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Abstract
Free space optical propagation affects many application areas, including communication, LIDAR, and directed energy. Optical turbulence causes dynamic changes in the propagated beam that can impact these applications. A primary measure of these effects is the optical scintillation index. In this work, we report on a comparison of experimental measurements of optical scintillation, conducted on a 16 km range across the Chesapeake Bay over a period of three months, to model predictions. Models for turbulence parameters were based on NAVSLaM, a Monin Obhukov similarity theory, and used environmental measurements taken on the range simultaneously with the scintillation measurements. These parameters were then used in two different classes of optical scintillation models: Extended Rytov theory, and wave optic simulation. We show that wave optics simulation matched the data much better than Extended Rytov theory, and that prediction of scintillation using environmental parameters is possible. In addition, we show that optical scintillation over water has different characteristics in stable versus unstable conditions.
1. Introduction
Optical propagation through the atmosphere underlies several applications, including free space optical communication (FSOC) [1,2], LIDAR [3] and directed energy [4]. Atmospheric effects on propagation divide into two broad categories: scattering effects, that typically depend on particulates in the atmosphere such as water droplets and other aerosols, and turbulence effects that depend on random index of refraction variations. Turbulence effects include beam wander and broadening [5], and optical scintillation. Scintillation, which causes millisecond-long surges and fades in the received optical power, is a particular challenge for FSOC since it can cause packet loss. However, these effects can be mitigated by choices in communication protocols, if the statistical characteristics of scintillation are understood [6]. These characteristics are determined by the scintillation index in combination with the distribution function of scintillation [7,8].
At a base level, optical scintillation depends on the strength and spatial spectrum of atmospheric index of refraction fluctuations. These fundamental optical turbulence characteristics are characterized by the refractive index structure parameter, Cn 2, which controls the strength of turbulence, together with the inner scale, l0, the outer scale, L0, and the turbulence spatial spectrum function, which control the distribution of size scales [9].
Given these turbulence parameters, the optical scintillation index can be calculated using models of optical propagation through random media. These models fall into two classes: Analytic models, which use approximations to allow the integrals describing propagation to be solved [10,11], and Monte Carlo approaches using wave optics simulation (WOS) [12–15]. The distribution function of scintillation can then either be derived directly from wave optics simulation [16], or use one of several proposed models, if an analytic approach is used [17].
The strength of turbulence depends on the height above the surface. Additionally, in the surface layer, the lowest part of the atmosphere, turbulence depends strongly on the type of terrain. Turbulence over land [18], urban areas [19], or water [20] are distinct from each other, and depend on different environmental parameters.
Optical scintillation over water is important to understand because the maritime environment offers several advantages for applications like FSOC including long lines of sight, and more stable turbulence conditions than over land. As a result, several demonstrations of ship-to-ship FSOC have been successfully conducted [2].
A long-standing goal has been to be able to predict the optical scintillation of a free space link given the characteristics of the transmit and receive terminals, the range, and the environmental conditions [21,22]. This would be desirable because turbulence parameters are not routinely measured globally, whereas weather conditions are. In addition, this would open up the possibility of combining weather forecasting models and scintillation theory to predict performance [20,23]. The ability to predict scintillation based on weather, combined with models of scattering loss due to rain, fog and other particulates, would allow global forecasting of FSOC availability [22].
Achieving this goal first requires being able to calculate atmospheric parameters, such as Cn 2, with environmental inputs. In the atmospheric surface layer, a variety of models have been proposed. For the maritime environment, a leading model for predicting optical turbulence parameters has been the Navy Surface Layer Atmospheric Model (NAVSLaM), developed by the US Naval Postgraduate School [24]. The model is based on Monin-Obhukov similarity theory and uses environmental inputs such as air and water temperature, wind speed and humidity to predict turbulence parameters. Several groups, have measured turbulence parameters over water in different locations, and using different techniques, and compared their measurements to NAVSLaM with good agreement under most conditions [25–27].
There is less certainty about whether optical propagation models can accurately predict the scintillation index, particularly in deep turbulence, which occurs at longer propagation distances. Scintillation increases with propagation length, and also Cn 2, until it reaches a peak in the so-called focusing regime, it then declines as further propagation reduces the coherence of the beam. It is important to understand deep turbulence, because, for horizontal links in the lower atmosphere, propagation of more than a few kilometers often produces deep turbulence. Many applications require propagation lengths of tens of kilometers.
Analytic models, such as Extended Rytov Theory (ERT) [11], do not agree with wave optic simulations, particularly in the focusing regime, where scintillation peaks. WOS predicts much larger peak values of scintillation than ERT [28–30]. While there have been a large number of theoretical studies of scintillation, there have been relatively few attempts to compare theory to experiment in strong turbulence, so it is not clear which model is correct.
For measurements of the scintillation index over land, the experimental data set that has been most often compared to theory is that of Consortini et al. [31]. This work measured the scintillation index, Cn 2 and the inner scale, over a 1.2 km range in conditions that spanned the range from weak to saturated turbulence. Flatté et al. [32]. successfully modeled this experiment using the measured atmospheric turbulence parameters and wave optics simulation. Wayne et al. [7] measured the scintillation index and aperture averaging effects on a 1 km link over land, and compared the measured scintillation index to one determined by fits to the data using lognormal and gamma-gamma distribution functions.
For links over water, we previously measured the scintillation index over the Chesapeake Bay in conditions ranging from weak to saturated, and showed that Extended Rytov theory did not match the data well [33]. Mosavi et al. [34] measured optical turbulence off the Atlantic coast, in weak turbulence, and compared their measurements to a wave optics simulation. They found good agreement, but had to adjust the value of Cn 2 from the experimentally measured value to achieve the agreement.
In addition, to these direct measurements of the scintillation index, Jaurez [35], and Lionis [36] have measured the performance of maritime FSOC links.
In this work we present optical scintillation measurements collected on a 16 km range over the Chesapeake Bay during a three-month period. We use simultaneous environmental measurements, in combination with NAVSLaM, to calculate values for Cn 2 and the inner scale. These turbulence parameters are then used in both Extended Rytov theory and wave optic simulation to calculate the scintillation index. The modelled scintillation index is compared to that measured on the range.
2. Measurements of scintillation across the Chesapeake Bay
For over ten years, the United States Naval Research Laboratory (NRL) has maintained a laser test range across the Chesapeake Bay [33]. As shown in Fig. 1, the range has sites on both sides of the bay, separated by 16 km. On the western side of the bay, at Chesapeake Beach, Maryland (CBD), there is a full optics laboratory situated on a cliff 30 meters above the water, while on the eastern side of the bay, at Tilghman Island, MD (TI), there is a smaller facility situated about 5 meters above the water. In both cases, the labs are less than 10 meters from the water’s edge, so the link is almost entirely over water. The TI facility is generally unmanned.
At TI, there is an automated laser transmitter. The transmit beam has a diameter of 5 cm, a beam divergence of 170 µradians (defined as 1/e2 full-width) and a power of 100 mW. The laser wavelength is 1550 nm. The transmitter has a fast steering mirror and a microprocessor-controlled pointing system. At the other end of the link, 16 km away, the transmitter produces essentially a spherical wave.
At CBD, there are a variety of optical instruments. The most important one for this work is a scintillation characterization system which consists of a set of three optical receivers with diameters of 1.2 cm, 2.5 cm and 5 cm. These receivers are fed by a single entrance aperture and beam splitters are used to divide the light among them. The receivers use InGaAs PIN photodetectors with 45 dB dynamic range. They are connected to digitizers with 24-bit range which collect data at 5 KHz. The different receive apertures in this system allow the measurement of aperture averaging, but for this work only the 1.2 cm diameter aperture was used.
Both CBD and TI are also connected by an RF data link. This allows feedback from the receivers at TI to drive the microprocessor-based pointing system at TI, keeping the transmit beam centered on the CBD receivers.
The scintillation collection system runs continuously. A computer running a MATLAB-based control program acquires the digitized intensity data from each receiver. Once an hour the transmit beam is turned off and the background level of each receiver is determined. The rest of the time the laser is on, and the transmitter at TI illuminates the receivers at CBD. The program collects the digitized intensity falling on each detector at CBD. Once each minute, the program subtracts the background, normalizes the signal, and calculates the variance. This produces the linear scintillation index, σI 2. In addition, the program takes the natural logarithm of the normalized signal and calculates its variance, producing the logarithmic scintillation index, σlnI 2. It is important to note that we do not calculate σlnI 2 by employing the oft-used relationship:
This relationship holds only when the distribution function of scintillation is lognormal, which is often not the case [37,38].
σI 2 and σlnI 2 are then logged each minute. In addition, once each minute, the power spectral density of the intensity fluctuations is also calculated and stored along with the mean power on the link. The rate of scintillation on the link varies with conditions, but is generally in the range 100-200 Hz. Thus, the one-minute scintillation samples represent approximately 10,000 independent samples of intensity fluctuations.
Once an hour, a five-minute long sample of the intensity, digitized at a 5 KHz rate, is also stored. This can be used for subsequent determination of the scintillation distribution function.
The power on the receivers is generally set so that the mean signal is approximately 15 dB below the saturation point of the detectors, and 30 dB above the noise floor.
While the configuration described above was used to collect the data presented in this work, previous data we have presented from the Chesapeake range had a different configuration and set of equipment [25,33]. In our previous work, the direction of the link was flipped, with the transmitter at CBD and the receivers at TI. In addition, the scintillation receivers had a lower dynamic range, of about 30 dB, with 16-bit digitization. The current configuration allows longer intensity data collects, and more sophisticated processing because of the greater capabilities at the CBD lab. The higher dynamic range of the receivers is especially important because the strong scintillation on this link can produce very deep fades. In addition, in our previous work the pointing of the transmitter was fixed, with periodic repointing. In the current system, pointing corrections are applied continuously to keep the transmit beam centered on the receive apertures.
In addition to optical data, a variety of environmental data is collected on the test range. There is a small weather station at CBD. In addition, two NOAA weather stations on the Chesapeake Bay, one to the north of the range at Thomas Point and one to the south at Goose’s Reef, provide an extensive set of weather data including air and water temperature, wind speed and pressure.
In previous work, we presented scintillation data on this range versus Cn 2 values determined by simultaneous measurements of the angle of arrival variation on the link [33], which can be related to the atmospheric structure constant through theory. In this work we employ a different approach, using the environmental data that we collect, and the NAVSLaM model, to determine both Cn 2 and the inner scale of turbulence, l0. This allows us to investigate whether optical scintillation can be predicted with just environmental and system parameters. In addition, using angle of arrival to determine Cn 2 is incompatible with the active pointing system used in this work, since this will produce an additional angle variation, not associated with the atmosphere.
In prior work we validated this approach by comparing Cn 2, as measured by angle of arrival (using a fixed-pointing beam), to that determined by NOAA data and NAVSLaM [25]. We found good agreement for most cases. We also found reasonably good agreement in calculated Cn 2 values using environmental data, taken at the same time, from both the NOAA Thomas Point Lighthouse station, and the NOAA Goose’s Reef buoy. This is despite the fact that Thomas Point is 40 km north of Goose’s Reef, attesting to relatively uniform conditions north-south along the bay.
Deviations between our measured value of Cn 2 and NAVSLaM did occur for values below about 5 x10−16 m2/3. At these low values, our angle-of-arrival based system was below its sensitivity limit. In addition, work by other groups, comparing NAVSLaM values to sonic anemometer readings, also showed that NAVSLaM becomes inaccurate in near-neutral conditions when the temperature difference between the air and the surface becomes small. This also occurs at values of Cn 2 in the low 10−16 m2/3 range. In general, NAVSLaM tends to predict values of Cn 2 that are lower than those measured by the sonic anenometers in this range [27]. Thus, low values of Cn 2 remain a regime of uncertainty.
NAVSLaM, also produces a value for the inner scale of turbulence, l0, using the relation [9,39],
Thus, using environmental data and NAVSLaM, we can calculate two of the fundamental turbulence parameters, as a function of time, on the range. In addition, we have the simultaneously measured value of σI 2 and σlnI 2 on the range. When the measurement interval for various values was different, the value of the less frequently sampled values was estimated using linear interpolation.
In this work, we used air temperature, water temperature, pressure and wind speed values from NOAA’s Thomas Point Lighthouse. This site’s instruments are at a height of 17 m, almost exactly the height at the midpoint of the link, which has the dominant weighting in scintillation calculations. NAVSLaM does consider the height of the measurements, in its calculations, and produces height varying profiles of turbulence parameters. So, the varying height of our slant path does not pose a problem. Relative humidity values were measured at CBD, because these values were not available from Thomas Point.
An example of a time series of some of the environmental data, the NAVSLaM calculated turbulence parameters, and the measured value of σlnI 2 is shown in Fig. 2.
Fig. 2. A sample time series of environmental data from the Thomas Point Lighthouse, the atmospheric structure constant and inner scale calculated by NAVSLaM using this data, and the simultaneously measured value of the log scintillation index on the range. The structure constant and inner scale are calculated at a height of 17 m above the water’s surface, which is the midpoint height of the link.
3. Optical propagation models
We used both analytic models and wave optic simulation to investigate whether the turbulence parameters produced by NAVSLaM using the measured environmental data could be used to accurately predict the measured scintillation indices.
Both these propagation models require a value of the outer scale of turbulence, which is not produced by NAVSLaM. In the surface layer, the outer scale is generally considered to be related to the height of the link above the surface, but there are many different estimates of its value. A common assumption is that the outer scale is approximately 40% of the height of the link [40]. We adopt that estimate in our calculations. The outer scale has relatively little effect on scintillation up to the peak value in the focusing regime, but it does affect the fall-off in scintillation in the saturation regime.
In addition to the fundamental turbulence parameters, optical propagation models also use a model for the spatial power spectra density (PSD) of the refractive index fluctuations. Most commonly, these models begin by assuming the Kolmogorov spectrum, but then introduce factors that modify this spectrum near spatial values at the inner and outer scale. The simplest such model that includes the effect of inner and outer scales is the modified von Karman model (MVK),
Experimental measurements over land by Champagne et al. [41] showed that there was a bump in the spatial PSD near values corresponding to the inner scale. Hill and Clifford [9] explained this bump and proposed a new spectrum that incorporated it. Andrews [42] developed an analytic approximation for this spectrum, called the modified atmospheric spectrum (MAS). This has the form,
Friehe et al. [43] measured turbulence over water. Based on these measurements over the Salton Sea, Hill also developed a marine version of the atmospheric spectrum [9]. An analytic version of the Marine spectrum (MAR) was proposed by Grayshan [44]. This has the form,
Normalized to the Kolmogorov spectra, these three choices of spectrum have different shapes, as shown in Fig. 3.
Fig. 3. The modified Von Karman (MVK), modified atmospheric spectrum (MAS) and Marine spectrum (MAR) spectrum normalized to the Kolmogorov spectrum and plotted versus spatial wavenumber times the inner scale.
There is no exact solution to laser beam propagation through the turbulent atmosphere, so all analytic approaches rely on approximations. One of the most commonly used analytic formulations, for strong turbulence, is Extended Rytov theory, which uses the Rytov approximation for weak fluctuations, and extends these solutions into the saturation regime by accounting for the loss of spatial coherence in strong scintillation [11,45]. ERT uses the modified atmospheric spectrum. Because the beam in our test range is very close to a spherical wave we use the spherical wave, strong fluctuation, formulation of ERT to calculate the scintillation index. Specifically, Eqs. 70–77 of Chapter 10 of Ref. [45], which also include the effects of aperture averaging. This formulation of ERT assumes constant values for the turbulence parameters over the path.
Analytic approaches to scintillation inherently involve approximations. An alternative is wave optics simulation, which uses split step Fourier techniques to simulate optical propagation through random index fluctuations [14]. Turbulence is represented by randomly generated phase screens placed in the middle of each propagation step. Monte Carlo techniques are used to simulate the inherent randomness of the atmosphere. The end result of a propagation is a set of solutions for the optical field that can be used to determine the final spatial profile of the beam for each Monte Carlo iteration. This can be used to determine optical scintillation as well as other optical beam parameters. WOS has been used for about thirty years to simulate optical scintillation [12], and over that time a variety of techniques have been developed to determine the required grid sizes, spacing of turbulence screens [46], and efficient and accurate generation of random turbulence screens that obey a given spatial PSD [47].
In this work we used the approach outlined by Schmidt [14]. We simulated propagation from our transmitter, 5 m above the water surface to our receiver, 30 m above the water surface, 16 km away. We used a sinc function to simulate a spherical wave. We used a grid spacing of 2.5 mm. The smallest value of the inner scale that commonly occurs on the link is 5 mm. So, this grid spacing is sufficient to sample the typical turbulence structure. The simulation size was 2048 × 2048 points.
Phase screens were randomly generated and used three subharmonics [47]. In a given simulation we started with values of the turbulence parameter, Cn 2, l0, and L0. We also chose one of the turbulence spatial PSD functions (Eq. (3 )–(5)). These were used to randomly generate the index of refraction fluctuations for the phase screens. We ran simulations with both constant values of the turbulence parameters, and also height-varying values of these parameters. The number of phase screen used in any simulation varied, depending on the strength of the turbulence. The spacing of the screens was set so that the Rytov variance in going from one screen to the next is less than 0.1.
We used the WOS to produce the final intensity pattern for each Monte Carlo iteration. To simulate the effects of aperture averaging in the actual experiment, which used a circular receiver with a 1.2 cm diameter, tiles of four adjacent pixels were summed together. Their total area was 1 cm2, similar to the detector. We used tiles from the central 10% (50 cm) of the simulation, and from all the Monte Carlo iterations. This resulted in approximately 50,000 intensity realizations for each simulation run. σI 2 was determined by calculating the variance of the normalized intensity values. σlnI 2 was determined by taking the natural logarithm of the normalized intensity and finding its variance. Convergence of the solution was monitored to verify that a sufficient number of Monte Carlo iterations were used.
We found, as several other groups had shown previously [29,30,48], that the wave optics simulation produced a much higher value of the peak scintillation index than Extended Rytov Theory. Figure 4 shows a comparison of calculations, using both approaches, for different values of the inner scale. σlnI 2 is plotted versus β0, where
Fig. 4. Comparison of Extended Rytov and wave optic simulation calculations of the logarithmic scintillation index as a function of turbulence strength. Constant values of the turbulence parameters are assumed over the range. The wavelength is 1550 nm, the link range is 16 km, the outer scale is 7 m, and varying values of the inner scale are shown. The receive diameter is 1.2 cm.
4. Comparison of experimental measurements to model calculations
Scintillation and environmental data were collected on the Chesapeake Bay range between mid-February to mid-May, 2022. The optical link generally ran continuously, but there were link shut-downs at various times due to conflicting tests, power outages, and other external conditions. Overall, the link was operational for 52 days, about 60% of the total test duration. Shut-downs were typically for 1-2 weeks at a time. In addition, even when the link was running, some data was rejected due to quality control. Data was rejected at times when the link was interrupted by poor weather, background checks, system recalibration, and when the tracking system lost lock. These outages ranged with a few minutes to several hours and were randomly distributed through the measurement period. This reduced the data by about another 50%.
Over this time period, there were a wide variety of environmental conditions. A histogram of these conditions, from measurements at the Thomas Point lighthouse, is shown in Fig. 5. The primary driver of optical scintillation over water is the air water temperature difference (AWTD). This varied from -8 °C to +8 °C, spanning unstable and stable conditions.
Fig. 5. Histograms of the environmental conditions on the test range during the collection period from February-May, 2022.
This environmental data was fed into NAVSLaM, and values for Cn 2 and l0 as a function of time were generated. Figure 6 shows histograms of the calculated turbulence values. Because the link is a slant path, going from 30 m above the water down to 5 m, the turbulence values will vary over the link range. NAVSLaM calculates the height dependence of these parameters. The values shown in these histograms are for a height of 17 meters above the water, the midpoint of the link. The most common Cn 2 values were in the range around 10−15 m-2/3, but values below 10−17 m-2/3 and into the mid 10−14 m-2/3 were also calculated. Inner scale values varied from 5 mm to 25 mm, though values above about 15 mm were uncommon.
Fig. 6. Histograms of the base 10 logarithm of the refractive index structure constant and the inner scale as calculated by NAVSLaM based on environmental measurements on the test range during the collection period from February-May, 2022.
The scintillation index, as well as other optical parameters was measured as described in Section 2. Both σI 2 and σlnI 2 were measured. We prefer using σlnI 2 as a measure of scintillation for several reasons. σI 2 is highly sensitive to large intensity spikes [29], because these values cause a large increase in the variance. However, σI 2 is relatively insensitive to deep fades. The sensitivity to high intensities causes the measured value of σI 2 to be quite sensitive to detector saturation, as has long been noted [49]. Changes in the average power of the link will change the effective saturation point of the detector relative to the mean power, thereby changing the measured value of σI 2. σlnI 2 has a much weaker dependence on high intensity spikes and on detector saturation [50]. Thus, it provides a measure of scintillation that is much less dependent on the average power in the link. In addition, from the point of view of free space optical communication, fades are generally more important than surges in determining link performance. σlnI 2 weights surges and fades equally, so its strength is a better indicator of the effects of scintillation on link quality.
As described in Section 2, scintillation was logged once a minute on the link. In analyzing the data, we averaged five successive one-minute samples to produce a value of scintillation at a five-minute interval. The standard deviation of these samples was used to provide a measure of the statistical variation of individual scintillation measurements. We can use the standard deviation of individual measurements in examining the spread in the measured data. It allows us to separate the statistical variation of the measurement from uncertainty in turbulence parameters, or model inaccuracies.
Figure 7 shows a histogram of the value of σlnI 2 over the measurement campaign. Values between 1.5 and 2.0 were very common because these are the saturation values of scintillation on the link. This is an important point for maritime optical systems working at ranges more than a few kilometers in the lower atmosphere. Low scintillation conditions may be uncommon because they require Cn 2 values of around 10−16 m-2/3, or less. On our 16 km link, scintillation generally peaks at values of Cn 2 between 1-2 × 10−15 m-2/3, which occurs at air water temperature differences of about 2°C. Low scintillation conditions occur only when the air water temperature difference is much less than 1°C. In most cases, this only occurs for brief periods of time.
Fig. 7. Histogram of the logarithmic scintillation index during the collection period from February-May, 2022.
Figure 8 shows all of the σlnI 2 measurements versus the air water temperature difference at the time the measurement was taken. As can be seen, there is a strong correlation between the AWTD and the scintillation index. Scintillation drops to a minimum when the AWTD is near zero. It then peaks at AWTD values of about ±2 °C, which correspond to the focusing regime. At higher temperature differences, scintillation drops due to saturation and aperture averaging.
Some other features can also be seen. The peak value of scintillation is slightly higher for negative values of the AWTD (unstable conditions) than for positive values (stable conditions).
Also, the relative spread in scintillation values for a given AWTD is higher for stable and neutral conditions than it is for unstable conditions. We have observed a higher variability of optical parameters in stable turbulence conditions before. Its shows up in both scintillation indices and in angle of arrival variation [33]. It may be associated with stratification in the atmosphere in stable conditions, which can produce layers with different turbulence parameters [51].
In near-neutral conditions, variability may be produced because these conditions correspond to the linear, weak turbulence regime. The scintillation index has its highest sensitivity to turbulence parameters here, so any differences between the AWTD at the Thomas Point lighthouse and that of the beam path will have their greatest effect on data variability in this regime. Higher AWTD values correspond to either peak, or saturated, conditions, in which the sensitivity of scintillation to turbulence parameters is weaker.
The scintillation index shows an empirical relationship with the air water temperature difference, but what we want is a model that will predict the value of σlnI 2. For this we turn to NAVSLaM and the propagation models.
To compare the models to the data, we apply the following process: For each five-minute time interval over the measurement campaign, we find the measured value of σlnI 2. We use the simultaneous environmental parameters as inputs to NAVSLaM which produces profiles, versus height, of both Cn 2 and l0. We reference our data to NAVSLaM values of Cn 2 and l0 at 17 m, the midpoint height of the link. This mid-point Cn 2 value is used to calculate β0. We plot the measured value of σlnI 2 versus β0, and color code the point based on the calculated value of l0.
We then produce model curves to compare to the measured data. For Extended Rytov Theory, which assumes constant values of the environmental parameters over the path, we assume the outer scale is 7 m, about 40% of the midpoint height of the path. We then pick a value of the inner scale and calculate the scintillation index as a function of Cn 2. As with the outer scale, these are assumed to represent the values at the mid-point height of the path. The effects of aperture averaging of our 1.2 cm receiver are also included. The calculated curves can then be plotted versus β0, for comparison to the data.
Using the wave optics simulation, we can take the actual height variation of the turbulence parameters, as predicted by NAVSLaM, into account. To generate the WOS curves, we start with inputs of Cn 2 and l0 that represent the values of these parameters at a height of 17 m. We then adjust for the height of the link over the slant path. As we generate phase screens over the link, we calculate the link height. We then use the height profile predicted by NAVSLaM to adjust the values of Cn 2 in that screen relative to the value at 17 m height. This relative variation depends almost entirely on whether conditions are stable or unstable. For stable conditions, the height variation is approximately z-2/3, and for unstable conditions it is about z-4/3. Over the height range of our link, the variation in the value of the inner scale is very weak, so we assume it to be constant with the value at 17 m. The outer scale is again assumed to be 40% of the height of the link. The effects of aperture averaging are included as described in Section 3. Because the height variation depends on whether the conditions are stable or unstable, we do a WOS for both cases. In addition, for comparison, we also did WOS assuming constant values of Cn 2, l0 and L0 over the path. These produced similar results to those that took the height variation into account, and would certainly be an approach better suited to generalizing the results. Each wave optics simulation produces one value of the scintillation index for a given input of Cn 2 and the inner scale. By running WOS for a range of Cn 2 values, we can, as with ERT, produce a curve of the predicted σlnI 2 versus β0 for a given value of the inner scale. We ran WOS for the three turbulence spectra described in Section 3. As with the experimental results, we can estimate the statistical uncertainty of the WOS. In parallel, with our experimental approach, we break the simulation into five segments with 10,000 points each. We determine the scintillation index of each segment, and find the standard deviation of the segments to estimate the statistical error.
We divide the results into those collected under stable and unstable conditions. Figure 9 shows the case of stable conditions (AWTD > 0), and Fig. 10 shows unstable conditions (AWTD < 0). Each figure shows measured values of σlnI 2 plotted versus β0 with each dot color coded for the inner scale value. For readability, we do not show the standard deviation of the individual measurements for each data point, but, we do show typical values of this error in the box at the top of the graph.
Fig. 9. The logarithmic scintillation index, as calculated from WOS, is plotted versus β0, for varying choices of turbulence spectra under stable atmospheric conditions. The WOS were carried out for an inner scale of 10 mm. Also shown are experimental data points with the calculated inner scale value being depicted by the color scale to the right. Typical standard deviations of individual data points at various values of β0 are shown in the box at the top of the plot.
Fig. 10. The logarithmic scintillation index, as calculated from WOS, is plotted versus β0, for varying choices of turbulence spectra under unstable atmospheric conditions. The WOS were carried out for an inner scale of 10 mm. Also shown are experimental data points with the calculated inner scale value being depicted by the color scale to the right. Typical standard deviations of individual data points at various values of β0 are shown in the box at the top of the plot.
These figures also show model calculations. These calculations use an inner scale value of 10 mm, which is about the middle range of the values calculated by NAVSLaM based on the environmental data. We compare four different propagation models: Extended Rytov, which uses the modified atmospheric spectrum, and wave optics simulations using the modified Von Karman, modified atmospheric and marine spectra. The WOS use height varying values of Cn 2. The dots on the model curves show the values at which σlnI 2 was calculated. The WOS show the estimated statistical error at each point. Interpolation was used to connect the points.
We note several features of the scintillation data: The calculated values of the inner scale tend to be larger for lower values of β0, with values generally between 10-15 mm, and smaller in the saturation regime, with values between 5-8 mm. This is similar to behavior noted by Consortini over land [31]. The peak value of σlnI 2 occurs at values of β0 around 1.5. As was seen in Fig. 8, the peak scintillation value is slightly higher for unstable conditions than stable.
Measured scintillation values for a given value of β0 exhibit a spread that is greater than the standard deviation of individual experimental data points obtained, as described above, from five successive one-minute samples. Some of this spread can be attributed to differences in the inner scale. Higher values of the inner scale generally lead to higher scintillation values. However, this trend is not completely consistent in the data, and does not explain all of the spread. In the saturation regime, the data taken under stable conditions exhibits a higher spread, than that taken under unstable conditions. As we discussed above, this higher variability under stable atmospheric conditions may be because of stratification, and is observed in other data.
Additional factors may contribute to the data spread at values of β0 below about 1.0. These correspond to Cn 2 less than about 5 × 10−16, and small AWTD. This is the regime in which NAVSLaM tends to underestimate Cn 2. This can result in some points corresponding to higher turbulence being placed at values of β0 that are too low. This shows up in the plots as a wide spread in σlnI 2 at low β0 values. These low values of β0 will include points that were actually taken in low turbulence, along with points taken in higher turbulence and mis-assigned there by NAVSLaM. In addition, as we discussed earlier, differences in the AWTD as measured by the NOAA station and the actual AWTD on the path will also contribute to the spread in values.
In the comparison of the various propagation models to the data, we see that for weak turbulence (β0 < 0.5), all the models agree with each other, and go through the mid-point of the data. This is unsurprising as most models agree in weak turbulence.
However, as we move out of the weak turbulence regime, to the focusing regime, Extended Rytov Theory consistently produces a smaller value of σlnI 2, and also peaks at a larger value of β0, than the experimental data. For data taken under stable conditions, all of the wave optics simulations produce scintillation peak values, and locations, much closer to what the data shows. These simulations included the slant path variation of Cn 2, using NAVSLaM’s estimate for height variation in stable conditions. Given the spread of the data, no one spectrum can be said to be a better match in the focusing regime. For data taken under unstable conditions, we ran a different set of simulations using NAVSLaM’s estimate for height variation in unstable conditions. Here we see that the WOS produced a lower value of the peak scintillation than is seen in the data, although it is much closer than that produced by ERT.
As we move into the saturation regime, we see that ERT again fails to reproduce the shape of the data curve, falling off more slowly than the data. The wave optics simulation curves are much closer to the data. In the saturation regime, the spread of the data is less, and we can see good agreement particularly with the modified atmospheric spectrum.
In Fig. 11 and 12 we compare the data to WOS using the modified atmospheric spectrum, but with varying inner scale values. For weak turbulence, there is relatively little difference in these curves, but they do differ in the focusing and saturation regimes.
Fig. 11. The logarithmic scintillation index, as calculated from WOS, is plotted versus β0, using the modified atmospheric spectrum, for varying choice of inner scale under stable atmospheric conditions. Also shown are experimental data points with the calculated inner scale value being depicted by the color scale to the right. Typical standard deviations of individual data points at various values of β0 are shown in the box at the top of the plot.
Fig. 12. The logarithmic scintillation index, as calculated from WOS, is plotted versus β0, using the modified atmospheric spectrum, for varying choice of inner scale under unstable atmospheric conditions. Also shown are experimental data points with the calculated inner scale value being depicted by the color scale to the right. Typical standard deviations of individual data points at various values of β0 are shown in the box at the top of the plot.
For data taken under stable conditions, inner scale values in the focusing regime range from about 7-15 mm, and the WOS curves for 5, 10, and 15 mm, cover most of the observed points. In the saturation regime, inner scale values range from 5-8 mm and the WOS curves for 5 and 10 mm inner scale cover most of the points.
For data taken under unstable conditions, the range of inner scale values is the same, but we see that the WOS curves corresponding to these values are generally somewhat lower than the observed data. In the focusing regime, WOS using an inner scale of 20 mm would approximately match the peak value of scintillation that is observed, but the peak occurs at a higher value of β0. Also, this inner scale value is about twice what NAVSLaM calculates. Since we do not have an independent measurement of the inner scale, it is certainly possible that the NAVSLaM calculation is not correct. However, since the WOS using the NAVSLaM inner scale value do seem to match up in stable conditions, we would have to assume an error that occurs in one case, but not the other.
On the whole the data in Fig. 9 –12 show that the combination of environmental data, NAVSLaM, and wave optics simulation can do a reasonable job of matching the observations under stable conditions. Under unstable conditions, the models match well in weak and saturated turbulence, but underestimate the scintillation index by a small amount in the focusing regime.
Whether the predictions we show here match the data well enough to be useful, depends on the application, but we can give a concrete example. In free space optical communication, the principal effect of scintillation is to cause packet loss. In a simple communication link (without retransmission), when a fade takes the received optical power below the detector sensitivity, the packet is lost. The packet error rate (PER) then will depend on the scintillation index, the assumed scintillation distribution function, and the margin in the link.
As an example, we assume the distribution function is lognormal. In that case the probability of a fade below detector sensitivity, Pfa, as a function of margin, and scintillation index is [52],
The probability of fade equates to the packet error rate, PER, of the link. For a given desired PER and scintillation index, this equation can be solved numerically for the amount of link margin needed for the desired quality of service. In Table 1 we show the measured scintillation index for a small band of values of β0 in the focusing regime, and for two bands of inner scale values. We also show the link margin, needed to maintain a packet error of 1%, calculated from the scintillation index using Eq. (7). The uncertainty values of this link margin are derived from the standard deviation of the σlnI 2 data points that fall within the bands of β0 and l0. We compare this to the scintillation index calculated by wave optics simulation, for the same values of β0 and l0, using the modified atmospheric spectrum, and to the link margin calculated from this scintillation index. We do the same for Extended Rytov Theory. We see that in the stable regime, WOS predicts the link margin, derived from the measured data, to better than 1 dB. In the unstable regime, WOS simulation predicts the measured value to within slightly more than 1 dB. In both cases the WOS predictions fall within the uncertainty in the measurement. For many purposes, such as predicting FSOC availability, this accuracy would be more than sufficient. By contrast ERT is off by about 5 dB, which would not be accurate enough to predict availability.
Table 1. Comparison of measurements to theory in the focusing regime
5. Conclusions
Our goal in this work has been to investigate if it is possible to predict, with useful precision, optical scintillation over water using environmental and system parameters, together with models for atmospheric turbulence parameters and optical propagation through turbulent media. We collected long-term scintillation and environmental data on an optical link that is representative of ship-to-ship, or ship-to-shore links. Since we, and others, had previously shown that models, such as NAVSLaM, could predict turbulence parameters given environmental parameters, our focus here has been on using those parameters in conjunction with optical propagation models to predict scintillation.
There are some inherent limitations in this approach. We used data from NOAA weather stations, which are physically displaced in position from our beam path, so the values we derive for Cn 2 and l0 may not always correspond to those present on the link. This may contribute to the spread in scintillation values for a given set of turbulence parameters. While having environmental data taken on multiple points along the link range would almost certainly have produced a more accurate data set, it is also true that for practical applications such fine-grained data is unlikely to be available.
It would also be desirable to have better values for Cn 2 when turbulence is weak. For predicting scintillation with environmental data, this would require higher accuracy from NAVSLaM in near neutral conditions. This might be possible using either empirical corrections to the model based on experimental measurements, or by applying machine learning techniques to supplement the Monin-Obhukov theory [53]. Better experimental path-integrated values for Cn 2 may also be possible. Our link range of 16 km exceeds the maximum range of conventional scintillometers. In our past work, we used angle of arrival variation to measure Cn 2, but that was limited to values of about 10−15 m-2/3 or larger. In addition, the active pointing used in our current measurement of the scintillation index does not allow simultaneous use of angle of arrival measurements for turbulence monitoring. In future work, it might be possible to interleave Cn 2 and scintillation index measurements. Higher resolution cameras and more sophisticated centroiding techniques may allow higher sensitivity for weak turbulence conditions.
Our data shows that wave optics simulations predict scintillation values much closer to our data than Extended Rytov Theory, particularly in the focusing regime. We compared three different turbulence spectra. Two of them include the “Hill bump” and one did not. The scintillation indices predicted by these spectra differ, but not strongly. On the whole the modified atmospheric spectrum seemed to fit the data the best. However, given the spread in the data, a firm conclusion cannot be drawn.
We see a difference in the peak value of the scintillation index in unstable conditions as compared to stable. In addition, WOS fits the stable data somewhat better in the focusing regime. The difference between the model and experiment cannot be explained by an error, systematic or random, in our value of Cn 2 since this would simply move the peak value of scintillation, not make it larger. It is conceivable that the discrepancy could be explained by an error in the calculated value of the inner scale. However, there is another possibility. All of the turbulence spectra we evaluated are essentially Kolmogorov in nature. Recently, Bos has shown, using wave optics simulation, that non-Kolomogorov turbulence spectra can produce higher peak values of scintillation [54]. Friehe’s original study of turbulence spectra over water consisted of measurements over the Salton sea in stable conditions, but also measurements on the FLIP platform off the coast of California under unstable conditions. The FLIP measurements were not used by Hill in his marine spectrum because they did not show an inertial range. Recently, new measurements, also using the FLIP platform, have also shown non-Kolmogorov behavior in the unstable regime [55]. It is possible then, that non-Kolmogorov wave optics simulations could better match the unstable data. Anisotropy of turbulence, in combination with non-Kolmogorov behavior [56,57], may also play a part in the differences we observe between stable and unstable conditions. We have observed anisotropy in angle of arrival variations on our link.
Overall, we believe that our results show that prediction of optical scintillation over water is possible. This is an important step in predicting the performance of maritime free space optical systems. For free space optical systems, an additional required step is the determination of the proper scintillation distribution function. The data we have taken on the Chesapeake Bay range includes long runs of intensity time series that can be used to compare to different proposed distribution functions. We plan to address this in future work.
Funding
Office of Naval Research (NRL Base 6.1).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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