Abstract

The boresight and atmospheric jitter errors in a satellite tracking experiment are currently estimated by matching the probability density function (PDF) of the received signal counts with a set of PDFs of the signal for several combinations of jitter and boresight errors and then the best choice of jitter and boresight error is accepted via the χ-square test. Here a technique that can estimate atmospheric beam jitter and boresight error directly in a satellite active tracking experiment using the moments of the returns off the satellites is proposed. That is, we use the theoretical PDF for the signal return from a small target and compute the corresponding theoretical PDF moments. We can then form a few equations from these moments with only two unknowns, namely, the jitter and boresight. Solving for the unknowns is then unambiguous and very rapid. The method is valid for small physical cross-section targets and has been verified by using simulation and experimental data. Extending the case to asymmetric jitter and asymmetric boresight is possible.

© 2007 Optical Society of America

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References

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  2. G. R. Osche, "Single- and multiple-pulse noncoherent detection statistics associated with partially developed speckle," Appl. Opt. 39, 4255-4262 (2000).
  3. G. Lukesh, S. Chandler, and D. Voelz, "Estimation of laser system pointing performance by use of statistics of return photons," Appl. Opt. 39, 1359-1371 (2000).
  4. G. Lukesh, S. Chandler, and C. Barnard, "Estimation of satellite laser optical cross section: a comparison of simulation and field results," Proc. SPIE 4167, 53-63 (2001).
    [CrossRef]
  5. G. Lukesh, S. Chandler, and D. Voelz, "Estimation of laser system pointing performance by the statistics of return signal," Proc. SPIE 3494, 111-121 (1998).
    [CrossRef]
  6. D. Voelz, "Flood Beam Experiments I," PL-TR-96-1162, Air Force Research Laboratory, Kirtland Air Force Research Laboratory, Kirtland AFB, NM, 1996.
  7. E. Caudill, D. G. Voelz, G. Lukesh, and S. Chandler, "Satellite Laser Cross Sections from the Floodbeam Experiments," Air Force Research Laboratory, Kirtland Air Force Research Laboratory, Kirtland AFB, NM, 1998.
  8. B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, "Laser beam uplink jitter and boresight estimation for a point source target," SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004 (unpublished). For information of readers, SPIE digital library does not provide a copy of this paper.
  9. J. F. Riker, "TEM 2004-07: Beam diameter at distant targets illuminated from Maui," Air Force Research Laboratory, Kirtland Air Force Research Laboratory, Kirtland AFB, NM, 3 April 2004. A related summary and results can also be found in J. F. Riker, "Validation of active track Gaussian beam propagation and target signature prediction," in Laser Weapons Technology III, William E. Thompson and Paul H. Merritt (Ed.), Proc. SPIE , Vol. 4724, 45-56, (2002).
    [CrossRef]
  10. J. F. Riker, "Active tracking lasers for precision target stabilization," keynote paper, SPIE Aerosense Conference, Orlando, FL, 21 April, 2003.
  11. J. F. Riker, P. Merritt, and J. T. Roark, "Laser requirements for active tracking," OSA Conference on non-astronomical adaptive optics," Munich, Germany, 16 June 1997.
  12. T. S. Ross and W. P. Latham, "Appropriate measures and consistent standard for high-energy laser beam quality," J. Directed Energy 2, 22-58 (2006).
  13. J. F. Riker, "TEM 2003-27: Exact tilt Strehl for degraded Gaussian beams," 30 September 2003. See also: J. F. Riker, "Active track optical cross sections in the presence of local tilt," SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004.
  14. R. K. Tyson and B. W. Frazier, Field Guide to Adaptive Optics, (SPIE Press, June 2004, the relevant equations are in pages 55-56, equations for brightness page 55, Strehl ratio with wave front error and jitter page 55, and Gaussian growth spot with turbulence page 56).
  15. J. W. Goodman, Statistical Optics (Wiley, 1985).
  16. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1984).
  17. I. S. Gradshteyn and I. M. Rhyzak, Table of Integrals, Series and Products (Academic Press, 4th edition, 1965).
  18. J. M. Mendel, "Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications," Proc. IEEE 79, 278-305 (1991).
    [CrossRef]
  19. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, "Polarized pulse waves in random discrete scatterers," Appl. Opt. 40, 5495-5502 (2001).

2006

T. S. Ross and W. P. Latham, "Appropriate measures and consistent standard for high-energy laser beam quality," J. Directed Energy 2, 22-58 (2006).

2002

J. F. Riker, "TEM 2004-07: Beam diameter at distant targets illuminated from Maui," Air Force Research Laboratory, Kirtland Air Force Research Laboratory, Kirtland AFB, NM, 3 April 2004. A related summary and results can also be found in J. F. Riker, "Validation of active track Gaussian beam propagation and target signature prediction," in Laser Weapons Technology III, William E. Thompson and Paul H. Merritt (Ed.), Proc. SPIE , Vol. 4724, 45-56, (2002).
[CrossRef]

2001

G. Lukesh, S. Chandler, and C. Barnard, "Estimation of satellite laser optical cross section: a comparison of simulation and field results," Proc. SPIE 4167, 53-63 (2001).
[CrossRef]

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, "Polarized pulse waves in random discrete scatterers," Appl. Opt. 40, 5495-5502 (2001).

2000

1998

G. Lukesh, S. Chandler, and D. Voelz, "Estimation of laser system pointing performance by the statistics of return signal," Proc. SPIE 3494, 111-121 (1998).
[CrossRef]

1994

1991

J. M. Mendel, "Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications," Proc. IEEE 79, 278-305 (1991).
[CrossRef]

Appl. Opt.

J. Directed Energy

T. S. Ross and W. P. Latham, "Appropriate measures and consistent standard for high-energy laser beam quality," J. Directed Energy 2, 22-58 (2006).

Proc. IEEE

J. M. Mendel, "Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications," Proc. IEEE 79, 278-305 (1991).
[CrossRef]

Proc. SPIE

J. F. Riker, "TEM 2004-07: Beam diameter at distant targets illuminated from Maui," Air Force Research Laboratory, Kirtland Air Force Research Laboratory, Kirtland AFB, NM, 3 April 2004. A related summary and results can also be found in J. F. Riker, "Validation of active track Gaussian beam propagation and target signature prediction," in Laser Weapons Technology III, William E. Thompson and Paul H. Merritt (Ed.), Proc. SPIE , Vol. 4724, 45-56, (2002).
[CrossRef]

G. Lukesh, S. Chandler, and C. Barnard, "Estimation of satellite laser optical cross section: a comparison of simulation and field results," Proc. SPIE 4167, 53-63 (2001).
[CrossRef]

G. Lukesh, S. Chandler, and D. Voelz, "Estimation of laser system pointing performance by the statistics of return signal," Proc. SPIE 3494, 111-121 (1998).
[CrossRef]

Other

D. Voelz, "Flood Beam Experiments I," PL-TR-96-1162, Air Force Research Laboratory, Kirtland Air Force Research Laboratory, Kirtland AFB, NM, 1996.

E. Caudill, D. G. Voelz, G. Lukesh, and S. Chandler, "Satellite Laser Cross Sections from the Floodbeam Experiments," Air Force Research Laboratory, Kirtland Air Force Research Laboratory, Kirtland AFB, NM, 1998.

B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, "Laser beam uplink jitter and boresight estimation for a point source target," SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004 (unpublished). For information of readers, SPIE digital library does not provide a copy of this paper.

J. F. Riker, "Active tracking lasers for precision target stabilization," keynote paper, SPIE Aerosense Conference, Orlando, FL, 21 April, 2003.

J. F. Riker, P. Merritt, and J. T. Roark, "Laser requirements for active tracking," OSA Conference on non-astronomical adaptive optics," Munich, Germany, 16 June 1997.

J. F. Riker, "TEM 2003-27: Exact tilt Strehl for degraded Gaussian beams," 30 September 2003. See also: J. F. Riker, "Active track optical cross sections in the presence of local tilt," SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004.

R. K. Tyson and B. W. Frazier, Field Guide to Adaptive Optics, (SPIE Press, June 2004, the relevant equations are in pages 55-56, equations for brightness page 55, Strehl ratio with wave front error and jitter page 55, and Gaussian growth spot with turbulence page 56).

J. W. Goodman, Statistical Optics (Wiley, 1985).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1984).

I. S. Gradshteyn and I. M. Rhyzak, Table of Integrals, Series and Products (Academic Press, 4th edition, 1965).

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Figures (6)

Fig. 1
Fig. 1

Time Series of the simulated signal received when the transmitter diameter is 10   cm with jitter of 1, 5, and 10 μrad (respectively, from left to right). Top row, zero boresight error; bottom row, boresight error of 3 μrad.

Fig. 2
Fig. 2

Simulated PDFs of the signal when we use 200 MATLAB bins (with the maximum number of photons specifying the bin size). All 10 values of jitter (1 through 10 in steps of 1 μrad) are included. Top figures are at boresight error of 0 μrad and the bottom figures are at a boresight error value of 4 μrad. Transmitter aperture is 10   cm .

Fig. 3
Fig. 3

PDF of the signal counts when the beam jitter is changed from 1 to 6 μrad with boresight error = 0 μrad. At higher jitter values, it has the exponential statistics as in the right-hand side figure above (PDF large at low values of signal count). Transmitter aperture diameter is 30   cm .

Fig. 4
Fig. 4

Top row, comparison of simulated and extracted beam jitter values at several boresight values (see legend) for an aperture diameter of 10   cm . Bottom row, comparison of simulated and extracted boresight error values for beam jitter values of 1 to 9 μrad.

Fig. 5
Fig. 5

Comparison of extracted (y-axis) and simulated (x-axis) boresight values for a transmitter size of 30   cm at three boresight values (legend). Excellent extraction is observed as in this case, the Fried coherence parameter is changed with the beam jitter, taking into account their relation for a point source.

Fig. 6
Fig. 6

PDFs of the signal counts of two satellite objects, collected at USAF AMOS site on the same day at about the same time.

Equations (121)

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D D L = 2 w D L = D T X [ 1 + ( 4 λ z π D T X 2 ) 2 ] 1 / 2 exact   diffraction-limited   beam   diameter , I ( r ) = 8 P π D D L 2 exp ( 2 r 2 w D L 2 ) exact   Gaussian   irradiance   profile .
D T X
D T X
w D L
D A T M = 2 w A T M = r 0 , λ [ 1 + ( 4 λ z π r 0 , λ 2 ) 2 ] 1 / 2 .
r 0 , λ
D total = [ β 2 D D L 2 + D A T M 2 ] 1 / 2 = 2 w total
I max = 2 P laser π w total 2 τ a sec ψ τ opt , T X = 8 P laser π D total 2 τ a sec ψ τ opt , T X ,
I max
P laser
τ opt , T X
τ a sec ψ
( b x , b y )
( j x , j y )
I ( x , y ) = I max exp { 8 D Total 2 [ ( x b x z j x ( t ) z ) 2 + ( y b y z j y ( t ) z ) 2 ] }
N p h = χ I ( x , y ) Ω R X τ a sec ψ τ o , R X ( Δ t λ h c )
N p h = α I ( x , y )
α = χ Ω R X τ a sec ψ τ o , R X ( Δ t λ h c ) ,
N p h
Ω R X
τ o , R X
Δ t
τ a sec ψ
m 2
N p h
x = y = 0
N p h
( x 0 , y 0 )
p ( x , y ) = 1 2 π σ j 2 z 2 × exp [ ( ( x x 0 j x ( t ) z ) 2 + ( y y 0 j y ( t ) z ) 2 2 σ j 2 z 2 ) ] .
x 0 = b x z
y 0 = b y z
( σ j 2 )
u = x x 0 j x z , d u = d x , v = y y 0 j y z , d v = d ,
Forward   transformation Inverse   transformation r 2 = u 2 + v 2 , u = r cos θ x 0 , θ = tan 1 ( v / u ) , v = r sin θ y 0 ,
p r θ ( r , θ ) = J p u v ( u , v ) p r θ ( r , θ ) = r 2 π σ j 2 z 2 exp { r 2 + b 0 2 z 2 2 r b 0 z cos ( θ ) 2 σ j 2 z 2 } .
b 0 z = ( b x z ) 2 + ( b y z ) 2 .
p r ( r ) = 0 2 π p r θ ( r , θ ) d θ = 0 2 π r 2 π σ j 2 z 2 exp { r 2 + b 0 2 z 2 2 r b 0 z cos ( θ ) 2 σ j 2 z 2 } d θ = r σ j 2 z 2 exp { r 2 + b 0 2 z 2 2 σ j 2 z 2 } I 0 ( r b 0 z σ j 2 z 2 ) ,
I 0 ( x )
p θ ( θ )
p r ( r )
1 / m
p r ( r )
( b 0 )
0 p r ( r ) d r = 0 r σ j 2 z 2 exp { r 2 + b 0 2 z 2 2 σ j 2 z 2 } I 0 ( r b 0 z σ j 2 z 2 ) d r = 1 .
( N p h )
N p h = α I max exp ( γ r 2 )
α = χ Ω R X τ a sec ψ τ o , R X ( Δ t λ h c )   and   γ = 8 D total 2 .
N p h
N p h
N p h
N p h = f ( r ) = α I max exp ( γ r 2 )
r = ( 1 γ ln N p h α I max ) 1 / 2 = f 1 ( N p h ) ,
d N p h d r = α I max exp ( γ r 2 ) ( 2 γ r ) = 2 γ r N p h
| d N p h d r | = + 2 γ N p h ( 1 γ ln N p h α I max ) 1 / 2 .
N p h
p N p h ( N p h ) = p r ( r = f 1 ( N p h ) ) 2 γ N p h ( 1 γ ln N p h α I max ) 1 / 2 = 1 2 γ σ j 2 z 2 N p h ( N p h α I max ) 1 / 2 γ σ j 2 z 2 exp [ b 0 2 2 σ j 2 ] × I 0 ( ( 1 γ ln N p h α I max ) 1 / 2 b 0 σ j 2 z ) .
δ = 2 γ σ j 2 z 2
p N p h ( N p h ) = 1 δ α I max ( N p h α I max ) 1 + 1 / δ exp [ b 0 2 2 σ j 2 ] × I 0 ( ( 1 γ ln N p h α I max ) 1 / 2 b 0 σ j 2 z ) .
b 0 = 0
N p h p = 0 α I max N p h p p N p h ( N p h ) d N p h ,
x = 1 γ ln N p h α I max N p h = α I max exp ( γ x ) ,
N p h p = exp [ b 0 2 z 2 2 σ j 2 z 2 ] ( α I max ) p γ δ 0 d x × exp [ γ ( p + 1 / δ ) x ] I 0 ( x b 0 z σ j 2 z 2 ) .
N p h
α I max
N p h p = exp [ b 0 2 z 2 2 σ j 2 z 2 ] ( α I max ) p 1 b 0 z [ γ ( p + 1 / δ ) ] 1 / 2 × exp [ b 0 2 z 2 8 σ j 4 z 4 1 γ ( p + 1 / δ ) ] × M 1 / 2 , 0 [ b 0 2 z 2 4 σ j 4 z 4 1 γ ( p + 1 / δ ) ] ,
M 1 / 2 , 0 ( x ) = exp ( x / 2 ) x 1 / 2 Φ ( 1 , 1 , x ) = exp ( x / 2 ) x 1 / 2 ,
N p h p = ( α I max ) p [ ( 2 p σ j 2 z 2 γ + 1 ) ] 1 × exp [ p γ b 0 2 z 2 ( 1 + 2 p γ σ j 2 z 2 ) ] .
N p h p = ( α I max ) p [ ( 2 p σ j 2 z 2 γ + 1 ) ] 1 ,
N p h 1 = α I max / ( 1 + 2 γ σ j 2 z 2 ) ,
N p h 2 = ( α I max ) 2 ( 1 + 4 γ σ j 2 z 2 ) ,
N p h 2 N p h 2 = ( 1 + 2 γ σ j 2 z 2 ) 2 ( 1 + 4 γ σ j 2 z 2 ) .
N p h p N p h p = ( 1 + 2 γ σ j 2 z 2 ) p ( 1 + 2 p γ σ j 2 z 2 ) .
p = 2
N p h p N p h p = ( 1 + 2 σ j 2 z 2 γ ) p ( 1 + 2 p σ j 2 z 2 γ ) × exp [ 2 b 0 2 z 2 γ 2 σ j 2 z 2 p ( p 1 ) ( 1 + 2 p σ j 2 z 2 γ ) ( 1 + 2 σ j 2 z 2 γ ) ] .
log [ N p h p N p h p ] p log ( 1 + 2 σ j 2 z 2 γ ) + log ( 1 + 2 p σ j 2 z 2 γ ) = [ 2 b 0 2 z 2 γ 2 σ j 2 z 2 p ( p 1 ) ( 1 + 2 p σ j 2 z 2 γ ) ( 1 + 2 σ j 2 z 2 γ ) ] .
log [ N p h q N p h q ] q log ( 1 + 2 σ j 2 z 2 γ ) + log ( 1 + 2 q σ j 2 z 2 γ ) = [ 2 b 0 2 z 2 γ 2 σ j 2 z 2 q ( q 1 ) ( 1 + 2 q σ j 2 z 2 γ ) ( 1 + 2 σ j 2 z 2 γ ) ] .
σ j 2 z 2
log [ N p h p N p h p ] p log ( 1 + 2 σ j 2 z 2 γ ) + log ( 1 + 2 p σ j 2 z 2 γ ) log [ N p h p N p h p ] q log ( 1 + 2 σ j 2 z 2 γ ) + log ( 1 + 2 q σ j 2 z 2 γ ) = p ( p 1 ) q ( q 1 ) × ( 1 + 2 q σ j 2 z 2 γ ) ( 1 + 2 p σ j 2 z 2 γ ) ,
σ j 2 z 2 γ
σ j 2
r 0
r 0 = ( 0.3399 D 1 / 3 / σ j 2 ) 3 / 5
σ j
σ j 2 = 0
D T X = 0.1
D R X = 3.6
θ = 30
z = 1 e 6
τ a t m , z = 0.7
τ 0 , o p t = 0.5
= τ o , R X = 0.5
λ = 1.064 e 6
= β = 1.5
( χ ) = 0.1
( m 2 / sr )
Δ t = 0.001
= b x
= b y
( σ j ) = 1 e 6
( D T X )
( b x = b y )
( σ j )
b 0 = b x 2 + b y 2
D T X = 10   cm
( 20   μr )
10 μrad
30   cm
10   cm
30   cm
10   cm
p = 2
q = 3
r 0
r 0 = ( 0.3399 D 1 / 3 / σ j 2 ) 3 / 5
120   cm
120   cm
10   cm
10   cm
30   cm
10   cm
30   cm

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