Abstract

A detailed study of the lidar equation has been made for both scattering and fluorescent targets. Allowance has been made for the effects of finite excited state lifetime, optical depth, laser pulse duration, detector integration period, and laser pulse shape. Analytical solutions have been obtained, and graphical solutions are also presented to aid in evaluating the magnitude of the correction factor appropriate to several cases of interest.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. B. Mumola, O. Jarrett, C. A. Brown, NASA Conference on the Use of Lasers for Hydrographic Studies, Wallops Island, September 1973, NASA SP-375, pp. 137–145.
  2. H. H. Kim, Appl. Opt. 12, 1454 (1973).
    [CrossRef] [PubMed]
  3. R. M. Measures, G. Pilon, Opto-Electron. 4, 141 (May1972).
    [CrossRef]
  4. H. Kildal, R. L. Byer, Proc. IEEE 59, 1644 (1971).
    [CrossRef]
  5. R. M. Measures, J. Garlick, W. R. Houston, D. G. Stephenson, Can. J. Remote Sensing, 1, 95 (Nov.1975).
  6. R. M. Measures, W. R. Houston, D. G. Stephenson, Opt. Eng. 13, 494 (1974).
    [CrossRef]
  7. J. F. Fantasia, H. C. Ingrao, “Development of an Experimental Airborne Laser Remote Sensing System for the Detection and Classification of Oil Spills,” in Proceedings of the 9th International Symposium on Remote Sensing of the Environment, 15–19 April1974, paper 10700-1-K, pp. 1711–1745.
  8. R. M. Measures, W. Houston, M. Bristow, Can. Aeronaut. Space J. 19, 501 (1973).

1975 (1)

R. M. Measures, J. Garlick, W. R. Houston, D. G. Stephenson, Can. J. Remote Sensing, 1, 95 (Nov.1975).

1974 (1)

R. M. Measures, W. R. Houston, D. G. Stephenson, Opt. Eng. 13, 494 (1974).
[CrossRef]

1973 (2)

R. M. Measures, W. Houston, M. Bristow, Can. Aeronaut. Space J. 19, 501 (1973).

H. H. Kim, Appl. Opt. 12, 1454 (1973).
[CrossRef] [PubMed]

1972 (1)

R. M. Measures, G. Pilon, Opto-Electron. 4, 141 (May1972).
[CrossRef]

1971 (1)

H. Kildal, R. L. Byer, Proc. IEEE 59, 1644 (1971).
[CrossRef]

Bristow, M.

R. M. Measures, W. Houston, M. Bristow, Can. Aeronaut. Space J. 19, 501 (1973).

Brown, C. A.

P. B. Mumola, O. Jarrett, C. A. Brown, NASA Conference on the Use of Lasers for Hydrographic Studies, Wallops Island, September 1973, NASA SP-375, pp. 137–145.

Byer, R. L.

H. Kildal, R. L. Byer, Proc. IEEE 59, 1644 (1971).
[CrossRef]

Fantasia, J. F.

J. F. Fantasia, H. C. Ingrao, “Development of an Experimental Airborne Laser Remote Sensing System for the Detection and Classification of Oil Spills,” in Proceedings of the 9th International Symposium on Remote Sensing of the Environment, 15–19 April1974, paper 10700-1-K, pp. 1711–1745.

Garlick, J.

R. M. Measures, J. Garlick, W. R. Houston, D. G. Stephenson, Can. J. Remote Sensing, 1, 95 (Nov.1975).

Houston, W.

R. M. Measures, W. Houston, M. Bristow, Can. Aeronaut. Space J. 19, 501 (1973).

Houston, W. R.

R. M. Measures, J. Garlick, W. R. Houston, D. G. Stephenson, Can. J. Remote Sensing, 1, 95 (Nov.1975).

R. M. Measures, W. R. Houston, D. G. Stephenson, Opt. Eng. 13, 494 (1974).
[CrossRef]

Ingrao, H. C.

J. F. Fantasia, H. C. Ingrao, “Development of an Experimental Airborne Laser Remote Sensing System for the Detection and Classification of Oil Spills,” in Proceedings of the 9th International Symposium on Remote Sensing of the Environment, 15–19 April1974, paper 10700-1-K, pp. 1711–1745.

Jarrett, O.

P. B. Mumola, O. Jarrett, C. A. Brown, NASA Conference on the Use of Lasers for Hydrographic Studies, Wallops Island, September 1973, NASA SP-375, pp. 137–145.

Kildal, H.

H. Kildal, R. L. Byer, Proc. IEEE 59, 1644 (1971).
[CrossRef]

Kim, H. H.

Measures, R. M.

R. M. Measures, J. Garlick, W. R. Houston, D. G. Stephenson, Can. J. Remote Sensing, 1, 95 (Nov.1975).

R. M. Measures, W. R. Houston, D. G. Stephenson, Opt. Eng. 13, 494 (1974).
[CrossRef]

R. M. Measures, W. Houston, M. Bristow, Can. Aeronaut. Space J. 19, 501 (1973).

R. M. Measures, G. Pilon, Opto-Electron. 4, 141 (May1972).
[CrossRef]

Mumola, P. B.

P. B. Mumola, O. Jarrett, C. A. Brown, NASA Conference on the Use of Lasers for Hydrographic Studies, Wallops Island, September 1973, NASA SP-375, pp. 137–145.

Pilon, G.

R. M. Measures, G. Pilon, Opto-Electron. 4, 141 (May1972).
[CrossRef]

Stephenson, D. G.

R. M. Measures, J. Garlick, W. R. Houston, D. G. Stephenson, Can. J. Remote Sensing, 1, 95 (Nov.1975).

R. M. Measures, W. R. Houston, D. G. Stephenson, Opt. Eng. 13, 494 (1974).
[CrossRef]

Appl. Opt. (1)

Can. Aeronaut. Space J. (1)

R. M. Measures, W. Houston, M. Bristow, Can. Aeronaut. Space J. 19, 501 (1973).

Can. J. Remote Sensing (1)

R. M. Measures, J. Garlick, W. R. Houston, D. G. Stephenson, Can. J. Remote Sensing, 1, 95 (Nov.1975).

Opt. Eng. (1)

R. M. Measures, W. R. Houston, D. G. Stephenson, Opt. Eng. 13, 494 (1974).
[CrossRef]

Opto-Electron. (1)

R. M. Measures, G. Pilon, Opto-Electron. 4, 141 (May1972).
[CrossRef]

Proc. IEEE (1)

H. Kildal, R. L. Byer, Proc. IEEE 59, 1644 (1971).
[CrossRef]

Other (2)

J. F. Fantasia, H. C. Ingrao, “Development of an Experimental Airborne Laser Remote Sensing System for the Detection and Classification of Oil Spills,” in Proceedings of the 9th International Symposium on Remote Sensing of the Environment, 15–19 April1974, paper 10700-1-K, pp. 1711–1745.

P. B. Mumola, O. Jarrett, C. A. Brown, NASA Conference on the Use of Lasers for Hydrographic Studies, Wallops Island, September 1973, NASA SP-375, pp. 137–145.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Essential elements of an environmental laser–sensor (LIDAR).

Fig. 2
Fig. 2

Optical interaction processes and appropriate range of cross sections of relevance to laser environmental sensing.

Fig. 3
Fig. 3

Spatial resolution for scattering phenomena as seen from space–time diagram of propagating rectangular shaped laser pulse.

Fig. 4
Fig. 4

Space–time view of laser pulse propagation and excitation of fluorescent target medium in the case of a rectangular shaped laser pulse.

Fig. 5
Fig. 5

Boundary model for LIDAR equation.

Fig. 6
Fig. 6

Variation of the optical thin correction factor γ(Z), with the normalized penetration Z into a fluorescent target for several values of T (ratio of laser pulse duration to fluorescence lifetime, τl/τ) for a rectangular shaped laser pulse. R represents the range of the laser pulse leading edge, Ro the range of the target boundary, and L the laser pulse length (l/2). (a) τd/τl = 0.2 and for (b) τd/τl = 1.0. Laser pulse length to attenuation length, ∊L = 0.005 for each case.

Fig. 7
Fig. 7

Variation of optically thick correction factor γ ¯ (Z), with normalized penetration Z, into a fluorescent target for several values of T and a rectangular shaped laser pulse.

Fig. 8
Fig. 8

Space time view of laser pulse propagation and excitation of fluorescent target medium in the case of a realistic shaped laser pulse.

Fig. 9
Fig. 9

Variation of the optical thin correction factor γ(Z*) with the normalized penetration Z* into a fluorescent target for several values of T* (ratio of laser pulse duration to fluorescence lifetime, τl*/τ) for a realistic shaped laser pulse. R represents the range of the laser pulse leading edge, Ro the range of the target boundary, and L the laser pulse length (l*/2). (a) τd/τl* = 0.2 and for (b) τd/τl* = 1.0. Laser pulse length to attenuation length, ∊L = 0.005 for each.

Fig. 10
Fig. 10

Variation of the optically thick correction factor γ ¯ (Z*) with the normalized penetration Z* into a fluorescent target for several values of T*. τd/τl* = 0.2 for each case but (a) ∊L = 25 and (b) ∊L = 2.5.

Fig. 11
Fig. 11

Comparison between exact solution and approximate solution for the optically thick correction factor.

Equations (57)

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

Δ P ( λ , R ) = T r ( λ ) [ ( A r ) / ( R 2 ) ] T ( λ , R ) ξ ( R ) A E ( R ) W ( λ , R ) Δ R Δ λ ,
W ( λ , R ) = i N i ( R ) d d Ω σ i s ( λ l ) L i s ( λ ) I ( R ) ,
W ( λ , R ) = i N i * ( R ) h c L i F ( λ ) 4 π λ τ RAD i ,
P ( λ , R ) = A r 0 R ξ ( R ) A E ( R ) d R R 2 × Δ λ r T r ( λ ) T ( λ , R ) W ( λ , R ) d λ
P ( λ s , R ) = A r T r ( λ s ) × o R ξ ( R ) A E ( R ) T ( λ s , R ) i N i ( R ) d d Ω σ i s ( λ l ) I ( R ) d R R 2 .
P ( λ , R ) = T r ( λ ) A r R 2 T ( λ , R ) ξ ( R ) A E ( R ) i N i ( R ) d d Ω σ i s ( λ l ) I ( R ) c τ l 2 ,
I ( R ) = E l T ( λ l , R ) τ l A E ( R ) ,
T ( λ l , R ) exp [ - o R ( λ l , R ) d R ] ,
T ( λ , R ) exp [ - o R ( λ , R ) d R ] ,
T ( R ) T ( λ l , R ) T ( λ , R ) = exp { - o R [ ( λ l , R ) + ( λ , R ) ] d R } ,
E ( λ , R ) = t t + τ d P ( λ , R ) d t .
E ( λ , R ) = E l T r ( λ ) T ( R ) ξ ( R ) A r R 2 i N i ( R ) d d Ω σ i s ( λ l ) c τ d 2 .
E ( λ , R ) = E l T r ( λ ) T ( R ) ξ ( R ) A r R 2 N ( R ) σ s ( λ l ) 4 π c τ d 2 ,
I ( R , t ) = E l T ( λ l , R ) A E ( R ) j ( t ) ,
0 j ( t ) d t = 1.
E ( λ , R ) = E l T r ( λ ) A r × t t + τ d d t o R ξ ( R ) T ( R ) i N i ( R ) d d Ω σ i s ( λ l ) j ( R ) d R R 2 ,
E ( λ , R ) E l T r ( λ ) A r i N i σ i s ( λ l ) 4 π × t t + τ d d t R o R ξ ( R ) T ( R ) R 2 j ( R ) d R
E ( λ , R ) ~ E l T r ( λ ) A r R 2 ξ ( R ) T ( R ) i N i σ i s ( λ l ) 4 π × t t + τ d d t R o R j ( R ) d R .
d d t N * ( R , t ) = λ l σ A ( λ l ) h c N ( R , t ) I ( R , t ) - N * ( R , t ) τ ,
N * ( R , t ) = λ l N o ( R ) σ A ( λ l ) h c exp { - t / τ } o t I ( R , x ) exp ( x / τ ) d x ,
W ( λ , R ) = N o ( R ) σ A ( λ l ) λ l L F ( λ ) 4 π τ RAD λ exp ( - t / τ ) × o t I ( R , x ) exp ( x / τ ) d x ,
E ( λ , R ) = A r t t + τ d d t o R d R A E ( R ) ξ ( R ) N o ( R ) 4 π τ R r 2 exp ( - t / τ ) × o t I ( R , x ) exp ( x / τ ) d x × Δ λ r T ( λ , R ) T r ( λ ) σ F ( λ l ) L F ( λ ) d λ ,
σ F ( λ l ) λ l σ A ( λ l ) ϕ F / λ ,
ϕ F τ / τ RAD
T ( λ , R ) σ F ( λ l ) L F ( λ ) K r ( λ ) ,
K r ( λ ) Δ λ r T r ( λ ) d λ .
E ( λ , R ) = A r K r ( λ ) σ F ( λ l ) 4 π τ L F ( λ ) × t t + τ d d t d R A E ( R ) ξ ( R ) T ( λ , R ) N o ( R ) R 2 exp ( - t / τ ) × o t I ( R , x ) exp ( x / τ ) d x .
E ( λ , R ) = E l T ( R o ) K r ( λ ) ξ A r N o σ F ( λ l ) L F ( λ ) 4 π τ l H ( R ) ,
H ( R ) = t t + τ d d t ( R o R * [ 1 - exp ( - τ l / τ ) ] exp [ - ( R - R o ) - ( t - τ l - 2 R / c ) τ ] d R R 2 + R * R { 1 - exp [ - ( t - 2 R / c ) / τ ] } exp [ - ( R - R o ) ] d R R 2 ) ,
E ( λ , R ) = E l T ( R o ) K r ( λ ) ξ A r N o σ F ( λ l ) L F ( λ ) 4 π R 2 c τ d 2 × γ ( R ) exp [ - ( R - R o ) ] .
γ ( R ) 2 τ l R 2 H ( R ) exp [ ( R - R o ) ] / c τ d ,
γ ( R ) = 1 Q ζ ( 1 - A ) × ( 1 ζ ψ ( ζ Q ) - A T ψ ( T ) ψ ( T Q ) exp { ( 1 - A ) [ 1 - ( R - R o ) / L ] T } ) ,
γ ( z ) = ( 1 - z ) G 1 ( z ) Q ζ + G 2 ( z ) Q ζ ( 1 - A ) exp [ ζ ( x - 1 ) ] ,
G 1 ( z ) = [ 1 - ψ [ ζ ( 1 - z ) ] exp ( - ζ z ) ζ ( 1 - z ) ( 1 - A ) + A ψ [ T ( 1 - z ) ] exp ( - T z ) T ( 1 - z ) ( 1 - A ) ] exp ( ζ z )
G 2 ( z ) = ψ ( ζ ) ψ [ ζ ( z + Q - 1 ) ] ζ - A ψ ( T ) ψ [ T ( z + Q - 1 ) ] T .
γ ( z ) = 1 ζ [ 1 - ψ ( ζ Q ) exp ( - ζ z ) ζ Q ( 1 - A ) + A ψ ( T Q ) exp ( - T z ) T Q ( 1 - A ) ] exp ( ζ z ) .
d E ( λ , R ) = E l T ( R o ) K r ( λ ) ξ A r N o σ F ( λ l ) L F ( λ ) 4 π R 2 J ( R ) d t ,
J ( R ) R o R d R exp [ - ( R - R o ) ] exp ( - t / τ ) × o t j ( x ) exp ( x / τ ) d x
E ( λ , R ) = E l T ( R o ) K r ( λ ) ζ A r N o σ F ( λ l ) L F ( λ ) 4 π R 2 t t + τ d J ( R ) d t .
j ( t ) = [ ( t / τ o ) n exp ( - t / τ o ) ] / [ τ o Γ ( n + 1 ) ] ,
E ( λ , R ) = E o [ ( 1 - α ) c τ o / 2 τ o Γ ( n + 1 ) α n + 1 ] t t + τ d d t o z o exp [ - ζ o z o - ( z o - z o ) ( 1 - α ) ] d z o o ( z o - z o ) α y n exp ( - y ) d y ,
E o E l T ( R o ) K r ( λ ) ξ A r N o σ F ( λ l ) L F ( λ ) 4 π R 2 , α ( 1 - τ o τ ) , ζ o c τ o / 2
z o 2 ( R - R o ) c τ o
E ( λ , R ) = E l T ( R o ) K r ( λ ) ξ A r N o σ F ( λ l ) L F ( λ ) 4 π R 2 × γ ( R ) c τ d 2 exp [ - ( R - R o ) ] .
γ ( z o ) = ( 1 - α ) Q o α 3 Γ ( 3 ) { S 1 ζ o ψ ( Q ζ o ) + W ( z o , Q o ) exp ( - β z o ) - S 5 ψ [ Q o ( 1 - α ) ] ( 1 - α ) × exp [ ( α - β ) z o ] } ,
W ( z o , Q o ) = S 2 ψ ( Q o ) + S 3 U ( z o , Q o ) + S 4 V ( z o , Q o )
Q o τ d / τ o , β ( 1 - ζ o ) , Γ ( 3 ) = 2 , S 1 = 2 α 3 β 3 ( β - α ) , S 2 = 2 ( β 2 + α β + α 2 ) β 3 , S 3 = α 2 β , S 4 = 2 ( α β + α 2 ) β 2 , S 5 = 2 ( β - α ) , U ( z o , Q o ) = [ ( z o + 1 ) 2 + 1 ] [ 1 - M ( z o , Q o ) exp ( - Q o ) ] , M ( z o , Q o ) = 1 + Q o ( Q o + 2 z o + 2 ) ( z o + 1 ) 2 + 1 , V ( z o , Q o ) = ( z o + 1 ) [ 1 - K ( z o , Q o ) exp ( - Q o ) ] , K ( z o , Q o ) = 1 + Q o z o + 1 ,
γ ¯ ( z * ) γ ( z * ) exp ( - L * z * ) .
γ ¯ ( z * ) [ 1 / ( ζ * Q * ) ] G ¯ ( z * ) ,
G ( z * ) = T * ( 1 - T * ) 3 × [ ψ ( Q * T * ) T * exp ( - z * T * ) - W ¯ ( z * , Q * ) exp ( - z * ) ] ,
W ¯ ( z * , Q * ) = ψ ( Q * ) + ( 1 - T * ) 2 2 U ( z * , Q * ) + ( 1 - T * ) V ( z * , Q * ) .
E ( λ , R ) = E l T ( R o ) K r ( λ ) ξ A r N o σ F ( λ l ) L F ( λ ) G ¯ ( R ) 4 π R 2 [ ( λ l ) + ( λ ) ] .
E ( λ , R ) = E l T ( R o ) K r ( λ ) ξ A r R 2 F ( λ , λ l ) G ¯ ( R ) ,
F ( λ , λ l ) = λ l ϕ F L F ( λ ) 4 π λ .
E ( λ , R o ) = E l T ( R o ) K r ( λ ) ξ A r R 2 F ( λ , λ l ) ,
E ( λ , R ) = E l T ( R o ) ξ A r R 2 i N i σ i s ( λ l ) T r ( λ ) 4 π × [ γ ( R ) ] τ = 0 c τ d 2 exp [ - ( R - R o ) ] ,
E ( λ , R ) = E l T ( R o ) ξ A r R 2 N o σ F ( λ l ) 4 π × L F ( λ ) K r ( λ ) γ ( R ) c τ l d 2 exp [ - ( R - R o ) ] .

Metrics