Abstract

The receiving efficiency η as a function of range z is investigated for pulsed coherent lidars using a theory that relates η(z) to the transmitted laser intensity and the point-source receiving efficiency ηs(r,z). The latter can be calculated either by a forward method, or by a backward method that employs the back-propagated local oscillator (BPLO) approach. The BPLO method is efficient and accurate provided that cascaded diffraction effects inside the lidar system are properly taken into account. The theory is applied to the ideal case to examine the optimization of the system when both transmitted and BPLO fields at the antenna are Gaussian, including optimum telescope aperture.

© 1990 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

1988 (1)

1987 (2)

1985 (1)

M. J. Post, “Atmospheric Infrared Scattering Profiles: Interpretation of Statistical and Temporal Properties,” NOAA Technical Memorandum ERL WPL-122 (1985).

1982 (2)

1981 (1)

1979 (1)

1974 (1)

1971 (1)

1966 (1)

A. E. Siegman, “The Antenna Properties of Optical Heterodyne Receivers,” Appl. Opt. Vol. 5, 1588–1594 (1966).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th Ed. (Pergamon, New York, 1986).

Degnan, J. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” Laser Speckle, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975).
[CrossRef]

Hanson, S. G.

Horrigan, F. A.

Klein, B. J.

Menzies, R. T.

Post, M. J.

M. J. Post, “Atmospheric Infrared Scattering Profiles: Interpretation of Statistical and Temporal Properties,” NOAA Technical Memorandum ERL WPL-122 (1985).

Rye, B. J.

Shapiro, J. H.

Siegman, A. E.

A. E. Siegman, “The Antenna Properties of Optical Heterodyne Receivers,” Appl. Opt. Vol. 5, 1588–1594 (1966).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

Sonnenschein, C. M.

Tratt, D. M.

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th Ed. (Pergamon, New York, 1986).

Yura, H. T.

Appl. Opt. (7)

Appl. Opt. Vol. (1)

A. E. Siegman, “The Antenna Properties of Optical Heterodyne Receivers,” Appl. Opt. Vol. 5, 1588–1594 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

NOAA Technical Memorandum ERL WPL-122 (1)

M. J. Post, “Atmospheric Infrared Scattering Profiles: Interpretation of Statistical and Temporal Properties,” NOAA Technical Memorandum ERL WPL-122 (1985).

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics, 6th Ed. (Pergamon, New York, 1986).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” Laser Speckle, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

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Figures (6)

Fig. 1
Fig. 1

Scattering plane (S), antenna plane (A), and detector plane (D).

Fig. 2
Fig. 2

Conceptual optical layout of a receiving system in a coherent lidar.

Fig. 3
Fig. 3

ηM and ρM as functions of the Fresnel number F = π R a 2 / ( λ z ) for an afocal system. 1, ηM;2, ρM.

Fig. 4
Fig. 4

Y(F) = M(F) for afocal system. The dashed line is the tangent line at F = 0.

Fig. 5
Fig. 5

ηM as a function of F and g in general cases.

Fig. 6
Fig. 6

Y = M as a function of F and g in general cases.

Tables (1)

Tables Icon

Table I The Maximum Antenna Radius RM and Optimum Antenna Radius Ropt as Functions of Wavelength and Maximum Detection Range zM

Equations (60)

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P ac = i s ( z ) 2 R = T s T a ( z ) e 2 η d M R ( n ν ) 2 × [ A D V s ( z , P ) V L * ( P ) d P ] [ A D V s * ( z , P ) V L ( P ) d P ] ,
P ac ( z ) = T s e 2 η d 2 M R c E 0 P L 2 ( h ν ) 2 T a ( z ) β ( z ) Ω ( z ) η ( z ) ,
η ( z ) = [ A D V s ( z , P ) V L * ( P ) d P ] [ A D V s * ( z , P ) V L ( P ) d P ] P L P s ,
P s ( z ) = c E 0 2 β ( z ) Ω ( z ) ,
η ( z ) = [ A D V sn ( z , P ) V Ln * ( P ) d P ] [ A D V sn * ( z , P ) V Ln ( P ) d P ] ,
η ( z ) = P d ( z ) P s [ A D V s ( z , P ) V L * ( P ) d P ] [ A D V s * ( z , P ) V L ( P ) d P ] P L P d ( z ) = η p η mix ,
R s ( z ) = η ( z ) z 2 .
V s ( P , z ) = 0 z s U t ( Q , z ) B ( Q , z ) K ( Q , P , z ) d z d Q ,
U t ( Q , z ) w ( z ) = U t n ( Q , z ) ,
K n ( Q , P , z ) = K ( Q , P , z ) Ω = K n ( Q , P , z ) A R z ,
V s ( P , z ) = Ω z 1 z w ( z ) d z s U tn ( Q , z ) B ( Q , z ) K n ( Q , P , z ) d Q ,
V Ln ( P ) = V L ( P ) P L ,
η ( z ) = Λ ( z ) Λ * ( z ) c E 0 β ( z ) / 2 ,
Λ ( z ) = A D [ z 1 z w ( z ) d z s U tn ( Q , z ) B ( Q , z ) K n ( Q , P , z ) d z d Q ] × V Ln * ( P ) d P .
G ( Q , z ) = A D K n ( Q , P , z ) V Ln * ( P ) d P .
η ( z ) = [ z 1 z w ( z ) d z s B ( Q , z ) F ( Q , z ) d Q ] [ z 1 z w ( z ) d z s B * ( Q , z ) F * ( Q , z ) d Q ] c E 0 β ( z ) / 2 = z 1 z z 1 z s s w ( z ) w ( z ) B ( Q , z ) B * ( Q , z ) F ( Q , z ) F * ( Q , z ) d Q d Q d z d z c E 0 β ( z ) / 2 .
B ( Q , z ) B * ( Q , z ) = β ( Q , z ) δ ( z , z ) δ ( Q , Q ) ,
η ( z ) = β ( z ) c E 0 β ( z ) / 2 z 1 z P 0 ( z ) d z s I tn ( Q , z ) G ( Q , z ) 2 d Q = s I tn ( Q , z ) η s ( Q , z ) d Q ,
z 1 z P 0 ( z ) d z = z 1 z w ( z ) 2 d z = 0 τ P 0 ( t ) c d t / 2 = c E 0 / 2 ,
I tn ( Q , z ) = U tn ( Q , z ) U tn * ( Q , z ) ,
η s ( Q , z ) = G ( Q , z ) G * ( Q , z ) = | A D K n ( Q , P , z ) V Ln * ( P ) d P | 2
η ( z ) = s I tn ( Q , z ) d Q | A D K n ( Q , P , z ) V Ln * ( P ) d P | 2 = A D A D [ s I tn ( Q , z ) d Q K n ( Q , P , z ) K n * ( Q , P , z ) ] × V Ln * ( P ) V Ln ( P ) d P d P .
V sn ( P , z ) V sn * ( P , z ) = s I tn ( Q , z ) K n ( Q , P , z ) K n * ( Q , P , z ) d Q
η ( z ) = s I tn ( Q , z ) d Q | A D K n ( Q , P , z ) V Ln * ( P ) d P | 2 = s d Q | A D I tn ( Q , z ) K n ( Q , P , z ) V Ln * ( P ) d P | 2 = s d Q | A D V ¯ sn ( Q , P , z ) V Ln * ( P ) d P | 2 ,
U s ( Q , Q A ) = exp ( j k z ) z exp [ j π λ z ( r - r A ) 2 ] ,
U sn ( Q , Q A ) = exp ( j k z ) A R exp [ j π λ z ( r - r A ) 2 ] .
U ( G ) = U 0 ( F ) H ( F , G ) d F ,
H ( F , G ) = j λ R exp [ j π ( r G - r F ) 2 λ R ] ,
exp [ - j π r 2 λ f ] ,
K n ( Q , P , z ) = L H L ( P L , P ) d P L × I N H N ( P N , P L ) d P N B H B ( Q B , P 1 ) d Q B × c H c ( Q c , Q B ) d Q c A H A ( Q A , Q c ) U sn ( Q , Q A ) d Q A .
G ( Q , z ) = A D V Ln * ( P ) [ L H L ( P L , P ) d P L × I N H N ( P N , P L ) d P N B H B ( Q B , P 1 ) d Q B × c H c ( Q c , Q B ) d Q c A H A ( Q A , Q c ) U sn ( Q , Q A ) d Q A ] d P .
G ( Q , z ) = λ z j A R A H ( Q A , Q ) d Q A c H A ( Q c , Q A ) d Q c B H c ( Q B , Q c ) d Q B × I 1 H B ( P 1 , Q B ) d P 1 L H N ( P L , P N ) d P L A D H L ( P , P L ) V Ln * ( P ) d P .
η s ( Q , z ) = α U Ln ( Q , z ) 2 = ( λ z ) 2 A R I Ln ( Q , z ) .
G ( Q , z ) = A H ( Q A , Q ) W ( Q A ) d Q A ,
W ( Q A ) = c H A ( Q c , Q A ) d Q c B H c ( Q B , Q c ) d Q B I 1 H B ( P 1 , Q B ) d P 1 × L H N ( P L , P N ) d P L A D H L ( P , P L ) V Ln * ( P ) d P
U ta ( r a ) = 2 π R T 2 exp ( - r a 2 R T 2 ) ,
U La ( r a ) = 2 π R L 2 exp ( - r a 2 R L 2 ) ,
U tn ( r , z ) = 2 π λ z 0 R a U ta ( r a ) exp ( j π r a 2 λ z e ) J 0 ( 2 π r r a λ z ) r a d r a ,
U Ln ( r , z ) = 2 π λ z 0 R a U La ( r a ) exp ( j π r a 2 λ z e ) J 0 ( 2 π r r a λ z ) r a d r a ,
U tn ( r , z ) = 8 π λ z R T 0 R a exp [ ( - 1 R T 2 + j π λ z e ) r a 2 ] J 0 ( 2 π r r a λ z ) r a d r a .
U tn ( y , z ) = 8 π R a 2 λ z R T 0 1 exp [ ( - 1 ρ T 2 + j F e ) x 2 ] J 0 ( 2 F x y ) x d x .
I tn ( y , z ) = 8 F λ z ρ T 2 { 0 1 exp [ ( - 1 ρ T 2 + j F e ) x 2 ] J 0 ( 2 F x y ) x d x } 2 .
I Ln ( y , z ) = 8 F λ z ρ L 2 { 0 1 exp [ ( - 1 ρ L 2 + j F e ) x 2 ] J 0 ( 2 F x y ) x d x } 2 .
A ( ξ , F , F e , y ) = { 0 1 exp [ ( - ξ + j F e ) x 2 ] J 0 ( 2 F x y ) x d x } 2 .
η ( z ) = ( λ z ) 2 A R [ 2 π 0 I tn ( r , z ) I Ln ( r , z ) r d r ] = 128 F 2 ρ T 2 ρ L 2 0 A ( ρ T - 2 , F , F e , y ) A ( ρ L - 2 , F , F e , y ) y d y = 128 F 2 a b 0 A ( a , F , F e , y ) A ( b , F , F e , y ) y d y ,
η ( z ) = η ( F , F e , a , b ) .
{ η a = 128 F 2 b 0 A ( b , F , F e , y ) × [ A ( a , F , F e , y ) + a A ( a , F , F e , y ) a ] y d y = 0 , η b = 128 F 2 a 0 A ( a , F , F e , y ) × [ A ( b , F , F e , y ) + b A ( b , F , F e , y ) b ] y d y = 0.
{ 0 A ( b m , F , F e , y ) [ A ( a m , F , F e , y ) + a m A ( a , F , F e , y ) a | a = a m ] y d y = 0 , 0 A ( a m , F , F e , y ) [ A ( b m , F , F e , y ) + b m A ( b , F , F e , y ) b | b = b m ] y d y = 0.
η M = 128 F 2 [ c ( F , F e ) ] 2 0 [ A ( c , F , F e , y ) ] 2 y d y = η M ( F , F e ) ,
η M = η M ( F , g ) .
S ( R a ) = π R a 2 z 2 η ( z ) = Ω η ( z ) ,
S ( R a ) = π R a 2 z M 2 η M ( F ) = λ z M F η M ( F ) = λ z M Y ( F ) ,
Y ( F ) = F η M ( F ) .
π R M 2 λ z M = 2.47 ,
R M = 0.89 λ z M .
R opt = 0.546 λ z M .
Y opt = 0.355 , S opt = 0.355 λ z M .
SNR opt 0.355 λ n - 1 z M .
S = 0.4 λ F z M = 0.4 Ω ,
Δ g = Δ F e F = z M Δ ( 1 z e ) = - z M f 2 = - L δ ,

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