Abstract

A maximum-likelihood estimator used to determine boresight and jitter performance of a laser pointing system has been derived. The estimator is based on a Gaussian jitter model and uses a Gaussian far-field irradiance profile. The estimates are obtained using a set of return shots from the intended target. An experimental setup with a He–Ne laser and steering mirrors is used to study the performance of the proposed method. Both Monte Carlo simulations and experimental results demonstrate excellent performance of the estimator. Our study shows that boresight estimation is more challenging than jitter estimation when both quantities are estimated. Furthermore, their estimation performance improves with an increase in the number of shots. The experimental results are found to agree well with the simulation results.

© 2006 Optical Society of America

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References

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  1. 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]
  2. A. Erteza, "Boresighting a Gaussian beam on a specular target point: a method using conical scan," Appl. Opt. 15, 656-660 (1976).
    [CrossRef] [PubMed]
  3. P. S. Neelakantaswarmy and A. Rajaratram, "Boresight error in the conical scan method of autoboresighting a laser beam on a specular point-target," Appl. Opt. 21, 3607-3612 (1982).
    [CrossRef]
  4. S. Arnon, "Use of satellite natural vibrations to improve performance of free-space satellite laser communication," Appl. Opt. 37, 5031-5036 (1998).
    [CrossRef]
  5. G. W. Lukesh, S. M. Chandler, and D. G. Voelz, "Analysis of satellite laser optical cross sections from the active imaging testbed," in Proc. SPIE 4538, 24-33 (2002).
    [CrossRef]
  6. G. W. Lukesh and S. M. Chandler, "Non-imaging active system determination of target shape through a turbulent medium," in Proc. SPIE 4167, 111-119 (2000).
  7. S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
    [CrossRef]
  8. S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in Proc. SPIE 5552, 114-122 (2004).
    [CrossRef]
  9. L. L. Scharf, Statistical Signal Processing Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).
  10. D. G. Luenberger, Linear and Nonlinear Programming (Kluwer Academic, 2004).

2004 (2)

S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
[CrossRef]

S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in 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," in Proc. SPIE 4538, 24-33 (2002).
[CrossRef]

2000 (2)

G. W. Lukesh and S. M. Chandler, "Non-imaging active system determination of target shape through a turbulent medium," in 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)

1982 (1)

1976 (1)

Arnon, S.

Basu, S.

S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in Proc. SPIE 5552, 114-122 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
[CrossRef]

Chandler, S.

Chandler, S. M.

S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
[CrossRef]

S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in 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," in 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," in Proc. SPIE 4167, 111-119 (2000).

Erteza, A.

Luenberger, D. G.

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

Lukesh, G.

Lukesh, G. W.

S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
[CrossRef]

S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in 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," in 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," in Proc. SPIE 4167, 111-119 (2000).

Neelakantaswarmy, P. S.

Rajaratram, A.

Scharf, L. L.

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

Sjogren, J.

S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in Proc. SPIE 5552, 114-122 (2004).
[CrossRef]

S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
[CrossRef]

Voelz, D.

S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
[CrossRef]

S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in Proc. SPIE 5552, 114-122 (2004).
[CrossRef]

Voelz, D. G.

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

Appl. Opt. (4)

Proc. SPIE (4)

G. W. Lukesh, S. M. Chandler, and D. G. Voelz, "Analysis of satellite laser optical cross sections from the active imaging testbed," in 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," in Proc. SPIE 4167, 111-119 (2000).

S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility," in Proc. SPIE 5552, 105-113 (2004).
[CrossRef]

S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, "Model-based beam control for illumination of remote objects. Part II: Laboratory testbed," in Proc. SPIE 5552, 114-122 (2004).
[CrossRef]

Other (2)

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

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

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Figures (9)

Fig. 1
Fig. 1

Laser pointing system.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

Estimates on the jitter–boresight plane for ten random cases each with N = 20 observations.

Fig. 4
Fig. 4

Estimates on the jitter–boresight plane for ten random cases each with N = 100 observations.

Fig. 5
Fig. 5

Jitter estimation performance of ML and key ratio methods with increasing number of observations in the presence of zero boresight error. The true jitter value is 1.5.

Fig. 6
Fig. 6

Estimation performance of the ML method with increasing number of observations in the presence of boresight error. The true jitter value is 0.8 and the boresight is 1.0.

Fig. 7
Fig. 7

Jitter estimation performance of ML and key ratio methods in the presence of zero boresight. The estimates are obtained using N = 50 samples.

Fig. 8
Fig. 8

Estimation performance of ML and key ratio methods for a boresight of 1.2. The estimates are obtained using N = 50 samples.

Fig. 9
Fig. 9

Estimation performance of ML and key ratio methods for a jitter of 0.4. The estimates are obtained using N = 50 samples.

Equations (16)

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r [ n ] = K exp ( ( ( x [ n ] + A ) 2 + y 2 [ n ] ) 2 Ω 2 ) ,
n = 1 , 2 , , N ,
p ( x [ n ] , y [ n ] ) = 1 2 πσ j 2 exp ( ( x 2 [ n ] + y 2 [ n ] ) 2 σ j 2 ) ,
z [ n ] = 2 Ω 2 log ( K r [ n ] ) ,
p ( z ˜ [ n ] ) = 1 2 exp ( 1 2 ( z ˜ [ n ] + λ ) ) I 0 ( λ z ˜ [ n ] ) u ( z ˜ [ n ] ) ,
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 ( z ) = 1 ( 2 σ j 2 ) N exp ( 1 2 σ j 2 ( N A 2 + n = 1 N z [ n ] ) ) n = 1 N I 0 ( A σ j 2 z [ n ] ) u ( z [ n ] ) .
p ( r ; K , A , σ j ) = ( Ω σ j ) 2 N i = 1 N ( 1 r [ i ] ) exp ( 1 2 σ j     2 ( N A 2 + 2 N Ω 2 log K 2 Ω 2 n = 1 N log r [ n ] ) ) n = 1 N I 0 ( A σ j     2 2 Ω 2 log ( K / r [ n ] ) ) × u ( 2 Ω 2 log ( K / r [ n ] ) ) .
log p ( r ; K , A , σ j ) = 2 N log Ω n = 1 N log r [ n ] 2 N log σ j 1 2 σ j     2 ( A 2 N + 2 Ω 2 n = 1 N log ( K / r [ n ] ) )
+ n = 1 N log ( I 0 ( A σ j 2 2 Ω 2 log ( K / r [ n ] ) ) ) .
A 1 N n = 1 N I 1 ( A σ j     2 2 Ω 2 log ( K / r [ n ] ) ) I 0 ( A σ j     2 2 Ω 2 log ( K / r [ n ] ) ) × 2 Ω 2 log ( K / r [ n ] ) = 0 ,
A 2 2 σ j     2 + 2 Ω 2 N n = 1 N log ( K / r [ n ] ) 2 A N n = 1 N I 1 ( A σ j     2 2 Ω 2 log ( K / r [ n ] ) ) I 0 ( A σ j     2 2 Ω 2 log ( K / r [ n ] ) ) 2 Ω 2 log ( K / r [ n ] ) = 0.
A 2 + 2 σ j     2 = 2 Ω 2 N n = 1 N log ( K / r [ n ] ) .
σ ^ j = Ω 2 N n = 1 N log ( K r [ n ] ) .
σ ^ j = Ω 2 ( β 2 + β β 2 + 1 ) ,
MSE  =   1 M i = 1 M | q q ^ i | 2 ,

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