Abstract

A simple model for the propagation of laser beams in a turbulent atmosphere is presented. Closed analytical expressions, suitable for system research and real-time analysis of field experiments, are obtained for the probability of the intercepted power to exceed a certain threshold level. A hierarchy of approximations is described and the validity range of each is discussed.

© 1982 Optical Society of America

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References

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  1. For example, R. A. McClatchey, J. E. A. Selby, Environmental Research Paper 460, AFCRL-TR-74-0003 (Air Force Cambridge Research Laboratories, Bedford, Mass., 1974).
  2. R. Esposito, Proc. IEEE 55, 1533 (1967).
    [CrossRef]
  3. D. L. Fried, Appl. Opt. 12, 422 (1973).
    [CrossRef] [PubMed]
  4. P. J. Titterton, Appl. Opt. 12, 423 (1973).
    [CrossRef] [PubMed]
  5. A. A. Taklaya, Sov. J. Quantum Electron. 7, 517 (1977).
    [CrossRef]
  6. A. A. Taklaya, Sov. J. Quantum Electron. 8, 85 (1978).
    [CrossRef]
  7. R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [CrossRef]
  8. M. Tamir, U. Halavee, E. Azoulay, Appl. Opt. 20, 734 (1981).
    [CrossRef] [PubMed]
  9. R. L. Fante, Proc. IEEE 63, 1669 (1975).
    [CrossRef]
  10. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  11. A. Ishimaru, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).
  12. We have found experimentally that Eq. (14) is accurate for values of σ12 slightly >0.5 (i.e., even when the probability distribution function deviates from the lognormal distribution). Since in practice one is interested in the cumulative distribution, this fact is of great importance.
  13. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  14. For example, J. Abele, H. Raidt, W. Jessen, R. Kirschmer, in AGARD Conf. Proc. 238, (1978).

1981 (1)

1978 (2)

For example, J. Abele, H. Raidt, W. Jessen, R. Kirschmer, in AGARD Conf. Proc. 238, (1978).

A. A. Taklaya, Sov. J. Quantum Electron. 8, 85 (1978).
[CrossRef]

1977 (1)

A. A. Taklaya, Sov. J. Quantum Electron. 7, 517 (1977).
[CrossRef]

1975 (1)

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

1973 (2)

1970 (1)

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

1967 (1)

R. Esposito, Proc. IEEE 55, 1533 (1967).
[CrossRef]

Abele, J.

For example, J. Abele, H. Raidt, W. Jessen, R. Kirschmer, in AGARD Conf. Proc. 238, (1978).

Azoulay, E.

Esposito, R.

R. Esposito, Proc. IEEE 55, 1533 (1967).
[CrossRef]

Fante, R. L.

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

Fried, D. L.

Halavee, U.

Ishimaru, A.

A. Ishimaru, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).

Jessen, W.

For example, J. Abele, H. Raidt, W. Jessen, R. Kirschmer, in AGARD Conf. Proc. 238, (1978).

Kirschmer, R.

For example, J. Abele, H. Raidt, W. Jessen, R. Kirschmer, in AGARD Conf. Proc. 238, (1978).

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

McClatchey, R. A.

For example, R. A. McClatchey, J. E. A. Selby, Environmental Research Paper 460, AFCRL-TR-74-0003 (Air Force Cambridge Research Laboratories, Bedford, Mass., 1974).

Raidt, H.

For example, J. Abele, H. Raidt, W. Jessen, R. Kirschmer, in AGARD Conf. Proc. 238, (1978).

Selby, J. E. A.

For example, R. A. McClatchey, J. E. A. Selby, Environmental Research Paper 460, AFCRL-TR-74-0003 (Air Force Cambridge Research Laboratories, Bedford, Mass., 1974).

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Taklaya, A. A.

A. A. Taklaya, Sov. J. Quantum Electron. 8, 85 (1978).
[CrossRef]

A. A. Taklaya, Sov. J. Quantum Electron. 7, 517 (1977).
[CrossRef]

Tamir, M.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Titterton, P. J.

AGARD Conf. Proc. (1)

For example, J. Abele, H. Raidt, W. Jessen, R. Kirschmer, in AGARD Conf. Proc. 238, (1978).

Appl. Opt. (3)

Proc. IEEE (3)

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

R. Esposito, Proc. IEEE 55, 1533 (1967).
[CrossRef]

Sov. J. Quantum Electron. (2)

A. A. Taklaya, Sov. J. Quantum Electron. 7, 517 (1977).
[CrossRef]

A. A. Taklaya, Sov. J. Quantum Electron. 8, 85 (1978).
[CrossRef]

Other (5)

For example, R. A. McClatchey, J. E. A. Selby, Environmental Research Paper 460, AFCRL-TR-74-0003 (Air Force Cambridge Research Laboratories, Bedford, Mass., 1974).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

A. Ishimaru, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).

We have found experimentally that Eq. (14) is accurate for values of σ12 slightly >0.5 (i.e., even when the probability distribution function deviates from the lognormal distribution). Since in practice one is interested in the cumulative distribution, this fact is of great importance.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

F ( p p 1 ) vs p1 as a function of C n 2. The calculations were performed for a Nd laser (λ = 1.06μ) with a beam divergence of 0.2 mrad and for L = 1500 m. The beam radius was calculated according to Ref. 9.

Fig. 2
Fig. 2

F ( p p 1 ) vs p1 as a function of the receiver radius r. The values of the laser parameters and range are the same as in Fig. 1.

Fig. 3
Fig. 3

F ( p p 1 ) vs p1 as a function of the offset ρ0. The values of the laser parameters and range are the same as in Fig. 1.

Equations (32)

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F ( p p 1 ) = 0 d ρ H ( ρ , ρ 0 ) F ( p p 1 ρ ) ,
H ( ρ , ρ 0 ) = dxd y h ( x ) h ( y ) δ ( x 2 + y 2 ρ 2 ) ,
h ( x ) = ( 2 π σ ω 2 ) 1 / 2 exp ( x 2 / 2 σ ω 2 ) ,
h ( y ) = ( 2 π σ ω 2 ) 1 / 2 exp [ ( y ρ 0 ) 2 / 2 σ ω 2 ] .
H ( ρ , ρ 0 ) = ( ρ / σ ω 2 ) exp [ ( ρ ω 2 + ρ 2 ) / 2 σ ω 2 ] I 0 ( ρ ρ 0 / σ ω 2 ) ,
F ( p p 1 ρ ) = p 1 p t dpf ( p ρ ) ,
f ( p ρ ) = ( 2 π p 2 σ ln p 2 ) 1 / 2 exp [ ( ln p ln p + σ ln p 2 / 2 ) 2 / 2 σ ln p 2 ] ,
p ( ρ ) = 1 Q ( ρ ̂ , r ̂ ) ,
Q ( α , β ) = β t exp ( t 2 + α 2 2 ) I 0 ( α t ) d t .
H ( p ) = H [ ρ ( p ) ] ρ p .
p 2 p t r ̂ 2 exp [ 2 ρ ̂ 2 / ( 4 + r ̂ 2 ) ] / ( 4 + r ̂ 2 ) .
H ( p ) = H 0 ( p ) exp [ ρ 0 2 / ( 2 σ ω 2 ) I 0 [ ρ ( p ) ρ 0 / σ ω 2 ] ,
H 0 ( p ) = ( 2 σ ω 2 / r 2 ) A A + 1 p A 1 ,
F ( p p 1 ρ ) = Q ( a 1 ρ 2 + b 1 ) ,
a 1 = [ 1 2 σ ln p ( 4 R 2 + r 2 ) ] 1 ,
b 1 = { ln [ p 1 ( 4 R 2 + r 2 ) / 2 p t r 2 ] + 1 2 σ ln p 2 } / σ ln p 2 ,
Q ( x ) = ( 2 π ) 1 / 2 x d t exp ( t 2 / 2 ) .
F ( p p 1 ) = σ ω 2 exp ( ρ 0 2 / 2 σ ω 2 ) 0 d ρ exp ( ρ / 2 σ ω 2 ) × I 0 ( ρ 0 ρ / σ ω 2 ) Q ( a 1 ρ 2 + b 1 ) ρ .
F ( p p 1 ) = q 2 exp ( 1 4 q ) 0 d t exp ( q t ) I 0 ( t 1 / 2 ) [ 1 erf ( a t + b ) ] = 1 2 q 2 exp ( 1 4 q ) × 0 d t exp ( q t ) I 0 ( t 1 / 2 ) erf ( a t + b ) ,
I 0 ( t 1 / 2 ) = n = 0 6 a n t n ( a 0 = 1 ) .
F ( p p 1 ) = 1 2 q 2 exp ( 1 4 q ) n = 0 6 a n 0 d t × exp ( q t ) t n erf ( a t + b ) .
L [ t n f ( t ) ] = ( 1 ) n d n d q n g ( q ) ,
L [ f ( t ) ] = g ( q ) .
g ( q ) 0 d t exp ( q t ) erf ( a t + b ) = 1 a exp ( q b a ) b d x exp ( q x a ) erf ( x ) = q 1 erf ( b ) + q 1 exp ( b 2 ) × exp [ ( b + q / 2 a ) 2 erfc ( b + q / 2 a ) ] ,
( 1 ) n d n d q n [ g ( q ) ] = n ! erf ( b ) / q n + 1 + ( 1 ) n exp ( b 2 ) ( 2 a ) n 1 × d n d x n [ exp ( x 2 ) erfc ( x ) / ( x b ) ] .
d n d x n [ u ( x ) υ ( x ) w ( x ) ] = k = 0 n ( n k ) u ( k ) ( x ) t = 0 n k ( n k l ) υ ( l ) ( x ) × w ( n k l ) ( x ) .
u ( x ) erfc ( x ) , υ ( x ) exp ( x 2 ) , w ( x ) ( x b ) 1 ,
u ( k ) = ( 1 ) k 2 π exp ( x 2 ) H k 1 ( x ) , υ ( l ) = exp ( x 2 ) H l + ( x ) ,
F ( p p 1 ) 1 2 1 2 exp ( ρ 0 2 2 σ ω 2 ) n = 0 6 n ! a n ( 2 ρ 0 2 2 σ ω 2 ) n × { erf ( b 1 2 ) + 2 π exp ( b 1 2 2 ) k = 0 n H k 1 [ ( b 1 + 1 2 σ ω 2 a 1 ) / 2 ] × ( 1 2 2 σ ω 2 a 1 ) k / k ! l = 0 n k ( 1 ) l H l + [ ( b 1 + 1 2 σ ω 2 a 1 ) / 2 ] × ( 1 2 2 σ ω 2 a 1 ) l / l ! } .
a 1 ( 2 R 2 σ ln p ) 1 ,
b 1 ln ( p 1 / p a ) / σ ln p + σ ln p / 2
F ( p p 1 ) 1 2 1 2 exp ( ρ 0 2 2 σ ω 2 ) n = 0 6 n ! a n ( 2 ρ 0 2 2 σ ω 2 ) n × { erf [ ln ( p 1 p a ) / 2 σ ln p + σ ln p / 2 2 ] + ( p 1 p a ) R 2 / σ ω 2 × exp ( σ ln p 2 R 2 / 2 σ ω 2 + σ ln p 2 R 4 / 2 σ ω 4 ) × erfc [ ln ( p 1 p a ) / 2 σ ln p + σ ln p / 2 2 + σ ln p R 2 / 2 σ ω 2 ] } .

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