## Performance of synchronous optical receivers using atmospheric compensation techniques

Optics Express, Vol. 16, Issue 18, pp. 14151-14162 (2008)

http://dx.doi.org/10.1364/OE.16.014151

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### Abstract

We model the impact of atmospheric turbulence-induced phase and amplitude fluctuations on free-space optical links using synchronous detection. We derive exact expressions for the probability density function of the signal-to-noise ratio in the presence of turbulence. We consider the effects of log-normal amplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillator shot noise, for both passive receivers and those employing active modal compensation of wave-front phase distortion. We compute error probabilities for M-ary phase-shift keying, and evaluate the impact of various parameters, including the ratio of receiver aperture diameter to the wave-front coherence diameter, and the number of modes compensated.

© 2008 Optical Society of America

## 1. Introduction

1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE **55**, 57–67 (1967). [CrossRef]

2. D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. **QE-3**, 213–221 (1967). [CrossRef]

3. J. H. Churnside and C. M. McIntyre, “Signal current probability distribution for optical heterodyne receivers in the turbulent atmosphere. 1: Theory,” Appl. Opt. **17**, 2141–2147 (1978). [CrossRef] [PubMed]

4. J. H. Churnside and C. M. McIntyre, “Heterodyne receivers for atmospheric optical communications,” Appl. Opt. **19**, 582–590 (1980). [CrossRef] [PubMed]

5. K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communication links,” Appl. Opt. **25**, 1817–1825 (1986). [CrossRef] [PubMed]

6. N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt. **46**, 7218–7226 (2007). [CrossRef] [PubMed]

7. A. Belmonte, “Influence of atmospheric phase compensation on optical heterodyne power measurements,” Opt. Express **16**, 6756–6767 (2008). [CrossRef] [PubMed]

## 2. First-order statistics in optical homodyne or heterodyne detection

8. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express **16**, 753–791 (2008). [CrossRef] [PubMed]

*A*is the amplitude without the effect of turbulence, and

_{s}*χ*(

*r*) and

*ϕ*(

*r*) represent the logamplitude fluctuations (scintillation) and phase variations (aberrations), respectively, introduced by atmospheric turbulence. In the heterodyne downconverter, the information-carrying photocurrent

*i*at the output of the balanced receiver is:

_{s}*η*is the quantum efficiency of the photodetector,

*A*

_{0}is the amplitude of the local oscillator, and Δ

*f*and Δ

*ϕ*are, respectively, the differences between the frequencies and phases of the signal and local oscillator. The circular receiving aperture of diameter

*D*is defined by the aperture function

*W*(

*r*), which equals unity for |

*r*|≤

*D*/2, and equals zero for |

*r*|>

*D*/2. We can rewrite the cosine using cos(

*u*-

*v*)=cos

*u*cos

*v*+sin

*u*sin

*v*, obtaining

*f*oscillations of Eq. (3), the average detected signal power

*α*and

_{r}*α*represent integrals over the collecting aperture of the real and imaginary parts, respectively, of the normalized optical field reaching the receiver. These real and imaginary parts can be considered as the components of a complex random phasor.

_{i}*e*is the electronic charge. The SNR per unit bandwidth

*α*

^{2}=

*α*

^{2}

*+*

_{r}*α*

^{2}

*, a random scale factor representing the effect of both the amplitude and phase fluctuations of the optical field. The statistical properties of the random variable*

_{i}*α*

^{2}, with mean

*p*

*α*

^{2}(

*α*

^{2}), provide a statistical characterization of the SNR γ. Using the average SNR

*α*

^{2}, we study how amplitude and phase fluctuations of the optical field define the statistics of the fading intensity

*α*

^{2}=

*α*

^{2}

*+*

_{r}*α*

^{2}

*. We note that the two random magnitudes*

_{i}*α*

*and*

_{r}*α*

*are expressed in Eq. (5) as integrals over the aperture and, hence, are the sums of contributions from each point in the aperture. In order to proceed with the analysis, we consider a statistical model in which these continuous integrals are expressed as finite sums over*

_{i}*N*statistically independent cells in the aperture:

*χk*is the log-amplitude and

*ϕ*is the phase of the kth statistically independent cell. A similar approach has been used to analyze the statistics of the Strehl ratio [9

_{k}9. M. P. Cagigal and V. F. Canales, “Speckle statistics in partially corrected wave fronts,” Opt. Lett. **23**, 1072–1074 (1998). [CrossRef]

*N*, the number of independent cells, is large enough, we can consider that

*α*and

_{r}*α*asymptotically approach jointly normal random variables:

_{i}*α*̄

*,*

_{r}*α*̄

*and*

_{i}*σ*

^{2}

*,*

_{r}*σ*

^{2}

*are the means and variances of*

_{i}*α*

*,*

_{r}*α*

*, which are required to evaluate the joint PDF. To estimate these means and variances, we recall that*

_{i}*α*

*and*

_{r}*α*

*can be considered as the real and imaginary parts of a random phasor. Hence, it is possible to evaluate the means and variances of*

_{i}*α*

*and*

_{r}*α*

*by using the classical statistical model for speckle with a non-uniform distribution of phases [10]. After some algebraic manipulation, mean values can be obtained as*

_{i}*M*

*(*

_{ϕ}*ω*) is the characteristic function of the phase, i.e., the Fourier transform of its PDF. We point out that because

*α*,

_{r}*α*result from atmospheric turbulence, we can consider phases

_{i}*ϕ*that obey zero-mean Gaussian statistics:

_{k}11. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1976). [CrossRef]

*σ*

^{2}

*were extended to consider modal compensation of atmospheric phase distortion. In such modal compensation, Zernike polynomials are widely used as basis functions because of their simple analytical expressions and their correspondence to classical aberrations [12]. It is known that the residual phase variance after modal compensation of*

_{ϕ}*J*Zernike terms is given by

*D*is normalized by the wavefront coherence diameter

*r*

_{0}, which describes the spatial correlation of phase fluctuations in the receiver plane [1

1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE **55**, 57–67 (1967). [CrossRef]

*C*depends on

_{J}*J*[11

11. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1976). [CrossRef]

*J*=3, 6, 10 and 20, respectively. Ideally, it is desirable to choose

*J*large enough that the residual variance Eq. (15) becomes negligible.

*χ*are normal random variables [13

_{k}13. J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Science **10**, 59–70 (1975). [CrossRef]

*β*≡exp

^{2}(

*χ*-

_{k}*χ*̄) has a mean given by exp(2

*σ*

^{2}

*). The log-amplitude variance σ2χ is often expressed as a scintillation index*

_{χ}*σ*

^{2}

*=exp(4σ*

_{β}^{2}

*)-1.*

_{χ}*N*present in the aperture. An analytical expression to estimate

*N*can be defined by

*W*(

*r*) again characterizes the collecting aperture with area

*S*=(π/4)

*D*

^{2}. Here,

*C*(

*r*) is the coherence function describing the wavefront distortion introduced by atmospheric turbulence [1

1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE **55**, 57–67 (1967). [CrossRef]

*r*

_{0}, i.e.,

*D*»

*r*

_{0}. In this case,

*N*=[1.007 (

*r*

_{0}/

*D*)

^{2}]

^{-1}. To a good approximation, the aperture can be considered to consist of (

*D*/

*r*

_{0})

^{2}independent cells, each of diameter

*r*

_{0}. In the opposite extreme of an aperture much smaller than the coherence diameter,

*D*«

*r*

_{0}, we have C(

*r*)≈1 and

*N*=1. This result indicates that as the aperture gets smaller, the number of cells approaches unity. Values of

*N*<1 are not possible. An analytical expression for

*N*valid for all aperture diameters is given by

*a*,

*x*) is the lower incomplete gamma function. Equation (22) will be used to estimate the value of

*N*needed to evaluate Eq. (17).

*α*

^{2}and phase

*θ*can be found by substituting

*α*

^{2}, we integrate with respect to

*θ*:

*α*

*and*

_{r}*α*

*are jointly normal random variables, it is possible to obtain the mean and variance of the intensity fading as*

_{i}*σ*

^{2}

*and*

_{ϕ}*σ*

^{2}

*are small but the log-amplitude variance*

_{χ}*σ*

^{2}

*can be neglected in comparison to the effect of phase aberrations. From Eq. (17), we obtain*

_{χ}*σ*

^{2}

*=0, α*

_{r}*̄*

*=1-*

_{r}*σ*

^{2}

*/2, and*

_{ϕ}*σ*

^{2}

*=*

_{i}*σ*

^{2}

*/*

_{ϕ}*N*. In this case, the fading intensity

*α*

^{2}is defined as the sum of a constant (coherent) term

*α*

*with amplitude*

_{r}*α*̄

*and a random (incoherent) term*

_{r}*α*

*with zero mean and variance*

_{i}*σ*

^{2}

*. Such a random variable, defined as the sum of a known dominant phasor plus a random phasor sum, is characterized by a Rice PDF [10]:*

_{i}*σ*

^{2}=

*σ*

^{2}

*. The parameter*

_{i}*a*

^{2}represents the coherent intensity that dominates over the fluctuating residual halo, whose intensity is represented by 2

*σ*

^{2}. It is convenient to express the basic Rice PDF (24) in terms of the mean

*r*≡2

*σ*

^{2}/

*a*

^{2}, a measure of the strength of the residual halo to the coherent component:

*r*as

*σ*

^{2}

*and the log-amplitude variance*

_{ϕ}*σ*

^{2}

*need to be considered. We can expect the general marginal PDF Eq. (23) to behave like the Rice PDF Eq. (25) provided that we can find a set of equivalent Rice parameters (*

_{χ}*r*) that makes Eq. (17) and Eq. (19) have identical mean and variance. Comparing the means and variances of the two distributions, the required values of

*r*can be computed as functions of

*α*̄

*,*

_{r}*σ*

^{2}

*, and*

_{r}*σ*

^{2}

*through the relations*

_{i}*r*can be obtained from Eq. (26) as

*α*̄

*and variances*

_{r}*σ*

^{2}

*,*

_{r}*σ*

^{2}

*are obtained with the help of Eq. (17).*

_{r}*r*ranges between 0 and ∞. It can be shown that when the dominant term is very weak (

*r*→0), intensity fading

*α*

^{2}becomes negative-exponential-distributed, just as in a speckle pattern. Likewise, when the dominant term is very strong (

*r*→∞), the density function becomes highly peaked around the mean value

*r*is large, it can be shown that the PDF of

*α*

^{2}is, except for a skewness factor

*α*, approximately Gaussian with mean

*α*

^{2}, and applying Eq. (8), we find that the SNR γ is described by a noncentral chi-square distribution with two degrees of freedom:

*γ*

_{0}and the parameter

6. N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt. **46**, 7218–7226 (2007). [CrossRef] [PubMed]

## 3. Performance of coherent receivers

*α*

^{2}, a random variable with PDF

*α*

^{2}. Hence, the instantaneous SNR γ is proportional to

*α*

^{2}. The symbol-error probability (SEP)

*p*(

_{s}*E*) of an ideal coherent receiver is obtained by averaging the SEP conditioned on the SNR γ over the PDF of the instantaneous SNR,

*p*

*(*

_{γ}*γ*):

*M*-ary phase-shift keying (

*M*-PSK) with ideal coherent detection based on maximum-likelihood principles. In this case, the SEP conditioned on the instantaneous SNR is given by [14

14. R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. **COM-30**, 1828–1841 (1982). [CrossRef]

^{2}θ≤1. The integral in Eq. (30) is reduced to

*M*-PSK receiver:

_{0}per symbol, the receiver aperture diameter

*D*, the number of spatial modes

*J*removed by the compensation system, and the strength of atmospheric turbulence. Turbulence is quantified by two parameters: the phase coherence length

*r*

_{0}and the scintillation index

*σ*

^{2}

_{β}. We consider two nonzero values of the scintillation index. The value

*σ*

^{2}

*=0.3 corresponds to relatively low scintillation levels, while*

_{β}*σ*

^{2}

*=1 corresponds to strong scintillation, but still below the saturation regime. When the turbulence reaches the saturation regime, wavefront distortion becomes so severe that it would be unrealistic to consider phase compensation. In most practical free-space links, amplitude fluctuations are not saturated [15*

_{β}15. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, *Laser Beam Scintillation with Applications* (SPIE Press, 2001). [CrossRef]

_{0}. Figure 1(a) shows the performance for different values of the normalized aperture diameter

*D*/

*r*

_{0}, while Fig. 1(b) shows the performance for different values of

*J*, the number of modes compensated. We assume no scintillation,

*σ*

^{2}

*=0, so the effect of turbulence is simply to reduce the coherence length*

_{β}*r*

_{0}. For a fixed aperture diameter

*D*, as

*r*

_{0}is reduced, the normalized aperture diameter

*D*/

*r*

_{0}increases, and turbulence reduces the heterodyne or homodyne downconversion efficiency. Even using a relatively small normalized aperture diameter

*D*/

*r*

_{0}=1, turbulence introduces more than a 15-dB performance penalty at 10

^{-3}SEP. When phase correction is used, as in Fig. 1(b), in most situations, compensation of just a few modes yields a substantial performance improvement. Compensation of

*J*=20 modes yields significant improvement for even the largest normalized apertures considered. For example, for a normalized aperture

*D*/

*r*

_{0}=10, at a SEP=10

^{-3}, the SNR penalty is just over 5 dB. This value should be contrasted with the 15-dB penalty observed in Fig. 1(a) for

*D*/

*r*

_{0}=1 when

*J*=0, i.e., no modes are compensated.

*r*

_{0}. Figure 2(a) shows the performance for different values of the scintillation index

*σ*

^{2}

_{β}, while Fig. 2(b) shows the performance for different values of

*J*, the number of modes compensated. In all cases presented, the turbulence-free SNR has a value γ

_{0}=10 dB. The aperture diameter is fixed in every plot to

*D*=10 cm. As we observe in Fig. 2(a), for strong turbulence, corresponding to small values of

*r*

_{0}, the SEP is substantially independent of the scintillation index

*σ*

^{2}

_{β}. In this regime, phase distortions have a large impact, and high-order phase corrections may be required. We note that in this regime of strong turbulence, the coherent part of

*α*

^{2}is very weak,

*r*→0, and intensity fading becomes negative-exponential distributed, i.e.

*p*(

_{r}*γ*)=1/

*γ*̄exp(-

*γ*/

*γ*̄). In this case, SEP Eq. (31) reduces to

*r*

_{0}) considered,

*γ*̄→0 and the SEP Eq. (34) asymptotes to a maximum value (

*M*-1)/

*M*. In the plots shown, where

*M*=4, we have

*p*(

_{s}*E*)→3/4.

*D*/

*r*

_{0}for a constant phase coherence length

*r*

_{0}and constant scintillation index

*σ*

^{2}

_{β}. For the smallest aperture diameter considered, the turbulence-free SNR has a value γ

_{0}=10 dB. For any other aperture diameter, the value of γ

_{0}is proportional to

*D*

^{2}. Figure 3 illustrates the concept of an optimal aperture diameter in coherent free-space links. This optimal aperture diameter, which minimizes the SEP, exhibits two different regimes in our studies. For relatively small apertures, amplitude scintillation is dominant, and performance is virtually unaffected by wavefront phase correction. When the aperture is larger, phase distortion becomes dominant, and high-order phase correction may be needed to improve performance to acceptable levels. In [7

7. A. Belmonte, “Influence of atmospheric phase compensation on optical heterodyne power measurements,” Opt. Express **16**, 6756–6767 (2008). [CrossRef] [PubMed]

*J*=0), and the performance is shown for different values of the scintillation index

*σ*

^{2}

_{β}. Here, the optimal normalized aperture diameter is close to

*D*/

*r*

_{0}=0.3, and increases slightly with increasing scintillation index. In any case, when

*σ*

^{2}

_{β}becomes too large, the optimization is of little practical significance. In Fig. 3(b), we consider strong scintillation,

*σ*

^{2}

_{β}=1, and show the performance for different values of

*J*, the number of modes compensated. As we increase

*J*, the optimized value of

*D*/

*r*

_{0}increases, and the optimized SEP improves substantially. Even for such strong scintillation, with compensation of

*J*=20 modes and optimized

*D*/

*r*

_{0}, excellent SEP performance is obtained. In this case, the optimized

*D*/

*r*

_{0}is rather large (close to 4). For the larger values of

*D*/

*r*

_{0}considered in these plots, the coherent part of

*α*

^{2}is very weak (

*r*→0) and fading intensity becomes negative-exponential-distributed, such that the SEP is described by Eq. (34). In Fig. 3(a), when large normalized apertures

*D*/

*r*

_{0}are considered, the SEP becomes independent of the scintillation index

*σ*

^{2}

_{β}, and tends toward an asymptotic value that is independent of normalized aperture diameter

*D*/

*r*

_{0}. In Fig. 3(b), at large values of

*D*/

*r*

_{0}, the SEPs also tend toward asymptotic values, independent of normalized aperture diameter

*D*/

*r*

_{0}, which depend only weakly on the scintillation index

*σ*

^{2}

_{β}.

## 4. Conclusions

*M*-PSK with additive white Gaussian noise, and have used them to study the effect of various parameters on performance, including signal level, aperture diameter, turbulence strength, and the number of modes compensated. We have separately quantified the effects of amplitude fluctuations and wavefront phase distortion on system performance, and have identified two different regimes of turbulence, depending on the receiver aperture diameter normalized to the coherence diameter of the wavefront phase. When the normalized aperture diameter is relatively small, amplitude scintillation dominates and, as phase fluctuations have little impact, performance is virtually independent of the number of modes compensated. When the normalized aperture is larger, amplitude fluctuations become negligible, and phase fluctuations become dominant, so that high-order phase compensation may be needed to improve performance to acceptable levels. We have found that for most typical link designs, wavefront phase fluctuations are the dominant impairment, and compensation of a modest number of modes can reduce performance penalties by several decibels.

## Acknowledgment

## References and links

1. | D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE |

2. | D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. |

3. | J. H. Churnside and C. M. McIntyre, “Signal current probability distribution for optical heterodyne receivers in the turbulent atmosphere. 1: Theory,” Appl. Opt. |

4. | J. H. Churnside and C. M. McIntyre, “Heterodyne receivers for atmospheric optical communications,” Appl. Opt. |

5. | K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communication links,” Appl. Opt. |

6. | N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt. |

7. | A. Belmonte, “Influence of atmospheric phase compensation on optical heterodyne power measurements,” Opt. Express |

8. | E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express |

9. | M. P. Cagigal and V. F. Canales, “Speckle statistics in partially corrected wave fronts,” Opt. Lett. |

10. | J. W. Goodman, |

11. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

12. | M. Born and E. Wolf, |

13. | J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Science |

14. | R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. |

15. | L. C. Andrews, R. L. Phillips, and C. Y. Hopen, |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.3640) Atmospheric and oceanic optics : Lidar

(030.6600) Coherence and statistical optics : Statistical optics

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Atmospheric and oceanic optics

**History**

Original Manuscript: June 17, 2008

Revised Manuscript: July 23, 2008

Manuscript Accepted: August 20, 2008

Published: August 26, 2008

**Citation**

Aniceto Belmonte and Joseph Khan, "Performance of synchronous optical receivers
using atmospheric compensation techniques," Opt. Express **16**, 14151-14162 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14151

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### References

- D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967). [CrossRef]
- D. L. Fried, "Atmospheric modulation noise in an optical heterodyne receiver," IEEE J. Quantum Electron. QE-3, 213-221 (1967). [CrossRef]
- J. H. Churnside and C. M. McIntyre, "Signal current probability distribution for optical heterodyne receivers in the turbulent atmosphere. 1: Theory," Appl. Opt. 17, 2141-2147 (1978). [CrossRef] [PubMed]
- J. H. Churnside and C. M. McIntyre, "Heterodyne receivers for atmospheric optical communications," Appl. Opt. 19, 582-590 (1980). [CrossRef] [PubMed]
- K. A. Winick, "Atmospheric turbulence-induced signal fades on optical heterodyne communication links," Appl. Opt. 25, 1817-1825 (1986). [CrossRef] [PubMed]
- N. Perlot, "Turbulence-induced fading probability in coherent optical communication through the atmosphere," Appl. Opt. 46, 7218-7226 (2007). [CrossRef] [PubMed]
- A. Belmonte, "Influence of atmospheric phase compensation on optical heterodyne power measurements," Opt. Express 16, 6756-6767 (2008). [CrossRef] [PubMed]
- E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16, 753-791 (2008). [CrossRef] [PubMed]
- M. P. Cagigal and V. F. Canales, "Speckle statistics in partially corrected wave fronts," Opt. Lett. 23, 1072-1074 (1998). [CrossRef]
- J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Ben Roberts & Company, 2007).
- R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999)
- J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Science 10, 59-70 (1975). [CrossRef]
- R. F. Pawula, S. O. Rice, and J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982). [CrossRef]
- L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]

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