Abstract

A general procedure for the analysis of the diffraction capabilities of a three-dimensional low-efficiency hologram is presented. Tightly coupled to a description of the hologram in terms of its three-dimensional spatial Fourier modes, the procedure uses the angular spectrum theory for decomposing the reading light into plane waves. The convolution in the Fourier domain between the two Fourier distributions produces the angular spectrum of the diffracted light. Of special interest are the angular moments of the diffracted light, as a means of detecting the average orientation and distortion of the fringes in the hologram. These vary, in the case studied here, as a consequence of the uneven motion of the supporting media, which subjects the hologram to convection, rotation, and deformation. Diffusion takes place simultaneously, reducing the hologram modulation. Computed time evolutions of diffracted spots from holograms deformed by simple case flows are presented. In this paper we deal only with the readout process of deformed holograms; the effect of the fluid motion on the hologram is studied in detail in the first part of this series [ J. Opt. Soc. Am. A 5, 1287 ( 1988)].

© 1988 Optical Society of America

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References

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  1. J. C. Agüí, J. Hesselink, “Holograms in motion. I. Effect of fluid motion on volume holograms,” J. Opt. Soc. Am. A 5, 1287–1296 (1988).
    [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. J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2698–2710 (1971).
    [CrossRef] [PubMed]
  4. H. W. Holloway, R. A. Ferrante, “Computer analysis of holographic systems by means of vector ray tracing,” Appl. Opt. 20, 2081–2084 (1981).
    [CrossRef] [PubMed]
  5. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Diffraction characteristics of three-dimensional crossed-beam gratings,”J. Opt. Soc. Am. 70, 437–442 (1980).
    [CrossRef]
  6. P. Russel, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta, 26, 329–347 (1979).
    [CrossRef]
  7. A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,”J. Opt. Soc. Am. 67, 545–550 (1977).
    [CrossRef]
  8. J. C. Agüí, J. Hesselink, “Holographic measurements of velocity gradients in fluids,” Bull. Am. Phys. Soc. 32, 2106 (1987).
  9. H. M. Smith, Principles of Holography (Wiley-Interscience, New York, 1969).
  10. L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).
  11. M. VanDyke, Perturbation Methods in Fluid Mechanics (Parabolic, Stanford, Calif., 1975).
  12. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. Here a slight deviation from the standard definition of the angular spectrum14 is taken, since no clear axis of propagation is present in this case. The modification consists of not projecting the sphere in which the Ki vector lies upon a plane normal to the propagation direction. In this way, the arbitrary choice of an artificial propagation direction is avoided, as is the computational difficulty arising from rays that may be nearly parallel to the projection plane. In this new twist, the angular spectrum becomes almost undistinguishable from the three-dimensional Fourier transform.
  15. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  16. R. N. Bracewell, Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  17. G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974).

1988 (1)

1987 (1)

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]

1981 (1)

1980 (1)

1979 (1)

P. Russel, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta, 26, 329–347 (1979).
[CrossRef]

1977 (1)

1971 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Agüí, J. C.

J. C. Agüí, J. Hesselink, “Holograms in motion. I. Effect of fluid motion on volume holograms,” J. Opt. Soc. Am. A 5, 1287–1296 (1988).
[CrossRef]

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

Bracewell, R. N.

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

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]

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

Ferrante, R. A.

Gaylord, T. K.

Goodman, J. W.

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

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, “Holograms in motion. I. Effect of fluid motion on volume holograms,” J. Opt. Soc. Am. A 5, 1287–1296 (1988).
[CrossRef]

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

Holloway, H. W.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Latta, J. N.

Magnusson, R.

Moharam, M. G.

Russel, P.

P. Russel, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta, 26, 329–347 (1979).
[CrossRef]

Siegman, A. E.

Smith, H. M.

H. M. Smith, Principles of Holography (Wiley-Interscience, New York, 1969).

Solymar, L.

P. Russel, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta, 26, 329–347 (1979).
[CrossRef]

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

VanDyke, M.

M. VanDyke, Perturbation Methods in Fluid Mechanics (Parabolic, Stanford, Calif., 1975).

Whitham, G. B.

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

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Bull. Am. Phys. Soc. (1)

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

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. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

P. Russel, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta, 26, 329–347 (1979).
[CrossRef]

Other (8)

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

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

H. M. Smith, Principles of Holography (Wiley-Interscience, New York, 1969).

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981).

M. VanDyke, Perturbation Methods in Fluid Mechanics (Parabolic, Stanford, Calif., 1975).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

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

Here a slight deviation from the standard definition of the angular spectrum14 is taken, since no clear axis of propagation is present in this case. The modification consists of not projecting the sphere in which the Ki vector lies upon a plane normal to the propagation direction. In this way, the arbitrary choice of an artificial propagation direction is avoided, as is the computational difficulty arising from rays that may be nearly parallel to the projection plane. In this new twist, the angular spectrum becomes almost undistinguishable from the three-dimensional Fourier transform.

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Figures (9)

Fig. 1
Fig. 1

Geometrical interpretation of the Bragg condition (a) as imposed by a given irradiated light Ki, where there is reconstruction for all Kg lying upon the dashed sphere, and (b) as imposed by a given hologram Fourier mode Kg, where there is reconstruction for all the plane waves Ki lying in a cone centered around Kg.

Fig. 2
Fig. 2

Discretization of the incoming light angular spectrum into patches upon the sphere of radius 2π/λ.

Fig. 3
Fig. 3

Perspective plot of the values of the Fourier transform of the hologram in a section through the symmetry plane K2 = 0, with antisymmetric flow. The size ratio hf/h is 1.0t = tU0/hf = 0.1. Data are from part I.1

Fig. 4
Fig. 4

Reading geometry. The thin arrow denotes the time evolution of the hologram Fourier distribution. (a) The hologram moves almost normally to the Bragg-reconstruction sphere. (b) The hologram moves tangentially to the sphere.

Fig. 5
Fig. 5

Section at the central azimuth of the diffracted spot, with diffracted field intensity and antisymmetric flow. The reading geometry is as in Fig. 4(b) with no diffusion. (a), (d) hf/h = 3.0; (b), (e) hf/h = 1.0; (c), (f) hf/h = 0.3.

Fig. 6
Fig. 6

Section at the central azimuth of the diffracted spot, with diffracted field intensity and symmetric flow. The reading geometry is as in Fig. 4(b) with no diffusion. (a), (d) hf/h = 3.0; (b), (e) hf/h = 1.0; (c), (f) hf/h = 0.3.

Fig. 7
Fig. 7

Evolution of the angular moments of the diffracted light. (a) Center of gravity, with antisymmetric flow. (b) Characteristic width, with (solid lines) antisymmetric flow and (dashed lines) symmetric flow. □, hf/h = 3.0; ○, hf/h = 1.0; Δ, hf/h = 0.3.

Fig. 8
Fig. 8

Evolution of diffusive holograms. Parameters are as in Figs. 5(b) and 5(e). (a), (d) α = 0.0005; (b), (e) α = 0.001; (c), (f) α = 0.005.

Fig. 9
Fig. 9

Evolution of holograms under the reading geometry of Fig. 4(a). Parameters are equivalent to those given for Figs. 5(a) and 5(d).

Equations (39)

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2 E γ 2 E = 0 ,
γ 2 = β 2 + 2 i α β 4 κ β F ( x , t ) ,
κ = r 1 β 4 r 0 + i σ 1 4 ( μ / 0 r 0 ) 1 / 2 ,
E = E 0 + E 1 κ + E 2 κ 2 + .
2 E 0 ( β 2 + 2 i α β ) E 0 = 0 ,
2 E 1 ( β 2 + 2 i α β ) E 1 = 4 β F ( x , t ) E 0 .
E d ( x , t ) = κ β π x F ( x , t ) E i ( x ) exp ( K | x x | ) | x x | d x .
E d ( x , t ) = κ β π e i K x x x F ( x , t ) , E i ( x ) exp ( i K n x ) d x ,
E d ( x ) ( x ) = i K ( 2 π ) 1 / 2 E ˆ d ( K d ) e i K x x .
E ˆ d ( K d , t ) = i ( 2 / π ) 1 / 2 κ x exp ( i K d x ) E i ( x ) F ( x , t ) d x .
E i ( x ) = ( 2 π ) 3 / 2 K i E ˆ i ( K i ) exp ( i K i x ) d K i , F ( x ) = ( 2 π ) 3 / 2 K g F ˆ ( K g ) exp ( i K g x ) d K g .
E ˆ d ( K d , t ) = A κ ( 2 π ) 3 K i d K i E ˆ i ( K i ) K g d K g F ˆ ( K g , t ) × x exp [ i ( K i + K g K d ) x ] d x ,
E ˆ d ( K d , t ) = A κ K i E ˆ i ( K i ) F ˆ ( K d K i , t ) d K i .
κ F ˆ ( K g , t ) = κ W W ˆ ( K g , t ) .
E ˆ ( K i ) = π w r 2 exp [ ( w r 2 | K ¯ i K ¯ i 0 | ) 2 ] E 0 .
w r K Δ θ a 1 , w r K Δ θ e 1.
K Δ θ a Δ K g 1 , K Δ θ e Δ K g 1.
E ( x , t ) = ( 2 π ) 3 / 2 K d E ˆ ( K d , t ) exp ( i K d x ) d K d ,
E ( x , y , d , t ) = K d E ˆ ( K d , t ) exp ( i K d 3 d ) × exp [ i ( K d 1 x + K d 2 y ) ] d K d ,
I ( x , y , d , t ) = E ( x , y , d , t ) E * ( x , y , d , t ) = K d d K d E ˆ ( K d , t ) exp ( i K d 3 d ) × K d d K d E ˆ * ( K d , t ) exp ( i K d 3 d ) × exp { i [ ( K d 1 K d 1 ) x + ( K d 2 K d 2 ) y ] } .
M 0 ( t ) = x , y I ( x , y , d , t ) d x d y = K d d K d E ˆ ( K d , t ) K d d K d E ˆ * ( K d , t ) exp [ i ( K 3 K 3 ) d ] × x , y exp { i [ ( K d 1 K d 1 ) x + ( K d 2 K d 2 ) y ] } d x d y .
M 0 ( t ) = K d | E ˆ ( K d , t ) | 2 d K d .
M 1 ( t ) = x , y I ( x , y , d , t ) x d x d y = K d d K d E ˆ ( K d , t ) K d d K d E ˆ * ( K d , t ) × exp [ i ( K d 3 K d 3 ) d ] { x x exp [ i ( K d 1 K d 1 ) x ] d x } × { y exp [ i ( K d 2 K d 2 ) y ] d y } .
M 1 = i ( 2 π ) 2 K d E ˆ ( K d , t ) exp { i K d [ 1 ( K d 1 / K ) 2 ( K d 2 / K ) 2 ] 1 / 2 } K d 1 ( E ˆ * ( K d , t ) exp { i K d × [ 1 ( K d 1 / K ) 2 ( K d 2 / K ) 2 ] 1 / 2 } ) d K d .
E ˆ ( K d , t ) = | E ˆ ( K d , t ) | exp [ i Φ ( K d , t ) ] .
M 1 ( t ) = ( 2 π ) 2 K d | E ˆ ( K d , t ) | 2 { Φ ( K d , t ) K d 1 + d K d 1 / K [ 1 ( K d 1 / K ) 2 ( K d 2 / K ) 2 ] 1 / 2 } d K d ,
E ( x ) = ( 2 π ) 2 / 3 K Ê ( K ) exp ( i K x ) d K .
x = R K d K .
K d = K ( cos θ 0 cos ϕ 0 ; cos θ 0 sin ϕ 0 ; sin θ 0 ) , K = K ( cos θ cos ϕ ; cos θ sin ϕ ; sin θ ) ,
E ( x ) = ( 2 π ) 2 / 3 K 2 E ˆ ( θ , ϕ ) cos θ exp [ i Ψ ( θ , ϕ ) R ] d θ d ϕ ,
Ψ ( θ , ϕ ) = K [ cos θ 0 cos θ cos ϕ 0 cos ϕ + cos θ 0 cos θ sin ϕ × sin ϕ 0 + sin θ 0 sin θ ] .
Ψ θ = Ψ ϕ = 0 ,
( θ = θ 0 , ϕ = ϕ 0 ) ( θ = θ 0 , ϕ = ϕ 0 + π ) .
E ( x ) = ( 2 π ) 2 / 3 K 2 m = 1,2 E ( θ m , ϕ m ) cos θ m exp [ i Ψ ( θ m , ϕ m ) R ] × exp { i R [ ( 2 Ψ θ 2 ) m ( θ θ m ) 2 2 + ( 2 Ψ ϕ 2 ) m × ( ϕ ϕ m ) 2 2 + ( 2 Ψ ϕ θ ) m ( ϕ ϕ m ) ( θ θ m ) ] } d θ d ϕ .
( 2 π R ( | 2 Ψ θ 2 | | 2 Ψ ϕ 2 | ) 1 / 2 × exp { i π 4 [ sign ( 2 Ψ θ 2 ) + sign ( 2 Ψ ϕ 2 ) ] } ) stationary point .
for m = 1 , Ψ ( θ 0 , ϕ 0 ) = K , 2 Ψ θ 2 = K , 2 Ψ ϕ 2 = K cos 2 θ 0 , 2 Ψ θ ϕ = 0 ; for m = 1 , Ψ ( θ 0 , ϕ 0 ) = K , 2 Ψ θ 2 = K , 2 Ψ ϕ 2 = K cos 2 θ 0 , 2 Ψ θ ϕ = 0 ,
E ( x ) = i K ( 2 π ) 1 / 2 [ E ˆ ( K d ) e i K R R E ˆ ( K d ) e i K R R ] .
E ( x ) ( | x | ) = i K ( 2 π ) 1 / 2 E ˆ ( K d ) e i K x x .
E ˆ d ( K d , t ) = i ( 2 π ) 1 / 2 κ x exp ( i K d x ) E ( x ) F ( x , t ) d x ,

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