Outage Performance of Uplink Pre-Amplified FSO Links Over Turbulence, Beam Wander, and Pointing Errors

We analytically derive the outage probability (OP) estimation of a ground-to-satellite pre-amplified free-space optical (FSO) link. Our derivations analysis, for the first time, takes into account all the probabilistic impairments, i.e., atmospheric turbulence, beam wander and pointing errors (PEs). Novel, analytical closed-form expressions are extracted for the probability density function (PDF) and the cumulative distribution function (CDF) of the optical signal-to-noise ratio (OSNR) at the receiver pre-amplifier output. Furthermore, we conduct the asymptotic outage analysis in the high-OSNR regime, which provides insightful results for the uplink amplified FSO systems. Numerical OP results are presented for the case of a ground FSO link to a Geostationary Earth orbit (GEO) satellite under various turbulence conditions and misalignment scenarios. Finally, the whole outage analysis is verified by Monte Carlo simulations.


I. INTRODUCTION
O PTICAL satellite links are envisioned to play a key role in the future satellite networks and be integrated seamlessly with the terrestrial fiber networks [1].Today, the backbone of satellite networks is being designed to rely on very-high-speed free-space optical (FSO) links [2], establishing uplink/downlink ground-to-space links as well as inter-satellite optical links and capitalizing on the same technology developed for fiber communications [1].
Among those, the most challenging links are the feeder uplinks, e.g., from an optical ground station (OGS) to a satellite.Those feeder links can suffer from major impairments due to the atmospheric layer.Even in good visibility conditions, atmospheric turbulence can provoke severe irradiance fluctuations in the received signal.Refractive and diffractive phenomena lead to undesired scintillation, spreading, and beam wandering [3], [4], [5].In addition, feeder FSO systems require accurate alignment, so that any pointing jitter can significantly affect the system performance.As the diffractionlimited beam divergence angle is typically on the order of µrads, a pointing error (PE) higher than around 1 µrad severely impairs the link performance [6], [7].
Thus far, the performance of uplink FSO feeder links has been investigated in several letters.Firstly, the probability density function (PDF) under the assumption of two independent random variables (RVs) was derived [8], for various turbulence models and beam wander pointing jitter.The biterror rate (BER) of heterodyne detection schemes and the probability of fade for an uplink system were investigated [4], [5], under the composite channel models of the Gamma (G) and Gamma-Gamma (GG) distributions with beam wander as two independent factors.The BER performance of an uplink FSO link with intensity modulation/direct detection (IM/DD) schemes was studied under the GG turbulence and beam wander-induced pointing error [9].The development of a fading model with three independent RVs for GG-modeled turbulence, beam wander and PEs in a horizontal link, was firstly launched in [10].The same approach was then applied to an uplink FSO system with G-modeled turbulence, beam wander and PEs [11].
However, future high-throughput FSO systems are being designed to encompass an optical erbium-doped fiber amplifier (EDFA) as a booster at the transmitter (TX) and, most importantly, as a pre-amplifier at the receiver (RX) [12].EDFAs, which played a key role in high-speed fiber links, change the power-budget rules and make the power-sensitivity of the RX unimportant; when using them, the thermal and shot noises become negligible and the dominant source of noise is the amplified spontaneous emission (ASE), through the signal-ASE and ASE-ASE beating noise.Hence, in this regime the system performance is only determined by the optical signalto-noise ratio (OSNR) at the pre-amplifier output [13]; this must exceed a minimum required value, which depends on the modulation format, bit rate and type of forward error correction (FEC) at the RX [13].
In this letter, we develop an analytical framework for the outage probability (OP) estimation of a ground-to-satellite preamplified FSO uplink.We take into account jointly the effects of atmospheric turbulence, beam wander, and PEs, each one treated as an independent RV.Under this assumption, we are able to independently assess the severity of each effect.The effect of scintillation is modelled by the G and GG distribution models, in order to provide a comprehensive assessment of every turbulence scenario.Specifically, in the weak turbulence regime, i.e. small zenith angles, the G distribution model is employed, while the GG model is selected for moderate to strong turbulence conditions [3], [4].For the statistical evaluation of PEs, we employ the PE model of Toyoshima et al. [6], which is very accurate and appropriate for uplink laser communications and combined with the G and GG distribution models for the first time.For the beam wander effect, we consider the tracked beam case, i.e. tip-tilt adaptive optics (AO) pre-compensation [14], [15], and evaluate its impact as an independent source of spatial jitter by means of the rootmean-square (RMS) angular jitter.To the best of the authors' knowledge, this is the first attempt to evaluate the performance of an uplink FSO system with pre-amplification under the composite channel of G and GG distributions with beam wander and PEs, unlike the cases of [4], [5], and [9] where no PEs are considered or [11] where only the G distribution model is employed without pre-amplification.We evaluate its OP by means of new and tractable closed-form expressions for the PDF and the cumulative distribution function (CDF) of the OSNR at the pre-amplifier output.Finally, asymptotic analysis in the high-OSNR regime is conducted, providing insightful results for such laser communications systems.

II. SYSTEM MODEL
In an amplified FSO system, the TX emits a signal, which is amplified by a booster EDFA and launched into free space by the TX telescope with average transmitted power P 0 , as shown in Fig. 1.The received beam is collected by a second telescope, and P in power is coupled into a single-mode fiber, amplified by a low-noise EDFA pre-amplifier and detected by a DD receiver or an array of balanced detectors.
As P in is affected by the various effects described above, the composite channel coefficient I can be represented as a product of four independent factors, i.e.I = I l I t I b I p .Among them, I l includes all the fixed losses of the link, i.e., extinction effects (absorption, scattering), losses due to free-space loss, TX/RX antenna gains and optics losses, fiber coupling loss etc. [3], [12], [16].The other terms correspond to the three RVs of the channel, where I t represents the scintillation effect, I b the beam wander, and I p the PEs [3], [4], [6].For the sake of simplicity, we define the RV I ′ t as I ′ t = I l I t .As known, the instantaneous OSNR at the EDFA output is related to the instantaneous power P in = P 0 I as [17] OSNR where G is the EDFA gain, N F is the pre-amplifier noise figure, h is Planck's constant, ν is the optical carrier frequency and ∆ν is the resolution bandwidth, usually fixed at 12.5 GHz (0.1 nm) [13], without considering background noise. 1 Since the power fluctuations do not move the EDFA into saturation, the ASE spectral density and the EDFA gain are both constant, therefore the irradiance fluctuations produce a time-varying OSNR [17].Note also that, in case of a wavelength division multiplexing (WDM) system, P 0 I is the per-channel input power.Finally, the average OSNR is given by where E{•} denotes the expected value of the enclosed, with E{I} = E{I ′ t }E{I b }E{I p }.We highlight that modern optical systems exploit FEC; therefore, it is known that error-free transmission can be achieved at any OSNR value greater than a threshold OSNR (OSNR th ), which can be linked directly to standard pre-FEC BER targets [13].
From the OGS to the satellite, the optical signal experiences the degradation effects of atmospheric turbulence.The G distribution is generally considered an accurate model for the uplink irradiance fluctuations for the weak turbulence regime [3].Its mathematical representation is given as [3], [4] where m is the G distribution parameter defined as m = exp σ 2 ln X + σ 2 ln Y − 1 −1 [3, Eq. ( 24)], with σ 2 ln X and σ 2 ln Y denoting the small-scale and large-scale log-irradiance variances, related to the wavelength λ, the link distance L, the C 2 n (h) turbulence profile and the beam parameters.The link distance is calculated as L = (H − H OGS ) sec(ζ), where H OGS is the OGS altitude and H, ζ are the altitude and zenith angle of the satellite, respectively [3].
For the moderate to strong turbulence regime, it is well-known that the GG distribution accurately models the irradiance fluctuations.Its PDF is given as [3] with [3, Eq. ( 22)].The mean irradiance E{I t } is calculated as [5, Eq. ( 13)].For the C 2 n (h), we use the modified Hufnagel-Valley model, which takes into account the OGS altitude above sea level [16].
Considering a Gaussian beam, where the azimuth and elevation angles are affected by independent Gaussian angular jitters with zero mean and identical angular variances, σ 2 θ,j , the angular pointing error, θ, follows a Rayleigh distribution [6], where j ∈ {p, b} stands for the case of misalignment either due to pointing errors or beam wander.Therefore, the instantaneous normalized irradiance for PEs, I p , or beam wander, I b , follows a beta distribution [6], given by where q = θ 2 div /4σ 2 θ,p , and β w = θ 2 div /4σ 2 θ,b , respectively.The expected values of I p and I b are equal to E{I p } = q (q + 1) −1 and E{I b } = β w (β w + 1) −1 [6].The half-angle Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
beam divergence, θ div , is evaluated through the expression [19, Eq. ( 14)], and is connected with the beam spot radius W 0 at the TX and the diameter D T X of the outer circular aperture of the TX telescope.
The PEs, which have a strong impact on the system performance, including the effect of wind speed, mechanical vibrations and errors in the pointing and tracking subsystem.All contribute to an RMS fashion, which amounts to a total pointing jitter, σ θ,p , ranging from 0.5 to 2 µrads for weak to strong influence [6], [7].
In addition, beam wander causes further beam movement.The RMS radial displacement due to beam wander, considering the case of a tracked beam where the Zernike tilt can ideally be subtracted [14], [15], is calculated as where r 2 c is the RMS beam wander displacement, calculated as r 2 c ≈ 0.54 L 2 (λ/2W 0 ) 2 (2W 0 /r 0 ) 5/3 for H ≫ 20 km [3], and T 2 z = 0.3641 (λ/D R ) 2 (D R /r 0 ) 5/3 is the Zernike tilt angle variance [14], with D R the receiver aperture diameter of the wave-front sensor and r 0 the Fried's parameter.The RMS angular displacement due to beam wander is defined as

III. OUTAGE PROBABILITY ANALYSIS A. Exact Closed-Form Expressions
The outage probability of the amplified uplink can be evaluated by having the CDF F OSNR (x) of the time-varying OSNR.As a first step, we calculate the PDF f I (I) of the composite channel coefficient I, by using the multiplicative property of the Mellin transform (MT) [20].Considering that the composite channel coefficient is I = I ′ t I b I p , the product of the MTs of their PDFs is given by M (f where M (f X (x) | s) denotes the MT of the PDF f X (x) and s is a complex number in the s-domain.First, we calculate the MT for the G distribution, Eq. ( 3), which is obtained as The MT for the two beta distributions of Eqs. ( 6) for PEs and beam wander are calculated according to [20] Consequently, we take the inverse MT of (9) as f I (I) =  10) and (11), we arrive at By noticing this Mellin-Barnes integral, the inverse MT is solved by using consecutively [21,Eqs. (6.422.19),(9.31.2), (9.31.5)], and calculating the PDF, f I (I), in a closed-form expression, as where G m,n p,q (z| a p ; b q ) denotes the Meijer's G-function [22,Eq. (9.301)].Following a RV transformation, we obtain the PDF of OSNR as follows The CDF of OSNR is derived in a closed-form solution, by means of the formula in [22, Eq. ( 26)] as Thus, the OP can be evaluated at a specified OSNR th , as OP = F OSNR,G (OSNR th ) [13].
Concerning the case of the GG distribution with beam wander and PEs, we follow similar steps as analysed above.Specifically, the MT of the GG distribution, Eq. ( 4), is given as [10] Replacing ( 16) into (9) and using Eqs.(11), the inverse MT of the product can readily be calculated following the steps as in (12).Thus, the PDF of the total channel coefficient I, for the case of GG model, is calculated as The PDF of OSNR for the GG turbulence case with beam wander and PEs is derived accordingly as while the corresponding CDF of OSNR for the GG model is deduced as

B. Asymptotic Analysis
In addition to the closed-form expressions of Eqs. ( 15) and ( 19), we provide an asymptotic analysis, which is valid when OSNR → ∞.In order to derive an asymptotic approximation of (15), we use the expansion formula for the Meijer's Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
G-function in terms of the generalized hypergeometric function q F p (a q , b p ; z) as given in [21, Eq. (9.303)], where When OSNR → ∞, i.e. z G ≪ 1, then q F p (a q , b p ; z G ) → 1 and ( 15) is approximated as follows Following the same procedure as described above, we obtain the asymptotic expression of ( 19) as follows where z GG is equal to From Eqs. ( 21) and ( 22), we notice the asymptotic behaviour of the OP, as OP ≈ O c OSNR −O d , where O d denotes the outage diversity order.Thus, the outage diversity order, for the G and GG turbulence models, is given as IV. NUMERICAL RESULTS With the aid of the closed-form expressions of ( 15), (19) and the asymptotic expressions of ( 21) and (22), we provide numerical results for the OP estimation of an uplink-to-Geostationary Earth orbit (GEO) satellite (altitude H = 35786 km) FSO link.For all the results, we consider OSNR th = 10 dB.We assume an optical wavelength λ = 1.55 µm, the beam spot radius W 0 = 122 mm, TX telescope and sensor diameters D T X , D R = 270 mm, and the phase front radius F 0 → ∞.The RMS wind speed is υ rms = 21 m/s and we assume moderate and strong ground level turbulence with C 2 n (0) = 1.7 × 10 −14 m −2/3 and C 2 n (0) = 1.7 × 10 −13 m −2/3 , considered as realistic nighttime and daytime conditions, respectively.We assume two zenith angles where for ζ = 30 o the G model is employed, whilst for ζ = 60 o the GG distribution is used.All the numerical results  are accompanied by extensive Monte Carlo simulations using 2 × 10 6 realizations.
In Fig. 2 the OP is shown versus OSNR for the case of an OGS at H OGS = 0 km.The blue curves refer to weak PEs, whilst the red curves refer to quite high PEs.Also, different ζ values are assumed, i.e. 30 o (triangles) and 60 o (diamonds).It can clearly be noticed that very low OP is obtained at quite acceptable OSNR values (e.g.20 dB).Furthermore, the turbulence impact is predominant when PEs are low, observing the remarkable impact of ζ.If PEs are high, its statistical impact dominates over all other effects and much higher OSNR values are needed to achieve the same outage performance.All these curves are accompanied by asymptotic results, plotted in the whole range of OSNR by using only the dominant term J χ,χ which in turn determines the diversity order in the high-OSNR regime.We can clearly notice the tightness of the asymptotics even in the low to medium OSNR regime, emulating perfectly the closed-form results.In Fig. 3 the OP results are illustrated for similar conditions, but for H OGS = 1 km.In this case, we obtain a lower impact of atmospheric effects.This is clearly noticeable, however, only for weak PEs of blue curves.For any OSNR value, the corresponding OP is acceptable for both values of ζ.The asymptotic curves follow firmly the closed-form ones with the use of J 2,G and J 2,GG terms, respectively.However, the OP corresponding to the red curves (σ θ,p = 2 µrad) shows only minor improvements between the two zenith angles.This confirms that in this regime the system outage is largely determined by PEs, verified also by the asymptotic terms of J 3,G and J 3,GG in the whole OSNR range.Finally, Figs. 4 and 5, present the OP results considering strong ground level turbulence, which correspond to realistic daytime conditions, for the cases of OGS height at H OGS = 0 km and H OGS = 1 km, respectively.For the case of H OGS = 0 km (Fig. 4), we notice the performance degradation across all the range of OSNR, where neither low nor high satellite-zenith-angle FSO uplinks can be implemented.The asymptotic results are plotted by using only the J 2,G and J 2,GG terms, from low to high OSNR, revealing the tremendous impact of beam wander effect under such conditions.On the other hand, in the case of H OGS = 1 km (Fig. 5), it is clearly observed that the OP performance can be improved and retained at acceptable levels.Again, under strong PEs influence the performance is strongly affected and OSNR values above 27 dB are required for acceptable performance.For the case of weak PEs, we observe how critical becomes the influence of ζ on the uplink FSO system performance.The asymptotic results are plotted by using all the terms of ( 21) and ( 22) and some deviations are noticed, especially in the low OSNR regime.
V. CONCLUSION This letter provides a comprehensive and analytical performance estimation of an optically pre-amplified ground-tosatellite FSO link by means of novel and tractable closed-form and asymptotic OP analysis in terms of the OSNR at the receiver end.We take into account the G and the GG-modeled turbulence in conjunction with beam wander and PEs, each one of them as an independent RV.The derived mathematical framework can be employed and extended to a plethora of uplink scenarios, including the design of WDM-FSO systems by means of a few simple generalizations.

Fig. 1 .
Fig. 1.Block diagram of the uplink ground-to-satellite pre-amplified FSO communication system.
denotes the modified Bessel function of the second kind with a, b being equal to a = exp σ 2 ln X − 1 −1 and b = exp σ 2 ln Y − 1 −1 M (f I (I) | s) I −s ds, and by virtue of Eqs. (