Abstract

A mathematical model of the thermally induced nonlinear propagation of a laser beam in an absorbing fluid medium is developed. The theory includes, identifies, and clearly assesses the relative importance of the different regimes due to conduction, free convection, and/or forced convection cooling. The derived equations of propagation explicitly account for the diffraction effects and permit a straightforward estimate of their relative influence with respect to the thermal distortion. More specifically, similarity parameters or scaling laws are deduced from the governing equations. There is a total of five of these parameters, but it frequently happens in practical applications that the actual number is reduced to only one. Finally, sample numerical results are presented to illustrate some of the implications connected with the proposed model, and the qualitative agreement with published experimental data is very gratifying.

© 1973 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. Glass, “Propagation Effects at High Intensity,” presented at a series of lectures on Laser Propagation Through The Atmosphere, Dept of Elect. Eng., Ohio State University, Columbus (1967).
  2. G. A. Askaryan, V. B. Studenov, JETP Lett. 10, 71 (1969).
  3. G. A. Askaryan, V. G. Mikhalevich, V. B. Studenov, G. P. Shipulo, Sov. Phys. JETP 32, 1036 (1971).
  4. W. R. Callen, B. G. Huth, R. H. Pantell, Appl. Phys. Lett. 11, 103 (1967).
    [CrossRef]
  5. R. L. Carman, P. L. Kelley, Appl. Phys. Lett. 12, 241 (1968).
    [CrossRef]
  6. R. A. Chodzko, S. C. Lin, AIAA J. 9, 1105 (1971).
    [CrossRef]
  7. F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 14, 52 (1969).
    [CrossRef]
  8. F. G. Gebhardt, D. C. Smith, Appl. Opt. 11, 244 (1972).
    [CrossRef] [PubMed]
  9. J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17, 220 (1970).
    [CrossRef]
  10. H. Kleiman, R. W. O’Neil, J. Opt. Soc. Am. 61, 12 (1971).
    [CrossRef]
  11. E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13, 369 (1968).
    [CrossRef]
  12. L. Sica, E. A. McLean, U.S. Naval Research Laboratory, Report 7107 (1970).
  13. J. W. Tucker, R. N. DeWitt, U.S. Naval Research Laboratory, Report 7038 (1969).
  14. D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
    [CrossRef]
  15. J. W. Tucker, H. Hancock, U.S. Naval Research Laboratory, Report 7096 (1970).
  16. H. B. Rosenstock, J. H. Hancock, Appl. Opt. 10, 1299 (1971).
    [CrossRef] [PubMed]
  17. P. M. Livingston, Appl. Opt. 10, 426 (1971).
    [CrossRef] [PubMed]
  18. J. N. Hayes, U.S. Naval Research Laboratory, Report 7213 (1971).
  19. J. N. Hayes, P. B. Ulrich, A. H. Aitken, Appl. Opt. 11, 257 (1972).
    [CrossRef] [PubMed]
  20. A. D. Wood, M. Camac, E. T. Gerry, Appl. Opt. 10, 1877 (1971).
    [CrossRef] [PubMed]
  21. F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
    [CrossRef]
  22. J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 1587 (1970).
    [CrossRef]

1972 (3)

1971 (6)

1970 (2)

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17, 220 (1970).
[CrossRef]

J. Wallace, M. Camac, J. Opt. Soc. Am. 60, 1587 (1970).
[CrossRef]

1969 (3)

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 14, 52 (1969).
[CrossRef]

G. A. Askaryan, V. B. Studenov, JETP Lett. 10, 71 (1969).

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

1968 (2)

R. L. Carman, P. L. Kelley, Appl. Phys. Lett. 12, 241 (1968).
[CrossRef]

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13, 369 (1968).
[CrossRef]

1967 (1)

W. R. Callen, B. G. Huth, R. H. Pantell, Appl. Phys. Lett. 11, 103 (1967).
[CrossRef]

Aitken, A. H.

Askaryan, G. A.

G. A. Askaryan, V. G. Mikhalevich, V. B. Studenov, G. P. Shipulo, Sov. Phys. JETP 32, 1036 (1971).

G. A. Askaryan, V. B. Studenov, JETP Lett. 10, 71 (1969).

Avizonis, P. V.

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17, 220 (1970).
[CrossRef]

Callen, W. R.

W. R. Callen, B. G. Huth, R. H. Pantell, Appl. Phys. Lett. 11, 103 (1967).
[CrossRef]

Camac, M.

Carman, R. L.

R. L. Carman, P. L. Kelley, Appl. Phys. Lett. 12, 241 (1968).
[CrossRef]

Chodzko, R. A.

R. A. Chodzko, S. C. Lin, AIAA J. 9, 1105 (1971).
[CrossRef]

DeWitt, R. N.

J. W. Tucker, R. N. DeWitt, U.S. Naval Research Laboratory, Report 7038 (1969).

Gebhardt, F. G.

F. G. Gebhardt, D. C. Smith, Appl. Opt. 11, 244 (1972).
[CrossRef] [PubMed]

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
[CrossRef]

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 14, 52 (1969).
[CrossRef]

Gerry, E. T.

Glass, A. J.

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13, 369 (1968).
[CrossRef]

A. J. Glass, “Propagation Effects at High Intensity,” presented at a series of lectures on Laser Propagation Through The Atmosphere, Dept of Elect. Eng., Ohio State University, Columbus (1967).

Hancock, H.

J. W. Tucker, H. Hancock, U.S. Naval Research Laboratory, Report 7096 (1970).

Hancock, J. H.

Hayes, J. N.

J. N. Hayes, P. B. Ulrich, A. H. Aitken, Appl. Opt. 11, 257 (1972).
[CrossRef] [PubMed]

J. N. Hayes, U.S. Naval Research Laboratory, Report 7213 (1971).

Hogge, C. B.

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17, 220 (1970).
[CrossRef]

Huth, B. G.

W. R. Callen, B. G. Huth, R. H. Pantell, Appl. Phys. Lett. 11, 103 (1967).
[CrossRef]

Kelley, P. L.

R. L. Carman, P. L. Kelley, Appl. Phys. Lett. 12, 241 (1968).
[CrossRef]

Kenemuth, J. R.

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17, 220 (1970).
[CrossRef]

Kleiman, H.

Lin, S. C.

R. A. Chodzko, S. C. Lin, AIAA J. 9, 1105 (1971).
[CrossRef]

Livingston, P. M.

McLean, E. A.

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13, 369 (1968).
[CrossRef]

L. Sica, E. A. McLean, U.S. Naval Research Laboratory, Report 7107 (1970).

Mikhalevich, V. G.

G. A. Askaryan, V. G. Mikhalevich, V. B. Studenov, G. P. Shipulo, Sov. Phys. JETP 32, 1036 (1971).

O’Neil, R. W.

Pantell, R. H.

W. R. Callen, B. G. Huth, R. H. Pantell, Appl. Phys. Lett. 11, 103 (1967).
[CrossRef]

Rosenstock, H. B.

Shipulo, G. P.

G. A. Askaryan, V. G. Mikhalevich, V. B. Studenov, G. P. Shipulo, Sov. Phys. JETP 32, 1036 (1971).

Sica, L.

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13, 369 (1968).
[CrossRef]

L. Sica, E. A. McLean, U.S. Naval Research Laboratory, Report 7107 (1970).

Smith, D. C.

F. G. Gebhardt, D. C. Smith, Appl. Opt. 11, 244 (1972).
[CrossRef] [PubMed]

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
[CrossRef]

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 14, 52 (1969).
[CrossRef]

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

Studenov, V. B.

G. A. Askaryan, V. G. Mikhalevich, V. B. Studenov, G. P. Shipulo, Sov. Phys. JETP 32, 1036 (1971).

G. A. Askaryan, V. B. Studenov, JETP Lett. 10, 71 (1969).

Tucker, J. W.

J. W. Tucker, H. Hancock, U.S. Naval Research Laboratory, Report 7096 (1970).

J. W. Tucker, R. N. DeWitt, U.S. Naval Research Laboratory, Report 7038 (1969).

Ulrich, P. B.

Wallace, J.

Wood, A. D.

AIAA J. (1)

R. A. Chodzko, S. C. Lin, AIAA J. 9, 1105 (1971).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (6)

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 20, 129 (1972).
[CrossRef]

E. A. McLean, L. Sica, A. J. Glass, Appl. Phys. Lett. 13, 369 (1968).
[CrossRef]

J. R. Kenemuth, C. B. Hogge, P. V. Avizonis, Appl. Phys. Lett. 17, 220 (1970).
[CrossRef]

F. G. Gebhardt, D. C. Smith, Appl. Phys. Lett. 14, 52 (1969).
[CrossRef]

W. R. Callen, B. G. Huth, R. H. Pantell, Appl. Phys. Lett. 11, 103 (1967).
[CrossRef]

R. L. Carman, P. L. Kelley, Appl. Phys. Lett. 12, 241 (1968).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. C. Smith, IEEE J. Quantum Electron. QE-5, 600 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

JETP Lett. (1)

G. A. Askaryan, V. B. Studenov, JETP Lett. 10, 71 (1969).

Sov. Phys. JETP (1)

G. A. Askaryan, V. G. Mikhalevich, V. B. Studenov, G. P. Shipulo, Sov. Phys. JETP 32, 1036 (1971).

Other (5)

J. W. Tucker, H. Hancock, U.S. Naval Research Laboratory, Report 7096 (1970).

L. Sica, E. A. McLean, U.S. Naval Research Laboratory, Report 7107 (1970).

J. W. Tucker, R. N. DeWitt, U.S. Naval Research Laboratory, Report 7038 (1969).

J. N. Hayes, U.S. Naval Research Laboratory, Report 7213 (1971).

A. J. Glass, “Propagation Effects at High Intensity,” presented at a series of lectures on Laser Propagation Through The Atmosphere, Dept of Elect. Eng., Ohio State University, Columbus (1967).

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 (5)

Fig. 1
Fig. 1

Diagram illustrating various regimes of horizontal propagation in a quiescent atmosphere at sea level of a cw-CO2 laser beam.

Fig. 2
Fig. 2

Predicted steady state intensity distributions for an initially Gaussian and cylindrically symmetric beam propagating under dominant conduction cooling. The Fresnel number F is set equal to 100, and the net energy depletion is assumed negligible. The radius a is defined as the 1/e point of the initial Gaussian intensity distribution.

Fig. 3
Fig. 3

Predicted steady state distributions for an initially Gaussian beam of planar geometry propagating under dominant forced convection cooling. The wind is out of the minus y ¯ axis. The Fresnel number is 100, and the net energy depletion is neglected. The parameter a is defined as in Fig. 2.

Fig. 4
Fig. 4

Curves illustrating the Fresnel number dependence of the predicted steady state axial peak intensity distribution of an initially planar and Gaussian beam subjected to crosswinds or dominant forced convection.

Fig. 5
Fig. 5

Curves illustrating the Fresnel number dependence of the predicted steady state lateral deflection, vs z ¯, of the peak intensity of an initially planar and Gaussian beam subjected to crosswinds or dominant forced convection.

Equations (59)

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

( δ ρ / δ t ) + · ρ V = 0 ;
ρ ( δ V / δ t ) + ρ V · V = ρ g - p ;
ρ ( δ h / δ t ) + ρ V · h = · ( κ θ ) + ( δ p / δ t ) + V · p + I / L .
ρ 0 = cst . ; θ 0 = cst . ; V 0 = cst . ; h 0 = ( 1 / ρ 0 ) ρ 0 = g .
p 1 = 0.
ρ 1 = - ρ 0 β θ 1 , h 1 = c p θ 1 .
· V g = 0 ,
( δ V g / δ t ) + ( V 0 + V g ) · V g = - β θ 1 g ,
( δ θ 1 / δ t ) + ( V 0 + V g ) · θ 1 = ( κ / ρ 0 c p ) 2 θ 1 + ( I / ρ 0 c p L ) .
V g · V g = - β θ 1 g , V g · θ 1 = I / ( ρ 0 c p L ) .
U 2 / a = β g 0 ( θ 1 ) , ( U / a ) 0 ( θ 1 ) = P / ( π ρ 0 c p a 2 L ) .
U = [ β g P / ( π ρ 0 c p L ) ] 1 / 3 .
t r = ( π ρ 0 c p a 2 L ) / β P ; t d = ( ρ 0 c p a 2 ) / κ ; t g = [ ( π ρ 0 c p a 3 L ) / β P g ] 1 / 3 ; t w = a / W ; γ = t c / t r ; I ¯ = ( π a 2 / P ) I ; θ ¯ 1 = ( β θ 1 / γ ) ; V ¯ g = V g / U ; V ¯ 0 = V 0 / W ; ¯ = a ; t ¯ = t / t c ; z ¯ = z / l ,
¯ · V ¯ g = 0 ,
δ V ¯ g δ t ¯ + t c t w V ¯ 0 · ¯ V ¯ g + t c t g V ¯ g · ¯ V ¯ g = - ( t c t g ) 2 θ ¯ 1 ( g / g ) ,
δ θ ¯ 1 δ ¯ t + t c t w V ¯ 0 · ¯ θ ¯ 1 + t c t g V ¯ g · ¯ θ ¯ 1 = t c t d ¯ 2 θ ¯ 1 + I ¯
2 E - μ c 2 δ 2 E δ t 2 - 2 c 2 δ ( μ ) δ t δ E δ t - E c 2 δ 2 ( μ ) δ t 2 + [ E · ln ( μ ) ] = 0 ,
( μ ) 1 / 2 = n + i α ,
E = A exp [ i k z - i ω t ] ,
k z = 0 s k l d s ,
l 2 π / k 0 ,
t c 2 π / ω .
n = n 0 + n 1 .
n 1 = - σ θ 1 ,
N = σ γ / β .
σ γ / n 0 β 1.
k 2 = k 0 2 + 2 k 0 k 1 + .
ω 2 n 2 / c 2 = k 0 2 + 2 k 0 2 ( n 1 / n 0 ) + .
K = k 0 ( σ γ / n 0 β ) .
l 2 a 2 , a 2 π / k 0 ,
z ¯ A ¯ ( δ k ¯ 1 / δ z ¯ ) + A ¯ k ¯ 1 - ( i / k 0 l ) ( n 0 β / σ γ ) ( δ A ¯ / δ z ¯ ) - ( i l / k 0 a 2 ) z ¯ ( ¯ k ¯ 1 ) ( ¯ A ¯ ) - ( i l / 2 k 0 a 2 ) A ¯ z ¯ ( ¯ 2 k ¯ 1 ) + ( l 2 / 2 a 2 ) ( σ γ / n 0 β ) A ¯ z ¯ 2 ( ¯ k ¯ 1 ) 2 = - A ¯ θ ¯ 1 + i α n 0 ( n 0 β / σ γ ) A ¯ + ( 1 / 2 k 0 2 a 2 ) ( n 0 β / σ γ ) ¯ 2 A ¯ .
If we let l 2 = a 2 / ( n 0 β / σ γ ) ,
and define , F k 0 a 2 / l ,
z ¯ A ¯ ( δ k ¯ 1 / δ z ¯ ) + A ¯ k ¯ 1 + 1 2 A ¯ z ¯ 2 ( ¯ k ¯ 1 ) 2 - i F δ A ¯ δ z - i F z ¯ ( ¯ k ¯ 1 ) · ( ¯ A ¯ ) - i 2 F A ¯ z ¯ ( ¯ k ¯ 1 ) = - A ¯ θ ¯ 1 + i ( α / n 0 ) ( l 2 / a 2 ) A ¯ + ( 1 / 2 F 2 ) ¯ 2 A ¯ .
z ¯ δ k ¯ 1 δ z ¯ + k ¯ 1 + 1 2 z ¯ 2 ( ¯ k ¯ 1 ) 2 = - θ ¯ 1 + i α n 0 l 2 a 2 ,
δ A ¯ δ z ¯ + z ¯ ( ¯ k ¯ 1 ) · ( ¯ A ¯ ) + 1 2 z ¯ A ¯ ( 2 k ¯ 1 ) = i 2 F ¯ 2 A ¯ .
z ¯ ( δ k ¯ r / δ z ¯ ) + k ¯ r + 1 2 z ¯ 2 [ ( ¯ k ¯ r ) 2 - ( ¯ k ¯ i ) 2 ] = - θ ¯ 1 ,
z ¯ ( δ k ¯ i / δ z ¯ ) + k ¯ i + z ¯ 2 ( ¯ k ¯ r ) · ( ¯ k ¯ i ) = ( α / n 0 ) ( l 2 / a 2 ) .
k ¯ i = ( l 2 / a 2 ) ( α / n 0 ) = constant
z ¯ ( δ k ¯ r / δ z ¯ ) + k ¯ r + 1 2 z ¯ 2 ( ¯ k ¯ r ) 2 = - θ ¯ 1 ,
δ A ¯ δ z + z ¯ ( ¯ k ¯ r ) · ( ¯ A ¯ ) + 1 2 z ¯ A ¯ ( ¯ 2 k ¯ r ) = i 2 F ¯ 2 A ¯ .
E ¯ s = ϕ ¯ exp [ i k z - i ω t ] .
δ ϕ ¯ δ z + z ¯ ( ¯ k ¯ r ) · ( ¯ ϕ ¯ ) + 1 2 z ¯ ϕ ¯ ( ¯ 2 k ¯ r ) = i 2 F ¯ 2 ϕ ¯ .
I ¯ = E ¯ s E ¯ s * = ϕ ¯ ϕ ¯ * exp ( - 2 k i z ) .
- d x d y ϕ ¯ ϕ ¯ * = constant with respect to z ¯ .
α = n 0 / 2 k 0 L .
¯ · V g = 0 ,
δ V ¯ g δ t ¯ + t c t w V ¯ 0 · ¯ V g + t c t g V ¯ g · ¯ V ¯ g = - ( t c t g ) 2 θ ¯ 1 g g ,
δ θ ¯ 1 δ t ¯ + t c t w V ¯ 0 · ¯ θ ¯ 1 + t c t g V ¯ g · ¯ θ ¯ 1 = t c t d ¯ 2 θ ¯ 1 + I ¯ ,
δ δ z ¯ ( k ¯ r z ¯ ) + 1 2 [ ¯ ( k ¯ r z ¯ ) ] 2 = - θ ¯ 1 ,
δ ϕ ¯ δ z ¯ + [ ¯ ( k ¯ r z ¯ ) ] · [ ¯ ϕ ¯ ] + 1 2 ϕ ¯ ¯ 2 ( k ¯ r z ¯ ) = i 2 F ¯ 2 ϕ ¯ ,
I ¯ = ϕ ¯ ϕ ¯ * exp [ - ( l / L ) z ¯ ] .
t c t s and / or t a ,
γ 2 1 ,
σ γ / n 0 β 1 ,
t d t g and t w , t g t d and t w , t w t d and t g .
δ ϕ ¯ / δ η ¯ = ( i / 2 ) ¯ 2 ϕ ¯ ,
( d r / d s ) e z + ( a / l ) ¯ ( k ¯ r z ¯ ) ,
· [ I ( d r / d s ) ] = - I / L .

Metrics