Abstract

This paper is concerned with the propagation of laser light through a slab of a randomly varying medium. A theoretical analysis is presented which relates the spectrum of the recorded-intensity field some distance downstream of the medium to the spectrum of the index-of-refraction field. For a homogeneous and isotropic random field, the 3-D spectrum of the medium is obtained from the 2-D spectrum of the photograph by dividing each component of the spectrum by the frequency raised to the fourth power. Free-space propagation outside the random medium is accounted for by a scaling factor. Experimental results are presented which support the theoretical analysis. The nonintrusive diagnostic technique presented here is applicable to photographs which contain partially developed caustic networks.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. L. Brown, A. Roshko, J. Fluid Mech. 64, 775 (1974).
    [CrossRef]
  2. M. S. Uberoi, L. S. G. Kovasznay, J. Appl. Phys. 26, 19 (1954).
    [CrossRef]
  3. L. S. Taylor, AIAA J. 8, 1284 (1969).
  4. V. I. Tatarski, Wave Propagation In A Random Medium (McGraw-Hill, New York, 1960).
  5. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
  6. L. Hesselink, in Proceedings, Eleventh International Symposium on Shock Tubes and Waves, Seattle, Wash., July 1977.
  7. L. Hesselink, J. Fluid Mechanics, in preparation; B. Sturtevant, L. Hesselink, B. S. White, V. Kulkarny, C. Catherasoo, “Propagation of Shock Waves through Nonuniform and Random Media,” in Proceedings, Thirteenth International Symposium on Shock Tubes and Waves, July 1981, Niagara Falls, N.Y.
  8. M. V. Berry, Upstill, Prog. Opt. 18, 257 (1980).
    [CrossRef]
  9. V. Kulkarny, B. S. White, Phys. Fluids 25, 1770 (1982).
    [CrossRef]
  10. B. S. White, “The Stochastic Caustic,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. submitted.
  11. R. Buckley, Aust. J. Phys. 24, 351 (1971).
  12. R. Buckley, Aust. J. Phys. 24, 373 (1971).
  13. M. V. Berry, J. Phys. A. 10, 2061 (1977).
    [CrossRef]
  14. E. Jakeman, J. G. McWhirter, J. Phys. A. 10, 1599 (1977).
    [CrossRef]
  15. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960).
    [CrossRef]
  16. M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 439 (1958).
    [CrossRef]
  17. M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 91 (1951).
  18. M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 27 (1960).
    [CrossRef]
  19. Y. A. Kravtsov, Zh. Eksp. Teor. Fiz. 55, 798 (1969) [Soviet Phys. JETP 23, 413 (1969)].

1982

V. Kulkarny, B. S. White, Phys. Fluids 25, 1770 (1982).
[CrossRef]

1980

M. V. Berry, Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

1977

M. V. Berry, J. Phys. A. 10, 2061 (1977).
[CrossRef]

E. Jakeman, J. G. McWhirter, J. Phys. A. 10, 1599 (1977).
[CrossRef]

1974

G. L. Brown, A. Roshko, J. Fluid Mech. 64, 775 (1974).
[CrossRef]

1971

R. Buckley, Aust. J. Phys. 24, 351 (1971).

R. Buckley, Aust. J. Phys. 24, 373 (1971).

1969

Y. A. Kravtsov, Zh. Eksp. Teor. Fiz. 55, 798 (1969) [Soviet Phys. JETP 23, 413 (1969)].

L. S. Taylor, AIAA J. 8, 1284 (1969).

1960

M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960).
[CrossRef]

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 27 (1960).
[CrossRef]

1958

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 439 (1958).
[CrossRef]

1954

M. S. Uberoi, L. S. G. Kovasznay, J. Appl. Phys. 26, 19 (1954).
[CrossRef]

1951

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 91 (1951).

Berry, M. V.

M. V. Berry, Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

M. V. Berry, J. Phys. A. 10, 2061 (1977).
[CrossRef]

Brown, G. L.

G. L. Brown, A. Roshko, J. Fluid Mech. 64, 775 (1974).
[CrossRef]

Buckley, R.

R. Buckley, Aust. J. Phys. 24, 351 (1971).

R. Buckley, Aust. J. Phys. 24, 373 (1971).

Hesselink, L.

L. Hesselink, in Proceedings, Eleventh International Symposium on Shock Tubes and Waves, Seattle, Wash., July 1977.

L. Hesselink, J. Fluid Mechanics, in preparation; B. Sturtevant, L. Hesselink, B. S. White, V. Kulkarny, C. Catherasoo, “Propagation of Shock Waves through Nonuniform and Random Media,” in Proceedings, Thirteenth International Symposium on Shock Tubes and Waves, July 1981, Niagara Falls, N.Y.

Jakeman, E.

E. Jakeman, J. G. McWhirter, J. Phys. A. 10, 1599 (1977).
[CrossRef]

Kovasznay, L. S. G.

M. S. Uberoi, L. S. G. Kovasznay, J. Appl. Phys. 26, 19 (1954).
[CrossRef]

Kravtsov, Y. A.

Y. A. Kravtsov, Zh. Eksp. Teor. Fiz. 55, 798 (1969) [Soviet Phys. JETP 23, 413 (1969)].

Kulkarny, V.

V. Kulkarny, B. S. White, Phys. Fluids 25, 1770 (1982).
[CrossRef]

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 27 (1960).
[CrossRef]

M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960).
[CrossRef]

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 439 (1958).
[CrossRef]

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 91 (1951).

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, J. Phys. A. 10, 1599 (1977).
[CrossRef]

Roshko, A.

G. L. Brown, A. Roshko, J. Fluid Mech. 64, 775 (1974).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation In A Random Medium (McGraw-Hill, New York, 1960).

Taylor, L. S.

L. S. Taylor, AIAA J. 8, 1284 (1969).

Uberoi, M. S.

M. S. Uberoi, L. S. G. Kovasznay, J. Appl. Phys. 26, 19 (1954).
[CrossRef]

Upstill,

M. V. Berry, Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

White, B. S.

V. Kulkarny, B. S. White, Phys. Fluids 25, 1770 (1982).
[CrossRef]

B. S. White, “The Stochastic Caustic,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. submitted.

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

AIAA J.

L. S. Taylor, AIAA J. 8, 1284 (1969).

Aust. J. Phys.

R. Buckley, Aust. J. Phys. 24, 351 (1971).

R. Buckley, Aust. J. Phys. 24, 373 (1971).

J. Appl. Phys.

M. S. Uberoi, L. S. G. Kovasznay, J. Appl. Phys. 26, 19 (1954).
[CrossRef]

J. Fluid Mech.

G. L. Brown, A. Roshko, J. Fluid Mech. 64, 775 (1974).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. A.

M. V. Berry, J. Phys. A. 10, 2061 (1977).
[CrossRef]

E. Jakeman, J. G. McWhirter, J. Phys. A. 10, 1599 (1977).
[CrossRef]

Phys. Fluids

V. Kulkarny, B. S. White, Phys. Fluids 25, 1770 (1982).
[CrossRef]

Proc. Comb. Phil. Soc.

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 439 (1958).
[CrossRef]

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 91 (1951).

M. S. Longuet-Higgins, Proc. Comb. Phil. Soc. 54, 27 (1960).
[CrossRef]

Prog. Opt.

M. V. Berry, Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

Zh. Eksp. Teor. Fiz.

Y. A. Kravtsov, Zh. Eksp. Teor. Fiz. 55, 798 (1969) [Soviet Phys. JETP 23, 413 (1969)].

Other

B. S. White, “The Stochastic Caustic,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. submitted.

V. I. Tatarski, Wave Propagation In A Random Medium (McGraw-Hill, New York, 1960).

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

L. Hesselink, in Proceedings, Eleventh International Symposium on Shock Tubes and Waves, Seattle, Wash., July 1977.

L. Hesselink, J. Fluid Mechanics, in preparation; B. Sturtevant, L. Hesselink, B. S. White, V. Kulkarny, C. Catherasoo, “Propagation of Shock Waves through Nonuniform and Random Media,” in Proceedings, Thirteenth International Symposium on Shock Tubes and Waves, July 1981, Niagara Falls, N.Y.

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

Optical configuration.

Fig. 2
Fig. 2

Experimental apparatus for generating the random medium.

Fig. 3
Fig. 3

Representative shadowgraph photographs.

Fig. 4
Fig. 4

Power spectral density curves.

Fig. 5
Fig. 5

Shadowgraph photograph obtained at the exit window (background noise not removed).

Equations (33)

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

Δ u ^ + k 2 ( 1 + μ ) 2 u ^ = 0 ,
u ^ ( x , y , 0 ) = 1.
I ( x , y , z ) = B 0 T u ^ ( t , x , y , z ) 2 d t = B T u ^ 2 ,
I ( x , y , z ) = B t - T t U ( t - t ) exp [ - β ( t - t ) ] u ^ ( t , x , y , z ) 2 d t = B β [ 1 - exp ( - β T ) ] u ^ 2 = B T u ^ 2 ,
= 3 × 10 - 5             ω = 3 × 10 15             k = 10 5 cm - 1 c = 3 × 10 10 cm / sec             D = 26 cm .
u ^ ~ exp [ i k ϕ ( x , y , z ) ] n = 0 k - n V ( n ) ( x , y , z ) ,
I ( x , y , D ) = A 2 B T { 1 - α k z D d z [ ( D ^ - z ) ( 2 x 2 + 2 y 2 ) + z ] μ ( x , y , z ) } + 0 ( 1 k 2 ) ,
E [ I ( x , y ) ] = A 2 B T ,
R I ( ρ ) = E { [ I ( x 1 , y 1 ) - A 2 B T ] [ I ( x 2 , y 2 ) - A 2 B T ] } = ( α A 2 B T k ) 2 0 D d z 1 0 D d z 2 × [ ( D ^ - z 1 ) ( 2 x 1 2 + 2 y 1 2 ) + z 1 ] × [ ( D ^ - z 2 ) ( 2 x 2 2 + 2 y 2 2 ) + z 2 ] × { E [ μ ( x 1 , y 1 , z 1 ) μ ( x 2 , y 2 , z 2 ) ] } .
Δ ρ = 1 ρ ρ ρ ρ .
R I ( ρ ) = β 0 D d z 1 0 D d z 2 [ ( D ^ - z 1 ) Δ ρ + z 1 ] × [ ( D ^ - z 2 ) Δ ρ + z 2 ] R [ ρ 2 + ( z 1 - z 2 ) 2 ] .
R I ( ρ ) = β D ^ - D D ^ d z 1 D ^ - D D ^ d z 2 ( z 1 Δ ρ - z 1 ) ( z 2 Δ ρ - z 2 ) × R [ ρ 2 + ( z 1 - z 2 ) 2 ] .
R I ( ρ ) = 2 β Δ ρ 2 D ^ - D D ^ d z 1 D ^ - D z 1 d z 2 z 1 z 2 R [ ρ 2 + ( z 1 - z 2 ) 2 ] - 2 β Δ ρ D ^ - D D ^ d z 1 D ^ - D D ^ d z 2 z 1 z 2 R [ ρ 2 + ( z 1 - z 2 ) 2 ] + β D ^ - D D ^ d z 1 D ^ - D D ^ d z 2 2 z 1 z 2 R [ ρ 2 + ( z 1 - z 2 ) 2 ] .
( ξ = z 1 + z 2 z = z 1 - z 2 ) and [ z 1 = ½ ( ξ + z ) z 2 = ½ ( ξ - z ) ] .
J = ( ξ , z ) ( z 1 , z 2 ) = | 1 1 1 - 1 | = - 2.
R I ( ρ ) = 2 β Δ ρ 2 0 D d z 2 D ^ - 2 D + z 2 D ^ - z d ξ 1 4 ( ξ 2 - z 2 ) · 1 2 R ( ρ 2 + z 2 ) - 2 β Δ ρ D ^ - D D ^ d z 1 z 1 { R [ ρ 2 + ( D ^ - z 1 ) 2 ] - R [ ρ 2 + ( D ^ - D - z 1 ) 2 ] } + β D ^ - D D ^ d z 1 z 1 { R [ ρ 2 + ( D ^ - z 1 ) 2 ] - R [ ρ 2 + ( D ^ - D - z 1 ) 2 ] } .
R I ( ρ ) = 1 4 β Δ ρ 2 0 D d z ( ξ 3 3 - z 2 ξ ) ξ = 2 D ^ - 2 D + z 2 D ^ - z R ( ρ 2 + z 2 ) - 2 β Δ ρ [ 0 D d z ( D ^ - z ) R ( ρ 2 + z 2 ) - 0 D d z ( z + D ^ - D ) R ( ρ 2 + z 2 ) ] + β [ R ( ρ ) - R ( ρ 2 + D 2 ) - R ( ρ 2 + D 2 ) + R ( ρ ) ]
R I ( ρ ) = β Δ ρ 2 0 D d z [ ( D - z ) 2 ( 2 D + z ) 3 + 2 D ^ ( D ^ - D ) ( D - z ) ] R ( ρ 2 + z 2 ) - 2 β Δ ρ 0 D d z ( D - 2 z ) R ( ρ 2 + z 2 ) + 2 β [ R ( ρ ) - R ( ρ 2 + D 2 ) ] ,
β = ( α A 2 B T k ) 2 , R ( r ) = E [ μ ( x ) μ ( x + r ) ] , Δ ρ = 1 ρ ρ ρ ρ .
S ( q ) = γ S I ( q ) q 4 + q K ( q , ξ ) S ( ξ ) d ξ ,
K ( q , ξ ) = ξ [ h 1 ( ξ 2 - q 2 ) + 1 q 2 h 2 ( ξ 2 - q 2 ) + 1 q 4 h 3 ( ξ 2 - q 2 ) ] , h 1 ( q ) = γ q 0 D J 1 ( z q ) [ z ( D - z ) 2 3 ( 2 D + z ) + 2 z D ^ ( D ^ - z ) ] d z , h 2 ( q ) = 2 γ q 0 D J 1 ( z q ) z ( D - 2 z ) d z , h 3 ( q ) = - 2 D γ q J 1 ( D q ) , γ = [ D 4 4 + D 2 D ^ ( D ^ - D ) ] - 1 ,
- 3 S I ( q ) 2 [ 1 + 3 δ ( 1 + δ ) ] D 3 q 3 = 0 q S ( ξ 2 + q 2 ) ξ 2 + q 2 d ξ ,
S 3 ( q ) = 4 π q 0 ρ sin ( ρ q ) R ¯ ( ρ ) d ρ ,
S 3 ( q ) = 6 π S I ( q ) [ 1 + 3 δ ( 1 + δ ) ] D 3 q 4 .
S M = S R + Δ S = α q 4 S + Δ S ,
S I I I = S M / q 4 = S + Δ S / q 4 .
s = 2 / 3 σ 2 / 3 z ,
σ 2 = 6 0 1 r r [ 1 r r ] R ( r ) d r ,
R = exp ( - r 2 2 l 2 ) ,
f = 0.7146 l 2 / 3 .
d focus = { [ 1 / 8 ( π ) 1 / 2 ] l 3 / D 2 } 1 / 2 .
z k L 2 L 4 ϕ 0 ,
L n - n = ( - 1 r r r r ) 1 / 2 n [ R ( r ) ] r = 0 ,

Metrics