Abstract

Autonomous beam alignment for coherent Doppler lidar requires accurate information about optical misalignment and optical aberrations. A multielement heterodyne detector provides the required information without a loss in overall system performance. The effects of statistical variations from the random backscattered field (speckle field) are determined with computer simulations for both ground-based operation with a fixed calibration target and for space-based operation with random target backscatter.

© 1999 Optical Society of America

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1998 (1)

1997 (1)

1996 (4)

R. M. Huffaker, R. M. Hardesty, “Remote sensing of atmospheric wind velocities using solid-state and CO2 coherent laser systems,” Proc. IEEE 84, 181–204 (1996).
[Crossref]

J. M. Vaughan, K. O. Steinvall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–225 (1996).
[Crossref]

R. G. Frehlich, “Simulation of coherent Doppler lidar performance in the weak signal regime,” J. Atmos. Oceanic Technol. 13, 646–658 (1996).
[Crossref]

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 25, 5999–6009 (1996).
[Crossref]

1995 (2)

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

S. M. Hannon, S. W. Henderson, “Wind measurement applications of coherent lidar,” Rev. Laser Eng. 23, 124–130 (1995).
[Crossref]

1994 (7)

S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
[Crossref]

R. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
[Crossref]

G. Constant, R. Foord, P. A. Forrester, J. M. Vaughan, “Coherent laser radar and the problem of aircraft wake vortices,” J. Mod. Opt. 41, 2153–2173 (1994).
[Crossref]

G. E. Johnson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
[Crossref]

R. G. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 1217–1230 (1994).
[Crossref]

W. Pichler, W. R. Leeb, “Target-plane intensity approximation for apertured Gaussian beams applied to heterodyne backscatter lidar systems,” Appl. Opt. 33, 4761–4770 (1994).
[Crossref] [PubMed]

A. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler lidar with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
[Crossref] [PubMed]

1993 (7)

R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
[Crossref] [PubMed]

J. G. Hawley, R. Targ, S. W. Henderson, C. P. Hale, M. J. Kavaya, D. Moerder, “Coherent launch-site atmospheric wind sounder: theory and experiment,” Appl. Opt. 32, 4557–4568 (1993).
[Crossref] [PubMed]

R. G. Frehlich, “Optimal local oscillator field for a monostatic coherent laser radar with a circular aperture,” Appl. Opt. 32, 4569–4577 (1993).
[Crossref] [PubMed]

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[Crossref]

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2-µm using solid-state lasers,” IEEE Trans. Geosci. Remote Sens. 31, 4–15 (1993).
[Crossref]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
[Crossref]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans. Geosci. Remote Sens. 31, 28–35 (1993).
[Crossref]

1992 (6)

1991 (5)

1990 (3)

1989 (4)

M. J. Kavaya, S. W. Henderson, J. R. Magee, C. P. Hale, R. M. Huffaker, “Remote wind profiling with a solid-state Nd:YAG coherent lidar system,” Opt. Lett. 14, 776–778 (1989).
[Crossref] [PubMed]

J. C. Petheram, G. Frohbeiter, A. Rosenberg, “Carbon dioxide Doppler lidar wind sensor on a space station polar platform,” Appl. Opt. 28, 834–839 (1989).
[Crossref] [PubMed]

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurements of profiles of turbulence and momentum flux,” J. Atmos. Oceanic Technol. 6, 809–819 (1989).
[Crossref]

R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[Crossref]

1988 (2)

1987 (1)

1986 (2)

R. T. Menzies, “Doppler lidar atmospheric wind sensors: a comparative performance evaluation for global measurement applications from earth orbit,” Appl. Opt. 25, 2546–2553 (1986).
[Crossref] [PubMed]

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[Crossref]

1985 (2)

M. J. Post, W. D. Neff, “Doppler lidar wind measurements in a narrow valley,” Bull. Am. Meteorl. Soc. 67, 274–281 (1985).
[Crossref]

J. H. Shapiro, “Precise comparison of experimental and theoretical SNR’s in CO2 laser heterodyne systems: comments,” Appl. Opt. 24, 1245 (1985).
[Crossref] [PubMed]

1984 (1)

1982 (1)

1981 (1)

1979 (2)

B. J. Rye, “Antenna parameters for incoherent backscatter heterodyne lidar,” Appl. Opt. 18, 1390–1398 (1979).
[Crossref] [PubMed]

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[Crossref]

1976 (1)

1972 (1)

T. R. Lawrence, D. J. Wilson, C. E. Craven, I. P. Jones, R. M. Huffaker, J. A. L. Thomson, “A laser velocimeter for remote wind sensing,” Rev. Sci. Instrum. 43, 512–518 (1972).
[Crossref]

Anderson, J. R.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

Atlas, R. M.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

Baker, W. E.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

Banta, R. M.

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[Crossref]

Bilbro, J. W.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984).

Bowdle, D. A.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

Bowles, R. L.

Brown, R. A.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

Bruns, D. L.

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2-µm using solid-state lasers,” IEEE Trans. Geosci. Remote Sens. 31, 4–15 (1993).
[Crossref]

Capron, B. A.

Castellanos, D. C.

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 25, 5999–6009 (1996).
[Crossref]

Chan, K. P.

Clark, T. L.

S. M. Hannon, T. L. Clark, “Lidar measurement of windshear and turbulence and comparison with a predictive fine-mesh mesoscale model,” in Air Traffic Control Technologies II, R. G. Otto, J. Lenz, R. Targ, eds., Proc. SPIE2737, 151–161 (1996).
[Crossref]

Constant, G.

G. Constant, R. Foord, P. A. Forrester, J. M. Vaughan, “Coherent laser radar and the problem of aircraft wake vortices,” J. Mod. Opt. 41, 2153–2173 (1994).
[Crossref]

Costello, T. P.

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 25, 5999–6009 (1996).
[Crossref]

Craven, C. E.

T. R. Lawrence, D. J. Wilson, C. E. Craven, I. P. Jones, R. M. Huffaker, J. A. L. Thomson, “A laser velocimeter for remote wind sensing,” Rev. Sci. Instrum. 43, 512–518 (1972).
[Crossref]

Cupp, R. E.

M. J. Post, R. E. Cupp, “Optimizing a pulsed Doppler lidar,” Appl. Opt. 29, 4145–4158 (1990).
[Crossref] [PubMed]

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurements of profiles of turbulence and momentum flux,” J. Atmos. Oceanic Technol. 6, 809–819 (1989).
[Crossref]

Dabas, A.

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics, (Springer-Verlag, New York, 1975).
[Crossref]

DiMarzio, C.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[Crossref]

Eberhard, W. L.

T. Gal-chen, M. Xu, W. L. Eberhard, “Estimations of atmospheric boundary layer fluxes and other turbulence parameters from Doppler lidar data,” J. Geophys. Res. 97, 409–418 (1992).

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurements of profiles of turbulence and momentum flux,” J. Atmos. Oceanic Technol. 6, 809–819 (1989).
[Crossref]

Emmitt, G. D.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

M. J. Kavaya, G. D. Emmitt, “The Space Readiness Coherent Lidar Experiment (SPAR-CLE) Space Shuttle Mission,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 2–11 (1998).
[Crossref]

Fink, D.

Fitzjarrald, D.

J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
[Crossref]

Flamant, P. H.

J. M. Vaughan, K. O. Steinvall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–225 (1996).
[Crossref]

A. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler lidar with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
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G. Constant, R. Foord, P. A. Forrester, J. M. Vaughan, “Coherent laser radar and the problem of aircraft wake vortices,” J. Mod. Opt. 41, 2153–2173 (1994).
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G. Constant, R. Foord, P. A. Forrester, J. M. Vaughan, “Coherent laser radar and the problem of aircraft wake vortices,” J. Mod. Opt. 41, 2153–2173 (1994).
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R. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of winds in the weak signal regime,” Appl. Opt. 36, 3491–3499 (1997).
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R. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
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R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
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Frohbeiter, G.

Gal-chen, T.

T. Gal-chen, M. Xu, W. L. Eberhard, “Estimations of atmospheric boundary layer fluxes and other turbulence parameters from Doppler lidar data,” J. Geophys. Res. 97, 409–418 (1992).

Gatt, P.

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 25, 5999–6009 (1996).
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R. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
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Hannon, S. M.

S. M. Hannon, S. W. Henderson, “Wind measurement applications of coherent lidar,” Rev. Laser Eng. 23, 124–130 (1995).
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S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
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S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2-µm using solid-state lasers,” IEEE Trans. Geosci. Remote Sens. 31, 4–15 (1993).
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S. M. Hannon, T. L. Clark, “Lidar measurement of windshear and turbulence and comparison with a predictive fine-mesh mesoscale model,” in Air Traffic Control Technologies II, R. G. Otto, J. Lenz, R. Targ, eds., Proc. SPIE2737, 151–161 (1996).
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R. M. Huffaker, R. M. Hardesty, “Remote sensing of atmospheric wind velocities using solid-state and CO2 coherent laser systems,” Proc. IEEE 84, 181–204 (1996).
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W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
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B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
[Crossref]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans. Geosci. Remote Sens. 31, 28–35 (1993).
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Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 1: Theory,” Appl. Opt. 29, 4111–4119 (1990).
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Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 2: Applications,” Appl. Opt. 29, 4120–4132 (1990).
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R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
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Hawley, J. G.

Healy, K. R.

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurements of profiles of turbulence and momentum flux,” J. Atmos. Oceanic Technol. 6, 809–819 (1989).
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Heimmermann, D. A.

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 25, 5999–6009 (1996).
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R. Frehlich, S. Hannon, S. Henderson, “Coherent Doppler lidar measurements of winds in the weak signal regime,” Appl. Opt. 36, 3491–3499 (1997).
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R. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
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Huffaker, A. V.

Huffaker, M. R.

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R. M. Huffaker, R. M. Hardesty, “Remote sensing of atmospheric wind velocities using solid-state and CO2 coherent laser systems,” Proc. IEEE 84, 181–204 (1996).
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R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1991).
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G. E. Johnson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
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J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
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T. R. Lawrence, D. J. Wilson, C. E. Craven, I. P. Jones, R. M. Huffaker, J. A. L. Thomson, “A laser velocimeter for remote wind sensing,” Rev. Sci. Instrum. 43, 512–518 (1972).
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J. W. Bilbro, C. DiMarzio, D. Fitzjarrald, S. Johnson, W. Jones, “Airborne Doppler lidar measurements,” Appl. Opt. 25, 2952–2960 (1986).
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W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
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McElroy, J.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
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W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
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R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
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D. M. Tratt, R. T. Menzies, “Unstable resonator antenna properties in coherent lidar applications: a comparative study,” Appl. Opt. 27, 3645–3649 (1988).
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R. T. Menzies, “Doppler lidar atmospheric wind sensors: a comparative performance evaluation for global measurement applications from earth orbit,” Appl. Opt. 25, 2546–2553 (1986).
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Molinari, J. E.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
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M. J. Post, W. D. Neff, “Doppler lidar wind measurements in a narrow valley,” Bull. Am. Meteorl. Soc. 67, 274–281 (1985).
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R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
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Paegle, J.

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
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Petheram, J. C.

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Post, M. J.

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W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
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Rye, B. J.

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J. M. Vaughan, K. O. Steinvall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–225 (1996).
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Stickley, C. M.

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 25, 5999–6009 (1996).
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Sugimoto, N.

Suni, P. J. M.

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2-µm using solid-state lasers,” IEEE Trans. Geosci. Remote Sens. 31, 4–15 (1993).
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Targ, R.

Thomson, J. A.

S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
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Thomson, J. A. L.

T. R. Lawrence, D. J. Wilson, C. E. Craven, I. P. Jones, R. M. Huffaker, J. A. L. Thomson, “A laser velocimeter for remote wind sensing,” Rev. Sci. Instrum. 43, 512–518 (1972).
[Crossref]

Tratt, D. M.

Vaughan, J. M.

J. M. Vaughan, K. O. Steinvall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–225 (1996).
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G. Constant, R. Foord, P. A. Forrester, J. M. Vaughan, “Coherent laser radar and the problem of aircraft wake vortices,” J. Mod. Opt. 41, 2153–2173 (1994).
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Wang, J. Y.

Weeks, A. R.

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 25, 5999–6009 (1996).
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J. M. Vaughan, K. O. Steinvall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–225 (1996).
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T. R. Lawrence, D. J. Wilson, C. E. Craven, I. P. Jones, R. M. Huffaker, J. A. L. Thomson, “A laser velocimeter for remote wind sensing,” Rev. Sci. Instrum. 43, 512–518 (1972).
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T. Gal-chen, M. Xu, W. L. Eberhard, “Estimations of atmospheric boundary layer fluxes and other turbulence parameters from Doppler lidar data,” J. Geophys. Res. 97, 409–418 (1992).

Yuen, E. H.

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2-µm using solid-state lasers,” IEEE Trans. Geosci. Remote Sens. 31, 4–15 (1993).
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H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
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D. M. Tratt, “Optimizing coherent lidar performance with graded-reflectance laser resonator optics,” Appl. Opt. 31, 4233–4239 (1992).
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A. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler lidar with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
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Bull. Am. Meteorl. Soc. (1)

M. J. Post, W. D. Neff, “Doppler lidar wind measurements in a narrow valley,” Bull. Am. Meteorl. Soc. 67, 274–281 (1985).
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Bull. Am. Meteorol. Soc. (1)

W. E. Baker, G. D. Emmitt, P. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar measured winds from space: an essential component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (3)

S. W. Henderson, P. J. M. Suni, C. P. Hale, S. M. Hannon, J. R. Magee, D. L. Bruns, E. H. Yuen, “Coherent laser radar at 2-µm using solid-state lasers,” IEEE Trans. Geosci. Remote Sens. 31, 4–15 (1993).
[Crossref]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
[Crossref]

B. J. Rye, R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation,” IEEE Trans. Geosci. Remote Sens. 31, 28–35 (1993).
[Crossref]

J. Atmos. Oceanic Technol. (3)

W. L. Eberhard, R. E. Cupp, K. R. Healy, “Doppler lidar measurements of profiles of turbulence and momentum flux,” J. Atmos. Oceanic Technol. 6, 809–819 (1989).
[Crossref]

R. Frehlich, S. Hannon, S. Henderson, “Performance of a 2-µm coherent Doppler lidar for wind measurements,” J. Atmos. Oceanic Technol. 11, 1517–1528 (1994).
[Crossref]

R. G. Frehlich, “Simulation of coherent Doppler lidar performance in the weak signal regime,” J. Atmos. Oceanic Technol. 13, 646–658 (1996).
[Crossref]

J. Atmos. Sci. (2)

R. M. Banta, L. D. Olivier, D. H. Levinson, “Evolution of the Monterey Bay sea-breeze as observed by pulsed Doppler lidar,” J. Atmos. Sci. 50, 3959–3982 (1993).
[Crossref]

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[Crossref]

J. Geophys. Res. (1)

T. Gal-chen, M. Xu, W. L. Eberhard, “Estimations of atmospheric boundary layer fluxes and other turbulence parameters from Doppler lidar data,” J. Geophys. Res. 97, 409–418 (1992).

J. Mod. Opt. (3)

S. M. Hannon, J. A. Thomson, “Aircraft wake vortex detection and measurement with pulsed solid-state coherent laser radar,” J. Mod. Opt. 41, 2175–2196 (1994).
[Crossref]

G. Constant, R. Foord, P. A. Forrester, J. M. Vaughan, “Coherent laser radar and the problem of aircraft wake vortices,” J. Mod. Opt. 41, 2153–2173 (1994).
[Crossref]

R. G. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 1217–1230 (1994).
[Crossref]

Opt. Acta (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[Crossref]

Opt. Lett. (3)

Proc. IEEE (4)

R. M. Huffaker, R. M. Hardesty, “Remote sensing of atmospheric wind velocities using solid-state and CO2 coherent laser systems,” Proc. IEEE 84, 181–204 (1996).
[Crossref]

J. M. Vaughan, K. O. Steinvall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–225 (1996).
[Crossref]

R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[Crossref]

G. E. Johnson, “Constructions of particular random processes,” Proc. IEEE 82, 270–285 (1994).
[Crossref]

Rev. Laser Eng. (1)

S. M. Hannon, S. W. Henderson, “Wind measurement applications of coherent lidar,” Rev. Laser Eng. 23, 124–130 (1995).
[Crossref]

Rev. Sci. Instrum. (1)

T. R. Lawrence, D. J. Wilson, C. E. Craven, I. P. Jones, R. M. Huffaker, J. A. L. Thomson, “A laser velocimeter for remote wind sensing,” Rev. Sci. Instrum. 43, 512–518 (1972).
[Crossref]

Other (5)

S. M. Hannon, T. L. Clark, “Lidar measurement of windshear and turbulence and comparison with a predictive fine-mesh mesoscale model,” in Air Traffic Control Technologies II, R. G. Otto, J. Lenz, R. Targ, eds., Proc. SPIE2737, 151–161 (1996).
[Crossref]

M. J. Kavaya, G. D. Emmitt, “The Space Readiness Coherent Lidar Experiment (SPAR-CLE) Space Shuttle Mission,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 2–11 (1998).
[Crossref]

J. C. Dainty, Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics, (Springer-Verlag, New York, 1975).
[Crossref]

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

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984).

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Figures (21)

Fig. 1
Fig. 1

Geometry and coordinate system of a five-element detector with elements numbered from m = 1 to 5.

Fig. 2
Fig. 2

Simulations of the random backscattered field E S (w, L) in the focal plane of a perfectly aligned coherent Doppler lidar with effective focal length L and the parameters for the optimal Wang design. The real and imaginary parts of the complex scalar field are denoted by a solid curve and a dashed curve, respectively. The transverse coordinate w on the detector surface is normalized by λL/ a where a is the radius of the telescope aperture. A misalignment angle θ will translate the random pattern an amount w = θL with respect to the optimal LO field E LO(w, L).

Fig. 3
Fig. 3

Performance of the multielement detector of Fig. 1 as a function of the central radius R 1 and the annular ring radius Δr = R 2 - R 1. The lidar parameters are the optimal Wang design (a L = 1.763) with perfect alignment for a uniform diffuse target in the far field or focus. The system efficiency η S , heterodyne efficiency η H , and the direct detection efficiency ηDT of the central detector (m = 1) are shown as solid curves. The system efficiency of the annular rings (m = 2–5) is represented by dashed curves.

Fig. 4
Fig. 4

Performance of the multielement detector of Fig. 1 with the same parameters of Fig. 3 except that the LO field at the aperture plane is half of the size of the optimal Wang design (a L = 3.5), i.e., the LO beam on the detector surface is twice the optimal size [see Eq. (48)].

Fig. 5
Fig. 5

Performance of the multielement detector of Fig. 1 for two different detector designs as a function of various misalignment angles θ i for a diffuse target in the far field and no optical aberrations. The system efficiency η S (m = 1) of the central detector is shown as a solid curve for θ i = |(θ x , θ y )|. The dotted curve is η S (m = 2) for θ i = θ x and θ y = 0, the dashed curve is η S (m = 3) for θ i = θ x and θ y = 0, the dash–dot curve is η S (m = 2) for θ x = θ i and θ y = θ i , and the dash–dot–dot curve is η S (m = 3) for θ x = θ i and θ y = θ i . The performance of other detector elements can be extracted using symmetry.

Fig. 6
Fig. 6

Performance of misalignment estimates for detector design 1 [R 1 a/(λL) = 0.5, R 2 a/(λL) = 1.0] by use of N = 100 lidar shots from a calibration hard target in the far field and no aberrations. The mean (open circles) and standard deviation (error bars) of 10,000 simulated estimates are shown for angular misalignment chosen as integral multiples of 0.1θ x /λ and 0.1θ y /λ. The contours of 0.5 dB (solid curve), 1.0 dB (dotted curve), 2.0 dB (dashed curve), and 3 dB (dot–dash curve) loss of SNR are also shown.

Fig. 7
Fig. 7

Performance of misalignment estimates for detector design 2 [R 1 a/(λL) = 0.6, R 2 a/(λL) = 1.1] by use of N = 100 lidar shots from a calibration hard target in the far field and no aberrations. The mean (open circles) and standard deviation (error bars) of 10,000 simulated estimates are shown for angular misalignment chosen as integral multiples of 0.1θ x /λ and 0.1θ y /λ. The contours of 0.5 dB (solid curve), 1.0 dB (dotted curve), 2.0 dB (dashed curve), and 3 dB (dot–dash curve) loss of SNR are also shown.

Fig. 8
Fig. 8

Standard deviation SD[aθ i /λ] of misalignment estimates as a function of misalignment angle θ i for detector design 1 by use of N = 100 and N = 400 lidar shots from a calibration hard target in the far field and no aberrations. The solid curve is SD(aθ x /λ) as a function of aθ x /λ) for θ y = 0. The dotted curve is SD(aθ y /λ) as a function of aθ x /λ for θ y = 0. The dashed curve is SD(aθ x /λ) as a function of aθ/λ for θ x = θ y and θ = |(θ x , θ y )|.

Fig. 9
Fig. 9

Standard deviation SD(aθ i /λ) of misalignment estimates as a function of misalignment angle θ i for detector design 2 by use of N = 100 and N = 400 lidar shots from a calibration hard target in the far field and no aberrations. The solid curve is SD(aθ x /λ) as a function of aθ x /λ for θ y = 0. The dotted curve is SD(aθ y /λ) as a function of aθ x /λ for θ y = 0. The dashed curve is SD(aθ x /λ) as a function of aθ/λ for θ x = θ y and θ = |(θ x , θ y )|.

Fig. 10
Fig. 10

Performance of defocus estimates for detector design 1 with no misalignment and a calibration hard target in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of defocus a defocus are shown as solid curves. The standard deviation SD(a defocus) of the defocus estimates is shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the defocus estimates is indicated by a dot–dash curve for N = 100 and dot–dot–dash curve for N = 400 lidar shots.

Fig. 11
Fig. 11

Performance of defocus estimates for detector design 2 with no misalignment and a calibration hard target in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of defocus a defocus are shown as solid curves. The standard deviation SD(a defocus) of the defocus estimates is shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the defocus estimates are indicated by a dot–dash curve for N = 100 and a dot–dot–dash curve for N = 400 lidar shots.

Fig. 12
Fig. 12

Performance of estimates of spherical aberration for detector design 1 with no misalignment and a calibration hard target in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of spherical aberration a spherical are shown as solid curves. The standard deviation SD(a spherical) of the estimates of spherical aberration is shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the estimates of spherical aberration is indicated by a dot–dash curve for N = 100 and a dot–dot–dash curve for N = 400 lidar shots.

Fig. 13
Fig. 13

Performance of estimates of spherical aberration for detector design 2 with no misalignment and a calibration hard target in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of spherical aberration a spherical is shown as solid curves. The standard deviation SD(a spherical) of the estimates of spherical aberration is shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the estimates of spherical aberration is indicated by a dot–dash curve for N = 100 and a dot–dot–dash curve for N = 400 lidar shots.

Fig. 14
Fig. 14

Performance of misalignment estimates for detector design 1 using N = 100 lidar shots from a space-based platform in the far field and no aberrations. The mean (open circles) and standard deviation (error bars) of 10,000 simulated estimates are shown for angular misalignment chosen as integral multiples of 0.1θ x /λ and 0.1θ y /λ. The contours of 0.5 dB (solid curve), 1.0 dB (dotted curve), 2.0 dB (dashed curve), and 3 dB (dot–dash curve) loss of SNR are also shown.

Fig. 15
Fig. 15

Performance of misalignment estimates for detector design 2 using N = 100 lidar shots from a space-based platform in the far field and no aberrations. The mean (open circles) and standard deviation (error bars) of 10,000 simulated estimates are shown for angular misalignment chosen as integral multiples of 0.1θ x /λ and 0.1θ y /λ. The contours of 0.5 dB (solid curve), 1.0 dB (dotted curve), 2.0 dB (dashed curve), and 3 dB (dot–dash curve) loss of SNR are also shown.

Fig. 16
Fig. 16

Standard deviation SD(aθ i /λ) of misalignment estimates as a function of misalignment angle θ i for detector design 1 by use of N = 100 and N = 400 lidar shots from a space-based platform in the far field and no aberrations. The solid curve is SD(aθ x /λ) as a function of aθ x /λ for θ y = 0. The dotted curve is SD(aθ y /λ) as a function of aθ x /λ for θ y = 0. The dashed curve is SD(aθ x /λ) as a function of aθ/λ for θ x = θ y and θ = |(θ x , θ y )|.

Fig. 17
Fig. 17

Standard deviation SD(aθ i /λ) of misalignment estimates as a function of misalignment angle θ i for detector design 2 using N = 100 and N = 400 lidar shots from a space-based platform in the far field and no aberrations. The solid curve is SD(aθ x /λ) as a function of aθ x /λ for θ y = 0. The dotted curve is SD(aθ y /λ) as a function of aθ x /λ for θ y = 0. The dashed curve is SD(aθ x /λ) as a function of aθ/λ for θ x = θ y and θ = |(θ x , θ y )|.

Fig. 18
Fig. 18

Performance of defocus estimates for detector design 1 with no misalignment and a space-based platform in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of defocus a defocus are shown as solid curves. The standard deviation SD(a defocus) of the defocus estimates is shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the defocus estimates is indicated by a dot–dash curve for N = 100 and a dot–dot–dash curve for N = 400 lidar shots.

Fig. 19
Fig. 19

Performance of defocus estimates for detector design 2 with no misalignment and a space-based platform in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of defocus a defocus are shown as solid curves. The standard deviation SD(a defocus) of the defocus estimates are shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the defocus estimates is indicated by a dot–dash curve for N = 100 and a dot–dot–dash curve for N = 400 lidar shots.

Fig. 20
Fig. 20

Performance of estimates of spherical aberration a spherical for detector design 1 with no misalignment and a space-based platform in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of a spherical are shown as solid curves. The standard deviation SD(a spherical) of the estimates of spherical aberration is shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the estimates of spherical aberration are indicated by a dot–dash curve for N = 100 and a dot–dot–dash curve for N = 400 lidar shots.

Fig. 21
Fig. 21

Performance of estimates of spherical aberration a spherical for detector design 2 with no misalignment and a space-based platform in the far field. The system efficiency η S (m = 1) for the central detector and the annular rings η S (m = 2) as a function of a spherical are shown as solid curves. The standard deviation SD(a spherical) of the estimates of spherical aberration is shown as a dotted curve for N = 100 lidar shots and as a dashed curve for N = 400. The bias of the estimates of spherical aberration is indicated by a dot–dash curve for N = 100 and a dot–dot–dash curve for N = 400 lidar shots.

Equations (53)

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ESw, L, t=- Gfw, v, LWvESv, 0, tdv,
Gfw, v, L=k2πiL expik2Lv-w2,
iSt, m=2eGDmηQmhν Re- w, mESw, L, t×ELO*w, Lexp2πiΔft+iϕdw,
iN2t, m=2eGDmBIdcm
=2e2GDm2ηQmBhν- w, m×|ELOw, L|2dw,
Xt, m=iSt, miN2t, m1/2.
SNRt, m=X2t, m=iS2t, miN2t, m,
ELOw, L=PLOeLOw, L,
- |eLOw, L|2dw=1.
Xt, m=2ηQm1/2hνB1/2 Re- w, mESw, L, t×gLO*w, L, mexp2πiΔft+iϕdw,
gLOw, L, m=eLOw, L, m- w, m|eLOw, L|2dw1/2.
Bmnt=Xt, mXt, n,
Bmnt=ηQmηQn1/2hνB Re-- w1, mw2, n×ESw1, L, tES*w2, L, t×gLO*w1, L, mgLOw2, L, ndw1dw2.
Bmnt=ηQmηQn1/2hνB Re-- ESv1, 0, t×ES*v2, 0, t×eBPLOv1, 0, meBPLO*v2, 0, ndv1dv2,
eBPLOv, 0, m=Wv- Gfw, v, Lw, m×gLO*w, L, mdw.
ESv1, 0, tES*v2, 0, t=λ2K2RPLt-2R/c×- eTu1, 0eT*u2, 0ρp×Gp, u1, RG*p, u2, R×Gp, v1, RG*p, v2, Rdu1du2dp,
eTu, 0=WueLu, 0,
- |eLu, 0|2du=1.
Bmnt=λ2K2RPLt-2R/cηQmηQn1/2hνB×Re- ρpjTp, ReBPLOp, R, m×eBPLO*p, R, ndp,
jTp, R=|eTp, R|2,
eTp, R=- Gp, u, ReTu, 0du,
eBPLOp, R, m=- Gp, u, ReBPLOu, 0, mdu.
Bmnt=K2RPLt-2R/cρARηQmηQn1/2hνBR2×ηSm, n,
AR=- |Wu|2du,
ηSm, n=λ2R2AR Re- jTp, ReBPLOp, R, m×eBPLO*p, R, ndp.
SNRt, m=Bmmt=PDt, mηHm/hνB,
PDt, m=ηQm- w, m|ESw, L, t|2dw
ηHm=iS2t, m2IdcmISt, m,
ISt, m=eGDmhν PDt, m
PDt, m=k2ηQm2πL2-- Wv1W*v2×Ykv1-v2/L), mexp-ik2L2v12-v22×ESv1, 0, tES*v2, 0, tdv1dv2,
Yv, m=- w, mexp-iw·vdw.
ESv1, 0, tES*v2, 0, t=K2RPLt-2R/cρR2×exp-ik2Rv12-v22×Qv1-v2,
Qv=- jTp, Rexp-i kRp·vdp
PDt, m=K2RPLt-2R/cARρR-2ηQmηDm,
ηDm=k2AR2πL2-- Wv1W*v2×Ykv1-v2/L), mexp-ik2L2v12-v22×exp-ik2R2v12-v22Qv1-v2dv1dv2.
ηDm=Q0=- jTp, Rdp=TT,
ηSm, m=ηHmηDm=TTηHmηDTm,
ηSm, m=ηHmTT.
Um, k=- X2t, m, kdt,
Um, k=- X2t, m, kdt=- Bmmtdt=K2RULρARηQmhνBR2 ηSm, m,
Um, k=|Am, k+iBm, k|2=A2m, k+B2m, k,  Am, k=Bm, k=Am, kBm, k=0,  A2m, k=B2m, k=ηSm, m/2,
UNm, k=Um, kNdetn=1Ndet Un, k,
e2Φ=k=1Nm=1NdetUm, k-Um, Φ2varUm, Φ.
eTu, 0=1πσL exp-u2/2σL2-iku2/2FT
eBPLOv, 0=1πσLO exp-v2/2σLO2-ikv2/2FBPLO.
Wv=Avexp2πiΘv,
Wv=Avexp-ikv2/2L
eLOw, L=1πσLOw exp-w2/2σLOw2+ikw2/2L,
σLOw=LkσLO
Θr=atiltr/a=θr/λ,
atilt=aθ/λ,
Θr=adefocusr2/a2,
Θr=asphericalr4/a4,

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