Abstract

The problem of estimating mechanical boresight and jitter performance of a laser pointing system in the presence of atmospheric turbulence is considered. A novel estimator based on maximizing an average probability density function (pdf) of the received signal is presented. The proposed estimator uses a Gaussian far-field mean irradiance profile, and the irradiance pdf is assumed to be lognormal. The estimates are obtained using a sequence of return signal values from the intended target. Alternatively, one can think of the estimates being made by a cooperative target using the received signal samples directly. The estimator does not require sample-to-sample atmospheric turbulence parameter information. The approach is evaluated using wave optics simulation for both weak and strong turbulence conditions. Our results show that very good boresight and jitter estimation performance can be obtained under the weak turbulence regime. We also propose a novel technique to include the effect of very low received intensity values that cannot be measured well by the receiving device. The proposed technique provides significant improvement over a conventional approach where such samples are simply ignored. Since our method is derived from the lognormal irradiance pdf, the performance under strong turbulence is degraded. However, the ideas can be extended with appropriate pdf models to obtain more accurate results under strong turbulence conditions.

© 2007 Optical Society of America

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References

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  1. 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] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. 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]
  5. S. Arnon, "Use of satellite natural vibrations to improve performance of free-space satellite laser communication," Appl. Opt. 37, 5031-5036 (1998).
    [CrossRef]
  6. 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]
  7. G. W. Lukesh and S. M. Chandler, "Non-imaging active system determination of target shape through a turbulent medium," Proc. SPIE 4167, 111-119 (2000).
  8. 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] [PubMed]
  9. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (The Society of Photo-Optical Instrumentation Engineers, 2001).
    [CrossRef]
  10. J. H. Churnside and R. G. Frehlich, "Experimental evaluation of lognormal modulated Rician and IK models of optical scintillation in the atmosphere," J. Opt. Soc. Am. A 6, 1760-1766 (1989).
    [CrossRef]
  11. 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]
  12. G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994).
  14. D. G. Luenberger, Linear and Nonlinear Programming (Kluwer Academic, 2004).
  15. L. L. Scharf, Statistical Signal Processing Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).
  16. C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice Hall, 1992).
  17. S. M. Kay, Fundamentals of Statistical Signal Processing Estimation Theory (Prentice Hall, 1993).

2006 (2)

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)

G. W. Lukesh and S. M. Chandler, "Non-imaging active system determination of target shape through a turbulent medium," Proc. SPIE 4167, 111-119 (2000).

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]

1998 (1)

1989 (1)

1982 (1)

1976 (1)

Appl. Opt. (5)

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 (2)

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]

G. W. Lukesh and S. M. Chandler, "Non-imaging active system determination of target shape through a turbulent medium," Proc. SPIE 4167, 111-119 (2000).

Other (7)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (The Society of Photo-Optical Instrumentation Engineers, 2001).
[CrossRef]

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994).

D. G. Luenberger, Linear and Nonlinear Programming (Kluwer Academic, 2004).

L. L. Scharf, Statistical Signal Processing Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice Hall, 1992).

S. M. Kay, Fundamentals of Statistical Signal Processing Estimation Theory (Prentice Hall, 1993).

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Figures (8)

Fig. 1
Fig. 1

Laser pointing system.

Fig. 2
Fig. 2

Jitter estimation performance under weak turbulence ( C n 2 = 10 16 m 2 / 3 ) . Low, medium, and high boresight refer to A = 0.03906, 0.07812, and 0.11719 m, respectively.

Fig. 3
Fig. 3

Boresight estimation performance under weak turbulence ( C n 2 = 10 16 m 2 / 3 ) . Low, medium, and high jitter refer to σ j = 0.0332, 0.0664, and 0.0996 m, respectively.

Fig. 4
Fig. 4

Jitter estimation performance under strong turbulence ( C n 2 = 10 14 m 2 / 3 ) . Low, medium, and high boresight refer to A = 0.03906, 0.07812, and 0.11719 m, respectively.

Fig. 5
Fig. 5

Boresight estimation performance under strong turbulence ( C n 2 = 10 14 m 2 / 3 ) . Low, medium, and high jitter refer to σ j = 0.0891, 0.1486, and 0.2070 m, respectively.

Fig. 6
Fig. 6

Jitter estimation performance under weak turbulence ( C n 2 = 10 16 m 2 / 3 ) . Low and high boresight refer to A = 0.03906 and 0.11719 m, respectively. Jitter is σ j = 0.0332 m.

Fig. 7
Fig. 7

Boresight estimation performance under weak turbulence ( C n 2 = 10 16 m 2 / 3 ) . Low and high boresight refer to A = 0.03906 and 0.11719 m, respectively. Jitter is σ j = 0.0332 m.

Fig. 8
Fig. 8

Jitter estimation performance under zero boresight for C n 2 = 10 16 m 2 / 3 . Low jitter corresponds to σ j = 0.0442 m and high jitter corresponds to σ j = 0.0996 m.

Tables (1)

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Table 1 Wave Optics Simulation Parameters

Equations (26)

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s [ n ] = s ¯ [ n ] e 2 X ,
s ¯ [ n ] = K T exp ( ( ( x [ n ] + A ) 2 + y 2 [ n ] ) 2 Ω T 2 ) , n = 1 , 2 , … ,  N ,
p ( x [ n ] , y [ n ] ) = 1 2 π σ j 2 exp ( ( x 2 [ n ] + y 2 [ n ] ) 2 σ j 2 ) ,
2 Ω T 2 ln ( K T / s [ n ] ) = ( x [ n ] + A ) 2 + y 2 [ n ] 4 Ω T 2 X .
p ( z [ n ] ) = 1 2 σ j 2 exp ( 1 2 σ j 2 ( z [ n ] + A 2 ) ) I 0 ( A σ j 2 z [ n ] ) u ( z [ n ] ) ,
p ( c [ n ] ) = 1 32 π Ω T 4 σ X 2 exp ( ( c [ n ] + 4 Ω T 2 m X ) 2 32 Ω T 4 σ X 2 ) .
p ( q | [ n ] | m X , σ X ) = 1 2 σ j 2 32 π Ω T 4 σ X 2 × exp ( A 2 2 σ j 2 ( q [ n ] + 4 Ω T 2 m X ) 2 32 Ω T 4 σ X 2 ) × 0 exp ( g 2 32 Ω T 4 σ X 2 ( 1 2 σ j 2 q [ n ] + 4 Ω T 2 m X 16 Ω T 4 σ X 2 ) g ) × I 0 ( A σ j 2 g ) d g .
I 0 ( A σ j 2 g ) = k = 0 1 ( k ! ) 2 ( A 2 σ j 2 ) 2 k g k .
k = 0 1 ( k ! ) 2 ( A 2 σ j 2 ) 2 k 0 g k exp ( g 2 32 Ω T 4 σ X 2 ( 1 2 σ j 2 q [ n ] + 4 Ω T 2 m X 16 Ω T 4 σ X 2 ) g ) d g .
0 x v 1 exp ( β x 2 γ x ) d x = 1 ( 2 β ) v / 2 Γ ( v ) × exp ( γ 2 8 β ) D v ( γ 2 β ) ,
( 4 Ω T 2 σ X ) ( k + 1 ) Γ ( k + 1 ) exp ( ( Ω T 2 σ X σ j 2 q [ n ] + 4 Ω T 2 m X 8 Ω T 2 σ X ) 2 ) × D k 1 ( 2 Ω T 2 σ X σ j 2 q [ n ] + 4 Ω T 2 m X 4 Ω T 2 σ X ) ,
p ( q | [ n ] | m X , σ X ) = 1 2 σ j 2 32 π Ω T 4 σ X 2 exp ( A 2 2 σ j 2 + Ω T 4 σ X 2 σ j 4 q [ n ] + 4 Ω T 2 m X 4 σ j 2 ( q [ n ] + 4 Ω T 2 m X ) 2 64 Ω T 4 σ X 2 ) × k = 0 1 k ! ( A 2 σ j 2 ) 2 k ( 4 Ω T 2 σ X ) ( k + 1 ) × D k 1 ( 2 Ω T 2 σ X σ j 2 q [ n ] + 4 Ω T 2 m X 4 Ω T 2 σ X ) .
p ( s | [ n ] | m X , σ X ) = 1 2 π ( Ω T σ j ) 2 1 s [ n ] exp ( A 2 2 σ j 2 + Ω T 4 σ X 2 σ j 4 Ω T 2 m X σ j 2 Ω T 2 2 σ j 2 ln ( K T s [ n ] ) ( m X 2 σ X + 1 4 σ X ln ( K T s [ n ] ) ) 2 ) k = 0 1 k ! ( A Ω T σ X σ j 2 ) 2 k × D k 1 ( 2 Ω T 2 σ X σ j 2 m X σ X 1 2 σ X ln ( K T s [ n ] ) ) .
θ ^ = arg max θ ln p ( s ; θ ) ,
θ ^ = arg max θ n = 1 N ln p ( s [ n ] ; θ ) = arg max θ n = 1 N ln ( p ( s [ n ] ; θ m X , σ X ) p ( m X , σ X ) × d m X d σ X ) ,
θ ^ arg max θ n = 1 N ln ( i j p ( s [ n ] ; | θ | m X i , σ X j ) ) ,
D 1 ( z ) = π 2 exp ( z 2 / 4 ) erfc ( z 2 ) ,
D 2 ( z ) = π 2 exp ( z 2 / 4 ) ( 2 π exp ( z 2 / 2 ) z   erfc ( z 2 ) ) .
P δ = P [ s [ n ] s min m X , σ X ] p ( m X , σ X ) d m X d σ X 1 M h k , l 0 x max 1 8 π σ j 4 σ X l 2 exp ( 1 2 σ j 2 ( z + A 2 ) ) × I 0 ( A σ j 2 z ) exp ( 1 2 σ X l 2 ( x m X k ) 2 ) d x d z ,
x max = 0.5 ln ( s min K T ) + z 4 Ω T 2 .
P δ = 1 M h k , l 0 ( 1 4 σ j 2 ) exp ( 1 2 σ j 2 ( z + A 2 ) ) I 0 ( A σ j 2 z ) × erfc ( 0.5 ln ( s min K T ) + z 4 Ω T 2 m X k 2 σ X l ) d z .
θ ^ = arg max θ ( n = 1 M ln p ( s [ n ] ; θ ) + ( N M ) ln P δ ) .
p ( q | [ n ] | m X , σ X ) = 1 2 σ j 2 32 π Ω T 4 σ X 2 exp ( ( q [ n ] + 4 Ω T 2 m X ) 2 32 Ω T 4 σ X 2 ) × 0 exp ( g 2 32 Ω T 4 σ X 2 g ( 1 2 σ j 2 q [ n ] + 4 Ω T 2 m X 16 Ω T 4 σ X 2 ) ) d g .
p ( q | [ n ] | m X , σ X ) = 1 4 σ j 2 exp ( 2 Ω T 4 σ X 2 σ j 4 1 2 σ j 2 ( q [ n ] + 4 Ω T 2 m X ) ) × erfc ( 2 Ω T 2 σ X σ j 2 m X 2 σ X q [ n ] 4 2 Ω T 2 σ X ) ,
p ( s | [ n ] | m X , σ X ) = Ω T 2 2 σ j 2 ( 1 s [ n ] ) exp ( 2 Ω T 4 σ X 2 σ j 4 1 2 σ j 2 ( 4 Ω T 2 m X + 2 Ω T 2 ln ( K T s [ n ] ) ) ) × erfc ( 2 Ω T 2 σ X σ j 2 m X 2 σ X 1 2 2 σ X × ln ( K T s [ n ] ) ) .
MSE = 1 M i = 1 M | q q ^ i | 2 ,

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