Abstract

With the aid of the van Cittert–Zernike theorem we develop an analytical expression for the ensemble-averaged heterodyne mixing efficiency in coherent lidar receivers that are looking at a diffuse target that is in the receiver’s far field. Our extremely simple and straightforward analysis shows that the dependence of the mixing efficiency on the receive aperture size d R first follows a parabolic decrease and later approaches a (d R)-2 function. As a consequence, the signal-to-noise ratio does not increase proportionally to the aperture area but saturates. For the system model chosen, the heterodyne mixing efficiency exhibits the same functional dependence on the lidar geometry as the reciprocal of the number of speckle cells within the receive aperture.

© 1998 Optical Society of America

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References

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  1. Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 1. Theory,” Appl. Opt. 29, 4111–4119 (1990).
    [CrossRef] [PubMed]
  2. J. Y. Wang, “Detection efficiency of coherent optical radar,” Appl. Opt. 23, 3421–3427 (1984).
    [CrossRef] [PubMed]
  3. R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  4. W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
    [CrossRef]
  5. 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]
  6. 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. 35, 5999–6009 (1996).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965).
  8. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  9. B. J. Klein, J. J. Degnan, “Optical antenna gain. 1. Transmitting antennas,” Appl. Opt. 13, 2134–2140 (1974).
    [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. J. Y. Wang, “Optimum truncation of a lidar transmitted beam,” Appl. Opt. 27, 4470–4474 (1988).
    [CrossRef] [PubMed]
  13. More specifically, the factor given as 6/2 (=4.24) in relation (22) increases continually from 4.01 to 4.70 if dT/2WT increases from 1.0 to 2.0.
  14. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle, J. C. Dainty, ed. (Springer-Verlag, New York, 1975).
    [CrossRef]
  15. A. D. Dabas, P. H. Flamant, P. Salamitou, “Characterization of pulsed coherent Doppler LIDAR with the speckle effect,” Appl. Opt. 33, 6524–6532 (1994).
    [CrossRef] [PubMed]

1996 (2)

1995 (1)

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

1994 (2)

1993 (1)

1991 (1)

1990 (1)

1988 (1)

1984 (1)

1974 (1)

Anderson, J. R.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Atlas, R. M.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Baker, W. E.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965).

Bowdle, D. A.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Brown, R. A.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Castellanos, D. C.

Costello, T. P.

Dabas, A. D.

Degnan, J. J.

Emmitt, G. D.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Flamant, P. H.

Frehlich, R. G.

Gatt, P.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle, J. C. Dainty, ed. (Springer-Verlag, New York, 1975).
[CrossRef]

Hardesty, R. M.

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]

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 1. Theory,” Appl. Opt. 29, 4111–4119 (1990).
[CrossRef] [PubMed]

Heimmermann, D. A.

Huffaker, R. M.

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]

Kavaya, M. J.

Klein, B. J.

Krishnamurti, T. N.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Leeb, W. R.

Lorenc, A. C.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

McElroy, J.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Menzies, R. T.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Molinari, J. E.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Paegle, J.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Pichler, W.

Post, M. J.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 1. Theory,” Appl. Opt. 29, 4111–4119 (1990).
[CrossRef] [PubMed]

Robertson, F.

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Salamitou, P.

Stickley, C. M.

Wang, J. Y.

Weeks, A. R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965).

Zhao, Y.

Appl. Opt. (9)

B. J. Klein, J. J. Degnan, “Optical antenna gain. 1. Transmitting antennas,” Appl. Opt. 13, 2134–2140 (1974).
[CrossRef] [PubMed]

J. Y. Wang, “Detection efficiency of coherent optical radar,” Appl. Opt. 23, 3421–3427 (1984).
[CrossRef] [PubMed]

J. Y. Wang, “Optimum truncation of a lidar transmitted beam,” Appl. Opt. 27, 4470–4474 (1988).
[CrossRef] [PubMed]

Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving efficiency of monostatic pulsed coherent lidars. 1. Theory,” Appl. Opt. 29, 4111–4119 (1990).
[CrossRef] [PubMed]

R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[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]

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

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. 35, 5999–6009 (1996).
[CrossRef] [PubMed]

Bull. Am. Meteorol. Soc. (1)

W. E. Baker, G. D. Emmitt, F. 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: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Proc. IEEE (1)

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]

Other (4)

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

More specifically, the factor given as 6/2 (=4.24) in relation (22) increases continually from 4.01 to 4.70 if dT/2WT increases from 1.0 to 2.0.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle, J. C. Dainty, ed. (Springer-Verlag, New York, 1975).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Model of the lidar system.

Fig. 2
Fig. 2

Geometry underlying the van Cittert–Zernike theorem.

Fig. 3
Fig. 3

Heterodyne mixing efficiency m versus receive aperture size d R normalized to the spot size (expressed in multiples of λ/π) and to target distance R [upper abscissa axis, Eq. (17)] and versus the ratio of transmit and receive antenna areas [lower abscissa axis, Eq. (24)]. The squares show the results obtained in Ref. 2. The target is in the far field of the receiver, i.e., exp() = 1.

Fig. 4
Fig. 4

Normalized ensemble-averaged SNR versus the ratio A R / A T . Normalization is done to SNR1, which is the SNR for equal receive and transmit apertures, A R = A T , and for the receiver in the far field, i.e., ψ ≡ 0 (equivalent to u = 0). The solid curve, which is valid for the far-field case u = 0, was calculated from relation (4) and Eq. (24). The dashed curve was obtained numerically for u = π/4 rad. For comparison, the dotted curve shows the SNR for a deterministic plane input field (also normalized to SNR1) as it would occur in an ideal communications receiver, where m = 1.

Equations (31)

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m =   E R E 0 * d A R 2   | E 0 | 2 d A R   | E R | 2 d A R =   E R 1 E R 2 * E 01 * E 02 d A R 1 d A R 2   | E 0 | 2 d A R   | E R | 2 d A R ,
m =   E R 1 E R 2 * d A R 1 d A R 2 | E R | 2 1 A R 2 .
SNR = P R m η m h ν B ,
SNR     mA R .
μ 12 = E R 1 E R 2 * | E R 1 | 2 | E R 2 | 2 ,
μ 12 p ,   q = exp j ψ - + - +   I ξ ,   η exp - jk p ξ + q η d ξ d η - + - +   I ξ ,   η d ξ d η .
k = 2 π λ ,
p = X 1 - X 2 R ,     q = Y 1 - Y 2 R ,
ψ = k 2 R X 1 2 - X 2 2 + Y 1 2 - Y 2 2 .
R     k P 1 P 2 max 2 ,
m =   μ 12 d A R 1 d A R 2 A R 2 .
I ξ ,   η     exp - ξ 2 + η 2 W 2 ,
μ 12 p ,   q = exp j ψ exp - w 2 p 2 + q 2 ,
w = W   k 2 ,
m = π R wd R 4 0 wd R R erf z d z 2 ,
erf z = 2 π 0 z exp - t 2 d t .
m = π R wd R erf wd R R - R wd R 2 × 1 - exp - wd R R 2 2 .
m = 1 - 1 3 wd R R 2 + 17 180 wd R R 4 - 29 1260 wd R R 6 ,     d R     R / w .
m = π R wd R - R wd R 2 2 ,     d R     R / w ,
m = π R wd R 2 ,     d R     R / w .
W 3.8 R kd T .
W 6 2 R kd T .
A R / A T = 4 π d R d T 2 .
m = 8 3 A T A R   erf 3 π 8 A R A T - 8 9 π A T A R × 1 - exp - 9 π 8 A R A T 2 .
u = kA T 4 R .
ψ max = kd R 2 4 R = A R A T kA T 4 R = A R A T   u .
m =   μ 12 exp - j ψ d A R 1 d A R 2 A R 2 ,
| μ 12 | =   exp - π R 2 2 S C p 2 + q 2 ,
S C = π R 2 2 w 2 .
m π R wd R = 1 M π 2 R wd R
m A T A R = 1 M A T 2 A R

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