Abstract

The signal-to-noise ratio (SNR) and heterodyne efficiency are investigated for coherent (heterodyne detection) laser radar under the Fresnel approximation and general conditions. This generality includes spatially random fields, refractive turbulence, monostatic and bistatic configurations, detector geometry, and targets. For the first time to our knowledge, the effects of atmospheric refractive turbulence are included by using the path-integral formulation. For general conditions the SNR can be expressed in terms of the direct detection power and a heterodyne efficiency that can be estimated from the laser radar signal. For weak refractive turbulence (small irradiance fluctuations at the target) and under the Markov approximation, it is shown that the assumption of statistically independent paths is valid, even for the monostatic configuration. In the limit of large path-integrated refractive turbulence the SNR can become twice the statistically independent-path result. The effects of the main components of a coherent laser radar are demonstrated by assuming untruncated Gaussians for the transmitter, receiver, and local oscillator. The physical mechanisms that reduce heterodyne efficiency are identified by performing the calculations in the receiver plane. The physical interpretations of these results are compared with those obtained from calculations performed in the target plane.

© 1991 Optical Society of America

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1991 (1)

1990 (3)

1989 (1)

1988 (2)

1987 (10)

J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

R. G. Frehlich, “Space-time fourth moment of waves propagating in random media,” Radio Sci. 22, 481–490 (1987).
[CrossRef]

G. M. Ancellet, R. T. Menzies, “Atmospheric correlation-time measurements and effects on coherent Doppler lidar,” J. Opt. Soc. Am. A 4, 367–373 (1987).
[CrossRef]

K. Tanaka, N. Ohta, “Effects of tilt and offset of signal field on heterodyne efficiency,” Appl. Opt. 26, 627–632 (1987).
[CrossRef] [PubMed]

D. U. Fluckiger, R. J. Keyes, J. H. Shapiro, “Optical autodyne detection: theory and experiment,” Appl. Opt. 26, 318–325 (1987).
[CrossRef] [PubMed]

J. H. Churnside, S. F. Clifford, “Log-normal Rician probability density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
[CrossRef]

R. G. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
[CrossRef]

M. J. Kavaya, “Polarization effects on hard target calibration of lidar systems,” Appl. Opt. 26, 796–804 (1987).
[CrossRef] [PubMed]

T. Kobayashi, “Techniques for laser remote sensing of the environment,” Remote Sensing Rev. 3, 1–56 (1987).
[CrossRef]

D. K. Killinger, N. Menyuk, “Laser remote sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

1986 (4)

J. Y. Wang, “Lidar signal fluctuations caused by beam translation and scan,” Appl. Opt. 25, 2878–2885 (1986).
[CrossRef] [PubMed]

C. J. Leader, “Speckle effects on coherent laser radar detection efficiency,” Opt. Eng. 25, 644–649 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equations and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

1985 (2)

1984 (8)

1983 (5)

1982 (3)

1981 (5)

1980 (2)

1979 (4)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

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

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

R. L. Schwiesow, R. F. Calfee, “Atmospheric refractive effects on coherent lidar performance at 10.6 μm,” Appl. Opt. 18, 3911–3917 (1979).
[CrossRef] [PubMed]

1978 (2)

T. Takenaka, K. Tanaka, O. Fukumitsu, “Signal-to-noise ratio in optical heterodyne detection for Gaussian fields,” Appl. Opt. 17, 3466–3471 (1978).
[CrossRef] [PubMed]

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978)[Sov. Phys. JETP 48, 27–31 (1978)].

1977 (2)

V. A. Banakh, V. L. Mironov, “Phase approximation of the Huygens–Kirchhoff method in problems of laser-beam propagation in the turbulent atmosphere,” Opt. Lett. 1, 172–174 (1977).
[CrossRef] [PubMed]

V. U. Zavorotnyi, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Zh. Eksp. Teor. Fiz. 73, 481–497 (1977)[Sov. Phys. JETP 46, 252–260 (1977)].

1976 (1)

1975 (3)

1974 (2)

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]

1971 (1)

1970 (1)

1967 (3)

1966 (1)

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 51, 1350–1358 (1966).
[CrossRef]

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965)
[CrossRef]

Amzajerdian, F.

Ancellet, G. M.

Bachman, C. G.

C. G. Bachman, Laser Radar Systems and Techniques (Artech House, Dedham, Mass., 1979).

Banakh, V. A.

Beranek, R. G.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth and Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1062, 234–248 (1989).

Bilbro, J. W.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth and Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1062, 234–248 (1989).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980).

Bowles, R. L.

Boynton, F. P.

J. A. Thomson, F. P. Boynton, “Development of design procedures for coherent lidar measurements of atmospheric winds,” Final Rep., contract NOAA-03-7-022-35106, Rep. PD-B-77-137 (Physical Dynamics, Berkeley, Calif., June1977, revised Sept. 1977, Nov. 1977, and Jan. 1978).

Byer, R. L.

Calfee, R. F.

Capron, B. A.

Churnside, J. H.

Clifford, S. F.

Codona, J. L.

J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equations and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Cohen, S. C.

Coles, W. A.

W. A. Coles, R. G. Frehlich, “Simultaneous measurements of angular scattering and intensity scintillations in the atmosphere,” J. Opt. Soc. Am. 72, 1041–1048 (1982).
[CrossRef]

Collis, R. T. H.

R. T. H. Collis, P. B. Russell, “Lidar measurements of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E. D. Hinkley, ed. (Springer-Verlag, Berlin, 1976), Chap. 4.
[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]

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equations and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

Cupp, R. E.

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

Degnan, J. J.

Dickson, L. D.

DiMarzio, C. A.

C. A. DiMarzio, C. S. Lins, “Heterodyne SNR computations using orthogonal functions,” in Fifth Conference on Coherent Laser Radar: Technology and Applications, J. W. Bilbro, C. Werner, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1181, 176–185 (1989).
[CrossRef]

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L. E. Drain, The Laser Doppler Technique (Wiley-Interscience, Chichester, England, 1980).

Fante, R. L.

R. L. Fante, “Inner-scale size effect on the scintillations of light in the turbulent atmosphere,” J. Opt. Soc. Am. 73, 277–281 (1983).
[CrossRef]

R. L. Fante, “Wave propagation in random media: A systems approach,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1985), Vol. 22, pp. 341–398.
[CrossRef]

Fink, D.

Fitzjarrald, D. E.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth and Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1062, 234–248 (1989).

Flamant, P. H.

Flatté, S. M.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equations and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
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J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
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S. M. Flatté, “Wave propagation through random media: contributions from ocean acoustics,” Proc. IEEE 71, 1267–1294 (1983).
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Fluckiger, D. U.

Foord, R.

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R. G. Frehlich, “Conditions for optimal performance of mono-static coherent laser radar,” Opt. Lett. 15, 643–645 (1990).
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R. G. Frehlich, “Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
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J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

R. G. Frehlich, “Space-time fourth moment of waves propagating in random media,” Radio Sci. 22, 481–490 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equations and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
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W. A. Coles, R. G. Frehlich, “Simultaneous measurements of angular scattering and intensity scintillations in the atmosphere,” J. Opt. Soc. Am. 72, 1041–1048 (1982).
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Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–66 (1967).
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Fukumitsu, O.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965)
[CrossRef]

Grant, W. B.

W. B. Grant, “Laser Remote Sensing Techniques,” in Laser Spectroscopy and Its Applications, L. J. Radziemski, R. W. Solarz, J. A. Paisner, eds. (Dekker, New York, 1987), Chap. 8.

Hale, C. P.

Hall, F. F.

Haner, D. A.

Hanson, S.

Hardesty, R. M.

Harney, R. C.

Helstrom, C. W.

Henderson, S. W.

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equations and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Holmes, J. F.

Horrigan, F. A.

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). A list of unpublished errata for this paper can be obtained from the present authors.
[CrossRef] [PubMed]

Huffaker, R. M.

Hunt, J. M.

Jensen, A. S.

Jones, I. P.

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]

Jones, R.

Jones, W. D.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth and Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1062, 234–248 (1989).

Kane, T. J.

Katzir, A.

Kavaya, M. J.

Keeler, R. J.

Keller, V. W.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth and Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1062, 234–248 (1989).

Keyes, R. J.

Killinger, D. K.

D. K. Killinger, N. Menyuk, “Laser remote sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

Klein, B. J.

Klyatskin, V. I.

V. U. Zavorotnyi, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Zh. Eksp. Teor. Fiz. 73, 481–497 (1977)[Sov. Phys. JETP 46, 252–260 (1977)].

Kobayashi, T.

T. Kobayashi, “Techniques for laser remote sensing of the environment,” Remote Sensing Rev. 3, 1–56 (1987).
[CrossRef]

Lading, L.

Lawrence, T. R.

Leader, C. J.

C. J. Leader, “Speckle effects on coherent laser radar detection efficiency,” Opt. Eng. 25, 644–649 (1986).
[CrossRef]

Lins, C. S.

C. A. DiMarzio, C. S. Lins, “Heterodyne SNR computations using orthogonal functions,” in Fifth Conference on Coherent Laser Radar: Technology and Applications, J. W. Bilbro, C. Werner, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1181, 176–185 (1989).
[CrossRef]

Magee, J. R.

Mandel, L.

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R. M. Measures, “Fundamentals of Laser Remote Sensing,” in Laser Remote Chemical Analysis, R. M. Measures, ed. (Wiley, New York, 1988), Chap. 1.

R. M. Measures, Laser Remote Sensing. Fundamentals and Applications (Interscience, New York, 1984).

Menyuk, N.

D. K. Killinger, N. Menyuk, “Laser remote sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

Menzies, R. T.

Meyzonnette, J. L.

J. L. Meyzonnette, B. Remy, G. Saccomani, “Imaging CO2 laser radar with chirp pulse compression,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.783, 169–179 (1987).

Mironov, V. L.

Murty, R.

R. Murty, “Refractive turbulence effects on truncated Gaussian beam heterodyne lidar,” Appl Opt. 23, 2498–2502 (1984).
[CrossRef] [PubMed]

Naats, I. E.

V. E. Zuev, I. E. Naats, Inverse Problems of Lidar Sensing of the Atmosphere (Springer-Verlag, Berlin, 1983).

Ohta, N.

Oppenheim, U. P.

Perrine, B. S.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth and Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1062, 234–248 (1989).

Post, M. J.

Priestley, J. T.

Remy, B.

J. L. Meyzonnette, B. Remy, G. Saccomani, “Imaging CO2 laser radar with chirp pulse compression,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.783, 169–179 (1987).

Richter, R. A.

Ross, M.

M. Ross, Laser Receivers. Devices, Techniques, Systems (Wiley, New York, 1966).

Rudd, M. J.

B. M. Watrasiewicz, M. J. Rudd, Laser Doppler Measurements (Butterworth, London, 1976).

Russell, P. B.

R. T. H. Collis, P. B. Russell, “Lidar measurements of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E. D. Hinkley, ed. (Springer-Verlag, Berlin, 1976), Chap. 4.
[CrossRef]

Rye, B. J.

Saccomani, G.

J. L. Meyzonnette, B. Remy, G. Saccomani, “Imaging CO2 laser radar with chirp pulse compression,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.783, 169–179 (1987).

Saga, N.

Salzman, J.

Schwiesow, R. L.

Shapiro, J. H.

Siegman, A. E.

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 51, 1350–1358 (1966).
[CrossRef]

Sonnenschein, C. M.

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). A list of unpublished errata for this paper can be obtained from the present authors.
[CrossRef] [PubMed]

Takenaka, T.

Tanaka, K.

Targ, R.

Tatarskii, V. I.

V. U. Zavorotnyi, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Zh. Eksp. Teor. Fiz. 73, 481–497 (1977)[Sov. Phys. JETP 46, 252–260 (1977)].

V. I. Tatarskii, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. 28.
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V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).

Thomson, J. A.

J. A. Thomson, F. P. Boynton, “Development of design procedures for coherent lidar measurements of atmospheric winds,” Final Rep., contract NOAA-03-7-022-35106, Rep. PD-B-77-137 (Physical Dynamics, Berkeley, Calif., June1977, revised Sept. 1977, Nov. 1977, and Jan. 1978).

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]

van Vliet, K. M.

Vaughan, J. M.

Vodopia, S. N.

Wandzura, S.

Wandzura, S. M.

Wang, J. Y.

Watrasiewicz, B. M.

B. M. Watrasiewicz, M. J. Rudd, Laser Doppler Measurements (Butterworth, London, 1976).

Willetts, D. V.

Wilson, D. J.

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]

Wolf, E.

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]

H. T. Yura, “Optical heterodyne signal power obtained from finite sized sources of radiation,” Appl. Opt. 13, 150–157 (1974).
[CrossRef] [PubMed]

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978)[Sov. Phys. JETP 48, 27–31 (1978)].

V. U. Zavorotnyi, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Zh. Eksp. Teor. Fiz. 73, 481–497 (1977)[Sov. Phys. JETP 46, 252–260 (1977)].

V. I. Tatarskii, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. 28.
[CrossRef]

Zhao, Y.

Zhou, B.

Zuev, V. E.

V. E. Zuev, I. E. Naats, Inverse Problems of Lidar Sensing of the Atmosphere (Springer-Verlag, Berlin, 1983).

Appl Opt. (1)

R. Murty, “Refractive turbulence effects on truncated Gaussian beam heterodyne lidar,” Appl Opt. 23, 2498–2502 (1984).
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Appl. Opt. (38)

R. M. Hardesty, “Coherent DIAL measurement of range-resolved water vapor concentration,” Appl. Opt. 23, 2545–2553 (1984).
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J. Y. Wang, “Detection efficiency of coherent optical radar,” Appl. Opt. 23, 3421–3427 (1984).
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J. Y. Wang, “Lidar signal fluctuations caused by beam translation and scan,” Appl. Opt. 25, 2878–2885 (1986).
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J. Y. Wang, “Optimum truncation of a lidar transmitted beam,” Appl. Opt. 27, 4470–4474 (1988).
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Y. Zhao, M. J. Post, R. M. Hardesty, “Receiving efficiency of 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 pulsed coherent lidars. 2: Applications,” Appl. Opt. 29, 4120–4132 (1990).
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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).
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S. F. Clifford, S. Wandzura, “Monostatic heterodyne lidar performance: the effect of the turbulent atmosphere,” Appl. Opt. 20, 514–516 (1981);Appl. Opt. 20, 1502(E) (1981).
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R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, “Characteristics of coherent lidar returns from calibration targets and aerosols,” Appl. Opt. 20, 3763–3769 (1981).
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J. Y. Wang, “Heterodyne laser radar SNR from a diffuse target containing multiple glints,” Appl. Opt. 21, 464–475 (1982).
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B. J. Rye, “Primary aberration contribution to incoherent backscatter heterodyne lidar returns,” Appl. Opt. 21, 839–844 (1982).
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J. Salzman, A. Katzir, “Signal-to-noise ratio of heterodyne detection: matrix formalism,” Appl. Opt. 22, 888–890 (1983).
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J. Salzman, A. Katzir, “Heterodyne detection SNR: calculations with matrix formalism,” Appl. Opt. 23, 1066–1074 (1984).
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L. Lading, S. Hanson, A. S. Jensen, “Diffraction-limited lidars: the impact of refractive turbulence,” Appl. Opt. 23, 2492–2497 (1984).
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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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R. M. Huffaker, T. R. Lawrence, M. J. Post, J. T. Priestley, F. F. Hall, R. A. Richter, R. J. Keeler, “Feasibility studies for a global wind measuring satellite system (Windsat): analysis of simulated performance,” Appl. Opt. 23, 2523–2536 (1984).
[CrossRef] [PubMed]

H. T. Yura, “Optical heterodyne signal power obtained from finite sized sources of radiation,” Appl. Opt. 13, 150–157 (1974).
[CrossRef] [PubMed]

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

R. L. Schwiesow, R. F. Calfee, “Atmospheric refractive effects on coherent lidar performance at 10.6 μm,” Appl. Opt. 18, 3911–3917 (1979).
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R. L. Schwiesow, R. E. Cupp, “Calibration of a cw infrared Doppler lidar,” Appl. Opt. 19, 3168–3172 (1980).
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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).
[CrossRef] [PubMed]

J. H. Shapiro, “Precise comparison of experimental and theoretical SNRs in CO2 laser heterodyne systems: comments,” Appl. Opt. 24, 1245–1247 (1985).
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J. J. Degnan, B. J. Klein, “Optical antenna gain 2: Receiving antennas,” Appl. Opt. 13, 2397–2401 (1974).
[CrossRef] [PubMed]

D. Fink, “Coherent detection signal-to-noise,” Appl. Opt. 14, 689–690 (1975).
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S. C. Cohen, “Heterodyne detection: phase front alignment, beam spot size, and detector uniformity,” Appl. Opt. 14, 1953–1959 (1975).
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D. Fink, S. N. Vodopia, “Coherent detection SNR of an array of detectors” Appl. Opt. 15, 453–454 (1976).
[CrossRef] [PubMed]

T. Takenaka, K. Tanaka, O. Fukumitsu, “Signal-to-noise ratio in optical heterodyne detection for Gaussian fields,” Appl. Opt. 17, 3466–3471 (1978).
[CrossRef] [PubMed]

N. Saga, K. Tanaka, O. Fukumitsu, “Diffraction of a Gaussian beam through a finite aperture lens and the resulting heterodyne efficiency,” Appl. Opt. 20, 2827–2831 (1981).
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K. Tanaka, N. Saga, “Maximum heterodyne efficiency of optical heterodyne detection in the presence of background radiation,” Appl. Opt. 23, 3901–3904 (1984).
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K. Tanaka, N. Ohta, “Effects of tilt and offset of signal field on heterodyne efficiency,” Appl. Opt. 26, 627–632 (1987).
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M. J. Kavaya, R. T. Menzies, D. A. Haner, U. P. Oppenheim, P. H. Flamant, “Target reflectance measurements for calibration of lidar atmospheric backscatter data,” Appl. Opt. 22, 2619–2628 (1983).
[CrossRef] [PubMed]

M. J. Kavaya, R. T. Menzies, “Lidar aerosol backscatter measurements: systematic, modeling, and calibration error considerations,” Appl. Opt. 24, 3444–3453 (1985).
[CrossRef] [PubMed]

M. J. Kavaya, “Polarization effects on hard target calibration of lidar systems,” Appl. Opt. 26, 796–804 (1987).
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K. M. van Vliet, “Noise limitations in solid state detectors,” Appl. Opt. 6, 1145–1169 (1967).
[CrossRef] [PubMed]

J. M. Hunt, J. F. Holmes, F. Amzajerdian, “Optimum local oscillator levels for coherent detection using photoconductors,” Appl. Opt. 27, 3135–3141 (1988).
[CrossRef] [PubMed]

L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970).
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D. U. Fluckiger, R. J. Keyes, J. H. Shapiro, “Optical autodyne detection: theory and experiment,” Appl. Opt. 26, 318–325 (1987).
[CrossRef] [PubMed]

T. J. Kane, B. Zhou, R. L. Byer, “Potential for coherent Doppler wind velocity lidar using neodymium lasers,” Appl. Opt. 23, 2477–2481 (1984).
[CrossRef] [PubMed]

J. Math. Phys. (2)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment-equations and path-integral techniques for wave propagation in random media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (3)

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)

C. J. Leader, “Speckle effects on coherent laser radar detection efficiency,” Opt. Eng. 25, 644–649 (1986).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (4)

S. M. Flatté, “Wave propagation through random media: contributions from ocean acoustics,” Proc. IEEE 71, 1267–1294 (1983).
[CrossRef]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965)
[CrossRef]

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 51, 1350–1358 (1966).
[CrossRef]

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–66 (1967).
[CrossRef]

Radio Sci. (3)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21, 929–948 (1986).
[CrossRef]

J. L. Codona, R. G. Frehlich, “Scintillation from extended incoherent sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

R. G. Frehlich, “Space-time fourth moment of waves propagating in random media,” Radio Sci. 22, 481–490 (1987).
[CrossRef]

Remote Sensing Rev. (1)

T. Kobayashi, “Techniques for laser remote sensing of the environment,” Remote Sensing Rev. 3, 1–56 (1987).
[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]

Science (1)

D. K. Killinger, N. Menyuk, “Laser remote sensing of the atmosphere,” Science 235, 37–45 (1987).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (2)

V. U. Zavorotnyi, V. I. Klyatskin, V. I. Tatarskii, “Strong fluctuations of the intensity of electromagnetic waves in randomly inhomogeneous media,” Zh. Eksp. Teor. Fiz. 73, 481–497 (1977)[Sov. Phys. JETP 46, 252–260 (1977)].

V. U. Zavorotnyi, “Strong fluctuations of electromagnetic waves in a random medium with finite longitudinal correlation of the inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978)[Sov. Phys. JETP 48, 27–31 (1978)].

Other (23)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).

R. L. Fante, “Wave propagation in random media: A systems approach,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1985), Vol. 22, pp. 341–398.
[CrossRef]

V. I. Tatarskii, V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. 28.
[CrossRef]

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

J. L. Meyzonnette, B. Remy, G. Saccomani, “Imaging CO2 laser radar with chirp pulse compression,” in Laser Radar II, R. J. Becherer, R. C. Harney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.783, 169–179 (1987).

M. Ross, Laser Receivers. Devices, Techniques, Systems (Wiley, New York, 1966).

M. Ross, ed., Laser Applications (Academic, New York, 1971, 1974), Vols. 1 and 2.

B. M. Watrasiewicz, M. J. Rudd, Laser Doppler Measurements (Butterworth, London, 1976).

C. G. Bachman, Laser Radar Systems and Techniques (Artech House, Dedham, Mass., 1979).

L. E. Drain, The Laser Doppler Technique (Wiley-Interscience, Chichester, England, 1980).

D. K. Killinger, A. Mooradian, eds., Optical and Laser Remote Sensing (Springer-Verlag, Berlin, 1983).

V. E. Zuev, I. E. Naats, Inverse Problems of Lidar Sensing of the Atmosphere (Springer-Verlag, Berlin, 1983).

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). A list of unpublished errata for this paper can be obtained from the present authors.
[CrossRef] [PubMed]

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth and Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1062, 234–248 (1989).

R. T. Menzies, “Laser heterodyne detection techniques,” Chap. 7 in Ref. 1.

R. T. H. Collis, P. B. Russell, “Lidar measurements of particles and gases by elastic backscattering and differential absorption,” in Laser Monitoring of the Atmosphere, E. D. Hinkley, ed. (Springer-Verlag, Berlin, 1976), Chap. 4.
[CrossRef]

R. M. Measures, Laser Remote Sensing. Fundamentals and Applications (Interscience, New York, 1984).

W. B. Grant, “Laser Remote Sensing Techniques,” in Laser Spectroscopy and Its Applications, L. J. Radziemski, R. W. Solarz, J. A. Paisner, eds. (Dekker, New York, 1987), Chap. 8.

R. M. Measures, “Fundamentals of Laser Remote Sensing,” in Laser Remote Chemical Analysis, R. M. Measures, ed. (Wiley, New York, 1988), Chap. 1.

C. A. DiMarzio, C. S. Lins, “Heterodyne SNR computations using orthogonal functions,” in Fifth Conference on Coherent Laser Radar: Technology and Applications, J. W. Bilbro, C. Werner, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1181, 176–185 (1989).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Geometry for the coherent detection laser radar system. An actual system has an overlap of the transmitted and back propagated local oscillator beams at the target.

Fig. 2
Fig. 2

Depiction of coherent detection signal and power for a Gaussian pulse incident on a rigid hard target: (a) Signal current with no noise and normalized by the average dc 〈 Idc 〉. The average backscattered signal current (direct detection current) 〈Is(t)〉 is indicated by (––––). The i.f. signal is the oscillating component, (b)Intermediate frequency power with no noise (the square of the i.f. signal). The average laser radar power 〈iS2(t)〉 is indicated by (––––). (c) Same as (a) but with noise. (d) Same as (b) but with noise. The SNR is 20.0 at time 0.5.

Fig. 3
Fig. 3

Heterodyne efficiency η H and the SNR as a function of range R by using the statistically independent-path calculation for a monostatic laser radar system [see Eqs. (92), (189), and (190)] at a wavelength of 1.064-1 μm and focused at 1 km (FTE = FRE = F= 1 km). The system parameters are σ R = 10.0 cm, σl = σL0 = 7.07 cm, FLO = ∞, UT = 5 mJ, β(R) = β = 4 × 10−6m−1 sr−1, K(R) = 1.0, η Q = 0.5, and B = 50 MHz. The level of refractive turbulence Cn2 has the values Cn2 = 0(—),Cn2=10−14 m−2/3(⋯),Cn2 = 10−13 m−2/3 (----),Cn2 = 10−12 m−2/3(·-·-).

Fig. 4
Fig. 4

Heterodyne efficiency η H and the SNR as a function of range R for a collimated Gaussian monostatic laser radar system at a wavelength of 1.064 μm The system parameters are FTE = FRE = ∞, σ R = 10.0 cm, σl = σL0 = 7.07 cm, FLO = ∞, UT = 5 mJ, β = 4 × 10−6m−1 sr−1, K(R) = 1.0, η Q = 0.5, and B = 50 MHz. The level of refractive turbulence Cn2 has the values Cn2 = 0(—),Cn2=10−15 m−2/3, lf calculation [see Eqs. (84), (92), (167), and (200)] (⋯), lf plus hf calculation [see Eqs. (84), (92), (167), (169), (170), (189),(209), and (214)] (----); and Cn2 = 10−13 m−2/3, lf calculation (·-·-), lf plus hf calculation(··-··-).

Tables (2)

Tables Icon

Table I Assumptions for Calculations

Tables Icon

Table II Diversification of Coherent Laser Radar Theory

Equations (221)

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Ψ T ( u , z , t ) = E T ( u , z , t ) exp ( ikz i ω t ) ,
| Ψ T ( u , z , t ) | 2 d u = | E T ( u , z , t ) | 2 d u = P T ( t z / c ) ,
Ψ D ( w , L , t ) = E S ( w , L , t ) exp [ i ( k L ω t + θ S ) ] + E LO ( w , L , t ) exp [ i ( k L ω LO t ) ] ,
I ( t ) = G D e h ν D η Q ( w ) | Ψ D ( w , L , t ) | 2 d w ,
I ( t ) = I dc ( t ) + I S ( t ) + i S ( t ) ,
I dc ( t ) = G D e h ν D η Q ( w ) | E LO ( w , L , t ) | 2 d w = G D e h ν P LO D
I S ( t ) = G D e h ν D η Q ( w ) | E S ( w , L , t ) | 2 d w = G D e h ν P D ( t )
i s ( t ) = 2 G D e h ν Re D η Q ( w ) E S ( w , L , t ) E LO * ( w , L , t ) exp ( i Δ ω t + i θ S ) d w
P LO D ( t ) = D η Q ( w ) | E LO ( w , L , t ) | 2 d w
P D ( t ) = D η Q ( w ) | E S ( w , L , t ) | 2 d w
i S 2 ( t ) = 2 ( G D e h ν ) 2 D D η Q ( w 1 ) η Q ( w 2 ) M S ( w 1 , w 2 , L , t ) × M LO * ( w 1 , w 2 , L ) d w 1 d w 2 ,
M S ( w 1 , w 2 , L , t ) = E S ( w 1 , L , t ) E S * ( w 2 , L , t )
i N 2 = 2 G D e B I dc ,
SNR ( t ) = i S 2 ( t ) i N 2 = 1 h ν B P LO D D D η Q ( w 1 ) η Q ( w 2 ) M S ( w 1 , w 2 , L , t ) × M LO * ( w 1 , w 2 , L ) d w 1 d w 2 ,
η H ( t ) = i S ( t ) 2 I dc I S ( t ) = D D η Q ( w 1 ) η Q ( w 2 ) M S ( w 1 , w 2 , L , t ) M LO * ( w 1 , w 2 , L ) d w 1 d w 2 P D ( t ) P LO D .
I ( t ) = I dc + I S ( t )
SNR ( t ) = P D ( t ) h ν B η H ( t ) .
E S ( v , 0 , t ) = E S ( v , 0 , t ) W R ( v ) ,
E S ( w , L , t ) = E S ( v , 0 , t ) G f ( w ; v , L ) d v ,
G f ( w ; v , L ) = k 2 π i L exp [ i k 2 L ( w v ) 2 ]
i S 2 ( t ) = 2 [ G D e h ν ] 2 M S ( v 1 , v 2 , 0 , t ) M BPLO ( v 1 , v 2 , 0 ) d v 1 d v 2 ,
E BPLO ( v , 0 ) = W R ( v ) k 2 L 2 E LO * ( w , 0 ) Y [ ( v w ) k / L ] × exp [ i k 2 L ( υ 2 w 2 ) ] d w
Y ( κ ) = 1 ( 2 π ) 2 η Q ( w ) exp ( i κ w ) d w
SNR ( t ) = 1 h ν B P LO D M S ( v 1 , v 2 , 0 , t ) M BPLO ( v 1 , v 2 , 0 ) d v 1 v 2 .
E T ( p , R , t ) = E T ( u , 0 , t R / c ) G ( p ; u , R ) d u ,
E T ( u , 0 , t ) = E L ( u , 0 , t ) W T ( u ) ,
E S ( p , R , t ) = E T ( q , R , t ) V ( q , p ) d q ,
E S ( v , 0 , t ) = E T ( q , R , t R / c ) V ( q , p ) G ( p ; v , R ) d q d p
M S ( v 1 , v 2 , 0 , t ) = [ K ( R ) ] 2 E S ( v 1 , 0 , t ) E S * ( v 2 , 0 , t ) = [ K ( R ) ] 2 B ( q 1 , q 2 , p 1 , p 2 ) × E T ( q 1 , R , t R / c ) E T * ( q 2 , R , t R / c ) × G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) d q 1 d q 2 d p 1 d p 2 ,
B ( q 1 , q 2 , p 1 , p 2 ) = V ( q 1 , p 1 ) V * ( q 2 , p 2 )
K ( R ) = exp [ 0 R α ( z ) d z ]
M S ( v 1 , v 2 , 0 , t ) = [ K ( r ) ] 2 B ( q 1 , q 2 , p 1 , p 2 ) × M T ( u 1 , u 2 , 0 , t 2 R / c ) G ( q 1 ; u 1 , R ) G * ( q 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) × G * ( p 2 ; v 2 , R ) d q 1 d q 2 d p 1 d p 2 d u 1 d u 2 ,
M T ( u 1 , u 2 , 0 , t ) = E T ( u 1 , 0 , t ) E T * ( u 2 , 0 , t ) .
SNR ( t ) = [ K ( R ) ] 2 h ν B P LO D B ( q 1 , q 2 , p 1 , p 2 ) × M T ( u 1 , u 2 , 0 , t 2 R / c ) M BPLO ( v 1 , v 2 , 0 ) G ( q 1 ; u 1 , R ) × G * ( q 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) × d q 1 d q 2 d p 1 d p 2 d u 1 d u 2 d v 1 d v 2 ,
SNR ( t ) = [ K ( R ) ] 2 h ν B P LO D B ( q 1 , q 2 , p 1 , p 2 ) × E r ( q 1 , R , t R / c ) E T * ( q 2 , R , t R / c ) × E BPLO ( p 1 , R ) E BPLO * ( p 2 , R ) d q 1 d q 2 d p 1 d p 2
SNR ( t ) = [ K ( R ) ] 2 h ν B P LO D | V ( q 1 p 1 ) × E T ( q 1 , R , t R / c ) E BPLO ( p 1 , R ) d q 1 d p 1 | 2 .
V ( q 1 , p 1 ) = λ σ S 1 / 2 exp ( i θ s ) δ ( q 1 p ) δ ( p 1 p ) ,
B ( q 1 , q 2 , p 1 , p 2 ) = λ 2 σ S δ ( p 1 p ) δ ( p 2 p ) δ ( q 1 p ) δ ( q 2 p ) ,
M S ( v 1 , v 2 , 0 , t ) = λ 2 σ S [ K ( R ) ] 2 M T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p ; u 1 , R ) G * ( p ; u 2 , R ) G ( p ; v 1 , R ) G * ( p ; v 2 , R ) d u 1 d u 2 ,
SNR ( p , t ) = λ 2 σ S [ K ( R ) ] 2 h ν B P LO D J T ( p , R , t 2 R / c ) J BPLO ( p , R ) ,
J T ( p , R , t ) = | E T ( p , R , t ) | 2 , J BPLO ( p , R ) = | E BPLO ( p , R ) | 2
V ( q , p ) = r ( p ) δ ( q p ) exp ( 2 i k θ p p ) ,
B ( q 1 , q 2 , p 1 , p 2 ) = r ( p 1 ) r * ( p 2 ) exp [ 2 i k θ p ( p 1 p 2 ) ] δ ( q 1 p 1 ) δ ( q 2 p 2 ) ,
M S ( v 1 , v 2 , 0 , t ) = [ K ( R ) ] 2 r ( p 1 ) r * ( p 2 ) × exp [ 2 i k θ p ( p 1 p 2 ) ] M T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) × G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) d p 1 d p 2 d u 1 d u 2 ,
SNR ( t ) = [ K ( R ) ] 2 h ν B P LO D r ( p 1 ) r * ( p 2 ) exp [ 2 i k θ p ( p 1 p 2 ) ] × E T ( p 1 , R , t R / c ) E T * ( p 2 , R , t R / c ) × E BPLO ( p 1 , R ) E BPLO * ( p 2 , R ) d p 1 d p 2
B ( q 1 , q 2 , p 1 , p 2 ) = λ 2 ρ ( p 1 ) δ ( p 1 p 2 ) δ ( q 1 p 1 ) δ ( q 2 p 2 ) ,
M S ( v 1 , v 2 , 0 , t ) = λ 2 [ K ( R ) ] 2 ρ ( p ) × M T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p ; u 1 , R ) G * ( p ; u 2 , R ) G ( p ; v 1 , R ) × G * ( p ; v 2 , R ) d u 1 d u 2 d p .
SNR ( t ) = λ 2 [ K ( R ) ] 2 h ν B P LO D ρ ( p ) J T ( p , R , t 2 R / c ) J BPLO ( p , R ) d p
M S ( v , v , 0 , t ) = ρ R 2 [ K ( R ) ] 2 P T ( t 2 R / c ) .
V ( q , p ) = r ( p ) δ ( q + p ) ,
B ( q 1 , q 2 , p 1 , p 2 ) = r ( p 1 ) r * ( p 2 ) δ ( q 1 + p 1 ) δ ( q 2 + p 2 ) ,
M S ( v 1 , v 2 , 0 , t ) = [ K ( R ) ] 2 r ( p 1 ) r * ( p 2 ) × M T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) × G * ( p 2 ; v 2 , R ) d p 1 d p 2 d u 1 d u 2 ,
SNR ( t ) = [ K ( R ) ] 2 h ν B P LO D r ( p 1 ) r * ( p 2 ) × E T ( p 1 , R , t R / c ) E T * ( p 2 , R , t R / c ) × E BPLO ( p 1 , R ) E BPLO * ( p 2 , R ) d p 1 d p 2
M S ( v 1 , v 2 , 0 , t ) = λ 2 β ( p , R ) [ K ( R ) ] 2 × M T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p ; u 1 , R ) G * ( p ; u 2 , R ) G ( p ; v 1 , R ) × G * ( p ; v 2 , R ) d u 1 d u 2 d p d R ,
β ( p , R ) = σ S N ( σ S ; p , R ) d σ S
M S ( v , v , 0 , t ) = β ( R ) R 2 [ K ( R ) ] 2 P T ( t 2 R / c ) d R .
SNR ( t ) = λ 2 h ν B P LO D β ( p , R ) [ K ( R ) ] 2 × J T ( p , R , t 2 R / c ) J BPLO ( p , R ) d p d R ,
c 0 2 | E T ( x ) | 2 d x = P T ,
r ( t ) = h ν ( 2 P LO D ) 1 / 2 G D e η Q i S ( t ) ,
E L ( u , z , t ) = [ P L ( t ) ] 1 / 2 e L ( u , z , t ) ,
E LO ( v , z ) = [ P LO ] 1 / 2 e LO ( v , z ) ,
Y ( κ ) = η Q δ ( κ ) .
E BPLO ( v , 0 ) = η Q E LO * ( v , 0 ) W R ( v ) ,
P LO D = η Q P LO .
e T ( u , 0 , t ) = e L ( u , 0 , t ) W T ( u ) ,
e BPLO ( v , 0 ) = e LO * ( v , 0 ) W R ( v ) .
SNR ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 σ S h ν B c ( p , R , t ) ,
c ( p , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × m BPLO ( v 1 , v 2 , 0 ) G ( p ; u 1 , R ) G * ( p ; u 2 , R ) × G ( p ; v 1 , R ) G * ( p ; v 2 , R ) d u 1 d u 2 d v 1 d v 2
c ( p , R , t ) = λ 2 j T ( p , R , t R / c ) j BPLO ( p , R ) ,
j T ( p , R , t ) = | e T ( p , R , t ) | 2 , j BPLO ( p , R ) = | e BPLO ( p , R ) | 2
j T ( u , 0 , t ) d u = P T ( t ) / P L ( t ) = T T ( t ) ,
j BPLO ( v , 0 ) d v = T R ,
P D ( t ) = η Q | W R ( v ) | 2 M S ( v , v , 0 , t ) d v .
P D ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 σ S d ( p , R , t ) ,
d ( p , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) | W R ( v ) | 2 × G ( p ; u 1 , R ) G * ( p ; u 2 , R ) G ( p ; v , R ) G * ( p ; v , R ) d u 1 d u 2 d v
d ( p , R , t ) = λ 2 j T ( p , R , t R / c ) j R ( p , R ) ,
j R ( p , R ) = | W R ( v ) | 2 G ( p ; v , R ) G * ( p ; v , R ) d v
η H ( p , R , t ) = c ( p , R , t ) d ( p , R , t ) .
η H ( p , R , t ) = j T ( p , R , t R / c ) j BPLO ( p , R ) j T ( p , R , t R / c ) j R ( p , R ) .
SNR ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 ρ h ν B C ( R , t ) ,
C ( R , t ) = c ( p , R , t ) d p
C ( R , t ) = λ 2 j T ( p , R , t R / c ) j BPLO ( p , R ) d p .
P D ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 ρ D ( R , t ) ,
D ( R , t ) = d ( p , R , t ) d p = λ 2 j T ( p , R , t R / c ) j R ( p , R ) d p
η H ( R , t ) = C ( R , t ) D ( R , t ) .
η H ( R , t ) = j T ( p , R , t R / c ) j BPLO ( p , R ) d p j T ( p , R , t R / c ) j R ( p , R ) d p .
A COH ( R , t ) = R 2 C ( R , t ) = A RYE ( R , t ) T T ( t ) T R ,
A COH ( R , t ) = G A ( Φ , t ) d Φ ,
G A ( Φ , t ) = R 4 c ( Φ R , R , t ) ,
SNR ( t ) = η Q h ν B 0 P L ( t 2 R / c ) [ K ( R ) ] 2 β ( R ) C ( R , t ) d R .
P D ( t ) = η Q 0 P L ( t 2 R / c ) [ K ( R ) ] 2 β ( R ) D ( R , t ) d R .
η H ( R , t ) = P L ( t 2 R / c ) [ K ( R ) ] 2 β ( R ) C ( R , t ) d R P L ( t 2 R / c ) [ K ( R ) ] 2 β ( R ) D ( R , t ) d R .
SNR ( t ) η Q β ( R ) [ K ( R ) ] 2 c U L C ( R , t ) 2 h ν B ,
P D ( t ) η Q U L c 2 β ( R ) [ K ( R ) ] 2 D ( R , t ) ,
SNR ( t ) η Q P L ( t 2 R / c ) [ K ( R ) ] 2 | r | 2 h ν B C ( R , t ) ,
C ( R , t ) = exp [ 2 i k θ p ( p 1 p 2 ) ] c J ( p 1 , p 2 , R , t ) d p 1 d p 2 ,
c J ( p 1 , p 2 , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t ) × m BPLO ( v 1 , v 2 , 0 ) × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) × G * ( p 2 ; v 2 , R ) d u 1 d u 2 d v 1 d v 2
c J ( p 1 , p 2 , R , t ) = λ 2 e T ( p 1 , R , t R / c ) × e T * ( p 2 , R , t R / c ) e BPLO ( p 1 , R ) e BPLO * ( p 2 , R ) .
P D ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 | r | 2 × exp [ 2 i k θ p ( p 1 p 2 ) ] d J ( p 1 , p 2 , R , t ) d p 1 d p 2 ,
D ( R , t ) = d J ( p 1 , p 2 , R , t ) d p 1 d p 2 ,
d J ( p 1 , p 2 , R , t ) = λ 2 m t ( u 1 , u 2 , 0 , t ) | W R ( v ) | 2 × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) G ( p 1 ; v , R ) G * ( p 2 ; v , R ) d u 1 d u 2 d v
c J ( p 1 , p 2 , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × m BPLO ( v 1 , v 2 , 0 ) × G ( p 1 ; u 1 , R , ) G * ( p 2 ; u 2 , R , ) G ( p 1 ; v 1 , R ) × G * ( p 2 ; v 2 , R , ) d u 1 d u 2 d v 1 d v 2
c J ( p 1 , p 2 , R , t ) = λ 2 e T ( p 1 , R , t R / c ) × e T * ( p 2 , R , t R / c ) e BPLO ( p 1 , R ) e BPLO * ( p 2 , R ) .
d J ( p 1 , p 2 , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × | W R ( v ) | 2 G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) × G ( p 1 ; v , R ) G * ( p 2 ; v , R ) d u 1 d u 2 d v
SNR ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 h ν B ρ ( p ) c ( p , R , t ) d p .
P D ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 ρ ( p ) d ( p , R , t ) d p .
η H ( R , t ) = ρ ( p ) c ( p , R , t ) d p ρ ( p ) d ( p , R , t ) d p ,
SNR ( t ) = η Q h ν B 0 P L ( t 2 R / c ) [ K ( R ) ] 2 × β ( p , R ) c ( p , R , t ) d R d p .
P D ( t ) = η Q 0 P L ( t 2 R / c ) [ K ( R ) ] 2 β ( p , R ) d ( p , R , t ) d R d p .
η H ( R , t ) = 0 P L ( t 2 R / c ) [ K ( R ) ] 2 β ( p , R ) c ( p , R , t ) d R d p 0 P L ( t 2 R / c ) [ K ( R ) ] 2 β ( p , R ) d ( p , R , t ) d R d p ,
SNR ( t ) = η Q U L c [ K ( R ) ] 2 2 h ν B β ( p , R ) c ( p , R , t ) d p ,
P D ( t ) = η Q U L c 2 [ K ( R ) ] 2 β ( p , R ) d ( p , R , t ) d p .
SNR ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 h ν B r ( p 1 ) r * ( p 2 ) × exp [ 2 i k θ p ( p 1 p 2 ) ] c J ( p 1 , p 2 , R , t ) d p 1 d p 2 .
P D ( t ) = η Q P L ( t 2 R / c ) [ K ( R ) ] 2 × r ( p 1 ) r * ( p 2 ) × exp [ 2 i k θ p ( p 1 p 2 ) ] d J ( p 1 , p 2 , R , t ) d p 1 d p 2 .
η H ( R , t ) = r ( p 1 ) r * ( p 2 ) exp [ 2 i k θ p ( p 1 p 2 ) ] c J ( p 1 , p 2 , R , t ) d p 1 d p 2 r ( p 1 ) r * ( p 2 ) exp [ 2 i k θ p ( p 1 p 2 ) ] d J ( p 1 , p 2 , R , t ) d p 1 d p 2 ,
c ( p , R , t ) = λ 2 e T ( u 1 , 0 , t 2 R / c ) × e T * ( u 2 , 0 , t 2 R / c ) e BPLO ( v 1 , 0 ) e BPLO * ( v 2 , 0 ) × G f ( p ; u 1 , R ) G f * ( p ; u 2 , R ) G f ( p ; v 1 , R ) × G f * ( p ; v 2 , R ) d u 1 d u 2 d v 1 d v 2 .
D ( R , t ) = T T ( t ) Ω ( R ) ,
Ω ( R ) = A R / R 2 = λ 2 j R ( p , R )
A R = | W R ( v ) | 2 d v .
C ( R , t ) = Ω ( R ) T T ( t ) η H ( R , t ) = Ω ( R ) η S ( R , t ) ,
η S ( R , t ) = T T ( t ) η H ( R , t ) = A COH ( R , t ) A R = C ( R , t ) Ω ( R )
η H ( R , t ) = R 2 λ 2 A R T T ( t ) j T ( p , R , t R / c ) j BPLO ( p , R ) d p .
A COH ( R , t ) = O T ( s , R , t ) O BPLO * ( s , R ) d s ,
O T ( s , R , t ) = e T ( u , R , t ) e T * ( u s , R , t ) d u
e T ( u , R , t ) = e T ( u , 0 , t ) exp ( i k 2 R u 2 ) ,
O BPLO ( s , R ) = e BPLO ( v , R ) e BPLO * ( v , s , R ) d v
e BPLO ( v , R ) = e BPLO ( v , 0 ) exp ( i k 2 R υ 2 ) .
P LOD = k 2 L 2 P LO m LO ( v 1 , v 2 , 0 ) Y [ ( v 1 v 2 ) k / L ] × exp [ i k 2 L ( υ 1 2 υ 2 2 ) ] d v 1 d v 2
e BPLO ( v , 0 ) = k W R ( v ) e L O * ( w , 0 ) Y [ ( v w ) k / L ] exp [ i k 2 L ( υ 2 w 2 ) ] d w L [ m LO ( v 1 , v 2 , 0 ) Y [ ( v 1 v 2 ) k / L ] exp [ i k 2 L ( υ 1 2 υ 2 2 ) ] d v 1 d v 2 ] 1 / 2
η Q ( w ) = 1 on the detector surface , = 0 off the detector surface
d ( p , R , t ) = λ 2 k 2 L 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × W R ( v 1 ) W R * ( v 2 ) Y [ ( v 1 v 2 ) k / L ] × exp [ i k 2 L ( υ 1 2 υ 2 2 ) ] G ( p ; u 1 , R ) G * ( p ; u 2 , R ) × G ( p ; v 1 , R ) G * ( p ; v 2 , R ) d u 1 d u 2 d v 1 d v 2
d J ( p 1 , p 2 , R , t ) = λ 2 k 2 L 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × W R ( v 1 ) W R * ( v 2 ) Y [ ( v 1 v 2 ) k / L ] × exp [ i k 2 L ( υ 1 2 υ 2 2 ) ] G ( p 1 ; u 1 , R ) × G * ( p 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) × G * ( p 2 ; v 2 , R ) d u 1 d u 2 d v 1 d v 2
d J ( p 1 , p 2 , R , t ) = λ 2 k 2 L 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × W R ( v 1 ) W R * ( v 2 ) Y [ ( v 1 v 2 ) k / L ] × exp [ i k 2 L ( υ 1 2 υ 2 2 ) ] G ( p 1 ; u 1 , R ) × G * ( p 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) × G * ( p 2 ; v 2 , R ) d u 1 d u 2 d v 1 d v 2
c ( p , R , t ) = m S D ( p , v 1 , v 2 , 0 , t ) m BPLO ( v 1 , v 2 , 0 ) d v 1 d v 2 ,
m SD ( p , v 1 , v 2 , 0 , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p ; u 1 , R ) G * ( p ; u 2 , R ) G ( p ; v 1 , R ) G * ( p ; v 2 , R ) d u 1 d u 2
C ( R , t ) = m S ( v 1 , v 2 , 0 , t ) m BPLO ( v 1 , v 2 , 0 ) d v 1 d v 2 ,
m S ( v 1 , v 2 , 0 , t ) = m S D ( p , v 1 , v 2 , 0 , t ) d p
c J ( p 1 , p 2 , R , t ) = m S J ( p 1 , p 2 , v 1 , v 2 , 0 , t ) × m BPLO ( v 1 , v 2 , 0 ) d v 1 d v 2 ,
m S J ( p 1 , p 2 , v 1 , v 2 , 0 , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) × G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) d u 1 d u 2
m S J ( p 1 , p 2 , v 1 , v 2 , 0 , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) × G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) d u 1 d u 2 .
c 0 lf ( p , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × m BPLO ( v 1 , v 2 , 0 ) G ( p ; u 1 , R ) G * ( p ; u 2 , R ) × G ( p ; v 1 , R ) G * ( p ; v 2 , R ) d u 1 d u 2 d v 1 d v 2 ,
c 0 lf ( p , R , t ) = λ 2 j T ( p , R , t R / c ) j BPLO ( p , R ) .
d 0 lf ( p , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) | W R ( v ) | 2 × G ( p ; u 1 , R ) G * ( p ; u 2 , R ) × G ( p ; v , R ) G * ( p ; v , R ) d u 1 d u 2 d v 1 ,
d 0 lf ( p , R , t ) = λ 2 j T ( p , R , t R / c ) j R ( p , R ) .
c J 0 lf ( p 1 , p 2 , R , t ) = λ 2 × m T ( u 1 , u 2 , 0 , t 2 R / c ) m BPLO ( v 1 , v 2 , 0 ) × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) × G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) × d u 1 d u 2 d v 1 d v 2 ,
c J 0 lf ( p 1 , p 2 , R , t ) = λ 2 e T ( p 1 , R , t R / c ) e T * ( p 2 , R , t R / c ) × e BPLO ( p 1 , R ) e BPLO * ( p 2 , R ) .
d J 0 lf ( p 1 , p 2 , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) | W R ( v ) | 2 × G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) × G ( p 1 ; v , R ) G * ( p 2 ; v , R ) d u 1 d u 2 d v .
e T ( u , 0 , t ) = e BPLO ( u , 0 ) , j T ( p , R , t ) = j BPLO ( p , R ) ,
c ( p , R ) = λ 2 j T ( p , R ) 2 [ 1 + σ I 2 ( p , R ) ] ,
c 0 hf ( p , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × m BPLO ( v 1 , v 2 , 0 ) G ( p ; u 1 , R ) G * ( p ; v 2 , R ) × G ( p ; v 1 , R ) G * ( p ; u 2 , R ) d u 1 d u 2 d v 1 d v 2 ,
c 0 hf ( p , R ) = λ 2 | e T ( p , R , t R / c ) e BPLO ( p , R ) | 2 .
d 0 hf ( p , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t R / c ) | W R ( v ) | 2 × G ( p ; u 1 , R ) G * ( p ; v , R ) × G ( p ; v , R ) G * ( p ; u 2 , R ) d u 1 d u 2 d v .
c J 0 hf ( p 1 , p 2 , R , t ) = λ 2 × m T ( u 1 , u 2 , 0 , t 2 R / c ) m BPLO ( v 1 , v 2 , 0 ) × G ( p 1 ; u 1 , R ) G * ( p 2 ; v 2 , R ) × G ( p 1 ; v 1 , R ) G * ( p 2 ; u 2 , R ) d u 1 d u 2 d v 1 d v 2 ,
c J 0 hf ( p 1 , p 2 , R , t ) = λ 2 | e T ( p 1 , R , t R / c ) e BPLO ( p 2 , R ) | 2 .
d J 0 hf ( p 1 , p 2 , R , t ) = λ 2 m T ( u 1 , u 2 , 0 , t 2 R / c ) × | W R ( v ) | 2 G ( p 1 ; u 1 , R ) G * ( p 2 ; v , R ) × G ( p 1 ; v , R ) G * ( p 2 ; u 2 , R ) d u 1 d u 2 d v .
G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) = G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) G ( p 1 ; v 1 , R ) G * ( p 2 ; v 2 , R ) + G ( p 1 ; u 1 , R ) G * ( p 2 ; v 2 , R ) G ( p 1 ; v 1 , R ) G * ( p 2 ; u 2 , R ) .
c ( p , R , t ) = c 0 lf ( p , R , t ) + c 0 hf ( p , R , t ) ,
d ( p , R , t ) = d 0 lf ( p , R , t ) + d 0 hf ( p , R , t ) ,
c J ( p 1 , p 2 , R , t ) = c J 0 lf ( p 1 , p 2 , R , t ) + c J 0 hf ( p 1 , p 2 , R , t ) ,
d J ( p 1 , p 2 , R , t ) = d J 0 lf ( p 1 , p 2 , R , t ) + d J 0 hf ( p 1 , p 2 , R , t ) .
G ( p 1 ; u 1 , R ) G * ( p 2 ; u 2 , R ) = k 2 ( 2 π R ) 2 exp { i k 2 R [ ( p 1 u 1 ) 2 ( p 2 u 2 ) 2 ] } × exp { 1 2 0 R D [ ( u 1 u 2 ) ( 1 z / R ) + ( p 1 p 2 ) z / R , z ] d z } ,
D [ x , z ] = 4 π k 2 [ 1 cos ( κ x ) ] Φ n ( κ , κ z = 0 , z ) d κ
Φ n ( κ , κ z , z ) = 1 ( 2 π ) 3 B n ( r , s , z ) exp ( i κ r i κ z s ) d r d s ,
B n ( r , s , z ) = n ( p , z ) n ( p + r , z + s )
0 R D [ ( u 1 u 2 ) ( 1 z / R ) , z ] d z = [ | u 1 u 2 | ρ 0 ( R ) ] 5 / 3 ,
ρ 0 ( R ) = [ H k 2 0 R C n 2 ( z ) ( 1 z / R ) 5 / 3 d z ] 3 / 5
d 0 lf ( p , R , t ) = Ω ( R ) j T ( p , R , t R / c ) .
D 0 lf ( R , t ) = T T ( t ) Ω ( R ) ,
U ( R ) = R F 2 ρ 0 2 ( R ) = R S ( R ) ρ 0 ( R ) ,
C 0 ( R , t ) = C 0 lf ( R , t ) + U ( R ) 2 1 + U ( R ) 2 C 0 hf ( R , t ) ,
D 0 ( R , t ) = D 0 lf ( R , t ) + U ( R ) 2 1 + U ( R ) 2 D 0 hf ( R , t ) ,
e L ( u , 0 , t ) = ( π σ L 2 ) 1 / 2 exp ( u 2 2 σ L 2 i k u 2 2 F L ) ,
W T ( u ) = exp ( u 2 2 σ T 2 i k u 2 2 F T ) ,
e T ( u , 0 ) = ( π σ L 2 ) 1 / 2 exp ( u 2 2 σ TE 2 i k u 2 2 F TE ) ,
1 σ TE 2 = 1 σ L 2 + 1 σ T 2 ,
1 F TE = 1 F L + 1 F T ,
j T ( p , R ) = σ TE 2 π σ L 2 σ BT 2 ( R ) exp ( p 2 σ BT 2 ( R ) ) ,
σ BT 2 ( R ) = σ TE 2 ( 1 R / F TE ) 2 + R 2 k 2 σ TE 2 + 2 R 2 k 2 ρ 0 2 ( R ) .
m S 0 lf ( v 1 , v 2 , 0 ) = σ TE 2 σ L 2 R 2 exp [ i k R r s s 2 2 ρ 0 2 ( R ) s 2 σ BT 2 ( R ) k 2 4 R 2 ] ,
2 r = v 1 + v 2 , s = v 1 v 2
m ̂ ( v 1 , v 2 , 0 ) = m ( v 1 , v 2 , 0 ) [ m ( v 1 , v 1 , 0 ) m ( v 2 , v 2 , 0 ) ] 1 / 2 .
m ̂ S 0 lf ( v 1 , v 2 , 0 ) = exp [ i k R r s s 2 2 ρ ̂ 0 2 ( R ) ] ,
1 ρ ̂ 0 2 ( R ) = 1 ρ 0 2 ( R ) + σ BT 2 ( R ) k 2 2 R 2
m S 0 lf ( v , v , 0 ) = σ TE 2 σ L 2 R 2 ,
W R ( v ) = exp ( υ 2 2 σ R 2 i k υ 2 2 F R ) ,
e LO ( v , 0 ) = ( π σ LO 2 ) 1 / 2 exp ( υ 2 2 σ LO 2 i k υ 2 2 F LO ) ,
e BPLO ( v , 0 ) = ( π σ LO 2 ) 1 / 2 exp ( υ 2 2 σ RE 2 i k υ 2 2 F RE ) ,
1 σ RE 2 = 1 σ R 2 + 1 σ LO 2 ,
1 F RE = 1 F R 1 F LO .
C 0 lf ( R ) = π σ TE 2 σ RE 2 σ L 2 R 2 [ 1 4 + σ LO 2 4 σ R 2 + σ LO 2 4 σ TE 2 + ( 1 R F TE ) 2 k 2 σ TE 2 σ LO 2 4 R 2 + ( 1 R F RE ) 2 k 2 σ TE 2 σ LO 2 4 R 2 + σ LO 2 ρ 0 2 ( R ) ] 1 ,
η H 0 lf ( R ) = σ RE 2 σ R 2 [ 1 4 + σ LO 2 4 σ R 2 + σ LO 2 4 σ TE 2 + ( 1 R F TE ) 2 k 2 σ TE 2 σ LO 2 4 R 2 + ( 1 R F RE ) 2 k 2 σ RE 2 σ LO 2 4 R 2 + σ LO 2 ρ 0 2 ( R ) ] 1 ,
C 0 lf ( R ) = π σ TE 2 σ RE 2 σ L 2 σ LO 2 R 2 [ 1 4 σ RE 2 + 1 4 σ TE 2 + ( 1 R F TE ) 2 k 2 σ TE 2 4 R 2 + ( 1 R F RE ) 2 k 2 σ RE 2 4 R 2 + 1 ρ 0 2 ( R ) ] 1 .
j BPLO ( p , R ) = σ RE 2 π σ LO 2 π σ BR 2 ( R ) exp ( p 2 σ BR 2 ( R ) ) ,
σ BR 2 ( R ) = σ RE 2 ( 1 R / F RE ) 2 + R 2 k 2 σ RE 2 + 2 R 2 k 2 ρ 0 2 ( R ) ,
T T = σ TR 2 σ L 2 , T R = σ RE 2 σ LO 2 .
c 0 lf ( p , R ) = λ 2 T T T R π 2 σ BT 2 ( R ) σ BR 2 ( R ) exp ( p 2 σ BT 2 ( R ) p 2 σ BR 2 ( R ) ) .
c 0 lf ( R ) = λ 2 T T T R π [ σ BT 2 ( R ) + σ BR 2 ( R ) ] ,
C 0 lf ( R ) = D 0 lf ( R ) ,
σ LO opt = σ R [ 1 + k 2 R 2 σ R 2 σ B T 2 ( R ) + 2 σ R 2 ρ 0 2 ( R ) ] 1 / 4 ,
F LO opt = R F R R F R ,
η H ( R ) = 4 9 [ 1 + ( 1 R F ) 2 k 2 σ R 4 9 R 2 + 2 3 σ R 2 ρ 0 2 ( R ) ] 1 .
C 0 lf ( R ) = π σ LO 2 [ 1 4 + σ LO 2 4 σ TE 2 + ( 1 R F TE ) 2 k 2 σ TE 2 σ LO 2 4 R 2 + ( 1 R F RE ) 2 k 2 σ LO 4 4 R 2 + σ LO 2 ρ 0 2 ( R ) ] R 2 .
SNR = π η Q U T β c τ D 2 2 [ K ( R ) ] 2 8 h ν R 2 [ 1 + ( D 2 2 ρ 0 ) 2 + ( π D 2 2 4 λ R ) 2 ( 1 R F ) 2 ] ,
SNR = π η Q U T λ β D 2 2 [ K ( R ) ] 2 8 h B R 2 ,
m S 0 hf ( v 1 , v 2 , 0 ) = T T R 2 [ 1 + σ TE 2 / ρ 0 2 ( R ) ] × exp ( i k R { 1 ( 1 R / F TE ) / [ 1 ρ 0 2 ( R ) / σ TE 2 ] } r s s 2 σ TE 2 k 2 ( 1 R / F TE ) 2 4 R 2 [ 1 + σ TE 2 / ρ 0 2 ( R ) ] s 2 4 σ TE 2 r 2 σ TE 2 + ρ 0 2 ( R ) ) .
m S 0 hf ( v , v , 0 ) = T T R 2 [ 1 + σ TE 2 / ρ 0 2 ( R ) ] exp [ υ 2 σ TE 2 + ρ 0 2 ( R ) ] ,
m S 0 hf ( v , v , 0 ) = m S 0 lf ( v , v , 0 ) [ 1 + σ TE 2 / ρ 0 2 ( R ) ] exp [ υ 2 σ TE 2 + ρ 0 2 ( R ) ] .
m ̂ S 0 hf ( v 1 , v 2 , 0 ) = exp ( i k R r s { 1 ( 1 R / F TE ) / [ 1 + ρ 0 2 ( R ) / σ TE 2 ] } s 2 2 ρ ̂ 0 2 ( R ) ) ,
1 ρ ̂ 0 2 ( R ) = k 2 2 R 2 [ 1 + σ TE 2 / ρ 0 2 ( R ) ] [ σ TE 2 ( 1 R / F TE ) 2 + R 2 k 2 σ TE 2 ] ,
C 0 hf ( R ) = π T T T R R 2 { k 2 4 R 2 [ σ BT 2 ( R ) + σ BR 2 ( R ) ] + 1 4 ρ 0 2 ( R ) × [ σ RE 2 σ TE 2 + σ TE 2 σ RE 2 + k 2 σ TE 2 σ RE 2 ( F TE 1 F RE 1 ) 2 2 ] } 1 .
c 0 hf ( p , R ) = λ 2 T T T R π 2 σ 0 2 ( R ) σ 1 2 ( R ) exp [ p 2 / σ 0 2 ( R ) ] ,
σ 1 2 ( R ) = σ B T 2 ( R ) + σ B R 2 ( R ) + R S 2 ( R ) [ σ RE 2 σ TE 2 + σ TE 2 σ RE 2 + k 2 σ TE 2 σ RE 2 ( F TE 1 F RE 1 ) 2 2 ] ,
σ 0 2 ( R ) = R S 2 ( R ) + { R S 2 ( R ) [ σ BT 2 ( R ) + σ BR 2 ( R ) ] + σ BT 2 ( R ) σ BR 2 ( R ) } σ 1 2 .
C 0 hf ( R ) = λ 2 T T T R π σ 1 2 ( R ) ,
D 0 hf ( R ) = T T Ω ( R ) ρ 0 2 ( R ) [ ρ 0 2 ( R ) + σ TE 2 + σ R 2 ] .
D 0 hf ( R ) = T T Ω ( R ) = D 0 lf ( R ) .
D 0 hf ( R ) = D 0 lf ( R ) ρ 0 2 ( R ) σ TE 2 + σ R 2 .
SNR = η P T β ( π ) π R 2 h ν B 0 d L [ 1 + ( π R 2 λ L ) 2 ( 1 L f ) 2 ] L 2 ,
ψ = b 2 + a 2 [ 1 ( z / f ) ] 2 + ( z / k a ) 2 + ( z / k b ) 2 b 2 + a 2 [ 1 ( z / f ) ] 2 + ( z / k a ) 2 + ( z / k b ) 2 + ( 2 z / k ρ R ) 2 ,
i S ( t ) = G D e h ν D η Q ( w ) [ E S ( w , L , t ) E LO * ( w , L ) exp ( i Δ ω t + i θ S ) + E S * ( w , L , t ) E LO ( w , L ) exp ( i Δ ω t i θ S ) ] d w .
i S 2 ( t ) = ( G D e h ν ) 2 D D η Q ( w 1 ) η Q ( w 2 ) × [ E S ( w 1 , L , t ) E S * ( w 2 , L , t ) E LO * ( w 1 , L ) E LO ( w 2 , L ) + E S ( w 2 , L , t ) E S * ( w 1 , L , t ) E LO * ( w 2 , L ) E LO ( w 1 , L ) + E S ( w 1 , L , t ) E S ( w 2 , L , t ) E LO * ( w 1 , L ) E LO ( w 2 , L ) × exp ( 2 i Δ ω t + 2 i θ S ) + E S * ( w 1 , L , t ) E S * ( w 2 , L , t ) × E LO ( w 1 , L ) E LO ( w 2 , L ) exp ( 2 i Δ ω t 2 i θ S ) ] d w 1 d w 2 .

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