Abstract

We performed an experimental study on the effect of atmospheric turbulence on heterodyne and direct detection lidar at 1 μm, employing a pulsed Nd:YAG bistatic focused beam lidar that permitted simultaneous heterodyne and direct detection of the same lidar returns. The average carrier-to-noise ratio and statistical fluctuation level in the lidar return signals were measured in various experimental and atmospheric conditions. The results showed that atmospheric turbulence could reduce the effective receiver telescope diameter of the 1-μm heterodyne lidar to <5 cm at a relatively short range of ~450 m near the ground. The observed effective telescope aperture and heterodyne detection efficiency varied during the day as the atmospheric turbulence level changed. At this time, we are not able to compare our experimental lidar data to a rigorous atmospheric turbulence and lidar detection theory which includes independently variable transmitter, receiver, and detector geometry. It is interesting to note, however, that the observed limitation of the effective receiver aperture was similar in functional form with those predictions based on the heterodyne wavefront detection theory by D. L. Fried [ Proc. IEEE 55, 57– 67 ( 1967)] and the heterodyne lidar detection theory for a fixed monostatic system by S. F. Clifford and S. Wandzura [ Appl. Opt. 20, 514– 516 ( 1981)]. We have also applied such an effective receiver aperture limitation to predict the system performance for a heterodyne Ho lidar operating at 2 μm.

© 1991 Optical Society of America

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References

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  1. See, for example, Special Issue on Solid-State Lasers, IEEE J. Quantum Electron. QE-24 (June1988). For latest reports, see papers in Technical Digest, Topical Meeting on Advanced Solid-State Lasers (Optical Society of America, Washington, DC, 1990).
  2. N. Menyuk, D. K. Killinger, “Atmospheric Remote Sensing of Water Vapor, HCl, and CH4 Using a Continuously Tunable Co:MgF2 Laser,” Appl. Opt. 26, 3061–3065 (1987).
    [CrossRef] [PubMed]
  3. T. J. Kane, W. J. Kozlovsky, R. L. Byer, C. Byvik, “Coherent Laser Radar at 1.06 μm Using Nd:YAG Lasers,” Opt. Lett. 12, 239–241 (1987).
    [CrossRef] [PubMed]
  4. M. J. Kavaya, S. M. 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]
  5. N. Sugimoto, N. Sims, K. Chan, D. K. Killinger, “Eye-Safe 2.1-μm Ho Lidar for Measuring Atmospheric Density Profiles,” Opt. Lett. 15, 302–304 (1990).
    [CrossRef] [PubMed]
  6. K. P. Chan, D. K. Killinger, “Short-Pulse, Coherent Doppler Nd:YAG Lidar,” Opt. Eng. 30, 49–54 (1991).
    [CrossRef]
  7. R. T. Menzies, R. M. Hardesty, “Coherent Doppler Lidar for Measurements of Wind Field,” Proc. IEEE 77, 449–462 (1989).
    [CrossRef]
  8. V. I. Tatarski, “The Effects of the Turbulent Atmosphere on Wave Propagation,” in IPST Catalog 5319 (National Technical Information Service, Springfield, VA 22151, 1971).
  9. J. W. Strohbehn, Ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978), and references therein.
    [CrossRef]
  10. V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).
  11. I. Goldstein, P. A. Miles, A. Chabot, “Heterodyne Measurement of Light Propagation Through Atmospheric Turbulence,” Proc. IEEE 53, 1172–1180 (1965).
    [CrossRef]
  12. D. L. Fried, “Optical Heterodyne Detection of an Atmospherically Distorted Signal Wave Front,” Proc. IEEE 55, 57–67 (1967).
    [CrossRef]
  13. J. H. Churnside, C. M. McIntyre, “Signal Current Probability Distribution for Heterodyne Receiver in the Turbulent Atmosphere. 1: Theory,” Appl. Opt. 17, 2141–2147 (1978).
    [CrossRef] [PubMed]
  14. J. H. Churnside, C. M. McIntyre, “Signal Current Probability Distribution for Heterodyne Receiver in the Turbulent Atmosphere. 2: Experiment,” Appl. Opt. 17, 2148–2152 (1978).
    [CrossRef] [PubMed]
  15. P. A. Pincus, M. E. Fossey, J. Holmes, J. R. Kerr, “Speckle Propagation Through Turbulence: Experimental,” J. Opt. Soc. Am. 68, 760–762 (1978).
    [CrossRef]
  16. H. T. Yura, “Signal-to-Noise Ratio of Heterodyne Lidar Systems in the Presence of Atmospheric Turbulence,” Opt. Acta 26, 627–644 (1979).
    [CrossRef]
  17. R. L. Schwiesow, R. F. Calfee, “Atmosphere Refractive Effects on Coherent Lidar Performance at 10.6 μm,” Appl. Opt. 18, 3911–3917 (1979).
    [CrossRef] [PubMed]
  18. J. F. Holmes, M. H. Lee, J. R. Kerr, “Effect of the Log-Amplitude Covariance Function on the Statistics of Speckle Propagation through the Turbulent Atmosphere,” J. Opt. Soc. Am. 70, 355–360 (1980).
    [CrossRef]
  19. S. F. Clifford, S. Wandzura, “Monostatic Heterodyne Lidar Performance: The Effect of the Turbulent Atmosphere,” Appl. Opt. 20, 514–516 (1981); “Monostatic Heterodyne Lidar Performance: The Effect of the Turbulent Atmosphere, Correction,” Appl. Opt. 20, 1502 (1981).
    [CrossRef] [PubMed]
  20. J. H. Shapiro, B. A. Capron, R. C. Harney, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Appl. Opt. 20, 3292–3313 (1981).
    [CrossRef] [PubMed]
  21. B. J. Rye, “Refractive-Turbulence Contribution to Incoherent Backscatter Heterodyne Lidar Returns,” J. Opt. Soc. Am. 71, 687–691 (1981).
    [CrossRef]
  22. R. Murty, “Refractive Turbulence Effects on Truncated Gaussian Beam Heterodyne Lidar,” Appl. Opt. 23, 2498–2502 (1984).
    [CrossRef] [PubMed]
  23. Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving Efficiency of Monostatic Pulsed Coherent Lidars. 1: Theory,” Appl. Opt. 29, 4111–4119 (1980).
    [CrossRef]
  24. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1978).
  25. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), Chap. 2, pp. 9–75.
    [CrossRef]
  26. N. Sugimoto, K. P. Chan, D. K. Killinger, “Video Camera Measurement of Atmospheric Turbulence Using the Telescope Image of a Distant Light Source,” Appl. Opt. 30, 365–367 (1991).
    [CrossRef] [PubMed]
  27. J. C. Wyngard, Y. Izumi, S. A. Collins, “Behavior of the Refractive-Index-Structure Near the Ground,” J. Opt. Soc. Am. 61, 1646–1650 (1971).
    [CrossRef]
  28. D. K. Killinger, N. Menyuk, W. E. DeFeo, “Experimental Comparison of Heterodyne and Direct Detection for Pulsed Differential Absorption CO2 Lidar,” Appl. Opt. 22, 682–689 (1983).
    [CrossRef] [PubMed]
  29. S. C. Cohen, “Heterodyne Detection: Phase Front Alignment, Beam Spot Size, and Detector Uniformity,” Appl. Opt. 14, 1953–1959 (1975).
    [CrossRef] [PubMed]
  30. J. Y. Wang, “Detection Efficiency of Coherent Optical Radar,” Appl. Opt. 23, 3421–3427 (1984).
    [CrossRef] [PubMed]
  31. N. Sugimoto, K. P. Chan, D. K. Killinger, “Optimal Heterodyne Detection Array Size for 1-μm Coherent Lidar Propagation through Atmospheric Turbulence,” Appl. Opt. 30, this issue (1991).
    [CrossRef] [PubMed]
  32. R. T. Menzies, M. J. Kavaya, P. H. Flamant, D. A. Haner, “Atmospheric Aerosol Backscatter Measurements Using a Tunable Coherent CO2 Lidar,” Appl. Opt. 23, 2510–2517 (1984).
    [CrossRef] [PubMed]
  33. R. M. Hardesty, “Coherent DIAL Measurement of Range-Resolved Water Vapor Concentration,” Appl. Opt. 23, 2545–2553 (1984).
    [CrossRef] [PubMed]

1991 (3)

K. P. Chan, D. K. Killinger, “Short-Pulse, Coherent Doppler Nd:YAG Lidar,” Opt. Eng. 30, 49–54 (1991).
[CrossRef]

N. Sugimoto, K. P. Chan, D. K. Killinger, “Video Camera Measurement of Atmospheric Turbulence Using the Telescope Image of a Distant Light Source,” Appl. Opt. 30, 365–367 (1991).
[CrossRef] [PubMed]

N. Sugimoto, K. P. Chan, D. K. Killinger, “Optimal Heterodyne Detection Array Size for 1-μm Coherent Lidar Propagation through Atmospheric Turbulence,” Appl. Opt. 30, this issue (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

1988 (1)

See, for example, Special Issue on Solid-State Lasers, IEEE J. Quantum Electron. QE-24 (June1988). For latest reports, see papers in Technical Digest, Topical Meeting on Advanced Solid-State Lasers (Optical Society of America, Washington, DC, 1990).

1987 (2)

1984 (4)

1983 (1)

1981 (3)

1980 (2)

1979 (2)

H. T. Yura, “Signal-to-Noise Ratio of Heterodyne Lidar Systems in the Presence of Atmospheric Turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

R. L. Schwiesow, R. F. Calfee, “Atmosphere Refractive Effects on Coherent Lidar Performance at 10.6 μm,” Appl. Opt. 18, 3911–3917 (1979).
[CrossRef] [PubMed]

1978 (3)

1975 (1)

1971 (1)

1967 (1)

D. L. Fried, “Optical Heterodyne Detection of an Atmospherically Distorted Signal Wave Front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

1965 (1)

I. Goldstein, P. A. Miles, A. Chabot, “Heterodyne Measurement of Light Propagation Through Atmospheric Turbulence,” Proc. IEEE 53, 1172–1180 (1965).
[CrossRef]

Banakh, V. A.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).

Byer, R. L.

Byvik, C.

Calfee, R. F.

Capron, B. A.

Chabot, A.

I. Goldstein, P. A. Miles, A. Chabot, “Heterodyne Measurement of Light Propagation Through Atmospheric Turbulence,” Proc. IEEE 53, 1172–1180 (1965).
[CrossRef]

Chan, K.

Chan, K. P.

K. P. Chan, D. K. Killinger, “Short-Pulse, Coherent Doppler Nd:YAG Lidar,” Opt. Eng. 30, 49–54 (1991).
[CrossRef]

N. Sugimoto, K. P. Chan, D. K. Killinger, “Video Camera Measurement of Atmospheric Turbulence Using the Telescope Image of a Distant Light Source,” Appl. Opt. 30, 365–367 (1991).
[CrossRef] [PubMed]

N. Sugimoto, K. P. Chan, D. K. Killinger, “Optimal Heterodyne Detection Array Size for 1-μm Coherent Lidar Propagation through Atmospheric Turbulence,” Appl. Opt. 30, this issue (1991).
[CrossRef] [PubMed]

Churnside, J. H.

Clifford, S. F.

Cohen, S. C.

Collins, S. A.

DeFeo, W. E.

Flamant, P. H.

Fossey, M. E.

Fried, D. L.

D. L. Fried, “Optical Heterodyne Detection of an Atmospherically Distorted Signal Wave Front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Goldstein, I.

I. Goldstein, P. A. Miles, A. Chabot, “Heterodyne Measurement of Light Propagation Through Atmospheric Turbulence,” Proc. IEEE 53, 1172–1180 (1965).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), Chap. 2, pp. 9–75.
[CrossRef]

Hale, C. P.

Haner, D. A.

Hardesty, R. M.

Harney, R. C.

Henderson, S. M.

Holmes, J.

Holmes, J. F.

Huffaker, R. M.

Izumi, Y.

Kane, T. J.

Kavaya, M. J.

Kerr, J. R.

Killinger, D. K.

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1978).

Kozlovsky, W. J.

Lee, M. H.

Magee, J. R.

McIntyre, C. M.

Menyuk, N.

Menzies, R. T.

Miles, P. A.

I. Goldstein, P. A. Miles, A. Chabot, “Heterodyne Measurement of Light Propagation Through Atmospheric Turbulence,” Proc. IEEE 53, 1172–1180 (1965).
[CrossRef]

Mironov, V. L.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).

Murty, R.

Pincus, P. A.

Post, M. J.

Rye, B. J.

Schwiesow, R. L.

Shapiro, J. H.

Sims, N.

Sugimoto, N.

Tatarski, V. I.

V. I. Tatarski, “The Effects of the Turbulent Atmosphere on Wave Propagation,” in IPST Catalog 5319 (National Technical Information Service, Springfield, VA 22151, 1971).

Wandzura, S.

Wang, J. Y.

Wyngard, J. C.

Yura, H. T.

H. T. Yura, “Signal-to-Noise Ratio of Heterodyne Lidar Systems in the Presence of Atmospheric Turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

Zhao, Y.

Appl. Opt. (15)

N. Menyuk, D. K. Killinger, “Atmospheric Remote Sensing of Water Vapor, HCl, and CH4 Using a Continuously Tunable Co:MgF2 Laser,” Appl. Opt. 26, 3061–3065 (1987).
[CrossRef] [PubMed]

J. H. Churnside, C. M. McIntyre, “Signal Current Probability Distribution for Heterodyne Receiver in the Turbulent Atmosphere. 1: Theory,” Appl. Opt. 17, 2141–2147 (1978).
[CrossRef] [PubMed]

J. H. Churnside, C. M. McIntyre, “Signal Current Probability Distribution for Heterodyne Receiver in the Turbulent Atmosphere. 2: Experiment,” Appl. Opt. 17, 2148–2152 (1978).
[CrossRef] [PubMed]

R. L. Schwiesow, R. F. Calfee, “Atmosphere Refractive Effects on Coherent Lidar Performance at 10.6 μm,” Appl. Opt. 18, 3911–3917 (1979).
[CrossRef] [PubMed]

R. Murty, “Refractive Turbulence Effects on Truncated Gaussian Beam Heterodyne Lidar,” Appl. Opt. 23, 2498–2502 (1984).
[CrossRef] [PubMed]

Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving Efficiency of Monostatic Pulsed Coherent Lidars. 1: Theory,” Appl. Opt. 29, 4111–4119 (1980).
[CrossRef]

S. F. Clifford, S. Wandzura, “Monostatic Heterodyne Lidar Performance: The Effect of the Turbulent Atmosphere,” Appl. Opt. 20, 514–516 (1981); “Monostatic Heterodyne Lidar Performance: The Effect of the Turbulent Atmosphere, Correction,” Appl. Opt. 20, 1502 (1981).
[CrossRef] [PubMed]

J. H. Shapiro, B. A. Capron, R. C. Harney, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Appl. Opt. 20, 3292–3313 (1981).
[CrossRef] [PubMed]

N. Sugimoto, K. P. Chan, D. K. Killinger, “Video Camera Measurement of Atmospheric Turbulence Using the Telescope Image of a Distant Light Source,” Appl. Opt. 30, 365–367 (1991).
[CrossRef] [PubMed]

D. K. Killinger, N. Menyuk, W. E. DeFeo, “Experimental Comparison of Heterodyne and Direct Detection for Pulsed Differential Absorption CO2 Lidar,” Appl. Opt. 22, 682–689 (1983).
[CrossRef] [PubMed]

S. C. Cohen, “Heterodyne Detection: Phase Front Alignment, Beam Spot Size, and Detector Uniformity,” Appl. Opt. 14, 1953–1959 (1975).
[CrossRef] [PubMed]

J. Y. Wang, “Detection Efficiency of Coherent Optical Radar,” Appl. Opt. 23, 3421–3427 (1984).
[CrossRef] [PubMed]

N. Sugimoto, K. P. Chan, D. K. Killinger, “Optimal Heterodyne Detection Array Size for 1-μm Coherent Lidar Propagation through Atmospheric Turbulence,” Appl. Opt. 30, this issue (1991).
[CrossRef] [PubMed]

R. T. Menzies, M. J. Kavaya, P. H. Flamant, D. A. Haner, “Atmospheric Aerosol Backscatter Measurements Using a Tunable Coherent CO2 Lidar,” Appl. Opt. 23, 2510–2517 (1984).
[CrossRef] [PubMed]

R. M. Hardesty, “Coherent DIAL Measurement of Range-Resolved Water Vapor Concentration,” Appl. Opt. 23, 2545–2553 (1984).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

See, for example, Special Issue on Solid-State Lasers, IEEE J. Quantum Electron. QE-24 (June1988). For latest reports, see papers in Technical Digest, Topical Meeting on Advanced Solid-State Lasers (Optical Society of America, Washington, DC, 1990).

J. Opt. Soc. Am. (4)

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. Eng. (1)

K. P. Chan, D. K. Killinger, “Short-Pulse, Coherent Doppler Nd:YAG Lidar,” Opt. Eng. 30, 49–54 (1991).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (3)

R. T. Menzies, R. M. Hardesty, “Coherent Doppler Lidar for Measurements of Wind Field,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

I. Goldstein, P. A. Miles, A. Chabot, “Heterodyne Measurement of Light Propagation Through Atmospheric Turbulence,” Proc. IEEE 53, 1172–1180 (1965).
[CrossRef]

D. L. Fried, “Optical Heterodyne Detection of an Atmospherically Distorted Signal Wave Front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Other (5)

V. I. Tatarski, “The Effects of the Turbulent Atmosphere on Wave Propagation,” in IPST Catalog 5319 (National Technical Information Service, Springfield, VA 22151, 1971).

J. W. Strohbehn, Ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978), and references therein.
[CrossRef]

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1978).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), Chap. 2, pp. 9–75.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic of short-pulse Nd:YAG lidar system at 1 μm providing simultaneous heterodyne and direct detection of the same lidar return.

Fig. 2
Fig. 2

Oscilloscope displays showing a single-pulse heterodyne signal of lidar return from a hard target at a range of 450 m. The receiver telescope aperture size was (a) 5 and (b) 15 cm. Note that the 550-MHz heterodyne signal is superimposed on top of the direct detection, 8-ns long pulse envelope.

Fig. 3
Fig. 3

Measured average CNR for heterodyne (open circles) and direct detection (solid circles) of lidar returns from a hard target at a range of 450 m as a function of the receiver aperture area, which was varied by changing the diameter of (a) the receiver telescope aperture and (b) the collection aperture stops in front of the heterodyne and direct detection detector.

Fig. 4
Fig. 4

Measured normalized signal standard deviation of heterodyne (open circles) and direct detection (solid circles) lidar returns from a hard target at a range of 450 m as a function of the receiver telescope aperture area.

Fig. 5
Fig. 5

Measured variation of (a) average CNR and (b) normalized signal standard deviation of heterodyne (open circles) and direct detection (solid circles) lidar returns from a hard target at a range of 450 m as a function of time during the day.

Fig. 6
Fig. 6

Comparison of measured lidar CNR values (open circles) and theoretical predictions using the theories presented by Fried12 (solid line), and Clifford and Wandzura19 (dashed line).

Fig. 7
Fig. 7

Calculated results of (a) heterodyne lidar detection CNR in the presence of atmospheric turbulence and (b) the normalized value to the maximum CNR that a heterodyne lidar achieves with an effective receiver telescope aperture area of Aeff = π(3ρ0)2/4. The receiver telescope is varied over a range of 0–700 cm2, i.e., telescope diameter of 0–30 cm.

Fig. 8
Fig. 8

Calculated results of the maximum effective receiver telescope diameter (solid lines) and the practically useful telescope diameter (dashed lines) for a 1.06-μm Nd:YAG heterodyne lidar as a function of detection range. The atmospheric parameters C n 2 = 1 × 10 - 12 , 5 × 10−14, and 2 × 10−15 m−2/3 are assumed to be uniform along the horizontal lidar propagation path at a height of 1, 10, and 100 m, respectively.

Fig. 9
Fig. 9

Calculated results of the maximum effective receiver telescope diameter (solid lines) and the practically useful telescope diameter (dashed lines) for a 2.1-μm Ho heterodyne lidar as a function of detection range. The atmospheric parameters are the same as those used in Fig. 8.

Fig. 10
Fig. 10

Comparison of heterodyne and direct detection CNR for a ground-based (a) 1-μm Nd:YAG lidar and (b) 10-μm CO2 lidar, both using a 20-cm diam receiver telescope. The atmospheric parameter C n 2 = 5 × 10 - 14 m - 2 / 3 is assumed to be uniform along the lidar propagation path and two different electrical bandwidths of 10 and 500 MHz are shown.

Equations (12)

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i if 2 ¯ = ( Ω / 2 ) [ ( η e / h ν ) E s 0 E lo A r ] 2 ,
i n 2 ¯ = 2 e B ( η e P l / h ν ) ,
CNR H , 0 = i if 2 ¯ / i n 2 ¯ = Ω η P s / ( h ν B ) ,
ρ 0 = [ 2.91 k 2 0 R C n 2 ( z ) ( 1 - z / R ) 5 / 3 d z ] - 3 / 5 ,
F 0 ( U ) = ( 16 / π ) 0 1 u [ cos - 1 u - u ( 1 - u 2 ) 1 / 2 ] × exp [ - ( 1 / 2 ) ( U u ) 5 / 3 ] d u ,
CNR H , av ( d r , ρ 0 ) = i if 2 ¯ / i n 2 ¯ = CNR H , 0 F 0 ( U ) .
F 0 ( U ) [ 1 + ( U / 3 ) 2 ] - 1 ,
P s = [ P t ζ A r K exp ( - 2 α R ) ] / R 2 ,
CNR D = P s / ( NEP D L 2 ) + NEP A L 2 ) 1 / 2 ,
A eff = F 0 ( U ) A r ,
CNR H , av ( d r , ρ 0 ) = Ψ CNR H , 0 d r = ,
Γ = ( η / h ν B ) F 0 ( ρ 0 , d r ) G ( NEP D L 2 + NEP A L 2 ) 1 / 2 ,

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