Abstract

The antenna and beam geometry of lidar systems employing heterodyne reception of incoherent backscatter signals are discussed. Particular emphasis is placed on systems where the target extends uniformly across the transmitted beam using topographic targets or atmospheric backscatter. The geometry is assumed to be circularly symmetrical, but otherwise arbitrary obscurations are permitted. The effects of atmospheric scintillation are neglected. Parameters are defined which characterize the system efficiency, and the conditions under which these parameters may be maximized are considered.

© 1979 Optical Society of America

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  1. This assumption is justified by the central limit theorem if there are many particles in the scattering volume and if the signal is sampled over times long compared with its coherence time. Relevant discussions in the cases where one of these assumption is not valid are given, respectively, by E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978) and by L. Mandel, Phys. Rev. 181, 75 (1969).
    [Crossref]
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).
  3. L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [Crossref]
  4. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [Crossref]
  5. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [Crossref]
  6. A. T. Forrester, R. A. Gudmundsen, P. E. Johnson, Phys. Rev. 99, 1691 (1955).
    [Crossref]
  7. H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 123 (1970).
  8. A. E. Siegman, Appl. Opt. 5, 1588 (1966).
    [Crossref] [PubMed]
  9. C. M. Sonnenschein, S. A. Horrigan, Appl. Opt. 10, 1600 (1971).
    [Crossref] [PubMed]
  10. J. J. Degnan, B. J. Klein, Appl. Opt. 13, 2397 (1974).
    [Crossref] [PubMed]
  11. B. J. Klein, J. J. Degnan, Appl. Opt. 13, 2134 (1974).
    [Crossref] [PubMed]
  12. B. J. Klein, J. J. Degnan, Appl. Opt. 15, 977 (1976).
    [Crossref] [PubMed]
  13. F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, IEEE J. Quantum Electron. QE-3, 484 (1967).
    [Crossref]
  14. A. T. Forrester, J. Opt. Soc. Am. 51, 253 (1961).
    [Crossref]
  15. All integrals in this paper are between extreme limits, i.e., −∞ and ∞, or 0 and ∞ as appropriate, except for the double intergrals which have been characterized by the area of integration and where otherwise specified.
  16. B. J. Rye, “Basic Principles of Signal Acquisition in DAS Systems Employing Incoherent Backscatter,” Hull University, Department of Applied Physics, Internal Report (January1978).
  17. E. Jakeman, C. J. Oliver, E. R. Pike, Adv. Phys. 24, 349 (1975).
    [Crossref]
  18. F. Zernike, Physica V8, 785 (1938).
    [Crossref]
  19. See e.g., R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
  20. R. L. Byer, Opt. Quantum Electron. 7, 147 (1975).
    [Crossref]
  21. A. T. Forrester, Am. J. Phys. 24, 192 (1956).
    [Crossref]
  22. J. Harms, W. Lakmann, C. Weitkamp, Appl. Opt. 17, 1131 (1978).
    [Crossref] [PubMed]
  23. D. Fink, Appl. Opt. 14, 689 (1975).
    [Crossref] [PubMed]
  24. O. O. Andrade, B. J. Rye, J. Phys. D 7, 280 (1974).
    [Crossref]
  25. L. Mandel, E. Wolf, J. Opt. Soc. Am. 65, 413 (1975).
    [Crossref]
  26. S. C. Cohen, Appl. Opt. 14, 1953 (1975).
    [Crossref] [PubMed]

1978 (2)

This assumption is justified by the central limit theorem if there are many particles in the scattering volume and if the signal is sampled over times long compared with its coherence time. Relevant discussions in the cases where one of these assumption is not valid are given, respectively, by E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978) and by L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

J. Harms, W. Lakmann, C. Weitkamp, Appl. Opt. 17, 1131 (1978).
[Crossref] [PubMed]

1976 (1)

1975 (5)

1974 (3)

1971 (1)

1970 (1)

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 123 (1970).

1967 (2)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, IEEE J. Quantum Electron. QE-3, 484 (1967).
[Crossref]

1966 (1)

1965 (2)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

1961 (1)

1956 (1)

A. T. Forrester, Am. J. Phys. 24, 192 (1956).
[Crossref]

1955 (1)

A. T. Forrester, R. A. Gudmundsen, P. E. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

1938 (1)

F. Zernike, Physica V8, 785 (1938).
[Crossref]

Andrade, O. O.

O. O. Andrade, B. J. Rye, J. Phys. D 7, 280 (1974).
[Crossref]

Arams, F. R.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, IEEE J. Quantum Electron. QE-3, 484 (1967).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).

Bracewell, R.

See e.g., R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Byer, R. L.

R. L. Byer, Opt. Quantum Electron. 7, 147 (1975).
[Crossref]

Cohen, S. C.

Cummins, H. Z.

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 123 (1970).

Degnan, J. J.

Fink, D.

Forrester, A. T.

A. T. Forrester, J. Opt. Soc. Am. 51, 253 (1961).
[Crossref]

A. T. Forrester, Am. J. Phys. 24, 192 (1956).
[Crossref]

A. T. Forrester, R. A. Gudmundsen, P. E. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

Fried, D. L.

D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

Gudmundsen, R. A.

A. T. Forrester, R. A. Gudmundsen, P. E. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

Harms, J.

Horrigan, S. A.

Jakeman, E.

This assumption is justified by the central limit theorem if there are many particles in the scattering volume and if the signal is sampled over times long compared with its coherence time. Relevant discussions in the cases where one of these assumption is not valid are given, respectively, by E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978) and by L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

E. Jakeman, C. J. Oliver, E. R. Pike, Adv. Phys. 24, 349 (1975).
[Crossref]

Johnson, P. E.

A. T. Forrester, R. A. Gudmundsen, P. E. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

Klein, B. J.

Lakmann, W.

Mandel, L.

L. Mandel, E. Wolf, J. Opt. Soc. Am. 65, 413 (1975).
[Crossref]

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

Oliver, C. J.

E. Jakeman, C. J. Oliver, E. R. Pike, Adv. Phys. 24, 349 (1975).
[Crossref]

Pace, F. P.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, IEEE J. Quantum Electron. QE-3, 484 (1967).
[Crossref]

Peyton, B. J.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, IEEE J. Quantum Electron. QE-3, 484 (1967).
[Crossref]

Pike, E. R.

E. Jakeman, C. J. Oliver, E. R. Pike, Adv. Phys. 24, 349 (1975).
[Crossref]

Pusey, P. N.

This assumption is justified by the central limit theorem if there are many particles in the scattering volume and if the signal is sampled over times long compared with its coherence time. Relevant discussions in the cases where one of these assumption is not valid are given, respectively, by E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978) and by L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

Rye, B. J.

O. O. Andrade, B. J. Rye, J. Phys. D 7, 280 (1974).
[Crossref]

B. J. Rye, “Basic Principles of Signal Acquisition in DAS Systems Employing Incoherent Backscatter,” Hull University, Department of Applied Physics, Internal Report (January1978).

Sard, E. W.

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, IEEE J. Quantum Electron. QE-3, 484 (1967).
[Crossref]

Siegman, A. E.

Sonnenschein, C. M.

Swinney, H. L.

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 123 (1970).

Weitkamp, C.

Wolf, E.

L. Mandel, E. Wolf, J. Opt. Soc. Am. 65, 413 (1975).
[Crossref]

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).

Zernike, F.

F. Zernike, Physica V8, 785 (1938).
[Crossref]

Adv. Phys. (1)

E. Jakeman, C. J. Oliver, E. R. Pike, Adv. Phys. 24, 349 (1975).
[Crossref]

Am. J. Phys. (1)

A. T. Forrester, Am. J. Phys. 24, 192 (1956).
[Crossref]

Appl. Opt. (8)

IEEE J. Quantum Electron. (1)

F. R. Arams, E. W. Sard, B. J. Peyton, F. P. Pace, IEEE J. Quantum Electron. QE-3, 484 (1967).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. D (1)

O. O. Andrade, B. J. Rye, J. Phys. D 7, 280 (1974).
[Crossref]

Opt. Quantum Electron. (1)

R. L. Byer, Opt. Quantum Electron. 7, 147 (1975).
[Crossref]

Phys. Rev. (1)

A. T. Forrester, R. A. Gudmundsen, P. E. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

Phys. Rev. Lett. (1)

This assumption is justified by the central limit theorem if there are many particles in the scattering volume and if the signal is sampled over times long compared with its coherence time. Relevant discussions in the cases where one of these assumption is not valid are given, respectively, by E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978) and by L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

Physica (1)

F. Zernike, Physica V8, 785 (1938).
[Crossref]

Proc. IEEE (2)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

Prog. Opt. (1)

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 123 (1970).

Rev. Mod. Phys. (1)

L. Mandel, E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

Other (4)

See e.g., R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).

All integrals in this paper are between extreme limits, i.e., −∞ and ∞, or 0 and ∞ as appropriate, except for the double intergrals which have been characterized by the area of integration and where otherwise specified.

B. J. Rye, “Basic Principles of Signal Acquisition in DAS Systems Employing Incoherent Backscatter,” Hull University, Department of Applied Physics, Internal Report (January1978).

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Figures (2)

Fig. 1
Fig. 1

Geometry of the local oscillator beam.

Fig. 2
Fig. 2

The transmitted beam geometry for atmospheric backscatter lidar.

Equations (78)

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i ( ν ) 2 e 2 F ν { η Φ L δ ( τ ) + 2 Re ( η h ν ) 2 g S ( τ ) exp ( j ω τ ) × A D A D J R ( a 1 : a 2 ) J L * ( a 1 : a 2 ) d a 1 d a 2 } ,
[ ( Δ i 2 ) 2 ] 1 / 2 = [ i 4 - i 2 2 ] 1 / 2 2 1 / 2 i 2 ,
J R ( b 1 : b 2 ) = [ ( Φ R h ν ) / A R ] μ S ( β ) ,
J R ( a 1 : a 2 ) = ν ( a 1 ) ν * ( a 2 ) ( λ v ) 2 f R ( b 1 ) ν ( b 1 ) f R * ( b 2 ) v * ( b 2 ) J R ( b 1 : b 2 ) × exp [ - j k ( a 1 · b 1 - a 2 · b 2 ) / v ] d b 1 d b 2 ,
x ( y ) = exp ( j k y 2 2 x ) .
f R ( b ) = f R ( b ) f * ( b ) ,
J L ( b 1 : b 2 ) = f R * ( b 1 ) v * ( b 1 ) f R ( b 2 ) v ( b 2 ) ( λ v ) 2 f D ( a 1 ) v ( a 1 ) f D ( a 2 ) v * ( a 2 ) × J L ( a 1 : a 2 ) exp [ j k ( a 1 · b 1 - a 2 · b 2 ) / v ] d a 1 d a 2 ,
f D ( a 1 ) f D ( a 2 ) J R ( a 1 : a 2 ) J L * ( a 1 : a 2 ) d a 1 d a 2 = f R ( b 1 ) f R ( b 2 ) J R ( b 1 : b 2 ) J L * ( b 1 : b 2 ) d b 1 d b 2
J L ( a 1 : a 2 ) = Φ L h ν u L ( a 1 ) u L * ( a 2 ) = Φ L h ν u L ( a 1 ) u L ( a 2 ) exp [ j χ L ( a 1 , a 2 ) ] ,
χ L ( a 1 , a 2 ) = [ k ( a 1 2 - a 2 2 ) ] / [ 2 ( v + d ) ] ,
u L ( b ) = { [ v ( b ) ] / λ v } H σ ( b / v ) { f D v u L * ( a ) } ,
H x { g ( y ) } = g ( y ) exp ( - j 2 π x · y ) d y = 2 π g ( y ) J 0 ( 2 π x y ) y d y .
J L * ( b 1 : b 2 ) = Φ L h ν L f R ( b 1 ) u L ( b 1 ) f R * ( b 2 ) u L * ( b 2 ) ,
A R A R J R ( b 1 : b 2 ) J L * ( b 1 : b 2 ) d b 1 d b 2 = Φ R Φ L A R ( h ν ) 2 μ S ( β ) f R ( b 1 ) u L ( b 1 ) f R * ( b 2 ) u L * ( b 2 ) d b 1 d b 2 .
u L ( b ) 2 d b = f D ( a ) u L ( a ) 2 d a = 1.
J R ( b 1 : b 2 ) = exp ( - α r ) r ( b 1 ) r * ( b 2 ) r 2 I [ σ ( s / r ) ] exp ( - k s β / r ) d s = λ 2 exp ( - α r ) r ( b 1 ) r * ( b 2 ) H β { L [ σ ( s / r ) ] } ,
V T ( b , t ) = [ P 0 ( t ) c 0 ] 1 / 2 f T ( b ) u T ( b ) ,
u T ( b ) 2 d b = 1.
T T = f T ( b ) u T ( b ) 2 d b .
I T [ σ ( s / r ) , t + r c ] = c 0 | V T ( s , t + r c ) | 2 exp ( - α r ) = P 0 ( t ) λ 2 r 2 exp ( - α r ) H σ ( s / r ) { f T r u T ( b ) } 2 .
J R ( b 1 : b 2 ) = K P 0 ( t ) exp ( - 2 α r ) r 2 × r ( b 1 ) r * ( b 2 ) H β { H σ ( s / r ) { f T r u T ( b ) } 2 }
Φ R = 1 2 A R β S c exp ( - 2 α r ) r 2 N T ,
Φ R = ρ A R π h ν exp ( - 2 α r ) r 2 P T ,
μ S ( β ) = μ S ( β ) r ( b 1 ) r * ( b 2 ) ,
μ S ( β ) = 1 T T H β { H σ ( s / r ) { f T r u T ( b ) } 2 } = 1 T T f T r u T ( b ) * f T r * u T * ( b ) ,
| 1 r + 1 r 0 | λ A T ,
μ S ( β ) = 1 T T f T u T ( b ) * f T u T ( b ) ,
μ S ( β ) [ 1 / ( T T ) ] f T u T ( b ) * f T u T * ( b ) ,
A R A R J R ( b 1 : b 2 ) J L * ( b 1     b 2 ) d b 1 d b 2 = Φ R Φ L A R ( h ν ) 2 μ S ( β ) f R ( b 1 ) r ( b 1 ) u L ( b 1 ) f R * ( b 2 ) r * ( b 2 ) u L * ( b 2 ) d b 1 d b 2 = Φ R Φ L A R ( h ν ) 2 μ S ( β ) μ L ( β ) d β ,
μ L ( β ) = f R r u L ( b ) * f R * r * u L * ( b )
i ( ν ) 2 = e 2 η ( Φ L ) F ν [ δ ( τ ) + 2 η η h Φ R n 1 g S ( τ ) cos ω τ ]
n 1 = A R [ μ S ( β ) μ R ( β ) d β ] ,
η h = μ S ( β ) μ L ( β ) d β μ S ( β ) μ R ( β ) d β ,
μ R ( β ) = 1 A R f R ( b ) * f R ( b )
n 1 = A R μ S ( β ) d β = A r A C ,
μ S ( β ) μ R ( β ) d β μ R ( β ) d β = A R ,
N 1 = A R / [ μ S ( β ) 2 μ R ( β ) d β ] .
k b 2 2 ( 1 r - 1 f ) + arg u L ( b ) = 0
A C = μ S ( β ) d β = 1 T T f T r u T ( b ) * f T r * u T * ( b ) d β = 1 T T f T ( b ) r ( b ) u T ( b ) d b 2 .
f T ( b ) r ( b ) u T ( b ) d b 2 f T ( b ) r ( b ) 2 d b f T ( b ) u T ( b ) 2 d b = A T T T ,
A C A T .
A R n 1 = μ S ( β ) μ R ( β ) d β A T .
η a = ( T T A R ) / ( n 1 A T ) ,
A C Ω S ~ λ 2 ,
v u L * ( a ) = u L ( a ) ,
u L ( b ) = v ( b ) u L ( b ) , u L ( b ) = 1 λ v H σ ( b / v ) { f D ( a ) u L ( a ) } .
( 1 / v ) + ( 1 + u ) = 1 / f .
μ L b ( β ) = f R u L ( b ) * f R u L ( b ) ,
η b = [ μ S ( β ) μ L b ( β ) d β ] / [ μ S ( β ) μ R ( β ) d β ] .
f R ( b ) r ( b ) u L ( b ) = f R ( b ) r ( b ) u L ( b ) .
μ L c ( β ) = f R r u L ( b ) * f R * r * u L * ( b ) ,
η c = μ S ( β ) μ L c ( β ) d β μ S ( β ) μ L b ( β ) d β .
k b 2 2 | 1 u - 1 r | π 2 ,
| 1 u - 1 r | n 1 λ A R .
η d = [ μ S ( β ) μ L ( β ) d β ] / [ μ S ( β ) μ L c ( β ) d β ] .
k a 2 2 | 1 v - 1 v + d | π 2
| 1 v - 1 v + d | λ A D | .
i 2 ( ν ) = e 2 η Φ L F ν { δ ( τ ) + 2 η η a η b η d A T β S c × exp ( - 2 α r ) r 2 ρ π P 0 h ν g S ( τ ) cos ω τ } .
i 2 ( ν ) = e 2 η Φ L F ν { δ ( τ ) + η η a η b η c η d A T β S c × exp ( - 2 α r ) r 2 N 0 g S ( τ ) cos ω t } .
μ S ( β ) μ L ( β ) d β = μ S ( β ) 2 d β = H σ ( s / r ) { f T r u T ( b ) } 4 d σ ( s / r ) ,
η a = T T A T μ S ( β ) μ R ( β ) d β < 1 ,
η h = μ S ( β ) 2 d β μ S ( β ) μ R ( β ) d β .
i t = e δ ( t - t 1 ) δ ( a - a 1 ) d a 1 .
p ( a , t ) = [ η / ( h ν ) ] V * ( a , t ) V ( a , t ) ,
i t = i t p ( a , t ) d a d t = η e η ν A D J R ( a 1 : a 1 ) d a .
V * ( a 1 , t 1 ) V ( a 2 , t 2 ) = Γ ( 1 , 2 ) = J ( a 1 : a 2 ) g ( t 1 : t 2 ) .
i 2 = i t 2 - i t 2 ,
i t 2 = e 2 δ ( t - t 1 ) δ ( t - t 2 ) δ ( a - a 1 ) δ ( a - a 2 ) d a d a p ( a , a , t , t ) = ( η h ν ) 2 V * ( a , t ) V * ( a , t ) V ( a , t ) V ( a , t ) .
i t 2 = e ( η e h ν ) δ ( τ ) A D Γ ( 1 , 1 ) d 2 a 1 + ( η e h ν ) 2 A D A D Γ ( 1 , 2 ; 1 , 2 ) d a 1 d a 2 ,
Γ ( 1 , 1 ) = Γ L ( 1 , 1 ) + Γ R ( 1 , 1 ) , Γ ( 1 , 2 : 1 , 2 ) = Γ L ( 1 , 2 ) Γ L ( 2 , 2 ) + Γ R ( 1 , 1 ) Γ R ( 2 , 2 ) + 2 Γ L ( 1 , 1 ) Γ R ( 2 , 2 ) + 2 Re [ Γ R ( 1 , 2 ) Γ L ( 2 , 1 ) ] + Γ R ( 1 , 2 ) 2 .
Γ R ( 1 , 2 ) Γ L ( 2 , 1 ) = J R ( a 1 ; a 2 ) J L * ( a 1 ; a 2 ) g S ( τ ) exp ( j ω τ ) ,
H x { H y { g ( x ) } H y { h ( x ) } = g ( x ) * h ( x ) = 0 - π + π g ( x ) h ( x - x ) d x d ϕ ,
H y { g * ( x ) } = H y * { g ( x ) } ,
H x { H y { g ( x ) } 2 } = g ( x ) * g * ( x ) ,
g ( x i ) * h ( x 1 ) d x = 0 - π + π g ( x 1 ) h ( x 1 - x ) d x 1 d ϕ d x
g ( x 1 ) * h ( x 1 ) ] d x = g ( x 1 ) h ( x 2 ) d x 1 d x 2 .
f ( x ) [ g ( x 1 ) * h ( x 1 ) ] d x = f ( x ) g ( x 1 ) h ( x 2 ) d x 1 d x 2
g ( x 1 ) * g * ( x 1 ) d x = g ( x 1 ) d x 1 g * ( x 2 ) d x 2 = g ( x ) d x 2 ,

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