Coherent free space optics communications over the maritime atmosphere with use of adaptive optics for beam wavefront correction Ming Li1,2 and Milorad Cvijetic1,* 1
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
2
State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China *Corresponding author: [email protected] Received 12 November 2014; accepted 4 January 2015; posted 12 January 2015 (Doc. ID 226797); published 19 February 2015
We evaluate the performance of the coherent free space optics (FSO) employing quadrature array phase-shift keying (QPSK) modulation over the maritime atmosphere with atmospheric turbulence compensated by use of adaptive optics (AO). We have established a comprehensive FSO channel model for maritime conditions and also made a comprehensive comparison of performance between the maritime and terrestrial atmospheric links. The FSO links are modeled based on the intensity attenuation resulting from scattering and absorption effects, the log-amplitude fluctuations, and the phase distortions induced by turbulence. The obtained results show that the FSO system performance measured by the bit-error-rate (BER) can be significantly improved when the optimization of the AO system is achieved. Also, we find that the higher BER is observed in the maritime FSO channel with atmospheric turbulence, as compared to the terrestrial FSO systems if they experience the same turbulence strength. © 2015 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (060.1660) Coherent communications; (010.1330) Atmospheric turbulence; (010.1080) Active or adaptive optics. http://dx.doi.org/10.1364/AO.54.001453
1. Introduction
It is well recognized that free-space optical (FSO) communications can offer considerable advantages over radio frequency (RF) communications. The larger antenna gain of the FSO system has an impact on the smaller size, weight, and power of terminals. Also, the FSO systems operate over a large, unregulated, and license-free frequency spectrum. As compared to a microwave radio beam, the laser beam in an FSO channel has extremely narrow divergence so that the FSO system offers interference-free properties and immunity against eavesdropping. Some other advantages include possible use for quantum key distribution (QKD) applications, and more flexibility 1559-128X/15/061453-10$15.00/0 © 2015 Optical Society of America
in installation and eventual relocation. On the other hand, it is well known that the coherent detection scheme, when compared with the traditional intensity modulation direct detection (IM/DD) scheme, has the ability to suppress the impact of both background and thermal noise. Accordingly, the sensitivity of the coherent receiver is significantly higher than that of IM/DD, although the system complexity has been added. Furthermore, the coherent detection scheme is applicable for the multilevel modulation formats, such as M-array phase-shift keying (MPSK) and quadrature amplitude modulation (MQAM), thus offering a considerable increase in the spectral efficiency. All of these benefits outline the capability of coherent detection-based FSO systems as one of the most promising candidates for a number of applications. With increasing activities and communication needs in the maritime environment, coherent FSO 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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over the maritime atmosphere can be considered as a formidable candidate for information transfer for different purposes (such as the vessel-to-vessel or ship-to-satellite communications). However, the performance of coherent FSO can be highly degraded by the deleterious maritime atmospheric turbulence. To the best of our knowledge, the vast majority of coherent FSO channel analyses to date have only involved the terrestrial atmospheric links, while the specific analysis of the coherent FSO system behavior operating over the maritime atmospheric turbulence channel has not been reported. That also means that the improvement in performance has not been studied when an adaptive optics (AO) system is applied to mitigate the turbulence effects. As for terrestrial systems, we should mention that, based on the statistical description of terrestrial atmospheric turbulence, Belmonte and Khan [1] introduced a specific expression that represented the probability density function (PDF) of the symbolerror-rate (SER) of a coherent FSO system by using appropriate approximations. Also, the degradation in SER performance induced by turbulence effects and the improvement by the simple modal compensation in system performance were shown in [1]. The fading in the satellite-to-ground FSO uplink was studied in [2] when the beam scintillation and wander effects caused by the terrestrial atmospheric turbulence were considered. In parallel, the signal fading resulting from the terrestrial atmospheric turbulence was calculated in [3,4]. It was also demonstrated in [3,4] that the coherent FSO system has the potential for further improvement when the modal compensation and diversity technique are applied. Finally, in [5], the signal-to-noise ratio (SNR) due to fading induced by the terrestrial atmospheric turbulence was evaluated without and with the tilt correction employed. However, we should recognize that the maritime and terrestrial atmospheric environments are substantially different. As a result, the atmospheric turbulence over the sea and the land will initiate different random processes, thus having different effects on the coherent FSO system performance. In this paper, we will study the fading induced by the maritime atmospheric turbulence with respect to the coherent FSO system with quadrature array phase-shift keying (QPSK) modulation employed, and the improvement in the bit-error-rate (BER) performance when the AO compensation is applied. We will establish a comprehensive model and make a comparison between the maritime and terrestrial atmospheric links operating under the same conditions. Note that unlike [1], which took the assumptions in order to derive a specific expression for the PDF of the SNR, our results are based on the accurate numerical simulations without additional approximations. Furthermore, instead of the simple modal compensation, which was used in [1,3,4], we have designed a practical closed-loop AO system to perform the correction process in order to mitigate the turbulence effects. 1454
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The remainder of this paper is organized as follows. In Section 2, we will establish the models of the maritime and terrestrial FSO links, respectively, in terms of the transmittance and the refractive index fluctuation statistics. The transmittances of optical signals at wavelengths of 850 and 1550 nm are evaluated by the MODTRAN software, while the intensity scintillation is calculated by using analytical expressions. The turbulence-induced phase distortions are generated by the using Monte Carlo phase screen method and validated by comparison with the phase structure function. In Section 3, we present the evaluation of the BER function of coherent FSO with QPSK modulation. In order to minimize the BER, we have designed an optimum closed-loop AO system that helps to correct the wavefront phase distortions. In Section 4, we evaluate the BER performances of coherent FSO systems employing QPSK modulation. It was done both with and without the AO compensation in place and applied to both the maritime and terrestrial FSO channels experiencing air turbulence effects. Finally, some conclusions are presented in Section 5. 2. Analysis of Maritime and Terrestrial FSO Links
The laser beam propagating through FSO channels inevitably interacts with the aerosols and atmospheric molecules, which leads to scattering and absorption effects. These effects give rise to the attenuation of the signal carried by the laser beam [6]. On the other hand, the temperature gradient, wind, and changes of humidity generate the atmospheric turbulence, thus giving rise to the perturbation of the atmospheric refractive index. As a result, the wavefront phase distortions and amplitude fluctuations of the optical signal will be induced. Eventually, the turbulence characterizing the FSO channel will result in the system performance degradation. A. Air Transmittance
Before we proceed with the analysis of turbulence effects, let us verify the differences in the transmittance of FSO channels operating in maritime and terrestrial environments, respectively. During the laser beam propagation through the FSO channel, its intensity is attenuated due to scattering and absorption effects caused by atmospheric particles. The attenuation can be characterized by the transmittance, which is described by the Lambert–Beer law, given as τ I∕I 0 exp−αL;
(1)
where I and I 0 are the received and transmitted intensities, respectively; α represents the extinction coefficient of the atmospheric transmission media; and L is the propagation distance in the FSO link. A larger value of τ indicates lower attenuation. We can use the standard MODTRAN software package to find solutions of the radiative transfer equation, including the effects of molecular and
particulate absorption/emission and scattering for UV to IR wavelengths by choosing a specific atmospheric environment. Accordingly, we first verified the advantage of using wavelengths of 850 and 1550 nm in combination with specific beam trajectories. Specifically, a horizontal propagation path of 10 km and an observer height of 20 m are analyzed. Figure 1 shows the obtained transmittances related to the maritime and terrestrial atmospheric environments. As we can see, the value of τ at 1550 nm is greater than that at the wavelength of 850 nm for both atmospheric environments. Also, we verified that a smaller attenuation is obtained under maritime atmosphere conditions for both wavelengths. Therefore, we can assume that, from the point of view of transmittance, the maritime atmospheric link would provide a higher OSNR than the terrestrial atmospheric link for the two concerned wavelengths under ideal conditions when no air turbulence is present, while the wavelength of 1550 nm is the preferred one. However, in real conditions, there is the air turbulence impact and the situation will change.
B. Measurement of Atmospheric Turbulence
On the near-maritime surface, the variation of water vapor concentration and the existence of temperature gradient are the main cause of the atmospheric turbulence, which gives rise to random fluctuation of the atmospheric refractive index. It is commonly assumed that the atmospheric refractive index structure parameter, C2n (m−2∕3 ), which is directly associated with the turbulence strength, represents the measurement of the refractive index fluctuation [7]. Accordingly, measuring the parameter C2n is crucial to evaluate the turbulence effects in practice. For the maritime atmospheric turbulence, temperature gradient, humidity fluctuations, and the correlation between temperature and humidity fluctuations must be considered. In accordance with this, the C2n can be defined as [8] C2n A2T C2T 2AT Bq CTq B2q C2q ;
(2)
where AT and Bq are the coefficients that indicate the fluctuations of refractive index with temperature and humidity, respectively; C2T , CTq , and C2q are the temperature structure parameter, the specific humidity structure parameters, and the temperature-specific humidity cross-structure parameter, respectively. The bulk model [8], which is based on the Monin–Obukhov similarity theory [9], can be applied to measure the value of C2n. With this model, all parameters in Eq. (2) above can be characterized by the meteorological measurements. On the other side, the humidity fluctuations can be negligible for the terrestrial atmosphere. Thus, the terrestrial turbulence depends only on the temperature gradient and pressure. In such a case, C2n is represented by a simple expression, P 2 C2n 79 × 10−6 2 C2T ; T
(3)
where P and T are the atmospheric pressure and temperature. In practice, the C2n of the terrestrial atmospheric turbulence can be estimated in terms of the Hufnagel–Valley model [7], h exp − 1000 h h H exp − ; (4) 2.7 × 10−16 exp − 1500 100
C2n h 0.00594
Fig. 1. Transmittances at 850 and 1550 nm under (a) maritime atmospheric environment and (b) terrestrial atmospheric environment.
w 27
2
10−5 h10
where w is the wind speed (in m/s); H is a nominal value of C2n 0 at the ground level, and h is the altitude (in m). The refractive index structure parameter of maritime atmospheric turbulence has a more complicated measurement process as compared to that of terrestrial atmospheric turbulence. On the other hand, since the thermal capacity of water is larger than that of soil, the temperature difference between air and sea is smaller than that between the air and 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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ground. Accordingly, the value of C2n for the maritime atmospheric turbulence will be smaller than that for the terrestrial atmospheric turbulence [10]. Also, the C2n parameter is regarded as a constant for any horizontal signal path. For comparison sake, and without loss of generality, we assume that conditions that are created result in C2n 10−15 m−2∕3 for the both maritime and terrestrial atmospheric turbulence. Also, the outer and inner scale of turbulence under such a value of C2n are Louter 5 m, and linner 6 mm, respectively [11]. C.
Turbulence-Induced Wavefront Phase Distortions
Atmospheric turbulence causes a random fluctuation of the refractive index and distortions of the propagating laser beam wavefront. The refractive index power spectrum density (PSD) function can be used to characterize the statistics of the random process. An accurate refractive index PSD model, based on experimental data, has been proposed for the maritime atmospheric environment in [10,12], 7∕6 κ κ 2 κ 0.033C 1 − 0.061 2.836 ΦMar n n κH κH exp−κ 2 ∕κ2H ; × 2 κ κ 20 11∕6
for 0 ≤ κ < ∞;
(5)
where κ denotes spatial frequency; κ H 3.41∕linner and κ 0 2π∕Louter are associated with the inner and outer scale of turbulence, respectively. At the same time, several PSD models have been proposed for the terrestrial atmospheric turbulence. In our previous work [11,13], we showed that using the modified Von Karman PSD function is beneficial for analysis of wavefront distortions and eventual compensation by AO since it includes the effects of both low and high spatial frequency factors. We should outline that the basic Kolmogorov PSD version is more appropriate to be applied within the inertial range. The Von Karman PSD function can be expressed as 2 ΦTer n 0.033Cn
exp−κ2 ∕κ 2m ; κ 2 κ 20 11∕6
for 0 ≤ κ < ∞;
(6)
where κ m 5.92∕linner ; while the rest of the parameters are the same as in Eq. (5). Based on Eqs. (5) and (6), and multiplying by the factor 2πk2 L (where k and L are the wavenumber and propagation distance, respectively), we can calculate the PSD functions of turbulence-induced phase. For the maritime atmosphere, we have that 7∕6 κ κ −5∕3 1 − 0.061 2.836 ΦMar κ 0.49r ϕ 0 κH κH ×
exp−κ 2 ∕κ2H ; κ 2 κ 20 11∕6
for 0 ≤ κ < ∞;
(7)
where r0 0.423k2 C2n L−3∕5 is the atmospheric coherent diameter (also referred to as the Fried 1456
APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015
parameter). At the same time, the phase PSD function for the terrestrial atmospheric turbulence is given as −5∕3 ΦTer ϕ κ 0.49r0
exp−κ2 ∕κ 2m ; for 0 ≤ κ < ∞: κ 2 κ 20 11∕6
(8)
Next, we can utilize the Monte Carlo phase screen method [14] to accurately generate the turbulenceinduced phase; the distorted phase resulting from atmospheric turbulence is represented by a computergenerated array of the random sample points that have the statistics in accordance with the phase PSD function. Eventually, the random draws of the phase screen need to be validated by comparison with the theoretical phase structure function derived from the phase PSD function as Z DMar;Ter r ϕ
8π
∞ 0
κ
2
ΦMar;Ter κ ϕ
sin κr dκ; (9) 1− κr
where r is the spatial distance between two points over the phase screen. We determine the desirable phase screen by the ratio Δ max
jDMar;Ter rtheory − DMar;Ter rMC j ϕ ϕ rtheory g maxfDMar;Ter ϕ
;
(10)
rtheory and DMar;Ter rMC are the values where DMar;Ter ϕ ϕ calculated by Eq. (9) and the numerical data of the generated phase screen, respectively. If Δ < 15%, the phase screen will be considered as an accurate modeling of turbulence. As an example, the generated phase screen for the maritime atmospheric turbulence at the wavelength of 1550 nm, as well as the intensity distribution of transmitted and received beams after propagation through the modeled turbulent maritime air channel are shown in Fig. 2. D.
Turbulence-Induced Scintillation
The atmospheric turbulence causes not only the phase distortions of laser beams, but also induces the amplitude fluctuations. Therefore, the intensity of a laser beam (or irradiance) will fluctuate with time (scintillation effect). The irradiance fluctuation is characterized by the scintillation index, which is defined by σ 2I L
hI 2 Li − 1; hILi2
(11)
where hI 2 Li denotes the second momentum of the irradiance and hILi is the mean irradiance value. The scintillation index can be expressed as a function of the refractive index PSD function by using the Rytov method as [15]
11 11∕12 −1 σ 2I L 3.86σ 2R 1 Q−2 tan sin Q m m 6 11 −5∕6 ; (14) − Qm 6 where Qm 35.05 L∕kl2inner . If the scintillation is not in the saturation regime, the log-amplitude variance can be calculated by the scintillation index, σ 2χ L lnσ 2I L 1∕4:
(15)
3. BER of Coherent FSO with QPSK Modulation
The phase distortions and scintillation resulting from the atmospheric turbulence will increase the BER of the coherent FSO with MPSK modulation formats, thus leading to the performance degradation. Also, based on our previous work [11], we assume that the AO system can effectively mitigate the deleterious turbulence effects. Based on these facts, we will model an optimum AO system to correct the phase distortions and then evaluate the improvement in BER performance that is achieved. A. BER Evaluation of the FSO Coherent Detection System under Turbulence Effects
After the laser beam carrying signals propagates through the turbulent atmosphere, its optical field distribution becomes Fig. 2. (a) Generated phase screen by the Monte Carlo method; (b) screen validation showing Δ 12.41%, with propagation distance of 10 km, phase screen size of 240 × 240 with 0.001 m spacing, which is larger than the laser beam spot size of 0.1987 m with the beam waist of 0.0702 m; (c) optical signal intensity distribution for transmitted (left) and received signal (right).
σ 2I L
2 2
8π k L
Z 1Z 0
∞
0
2 Lκ z dκdz: κΦn κ 1 − cos k (12)
Now, by substituting Eqs. (5) and (6), the following expression for the scintillation index under maritime atmospheric turbulence can be obtained [12]: 8 2 sin 11 tan−1 QH 3 9 6 > > > > 4 −1 > > > < 1 Q−2 11∕12 6 − 0.051 sin 32tan1∕4 QH 7 > = 1Q 4 5 H H 2 2 ; σ I L 3.86σ R 5 −1 3.052 sin 4 tan QH > > > > 2 7∕24 > > 1Q > > H : ; −5∕6 −5.581QH (13)
Esig r; t Asig r; t expj2πf sig t ϕsig r; t · expχr · exp−jϕr;
(16)
where r is a two-dimensional vector in the aperture plane; Asig , f sig , and ϕsig are the amplitude, frequency, and the phase of the optical signals in the absence of atmospheric turbulence; χr and ϕr are the log-amplitude fluctuations and the phase distortions induced by the atmospheric turbulence, respectively. On the other hand, the local oscillator has the field distribution given as ELO r; t ALO r; t expj2πf LO t ϕLO r; t;
(17)
where ALO , f LO , and ϕLO are the amplitude, frequency, and the phase of the local oscillator. Eventually, at the heterodyne coherent receiver, the combination of the incoming optical signal and local oscillator light produces the intensity, Ir; t Esig r; t ELO r; t · Esig r; t ELO r; t ; (18)
where QH Lκ2H ∕k, while σ 2R 1.23C2n k7∕6 L11∕6 presents the Rytov variance. We can now compare Eq. (13) with the expression for the scintillation index under the terrestrial atmospheric turbulence [15], which is
where denotes the complex conjugate operation. As a result, the information after photodetection will be carried on photocurrent produced by the beating term given by 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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Z Wr expχr · cos2πΔf t Δϕ − ϕrdr R Wr expχr · cos2πΔf t Δϕ cos ϕrdr R ηALO Asig ; Wr expχr · sin2πΔf t Δϕ sin ϕrdr
isig ηALO Asig
where η is the quantum efficiency of the photodetector; Δf and Δϕ are the differences between the frequencies and phases of the optical signals and local light, respectively; Wr is the receiving aperture, Wr
1; r ≤ D∕2 ; 0; r > D∕2
(20)
where D is the diameter of the aperture. At the same time, the obtained signal power can be expressed by time averaging the value of the photocurrent square, 1 hi2sig i ηALO Asig 2 4
R Wrexpχr · cos ϕrdr 2
;
R Wrexpχr · sin ϕrdr 2 (21)
We can assume that the shot noise of the local oscillator is dominant in our coherent FSO model so that the SNR can be expressed as γ
hi2sig i hi2n i
R Wr expχr · cos ϕrdr
2 γ0
R
; Wr expχr · sin ϕrdr 2
γ 0 μ2r μ2i γ 0 μ2 ;
(22)
PSjγ
Wr expχr · cos ϕrdr; Z Wr expχr · sin ϕrdr:
Z
∞ 0
dγPSjγPγ γ;
(24)
where Pγ γ denotes the PDF of the SNR. As for PSjγ, the expression for the MPSK modulation can be presented as [16,17] 1458
−π∕2
π sin2 M dθ exp −γ : sin2 θ
Pγ γ
(25)
1 P 2 μ2 ; γ0 μ
(26)
where Pμ2 · is the PDF of μ2. By the large number theorem, we can assume that μr and μi are normal random variables. Accordingly, Pμ2 μ2 can be written as [1] Zπ 1 1 μ cos α − μ¯ r 2 dα exp − Pμ2 μ 4π σ r σ i −π 2σ 2r μ sin α − μ¯ i 2 ; (27) × exp − 2σ 2i 2
where μ¯ r and μ¯ i are the means of μr and μi , and σ 2r and σ 2i are the variances of μr and μi . We adopt the same assumption as in [1], that the integral expressions of μr and μi can be approximated by the sums over N statistically independent cells in the aperture, μr
N 1 X expχ m cos ϕm ; N m1
N 1 X μi expχ m sin ϕm ; N m1
(23)
Since the variations of amplitude and phase are random, μ, μr , and μi will also be random variables. In this sense, the SER PS for coherent detection can be expressed by averaging the conditional probability PSjγ of the SER, which is PS
π∕2−π∕M
(28)
where χ m and ϕm are the variations of amplitude and phase for the m-th statistically independent cell, respectively. Also, N can be calculated as [1]
Z
μi
Z
Now, by turning attention to Pγ γ, in accordance with Eq. (22), it can be written by using the Jacobian transformation as
where hi2n i denotes the ensemble averaging of noise power, γ 0 is the SNR in the absence of atmospheric turbulence, μ2 represents the fading factor including the amplitude fluctuations and phase distortions induced by the turbulence, while μr and μi have the integral forms, μr
1 π
(19)
APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015
N
8 D2
Z
D∕2 0
−1 1 rdr exp − Dϕ r : 2
(29)
Next, by using the classical statistical model [18], we have that
1 2 1 2 μ¯ r exp − σ χ exp − σ ϕ ; 2 2 μ¯ i 0; 1 1 exp−2σ 2ϕ − 2 exp−σ 2χ exp−σ 2ϕ ; 2N 1 σ 2i 1 − exp−2σ 2ϕ ; 2N σ 2r
(30)
where σ 2ϕ is the distorted phase variance. By substituting Eqs. (25)–(27) into Eq. (24), the SER becomes
2 3 2 q γ Zπ Z∞ ¯r γ 0 cos α − μ 1 1 1 5 dγ dα exp 4− PS γ 0 4π σ r σ i −π 2σ 2r 0 γ 2 Z π sin2 M γ sin α 1 π∕2−π∕M : dθ exp −γ × exp − 0 2 π −π∕2 sin2 θ 2σ i (31) Finally, the BER PB of the M-PSK format (in our specific case for the QPSK format where it is M 4) can be expressed through the approximated SER [19], PB PS∕log2 M:
(32)
As we can see from Eq. (31), for a given SNR value γ 0 in the absence of atmospheric turbulence, the BER value is determined by factors σ 2χ and σ 2ϕ representing the fading effect. B.
Fig. 3. (a) Lenslet distribution of Shack–Hartmann sensor and (b) actuator distribution of deformable mirror.
as shown in Fig. 4. Thus, the beacon beam with sampling information can be easily separated by a beam splitter. After the beacon beam is detected the Shack–Hartmann sensor, 370 points will be generated by the lenslet array within the pupil. Given that the coordinate of the ith point is (xi ; yi ), i 1 370, we have the wavefront error gradients,
T ∂W
∂W
∂W
∂W
C ; ;
; ; ;
: ∂x x1 ;y1 ∂x x370 ;y370 ∂y x1 ;y1 ∂y x370 ;y370
BER Evaluation When Turbulence Mitigation is Applied
Let us analyze BER performance under the same conditions as in previous section, but in the case when an optimally designed AO system is used to mitigate turbulence effects. The AO system has been successfully used to correct the phase distortions in numerous applications for many years. Based on that fact, an enhancement of BER performance is also expected when an AO system is deployed to compensate for the turbulence impact. Traditionally, the AO system consists of three components, which are: the Shack– Hartmann sensor, control system, and deformable mirror [20]. In our approach, we perform an optimization of the closed-loop AO system that will be applied. We assume that the Shack–Hartmann sensor is composed of a lenslet array with 22 × 22 size. The deformable mirror includes the actuators of the 12 × 12 array. Also, we assume that the control system has a gain of 0.3. Accordingly, the Shack–Hartmann sensor and the deformable mirror will have 370 lenses and 108 actuators over the circular apertures (which is a common case), respectively, as illustrated in Fig. 3. Since in our AO model the circular pupil is adopted, the Zernike polynomials would be the most appropriate tool to reconstruct the wavefront since they are orthogonal over a circle, so that the computation complexity is highly reduced. Now the reconstructed wavefront can be expressed by a sum of the Zernike polynomials as Wx; y
t X s1
as Zs x; y;
(34) Accordingly, the Zernike polynomial gradients can be written as 3 2
∂Z1 x;y
∂Zt x;y
∂x ∂x x1 ;y1 x1 ;y1 7 6 .. .. .. 7 6 7 6 .
. 7 6 ∂Z1 x;y . ∂Z x;y t
7 6 ∂x x 6 ∂x x 370 ;y370 370 ;y370 7
B 6 ∂Z1 x;y
(35) 7: ∂Zt x;y
7 6 ∂y ∂y x1 ;y1 x1 ;y1 7 6 7 6 .. .. .. 7 6 . . . 5 4
∂Z1 x;y
∂Zt x;y
∂y ∂y x ;y x ;y 370
370
370
370
Based on Eqs. (34) and (35), the coefficient matrix A a1 ; ; at T can be calculated by the least squares estimate method, A BT B−1 BT C:
(36)
The information of the reconstructed wavefront is now translated into the control signals. Next, the control system employs these signals to drive the actuators to alter the shape of the deformable mirror for distortion compensation. The correction rate of 1 kHz is set in our AO model. In order to reconstruct the wavefront more accurately, we will use 56
(33)
where as is the coefficient of polynomial basis function Zs ·, and t is the total number of polynomials. We can use a plane wave as the beacon beam wavelength λ2 to sample the atmosphere assuming that it is different from the optical signal wavelength λ1 ,
Fig. 4. Schematic of the maritime FSO over the turbulent atmosphere with the AO correction applied. 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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C2n 10−15 m−2∕3 ; the two different values of signal wavelengths (i.e., 850 and 1550 nm). Based on that, we can calculate the optical turbulence parameters, as presented in Table 1. As one can see from Table 1, the log-amplitude variance under maritime turbulence impact is larger than that caused by the impact of the terrestrial turbulence, which means that the laser beam will suffer from a stronger scintillation effect when the maritime FSO link is employed. Next, the statistically independent amount N of cells within the aperture, which is associated with the spot size of the laser beam at the detector, is calculated. It has been known that the diffractionlimited radius of the laser beam at the propagation distance z can be calculated as rz
Fig. 5. Wavefront phase distortions (a) without OA correction and (b) with AO correction.
Zernike polynomials (i.e., t 56). Figure 5 shows an example of the wavefront phase distortions without and with AO correction applied under the maritime atmospheric turbulence when the optical signal of wavelength λ1 1550 nm and the beacon beam of wavelength λ2 1500 nm are used. The other parameters are the same as those introduced in Section 2.A. As one can see, if there is no compensation, the wavefront phase distortions are distributed in the range [–4.6871, 3.8778] rad, with root mean square of 1.9289 rad. However, after the AO compensation takes place, the phase distortions are reduced substantially to fall within the range [–1.4738, 0.9568] rad with a root mean square of 0.2719 rad. With the calculated variances of the distorted phase and log-amplitude (i.e., σ 2χ and σ 2ϕ ) resulting from the turbulence, we can now evaluate the BER performance of the coherent FSO using M-PSK modulation formats by paying attention to the QPSK modulation case without and with AO in place by using Eqs. (30)–(32).
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(37)
where ω0 is the laser beam waist. For a given z, r has a minimum when ω0 λz∕π1∕2 . The calculated ω0 , r, and N are shown in the Table 2. As shown in Tables 1 and 2, four cases have been considered. For each case, we preformed the AO correction processes over 100 scenarios of turbulence. Based on the obtained values of distorted phase variance σ 2χ , we calculated the ensemble averages of BER when the turbulence-free SNR value γ 0 varies within the range [0,30] dB. For the sake of comparison, the BER performance in the absence of turbulence is also presented. As one can see from Fig. 6, the BER performance is severely degraded by the maritime atmospheric turbulence effects. Specifically, all values of BER within the entire range of γ 0 are even higher than the assumed forward-error-correction (FEC) limit of 10−3. However, after the AO system compensates for the phase distortions, the BER performance has been substantially improved by approaching the curve of turbulence-free BER. Now the values of BER are reduced to fall below 10−3 when γ 0 ≥ 12.50 dB and γ 0 ≥ 12.22 dB at 850 and 1550 nm, respectively. (At the same time we have calculated that γ 0 ≥ 11.88 dB and γ 0 ≥ 10.91 dB at 850 and 1550 nm, respectively, for the case of terrestrial turbulence). Moreover, we can observe that the BER Table 1.
Values of r 0 , σ R , and σ 2χ at 850 and 1550 m under Maritime and Terrestrial Atmospheric Turbulence
4. Numerical Results and Discussion
Based on the modeled maritime and terrestrial FSO links and an optimized AO system design, we performed a numerical evaluation of the BER performance of coherent FSO employing the QPSK modulation. It was done for both the maritime and terrestrial atmospheric turbulence conditions, in cases with and without AO compensation applied. As we mentioned, the following parameters are included in the calculations: the horizontal propagation path is 10 km; the outer and inner scale of turbulence are 5 m and 6 mm when the atmospheric refractive index structure parameter is
q ω20 λ2 z2 ∕π 2 ω20 ;
Maritime Atmospheric Turbulence
Atmospheric coherent diameter r0 (m) Rytov variance σ R Log-amplitude variance σ 2χ
Terrestrial atmospheric turbulence
at 850 nm
at 1550 nm
at 850 nm
at 1550 nm
0.0382
0.0785
0.0382
0.0785
2.7340
1.3564
2.7340
1.3564
0.3611
0.2337
0.3269
0.2130
Table 2.
Values of ω0 , r , and N at Wavelengths of 850 and 1550 nm under Maritime and Terrestrial Atmospheric Turbulence
Maritime Atmospheric Turbulence
ω0 (m) r (m) N
Terrestrial Atmospheric Turbulence
at 850 nm
at 1550 nm
at 850 nm
at 1550 nm
0.0520 0.0736 42
0.0702 0.0993 16
0.0520 0.0736 29
0.0702 0.0993 11
performances at 850 and 1550 nm are in a close range. This can be attributed to the fact that, although the turbulence effects are more severe at 850 nm than at 1550 nm, the laser beam divergence at 1550 nm is broader. Also, we can compare the BER performances of the maritime and terrestrial FSO links operating at the same wavelength, as illustrated for wavelength of 1550 nm and shown in Fig 7. As one can see from Fig. 7, the terrestrial air turbulence causes lower BER at both wavelengths
Fig. 6. BER performances for maritime atmospheric link without and with AO compensation at 850 and 1550 nm.
Fig. 7. BER performances without and with AO compensation under maritime atmospheric turbulence and terrestrial atmospheric turbulence operating at 1550 nm.
Fig. 8. BER performances under different propagation distances of 2, 4, 6, 8, and 10 km under maritime atmospheric turbulence (a) without AO compensation and (b) with AO compensation.
in the case of the same refractive index structure parameter (C2n 10−15 m−2∕3 ). It should be noticed that, in the practical case, the turbulence is stronger in the near-ground than the near-sea areas (due to a larger value of C2n ). Furthermore, the transmittance of maritime atmospheric media is higher than that of terrestrial atmospheric media. If we can use the same methodology to calculate the ensemble averages of BER for the other link distances (i.e., for 2, 4, 6, and 8) over the maritime atmospheric turbulence at 1550 nm. The results are shown in Fig. 8 and outline the high value of the optimized AO compensation method. As we can see, the area related to the BER of ≤10−3 increases significantly when the AO system is applied to mitigate the turbulence effects. This means a larger feasible set of propagation distances and a turbulence-free-like SNR can be achieved after the AO correction. As an example, when the propagation distance is 8 km, no error free transmission (defined by BER ≤10−3 ) can be achieved without AO correction, while it is possible with AO correction if γ 0 ≥ 11.10 dB. We should mention that the effects that degrade the performance of coherent FSO system include not only the atmospheric turbulence, but also the tracking error. Several researchers have studied the influence of tracking error on the FSO system performance in the terrestrial environment [21–23]. However, rather few investigations have reported the impact in the maritime environment. Physically, over the sea, the ships or vessels would be rocked more severely by the water wave compared with the terrestrial environment. This would be an important impact that should also be considered, which is the issue that we will investigate as our next step. Further on, the multiple-input-multiple-output 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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(MIMO) technique can be used to achieve a diversity gain, thus mitigating the turbulence effects and multipath fading [24,25]. Therefore, we can expect that the combination of the AO real-time correction and the MIMO would offer an even more promising scheme for the coherent FSO over the maritime environment, which we plan to investigate as well. 5. Conclusion
We have studied the BER performance of the coherent FSO with QPSK modulation over maritime atmospheric turbulence. A significant improvement of BER performance has been confirmed when an optimum closed-loop AO system is applied to compensate for the turbulence effects. We have also made a comprehensive comparison between the maritime and terrestrial FSO links operating in atmospheric turbulence conditions. By modeling the links based on the transmittance, the phase distortions, and the scintillation, we found that the optical signals will suffer a smaller attenuation in the maritime atmosphere as compared to the terrestrial atmosphere in both wavelength regions of interest (i.e., around 850 and 1550 nm). The turbulence-induced wavefront phase distortions were modeled by a generated phase screen based on the use of the Monte Carlo method, which is validated by comparison with the theoretical phase structure function. We designed an optimized closed-loop AO system and used it to perform the correction processes. Our simulation results show that the phase distortions can be substantially reduced after the AO correction is employed. The developed model of FSO links related to maritime and terrestrial conditions have been used in conjunction with the designed AO system to evaluate the system BER performance. A substantial/ dramatic improvement can be obtained with AO compensation, thus approaching to the turbulencefree BER curve defined by the FEC threshold of ∼10−3 . Under the same turbulence conditions, a higher BER is observed for the maritime FSO link as compared to the terrestrial FSO link when they have the same refractive index structure parameter (C2n 10−15 m−2∕3 ) characterizing air turbulence in spite of the fact that the maritime FSO link will suffer a smaller attenuation. M. Li acknowledges the support from the BUPT Excellent Ph.D. Students Foundation (Grant No. CX201333) and the program of China Scholarship Council (Grant No. 201306470039). References 1. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16, 14151–14162 (2008). 2. Y. Li, M. Li, Y. Poo, J. Ding, M. Tang, and Y. Lu, “Performance analysis of OOK, BPSK, QPSK modulation schemes in uplink of ground-to-satellite laser communication system under atmospheric fluctuation,” Opt. Commun. 317, 57–61 (2014). 3. S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2, 1087–1094 (2010).
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4. L. Zuo, Y. Ren, A. Dang, and G. Hong, “Performance of coherent BPSK systems using phase compensation and diversity techniques,” in Global Telecommunications Conference (IEEE, 2010), pp. 1–5. 5. N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt. 46, 7218–7226 (2007). 6. M. Li, P. Lu, Z. Yu, L. Yan, Z. Chen, C. Yang, and X. Luo, “Vector Monte Carlo simulations on atmospheric scattering of polarization qubits,” J. Opt. Soc. Am. A 30, 448–454 (2013). 7. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005). 8. F. Anctil and M. A. Donelan, “Air–water momentum flux observations over shoaling waves,” J. Phys. Oceanogr. 26, 1344–1353 (1996). 9. P. A. Frederickson, K. L. Davidson, C. R. Zeisse, and C. S. Bendall, “Estimating the refractive index structure parameter over the ocean using bulk methods,” J. Appl. Meteorol. 39, 1770–1783 (2000). 10. F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average C2n values in the marine environment,” J. Opt. Soc. Am. A 24, 3198–3206 (2007). 11. M. Li, Y. Takashima, X. Sun, Z. Yu, and M. Cvijetic, “Enhancement of channel capacity of OAM-based FSO link by correction of distorted wave-front under strong turbulence,” in Frontiers in Optics 2014, OSA Technical Digest (online) (Optical Society of America, 2014), paper FTh3B.6. 12. K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18, 173–184 (2008). 13. M. Li, M. Cvijetic, Y. Takashima, and Z. Yu, “Evaluation of channel capacity of the OAM-based FSO links with a precise assessment of turbulence impact,” in CLEO: 2014, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM3J.3. 14. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010). 15. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001). 16. M. Cvijetic, Coherent and Nonlinear Lightwave Communications (Artech House, 1996). 17. R. F. Pawula, S. O. Rice, and J. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. 30, 1828–1841 (1982). 18. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2010). 19. M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013). 20. R. K. Tyson, Principles of Adaptive Optics, 3rd ed. (CRC Press, 2010). 21. A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 20, 2096–2109 (2012). 22. A. García-Zambrana, C. Castillo-Vázquez, B. CastilloVázquez, and R. Boluda-Ruiz, “Bit detect and forward relaying for FSO links using equal gain combining over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 20, 16394–16409 (2012). 23. K. P. Peppas, A. N. Stassinakis, H. E. Nistazakis, and G. S. Tombras, “Capacity analysis of dual amplify-and-forward relayed free-space optical communication systems over turbulence channels with pointing errors,” J. Opt. Commun. Netw. 5, 1032–1042 (2013). 24. N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun. 7, 1744–1753 (2008). 25. A. A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60, 479–487 (2012).
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
2
State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China *Corresponding author: [email protected] Received 12 November 2014; accepted 4 January 2015; posted 12 January 2015 (Doc. ID 226797); published 19 February 2015
We evaluate the performance of the coherent free space optics (FSO) employing quadrature array phase-shift keying (QPSK) modulation over the maritime atmosphere with atmospheric turbulence compensated by use of adaptive optics (AO). We have established a comprehensive FSO channel model for maritime conditions and also made a comprehensive comparison of performance between the maritime and terrestrial atmospheric links. The FSO links are modeled based on the intensity attenuation resulting from scattering and absorption effects, the log-amplitude fluctuations, and the phase distortions induced by turbulence. The obtained results show that the FSO system performance measured by the bit-error-rate (BER) can be significantly improved when the optimization of the AO system is achieved. Also, we find that the higher BER is observed in the maritime FSO channel with atmospheric turbulence, as compared to the terrestrial FSO systems if they experience the same turbulence strength. © 2015 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (060.1660) Coherent communications; (010.1330) Atmospheric turbulence; (010.1080) Active or adaptive optics. http://dx.doi.org/10.1364/AO.54.001453
1. Introduction
It is well recognized that free-space optical (FSO) communications can offer considerable advantages over radio frequency (RF) communications. The larger antenna gain of the FSO system has an impact on the smaller size, weight, and power of terminals. Also, the FSO systems operate over a large, unregulated, and license-free frequency spectrum. As compared to a microwave radio beam, the laser beam in an FSO channel has extremely narrow divergence so that the FSO system offers interference-free properties and immunity against eavesdropping. Some other advantages include possible use for quantum key distribution (QKD) applications, and more flexibility 1559-128X/15/061453-10$15.00/0 © 2015 Optical Society of America
in installation and eventual relocation. On the other hand, it is well known that the coherent detection scheme, when compared with the traditional intensity modulation direct detection (IM/DD) scheme, has the ability to suppress the impact of both background and thermal noise. Accordingly, the sensitivity of the coherent receiver is significantly higher than that of IM/DD, although the system complexity has been added. Furthermore, the coherent detection scheme is applicable for the multilevel modulation formats, such as M-array phase-shift keying (MPSK) and quadrature amplitude modulation (MQAM), thus offering a considerable increase in the spectral efficiency. All of these benefits outline the capability of coherent detection-based FSO systems as one of the most promising candidates for a number of applications. With increasing activities and communication needs in the maritime environment, coherent FSO 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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over the maritime atmosphere can be considered as a formidable candidate for information transfer for different purposes (such as the vessel-to-vessel or ship-to-satellite communications). However, the performance of coherent FSO can be highly degraded by the deleterious maritime atmospheric turbulence. To the best of our knowledge, the vast majority of coherent FSO channel analyses to date have only involved the terrestrial atmospheric links, while the specific analysis of the coherent FSO system behavior operating over the maritime atmospheric turbulence channel has not been reported. That also means that the improvement in performance has not been studied when an adaptive optics (AO) system is applied to mitigate the turbulence effects. As for terrestrial systems, we should mention that, based on the statistical description of terrestrial atmospheric turbulence, Belmonte and Khan [1] introduced a specific expression that represented the probability density function (PDF) of the symbolerror-rate (SER) of a coherent FSO system by using appropriate approximations. Also, the degradation in SER performance induced by turbulence effects and the improvement by the simple modal compensation in system performance were shown in [1]. The fading in the satellite-to-ground FSO uplink was studied in [2] when the beam scintillation and wander effects caused by the terrestrial atmospheric turbulence were considered. In parallel, the signal fading resulting from the terrestrial atmospheric turbulence was calculated in [3,4]. It was also demonstrated in [3,4] that the coherent FSO system has the potential for further improvement when the modal compensation and diversity technique are applied. Finally, in [5], the signal-to-noise ratio (SNR) due to fading induced by the terrestrial atmospheric turbulence was evaluated without and with the tilt correction employed. However, we should recognize that the maritime and terrestrial atmospheric environments are substantially different. As a result, the atmospheric turbulence over the sea and the land will initiate different random processes, thus having different effects on the coherent FSO system performance. In this paper, we will study the fading induced by the maritime atmospheric turbulence with respect to the coherent FSO system with quadrature array phase-shift keying (QPSK) modulation employed, and the improvement in the bit-error-rate (BER) performance when the AO compensation is applied. We will establish a comprehensive model and make a comparison between the maritime and terrestrial atmospheric links operating under the same conditions. Note that unlike [1], which took the assumptions in order to derive a specific expression for the PDF of the SNR, our results are based on the accurate numerical simulations without additional approximations. Furthermore, instead of the simple modal compensation, which was used in [1,3,4], we have designed a practical closed-loop AO system to perform the correction process in order to mitigate the turbulence effects. 1454
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The remainder of this paper is organized as follows. In Section 2, we will establish the models of the maritime and terrestrial FSO links, respectively, in terms of the transmittance and the refractive index fluctuation statistics. The transmittances of optical signals at wavelengths of 850 and 1550 nm are evaluated by the MODTRAN software, while the intensity scintillation is calculated by using analytical expressions. The turbulence-induced phase distortions are generated by the using Monte Carlo phase screen method and validated by comparison with the phase structure function. In Section 3, we present the evaluation of the BER function of coherent FSO with QPSK modulation. In order to minimize the BER, we have designed an optimum closed-loop AO system that helps to correct the wavefront phase distortions. In Section 4, we evaluate the BER performances of coherent FSO systems employing QPSK modulation. It was done both with and without the AO compensation in place and applied to both the maritime and terrestrial FSO channels experiencing air turbulence effects. Finally, some conclusions are presented in Section 5. 2. Analysis of Maritime and Terrestrial FSO Links
The laser beam propagating through FSO channels inevitably interacts with the aerosols and atmospheric molecules, which leads to scattering and absorption effects. These effects give rise to the attenuation of the signal carried by the laser beam [6]. On the other hand, the temperature gradient, wind, and changes of humidity generate the atmospheric turbulence, thus giving rise to the perturbation of the atmospheric refractive index. As a result, the wavefront phase distortions and amplitude fluctuations of the optical signal will be induced. Eventually, the turbulence characterizing the FSO channel will result in the system performance degradation. A. Air Transmittance
Before we proceed with the analysis of turbulence effects, let us verify the differences in the transmittance of FSO channels operating in maritime and terrestrial environments, respectively. During the laser beam propagation through the FSO channel, its intensity is attenuated due to scattering and absorption effects caused by atmospheric particles. The attenuation can be characterized by the transmittance, which is described by the Lambert–Beer law, given as τ I∕I 0 exp−αL;
(1)
where I and I 0 are the received and transmitted intensities, respectively; α represents the extinction coefficient of the atmospheric transmission media; and L is the propagation distance in the FSO link. A larger value of τ indicates lower attenuation. We can use the standard MODTRAN software package to find solutions of the radiative transfer equation, including the effects of molecular and
particulate absorption/emission and scattering for UV to IR wavelengths by choosing a specific atmospheric environment. Accordingly, we first verified the advantage of using wavelengths of 850 and 1550 nm in combination with specific beam trajectories. Specifically, a horizontal propagation path of 10 km and an observer height of 20 m are analyzed. Figure 1 shows the obtained transmittances related to the maritime and terrestrial atmospheric environments. As we can see, the value of τ at 1550 nm is greater than that at the wavelength of 850 nm for both atmospheric environments. Also, we verified that a smaller attenuation is obtained under maritime atmosphere conditions for both wavelengths. Therefore, we can assume that, from the point of view of transmittance, the maritime atmospheric link would provide a higher OSNR than the terrestrial atmospheric link for the two concerned wavelengths under ideal conditions when no air turbulence is present, while the wavelength of 1550 nm is the preferred one. However, in real conditions, there is the air turbulence impact and the situation will change.
B. Measurement of Atmospheric Turbulence
On the near-maritime surface, the variation of water vapor concentration and the existence of temperature gradient are the main cause of the atmospheric turbulence, which gives rise to random fluctuation of the atmospheric refractive index. It is commonly assumed that the atmospheric refractive index structure parameter, C2n (m−2∕3 ), which is directly associated with the turbulence strength, represents the measurement of the refractive index fluctuation [7]. Accordingly, measuring the parameter C2n is crucial to evaluate the turbulence effects in practice. For the maritime atmospheric turbulence, temperature gradient, humidity fluctuations, and the correlation between temperature and humidity fluctuations must be considered. In accordance with this, the C2n can be defined as [8] C2n A2T C2T 2AT Bq CTq B2q C2q ;
(2)
where AT and Bq are the coefficients that indicate the fluctuations of refractive index with temperature and humidity, respectively; C2T , CTq , and C2q are the temperature structure parameter, the specific humidity structure parameters, and the temperature-specific humidity cross-structure parameter, respectively. The bulk model [8], which is based on the Monin–Obukhov similarity theory [9], can be applied to measure the value of C2n. With this model, all parameters in Eq. (2) above can be characterized by the meteorological measurements. On the other side, the humidity fluctuations can be negligible for the terrestrial atmosphere. Thus, the terrestrial turbulence depends only on the temperature gradient and pressure. In such a case, C2n is represented by a simple expression, P 2 C2n 79 × 10−6 2 C2T ; T
(3)
where P and T are the atmospheric pressure and temperature. In practice, the C2n of the terrestrial atmospheric turbulence can be estimated in terms of the Hufnagel–Valley model [7], h exp − 1000 h h H exp − ; (4) 2.7 × 10−16 exp − 1500 100
C2n h 0.00594
Fig. 1. Transmittances at 850 and 1550 nm under (a) maritime atmospheric environment and (b) terrestrial atmospheric environment.
w 27
2
10−5 h10
where w is the wind speed (in m/s); H is a nominal value of C2n 0 at the ground level, and h is the altitude (in m). The refractive index structure parameter of maritime atmospheric turbulence has a more complicated measurement process as compared to that of terrestrial atmospheric turbulence. On the other hand, since the thermal capacity of water is larger than that of soil, the temperature difference between air and sea is smaller than that between the air and 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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ground. Accordingly, the value of C2n for the maritime atmospheric turbulence will be smaller than that for the terrestrial atmospheric turbulence [10]. Also, the C2n parameter is regarded as a constant for any horizontal signal path. For comparison sake, and without loss of generality, we assume that conditions that are created result in C2n 10−15 m−2∕3 for the both maritime and terrestrial atmospheric turbulence. Also, the outer and inner scale of turbulence under such a value of C2n are Louter 5 m, and linner 6 mm, respectively [11]. C.
Turbulence-Induced Wavefront Phase Distortions
Atmospheric turbulence causes a random fluctuation of the refractive index and distortions of the propagating laser beam wavefront. The refractive index power spectrum density (PSD) function can be used to characterize the statistics of the random process. An accurate refractive index PSD model, based on experimental data, has been proposed for the maritime atmospheric environment in [10,12], 7∕6 κ κ 2 κ 0.033C 1 − 0.061 2.836 ΦMar n n κH κH exp−κ 2 ∕κ2H ; × 2 κ κ 20 11∕6
for 0 ≤ κ < ∞;
(5)
where κ denotes spatial frequency; κ H 3.41∕linner and κ 0 2π∕Louter are associated with the inner and outer scale of turbulence, respectively. At the same time, several PSD models have been proposed for the terrestrial atmospheric turbulence. In our previous work [11,13], we showed that using the modified Von Karman PSD function is beneficial for analysis of wavefront distortions and eventual compensation by AO since it includes the effects of both low and high spatial frequency factors. We should outline that the basic Kolmogorov PSD version is more appropriate to be applied within the inertial range. The Von Karman PSD function can be expressed as 2 ΦTer n 0.033Cn
exp−κ2 ∕κ 2m ; κ 2 κ 20 11∕6
for 0 ≤ κ < ∞;
(6)
where κ m 5.92∕linner ; while the rest of the parameters are the same as in Eq. (5). Based on Eqs. (5) and (6), and multiplying by the factor 2πk2 L (where k and L are the wavenumber and propagation distance, respectively), we can calculate the PSD functions of turbulence-induced phase. For the maritime atmosphere, we have that 7∕6 κ κ −5∕3 1 − 0.061 2.836 ΦMar κ 0.49r ϕ 0 κH κH ×
exp−κ 2 ∕κ2H ; κ 2 κ 20 11∕6
for 0 ≤ κ < ∞;
(7)
where r0 0.423k2 C2n L−3∕5 is the atmospheric coherent diameter (also referred to as the Fried 1456
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parameter). At the same time, the phase PSD function for the terrestrial atmospheric turbulence is given as −5∕3 ΦTer ϕ κ 0.49r0
exp−κ2 ∕κ 2m ; for 0 ≤ κ < ∞: κ 2 κ 20 11∕6
(8)
Next, we can utilize the Monte Carlo phase screen method [14] to accurately generate the turbulenceinduced phase; the distorted phase resulting from atmospheric turbulence is represented by a computergenerated array of the random sample points that have the statistics in accordance with the phase PSD function. Eventually, the random draws of the phase screen need to be validated by comparison with the theoretical phase structure function derived from the phase PSD function as Z DMar;Ter r ϕ
8π
∞ 0
κ
2
ΦMar;Ter κ ϕ
sin κr dκ; (9) 1− κr
where r is the spatial distance between two points over the phase screen. We determine the desirable phase screen by the ratio Δ max
jDMar;Ter rtheory − DMar;Ter rMC j ϕ ϕ rtheory g maxfDMar;Ter ϕ
;
(10)
rtheory and DMar;Ter rMC are the values where DMar;Ter ϕ ϕ calculated by Eq. (9) and the numerical data of the generated phase screen, respectively. If Δ < 15%, the phase screen will be considered as an accurate modeling of turbulence. As an example, the generated phase screen for the maritime atmospheric turbulence at the wavelength of 1550 nm, as well as the intensity distribution of transmitted and received beams after propagation through the modeled turbulent maritime air channel are shown in Fig. 2. D.
Turbulence-Induced Scintillation
The atmospheric turbulence causes not only the phase distortions of laser beams, but also induces the amplitude fluctuations. Therefore, the intensity of a laser beam (or irradiance) will fluctuate with time (scintillation effect). The irradiance fluctuation is characterized by the scintillation index, which is defined by σ 2I L
hI 2 Li − 1; hILi2
(11)
where hI 2 Li denotes the second momentum of the irradiance and hILi is the mean irradiance value. The scintillation index can be expressed as a function of the refractive index PSD function by using the Rytov method as [15]
11 11∕12 −1 σ 2I L 3.86σ 2R 1 Q−2 tan sin Q m m 6 11 −5∕6 ; (14) − Qm 6 where Qm 35.05 L∕kl2inner . If the scintillation is not in the saturation regime, the log-amplitude variance can be calculated by the scintillation index, σ 2χ L lnσ 2I L 1∕4:
(15)
3. BER of Coherent FSO with QPSK Modulation
The phase distortions and scintillation resulting from the atmospheric turbulence will increase the BER of the coherent FSO with MPSK modulation formats, thus leading to the performance degradation. Also, based on our previous work [11], we assume that the AO system can effectively mitigate the deleterious turbulence effects. Based on these facts, we will model an optimum AO system to correct the phase distortions and then evaluate the improvement in BER performance that is achieved. A. BER Evaluation of the FSO Coherent Detection System under Turbulence Effects
After the laser beam carrying signals propagates through the turbulent atmosphere, its optical field distribution becomes Fig. 2. (a) Generated phase screen by the Monte Carlo method; (b) screen validation showing Δ 12.41%, with propagation distance of 10 km, phase screen size of 240 × 240 with 0.001 m spacing, which is larger than the laser beam spot size of 0.1987 m with the beam waist of 0.0702 m; (c) optical signal intensity distribution for transmitted (left) and received signal (right).
σ 2I L
2 2
8π k L
Z 1Z 0
∞
0
2 Lκ z dκdz: κΦn κ 1 − cos k (12)
Now, by substituting Eqs. (5) and (6), the following expression for the scintillation index under maritime atmospheric turbulence can be obtained [12]: 8 2 sin 11 tan−1 QH 3 9 6 > > > > 4 −1 > > > < 1 Q−2 11∕12 6 − 0.051 sin 32tan1∕4 QH 7 > = 1Q 4 5 H H 2 2 ; σ I L 3.86σ R 5 −1 3.052 sin 4 tan QH > > > > 2 7∕24 > > 1Q > > H : ; −5∕6 −5.581QH (13)
Esig r; t Asig r; t expj2πf sig t ϕsig r; t · expχr · exp−jϕr;
(16)
where r is a two-dimensional vector in the aperture plane; Asig , f sig , and ϕsig are the amplitude, frequency, and the phase of the optical signals in the absence of atmospheric turbulence; χr and ϕr are the log-amplitude fluctuations and the phase distortions induced by the atmospheric turbulence, respectively. On the other hand, the local oscillator has the field distribution given as ELO r; t ALO r; t expj2πf LO t ϕLO r; t;
(17)
where ALO , f LO , and ϕLO are the amplitude, frequency, and the phase of the local oscillator. Eventually, at the heterodyne coherent receiver, the combination of the incoming optical signal and local oscillator light produces the intensity, Ir; t Esig r; t ELO r; t · Esig r; t ELO r; t ; (18)
where QH Lκ2H ∕k, while σ 2R 1.23C2n k7∕6 L11∕6 presents the Rytov variance. We can now compare Eq. (13) with the expression for the scintillation index under the terrestrial atmospheric turbulence [15], which is
where denotes the complex conjugate operation. As a result, the information after photodetection will be carried on photocurrent produced by the beating term given by 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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Z Wr expχr · cos2πΔf t Δϕ − ϕrdr R Wr expχr · cos2πΔf t Δϕ cos ϕrdr R ηALO Asig ; Wr expχr · sin2πΔf t Δϕ sin ϕrdr
isig ηALO Asig
where η is the quantum efficiency of the photodetector; Δf and Δϕ are the differences between the frequencies and phases of the optical signals and local light, respectively; Wr is the receiving aperture, Wr
1; r ≤ D∕2 ; 0; r > D∕2
(20)
where D is the diameter of the aperture. At the same time, the obtained signal power can be expressed by time averaging the value of the photocurrent square, 1 hi2sig i ηALO Asig 2 4
R Wrexpχr · cos ϕrdr 2
;
R Wrexpχr · sin ϕrdr 2 (21)
We can assume that the shot noise of the local oscillator is dominant in our coherent FSO model so that the SNR can be expressed as γ
hi2sig i hi2n i
R Wr expχr · cos ϕrdr
2 γ0
R
; Wr expχr · sin ϕrdr 2
γ 0 μ2r μ2i γ 0 μ2 ;
(22)
PSjγ
Wr expχr · cos ϕrdr; Z Wr expχr · sin ϕrdr:
Z
∞ 0
dγPSjγPγ γ;
(24)
where Pγ γ denotes the PDF of the SNR. As for PSjγ, the expression for the MPSK modulation can be presented as [16,17] 1458
−π∕2
π sin2 M dθ exp −γ : sin2 θ
Pγ γ
(25)
1 P 2 μ2 ; γ0 μ
(26)
where Pμ2 · is the PDF of μ2. By the large number theorem, we can assume that μr and μi are normal random variables. Accordingly, Pμ2 μ2 can be written as [1] Zπ 1 1 μ cos α − μ¯ r 2 dα exp − Pμ2 μ 4π σ r σ i −π 2σ 2r μ sin α − μ¯ i 2 ; (27) × exp − 2σ 2i 2
where μ¯ r and μ¯ i are the means of μr and μi , and σ 2r and σ 2i are the variances of μr and μi . We adopt the same assumption as in [1], that the integral expressions of μr and μi can be approximated by the sums over N statistically independent cells in the aperture, μr
N 1 X expχ m cos ϕm ; N m1
N 1 X μi expχ m sin ϕm ; N m1
(23)
Since the variations of amplitude and phase are random, μ, μr , and μi will also be random variables. In this sense, the SER PS for coherent detection can be expressed by averaging the conditional probability PSjγ of the SER, which is PS
π∕2−π∕M
(28)
where χ m and ϕm are the variations of amplitude and phase for the m-th statistically independent cell, respectively. Also, N can be calculated as [1]
Z
μi
Z
Now, by turning attention to Pγ γ, in accordance with Eq. (22), it can be written by using the Jacobian transformation as
where hi2n i denotes the ensemble averaging of noise power, γ 0 is the SNR in the absence of atmospheric turbulence, μ2 represents the fading factor including the amplitude fluctuations and phase distortions induced by the turbulence, while μr and μi have the integral forms, μr
1 π
(19)
APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015
N
8 D2
Z
D∕2 0
−1 1 rdr exp − Dϕ r : 2
(29)
Next, by using the classical statistical model [18], we have that
1 2 1 2 μ¯ r exp − σ χ exp − σ ϕ ; 2 2 μ¯ i 0; 1 1 exp−2σ 2ϕ − 2 exp−σ 2χ exp−σ 2ϕ ; 2N 1 σ 2i 1 − exp−2σ 2ϕ ; 2N σ 2r
(30)
where σ 2ϕ is the distorted phase variance. By substituting Eqs. (25)–(27) into Eq. (24), the SER becomes
2 3 2 q γ Zπ Z∞ ¯r γ 0 cos α − μ 1 1 1 5 dγ dα exp 4− PS γ 0 4π σ r σ i −π 2σ 2r 0 γ 2 Z π sin2 M γ sin α 1 π∕2−π∕M : dθ exp −γ × exp − 0 2 π −π∕2 sin2 θ 2σ i (31) Finally, the BER PB of the M-PSK format (in our specific case for the QPSK format where it is M 4) can be expressed through the approximated SER [19], PB PS∕log2 M:
(32)
As we can see from Eq. (31), for a given SNR value γ 0 in the absence of atmospheric turbulence, the BER value is determined by factors σ 2χ and σ 2ϕ representing the fading effect. B.
Fig. 3. (a) Lenslet distribution of Shack–Hartmann sensor and (b) actuator distribution of deformable mirror.
as shown in Fig. 4. Thus, the beacon beam with sampling information can be easily separated by a beam splitter. After the beacon beam is detected the Shack–Hartmann sensor, 370 points will be generated by the lenslet array within the pupil. Given that the coordinate of the ith point is (xi ; yi ), i 1 370, we have the wavefront error gradients,
T ∂W
∂W
∂W
∂W
C ; ;
; ; ;
: ∂x x1 ;y1 ∂x x370 ;y370 ∂y x1 ;y1 ∂y x370 ;y370
BER Evaluation When Turbulence Mitigation is Applied
Let us analyze BER performance under the same conditions as in previous section, but in the case when an optimally designed AO system is used to mitigate turbulence effects. The AO system has been successfully used to correct the phase distortions in numerous applications for many years. Based on that fact, an enhancement of BER performance is also expected when an AO system is deployed to compensate for the turbulence impact. Traditionally, the AO system consists of three components, which are: the Shack– Hartmann sensor, control system, and deformable mirror [20]. In our approach, we perform an optimization of the closed-loop AO system that will be applied. We assume that the Shack–Hartmann sensor is composed of a lenslet array with 22 × 22 size. The deformable mirror includes the actuators of the 12 × 12 array. Also, we assume that the control system has a gain of 0.3. Accordingly, the Shack–Hartmann sensor and the deformable mirror will have 370 lenses and 108 actuators over the circular apertures (which is a common case), respectively, as illustrated in Fig. 3. Since in our AO model the circular pupil is adopted, the Zernike polynomials would be the most appropriate tool to reconstruct the wavefront since they are orthogonal over a circle, so that the computation complexity is highly reduced. Now the reconstructed wavefront can be expressed by a sum of the Zernike polynomials as Wx; y
t X s1
as Zs x; y;
(34) Accordingly, the Zernike polynomial gradients can be written as 3 2
∂Z1 x;y
∂Zt x;y
∂x ∂x x1 ;y1 x1 ;y1 7 6 .. .. .. 7 6 7 6 .
. 7 6 ∂Z1 x;y . ∂Z x;y t
7 6 ∂x x 6 ∂x x 370 ;y370 370 ;y370 7
B 6 ∂Z1 x;y
(35) 7: ∂Zt x;y
7 6 ∂y ∂y x1 ;y1 x1 ;y1 7 6 7 6 .. .. .. 7 6 . . . 5 4
∂Z1 x;y
∂Zt x;y
∂y ∂y x ;y x ;y 370
370
370
370
Based on Eqs. (34) and (35), the coefficient matrix A a1 ; ; at T can be calculated by the least squares estimate method, A BT B−1 BT C:
(36)
The information of the reconstructed wavefront is now translated into the control signals. Next, the control system employs these signals to drive the actuators to alter the shape of the deformable mirror for distortion compensation. The correction rate of 1 kHz is set in our AO model. In order to reconstruct the wavefront more accurately, we will use 56
(33)
where as is the coefficient of polynomial basis function Zs ·, and t is the total number of polynomials. We can use a plane wave as the beacon beam wavelength λ2 to sample the atmosphere assuming that it is different from the optical signal wavelength λ1 ,
Fig. 4. Schematic of the maritime FSO over the turbulent atmosphere with the AO correction applied. 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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C2n 10−15 m−2∕3 ; the two different values of signal wavelengths (i.e., 850 and 1550 nm). Based on that, we can calculate the optical turbulence parameters, as presented in Table 1. As one can see from Table 1, the log-amplitude variance under maritime turbulence impact is larger than that caused by the impact of the terrestrial turbulence, which means that the laser beam will suffer from a stronger scintillation effect when the maritime FSO link is employed. Next, the statistically independent amount N of cells within the aperture, which is associated with the spot size of the laser beam at the detector, is calculated. It has been known that the diffractionlimited radius of the laser beam at the propagation distance z can be calculated as rz
Fig. 5. Wavefront phase distortions (a) without OA correction and (b) with AO correction.
Zernike polynomials (i.e., t 56). Figure 5 shows an example of the wavefront phase distortions without and with AO correction applied under the maritime atmospheric turbulence when the optical signal of wavelength λ1 1550 nm and the beacon beam of wavelength λ2 1500 nm are used. The other parameters are the same as those introduced in Section 2.A. As one can see, if there is no compensation, the wavefront phase distortions are distributed in the range [–4.6871, 3.8778] rad, with root mean square of 1.9289 rad. However, after the AO compensation takes place, the phase distortions are reduced substantially to fall within the range [–1.4738, 0.9568] rad with a root mean square of 0.2719 rad. With the calculated variances of the distorted phase and log-amplitude (i.e., σ 2χ and σ 2ϕ ) resulting from the turbulence, we can now evaluate the BER performance of the coherent FSO using M-PSK modulation formats by paying attention to the QPSK modulation case without and with AO in place by using Eqs. (30)–(32).
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(37)
where ω0 is the laser beam waist. For a given z, r has a minimum when ω0 λz∕π1∕2 . The calculated ω0 , r, and N are shown in the Table 2. As shown in Tables 1 and 2, four cases have been considered. For each case, we preformed the AO correction processes over 100 scenarios of turbulence. Based on the obtained values of distorted phase variance σ 2χ , we calculated the ensemble averages of BER when the turbulence-free SNR value γ 0 varies within the range [0,30] dB. For the sake of comparison, the BER performance in the absence of turbulence is also presented. As one can see from Fig. 6, the BER performance is severely degraded by the maritime atmospheric turbulence effects. Specifically, all values of BER within the entire range of γ 0 are even higher than the assumed forward-error-correction (FEC) limit of 10−3. However, after the AO system compensates for the phase distortions, the BER performance has been substantially improved by approaching the curve of turbulence-free BER. Now the values of BER are reduced to fall below 10−3 when γ 0 ≥ 12.50 dB and γ 0 ≥ 12.22 dB at 850 and 1550 nm, respectively. (At the same time we have calculated that γ 0 ≥ 11.88 dB and γ 0 ≥ 10.91 dB at 850 and 1550 nm, respectively, for the case of terrestrial turbulence). Moreover, we can observe that the BER Table 1.
Values of r 0 , σ R , and σ 2χ at 850 and 1550 m under Maritime and Terrestrial Atmospheric Turbulence
4. Numerical Results and Discussion
Based on the modeled maritime and terrestrial FSO links and an optimized AO system design, we performed a numerical evaluation of the BER performance of coherent FSO employing the QPSK modulation. It was done for both the maritime and terrestrial atmospheric turbulence conditions, in cases with and without AO compensation applied. As we mentioned, the following parameters are included in the calculations: the horizontal propagation path is 10 km; the outer and inner scale of turbulence are 5 m and 6 mm when the atmospheric refractive index structure parameter is
q ω20 λ2 z2 ∕π 2 ω20 ;
Maritime Atmospheric Turbulence
Atmospheric coherent diameter r0 (m) Rytov variance σ R Log-amplitude variance σ 2χ
Terrestrial atmospheric turbulence
at 850 nm
at 1550 nm
at 850 nm
at 1550 nm
0.0382
0.0785
0.0382
0.0785
2.7340
1.3564
2.7340
1.3564
0.3611
0.2337
0.3269
0.2130
Table 2.
Values of ω0 , r , and N at Wavelengths of 850 and 1550 nm under Maritime and Terrestrial Atmospheric Turbulence
Maritime Atmospheric Turbulence
ω0 (m) r (m) N
Terrestrial Atmospheric Turbulence
at 850 nm
at 1550 nm
at 850 nm
at 1550 nm
0.0520 0.0736 42
0.0702 0.0993 16
0.0520 0.0736 29
0.0702 0.0993 11
performances at 850 and 1550 nm are in a close range. This can be attributed to the fact that, although the turbulence effects are more severe at 850 nm than at 1550 nm, the laser beam divergence at 1550 nm is broader. Also, we can compare the BER performances of the maritime and terrestrial FSO links operating at the same wavelength, as illustrated for wavelength of 1550 nm and shown in Fig 7. As one can see from Fig. 7, the terrestrial air turbulence causes lower BER at both wavelengths
Fig. 6. BER performances for maritime atmospheric link without and with AO compensation at 850 and 1550 nm.
Fig. 7. BER performances without and with AO compensation under maritime atmospheric turbulence and terrestrial atmospheric turbulence operating at 1550 nm.
Fig. 8. BER performances under different propagation distances of 2, 4, 6, 8, and 10 km under maritime atmospheric turbulence (a) without AO compensation and (b) with AO compensation.
in the case of the same refractive index structure parameter (C2n 10−15 m−2∕3 ). It should be noticed that, in the practical case, the turbulence is stronger in the near-ground than the near-sea areas (due to a larger value of C2n ). Furthermore, the transmittance of maritime atmospheric media is higher than that of terrestrial atmospheric media. If we can use the same methodology to calculate the ensemble averages of BER for the other link distances (i.e., for 2, 4, 6, and 8) over the maritime atmospheric turbulence at 1550 nm. The results are shown in Fig. 8 and outline the high value of the optimized AO compensation method. As we can see, the area related to the BER of ≤10−3 increases significantly when the AO system is applied to mitigate the turbulence effects. This means a larger feasible set of propagation distances and a turbulence-free-like SNR can be achieved after the AO correction. As an example, when the propagation distance is 8 km, no error free transmission (defined by BER ≤10−3 ) can be achieved without AO correction, while it is possible with AO correction if γ 0 ≥ 11.10 dB. We should mention that the effects that degrade the performance of coherent FSO system include not only the atmospheric turbulence, but also the tracking error. Several researchers have studied the influence of tracking error on the FSO system performance in the terrestrial environment [21–23]. However, rather few investigations have reported the impact in the maritime environment. Physically, over the sea, the ships or vessels would be rocked more severely by the water wave compared with the terrestrial environment. This would be an important impact that should also be considered, which is the issue that we will investigate as our next step. Further on, the multiple-input-multiple-output 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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(MIMO) technique can be used to achieve a diversity gain, thus mitigating the turbulence effects and multipath fading [24,25]. Therefore, we can expect that the combination of the AO real-time correction and the MIMO would offer an even more promising scheme for the coherent FSO over the maritime environment, which we plan to investigate as well. 5. Conclusion
We have studied the BER performance of the coherent FSO with QPSK modulation over maritime atmospheric turbulence. A significant improvement of BER performance has been confirmed when an optimum closed-loop AO system is applied to compensate for the turbulence effects. We have also made a comprehensive comparison between the maritime and terrestrial FSO links operating in atmospheric turbulence conditions. By modeling the links based on the transmittance, the phase distortions, and the scintillation, we found that the optical signals will suffer a smaller attenuation in the maritime atmosphere as compared to the terrestrial atmosphere in both wavelength regions of interest (i.e., around 850 and 1550 nm). The turbulence-induced wavefront phase distortions were modeled by a generated phase screen based on the use of the Monte Carlo method, which is validated by comparison with the theoretical phase structure function. We designed an optimized closed-loop AO system and used it to perform the correction processes. Our simulation results show that the phase distortions can be substantially reduced after the AO correction is employed. The developed model of FSO links related to maritime and terrestrial conditions have been used in conjunction with the designed AO system to evaluate the system BER performance. A substantial/ dramatic improvement can be obtained with AO compensation, thus approaching to the turbulencefree BER curve defined by the FEC threshold of ∼10−3 . Under the same turbulence conditions, a higher BER is observed for the maritime FSO link as compared to the terrestrial FSO link when they have the same refractive index structure parameter (C2n 10−15 m−2∕3 ) characterizing air turbulence in spite of the fact that the maritime FSO link will suffer a smaller attenuation. M. Li acknowledges the support from the BUPT Excellent Ph.D. Students Foundation (Grant No. CX201333) and the program of China Scholarship Council (Grant No. 201306470039). References 1. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16, 14151–14162 (2008). 2. Y. Li, M. Li, Y. Poo, J. Ding, M. Tang, and Y. Lu, “Performance analysis of OOK, BPSK, QPSK modulation schemes in uplink of ground-to-satellite laser communication system under atmospheric fluctuation,” Opt. Commun. 317, 57–61 (2014). 3. S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2, 1087–1094 (2010).
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