Abstract

The knowledge of range performance versus atmospheric transmission, often given by the visibility, is critical for the design, use, and prediction of laser and passive electro-optic systems. I present a solution of the ladar–lidar equation based on Lambert’s W function. This solution will reveal the dependence of the maximum range on the system and target parameters for different atmospheric attenuations and will also allow us to take the signal statistics into account by studying the influence on the threshold signal-to-noise ratio. The method is also applicable to many range calculations for passive systems where the atmospheric loss can be approximated by an exponential term.

© 2008 Optical Society of America

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References

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  1. V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004).
    [CrossRef]
  2. Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).
  3. D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
    [CrossRef]
  4. H. M. Tulldahl and K. O. Steinvall, “Analytical waveform generation from small objects in lidar bathymetry,” Appl Opt. 38, 1021-1039 (1999).
    [CrossRef]
  5. G. R. Osche, Optical Detection Theory (Wiley, 2002).

1999 (1)

H. M. Tulldahl and K. O. Steinvall, “Analytical waveform generation from small objects in lidar bathymetry,” Appl Opt. 38, 1021-1039 (1999).
[CrossRef]

1996 (1)

Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).

1994 (1)

D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
[CrossRef]

Bonnier, D.

D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
[CrossRef]

Corless, Robert M.

Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).

Eichinger, W. E.

V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004).
[CrossRef]

Gonnet, G. H.

Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).

Hare, D. E. G.

Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).

Hutt, D. L.

D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
[CrossRef]

Jeffrey, D. J.

Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).

Knuth, D. E.

Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).

Kovalev, V. A.

V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004).
[CrossRef]

Larochelle, V. G.

D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
[CrossRef]

Mathieu, P.

D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
[CrossRef]

Osche, G. R.

G. R. Osche, Optical Detection Theory (Wiley, 2002).

Steinvall, K. O.

H. M. Tulldahl and K. O. Steinvall, “Analytical waveform generation from small objects in lidar bathymetry,” Appl Opt. 38, 1021-1039 (1999).
[CrossRef]

Thériault, J.-M.

D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
[CrossRef]

Tulldahl, H. M.

H. M. Tulldahl and K. O. Steinvall, “Analytical waveform generation from small objects in lidar bathymetry,” Appl Opt. 38, 1021-1039 (1999).
[CrossRef]

Appl Opt. (1)

H. M. Tulldahl and K. O. Steinvall, “Analytical waveform generation from small objects in lidar bathymetry,” Appl Opt. 38, 1021-1039 (1999).
[CrossRef]

Opt. Eng. (1)

D. L. Hutt, J.-M. Thériault, V. G. Larochelle, P. Mathieu, and D. Bonnier, “Estimating atmospheric extinction for eyesafe laser rangefinders,” Opt. Eng. 33, 3762-3277 (1994).
[CrossRef]

Other (3)

G. R. Osche, Optical Detection Theory (Wiley, 2002).

V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004).
[CrossRef]

Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math., 5, 329-359 (1996).

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Figures (10)

Fig. 1
Fig. 1

y = W ( x ) calculated in MATLAB.

Fig. 2
Fig. 2

y = W ( x ) and the serial expansion for a number of different terms n.

Fig. 3
Fig. 3

Lambert’s W function and its derivative d y / d x .

Fig. 4
Fig. 4

Example of maximum range calculations versus visibility using the parameters in Table 1 and the solution based on Lambert’s W function. Note the change in slope for R m ( V ) for different values of the laser energy affecting the parameter x.

Fig. 5
Fig. 5

Same as in Fig. 4 but for a small diffuse target. The x values in this example are small (2.52, 1.04, and 1.88, respectively, at 100 mJ pulse energy), which results in a rather small increase of maximum range R m versus visibility V.

Fig. 6
Fig. 6

Same as in Figs. 4, 5 but with a small retroreflecting target.

Fig. 7
Fig. 7

Maximum aerosol backscatter range for the systems in Table 1 with the exception of a larger receiver diameter ( 30 cm as compared with 8 cm ). Note the optimum range around 10 km visibility for the 1.06 and 1.54 μm lidars, while the 10.6 μm lidar has an optimum range for a visibility lower then 1 km .

Fig. 8
Fig. 8

Maximum bottom and water backscatter range obtained by the simplified range equation (14).

Fig. 9
Fig. 9

Maximum possible flight altitude R max as a function of atmospheric visibility and with different depth penetrations measured as the number of attenuation lengths (KD).

Fig. 10
Fig. 10

Maximum range for a link system versus visibility and with different peak laser power levels P L .

Tables (2)

Tables Icon

Table 1 System Parameters for the Hard Target and Atmospheric Lidar

Tables Icon

Table 2 Parameters for the Depth Sounding Lidar

Equations (20)

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SNR T = P r NEP = P L η G A r NEP R 2 exp ( 2 0 R σ ( R ' ) d R ' ) ,
SNR T = P r NEP = P L η G A r NEP R 2 A T 4 π ( R θ ) 2 exp ( 2 σ R ) ,
G A T = k A T 2 / λ 2 ,
W exp ( W ) = ( k σ R ) exp ( k σ R ) = x ,
x 1 = σ S 1 = σ η P L G A r NEP × SNR T .
x 2 = σ / 2 S 2 4 = ( σ / 2 ) [ η P L G A m A T 4 NEP × SNR T π θ 2 ] 1 / 4 .
W ( x ) = n = 1 ( 1 ) n 1 n ! x n .
W ( x ) = W ( x ) x [ 1 + W ( x ) ] if x 1.
C = 1 1 + W ( x ) ,
β = 0 . 00037 σ 2 + 0 . 018 σ + 0 . 0013 ,
SNR T = P r NEP = P L η tot G A r NEP ( n w R + D ) 2 exp [ 2 0 D K ( z ) d z ] T atm 2 ,
SNR T = P r NEP P L η tot G A r NEP ( n w R ) 2 exp ( 2 K D ) ,
D max = 1 2 K ln ( S w )
S w = P L η tot G A r NEP ( n w R ) 2 SNR T .
Δ D = ln ( F ) / 2 K .
x 1 * = σ S 1 * = σ R 2 S w exp ( 2 K D ) .
SNR T = P r NEP = P L η NEP A r 4 π ( R θ ) 2 exp ( σ R ) ,
R max = 2 σ W [ ( σ / 2 ) P L η A r 4 NEP π θ 2 SNR T ] .
SNR A r I target η exp ( σ R ) D * R 2 ( A d Δ f ) 1 / 2 ,
R max = 2 σ W [ ( σ / 2 ) I target η A r D * ( A d Δ f ) 1 / 2 SNR T ] .

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