Abstract

Strategic laser systems are subject to residual pointing errors arising from vibrations and atmospheric turbulence, estimates of which may allow improved system performance. Field data from the Air Force Research Laboratory Floodbeam Experiments suggested a linear relationship between the mean and standard deviation of the shot-by-shot signals and the jitter. An ideal analytic solution and Monte Carlo simulations confirmed this result for a relatively large number of returns. A refined approach using statistical χ2 techniques, which simultaneously estimates jitter and boresight, was developed to address results from satellite passes with relatively few returns and provides excellent jitter and boresight predictions.

© 2000 Optical Society of America

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References

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  1. R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lenless imaging,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
    [CrossRef]
  2. B. F. Campbell, L. Rubin, R. B. Holmes, “Synthetic-aperture imaging through an aberrating medium: experimental demonstration,” Appl. Opt. 34, 5932–5937 (1995).
    [CrossRef] [PubMed]
  3. D. G. Voelz, K. A. Bush, P. S. Idell, “Illumination coherence effects in laser-speckle imaging: modeling and experimental demonstration,” Appl. Opt. 36, 1781–1788 (1997).
    [CrossRef] [PubMed]
  4. D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
    [CrossRef]
  5. D. G. Voelz, ed., “Floodbeam experiment I,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1996).
  6. E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).
  7. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1996).
    [CrossRef]
  8. J. B. Shellan, “Numerical evaluation of the first six moments of the Strehl ratio in the presence of atmospheric turbulence,” (Optical Sciences Company, P.O. Box 25309, Anaheim, Calif. 92825, 1998).
  9. P. A. Lightsey, “Scintillation in ground-to-space and retroreflected beams,” Opt. Eng. 33, 2535–2543 (1994).
    [CrossRef]
  10. K. Bush, Logicon/RDA, 2600 Yale Blvd. SE, Albuquerque, N.M. 87106 (personal communication, 1999).
  11. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
    [CrossRef]
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1992).
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

1997 (1)

1996 (1)

1995 (1)

1994 (1)

P. A. Lightsey, “Scintillation in ground-to-space and retroreflected beams,” Opt. Eng. 33, 2535–2543 (1994).
[CrossRef]

Bush, K.

E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).

K. Bush, Logicon/RDA, 2600 Yale Blvd. SE, Albuquerque, N.M. 87106 (personal communication, 1999).

Bush, K. A.

Campbell, B. F.

Caudill, E.

E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).

Chandler, S.

E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).

Dean, D.

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1992).

Fried, D. L.

Gamiz, V.

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

Goodman, J. W.

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

Holmes, R. B.

Hutchin, R. A.

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lenless imaging,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
[CrossRef]

Idell, P. S.

Lightsey, P. A.

P. A. Lightsey, “Scintillation in ground-to-space and retroreflected beams,” Opt. Eng. 33, 2535–2543 (1994).
[CrossRef]

Lukesh, G.

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1992).

Richard, W.

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

Rider, D. B.

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

Rubin, L.

Schulze, K. J.

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

Shellan, J. B.

J. B. Shellan, “Numerical evaluation of the first six moments of the Strehl ratio in the presence of atmospheric turbulence,” (Optical Sciences Company, P.O. Box 25309, Anaheim, Calif. 92825, 1998).

Smith, R.

E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1992).

Voelz, D.

E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).

Voelz, D. G.

D. G. Voelz, K. A. Bush, P. S. Idell, “Illumination coherence effects in laser-speckle imaging: modeling and experimental demonstration,” Appl. Opt. 36, 1781–1788 (1997).
[CrossRef] [PubMed]

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

P. A. Lightsey, “Scintillation in ground-to-space and retroreflected beams,” Opt. Eng. 33, 2535–2543 (1994).
[CrossRef]

Other (9)

K. Bush, Logicon/RDA, 2600 Yale Blvd. SE, Albuquerque, N.M. 87106 (personal communication, 1999).

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1992).

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

D. B. Rider, D. G. Voelz, W. Richard, K. J. Schulze, V. Gamiz, D. Dean, G. Lukesh, “Statistical and radiometric measurements of coherently illuminated, non-augmented, low earth orbit satellites,” in Optics in Atmospheric Propagation and Random Phenomena, A. Kohnle, A. D. Devir, eds., Proc. SPIE2312, 193–201 (1994).
[CrossRef]

D. G. Voelz, ed., “Floodbeam experiment I,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1996).

E. Caudill, G. Lukesh, R. Smith, S. Chandler, D. Voelz, K. Bush, “Satellite laser cross sections from floodbeam experiments,” (Air Force Research Laboratory, Kirtland Air Force Base, N.M., 1998).

J. B. Shellan, “Numerical evaluation of the first six moments of the Strehl ratio in the presence of atmospheric turbulence,” (Optical Sciences Company, P.O. Box 25309, Anaheim, Calif. 92825, 1998).

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lenless imaging,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 161–168 (1993).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Time history of the total number of return photons (diamonds) collected during the FBE I at 20 elevations as the target satellite crossed the sky. The predicted curve is also displayed. For readability, the elevations are shown uniformly spaced.

Fig. 2
Fig. 2

Histogram of the number of occurrences of q from 0 to 20 includes 580 of the 589 FBE I data points.

Fig. 3
Fig. 3

Running statistics for FBE I data. The σ r and μ r curves cross at q ≈ 10.

Fig. 4
Fig. 4

Histograms for, a, j = 0.33 and, b, j = 0.67 show that for low jitter the returns cluster toward unity, whereas for high jitter the returns cluster toward zero. Note the vertical scale change.

Fig. 5
Fig. 5

Monte Carlo simulated running statistics with j = 0.67 for a small satellite.

Fig. 6
Fig. 6

Fundamental relationship between jitter and return photon statistics. Dotted line, analytic solution derived in Section 3. Note the weak dependence on the satellite sizes and shapes studied.

Fig. 7
Fig. 7

Jitter estimation standard deviation as a function of draws. A small satellite was used with j = 0.67, b = 0.0.

Fig. 8
Fig. 8

Effect of boresight on the jitter estimation method for ten values of boresight from 0 to 1.33 in steps of 0.13 from bottom to top.

Fig. 9
Fig. 9

Results of simulated data; 25 points were used. The 90% confidence swath encompasses cells from (0.60, 0.10) to (0.33, 0.63). The actual (j, b) of (0.40, 0.53) lies inside this swath at 98%.

Fig. 10
Fig. 10

Results of a wave-optics simulation of the FBE far-field pattern. The central lobe is intact.

Fig. 11
Fig. 11

Results of simulated data with typical FBE level atmospheric fluctuations; 25 points were used. The jitter–boresight bins were modeled without the fluctuations. The input (j, b) of (0.40, 0.53) lies within the 90% confidence swath which extends from (0.30, 0.40) to (0.50, 0.67). The maximum confidence of 98% occurs at (0.50, 0.40). This is evidence that the fluctuations do not corrupt the predictions.

Fig. 12
Fig. 12

Results from the Monte Carlo simulation of 256 satellite passes, each with 200 shots. The three accumulated regions, for pass statistics a = (0.13, 0.53), b = (0.13, 0.27), and c = (0.53, 0.27) illustrate the ability of the χ2 approach to distinguish both in jitter and boresight. Four confidence levels, 95%, 90%, 80%, and 68.3% (interior to exterior, from dark gray to light gray) are shown. The white dots locate the input (j, b).

Fig. 13
Fig. 13

Confidences associated with the FBE II data from a satellite pass on 20 November 1996. The peak occurs at (j, b) = (0.10, 0.60) with 96% confidence.

Fig. 14
Fig. 14

Confidence plot for the FBE I data collected on 20 November 1993. The peak occurs at (j, b) = (0.27, 0.43), with confidence of 98%.

Fig. 15
Fig. 15

Data from the FBE I of a satellite pass on 8 December 1993. The returns with magnitude above 5 (indicated by the horizontal line) are considered to be glints.

Fig. 16
Fig. 16

Confidence plot for the data shown in Fig. 15. All 25 returns are included. The peak confidence of 81% occurs at (0.17, 0.70).

Fig. 17
Fig. 17

Confidence plot for 19 of the returns shown in Fig. 15. The six with magnitude greater than 5 are omitted. The peak confidence is 95% and falls at (0.17, 0.47).

Tables (1)

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Table 1 Transmit, Satellite, and Receive Path Budgets

Equations (14)

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μr(q), mean for all normalized returns ranging from 0 to q;
σr(q), standard deviation for all normalized returns ranging from 0 to q.
K(q)=σr(q)/μr(q)
J(x, y)=12πσj2 exp-(x2+y2)2σj2,
Effx, y=exp-(x2+y2)2Ω2,
K2=varEffEff2,
Eff=-+-+ Effx, yJx, ydxdy,
varEff=Eff2-Eff2.
K=σj2Ω22σj2Ω2+1-1/2.
Ω=FWHM2ln 4,
K=8 ln 2 σj2FWHM216 ln 2 σj2FWHM2+11/2.
σjFWHM  14ln 20.36,
K2ln 2σjFWHM.
Effx, y=sinc2πDλ xsinc2πDλ y,

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