Abstract

The problem of beam-induced thermal modifications of a high powered cw laser beam is considered. The extent and type of modification depend principally upon the nature of the medium (liquid or air) and whether an external flow field is present. These conditions determine the dominant heat loss mechanism operating the fluid: forced convection, free convection, and diffusion and, thus, the beam distortions. All three cases are analyzed and figures presented, showing the type of distortion expected as determined within the framework of geometric optics.

© 1971 Optical Society of America

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References

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  1. K. A. Brueckner, S. Jorna, Phys. Rev. 164, 182 (1967).
    [CrossRef]
  2. S. A. Akmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal Self Actions of Laser Beams,” Unpublished.
  3. S. A. Akmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp Fiz. Nauk 93, 19 (1967).
  4. A. G. Litvak, JETP Lett. 4, 341 (1966).
  5. R. L. Carman, P. L. Kelly, Appl. Phys. Lett. 12, 241 (1968).
    [CrossRef]
  6. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
    [CrossRef]
  7. R. Y. Chiao, P. L. Kelly, E. Garmire, Phys. Rev. Lett. 17, 1158 (1966).
    [CrossRef]
  8. N. M. Kroll, J. Appl. Phys. 36, 34 (1965).
    [CrossRef]
  9. P. L. Kelly, Phys. Rev. Lett. 15, 1003 (1965).
    [CrossRef]
  10. V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).
  11. Y. R. Shen, N. Bloembergen, Phys. Rev. 147, 1787, (1965).
    [CrossRef]
  12. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [CrossRef]
  13. For example, A. Yariv, Quantum Electronics (Wiley, New York, 1968), Chap. 23 and 25.
  14. N. M. Kroll, P. L. Kelly, “Instabilities and Beam Spreading Due to Heat Deposition,” Appendix K, Rep. S-294, Laser Summer Study, Institute for Defense Analysis (1967).
  15. E. Gerry, private communication.
  16. A. J. Glass, “Thermal Blooming in Gases,” Research Inst. for Eng. Sci., Wayne State U., Detroit (unpublished).
  17. F. C. Gebhardt, D. C. Smith, Appl. Phys. Lett. 14, 52 (1969).
    [CrossRef]
  18. Work begun as a consultant to NRL.
  19. See, for example, G. N. Murphy, Ordinary Differential Equations and Their Solutions (Van Nostrand, Princeton, N.J.1960), p. 352.
  20. E. A. McLean, L. Sica, A. J. Glass, “Interferometric Observation of Absorption Induced Index Changes Associated with Thermal Blooming,” NRL Preprint (submitted to Appl. Phys. Lett.), Oct. 1969.
  21. F. G. Gebhardt, D. C. Smith, United Aircraft Corporation, have completed a first order treatment essentially identical to the above. Experimental evidence for the correctness of the calculation in the form of measured half-moon isoirradiances in a forthcoming article will be published in Applied Optics. Our calculations were done independently with no previous communication.
  22. Chia-Shun Yih, “Free Convection due to Boundary Sources,” in Fluid Models in Geophysics, No. 1956 (U.S. GPO, Washington, D.C., 1953), pp. 117–133.

1969 (1)

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

1968 (1)

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

1967 (2)

K. A. Brueckner, S. Jorna, Phys. Rev. 164, 182 (1967).
[CrossRef]

S. A. Akmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp Fiz. Nauk 93, 19 (1967).

1966 (3)

A. G. Litvak, JETP Lett. 4, 341 (1966).

R. Y. Chiao, P. L. Kelly, E. Garmire, Phys. Rev. Lett. 17, 1158 (1966).
[CrossRef]

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

1965 (5)

Y. R. Shen, N. Bloembergen, Phys. Rev. 147, 1787, (1965).
[CrossRef]

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

N. M. Kroll, J. Appl. Phys. 36, 34 (1965).
[CrossRef]

P. L. Kelly, Phys. Rev. Lett. 15, 1003 (1965).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Akmanov, S. A.

S. A. Akmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp Fiz. Nauk 93, 19 (1967).

S. A. Akmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal Self Actions of Laser Beams,” Unpublished.

Bespalov, V. I.

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Bloembergen, N.

Y. R. Shen, N. Bloembergen, Phys. Rev. 147, 1787, (1965).
[CrossRef]

Brueckner, K. A.

K. A. Brueckner, S. Jorna, Phys. Rev. 164, 182 (1967).
[CrossRef]

Carman, R. L.

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

Chiao, R. Y.

R. Y. Chiao, P. L. Kelly, E. Garmire, Phys. Rev. Lett. 17, 1158 (1966).
[CrossRef]

Garmire, E.

R. Y. Chiao, P. L. Kelly, E. Garmire, Phys. Rev. Lett. 17, 1158 (1966).
[CrossRef]

Gebhardt, F. C.

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

Gebhardt, F. G.

F. G. Gebhardt, D. C. Smith, United Aircraft Corporation, have completed a first order treatment essentially identical to the above. Experimental evidence for the correctness of the calculation in the form of measured half-moon isoirradiances in a forthcoming article will be published in Applied Optics. Our calculations were done independently with no previous communication.

Gerry, E.

E. Gerry, private communication.

Glass, A. J.

E. A. McLean, L. Sica, A. J. Glass, “Interferometric Observation of Absorption Induced Index Changes Associated with Thermal Blooming,” NRL Preprint (submitted to Appl. Phys. Lett.), Oct. 1969.

A. J. Glass, “Thermal Blooming in Gases,” Research Inst. for Eng. Sci., Wayne State U., Detroit (unpublished).

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Jorna, S.

K. A. Brueckner, S. Jorna, Phys. Rev. 164, 182 (1967).
[CrossRef]

Kelly, P. L.

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

R. Y. Chiao, P. L. Kelly, E. Garmire, Phys. Rev. Lett. 17, 1158 (1966).
[CrossRef]

P. L. Kelly, Phys. Rev. Lett. 15, 1003 (1965).
[CrossRef]

N. M. Kroll, P. L. Kelly, “Instabilities and Beam Spreading Due to Heat Deposition,” Appendix K, Rep. S-294, Laser Summer Study, Institute for Defense Analysis (1967).

Khokhlov, R. V.

S. A. Akmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp Fiz. Nauk 93, 19 (1967).

S. A. Akmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal Self Actions of Laser Beams,” Unpublished.

Krindach, D. P.

S. A. Akmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal Self Actions of Laser Beams,” Unpublished.

Kroll, N. M.

N. M. Kroll, J. Appl. Phys. 36, 34 (1965).
[CrossRef]

N. M. Kroll, P. L. Kelly, “Instabilities and Beam Spreading Due to Heat Deposition,” Appendix K, Rep. S-294, Laser Summer Study, Institute for Defense Analysis (1967).

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Litvak, A. G.

A. G. Litvak, JETP Lett. 4, 341 (1966).

McLean, E. A.

E. A. McLean, L. Sica, A. J. Glass, “Interferometric Observation of Absorption Induced Index Changes Associated with Thermal Blooming,” NRL Preprint (submitted to Appl. Phys. Lett.), Oct. 1969.

Migulin, A. V.

S. A. Akmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal Self Actions of Laser Beams,” Unpublished.

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Murphy, G. N.

See, for example, G. N. Murphy, Ordinary Differential Equations and Their Solutions (Van Nostrand, Princeton, N.J.1960), p. 352.

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Shen, Y. R.

Y. R. Shen, N. Bloembergen, Phys. Rev. 147, 1787, (1965).
[CrossRef]

Sica, L.

E. A. McLean, L. Sica, A. J. Glass, “Interferometric Observation of Absorption Induced Index Changes Associated with Thermal Blooming,” NRL Preprint (submitted to Appl. Phys. Lett.), Oct. 1969.

Smith, D. C.

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

F. G. Gebhardt, D. C. Smith, United Aircraft Corporation, have completed a first order treatment essentially identical to the above. Experimental evidence for the correctness of the calculation in the form of measured half-moon isoirradiances in a forthcoming article will be published in Applied Optics. Our calculations were done independently with no previous communication.

Sukhorukov, A. P.

S. A. Akmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp Fiz. Nauk 93, 19 (1967).

S. A. Akmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal Self Actions of Laser Beams,” Unpublished.

Talanov, V. I.

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

Yariv, A.

For example, A. Yariv, Quantum Electronics (Wiley, New York, 1968), Chap. 23 and 25.

Yih, Chia-Shun

Chia-Shun Yih, “Free Convection due to Boundary Sources,” in Fluid Models in Geophysics, No. 1956 (U.S. GPO, Washington, D.C., 1953), pp. 117–133.

Appl. Phys. Lett. (2)

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

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

J. Appl. Phys. (2)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[CrossRef]

N. M. Kroll, J. Appl. Phys. 36, 34 (1965).
[CrossRef]

JETP Lett. (2)

A. G. Litvak, JETP Lett. 4, 341 (1966).

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Phys. Rev. (2)

Y. R. Shen, N. Bloembergen, Phys. Rev. 147, 1787, (1965).
[CrossRef]

K. A. Brueckner, S. Jorna, Phys. Rev. 164, 182 (1967).
[CrossRef]

Phys. Rev. Lett. (2)

P. L. Kelly, Phys. Rev. Lett. 15, 1003 (1965).
[CrossRef]

R. Y. Chiao, P. L. Kelly, E. Garmire, Phys. Rev. Lett. 17, 1158 (1966).
[CrossRef]

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Usp Fiz. Nauk (1)

S. A. Akmanov, A. P. Sukhorukov, R. V. Khokhlov, Usp Fiz. Nauk 93, 19 (1967).

Other (10)

S. A. Akmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal Self Actions of Laser Beams,” Unpublished.

For example, A. Yariv, Quantum Electronics (Wiley, New York, 1968), Chap. 23 and 25.

N. M. Kroll, P. L. Kelly, “Instabilities and Beam Spreading Due to Heat Deposition,” Appendix K, Rep. S-294, Laser Summer Study, Institute for Defense Analysis (1967).

E. Gerry, private communication.

A. J. Glass, “Thermal Blooming in Gases,” Research Inst. for Eng. Sci., Wayne State U., Detroit (unpublished).

Work begun as a consultant to NRL.

See, for example, G. N. Murphy, Ordinary Differential Equations and Their Solutions (Van Nostrand, Princeton, N.J.1960), p. 352.

E. A. McLean, L. Sica, A. J. Glass, “Interferometric Observation of Absorption Induced Index Changes Associated with Thermal Blooming,” NRL Preprint (submitted to Appl. Phys. Lett.), Oct. 1969.

F. G. Gebhardt, D. C. Smith, United Aircraft Corporation, have completed a first order treatment essentially identical to the above. Experimental evidence for the correctness of the calculation in the form of measured half-moon isoirradiances in a forthcoming article will be published in Applied Optics. Our calculations were done independently with no previous communication.

Chia-Shun Yih, “Free Convection due to Boundary Sources,” in Fluid Models in Geophysics, No. 1956 (U.S. GPO, Washington, D.C., 1953), pp. 117–133.

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Figures (10)

Fig. 1
Fig. 1

Relative radial beam profile for a diffusion cooled beam.

Fig. 2
Fig. 2

Relative intensity profiles for diffusion cooling at several ranges.

Fig. 3
Fig. 3

K = Ξ I 0 π - 2 v η 0 d 0 [ Z - ( 1 - e - α z α ) ] 1 α. Relative isoirradiance contours for scaling parameter K = 0.01.

Fig. 4
Fig. 4

Relative isoirradiance contours for scaling parameter K = 0.2.

Fig. 5
Fig. 5

Relative isoirradiance contours for scaling parameter K = 0.6.

Fig. 6
Fig. 6

Relative isoirradiance contours for scaling parameter K = 1.00.

Fig. 7
Fig. 7

Relative isoirradiance contours for scaling parameter K = 1.8.

Fig. 8
Fig. 8

Peak relative power vs K.

Fig. 9
Fig. 9

Relative beam center deviation vs K.

Fig. 10
Fig. 10

Free convection distortion of A 1 kW/cm−2 beam at 250 cm.

Equations (117)

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S ( x , y , z ) = e ,             a constant .
u = S ,
( S ) 2 = η 2 = u 2
d u / d S = η .
s = ( z ^ + w ) / θ ,
θ 2 = 1 + w 2 .
( d / d z ) w = θ 2 ( ln η - w ( / z ) ( ln η ) ]
( d / d z ) ln θ = ( d / d z ) ( ln η ) = θ 2 ( / z ) ln η ,
· ( I s ) + α I = 0 ,
( d / d z ) ln ( I / θ ) + · w + α θ = 0
I = I ˜ ( θ / θ ˜ ) exp [ - α 0 z θ d z - 0 z ( · w ) d z ] .
( V · - a 2 ) δ η = - ( 1 2 ) Ξ I , steady - state .
a = λ / ( ρ 0 c p ) = 0.188 cm 2 / sec ( for air at STP )
Ξ = α η 0 ( η 0 2 - 1 ) ( γ - 1 ) ρ 0 v 2 s = 1.318 × 10 - 11 η 0 cm 2 / J .
2 ( / z ) w + w 2 = 2 η ,
( / η z ) I + I ( · w ) + w · I + α I = 0 ,
a 2 δ η = Ξ I / 2.
w = r ^ w r .
d / d z = / z .
h = ( 1 / r ) ( / r ) r w r .
d h d z h z = 1 r r r r δ η = Ξ I / 2 a .
d 2 h d z 2 2 h z 2 = Ξ 2 a ( - I h - α I )
d 2 h d z 2 + ( α + h ) d h d z = 0.
U = α + h .
U + U U = 0.
ln [ ( C 1 + U ) / ( C 1 - U ) ] = C 1 ( z + C 2 ) ,
U = C 1
C 2 = ( 1 / U ) ln [ ( U + U 0 ) / ( U - U 0 ) ] ,
h = - α ( h + α ) { [ ( h + h 0 ) / ( h - h 0 ) + 2 α ] e ( h + α ) z - 1 [ ( h + h 0 ) / ( h - h 0 ) + 2 α ] e ( h + α ) z + 1 } .
h = U z = 0 = 1 2 ( U 2 - U 2 ) z = 0 = Ξ 2 a I z = 0 .
U = h + α = [ ( Ξ I 0 / a ) + ( h 0 + α ) 2 ] 1 2 .
I / I 0 , max = N ( z ) ( 1 + A ) 2 e - r 2 - k z ( A + e - k z ) 2 ,
A = ( k + α ) / ( k - α ) ,
k = [ ( Ξ / a ) I 0 , max e - r 2 + α ) ] 1 2 ,
I / I 0 , max = exp ( - r 2 - α z ) ,             [ N ( z ) = 1 ] .
r 0 = [ ln ( Ξ 0 , max / a α 2 ) ] 1 2 ,
I 0 , max > 1.4 × 10 - 4 W / cm 2
A 1 and k [ ( Ξ I 0 , max / a ) e - r 2 ] 1 2 ,
I / I 0 , max = N ( z ) 4 exp ( - r 2 - H e - r 2 / 2 z ) / [ 1 + exp ( - H e - r 2 / 2 z ) ] 2 , H = ( Ξ I 0 , max / a ) 1 2 , N ( z ) = 1 2 ( H 2 z 2 4 ) [ H z 2 tan h ( H z 2 ) - log cosh ( H z 2 ) ] - 1 .
d d r ( I / I 0 , max ) = r [ - 1 + ( 1 2 H z e - r 2 / 2 ) tanh ( 1 2 H z e - r 2 / 2 ) ] = 0.
r u = 0 and r u = [ 2 ln ( H z / 2.4 ) ] 1 2 ,
H z > 2.4.
z > 2.4 / 2.64 × 10 - 4 = 9.05 × 10 3 cm
P e = r V / a = r V / 0.188.
V > 18.8 / r cm / sec ,
V · δ η = - 1 2 Ξ I .
q = z / d ,             u = x / d ,             and v = v / d
β = Ξ I 0 , max d 2 V η 0 = I 0 , max d 2 V × 1.318 × 10 - 11 ,
Q = - u exp [ - 0 q ( α d θ + · w ) d q ] I θ I 0 θ 0 d u .
d w / d q = - β θ 2 [ Q - w ( / q ) Q ] ,
( d / d q ) ln θ = - [ ( d Q / d q ) - θ 2 ( / q ) Q ] .
w = w 0 + β w 1 + β 2 w 2 + , θ = θ 0 + β θ 1 + , Q = Q 0 + β Q 1 + , d d q = q + w · = q + w 0 · + β w 1 · + = d / d q 0 + β w 1 · Δ +
d w 0 / d q 0 = 0 = d θ / d q 0 .
w 0 / q + w 0 ( / r ) = 0.
w 0 = ± r tan θ = ± c r ,
w 0 = ± c r 1 ± c q             + divergence - convergence ,
θ = 1 + w 0 2 = 1 + [ c 2 r 2 / ( 1 ± c q ) 2 ]
· w 0 = 2 c / 1 ± c q ,
0 q · w 0 d q = ln [ ( 1 ± c q ) 2 ] .
Q 0 = exp ( - v 2 - d α 0 q θ 0 d q ) ( 1 ± c q ) 2 ( - u e - t 2 d t ) ,
d d q 0 w 1 + w 1 · w 0 = - θ 0 2 ( Q 0 - w 0 q Q 0 )
d d q 0 θ 1 + w 1 · θ 0 = θ 0 ( d Q 0 d q 0 - θ 0 2 q Q 0 ) .
w 1 - θ 1 = 0 at q = 0.
( d / q ) w 1 = - Q 0 ,
Q 0 = e - α d q - v 2 - u e - t 2 d t .
· w 1 = π 1 2 ( 1 - e - α d q ) α d e - v 2 { 2 u e - u 2 π 1 2 - ( 2 v 2 - 1 ) [ ( 1 + erf ( u ) ] } .
I / I 0 = exp { - α d q - u 2 - v 2 - β π 1 2 e - v 2 × [ q α d - 1 ( α d ) 2 ( 1 - e - α d q ) ] [ 2 u π 1 2 e - u 2 - ( 2 v 2 - 1 ) F ] } ,
F = [ 1 + erf ( u ) ] .
u c = - β q 2 ( 1 - u c 2 ) e - u 2 c .
δ 2 F 2 F y 2 < 1 and δ F δ F y < 1 ,
( v x / x ) + ( v y / y ) = 0.
v x ( T / x ) + v y ( T / y ) = a ( 2 T / x 2 ) .
v x ( v y x ) + v y ( v y y ) = ν 2 v y 2 + g β T .
v y = ψ / x and v x = - ψ / y .
ψ y T x + ψ x T y = a 2 T x 2
- ψ y 2 ψ x 2 + ψ x 2 ψ y x = 2 x 2 ψ x + g β T .
v x = T x = x v y = 0 at x = 0
v y = T = 0 at x = ± .
x z plane dS · ( v T - a T ) = z 2 α I 0 , max d 0 c p π 0 x z plane ρ d ρ e - ρ 2 .
I = I 0 , max e - ρ 2
0 z d z - d x ( v y T - a T y ) = α z π I 0 , max d 0 c p .
v y > a | ln T y | .
- d x T ( x , y ) ψ x = π α I 0 , max d 0 c p .
ψ ( x , y ) = A y m U ( ξ ) ,
ξ = B x y - n ,
m U U - ( m - n ) U 2 = - ν B A U y - n - m + 1 - g β T A 2 B 2 y 1 + 2 n - 2 m .
n + m = 1
ν B / A = 1.
( 1 - n ) U U - ( 1 - 2 n ) U 2 = U - g β T ν 2 B 4 y 4 n - 1 .
θ ( ξ ) = g y 4 n - 1 β T ( x , y ) ν 2 B 4 .
( 1 - 2 n ) U 2 = ( 1 - n ) U U = U + θ ( ξ ) .
- ( 1 - n ) U θ + U θ ( 1 - 4 n ) = P θ .
P = a / ν .
ν 3 B 5 g β - d ξ y 2 - 5 n θ U = α π I 0 , max d 0 c p .
n = 2 5 ,
δ = B - 5 3 .
- d ξ θ ( ξ ) U ( ξ ) = g β π I 0 , max δ 3 / d 0 c p ν 3 .
3 U U - U 2 = - 5 U - 5 θ
- ( U θ + U θ ) = ( 5 3 ) P θ .
W = ( 1 5 ) ξ and Φ = 125 θ .
3 U ( W ) U ( W ) - U 2 ( W ) = - U ( W ) - Φ
U Φ + U Φ = - ( 1 3 ) P Φ ,
- d W Φ ( W ) U ( W ) = 125 ( β π α g I 0 , max δ 3 ) d 0 c p ν 3
Φ = H 4 5 F ; z = W H 1 5 ; U = u H 1 5 .
H = β g π I 0 , max d 0 c p α δ 3 ν 3 r 0 2 .
u F + u F = - ( 1 3 ) P F
3 u ( z ) u ( z ) - u 2 ( z ) = u ( z ) - F ,
u ( 0 ) = u ( 0 ) = F ( 0 ) = 0 ; u ( ± ) = F ( ± ) = 0.
u ( z ) = 2.24 tanh ( 1.863 z )
F ( z ) = 41.99 sech 2 ( 1.863 z ) .
δ η = - ( - 1 ) β T / 2.
δ η = - 9.74 × 10 - 5 ( r 0 / y ) 3 5 ( τ vis τ gr τ heat ) 1 2 sech 2 [ 0.3726 ( y / r 0 ) - 2 5 Z ] , Z = [ ( x / r 0 ) ( τ vis τ gr τ heat ) 1 5 ] .
τ g 1 = ( r 0 / g ) 1 2 ; τ = r 0 / ν ; and τ heat = d 0 c p / α I 0 , max
R = 2 δ η = { 4 B 2 y - 4 5 [ 1 - ( 3 2 ) sech 2 ( B x y - 2 5 ) ] + ( 24 25 ) y - 2 [ 1 - B 2 x 2 y - 4 5 ( 1 - 2 3 sech 2 ( B x y - 2 5 ) ) ] - ( 52 24 ) B x y - 2 5 tanh ( B x y - 2 5 ) } ,
I 0 , max = 1 kw / cm 2 , α = 8 × 10 - 7 cm - 1 , ν = 0.152 cm 2 sec - 1 , r 0 = 1 cm .
τ gr = 0.0319 ; τ heat = 262.0 ; τ vis = 6.583 ; A = - 1.703 × 10 - 7 ; and B = 1.503.
I / I 0 , max = exp [ - x 2 - y 2 - R ( x , y ) z 2 / 2 ] ,

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