Abstract

The use of airborne laser radar (lidar) to measure wind velocities and to detect turbulence in front of an aircraft in real time can significantly increase fuel efficiency, flight safety, and terminal area capacity. We describe the flight-test results for two coherent lidar airborne shear sensor (CLASS) systems and discuss their agreement with our theoretical simulations. The 10.6-μm CO2 system (CLASS-10) is a flying brassboard; the 2.02-μm Tm:YAG solid-state system (CLASS-2) is configured in a rugged, lightweight, high-performance package. Both lidars have shown a wind measurement accuracy of better than 1 m/s. © 1996 Optical Society of America

© 1996 Optical Society of America

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  1. J. A. Johnson, S. M. Hannon, “Wake vortex modeling for airborne and ground-based measurements using coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 63–78 (1995).
  2. R. Bogue, H. R. Bagley, D. C. Soreide, “Coherent lidar solution for the HSCT supersonic inlet unstart problem,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 79–93 (1995).
  3. S. M. Hannon, J. A. Thomson, S. W. Henderson, R. M. Huffaker, “Windshear, turbulence, and wake vortex characterization using pulsed solid-state coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 94–102 (1995).
  4. T. J. Kane, J. D. Kmetec, T. J. Wagener, “Flight test of a 2-μm laser radar system,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 103–108 (1995).
  5. R. Targ, J. G. Hawley, B. C. Steakley, L. L. Ames, “Airborne lidar wind detection at 2 μm,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 109–115 (1995).
  6. R. L. McGann, “Flight test results from a low-power Doppler optical air data sensor,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 116–125 (1995).
  7. J. W. Wilson, R. D. Roberts, C. Kessinger, J. McCarthy, “Microburst wind structure and evaluation of Doppler radar for airport windshear detection,” prepared for the Joint Airport Weather Studies Project (National Center for Atmospheric Research, Boulder, Colo., January1984).
  8. R. M. Oseguera, R. L. Bowles, P. A. Robinson, “Airborne in situ computation of the windshear hazard index,” presented at the 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nev., 6–9 January 1992, paper AIAA 92-0291.
  9. R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1992).
  10. R. M. Huffaker, “CO2 laser Doppler systems for the measurement of atmospheric winds and turbulence,” Atmos. Tech. NCAR Winter (National Center for Atmospheric Research, Boulder, Colo., 1974–1975), p. 71.
  11. J. W. Bilbro, “Atmospheric laser Doppler velocimetry: an overview,” Opt. Eng. 19, 533–542 (1980).
  12. R. Foord, R. Jones, J. M. Vaughan, D. V. Willetts, “Precise comparison of experimental and theoretical SNRs in CO2 laser heterodyne systems,” Appl. Opt. 22, 3787–3795 (1983).
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  14. P. Rabinowitz, S. Jacobs, R. Targ, G. Gould, “Homodyne detection of phase modulated light,” Proc. IRE p. 2365 (1962).
  15. C. M. Sonnenschein, F. A. Horrigan, “Signal-to-noise relationships for coaxial systems that heterodyne backscatter from the atmosphere,” Appl. Opt. 10, 1600–1604 (1971).
  16. R. G. Frehlich, “Effects of refractive turbulence on coherent laser radar,” Appl. Opt. 32, 2122–2139 (1993).
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  18. E. Uthe, N. Nielsen, W. Jamison, “Airborne plume and haze analyzer, ALPHA,” Bull. Am. Meteorol. Soc. 61, 1035–1043 (1980).
  19. P. A. Robinson, “An experimental and theoretical evaluation of an airborne coherent lidar windshear sensor,” presented at the AIAA Aircraft Design, Systems and Operations Meeting, Monterey, Calif., 11–13 August 1993, paper AIAA 93-3947.

1993 (1)

1992 (1)

R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1992).

1983 (1)

1980 (2)

J. W. Bilbro, “Atmospheric laser Doppler velocimetry: an overview,” Opt. Eng. 19, 533–542 (1980).

E. Uthe, N. Nielsen, W. Jamison, “Airborne plume and haze analyzer, ALPHA,” Bull. Am. Meteorol. Soc. 61, 1035–1043 (1980).

1971 (1)

Ames, L. L.

R. Targ, J. G. Hawley, B. C. Steakley, L. L. Ames, “Airborne lidar wind detection at 2 μm,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 109–115 (1995).

Bagley, H. R.

R. Bogue, H. R. Bagley, D. C. Soreide, “Coherent lidar solution for the HSCT supersonic inlet unstart problem,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 79–93 (1995).

Bilbro, J. W.

J. W. Bilbro, “Atmospheric laser Doppler velocimetry: an overview,” Opt. Eng. 19, 533–542 (1980).

Bogue, R.

R. Bogue, H. R. Bagley, D. C. Soreide, “Coherent lidar solution for the HSCT supersonic inlet unstart problem,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 79–93 (1995).

Bowles, R. L.

R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1992).

R. M. Oseguera, R. L. Bowles, P. A. Robinson, “Airborne in situ computation of the windshear hazard index,” presented at the 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nev., 6–9 January 1992, paper AIAA 92-0291.

Foord, R.

Frehlich, R. G.

Gould, G.

P. Rabinowitz, S. Jacobs, R. Targ, G. Gould, “Homodyne detection of phase modulated light,” Proc. IRE p. 2365 (1962).

Hall, F. F.

R. M. Hardesty, T. R. Lawrence, R. A. Richter, M. J. Post, F. F. Hall, R. M. Huffaker, “Ground-based coherent lidar measurement of tropospheric and stratospheric parameters,” in Coherent Infrared Radar Systems and Applications II, R. C. Harney, ed., Proc. SPIE 415, 85–91 (1983).

Hannon, S. M.

S. M. Hannon, J. A. Thomson, S. W. Henderson, R. M. Huffaker, “Windshear, turbulence, and wake vortex characterization using pulsed solid-state coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 94–102 (1995).

J. A. Johnson, S. M. Hannon, “Wake vortex modeling for airborne and ground-based measurements using coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 63–78 (1995).

Hardesty, R. M.

R. M. Hardesty, T. R. Lawrence, R. A. Richter, M. J. Post, F. F. Hall, R. M. Huffaker, “Ground-based coherent lidar measurement of tropospheric and stratospheric parameters,” in Coherent Infrared Radar Systems and Applications II, R. C. Harney, ed., Proc. SPIE 415, 85–91 (1983).

Hawley, J. G.

R. Targ, J. G. Hawley, B. C. Steakley, L. L. Ames, “Airborne lidar wind detection at 2 μm,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 109–115 (1995).

Henderson, S. W.

S. M. Hannon, J. A. Thomson, S. W. Henderson, R. M. Huffaker, “Windshear, turbulence, and wake vortex characterization using pulsed solid-state coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 94–102 (1995).

Horrigan, F. A.

Huffaker, R. M.

R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1992).

R. M. Huffaker, “CO2 laser Doppler systems for the measurement of atmospheric winds and turbulence,” Atmos. Tech. NCAR Winter (National Center for Atmospheric Research, Boulder, Colo., 1974–1975), p. 71.

S. M. Hannon, J. A. Thomson, S. W. Henderson, R. M. Huffaker, “Windshear, turbulence, and wake vortex characterization using pulsed solid-state coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 94–102 (1995).

R. M. Hardesty, T. R. Lawrence, R. A. Richter, M. J. Post, F. F. Hall, R. M. Huffaker, “Ground-based coherent lidar measurement of tropospheric and stratospheric parameters,” in Coherent Infrared Radar Systems and Applications II, R. C. Harney, ed., Proc. SPIE 415, 85–91 (1983).

Jacobs, S.

P. Rabinowitz, S. Jacobs, R. Targ, G. Gould, “Homodyne detection of phase modulated light,” Proc. IRE p. 2365 (1962).

Jamison, W.

E. Uthe, N. Nielsen, W. Jamison, “Airborne plume and haze analyzer, ALPHA,” Bull. Am. Meteorol. Soc. 61, 1035–1043 (1980).

Johnson, J. A.

J. A. Johnson, S. M. Hannon, “Wake vortex modeling for airborne and ground-based measurements using coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 63–78 (1995).

Jones, R.

Kane, T. J.

T. J. Kane, J. D. Kmetec, T. J. Wagener, “Flight test of a 2-μm laser radar system,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 103–108 (1995).

Kavaya, M. J.

R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1992).

Kessinger, C.

J. W. Wilson, R. D. Roberts, C. Kessinger, J. McCarthy, “Microburst wind structure and evaluation of Doppler radar for airport windshear detection,” prepared for the Joint Airport Weather Studies Project (National Center for Atmospheric Research, Boulder, Colo., January1984).

Kmetec, J. D.

T. J. Kane, J. D. Kmetec, T. J. Wagener, “Flight test of a 2-μm laser radar system,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 103–108 (1995).

Lawrence, T. R.

R. M. Hardesty, T. R. Lawrence, R. A. Richter, M. J. Post, F. F. Hall, R. M. Huffaker, “Ground-based coherent lidar measurement of tropospheric and stratospheric parameters,” in Coherent Infrared Radar Systems and Applications II, R. C. Harney, ed., Proc. SPIE 415, 85–91 (1983).

Lee, K. A.

R. W. Lee, K. A. Lee, “A poly-pulse-pair signal processor for coherent Doppler lidar,” in Digest of Topical Meeting on Coherent Lasar Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), pp. WA2-1–WA2-4.

Lee, R. W.

R. W. Lee, K. A. Lee, “A poly-pulse-pair signal processor for coherent Doppler lidar,” in Digest of Topical Meeting on Coherent Lasar Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), pp. WA2-1–WA2-4.

McCarthy, J.

J. W. Wilson, R. D. Roberts, C. Kessinger, J. McCarthy, “Microburst wind structure and evaluation of Doppler radar for airport windshear detection,” prepared for the Joint Airport Weather Studies Project (National Center for Atmospheric Research, Boulder, Colo., January1984).

McGann, R. L.

R. L. McGann, “Flight test results from a low-power Doppler optical air data sensor,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 116–125 (1995).

Nielsen, N.

E. Uthe, N. Nielsen, W. Jamison, “Airborne plume and haze analyzer, ALPHA,” Bull. Am. Meteorol. Soc. 61, 1035–1043 (1980).

Oseguera, R. M.

R. M. Oseguera, R. L. Bowles, P. A. Robinson, “Airborne in situ computation of the windshear hazard index,” presented at the 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nev., 6–9 January 1992, paper AIAA 92-0291.

Post, M. J.

R. M. Hardesty, T. R. Lawrence, R. A. Richter, M. J. Post, F. F. Hall, R. M. Huffaker, “Ground-based coherent lidar measurement of tropospheric and stratospheric parameters,” in Coherent Infrared Radar Systems and Applications II, R. C. Harney, ed., Proc. SPIE 415, 85–91 (1983).

Rabinowitz, P.

P. Rabinowitz, S. Jacobs, R. Targ, G. Gould, “Homodyne detection of phase modulated light,” Proc. IRE p. 2365 (1962).

Richter, R. A.

R. M. Hardesty, T. R. Lawrence, R. A. Richter, M. J. Post, F. F. Hall, R. M. Huffaker, “Ground-based coherent lidar measurement of tropospheric and stratospheric parameters,” in Coherent Infrared Radar Systems and Applications II, R. C. Harney, ed., Proc. SPIE 415, 85–91 (1983).

Roberts, R. D.

J. W. Wilson, R. D. Roberts, C. Kessinger, J. McCarthy, “Microburst wind structure and evaluation of Doppler radar for airport windshear detection,” prepared for the Joint Airport Weather Studies Project (National Center for Atmospheric Research, Boulder, Colo., January1984).

Robinson, P. A.

R. M. Oseguera, R. L. Bowles, P. A. Robinson, “Airborne in situ computation of the windshear hazard index,” presented at the 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nev., 6–9 January 1992, paper AIAA 92-0291.

P. A. Robinson, “An experimental and theoretical evaluation of an airborne coherent lidar windshear sensor,” presented at the AIAA Aircraft Design, Systems and Operations Meeting, Monterey, Calif., 11–13 August 1993, paper AIAA 93-3947.

Sonnenschein, C. M.

Soreide, D. C.

R. Bogue, H. R. Bagley, D. C. Soreide, “Coherent lidar solution for the HSCT supersonic inlet unstart problem,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 79–93 (1995).

Steakley, B. C.

R. Targ, J. G. Hawley, B. C. Steakley, L. L. Ames, “Airborne lidar wind detection at 2 μm,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 109–115 (1995).

Targ, R.

R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1992).

R. Targ, J. G. Hawley, B. C. Steakley, L. L. Ames, “Airborne lidar wind detection at 2 μm,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 109–115 (1995).

P. Rabinowitz, S. Jacobs, R. Targ, G. Gould, “Homodyne detection of phase modulated light,” Proc. IRE p. 2365 (1962).

Thomson, J. A.

S. M. Hannon, J. A. Thomson, S. W. Henderson, R. M. Huffaker, “Windshear, turbulence, and wake vortex characterization using pulsed solid-state coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 94–102 (1995).

Uthe, E.

E. Uthe, N. Nielsen, W. Jamison, “Airborne plume and haze analyzer, ALPHA,” Bull. Am. Meteorol. Soc. 61, 1035–1043 (1980).

Vaughan, J. M.

Wagener, T. J.

T. J. Kane, J. D. Kmetec, T. J. Wagener, “Flight test of a 2-μm laser radar system,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 103–108 (1995).

Willetts, D. V.

Wilson, J. W.

J. W. Wilson, R. D. Roberts, C. Kessinger, J. McCarthy, “Microburst wind structure and evaluation of Doppler radar for airport windshear detection,” prepared for the Joint Airport Weather Studies Project (National Center for Atmospheric Research, Boulder, Colo., January1984).

Appl. Opt. (1)

R. Targ, M. J. Kavaya, R. M. Huffaker, R. L. Bowles, “Coherent lidar airborne windshear sensor: performance evaluation,” Appl. Opt. 30, 2013–2026 (1992).

Appl. Opt. (3)

Bull. Am. Meteorol. Soc. (1)

E. Uthe, N. Nielsen, W. Jamison, “Airborne plume and haze analyzer, ALPHA,” Bull. Am. Meteorol. Soc. 61, 1035–1043 (1980).

Opt. Eng. (1)

J. W. Bilbro, “Atmospheric laser Doppler velocimetry: an overview,” Opt. Eng. 19, 533–542 (1980).

Other (13)

P. A. Robinson, “An experimental and theoretical evaluation of an airborne coherent lidar windshear sensor,” presented at the AIAA Aircraft Design, Systems and Operations Meeting, Monterey, Calif., 11–13 August 1993, paper AIAA 93-3947.

R. W. Lee, K. A. Lee, “A poly-pulse-pair signal processor for coherent Doppler lidar,” in Digest of Topical Meeting on Coherent Lasar Radar for Atmospheric Sensing (Optical Society of America, Washington, D.C., 1980), pp. WA2-1–WA2-4.

R. M. Hardesty, T. R. Lawrence, R. A. Richter, M. J. Post, F. F. Hall, R. M. Huffaker, “Ground-based coherent lidar measurement of tropospheric and stratospheric parameters,” in Coherent Infrared Radar Systems and Applications II, R. C. Harney, ed., Proc. SPIE 415, 85–91 (1983).

P. Rabinowitz, S. Jacobs, R. Targ, G. Gould, “Homodyne detection of phase modulated light,” Proc. IRE p. 2365 (1962).

R. M. Huffaker, “CO2 laser Doppler systems for the measurement of atmospheric winds and turbulence,” Atmos. Tech. NCAR Winter (National Center for Atmospheric Research, Boulder, Colo., 1974–1975), p. 71.

J. A. Johnson, S. M. Hannon, “Wake vortex modeling for airborne and ground-based measurements using coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 63–78 (1995).

R. Bogue, H. R. Bagley, D. C. Soreide, “Coherent lidar solution for the HSCT supersonic inlet unstart problem,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 79–93 (1995).

S. M. Hannon, J. A. Thomson, S. W. Henderson, R. M. Huffaker, “Windshear, turbulence, and wake vortex characterization using pulsed solid-state coherent lidar,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 94–102 (1995).

T. J. Kane, J. D. Kmetec, T. J. Wagener, “Flight test of a 2-μm laser radar system,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 103–108 (1995).

R. Targ, J. G. Hawley, B. C. Steakley, L. L. Ames, “Airborne lidar wind detection at 2 μm,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 109–115 (1995).

R. L. McGann, “Flight test results from a low-power Doppler optical air data sensor,” in Air Traffic Control Technologies, R. G. Otto, J. Lenz, eds., Proc. SPIE 2464, 116–125 (1995).

J. W. Wilson, R. D. Roberts, C. Kessinger, J. McCarthy, “Microburst wind structure and evaluation of Doppler radar for airport windshear detection,” prepared for the Joint Airport Weather Studies Project (National Center for Atmospheric Research, Boulder, Colo., January1984).

R. M. Oseguera, R. L. Bowles, P. A. Robinson, “Airborne in situ computation of the windshear hazard index,” presented at the 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nev., 6–9 January 1992, paper AIAA 92-0291.

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Figures (14)

Fig. 1
Fig. 1

NASA research aircraft and scanner used in CLASS-2 and CLASS-10 flight-test programs.

Fig. 2
Fig. 2

Schematic of the 2-μm transceiver: PZT, piezoelectric transducer; TEC, thermoelectric cooler.

Fig. 3
Fig. 3

System-level block diagram of CLASS-2 lidar as installed in NASA’s B/737 research aircraft: A/D, analog to digital.

Fig. 4
Fig. 4

CLASS-2 lidar LOS wind measurements, comparing lidar data with aircraft INS data; rms difference is less than 1 m/s (40-pulse average).

Fig. 5
Fig. 5

CLASS-2 lidar wind measurements, showing rms uncertainty and narrow-band SNR as a function of range.

Fig. 6
Fig. 6

Range azimuth scan for CLASS-2 lidar looking at a smoke cloud, showing backscatter intensity as a function of range to 2.5 km.

Fig. 7
Fig. 7

Range azimuth scan ±8.5°, with 75-m range resolution, showing outflow wind velocity measurements from a forest-fire smoke cloud.

Fig. 8
Fig. 8

SNR for a 2-m, 3-mJ lidar with a 10-cm telescope. Wind velocity is measured to 7 km, where the narrow-band SNR falls to −5 dB.

Fig. 9
Fig. 9

Standard deviation of LOS velocity measurements versus the narrow-band SNR.

Fig. 10
Fig. 10

Block diagram of coherent CO2 transceiver used on NASA’s B/737 research aircraft in wind-shear deployments at Denver and Orlando: PZT, piezoelectric transducer.

Fig. 11
Fig. 11

SNR versus range from a calibrated canvas target at 700 m, showing 16-pulse averages of return signal for transmit energies of 10, 3, and 0.3 mJ.

Fig. 12
Fig. 12

Narrow-band SNR’s as a function of range, measured with a 8-mJ CLASS-10 lidar and a 15-cm telescope at NASA Langley Research Center (see Ref. 15).

Fig. 13
Fig. 13

Cockpit display of the 10.6-μm CLASS-10 lidar, showing the intensity of returns to a range of 8 km at Denver in July 1992.

Fig. 14
Fig. 14

Cockpit display of the 10.6-μm CLASS-10 lidar, measuring the velocity of turbulent winds to a range of 8 km at Denver in July 1992.

Tables (2)

Tables Icon

Table 1 Base-Case Parameters for the 2- and 10-μm Lidar Systems

Tables Icon

Table 2 Advantages of the 2-μm Solid-State Lidar over the 10-μm CO2 System

Equations (5)

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

SNR n ( R ) = E η β ( z ) λ K 2 π D 2 8 ( 1 + f ) h B n R 2 ,
η = 0.4 η 1 + [ 1 - ( R / F ) ] 2 ( π D e 2 / 4 λ R ) 2 + ( D / 2 S 0 ) 2 ,
S 0 [ H k 2 0 R C n 2 ( R ) ( 1 - R R ) 5 / 3 d R ] - 3 / 5 .
SNR N ( R ) = P η ρ λ K 2 π D 2 4 ( 1 + f ) h c B N R 2 .
F = d V i / d t ( 1 / g ) - V z / V a ,

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