## Abstract

This paper describes a new methodology of estimating free-space optical communications link budgets to be expected in conditions of severe turbulence. The approach is derived from observing that the ability of an adaptive optics (AO) system to compensate turbulence along a path is limited by the transmitter and receiver Rayleigh range, proportional to the diameter of the optics squared and inverse of the wavelength of light utilized. The method uses the Fried parameter computed over the range outside of the transmitter and receiver Rayleigh ranges, to calculate the Strehl ratios that yield a reasonable prediction of the light impinging on the receiving telescope aperture and the power coupling into the fiber. Comparisons will be given between theory and field measurements. These comparisons show that AO is most effective within the Rayleigh ranges, or when an atmospheric gradient is present, and lesser so when the total range is much greater than the sum of the Rayleigh ranges.

© 2010 Optical Society of America

## 1. Introduction

There is a need for high-capacity communication links for the tactical level of warfare because of the increasing need to transmit and receive video and the aggregation of low-rate data sources from forward areas back to tactical operations centers [1]. These data requirements are placing increased demands on the throughput of current radio-frequency (RF) systems and RF satellite links. In addition, increasing commercial demands on satellite links and the RF spectrum require the incorporation of optical communication to relieve RF congestion and more efficiently use allocated RF capacity for the most critical traffic. Current technology is now mature enough to design and prototype an airborne reach-back system capable of transmitting data, voice, and video traffic over an IP-compatible network utilizing hybrid links of RF and free-space optical communications (FSOC) equipment. Discussions of this can be found in several papers in the literature [2, 3, 4, 5].

Successful communications link closure has been achieved through atmospheric turbulence by several authors over ranges from 10 to $150\text{}\mathrm{km}$ [2, 3, 4, 5, 6]. Elements of such an FSOC system include (i) $1.55\text{}\mathrm{\mu m}$ wavelength lasers with up to $10\text{}\mathrm{W}$ output; (ii) acquisition, tracking, and pointing systems with near- microradian ($\mathrm{\mu rad}$) accuracy; (iii) deformable mirrors that use measurements of the incoming optical phase fronts to compensate for atmospheric turbulence and other aberrations; and (iv) optical automatic gain control systems that protect the detector from damage from high optical power and provide full use of $40\text{}\mathrm{dB}$ of dynamic range [2, 3, 4, 5, 6]. Details of these components have been described separately [6]. The challenge today is to characterize the link degradation by turbulence that allows estimation of performance under various atmospheric conditions.

Given the trend today in FSOC, adaptive optics (AO) plays a key role in reducing signal degradation and focusing the light into fibers or erbium-doped fiber amplifiers. To aid the communications engineer, we propose a methodology to estimate FSOC performance over long ranges and through strong atmospheric turbulence.

This methodology builds on the assumption that the capacity of an AO system to compensate for turbulence along a path is limited by the Rayleigh range, a factor proportional to the square of the diameter of the optics, *d*, and inversely proportional to the wavelength, *λ*, of light utilized, because beyond that range, the optical system cannot resolve phase or amplitude changes smaller than its diameter [2, 6]. Specifically, we previously define the Rayleigh range, ${R}_{\mathrm{RR}}$, over which AO can compensate turbulence phase perturbations to be given by

## 2. Background: Diffraction Spreading

It is useful to begin our paper with a discussion of the spreading of the beam between the transmitter and the receiver. In the absence of atmosphere or other aberrations, the Fraunhofer equation gives us the peak irradiance in the far field, ${I}_{\mathrm{ff}0}$, at a range *R* and wavelength *λ*, as the area of the transmitter, ${A}_{\mathrm{TX}}$, times the transmitted power, ${P}_{\mathrm{TX}}$, divided by (the wavelength times the range) quantity squared:

## 3. Degradation of FSOC Links by Turbulence

When aberrations are induced by the atmosphere, the transmitted irradiance is reduced by a factor commonly called the Strehl ratio {(SR) [7], p. 7}. The modern definition of the SR is the ratio of the observed peak intensity at the detection plane of a telescope or other imaging system from a point source compared to the theoretical maximum peak intensity of a perfect imaging system working at the diffraction limit. In other words, it is a good measure of the quality of the incoming irradiance to an optical system ([8], p. 462). Mathematically, the SR is defined as

where ${\sigma}_{\phi}^{2}$ is the residual phase variance ([9], p. 50). For purposes of this paper, we find, under more general conditions pertaining to turbulent propagation, the SR can be rewritten as where*d*, as before, is the diameter of the optical systems and ${r}_{0}$ is the Fried parameter ([9], p. 50). The Fried parameter is generically defined as

*ζ*is the zenith angle, $k\equiv \text{wavenumber}=2\pi /\lambda $,

*h*is the height above the ground,

*L*is the length of the turbulent regime, and ${C}_{n}^{2}(z)$ is the refractive index structure function ([10], p. 47). In Section 5, we will specify the ${C}_{n}^{2}(z)$ model we will use in our calculations.

The Fried parameter is the atmospheric coherence length and is the distance over which the phase varies no more than $\pm \pi $. It is the largest effective di ameter for image resolution and for maximizing signal-to-noise ratio in coherent detection systems [10]. It also is related to the plane wave coherence radiance, ${\rho}_{0}$, by the relation

In terms of the measurements discussed in this report, the far-field beam diameter is much larger than the receiver aperture. As a result, we can use the Fraunhofer equation modified by the second definition of the SR, i.e., dependent on the ratio of $d/{r}_{0}$, to calculate the peak irradiance that is incident on an aperture, subject to the Raleigh range reduction. {The use of the compensation range limitation in these calculations assumes that the AO system has adequate spatial compensation and a frequency response greater than the Greenwood frequency ([11], p. 622).} In particular, we assume that the transmitter AO systems will correct all turbulent effects within ${R}_{\mathrm{RR}}$ from that transmitter aperture and near the receiver aperture, and the turbulent effects only come from the range between ${R}_{\mathrm{Tx}\mathrm{RR}}<r<L-{R}_{\mathrm{Rx}\mathrm{RR}}$, where *L* is the total distance between transmitter (Tx) and receiver (Rx). The total downlink and uplink received power, then, are proportional to the product of their respective SRs over the reduced residual distance, $L-{R}_{\mathrm{Tx}\mathrm{RR}}-{R}_{\mathrm{Rx}\mathrm{RR}}$. To see this, let us relate this discussion to two key performance parameters in optical laser communications systems analysis, power in the bucket (PIB) and power in the fiber (PIF).

The PIB analysis is a direct measure of the average irradiance at the receiver aperture. Thus, it is a measure of the far-field irradiance times the SR. In this case, it is the transmitter SR, with the turbulence weighted most heavily toward the transmitter (irradiance measure definition). On the other hand, the PIF is a measure of the ability of the receiver optics to couple light into the optical modem, which is the interface in laser communications systems. In the absence of degrading effects, the resulting focused beam from the receiver optics would couple light into the single-mode fiber with near perfect efficiency, i.e., the SR essentially would approach unity; otherwise, with degrading effects, the SR will be less than 1, reflecting suboptimal coupling into the fiber. This implies the PIF is approximately the PIB times the receiver SR, where that SR weighs most heavily the turbulence nearest the receiver (classical definition). If the distribution of the turbulence between the two systems was symmetric and the receiver and transmitter aperture diameters were equal, the receiver PIF would then be proportional to the SR squared; otherwise, it is the product of the transmitter and receiver SRs. This is the second important point of this paper.

## 4. Degradation from Atmospheric Absorption and Scattering

In addition to other optical transmittances, the power at the receiver is reduced by the absorption and scattering by molecules and aerosols along the path. *The Infrared Handbook* [12] provides estimates of the attenuation coefficients in the rural aerosol model and the maritime aerosol model as a function of wavelength and in the variation with altitude for moderate volcanic conditions. For a wavelength of $1.55\text{}\mathrm{\mu m}$, the rural aerosol model attenuation coefficient, *α*, is given as $0.036/\mathrm{km}$ and the maritime aerosol model is $0.120/\mathrm{km}$. The transmission over a constant altitude path is then ${\mathrm{e}}^{-\alpha R}$. When the altitude is not constant, we integrate this transmission over the path length. In the analysis to come, we recognize that the maritime attenuation is appreciably greater than the rural value, due largely to the presence of salt aerosols. When propagating over the brackish portion of the Chesapeake Bay, on the other hand, we use half the maritime value because the salt aerosols are less frequent.

## 5. Relationship of Real Turbulence to the Hufnagle-Valley Models

Degradation of diffraction-limited laser beams in the atmosphere occurs due to atmospheric turbulence and also turbulent airflow past the transmitter/ receiver apertures in airborne systems. The latter degradation is known as the aero-optic effect. A major question is whether one can effectively compensate for this degradation when it is created through long-range light propagation under high atmospheric turbulence and nonlaminar airflow conditions created by the use of protruding windows, pods, and external terminals. The intent of this section is to establish the basic ${C}_{n}^{2}$ model that will be used to define the various effects and key parameters of atmospheric degradation over long, arbitrary ranges and how they are related to the occurrence of real turbulence measured statistically and estimated during the various experiments described. In our proposed methodology, we use the Hufnagle–Valley (HV) 5/7 mode for ${C}_{n}^{2}$. Here, the term 5/7 means that for a wavelength of $0.5\text{}\mathrm{\mu m}$, the value of 5 represents a Fried parameter of $5\text{}\mathrm{cm}$ and the value of 7 represents an isoplanatic angle for a receiver on the ground looking up of $7\text{}\mathrm{\mu rad}$. The HV 5/7 model is described, for example, in Tyson ([8], p. 33).

Figure 2 compares multiples of the HV 5/7 model to annual Korean turbulence statistics, measured by the Air Force in 1999. In this figure, we plot multiples of $0.2\times $, $1\times $, and $5\times $ of the HV 5/7 model values against measured turbulence occurrence statistics of 15%, 50%, and 85%. The percentages in the legend reflect the amount of time during the year that the measured refractive index structure function, ${C}_{n}^{2}$, occurred. If this Korean data are representative of conditions to be expected by FSOC systems, one must be able to compensate for turbulence effects up to a $5\times $ HV 5/7 atmosphere, if not beyond, to be considered a high availability link. The subsequent anal ysis will reference the turbulence conditions as an equivalent multiple of HV 5/7 and thus suggest what percentage of the environments we can expect that the FSOC link will close.

## 6. Defining the Cumulative Distribution Function of Atmospheric Turbulence

Over the past five years, DARPA and the U.S. Air Force Research Laboratory (USAFRL) both have conducted field trials to generate data on which to validate optical link budgets. These include the AFRL-sponsored 2006/2008 Integrated RF/Optical Networked Tactical Targeting Networking Technologies (IRON-T2) static experiments over a $147\text{}\mathrm{km}$ path between Haleakala and Mauna Loa in Hawaii [13] and the 2008 AFRL-DARPA-sponsored Optical RF Communications Adjunct (ORCA) static experiments over a $10\text{}\mathrm{km}$ path at Campbell, California. Recently, validation of these estimates under dynamic conditions occurred in the ORCA Program using $70\text{}\mathrm{km}$ sea-level ground to low-altitude aircraft ($\mathrm{8,000}\text{}\mathrm{ft}$ above ground level) tests at the Patuxent River Naval Air Station (PAX), and in 50 to $200\text{}\mathrm{km}$ tests between Antelope Peak and a $\mathrm{26,000}\text{}\mathrm{ft}$ MSL altitude BAC1-11 aircraft at the Nevada Test and Training Range (NTTR). Let us begin our data analysis with the measurements from our two static experiments and then see how they can be used to define the cumulative distribution function (CDF) for atmospheric turbulence.

Figure 3 shows estimated results for the atmospheric conditions experienced during the referenced August 2008 $10\text{}\mathrm{km}$ Campbell experiments. The Rayleigh range estimated for this link is $4.5\text{}\mathrm{km}$ (see Fig. 1). In Fig. 3, we have computed the PIB and PIF derived for the resulting reduced turbulence range in decibels relative to a milliwatt or $0\text{}\mathrm{dB}\mathrm{mW}$. For our model comparison, we have modified the Fried parameter to reflect the receiver and transmitter coherence lengths of spherical waves and the reduced integration range; specifically, we write

The above characterization is for the average PIB and PIF at the detector. In general, one usually wants to know if the FSOC link is available during high turbulence conditions, e.g., the saturation regime, where the deep fade depths below the mean PIF occur. Given the above, let us now use the data from the static experiments to plot the CDFs for both PIB and PIF, and hopefully, derive relationships between the mean and the 99% fade depths of the PIB and PIF.

Figure 4 shows the CDF as a function of PIF for the Campbell experiments. It is clear in this log-log plot that the mean level and all higher order statistics are linearly related. The same relation holds for the PIB. This means we should be able to derive a linear mapping between the 99% and mean levels for these two measurements.

Figure 5 plots the transmitter and receiver SRs as a function of the mean PIB from the 2008 IRON-T2 Hawaii experiment [13]. This graph shows that there is a linear relationship, on a log-log plot, between the mean PIB and the two SRs. In other words, it demonstrates the PIB statistics follow a lognormal distribution, as expected.

From data such as that shown in Fig. 4, we can plot the 99% PIF as a function of the mean PIF and find that the former is nearly linearly proportional to the latter, with a constant of $-17$ to $-22\text{}\mathrm{dB}\text{}\mathrm{mW}$. An example from the IRON T2 tests in October 2008 is shown in Fig. 6 and another from the Campbell tests in August 2008 in Fig. 7, when appropriate curve fits to the data shown. Again, the data are well into the saturation region of the log amplitude variance or Rytov number (see, for example, in *The Infrared Handbook* pp. 6–21 [12]). By processing the available data, we conclude that the 99% fade depths over these long ranges are $19\text{}\mathrm{dB}\pm 2\text{}\mathrm{dB}$ and we can use this value in building our link budget.

## 7. Applying our Model to a FSOC Link Budget

With the above information, we are now in the position to create a link budget for a FSOC system and see how it compares to field measurements. Following normal engineering convention, it is convenient to construct the link budget using logarithmic val ues; specifically, in dBmW. Let us now look at more results from the 2008 IRON-T2 Hawaii tests [13].

As noted above, the Fried parameter, ${r}_{0}$, is important to our calculation. Unfortunately, it cannot be explicitly measured. However, video images of the beam near the Haleakala receiver allowed us to estimate ${r}_{0}$ to be frequently less than $20\text{}\mathrm{cm}$ by recognizing that the received speckle size should be on the order of ${r}_{0}$. The terrain around the Mauna Loa Volcano slopes gradually downward, and the beam is within $\mathrm{1,000}\text{}\mathrm{m}$ of the ground until it is $10\text{}\mathrm{km}$ away from the transmitter. The Haleakala Volcano wall is much steeper near the receiver and drops away below $\mathrm{1,000}\text{}\mathrm{m}$ at $3\text{}\mathrm{km}$ from the receiver. So to first order, one might expect a uniform value of the refractive index structure function like Campbell for most of the link, with a gradient existing near the Mona Loa terminal. To account for this situation, we can add a boundary layer near the transmitter corresponding to these beam heights and 100 times stronger than the value given by the HV 5/7 at these altitudes. That profile is shown in Fig. 8. This assumed profile produces a calculated ${r}_{0}$ of $16\text{}\mathrm{cm}$, which is consistent with the estimates indicated above.

Figure 9 shows the calculated link budgets for data taken during the 2008 Hawaii IRON-T2 tests referenced above. Two cases are shown in Fig. 9; the first case uses the normal $0.2\times $ HV 5/7 model shown in Fig. 1; the second uses the $0.2\times $ HV 5/7 and the aforementioned $100\times $ boundary layer. While the $0.2\times $ HV 5/7 model includes a significant boundary layer at altitudes closer to sea level, and the latter model assumes that boundary layer at the altitudes in the Hawaii tests, above 10,000 feet it is clear in the two cases that measurements and predictions show no significant difference in turbulence strength and calculated numbers, despite the presence of different ground effects near each end of the beam. This goes back to our previous point that the transmitter AO systems will correct all turbulent effects within ${R}_{\mathrm{RR}}$ from that transmitter aperture and near the receiver aperture, and all turbulent effects come from interval ${R}_{\mathrm{Tx}\mathrm{RR}}<r<L-{R}_{\mathrm{Rx}\mathrm{RR}}$. Here the Rayleigh ranges of $4.5\text{}\mathrm{km}$ only comprise a small portion of the $147\text{}\mathrm{km}$ total link separation.

The power exiting the transmitter aperture for Fig. 9 is estimated to be $14\text{}\mathrm{dB}\text{}\mathrm{mW}$. The spreading loss is calculated as described in Section 2. The HV 5/7 model, with a $100\times $ boundary layer, predicts that the ${C}_{n}^{2}$ between the transmitter on Mauna Loa and the receiver on Haleakala varies between $6\times {10}^{-16}$ and $8\times {10}^{-16}$. This would produce a transmitter coherence diameter, ${r}_{0}$, of $16\text{}\mathrm{cm}$ over the $147\text{}\mathrm{km}$ path length, leading to a transmitter Strehl loss of only $-2.0\text{}\mathrm{dB}$. The atmospheric absorption and scattering loss is estimated to be $5.0\text{}\mathrm{dB}$, using the approach described in Section 3, The AO of the transmitter could compensate for $0.1\text{}\mathrm{dB}$, using the calculation as described in Section 7. This leads to a predicted PIB of $-20.9\text{}\mathrm{dB}\text{}\mathrm{mW}$, compared to a measured $-19.0\text{}\mathrm{dB}\text{}\mathrm{mW}$. This is quite good agreement, given the assumptions involved.

Using reported values for the receiver transmittance and the circulator loss into the single-mode fiber and the Strehl loss of $-1.1\text{}\mathrm{dB}$, the link budget predicts a PIF of $-30.2\text{}\mathrm{dB}\text{}\mathrm{mW}$, compared to a measured value of $-29.4$. Using the $99\%\text{}\mathrm{dB}$ spread of $19\text{}\mathrm{dB}$, as described in Section 4, leads to an expected 99% fade power out of the fiber of $-49.2\text{}\mathrm{dB}\text{}\mathrm{mW}$ as compared to a measured value of $-47.5\text{}\mathrm{dB}\text{}\mathrm{mW}$.

## 8. More Link Budget Validation—NTTR and Patuxent River Naval Air Station Test Results

As described earlier, there also was ORCA testing between an aircraft at 9,000 to $10,\mathrm{000}\text{}\mathrm{ft}$ altitude and a sea-level ground site at the PAX and between an aircraft at $26,\mathrm{000}\text{}\mathrm{ft}$ and a mountain top at the NTTR in May of 2009. At the NTTR site, an independent estimate of ${r}_{0}$ was obtained from a Weather Research and Forecasting (WRF) model, as described in [16]. Those predictions are reproduced here in Fig. 10.

Figure 10 shows a more detailed look at the Fried parameter during all the NTTR tests. This picture shows the Fried parameter ${r}_{0}$ derived from WRF, plotted against the aircraft distance from Antelope Peak, for the six flights on 16–18 May, 2009. Values are for air-to-mountain to simulate air-to-air. (It should be noted that aero-optics effects are not included in the WRF modeling.) Each flight is shown in a different color. Although there is a large spread in ${r}_{0}$ values between different flights, each flight shows the correlation of ${r}_{0}$ with path length. Figure 11 shows the associated ground-level refractive index structure function, ${C}_{n}^{2}$, for the set of tests shown in Fig. 8. It is clear that the middle of the day had ${C}_{n}^{2}\sim 1\times {10}^{-12}\text{}{\mathrm{m}}^{-2/3}$; because strong turbulence is rated as ${C}_{n}^{2}\sim 1\times {10}^{-13}\text{}{\mathrm{m}}^{-2/3}$ or more ([14], p. 11), for most of the middle of that day, testing was accomplished with turbulence 10 times that seen in prior IRON-T2 testing [2].

Referring to Fig. 10, many of the flights had predicted strengths of turbulence of $25\times $ HV 5/7 model, which is five times the Korean 85% value of $5\times $ HV 5/7. Let us first look at the $5\times $ HV 5/7 ORCA data with prediction, and then how the other daytime data compare at this much higher value. Figure 12 shows predicted and measured PIB and PIF for data taken at NTTR on 17 May 2009, $18\text{:}54$ local time. Here PIB and PIF are median values of measurements and 99% PIF is derived from median PIF using the linear equation specified above. Once again, close agreement is shown between PIB and PIF predictions and measurements, all within $2\text{}\mathrm{dB}$.

Figure 13 shows predicted PIB and PIF and measured PIF for data taken under $25\times $ HV 5/7 conditions on 17 May 2009. As can be seen from the table in this figure, the PIF predictions for the $25\times $ HV 5/7 model, agree well with the measured results for most cases.

Note that the NTTR transmitter predicted AO gains are small, because the HV 5/7 model predicts weak turbulence at $\mathrm{26,000}\text{}\mathrm{ft}$. In fact, there was other evidence that the turbulent boundary layer around the aircraft may have contributed a 5 to $10\text{}\mathrm{dB}$ loss, as described in [6].

Finally, Fig. 14 shows predicted PIB and PIF and measured PIF for data taken under $3\times $ HV 5/7 conditions at PAX for 50 and $70\text{}\mathrm{km}$ ranges on 12 May 2009. The PAX results also show good agreement with the values predicted from the FSOC link budget. The low elevation angle in the PAX tests, between $2.25\xb0$ and $3.5\xb0$ produces an expected strong turbulence layer near the ground. This shows an expected 7 to $9\text{}\mathrm{dB}$ AO gain at the receiver, in good agreement with the measured results.

## 9. Summary

This paper described a new methodology of estimating the FSOC link budgets to be expected in conditions of severe turbulence. The approach is derived from observing that the ability of an AO system to compensate turbulence along a path is limited by the transmitter and receiver Rayleigh range, proportional to the diameter of the optics squared and inverse of the wavelength of light utilized. The method uses the Fried parameter over range outside the two Rayleigh ranges and inserts that value into the transmitter and receiver SR to yield reasonable prediction of the light impinging on the receiving telescope aperture and the power coupling into the fiber. Comparisons were given between theory and field measurements. These comparisons show that AO is most effective within the Rayleigh ranges or when an atmospheric gradient is present, e.g., the aero-optic effect, upslope wave effect, and lesser so when the total range is much greater than the sum of the Rayleigh ranges.

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