Uplink laser satellite-communication system performance for a Gaussian beam propagating through three-layer altitude spectrum of weak-turbulence

Optik 124 (2013) 2916–2919

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Uplink laser satellite-communication system performance for a Gaussian beam propagating through three-layer altitude spectrum of weak-turbulence Xiang Yi ∗ , Zeng-ji Liu, Peng Yue State Key Laboratory on Integrated Services Networks, Xidian University, Xi’an 710071, PR China

a r t i c l e

i n f o

Article history: Received 21 April 2012 Accepted 24 August 2012

Keywords: Atmospheric turbulence Non-Kolmogorov spectrum Uplink laser satellite communication Beam wander Bit-error rate analysis

a b s t r a c t Uplink laser satellite-communication (satcom) system performance has been extensively studied under the assumption that atmospheric turbulence statistics obey a Kolmogorov power-law spectrum model. Unfortunately recent evidence has indicated that Kolmogorov spectrum is sometimes incomplete to describe atmospheric turbulence statistics properly, in particular in the upper atmosphere. In this paper, we use a three-layer attitude spectrum model that exhibits non-Kolmogorov properties in portions of the troposphere and stratosphere. Using this spectrum in weak turbulence, we analyze the bit-error rate (BER) performance of an uplink laser satcom system with a collimated untracked Gaussian beam subject to scintillation and beam wander impairments. Our theoretical results show that the difference in BER between the three-layer altitude spectrum and the Kolmogorov spectrum is not appreciable for smaller transmitter beam radius, but is significant at larger transmitter beam radius. Furthermore, for a given laser wavelength or zenith angle, the optimum transmitter beam radius that can minimize the BER is very different between the two turbulence spectrums. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction The possibility of using high-data-rate optical transmitters/receivers for satellite communication (satcom) channels has generated interest over many years in laser communication systems for both ground-to-satellite (uplink) and satellite-to-ground (downlink) data links. However, by propagating laser beams through atmosphere, the performance of such optical communication links is highly vulnerable to atmospheric turbulence. Turbulence causes irradiance scintillations that induce deep random fades and consequently result in degradation of system performance [1–3]. Moreover, for a Gaussian beam on an uplink path to a satellite, turbulence-induced beam wander may increase the scintillation level by a significant amount [4–7], and therefore must be taken into account in the associated uplink laser sactom systems. Several recent studies on uplink laser sactom systems with consideration of beam-wander-induced scintillation have presented bit error rate (BER) performance analysis assuming the Kolmogorov spectrum for atmospheric turbulence [8–11]. Although the Kolmogorov spectrum has shown good agreement with experiment in the past, recent evidence has indicated great deviations from the Kolmogorov model in portions of the troposphere and stratosphere [12,13]. By using a general spectrum

∗ Corresponding author. Tel.: +86 29 88201007; fax: +86 29 88201007. E-mail address: [email protected] (X. Yi). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.08.082

exponent value in the range of 3–5 rather than a constant value of 11/3 (for Kolmogorov turbulence), the authors of [14] introduced a non-Kolmogorov spectrum model. Based on this model, the performance of laser communication systems for an optical wave propagating through turbulence along a horizontal or a vertical/slant path has been studied in [15–17]. However, propagation along a vertical/slant path requires a spectrum model to describe properly the varying of turbulence statistical characteristics as a function of altitude. In [18], a two-layer altitude spectrum model that exhibits Kolmogorov properties below the altitude of 6 km and non-Kolmogorov properties (with spectrum exponent value of 5) above 6 km was employed to analyze the impact of atmospheric turbulence on laser satcom system performance. But it was indicated in [19] that the two-layer altitude model is less accurate than a newly proposed three-layer altitude model. Thus in [20], some propagation properties associated with an uplink/downlink Gaussian beam through the turbulent atmosphere of the threelayer altitude model was studied. Also using the three-layer altitude model for atmospheric turbulence, the authors of [21] investigated fade statistics and average BER of downlink and uplink laser satcom systems under weak turbulence conditions. However, the investigation in [21] assumed an infinite spherical wave model for the uplink propagation, which cannot predict fades in irradiance induced by beam wander. Therefore in the present study, we analyze the average BER of uplink laser sactom systems for a finite Gaussian beam propagating through the three-layer altitude spectrum of weak turbulence.

X. Yi et al. / Optik 124 (2013) 2916–2919

2. The three-layer altitude spectrum The three-layer altitude spectrum model that is used to characterize atmospheric turbulence statistics has the following presentation [19]: ˚n (, ˛) = A(˛)ˇ−˛ ,

˛(h) =

˛1 1 + (h/H1 ) +

+

b1

˛3 · (h/H2 ) 1 + (h/H2 )

˛2 · (h/H1 ) 1 + (h/H1 )

b1

b1

·

1 1 + (h/H2 )

b2

b2

b2

,

(2)

where ˛1 = 11/3, ˛2 = 10/3 and ˛3 = 5 are spectrum exponent values for boundary layer Kolmogorov turbulence, free troposphere nonKolmogorov turbulence and free stratosphere non-Kolmogorov turbulence, respectively. H1 and H2 are boundary layer altitudes. b1 and b2 are numerical coefficients of the model that describe the flatness between layers. The exact values of H1 and H2 as well as coefficients b1 and b2 have not yet been determined. For the purpose of this study, H1 = 2 km, H2 = 8 km, b1 = 8, and b2 = 10 are used like [19]. As suggested in [19], ˇ also varies with h and takes the form

 k ˛/2−11/6 C 2 (h) n

ˇ(h) = 0.033

L

A(˛)

 k ˛/2−11/6 L

Cn2 (h)−˛ .

(4)

It is to be noted that the three-layer altitude spectrum expressed in Eq. (4) reduces to conventional Kolmogorov spectrum [1, Eq. (12.15)] when ˛ is fixed at 11/3. 3. Scintillation index We assume that the transmitted optical wave is a collimated lowest-order mode paraxial Gaussian-beam wave with unit amplitude. In the absence of turbulence, such a Gaussian beam received by at distance L from the transmitter can be described   the transmitter beam parameters 0 = 1 and 0 = 2L/ kW02 , where W0 represents the beam radius at the transmitter. Correspondingly, the beam that is incident upon the receiver  plane is described by the receiver beam parameters: = 0 / 20 + 20 , = 1 − ,



 H

= 82 k2 sec()Re

2 I,l (0, L)





(5)



exp

−

L k

∞

˚n (, ˛, h)



h0 2 2



0

− exp

 2 (r, L) = I,r

H



−

−



82 k2 sec() × exp



iL2 [1 − ( + i ) ] k

(6) ddh,

∞

˚n (, ˛, h)



h0 0 L2 2

k

(7) [I0 (2 r ) − 1]ddh,

where Re denotes the real part of the argument, I0 (x) is the modified Bessel function of the first kind, and = 1 − (h − h0 )/(H − h0 ) is the normalized distance variable for uplink propagation. If we insert the three-layer altitude spectrum Eq. (4) into the integrals in Eqs. (6) and (7), the resulting longitudinal and radial scintillation indices become 5/6

sec11/6 (),

5/6

sec11/6 ()

2 I,l (0, L) = 1.303 1 k7/6 (H − h0 )

2 I,r (r, L) = 1.303 2 k7/6 (H − h0 )

where



H

1 = Re ×





1 − ˛

r2 , W2

(9)

Cn2 (h)

2

h0

(8)

(˛−2)/2 (˛−2) − (˛−2)/2 [ + i(1 − )]

(˛−2)/2

dh, (10)



H

2 =

(2 − ˛)

1 − ˛ 2

h0

Cn2 (h) ˛/2−1 ˛−2 dh.

(11)

The behavior of irradiance scintillations under weak turbulence conditions for a collimated Gaussian beam on an uplink path to a satellite is actually a combination of atmospherically induced scintillation and that caused by appreciable beam wander [1, Section 12.6.2]. To account for the beam-wander-induced scintillation, we will adopt the approach developed in [1, Section 8.3]. As presented in [1, Section 8.3], the beam wander causes a widening of the long-term beam profile near the boresight that leads to a slightly “flattened” beam. For an untracked Gaussian beam the net result of the flattened beam profile is an “effective pointing error”  pe that creates an increase in the longitudinal scintillation index that is not accounted for in the conventional weak turbulence theory. The variance of the “effective pointing error” caused by beam wander is given by [1, Eq. (8.36)]

 H

2 = pe



∞



˚n (, ˛, h) 1 − exp

42 k2 sec()W 2 h0

0 2



and = 2L/ kW 2 , where W = W0 20 + 20 denotes the beam radius at the receiver [2]. Based on conventional weak scintillation theory [1,2], the scintillation index for an uplink Gaussian-beam wave to a satellite can be expressed as a sum of longitudinal and radial components in the form 2 2 I2 (r, L) = I,l (0, L) + I,r (r, L),

×

(3)

,

where k = 2/ is the wave number at the wavelength ; L is the propagation distance; Cn2 (h) is the conventional refractive index structure parameter and has the units of m−2/3 . For a vertical or a slant path, the total propagation distance to a satellite can be described by L = (H − h0 )sec(). Here, H is the satellite altitude, h0 denotes the height above the ground of the uplink transmitter, and  is the zenith angle. Substituting Eq. (3) into Eq. (1), we obtain ˚n (, ˛, h) = 0.033

where r represents the radial distance from the beam center that 2 2 varies from 0 to W, I,l (0, L) and I,r (r, L) are longitudinal and radial terms defined, respectively, by

(1)

where  is the magnitude of the wave number vector (in units of rad/m), ˛ is the spectrum exponent, ˇ is the general refractive index structure parameter with the units of m3−˛ , and A(˛) =  (˛ − 1) · cos(˛/2)/(42 ) with (·) being the Gamma function. Here, ˛ is assumed to change with the altitude h and defined by [19], Eq. (3)

2917





× exp[−2 W02 ( 0 + 0 ) ] 1 − exp

−2 r2



− L2 k

 (12)

ddh

where r = Cr /r0 . Cr is a scaling constant typically on the order of Cr ∼ 2 and r0 is the atmospheric coherence width of a reciprocal propagating point source form the receiver at distance L. It is known from [22] that the atmospheric coherence width of non-Kolmogorov turbulence is determined by the plane wave structure function Dpl (,˛) and the parameter C(˛) =



˛/2−1

2 8 [2/ (˛ − 2)]/(˛ − 2)

. Based on the weak turbulence

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theory for the slant paths [1, Section 8.7.2] and the three-layer altitude spectrum Eq. (4), Dpl (,˛) is derived by = −47.85k1/6 (H − h0 )

Dpl (, ˛)



H

× h0

−11/6

sec−5/6 ()



 (1 − ˛/2) k 2˛  (˛/2) (H − h0 ) sec()

˛/2

BER = Cn2 (h)˛−2 dh. (13)

For ˛ ≡ ˛(h) Eq. (2), r0 can be evaluated by numerically solving Dpl (r0 , ˛)/C(˛) = 1. Substituting Eq. (4) into Eq. (12), the “effective pointing error” variance based on the three-layer altitude spectrum is given by

 2 pe

= 1.303k−11/6 (H

×

2 − ˛ 2

− h0 )

23/6



(H − h ) sec() −˛/2 0

h0



W0˛−4 1 −

H

sec29/6 () r2 W02

2−˛/2 

k Cn2 (h) 2 dh.

1 + r2 W02

(14) 2 Replacing r in the radial scintillation index Eq. (9) with pe derived by Eq. (14), the longitudinal scintillation induced by beam wander can be written as 2 I,lbw (0, L) = 1.303 2 k7/6 (H − h0 )

5/6

sec11/6 ()

2 pe

W2

.

(15)

Following the same procedure as discussed in [1, Section 12.6.3], the radial scintillation index in the presence of beam wander is given by 2 I,rbw (r, L) = 1.303 2 k7/6 (H − h0 )

5/6

considered laser satcom system with the detection threshold IT can be expressed as [8,9]

sec11/6 ()(r − pe )2

U(r − pe ) W2 (16)

where U(x) is the unit step function. By combing Eqs. (8), (15), and (16), we obtain the scintillation index for a collimated untracked Gaussian beam in the presence of beam wander given by 2 2 2 I2 (r, L) = I,rbw (r, L) + I,lbw (0, L) + I,l (0, L), 0 ≤ r ≤ W.

(17)

1 2



0

IT

fI (I)dI =

1 erfc 4





2 −  2 (r, L)/2 0.23FT − 2r 2 /WLT I , √ 2I (r, L) (19)

where erfc(·) is the complementary error function and FT =  10log10 I (0, L) /IT is the fade threshold parameter with units of dB. By using Eq. (17) in Eq. (19), we can obtain the uplink BER based on the three-layer altitude spectrum. 5. Numerical results and discussions In this section we provide some numerical examples to show the influence of the three-layer altitude spectrum modeled turbulence on the uplink laser satcom system performance. For simplicity, we take geosynchronous orbit satellite (H = 38,500 km and h0 = 0) as an example. Moreover, we choose the prevalent H-V5/7 profile model for Cn2 (h) with upper atmospheric wind speed 21 m/s and ground-level Cn2 value 1.7 × 10−14 m−2/3 [2, Eq. (1)]. To calculate the scintillation index Eq. (17), ˛ ≡ ˛ (h) Eq. (2) is employed in Eqs. (10), (13) and (14). In Fig. 1(a) the uplink BER for a collimated beam through turbulence described by the three-layer altitude spectrum is shown as a function of W0 for a zenith angle zero and a given fade level FT = 5 dB. The transmitted beam operates at  = 1550 nm. The receiver is located on the optical axis of the beam (r = 0). For comparison purposes, the uplink BER based on the Kolmogorov spectrum is also illustrated. Note that the BER results for the three-layer altitude and Kolmogorov spectrums are basically the same only for W0 < 1 cm, and beyond that are vastly different. Also note that with increasing W0 , the uplink BER based on the three-layer altitude spectrum initially falls, reaches a minimum value and then increases. This general behavior of the BER is essentially the same behavior shown for the Kolmogorov spectrum. However, the optimum W0 (named Wopt ) that can minimize the BER is very different between the two spectrums. The reason for the phenomenon in Fig. 1(a) can be deduced from Fig. 1(b) in which the longitudinal scintillation index values (solid curves) corresponding to the BER curves in Fig. 1(a) are illustrated. The longitudinal scintillation

4. Average BER analysis Weak turbulence theory is sufficient for most of the calculations required in the analysis of a slant path provided the zenith angle is sufficiently small (less than 60◦ in most cases). Hence, our analysis here focus on weak turbulence environments where the irradiance statistics of an optical wave are usually assumed to be governed by the lognormal probability density function (PDF) model [1]. For a Gaussian-beam wave, the lognormal PDF model takes the form [1, Eq. (12.65)]

  fI (I) =

1

exp √ II (r, L) 2

−







2 ln(I/ I (0, L) ) + 2r 2 /WLT + I2 (r, L)/2

2I2 (r, L)

, I > 0,

(18)





where I (0, L) is the mean irradiance at the optical axis (r = 0) and WLT is the long-term spot radius. The main purpose of this work is to investigate how the three-layer altitude spectrum of atmospheric turbulence affects the performance of the uplink laser satcom system, and so, for simplicity, we consider an on-off-keyed system with sufficiently high signal-to-noise ratio. In this case, the average BER of the

Fig. 1. (a) Uplink BER, (b) longitudinal scintillation index as a function of W0 .

X. Yi et al. / Optik 124 (2013) 2916–2919

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with previous results obtained by using the Kolmogorov spectrum. It is shown that the difference of the BER between the two turbulence spectrums is not obvious for smaller transmitter beam radius, but is significant at larger transmitter beam radius. There also exits an optimum transmitter beam radius that can minimize the BER in the case of the three-layer altitude spectrum similar to that in the Kolmomogrov case. However, for a given wavelength or zenith angle, the optimum beam radius is very different between the two spectrum models. Acknowledgements Fig. 2. Longitudinal scintillation index against zenith angle at each optimum W0 for two typical wavelengths (850 nm and 1550 nm).

2 index is composed of the beam wander term I,lbw and the con2 . For interpretation purpose,  2 ventional scintillation term I,l I,lbw

2 (dashed line) based on the three-altitude (dotted line) and I,l spectrum are also plotted in Fig. 1(b) as a function of W0 along with previous results obtained based on the Kolmogorov spectrum [9]. Here we note that for smaller beam radiuses, there is no appreciable difference between the three-layer altitude spectrum and the Kolmogorov spectrum in predicting the longitudinal 2 , consistent scintillation index that is determined primarily by I,l with the result of the previous analysis involving spherical waves 2 for small beams is similar to [21]. This is due to the fact that I,l that for spherical waves [2]. With increasing W0 , the difference 2 and  2 between the two turbulence spectrums becomes of I,l I,lbw larger and larger, consequently leading to the increasing difference of the longitudinal scintillation index between the two spectrum models. Furthermore, the curves corresponding to the three-layer altitude spectrum mostly lie below those linked to the Kolmogorov spectrum. This phenomenon can be interpreted as follows: The longitudinal scintillation terms are caused primarily by high-altitude turbulence [2]. The three-layer altitude spectrum at high altitude (H > 8 km) obeys non-Kolmogorov power law of 5. As presented in [18], the turbulence spectrum with spectrum exponent of 5 predicts reduced scintillation than does the Kolmogorov spectrum. It is clear that in the case of the three-layer altitude spectrum, Wopt 2 2 with and I,l also results from the different varying trends of I,lbw W0 , which is similar to that in the Kolmogorov case. The difference in Wopt between the two turbulence spectrums is induced by the 2 difference between the two spectrum models in predicting I,lbw 2 . and I,l In Fig. 2 the longitudinal scintillation index of the uplink collimated beam is depicted against the zenith angle at each Wopt for two typical laser wavelengths 850 nm (dashed line) and 1550 nm (solid line). Both the Kolmogorov and three-layer altitude spectrum models are featured in this figure for the purpose of comparison. It can be seen that at each Wopt the three-layer altitude spectrum predicts a lower scintillation index as compared with that of the Kolmogorov spectrum. Wopt in the three-layer altitude spectrum case also changes with the laser wavelength and zenith angle. But for a given wavelength or zenith angle, the newly found Wopt is vastly larger than that previously discovered in the Kolmogorov case [9].

6. Conclusion In conclusion, this letter presents a study on the BER performance of an uplink laser satcom system for a collimated untracked Gaussian beam propagating through weak turbulence described by the three-layer altitude spectrum. The obtained results include the effects from scintillation as well as beam wander, and are compared

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