comparisons of the theoretical curves to the entirety of the 5 cm data exhibiting an SI > ∼0.2 consistently show that the GG PDF provides the best model representation as indicated by the RMS error values displayed in the plots. Interestingly we observe that as the SI dips below 0.2, whether it be from the small aperture (with minimal aperture averaging) or the larger 40 cm aperture, the NK model offers the best model for all the data collected. Representative examples of the low SI conditions for both aperture diameters are displayed in Fig. 9, where the “effective” SIs are less than 0.2. Overall, our data suggests a generalization that for effective SIs below ∼0.2, the NK PDF is the most appropriate model, regardless of the aperture size. It is unclear, however, whether the NK model best represents scintillation statistics for the large apertures with significant aperture averaging over a broader range of effective SIs beyond 0.2 since we had no data in that regime.
Fig. 8. Comparison of the probability distribution of scintillation data captured with 5 cm aperture telescope with PDF models over a range of SIs. Data and theory plotted in linear units.
Fig. 9. Comparison of the probability distribution of scintillation data captured with the 5 and 40 cm aperture telescope with SI values < 0.2. Data (blue curve) and PDF models plotted in log units. Weak scintillation conditions for the 5 cm aperture collects (upper plots) compared with aperture averaged results with the 40 cm aperture (lower plots). For the two lower 40 cm plots the corresponding SIs for the 5 cm simultaneous collects were 1.24 and 0.71, respectively. The Nakagami-m PDF is the best fit to the data in all cases.
7. Scintillation link penalty examples
The fluctuation of the received optical signal, scintillation, due to atmospheric turbulence results in a link penalty or a degradation in performance. It is interesting to compare the effect of the different PDF models for both apertures. In Fig. 10 we plot the BER for an example on-off key (OOK) link with a fiber pre-amplifier receiver versus the quantity Eb/N0 (energy per bit/noise power spectral density), which is effectively the signal-to-noise ratio of the signal at the input to the detector. (See Appendix B for derivation of the curves in Fig. 10.) In the figure we compare the predicted impact of signal fading, or scintillation, for the candidate PDFs considered in this paper for both apertures. As shown in the plots for the 5 cm aperture, there are significant differences in the predicted performances for the select PDFs at BER levels < 0.01. The lognormal distribution is the most benign whereas the Nakagami-m yields the greatest link penalty. The magnitude of the scintillation link penalty and the differences between the PDFs are significantly reduced for the 40 cm aperture where aperture averaging reduces the depths of the fades. As shown in Fig. 10, the penalty differences between the PDFs are most pronounced at larger SIs and diminishes as the SI decreases. Eventually at very low SIs the curves become indistinguishable. Finally, these results apply to other waveforms such as QPSK and PPM. Other factors, such as forward error correction and interleaving, play a role but are not considered for this basic illustration.
Fig. 10. Example impact of the scintillation on the BER performance of a fiber pre-amplified OOK link using the four candidate probability distributions for the 5 cm (left) and 40 cm (right). The dotted curve in each plot represents the BER performance in the absence of scintillation. The example SI values of 0.79 and 0.066 for the upper left and lower right plots were derived from data collected simultaneously with the 5 and 40 cm apertures, respectively, whereas the SI values for the other two plots were determined from different collects and are included for illustrative purposes. For the lowest SI value of 0.066, the gamma-gamma and the inverse gaussian gamma BER curves are virtually identical to the Nakagami-m BER curve and the lognormal BER curve, respectively, so only the LN and NK curves are shown.
8. Concluding remarks
Atmospheric induced scintillation of collected stellar light was evaluated over a wide range of elevation angles at our optical ground station in El Segundo, CA using a pair of 5 and 40 cm aperture receivers. This configuration enabled us to characterize the strength and statistics of the scintillation with substantial and minimal aperture averaging. The experimental results were evaluated against four candidate PDF models. Measurements and analyses using simultaneous data collects from disparate apertures for space-to-ground propagation are notably lacking in the literature. For the smaller aperture, the SI vs. elevation angle essentially followed the HV-5/7 model and path length scaling but exhibited a significant spread in SI values at lower elevation angles. While the standard HV-5/7 model may provide a good baseline for predicting scintillation strength, it clearly underestimated the severity experienced at our location, especially at low elevation angles.
A comparison between simultaneous collects with both apertures showed an order of magnitude reduction in the scintillation strength for the larger aperture. This result is notably smaller than would be predicted for this aperture pair based on a previous report in the literature. While the impact of aperture averaging on the scintillation strength is well known, its impact on the scintillation statistics is not. Further, it is generally assumed in the literature that the scintillation statistics are lognormal. We, however, observed behavior that was better represented by a GG distribution for a non-aperture averaging condition (5 cm aperture) where the SI was above 0.2. For lower SIs with the 5 cm aperture and for all data collected with the 40 cm aperture, which exhibited significant aperture averaging, the NK distribution was found to be a more suitable model. We believe this is the first time this distribution has been used in the literature to characterize optical scintillation statistics.
We further evaluated the predicted BER performance degradation of an example optical communications link in the presence of scintillation. The analysis showed that the link penalties are, as expected, dependent on the PDF model and that the differences were most pronounced at high SIs, which would tend to occur at low elevation angles and small apertures. As the SI decreases either by increasing the elevation angle or receiver aperture size, the link penalty differences between the candidate PDFs decreases. Thus, designers of space-to-ground optical communication systems must not only consider the scintillation strength, but also the distribution of fades to anticipate the link performance. Further consideration must be given to the variability of the atmospheric conditions for a given ground site which impacts the scintillation and wavefront distortion, a key factor beyond the scope of this paper.
Appendix A: signal power temporal autocorrelation
Here we consider weak to moderate scintillation conditions where theory based on the Rytov approximation is valid [10]. This was the case for May 15, 2020 data, where simultaneous measurements of stellar scintillation statistics were obtained (from Arcturus) for elevation angles between 29 and 37 degrees. The scaling relation for the aperture averaged temporal covariance autocorrelation function is given by Eq. (3.6) of Ref. [5]. For simultaneous measurements at a fixed elevation angle, the ratio of the correlation times for aperture radii ${a_1}\; $ and ${a_2}\; $ is independent of the wind speed normal to the line of sight and is given by
$C_n^2 $ is the index structure constant profile, and $h(s )$ is the height above ground at a distance s along the propagation path, which for a spherical earth is given by
where the mean radius of the earth ${R_E}$ is 6371 km, H is the height above ground of the telescope aperture, and $\vartheta $ is the elevation angle. For downlink propagation and weak to moderate scintillation conditions the predominant turbulent eddy size involved in producing scintillation is on the order of the Fresnel size, $\sqrt {Leff/k\; } $, where Leff is the wavelength independent effective scintillation propagation range [15].The signal power data temporal covariance, obtained simultaneously from both telescopes, is presented in Fig. 6. Examinations of these figures reveal that the correlation time for the 40 cm telescope is larger than for the 5 cm telescope. Physically, it is expected that the temporal correlation time increases for increasing aperture size. This is because spatial scales less than the order of the aperture dimensions are averaged out, and only spatial scales larger than the aperture contribute, which take a longer time to pass over the aperture. The measured ratios of the 40 cm to 5 cm telescope correlation times are presented in Table 5 along with the corresponding theoretical ratios obtained from Eq. (A.1). Because the measurements were obtained near sea level over flat terrain, the theoretical results were based on the often-used Hufnagel-Valley 5/7 turbulence profile [16]. Examination of Table 5 shows generally good agreement between our measurements at weak to moderate scintillation conditions and the theoretical predictions given in Ref. [5].
For all cases of practical concern, the optical wave front impinging on the turbulent atmosphere can be considered as a plane wave. In the Rytov approximation, it can be shown that the plane wave temporal covariance function, $C(\tau ),$ is given by
Appendix B: BER curves
In the presence of a fluctuating signal, the BER (bit error rate) is considered a conditional probability that must be averaged over the probability density function (PDF) of the signal fluctuations to obtain the unconditional BER. As a result, the BER curve in the presence of scintillation is expressed as
where x, is the ratio of the energy per bit, ${E_b}$, to the noise power spectral density, ${N_0}$, or ${E_b}/{N_0}$, and ${p_N}(u )$ is the PDF of the fluctuating signal power. Further, for on-off key link with a fiber pre-amplified detector, the term $BE{R_0}$ assumes the functional form shown in Eq. (B.2) below:Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be available on a limited basis from the authors upon request.
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