Abstract

The effect of flow motion on a volume hologram written in a fluid medium is studied. The hologram is written in a sensitized fluid by either thermal absorption or the photochromic effect or by other mechanisms. After being written, the hologram is deformed because of the motion of the supporting fluid. The effect of several representative fluid motions on the hologram deformation, i.e., a plane shear flow and a symmetric plane jet, is found by solving the associated convection-diffusion equation. The effect of diffusion on the fading of the hologram is also studied for several diffusion rates corresponding to different hologram-writing mechanisms. The result of the calculation is the determination of the new shape and intensity of the hologram as functions of time. The analysis is carried out entirely in the Fourier domain, where the convection–diffusion equation is solved more easily. The output is therefore given in terms of spatial Fourier components that represent the deformed and convected hologram. The use of volume holograms for the determination of velocity gradients corresponding to small-scale fluid motion is discussed. In this paper we describe the deformation process that the hologram undertakes under the effect of fluid motion. In part II of this series [ J. Opt. Soc. Am. A 5, 1297 ( 1988)] we describe a method for predicting the diffracting characteristics of strained holograms. That method directly incorporates the results presented here.

© 1988 Optical Society of America

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References

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  1. H. J. Eichler, P. Günter, D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
    [Crossref]
  2. M. Cloitre, E. Guyon, “Forced Rayleigh scattering in turbulent plane Poiseuille flows. I. Study of the transverse velocity-gradient component,”J. Fluid Mech. 164, 217–236 (1986).
    [Crossref]
  3. W. J. Tomlinson, “Volume holograms in photochromic materials,” Appl. Opt. 14, 2456–2467 (1975).
    [Crossref] [PubMed]
  4. J. C. Agüí, J. Hesselink, “Holographic measurements of velocity gradients in fluids,” Bull. Am. Phys. Soc. 32, 2106 (1987).
  5. J. C. Agüí, L. Hesselink, “Holograms in motion. II. Diffracting capabilities of strained holograms,” J. Opt. Soc. Am. A 5, 1297–1308 (1988).
    [Crossref]
  6. H. J. Eichler, “Forced light scattering at laser induced gratings—a method for investigation of optically excited solids,” Festkoerperprobleme 8, 241–263 (1978).
  7. J. C. Agüí, J. Hesselink, “Volume holography in flowing liquids,” J. Opt. Soc. Am. A 4(13), P76 (1987).
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–267 (1956).
    [Crossref]
  10. H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).
  11. G. H. Brown, ed., Photochromism, Vol. III of Techniques of Chemistry (Wiley-Interscience, New York, 1971).
  12. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).
  13. R. N. Bracewell, Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  14. R. S. Rogallo, “Numerical experiments in homogeneous turbulence,” Tech. Rep. Tm 81315 (National Aeronautics and Space Administration, Ames Research Center, Moffett Field, Calif., 1981).
  15. J. H. Ferziger, Numerical Methods for Engineering Applications (Wiley-Interscience, New York, 1981).

1988 (1)

1987 (2)

J. C. Agüí, J. Hesselink, “Volume holography in flowing liquids,” J. Opt. Soc. Am. A 4(13), P76 (1987).

J. C. Agüí, J. Hesselink, “Holographic measurements of velocity gradients in fluids,” Bull. Am. Phys. Soc. 32, 2106 (1987).

1986 (1)

M. Cloitre, E. Guyon, “Forced Rayleigh scattering in turbulent plane Poiseuille flows. I. Study of the transverse velocity-gradient component,”J. Fluid Mech. 164, 217–236 (1986).
[Crossref]

1978 (1)

H. J. Eichler, “Forced light scattering at laser induced gratings—a method for investigation of optically excited solids,” Festkoerperprobleme 8, 241–263 (1978).

1975 (1)

1956 (1)

J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–267 (1956).
[Crossref]

Agüí, J. C.

J. C. Agüí, L. Hesselink, “Holograms in motion. II. Diffracting capabilities of strained holograms,” J. Opt. Soc. Am. A 5, 1297–1308 (1988).
[Crossref]

J. C. Agüí, J. Hesselink, “Holographic measurements of velocity gradients in fluids,” Bull. Am. Phys. Soc. 32, 2106 (1987).

J. C. Agüí, J. Hesselink, “Volume holography in flowing liquids,” J. Opt. Soc. Am. A 4(13), P76 (1987).

Bracewell, R. N.

R. N. Bracewell, Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Cloitre, M.

M. Cloitre, E. Guyon, “Forced Rayleigh scattering in turbulent plane Poiseuille flows. I. Study of the transverse velocity-gradient component,”J. Fluid Mech. 164, 217–236 (1986).
[Crossref]

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Eichler, H. J.

H. J. Eichler, “Forced light scattering at laser induced gratings—a method for investigation of optically excited solids,” Festkoerperprobleme 8, 241–263 (1978).

H. J. Eichler, P. Günter, D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

Ferziger, J. H.

J. H. Ferziger, Numerical Methods for Engineering Applications (Wiley-Interscience, New York, 1981).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Günter, P.

H. J. Eichler, P. Günter, D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

Guyon, E.

M. Cloitre, E. Guyon, “Forced Rayleigh scattering in turbulent plane Poiseuille flows. I. Study of the transverse velocity-gradient component,”J. Fluid Mech. 164, 217–236 (1986).
[Crossref]

Hesselink, J.

J. C. Agüí, J. Hesselink, “Holographic measurements of velocity gradients in fluids,” Bull. Am. Phys. Soc. 32, 2106 (1987).

J. C. Agüí, J. Hesselink, “Volume holography in flowing liquids,” J. Opt. Soc. Am. A 4(13), P76 (1987).

Hesselink, L.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Lumley, J. L.

H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).

Pohl, D. W.

H. J. Eichler, P. Günter, D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

Ratcliffe, J. A.

J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–267 (1956).
[Crossref]

Rogallo, R. S.

R. S. Rogallo, “Numerical experiments in homogeneous turbulence,” Tech. Rep. Tm 81315 (National Aeronautics and Space Administration, Ames Research Center, Moffett Field, Calif., 1981).

Tennekes, H.

H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).

Tomlinson, W. J.

Appl. Opt. (1)

Bull. Am. Phys. Soc. (1)

J. C. Agüí, J. Hesselink, “Holographic measurements of velocity gradients in fluids,” Bull. Am. Phys. Soc. 32, 2106 (1987).

Festkoerperprobleme (1)

H. J. Eichler, “Forced light scattering at laser induced gratings—a method for investigation of optically excited solids,” Festkoerperprobleme 8, 241–263 (1978).

J. Fluid Mech. (1)

M. Cloitre, E. Guyon, “Forced Rayleigh scattering in turbulent plane Poiseuille flows. I. Study of the transverse velocity-gradient component,”J. Fluid Mech. 164, 217–236 (1986).
[Crossref]

J. Opt. Soc. Am. A (2)

J. C. Agüí, L. Hesselink, “Holograms in motion. II. Diffracting capabilities of strained holograms,” J. Opt. Soc. Am. A 5, 1297–1308 (1988).
[Crossref]

J. C. Agüí, J. Hesselink, “Volume holography in flowing liquids,” J. Opt. Soc. Am. A 4(13), P76 (1987).

Rep. Prog. Phys. (1)

J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–267 (1956).
[Crossref]

Other (8)

H. Tennekes, J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).

G. H. Brown, ed., Photochromism, Vol. III of Techniques of Chemistry (Wiley-Interscience, New York, 1971).

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

R. N. Bracewell, Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

R. S. Rogallo, “Numerical experiments in homogeneous turbulence,” Tech. Rep. Tm 81315 (National Aeronautics and Space Administration, Ames Research Center, Moffett Field, Calif., 1981).

J. H. Ferziger, Numerical Methods for Engineering Applications (Wiley-Interscience, New York, 1981).

H. J. Eichler, P. Günter, D. W. Pohl, Laser Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (8)

Fig. 1
Fig. 1

Plane flows considered and the relative size between hologram and flow. (a) Shear flow. (b) Symmetric flow. Flow is everywhere parallel to the x axis and varies along the z coordinate only. h represents the characteristic size of the hologram, and hf represents that of the flow.

Fig. 2
Fig. 2

Hologram geometry at initial time, sectioned to show the even spacing of fringes along the x direction. Characteristic dimensions along the axes are hx, hy, and hz. The Kg vector represents the grating direction and spacing. M fringes are contained normal to the x axis.

Fig. 3
Fig. 3

Contour values of the magnitude of the Fourier transform of the hologram in a section through the symmetry plane K2 = 0, with antisymmetric flow. K2 = 0. The size ratio hz/h is 3.0, 1.0, and 0.3. t = tU0/hf = 0., 0.05, and 0.1. No diffusion is permitted.

Fig. 4
Fig. 4

Evolution of the magnitude of the hologram distribution in Fourier space. Parameters are similar to those for Fig. 3, with symmetric flow.

Fig. 5
Fig. 5

(a) Displacement of the center of gravity of the magnitude of the hologram Fourier transform along the K3 direction. Values of hf/h: Δ, 3.0; ○, 1.0; □, 0.3. (b) Second K3 moment of the distribution around the center of gravity, with (solid lines) antisymmetric flow and (dashed lines) symmetric flow. Identical labeling for hf/h.

Fig. 6
Fig. 6

Comparison between the solution of the full convection–diffusion equation with (solid lines) α = 0.0 and (dashed lines) the Lagrangian approach [Eq. (16)]; hf/h1 = 1.0, t ¯ = 0.1.

Fig. 7
Fig. 7

Evolution of diffusive holograms. Parameters identical to central case of Fig. 3. hz/h = 1.0, α = 0.0, 0.0005, and 0.001 from bottom to top.

Fig. 8
Fig. 8

Evolution of the relative energy of the Fourier transforms under an antisymmetric flow. hf/h = 1. α values: □, 0.0; Δ, 0.0005; ⋄, 0.001; ○, 0.005.

Equations (29)

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W ( x , t ) t + u · W ( x , t ) α 2 W ( x , t ) = 0 ,
W ˆ ( K , t ) = ( 2 π ) 3 / 2 x W ( x , t ) exp ( i K · x ) d x ,
W ˆ ( K , t ) t i u ˆ j ( K , T ) [ K j W ˆ ( K , t ) ] + α K 2 W ˆ ( K , t ) = 0 ,
u ˆ j ( K , t ) [ K j W ˆ ( K , t ) ] = ( 2 π ) 3 / 2 K u ˆ j ( K , t ) ( K K ) j W ˆ ( K K , t ) d K .
U ( x , t ) = U 0 tanh ( z h f ) e ˆ x ,
u ˆ 1 ( K , t ) = i π 3 / 2 2 U 0 h f δ ( K 1 , K 2 ) cosech ( π 2 K 3 h f ) e ˆ k 1 ,
U ( x , t ) = U 0 exp [ ( z h f ) 2 ] e ˆ x ,
u ˆ 1 ( K , t ) = π 2 U 0 h f δ ( K 1 , K 2 ) exp [ ( K 3 h f 2 ) 2 ] e ˆ k 1 .
( t + α K 2 ) W ˆ ( K , t ) + C a U 0 K 1 h f × W ˆ ( K 1 , K 2 , K 3 ) F [ h f ( K 3 K 3 ) ] d K 3 = 0 ,
W ( x , t = 0 ) = W 0 exp [ ( x h x ) 2 ( y h y ) 2 ( z h z ) 2 ] cos ( K g x ) ,
W ˆ ( K , 0 ) = W 0 2 5 / 2 h x h y h z exp [ ( K 2 h y 2 ) 2 ( 2 ) 2 ] × { exp [ ( ( K 1 K g ) h x 2 ) 2 ] + exp [ ( ( K 1 + K g ) h x 2 ) 2 ] } .
[ t + α ( ξ 1 2 + ξ 2 2 + ξ 3 2 ) ] f ( ξ 1 , ξ 2 , ξ 3 , t ) + C a ξ 1 + f ( ξ 1 , ξ 2 , ξ 3 , t ) F ( ξ 3 ξ 3 ) d ξ 3 = 0 ,
f ( ξ 1 , ξ 2 , ξ 3 , t = 0 ) = exp [ ( ξ 3 z 2 ) 2 ( ξ 2 y 2 ) 2 ] × ( exp { [ ( ξ 1 ξ g ) x 2 ] 2 } + exp { [ ( ξ 1 ξ g ) x 2 ] 2 } ) ,
Δ f i , j ( n ) [ 1 + Δ t 2 ( ξ 1 2 + ξ 2 2 + ξ 3 2 ) i , j ] + C a Δ t 2 ( ξ 1 ) i Δ f i , k ( n ) λ k j = Δ t ( ξ 1 2 + ξ 2 2 + ξ 3 2 ) i , j f i j ( ) C a Δ t ( ξ 1 ) i f i , k ( n ) λ k j ,
+ f ( ξ 1 , ξ 2 , ξ 3 , n Δ t ) F ( ξ 3 ξ 3 ) d ξ 3 = N b N b f i , k ( n ) λ k j = f i , k ( n ) β k j Δ ξ F [ ( ξ 3 ) j ( ξ 3 ) k ] ,
D D t W ( x , t ) = 0 ,
W ( x , t ) = W ( x 0 , t = 0 ) ,
η = ( ν 3 ε ) 1 / 4 , τ = ( ν ε ) 1 / 2 ,
α = α ν h h f .
τ 1 α h f h ( 2 π M / x ) 2 ,
M n m = W ˆ ( K , t ) W ˆ * ( K , t ) K n K m d K ,
W ( x , Δ t ) = W ( x U ˆ ( x , 0 ) Δ t , 0 ) ,
M n m ( Δ t ) = i ( 2 π ) 3 d x W ( x , 0 ) d x W ( x , 0 ) K n K m × exp { i K [ U ˆ ( x , 0 ) U ˆ ( x , 0 ) ] Δ t } exp [ i K ( x x ) ] d K .
( M n m Δ t ) t = 0 = W ( x , 0 ) x n x m [ U i ( x , 0 ) W ( x , 0 ) x i ] d x .
γ M x z M x x .
U ( z ) / U 0 = a 0 + a 1 ( z / h f ) + a 2 ( z / h f ) 2 + a 3 ( z / h f ) 3 + ,
γ t = n = 1 , , ( 2 ) a n ( h z h f ) n n ! ! 2 n ,
U = U 0 ( h f h z ) 1 / 3 sin ( 2 π z h f ) ,
α α t = 2 π ( h f h z ) ( 2 / 3 ) exp ( π 2 h 2 2 h f 2 ) ,

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