Abstract

Optical returns from remote resident space-based objects such as satellites suffer from pointing and tracking errors. In a previously reported paper [Appl. Opt. 46, 5608 (2007)], we developed a moment-matching technique that used the statistics of time series of these optical returns to extract information about bore sight and symmetric beam jitter errors (symmetric here implies that the standard deviations of the jitter measured along two orthogonal axes, perpendicular to the line of sight, are equal). In this paper, we extend that method to cover the case of asymmetric beam jitter and bore sight. The asymmetric beam jitter may be due to the combination of symmetric atmospheric turbulence beam jitter and optical beam train jitter. In addition, if a tracking control system is operating, even the residual atmospheric tracking jitter could be asymmetric because the power spectrum is different for the slewing direction compared to the cross-track direction. Analysis of the problem has produced a set of nonlinear equations that can be reduced to a single but much higher-order nonlinear equation in terms of one of the jitter variances. After solving for that jitter, all the equations can be solved to extract all jitter and bore sight errors. The method has been verified by using simulations and then tested on experimental data. In order to develop this method, we derived analytical expressions for the probability density function and the moments of the received total intensity. The results reported here are valid for satellites of small physical cross section, or else those with retroreflectors that dominate the signal return. The results are, in general, applicable to the theory of noncircular Gaussian speckle with a coherent background.

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References

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  1. V. S. R. Gudimetla and J. F. Riker, “Moment-matching method for extracting beam jitter and bore sight in experiments with satellites of small physical cross section,” Appl. Opt. 46, 5608–5616 (2007).
    [CrossRef]
  2. D. Voelz, “Flood Beam Experiments I,” PL-TR-96-1162, Air Force Research Laboratory, Kirtland Air Force Base, N.Mex. (1996).
  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).
    [CrossRef]
  4. G. Lukesh, S. Chandler, and D. Voelz, “Estimation of laser system pointing performance via the statistics of return signal,” Proc. SPIE 3494, 111–121 (1998).
    [CrossRef]
  5. 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]
  6. G. Lukesh, S. Chandler, and D. Voelz, “Analysis of satellite laser optical cross sections from the Active Imaging Testbed,” Proc. SPIE 4538, 24–33 (2002).
    [CrossRef]
  7. S. Chandler and G. Lukesh, “The statistical analysis of received time-series signals from the laser illumination of remote objects through turbulence,” Proc. SPIE 6522, 65220D (2006).
    [CrossRef]
  8. 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]
  9. D. K. Borah and D. G. Voelz, “Cramer–Rao lower bounds on estimation of laser beam pointing parameters by use of return photon signal,” Opt. Lett. 31, 1029–1031 (2006).
    [CrossRef]
  10. D. K. Borah, D. G. Voelz, G. Lukesh, and S. Chandler, “Maximum likelihood parameter estimation of a laser system using return photons,” Proc. SPIE 6522, 65220J (2006).
    [CrossRef]
  11. D. K. Borah and D. G. Voelz, “Estimation of laser beam pointing parameters in the presence of atmospheric turbulence,” Appl. Opt. 46, 6010–6018 (2007).
    [CrossRef]
  12. J. F. Riker, “Beam diameter at distant targets illuminated from Maui,” TEM 2004-07, Air Force Research Laboratory, Kirtland Air Force Base, N.Mex. (3 April 2004).
  13. A related summary and results of can be found in J. F. Riker, “Validation of active track Gaussian beam propagation and target signature prediction,” Proc. SPIE 4724, 45–56(2002).
    [CrossRef]
  14. J. F. Riker, “Exact tilt Strehl for degraded Gaussian beams,” TEM 2003-27, Air Force Research Laboratory, Kirtland Air Force Base, N. Mex., 30 September 2003.
  15. J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
    [CrossRef]
  16. J. F. Riker, “Active track optical cross sections in the presence of local tilt,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004.
  17. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (SPIE, 2007).
  18. R. K. Tyson and B. W. Frazier, Field Guide for Adaptive Optics (SPIE, 2004).
  19. B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, “Laser beam uplink jitter and bore sight estimation for a point source target,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004. For information of readers, SPIE digital library does not provide a copy of this paper.
  20. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1978).
  21. V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
    [CrossRef]
  22. J. W. Goodman, Statistical Optics (Wiley, 1985), Chaps. 2 and 3.
  23. G. Arfken, Mathematical Methods for Physicists (Academic, 1985), Chap. 11, p. 613.
  24. M. Nakagami, “The m-distribution-a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960), pp. 3–36.
  25. T. L. Saty and J. Braun, Nonlinear Mathematics (Dover, 1984), Chap. 1.
  26. Y. Bard, Nonlinear Parameter Estimation (Academic, 1974).
  27. G. R. Osche, “Single- and multiple-pulse noncoherent detection statistics associated with partially developed speckle,” Appl. Opt. 39, 4255–4262 (2000).
    [CrossRef]
  28. 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]
  29. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill , 1984), Chap. 5, p. 156, Eq. 5-80.
  30. I. S. Gradshteyn and I. M. Rhyzhik, Tables of Integrals, Series and Products (Academic, 1980).

2009 (1)

2007 (2)

2006 (4)

D. K. Borah and D. G. Voelz, “Cramer–Rao lower bounds on estimation of laser beam pointing parameters by use of return photon signal,” Opt. Lett. 31, 1029–1031 (2006).
[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]

S. Chandler and G. Lukesh, “The statistical analysis of received time-series signals from the laser illumination of remote objects through turbulence,” Proc. SPIE 6522, 65220D (2006).
[CrossRef]

D. K. Borah, D. G. Voelz, G. Lukesh, and S. Chandler, “Maximum likelihood parameter estimation of a laser system using return photons,” Proc. SPIE 6522, 65220J (2006).
[CrossRef]

2004 (1)

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

2002 (2)

A related summary and results of can be found in J. F. Riker, “Validation of active track Gaussian beam propagation and target signature prediction,” Proc. SPIE 4724, 45–56(2002).
[CrossRef]

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

2001 (1)

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]

2000 (2)

1998 (1)

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

1996 (1)

V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
[CrossRef]

Archambeault, D.

B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, “Laser beam uplink jitter and bore sight estimation for a point source target,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004. For information of readers, SPIE digital library does not provide a copy of this paper.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1985), Chap. 11, p. 613.

Bard, Y.

Y. Bard, Nonlinear Parameter Estimation (Academic, 1974).

Barnard, C.

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]

Basu, S.

Borah, D. K.

Braun, J.

T. L. Saty and J. Braun, Nonlinear Mathematics (Dover, 1984), Chap. 1.

Brown, J.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Chandler, S.

S. Chandler and G. Lukesh, “The statistical analysis of received time-series signals from the laser illumination of remote objects through turbulence,” Proc. SPIE 6522, 65220D (2006).
[CrossRef]

D. K. Borah, D. G. Voelz, G. Lukesh, and S. Chandler, “Maximum likelihood parameter estimation of a laser system using return photons,” Proc. SPIE 6522, 65220J (2006).
[CrossRef]

G. Lukesh, S. Chandler, and D. Voelz, “Analysis of satellite laser optical cross sections from the Active Imaging Testbed,” Proc. SPIE 4538, 24–33 (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 use of statistics of return photons,” Appl. Opt. 39, 1359–1371 (2000).
[CrossRef]

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

Dainty, J. C.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1978).

Frazier, B. W.

R. K. Tyson and B. W. Frazier, Field Guide for Adaptive Optics (SPIE, 2004).

Fugate, R. Q.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1978).

J. W. Goodman, Statistical Optics (Wiley, 1985), Chaps. 2 and 3.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Rhyzhik, Tables of Integrals, Series and Products (Academic, 1980).

Gudimetla, V. S. R.

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

V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
[CrossRef]

Hoffman, W. C.

M. Nakagami, “The m-distribution-a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960), pp. 3–36.

Holcomb, T.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Holmes, R. B.

B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, “Laser beam uplink jitter and bore sight estimation for a point source target,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004. For information of readers, SPIE digital library does not provide a copy of this paper.

Kann, J.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Lowrey, W. H.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Lukesh, G.

S. Chandler and G. Lukesh, “The statistical analysis of received time-series signals from the laser illumination of remote objects through turbulence,” Proc. SPIE 6522, 65220D (2006).
[CrossRef]

D. K. Borah, D. G. Voelz, G. Lukesh, and S. Chandler, “Maximum likelihood parameter estimation of a laser system using return photons,” Proc. SPIE 6522, 65220J (2006).
[CrossRef]

G. Lukesh, S. Chandler, and D. Voelz, “Analysis of satellite laser optical cross sections from the Active Imaging Testbed,” Proc. SPIE 4538, 24–33 (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 use of statistics of return photons,” Appl. Opt. 39, 1359–1371 (2000).
[CrossRef]

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

Nakagami, M.

M. Nakagami, “The m-distribution-a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960), pp. 3–36.

Osche, G. R.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill , 1984), Chap. 5, p. 156, Eq. 5-80.

Rhyzhik, I. M.

I. S. Gradshteyn and I. M. Rhyzhik, Tables of Integrals, Series and Products (Academic, 1980).

Riker, J. F.

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

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

A related summary and results of can be found in J. F. Riker, “Validation of active track Gaussian beam propagation and target signature prediction,” Proc. SPIE 4724, 45–56(2002).
[CrossRef]

J. F. Riker, “Beam diameter at distant targets illuminated from Maui,” TEM 2004-07, Air Force Research Laboratory, Kirtland Air Force Base, N.Mex. (3 April 2004).

J. F. Riker, “Exact tilt Strehl for degraded Gaussian beams,” TEM 2003-27, Air Force Research Laboratory, Kirtland Air Force Base, N. Mex., 30 September 2003.

J. F. Riker, “Active track optical cross sections in the presence of local tilt,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004.

Roggemann, M. C.

B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, “Laser beam uplink jitter and bore sight estimation for a point source target,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004. For information of readers, SPIE digital library does not provide a copy of this paper.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (SPIE, 2007).

Saty, T. L.

T. L. Saty and J. Braun, Nonlinear Mathematics (Dover, 1984), Chap. 1.

Slavin, A. C.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Spinhirne, J. M.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Stribling, B. E.

B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, “Laser beam uplink jitter and bore sight estimation for a point source target,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004. For information of readers, SPIE digital library does not provide a copy of this paper.

Tuffli, A. L.

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

Tyson, R. K.

R. K. Tyson and B. W. Frazier, Field Guide for Adaptive Optics (SPIE, 2004).

Voelz, D.

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]

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

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).
[CrossRef]

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

D. Voelz, “Flood Beam Experiments I,” PL-TR-96-1162, Air Force Research Laboratory, Kirtland Air Force Base, N.Mex. (1996).

Voelz, D. G.

Appl. Opt. (5)

J. Lightwave Technol. (1)

Opt. Commun. (1)

V. S. R. Gudimetla, “Moments of the intensity of a non-circular Gaussian laser speckle in the diffraction field,” Opt. Commun. 130, 348–356 (1996).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (7)

J. F. Riker, R.Q. Fugate, T. Holcomb, J. Kann, W. H. Lowrey, A. C. Slavin, J. M. Spinhirne, A. L. Tuffli, and J. Brown, “Active tracking with moderate power lasers,” Proc. SPIE 5552, 123–132 (2004) .
[CrossRef]

G. Lukesh, S. Chandler, and D. Voelz, “Estimation of laser system pointing performance via the statistics of return signal,” Proc. SPIE 3494, 111–121 (1998).
[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, “Analysis of satellite laser optical cross sections from the Active Imaging Testbed,” Proc. SPIE 4538, 24–33 (2002).
[CrossRef]

S. Chandler and G. Lukesh, “The statistical analysis of received time-series signals from the laser illumination of remote objects through turbulence,” Proc. SPIE 6522, 65220D (2006).
[CrossRef]

A related summary and results of can be found in J. F. Riker, “Validation of active track Gaussian beam propagation and target signature prediction,” Proc. SPIE 4724, 45–56(2002).
[CrossRef]

D. K. Borah, D. G. Voelz, G. Lukesh, and S. Chandler, “Maximum likelihood parameter estimation of a laser system using return photons,” Proc. SPIE 6522, 65220J (2006).
[CrossRef]

Other (15)

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill , 1984), Chap. 5, p. 156, Eq. 5-80.

I. S. Gradshteyn and I. M. Rhyzhik, Tables of Integrals, Series and Products (Academic, 1980).

J. F. Riker, “Exact tilt Strehl for degraded Gaussian beams,” TEM 2003-27, Air Force Research Laboratory, Kirtland Air Force Base, N. Mex., 30 September 2003.

D. Voelz, “Flood Beam Experiments I,” PL-TR-96-1162, Air Force Research Laboratory, Kirtland Air Force Base, N.Mex. (1996).

J. F. Riker, “Active track optical cross sections in the presence of local tilt,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (SPIE, 2007).

R. K. Tyson and B. W. Frazier, Field Guide for Adaptive Optics (SPIE, 2004).

B. E. Stribling, M. C. Roggemann, D. Archambeault, and R. B. Holmes, “Laser beam uplink jitter and bore sight estimation for a point source target,” presented at the SPIE International Symposium on Defense and Security, Orlando, Florida, April 2004. For information of readers, SPIE digital library does not provide a copy of this paper.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1978).

J. W. Goodman, Statistical Optics (Wiley, 1985), Chaps. 2 and 3.

G. Arfken, Mathematical Methods for Physicists (Academic, 1985), Chap. 11, p. 613.

M. Nakagami, “The m-distribution-a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960), pp. 3–36.

T. L. Saty and J. Braun, Nonlinear Mathematics (Dover, 1984), Chap. 1.

Y. Bard, Nonlinear Parameter Estimation (Academic, 1974).

J. F. Riker, “Beam diameter at distant targets illuminated from Maui,” TEM 2004-07, Air Force Research Laboratory, Kirtland Air Force Base, N.Mex. (3 April 2004).

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Figures (2)

Fig. 1
Fig. 1

Comparison of extracted and simulated total x and y jitter values at several values of x and y bore sight combinations with 1 million samples in the time series. RHS figures refer to y jitter values and LHS figures refer to x jitter. (a) x bore sight, 1 μrad ; y bore sight, 2 μrad . (b) x bore sight, 2 μrad ; y bore sight, 3 μrad . (c) x bore sight, 3 μrad ; y bore sight, 1 μrad . (d) x bore sight, 4 μrad ; y bore sight, 3 μrad .

Fig. 2
Fig. 2

Comparison of extracted and simulated total x and y jitter values at x bore sight error = 3 μrad and y bore sight error = 1 μrad , with 10 million samples in the time series. RHS figures refer to y jitter values and LHS figures refer to x jitter. In comparison with Fig. 1c, y jitter extraction scatter is considerably reduced.

Tables (3)

Tables Icon

Table 1 Simulation Parameters

Tables Icon

Table 2 Summary of the Extracted Bore Sight Errors for the x and y Axes

Tables Icon

Table 3 Summary of the Extracted Bore Sight and Beam Jitter Errors in the x and y axes (in Microradians) from Satellite Experimental Data

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

D DL = 2 w DL = D TX [ 1 + ( 4 λ z π D TX 2 ) 2 ] 1 / 2 ,
I ( r ) = 8 P π D DL 2 exp ( 2 r 2 w DL 2 ) .
D ATM = 2 w ATM = r 0 [ 1 + ( 4 λ z π r 0 2 ) 2 ] 1 / 2 .
D total = [ β 2 D DL 2 + D ATM 2 ] 1 / 2 = 2 w total , I max = 2 P laser π w total 2 τ a sec ψ τ opt , TX = 8 P laser π D total 2 τ a sec ψ τ opt , TX .
σ j 2 = 0.3399 λ 2 D 1 / 3 r 0 5 / 3 .
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 ] } = I max exp { γ [ ( x b x z j x ( t ) z ) 2 + ( y b y z j y ( t ) z ) 2 ] } with     γ = 8 D total 2 N ph = χ I ( x , y ) Ω RX τ a sec ψ τ o , RX ( Δ t λ h c ) N ph = α I ( x , y ) α = χ Ω RX τ a sec ψ τ o , RX ( Δ t λ h c ) ,
p ( x , y ) = 1 2 π σ j x σ j y z 2 exp [ ( ( x x 0 j x ( t ) z ) 2 2 σ j x 2 z 2 + + ( y y 0 j y ( t ) z ) 2 2 σ j y 2 z 2 ) ] .
u = x j x z = r cos ( θ ) d u = d x , v = y j y z = r sin ( θ ) d v = d y .
r 2 = u 2 + v 2 , θ = tan 1 ( v / u ) .
p ( r , θ ) = r exp ( b 0 2 z 2 F 1 b 0 2 z 2 F 2 cos ( 2 ε ) ) 2 π σ j x σ j y z 2 exp [ r 2 F 1 r 2 cos ( 2 θ ) F 2 + r F 3 cos ( θ ) + r F 4 sin ( θ ) ] ,
F 1 = 1 4 z 2 ( 1 σ j x 2 + 1 σ j y 2 ) , F 2 = 1 4 z 2 ( 1 σ j x 2 1 σ j y 2 ) , F 3 = b 0 z cos ( ε ) σ j x 2 z 2 , F 4 = b 0 z sin ( ε ) σ j y 2 z 2 .
F 3 cos ( θ ) + F 4 sin ( θ ) = F 3 2 + F 4 2 cos ( θ β ) ,
tan ( β ) = F 4 F 3 = σ j x 2 σ j y 2 tan ( ε ) , F 3 2 + F 4 2 = b 0 z cos 2 ( ε ) σ j x 4 z 4 + sin 2 ( ε ) σ j y 4 z 4 .
p ( r , θ ) = r exp ( b 0 2 z 2 F 1 b 0 2 z 2 F 2 cos ( 2 ε ) ) 2 π σ j x σ j y z 2 exp [ r 2 F 1 r 2 cos ( 2 θ ) F 2 + r F 3 2 + F 4 2 cos ( θ β ) ] .
exp ( r 2 cos ( 2 θ ) F 2 ) = I 0 ( r 2 F 2 ) + 2 n = 1 ( 1 ) n I n ( r 2 F 2 ) cos ( 2 n θ ) ,
exp ( r F 3 2 + F 4 2 cos ( θ β ) ) = I 0 ( r F 3 2 + F 4 2 ) + 2 m = 1 I m ( r F 3 2 + F 4 2 ) cos ( m θ n β ) .
p ( r ) = r σ j x σ j y z 2 exp [ r 2 F 1 b 0 2 z 2 F 1 b 0 2 z 2 F 2 cos ( 2 ε ) ] [ I 0 ( r 2 F 2 ) I 0 ( r F 3 2 + F 4 2 ) + 2 n = 1 ( 1 ) n I n ( r 2 F 2 ) I 2 n ( r F 3 2 + F 4 2 ) cos ( 2 n β ) ] .
p ( N ph ) = 1 2 σ j x σ j y z 2 γ α I max exp [ b 0 2 z 2 F 1 b 0 2 z 2 F 2 cos ( 2 ε ) ] ( N ph α I max ) F 1 γ 1 [ I 0 ( ( 1 γ ln N ph α I max ) F 2 ) I 0 ( ( 1 γ ln N ph α I max ) 1 / 2 F 3 2 + F 4 2 ) + 2 m = 1 ( 1 ) m I m ( ( 1 γ ln N ph α I max ) F 2 ) I 2 m ( ( 1 γ ln N ph α I max ) 1 / 2 F 3 2 + F 4 2 ) cos ( 2 m β ) ] .
N ph p = 0 α I max N ph p p N ph ( N ph ) d N ph .
N ph = α I max exp ( γ r 2 ) , r = ( 1 γ ln N ph α I max ) 1 / 2 .
ϕ ( ω ) = ( 2 σ x 2 z 2 ω + 1 ) 1 / 2 exp ( b 0 x 2 2 σ x 2 z 2 ) exp [ b 0 x 2 2 σ x 2 z 2 1 2 σ x 2 z 2 ω + 1 ] × ( 2 σ y 2 z 2 ω + 1 ) 1 / 2 exp ( b 0 y 2 2 σ y 2 z 2 ) exp [ b 0 y 2 2 σ y 2 z 2 1 2 σ y 2 z 2 ω + 1 ] .
r 2 = σ x 2 ( 1 + m 1 ) + σ y 2 ( 1 + m 2 ) = k 1 , r 4 = r 2 2 + 2 σ x 4 ( 1 + 2 m 1 ) + 2 σ y 4 ( 1 + 2 m 2 ) = r 2 2 + 2 k 2 r 6 = r 2 3 + 12 r 2 k 2 + 8 [ σ x 6 ( 1 + 3 m 1 ) + σ y 6 ( 1 + 3 m 2 ) ] = r 2 3 + 6 r 2 k 2 + 8 k 3 r 8 = r 2 4 + 12 r 2 2 k 2 + 12 k 2 2 + 32 r 2 k 3 + 48 k 4 ,
k 1 = σ x 2 ( 1 + m 1 ) + σ y 2 ( 1 + m 2 ) , k 2 = σ x 4 ( 1 + 2 m 1 ) + σ y 4 ( 1 + 2 m 2 ) , k 3 = σ x 4 ( 1 + 3 m 1 ) + σ y 4 ( 1 + 3 m 2 ) , k 4 = σ x 4 ( 1 + 4 m 1 ) + σ y 4 ( 1 + 4 m 2 ) .
6 k 1 p q 6 q 2 4 q ( p 2 2 q ) 3 k 2 ( p 2 2 q ) + 2 ( p 4 + 2 q 2 4 p 2 q ) 2 k 3 p + 3 k 4 = 0 ,
q = 3 k 2 p 2 k 3 p 3 6 [ k 1 p ] 0.
r 2 = x 2 + y 2 .
p ( x ) = 1 2 π σ x 2 exp [ ( x b 0 x ) 2 σ x 2 2 ] .
ϕ ( ω ) = + exp ( ω x 2 ) 1 2 π σ x 2 exp [ ( x b 0 x ) 2 σ x 2 2 ] d x ,
ϕ ( ω ) = 1 2 π σ x 2 exp ( b 0 x 2 2 σ x 2 ) 2 1 / 2 ( j ω + 1 2 σ x 2 ) 1 / 2 exp [ b 0 x 2 8 σ x 4 1 ω + 1 2 σ x 2 ] D 1 ( b 0 x σ x 2 2 ω + 1 σ x 2 ) + 1 2 π σ x 2 exp ( b 0 x 2 2 σ x 2 ) 2 1 / 2 ( j ω + 1 2 σ x 2 ) 1 / 2 exp [ b 0 x 2 8 σ x 4 1 ω + 1 2 σ x 2 ] D 1 ( b 0 x σ x 2 2 ω + 1 σ x 2 ) .
D 1 ( z ) + D 1 ( z ) = 2 π 2 exp [ z 2 4 ] .
ϕ ( ω ) = ( 2 σ x 2 ω + 1 ) 1 / 2 exp ( b 0 x 2 2 σ x 2 ) exp [ b 0 x 2 2 σ x 2 1 2 σ x 2 ω + 1 ] .
ϕ ( ω ) = ( 2 σ x 2 ω + 1 ) 1 / 2 exp ( b 0 x 2 2 σ x 2 ) exp [ b 0 x 2 2 σ x 2 1 2 σ x 2 ω + 1 ] × ( 2 σ y 2 ω + 1 ) 1 / 2 exp ( b 0 y 2 2 σ y 2 ) exp [ b 0 y 2 2 σ y 2 1 2 σ y 2 ω + 1 ] .
k 1 = a ( 1 + m 1 ) + b ( 1 + m 2 ) , k 2 = a 2 ( 1 + 2 m 1 ) + b 2 ( 1 + 2 m 2 ) , k 3 = a 3 ( 1 + 3 m 1 ) + b 3 ( 1 + 3 m 2 ) , k 4 = a 4 ( 1 + 4 m 1 ) + b 4 ( 1 + 4 m 2 ) .
m 1 = 2 b k 1 b 2 2 a b k 2 + a 2 2 a ( b a ) ,
m 2 = 2 a k 1 a 2 2 a b k 2 + b 2 2 b ( a b ) .
m 1 = 4 b k 3 b 4 4 a 3 b 3 k 4 + 3 a 4 12 a 3 ( b a ) ,
m 2 = 4 a k 3 a 4 4 a b 3 3 k 4 + 3 b 4 12 b 3 ( a b ) .
12 a 2 b k 1 6 a 2 b 2 8 a 3 b 6 a 2 k 2 + 3 a 4 4 b k 3 + b 4 + 3 k 4 = 0.
12 b 2 a k 1 6 b 2 a 2 8 b 3 a 6 b 2 k 2 + 3 b 4 4 a k 3 + a 4 + 3 k 4 = 0.
6 k 1 a b + 3 k 2 ( b + a ) 2 k 3 ( a + b ) [ ( a + b ) 2 6 a b ] = 0.
a b = 3 k 2 ( a + b ) 2 k 3 ( a + b ) 3 6 [ k 1 ( a + b ) ] 0.
a + b = p .
q = a b = 3 k 2 p 2 k 3 p 3 6 [ k 1 p ] 0 ,
( a b ) = [ ( a + b ) 2 4 a b ] 1 / 2 = ± [ p 2 4 3 k 2 p 2 k 3 p 3 6 [ k 1 p ] ] 1 / 2 = ± [ 3 k 1 p 2 p 3 6 k 2 p + 4 k 3 3 ( k 1 p ) ] 1 / 2 .
a = p + [ 3 k 1 p 2 p 3 6 k 2 p + 4 k 3 3 ( k 1 p ) ] 1 / 2 2 , b = p [ 3 k 1 p 2 p 3 6 k 2 p + 4 k 3 3 ( k 1 p ) ] 1 / 2 2 .
12 k 1 a b ( a + b ) 12 ( a b ) 2 8 a b ( a 2 + b 2 ) 6 k 2 ( b 2 + a 2 ) + 4 ( b 4 + a 4 ) 4 k 3 ( b + a ) + 6 k 4 = 0.
12 k 1 p q 12 q 2 8 q ( p 2 2 q ) 6 k 2 ( p 2 2 q ) + 4 ( p 4 + 2 q 2 4 p 2 q ) 4 k 3 p + 6 k 4 = 0 ,

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