Abstract

An accurate pointing system is required in free-space optical (FSO) communication links. Low energy-transmission efficiency caused by pointing errors would decline the communication system’s performance. The statistics of the detected signal or return signal values could be used to estimate the pointing parameters, whereas atmospheric turbulence brings in serious challenges. A modified moment-matching estimation method is presented in this paper. The irradiance fluctuation caused by the atmospheric turbulence is considered, and the probability density function (PDF) in a weak turbulence condition is assumed to be lognormal. This modified approach is evaluated with wave-propagation simulation data and shows significant improvement over the conventional approach. The estimation accuracy and the properties of this new approach are also discussed. Although our method is based on lognormal irradiance PDF under a weak turbulence condition, the irradiance PDF would tend to be lognormal with aperture averaging effect under moderate to strong turbulence, and the ideas can be extended with appropriate PDF models to satisfy different conditions.

© 2012 Optical Society of America

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References

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  1. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27, 3965–3973 (2009).
    [CrossRef]
  2. G. Lukesh, S. Chandler, and D. G. Voelz, “Estimation of laser system pointing performance by use of statistics of return photons,” Appl. Opt. 39, 1359–1371 (2000).
    [CrossRef]
  3. D. K. Borah, D. Voelz, and S. Basu, “Maximum-likelihood estimation of a laser system pointing parameters by use of return photon counts,” Appl. Opt. 45, 2504–2509 (2006).
    [CrossRef]
  4. D. K. Borah and D. Voelz, “Estimation of laser beam pointing parameters in the presence of atmospheric turbulence,” Appl. Opt. 46, 6010–6018 (2007).
    [CrossRef]
  5. V. S. R. Gudimetla and J. F. Riker, “Moment-matching method for extracting beam jitter and boresight in experiments with satellites of small physical cross section,” Appl. Opt. 46, 5608–5616 (2007).
    [CrossRef]
  6. V. S. R. Gudimetla and J. F. Riker, “Moment-matching method for extraction of asymmetric beam jitters and bore sight errors in simulations and experiments with actively illuminated satellites of small physical cross section,” Appl. Opt. 50, 1124–1135 (2011).
    [CrossRef]
  7. G. W. Lukesh, S. M. Chandler, and D. G. Voelz, “Analysis of satellite laser optical cross sections from the active imaging testbed,” Proc. SPIE 4538, 24–33 (2002).
    [CrossRef]
  8. S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part I: theory and real-time feasibility,” Proc. SPIE 5552, 105–113 (2004).
    [CrossRef]
  9. S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part II: laboratory experiment,” Proc. SPIE 5552, 114–122 (2004).
    [CrossRef]
  10. S. Chandler and G. Lukesh, “Non-imaging detection of target shape and size,” Proc. SPIE 6234, 85–94 (2006).
    [CrossRef]
  11. D. K. Borah and D. Voelz, “Cramer-Rao lower bounds on estimation of laser system pointing parameters by use of the return photon signal,” Opt. Lett. 31, 1029–1031 (2006).
    [CrossRef]
  12. X. F. Li, The Principle and Technology of the Satellite-to-Ground Laser Communication Links (National Defense Industry, Beijing, 2007).
  13. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  14. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
    [CrossRef]
  15. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46, 2099–2108 (2007).
    [CrossRef]
  16. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef]
  17. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  18. H. X. Yan, S. S. Li, D. L. Zhang, and S. Chen, “Numerical simulation of an adaptive optics system with laser propagation in the atmosphere,” Appl. Opt. 39, 3023–3031 (2000).
    [CrossRef]
  19. L. Zhou, Y. Tan, and G. Ren, “One new quality of the maximum-likelihood estimation of laser pointing system by use of return photon counts,” Proc. SPIE 7843, 78430K (2010).
    [CrossRef]

2011 (1)

2010 (1)

L. Zhou, Y. Tan, and G. Ren, “One new quality of the maximum-likelihood estimation of laser pointing system by use of return photon counts,” Proc. SPIE 7843, 78430K (2010).
[CrossRef]

2009 (1)

2007 (3)

2006 (3)

2004 (2)

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part I: theory and real-time feasibility,” Proc. SPIE 5552, 105–113 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part II: laboratory experiment,” Proc. SPIE 5552, 114–122 (2004).
[CrossRef]

2002 (1)

G. W. Lukesh, S. M. Chandler, and D. G. Voelz, “Analysis of satellite laser optical cross sections from the active imaging testbed,” Proc. SPIE 4538, 24–33 (2002).
[CrossRef]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

2000 (2)

1999 (1)

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1975 (1)

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Andrews, L.

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Basu, S.

D. K. Borah, D. Voelz, and S. Basu, “Maximum-likelihood estimation of a laser system pointing parameters by use of return photon counts,” Appl. Opt. 45, 2504–2509 (2006).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part I: theory and real-time feasibility,” Proc. SPIE 5552, 105–113 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part II: laboratory experiment,” Proc. SPIE 5552, 114–122 (2004).
[CrossRef]

Borah, D. K.

Chandler, S.

Chandler, S. M.

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part II: laboratory experiment,” Proc. SPIE 5552, 114–122 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part I: theory and real-time feasibility,” Proc. SPIE 5552, 105–113 (2004).
[CrossRef]

G. W. Lukesh, S. M. Chandler, and D. G. Voelz, “Analysis of satellite laser optical cross sections from the active imaging testbed,” Proc. SPIE 4538, 24–33 (2002).
[CrossRef]

Chen, S.

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Gudimetla, V. S. R.

Hopen, C. Y.

Li, S. S.

Li, X. F.

X. F. Li, The Principle and Technology of the Satellite-to-Ground Laser Communication Links (National Defense Industry, Beijing, 2007).

Lukesh, G.

Lukesh, G. W.

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part I: theory and real-time feasibility,” Proc. SPIE 5552, 105–113 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part II: laboratory experiment,” Proc. SPIE 5552, 114–122 (2004).
[CrossRef]

G. W. Lukesh, S. M. Chandler, and D. G. Voelz, “Analysis of satellite laser optical cross sections from the active imaging testbed,” Proc. SPIE 4538, 24–33 (2002).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

Recolons, J.

Ren, G.

L. Zhou, Y. Tan, and G. Ren, “One new quality of the maximum-likelihood estimation of laser pointing system by use of return photon counts,” Proc. SPIE 7843, 78430K (2010).
[CrossRef]

Riker, J. F.

Siegman, A. E.

Sjogren, J.

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part II: laboratory experiment,” Proc. SPIE 5552, 114–122 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part I: theory and real-time feasibility,” Proc. SPIE 5552, 105–113 (2004).
[CrossRef]

Sziklas, E. A.

Tan, Y.

L. Zhou, Y. Tan, and G. Ren, “One new quality of the maximum-likelihood estimation of laser pointing system by use of return photon counts,” Proc. SPIE 7843, 78430K (2010).
[CrossRef]

Vetelino, F. S.

Voelz, D.

Voelz, D. G.

Yan, H. X.

Young, C.

Zhang, D. L.

Zhou, L.

L. Zhou, Y. Tan, and G. Ren, “One new quality of the maximum-likelihood estimation of laser pointing system by use of return photon counts,” Proc. SPIE 7843, 78430K (2010).
[CrossRef]

Appl. Opt. (8)

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef]

G. Lukesh, S. Chandler, and D. G. Voelz, “Estimation of laser system pointing performance by use of statistics of return photons,” Appl. Opt. 39, 1359–1371 (2000).
[CrossRef]

H. X. Yan, S. S. Li, D. L. Zhang, and S. Chen, “Numerical simulation of an adaptive optics system with laser propagation in the atmosphere,” Appl. Opt. 39, 3023–3031 (2000).
[CrossRef]

D. K. Borah, D. Voelz, and S. Basu, “Maximum-likelihood estimation of a laser system pointing parameters by use of return photon counts,” Appl. Opt. 45, 2504–2509 (2006).
[CrossRef]

F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46, 2099–2108 (2007).
[CrossRef]

V. S. R. Gudimetla and J. F. Riker, “Moment-matching method for extracting beam jitter and boresight in experiments with satellites of small physical cross section,” Appl. Opt. 46, 5608–5616 (2007).
[CrossRef]

D. K. Borah and D. Voelz, “Estimation of laser beam pointing parameters in the presence of atmospheric turbulence,” Appl. Opt. 46, 6010–6018 (2007).
[CrossRef]

V. S. R. Gudimetla and J. F. Riker, “Moment-matching method for extraction of asymmetric beam jitters and bore sight errors in simulations and experiments with actively illuminated satellites of small physical cross section,” Appl. Opt. 50, 1124–1135 (2011).
[CrossRef]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (5)

L. Zhou, Y. Tan, and G. Ren, “One new quality of the maximum-likelihood estimation of laser pointing system by use of return photon counts,” Proc. SPIE 7843, 78430K (2010).
[CrossRef]

G. W. Lukesh, S. M. Chandler, and D. G. Voelz, “Analysis of satellite laser optical cross sections from the active imaging testbed,” Proc. SPIE 4538, 24–33 (2002).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part I: theory and real-time feasibility,” Proc. SPIE 5552, 105–113 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects—part II: laboratory experiment,” Proc. SPIE 5552, 114–122 (2004).
[CrossRef]

S. Chandler and G. Lukesh, “Non-imaging detection of target shape and size,” Proc. SPIE 6234, 85–94 (2006).
[CrossRef]

Other (1)

X. F. Li, The Principle and Technology of the Satellite-to-Ground Laser Communication Links (National Defense Industry, Beijing, 2007).

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Figures (12)

Fig. 1.
Fig. 1.

Laser pointing system demonstration.

Fig. 2.
Fig. 2.

The intensity distribution of (a) a single frame and (b) average of 100,000 frames in the receiving plane.

Fig. 3.
Fig. 3.

Time series of the simulated signal with a point detector without any pointing error.

Fig. 4.
Fig. 4.

Simulated PDF of the log irradiance without any pointing error and the PDF of lognormal distribution.

Fig. 5.
Fig. 5.

Distribution of the pointing error estimates with traditional moment-matching method (dot). The simulation values (circle) are boresight=20.72μrad and jitter=11.61μrad. The mean estimations are boresight=28.89μrad and jitter=9.03μrad.

Fig. 6.
Fig. 6.

Distribution of the pointing error estimates with modified moment-matching method (dot). The simulation values (circle) are boresight=20.72μrad and jitter=11.61μrad. The mean estimations are boresight=21.05μrad and jitter=11.46μrad.

Fig. 7.
Fig. 7.

Boresight estimate variation as the jitter estimate changes.

Fig. 8.
Fig. 8.

Boresight estimates with estimated jitter and known jitter.

Fig. 9.
Fig. 9.

Pointing errors estimation performance. MSE of the jitter estimation, the boresight estimation with estimated jitter, and boresight estimation with known jitter.

Fig. 10.
Fig. 10.

Relationship between the mean value and the variance of the irradiance logarithm.

Fig. 11.
Fig. 11.

Pointing errors estimation performance with different jitter.

Fig. 12.
Fig. 12.

Pointing errors estimation performance with different number of samples.

Tables (1)

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Table 1. Simulation Parameters for Wave Propagation Through Turbulence

Equations (28)

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r(xj,yj)=Aexp(xj2+yj22Ω2),
p(xj,yj)=12πσj2exp(xj2+yj22σj2),
k=(r2r21)1/2,
k0.707σj/Ω.
r(xj,yj,B)=Aexp[(xj+B)2+yj22Ω2],
rnrn=(σ^j2+1)nnσ^j2+1exp[n(n1)B^2σ^j22(σ^j2+1)(nσ^j2+1)],
ln[rnrn]nln(σ^j2+1)+ln(nσ^j2+1)=n(n1)B^2σ^j22(σ^j2+1)(nσ^j2+1),
ln[rmrm]mln(σ^j2+1)+ln(mσ^j2+1)=m(m1)B^2σ^j22(σ^j2+1)(nσ^j2+1).
ln[rnrn]nln(σ^j2+1)+ln(nσ^j2+1)ln[rmrm]mln(σ^j2+1)+ln(mσ^j2+1)=n(n1)(mσ^j2+1)m(m1)(nσ^j2+1).
B^=4bz/D,σ^j=4jz/D,
ln[rnrn]nln(2j2z2γ+1)+ln(2nj2z2γ+1)ln[rmrm]mln(2j2z2γ+1)+ln(2mj2z2γ+1)=n(n1)(2mj2z2γ+1)m(m1)(2nj2z2γ+1).
χ=lnII0,
pχ(χ)=12πσχexp[(χmχ)22σχ2],
I(χ)=I0exp(χ).
I(χ)=I0eχpχ(χ)dχ,
σχ2=2mχ.
I(xj,yj,B,χ)=I0(xj,yj,B)exp(χ).
I0(xj,yj,B)=Aexp[(xj+B)2+yj22ΩT2],
In=In(xj,yj,χ;B)p(xj,yj,χ)dxjdyjdχ,
In=In(xj,yj,χ;B)p(xj,yj)p(χ)dxjdyjdχ.
In=Annσ^j2+1exp[nB^22(nσ^j2+1)]exp(nmχ+n22σχ2),
ln[InIn]=nln(σ^j2+1)ln(nσ^j2+1)+n(n1)B^2σ^j22(σ^j2+1)(nσ^j2+1)+12n(n1)σχ2.
σχ2=[lnIlIllln(σ^j2+1)+ln(lσ^j2+1)l(l1)B^2σ^j22(σ^j2+1)(lσ^j2+1)]2l(l1).
ln[InIn]=nln(σ^j2+1)ln(nσ^j2+1)+n(n1)l(l1)[lnIlIllln(σ^j2+1)+ln(lσ^j2+1)]+n(n1)(ln)B^2σ^j42(σ^j2+1)(nσ^j2+1)(lσ^j2+1),
(mn)(lσ^j2+1)l(l1)[lnIlIllln(σ^j2+1)+ln(lσ^j2+1)]=(ln)(mσ^j2+1)m(m1)[lnImImmln(σ^j2+1)+ln(mσ^j2+1)](lm)(nσ^j2+1)n(n1)[lnInInnln(σ^j2+1)+ln(nσ^j2+1)],
WT=Wv·1+1.33σ12(2LkWv2)5/6,
{x¯=x1z/ly¯=y1z/lz¯=z,
MSE=1Mi=1M|qq^i|2,

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