Abstract

We report a new digital shearing method for extracting the three-dimensional displacement vector data from double-exposure holograms. With this method we can manipulate both the phase and the amplitude of the recorded signal, which, like optical correlation analysis, is inherently immune to imaging aberration. However, digital shearing is not a direct digital implementation of optical correlation, and a considerable saving in computation time results. We demonstrate the power of the method by matlab simulation and discuss its performance with reference to optical analysis.

© 2003 Optical Society of America

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References

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  1. R. J. Adrian, “Particle imaging technique for experimental fluid dynamics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
    [CrossRef]
  2. K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, 61–72 (2002).
    [CrossRef]
  3. K. D. Hinsch, H. Heinrichs, A. Roshop, F. Dreesen, “Holographic and stereoscopic advances in 3D-PIV,” in Holographic Particle Image Velocimetry, E. P. Rood, ed. (American Society of Mechanical Engineers Division of Fluids Engineering, New York, 1993), Vol. 148, pp. 33–36.
  4. D. Barnhart, R. J. Adrian, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle-image velocimetry,” Appl. Opt. 33, 1005–1007 (1994).
    [CrossRef]
  5. J. M. Coupland, N. A. Halliwell, “Particle image velocimetry: three-dimensional fluid velocimetry measurements using holographic recording and optical correlation,” Appl. Opt. 31, 1005–1007 (1992).
    [CrossRef] [PubMed]
  6. J. M. Coupland, N. A. Halliwell, “Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,” Proc. R. Soc. London Ser. A 453, 1053–1066 (1997).
    [CrossRef]
  7. D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Object conjugate reconstruction (OCR): a step forward in holographic metrology,” Proc. R. Soc. London Ser. A 458, 2083–2097 (2002).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

2002 (2)

K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, 61–72 (2002).
[CrossRef]

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Object conjugate reconstruction (OCR): a step forward in holographic metrology,” Proc. R. Soc. London Ser. A 458, 2083–2097 (2002).
[CrossRef]

1997 (1)

J. M. Coupland, N. A. Halliwell, “Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,” Proc. R. Soc. London Ser. A 453, 1053–1066 (1997).
[CrossRef]

1994 (1)

D. Barnhart, R. J. Adrian, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle-image velocimetry,” Appl. Opt. 33, 1005–1007 (1994).
[CrossRef]

1992 (1)

1991 (1)

R. J. Adrian, “Particle imaging technique for experimental fluid dynamics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Adrian, R. J.

D. Barnhart, R. J. Adrian, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle-image velocimetry,” Appl. Opt. 33, 1005–1007 (1994).
[CrossRef]

R. J. Adrian, “Particle imaging technique for experimental fluid dynamics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Barnhart, D.

D. Barnhart, R. J. Adrian, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle-image velocimetry,” Appl. Opt. 33, 1005–1007 (1994).
[CrossRef]

Barnhart, D. H.

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Object conjugate reconstruction (OCR): a step forward in holographic metrology,” Proc. R. Soc. London Ser. A 458, 2083–2097 (2002).
[CrossRef]

Coupland, J. M.

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Object conjugate reconstruction (OCR): a step forward in holographic metrology,” Proc. R. Soc. London Ser. A 458, 2083–2097 (2002).
[CrossRef]

J. M. Coupland, N. A. Halliwell, “Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,” Proc. R. Soc. London Ser. A 453, 1053–1066 (1997).
[CrossRef]

J. M. Coupland, N. A. Halliwell, “Particle image velocimetry: three-dimensional fluid velocimetry measurements using holographic recording and optical correlation,” Appl. Opt. 31, 1005–1007 (1992).
[CrossRef] [PubMed]

Dreesen, F.

K. D. Hinsch, H. Heinrichs, A. Roshop, F. Dreesen, “Holographic and stereoscopic advances in 3D-PIV,” in Holographic Particle Image Velocimetry, E. P. Rood, ed. (American Society of Mechanical Engineers Division of Fluids Engineering, New York, 1993), Vol. 148, pp. 33–36.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Halliwell, N. A.

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Object conjugate reconstruction (OCR): a step forward in holographic metrology,” Proc. R. Soc. London Ser. A 458, 2083–2097 (2002).
[CrossRef]

J. M. Coupland, N. A. Halliwell, “Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,” Proc. R. Soc. London Ser. A 453, 1053–1066 (1997).
[CrossRef]

J. M. Coupland, N. A. Halliwell, “Particle image velocimetry: three-dimensional fluid velocimetry measurements using holographic recording and optical correlation,” Appl. Opt. 31, 1005–1007 (1992).
[CrossRef] [PubMed]

Heinrichs, H.

K. D. Hinsch, H. Heinrichs, A. Roshop, F. Dreesen, “Holographic and stereoscopic advances in 3D-PIV,” in Holographic Particle Image Velocimetry, E. P. Rood, ed. (American Society of Mechanical Engineers Division of Fluids Engineering, New York, 1993), Vol. 148, pp. 33–36.

Hinsch, K. D.

K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, 61–72 (2002).
[CrossRef]

K. D. Hinsch, H. Heinrichs, A. Roshop, F. Dreesen, “Holographic and stereoscopic advances in 3D-PIV,” in Holographic Particle Image Velocimetry, E. P. Rood, ed. (American Society of Mechanical Engineers Division of Fluids Engineering, New York, 1993), Vol. 148, pp. 33–36.

Papen, G. C.

D. Barnhart, R. J. Adrian, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle-image velocimetry,” Appl. Opt. 33, 1005–1007 (1994).
[CrossRef]

Roshop, A.

K. D. Hinsch, H. Heinrichs, A. Roshop, F. Dreesen, “Holographic and stereoscopic advances in 3D-PIV,” in Holographic Particle Image Velocimetry, E. P. Rood, ed. (American Society of Mechanical Engineers Division of Fluids Engineering, New York, 1993), Vol. 148, pp. 33–36.

Annu. Rev. Fluid Mech. (1)

R. J. Adrian, “Particle imaging technique for experimental fluid dynamics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Appl. Opt. (2)

J. M. Coupland, N. A. Halliwell, “Particle image velocimetry: three-dimensional fluid velocimetry measurements using holographic recording and optical correlation,” Appl. Opt. 31, 1005–1007 (1992).
[CrossRef] [PubMed]

D. Barnhart, R. J. Adrian, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle-image velocimetry,” Appl. Opt. 33, 1005–1007 (1994).
[CrossRef]

Meas. Sci. Technol. (1)

K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, 61–72 (2002).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

J. M. Coupland, N. A. Halliwell, “Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,” Proc. R. Soc. London Ser. A 453, 1053–1066 (1997).
[CrossRef]

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Object conjugate reconstruction (OCR): a step forward in holographic metrology,” Proc. R. Soc. London Ser. A 458, 2083–2097 (2002).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

K. D. Hinsch, H. Heinrichs, A. Roshop, F. Dreesen, “Holographic and stereoscopic advances in 3D-PIV,” in Holographic Particle Image Velocimetry, E. P. Rood, ed. (American Society of Mechanical Engineers Division of Fluids Engineering, New York, 1993), Vol. 148, pp. 33–36.

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Figures (13)

Fig. 1
Fig. 1

OCR recording geometry.

Fig. 2
Fig. 2

OCR reconstruction geometry.

Fig. 3
Fig. 3

Wave-vector geometry.

Fig. 4
Fig. 4

Far-field fringe pattern as a function of wave-vector components.

Fig. 5
Fig. 5

Phase-only map of the curved wave front (modulo 2π).

Fig. 6
Fig. 6

Sheared phase distribution (modulo 2π).

Fig. 7
Fig. 7

Sheared phase distribution with a variable change.

Fig. 8
Fig. 8

Modulus FFT of Fig. 7.

Fig. 9
Fig. 9

Original phase-only map (Fig. 5) with curvature (due to s z displacement) subtracted.

Fig. 10
Fig. 10

Modulus of FFT of Fig. 9.

Fig. 11
Fig. 11

Comparison between the computed and the preassigned displacement in the z direction. (Preassigned displacements in the x and y directions are 10 μm.)

Fig. 12
Fig. 12

Comparison between the computed and the preassigned displacement in the x direction. (Preassigned displacements in the y and z directions are 10 and 15 μm, respectively.)

Fig. 13
Fig. 13

Comparison between the computed and the preassigned displacement in the y direction. (Preassigned displacements in the x and z directions are 10 and 15 μm, respectively.)

Tables (1)

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Table 1 Independence of x, y, and z Displacement Measurements

Equations (23)

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Sk=-+ Urexp-2πjk·rdr,
Ur=-+ Skexp2πjk·rdk.
Br=exp2πjλαrx+βry+γrz,
α=λkx, β=λky, γ=λkz,
|k|2=1/λ2.
Pk=|Sk|2.
kx=xλ-1x2+y2+f2-1/2,
ky=yλ-1x2+y2+f2-1/2.
Ur=U1r+U2r.
U2r=expjϕU1r-s,
Ur=U1r+expjϕU1r-s,
Pk=-+ Urexp-2πjk·rdr2 =2P1k1+cos2πk·s-ϕ,
Rr=-+ Pkexp2πjk·rdk=R1s*2δr+expjϕδr-s +exp-jϕδr+s,
Pkx, ky=2P1kx, ky1+cos2πkxsx+2πkysy+2πszλ-2-kx2-ky21/2-ϕ.
Pkx, ky=2+exp-j2πkxsix-ϕ×exp-2πjkxspx+kyspy+spz1/2 +expj2πkxsix-ϕ ×exp2πjkxspx+kyspy+spz1/2,
Wkx, ky=exp-2πjkxspx+kyspy+spzλ-2-kx2-ky21/2.
Pwkx, ky=Wkx-Δkx, kyW*kx+Δkx, ky=exp-2πj-2Δkxspx+spzλ-2-kx-Δkx2-ky21/2-spzλ-2-kx+Δkx2-ky21/2,
θ=-2π-2Δkxspx+spz2Δkxkxλ-2-Δkx2-ky21/2 + 1+Δkx2λ-2-Δkx2-ky2Δkxkx3λ-2-Δkx2-ky23/2 + high-order terms.
Pwkx, ky=exp-2πjθs×exp-2πj2Δkxkxspzλ-2-Δkx2-ky21/2,
Kx=2Δkxkxλ-2-Δkx2-ky21/2, Ky=ky,
PwzKx, Ky=Pwλ-2-Δkx2-ky21/2Kx2Δkx, Ky=exp-2πjspzKx+θs.
Rwzrx, ry= PwzKx, Kyexp2πjKxrx+KyrydKxdKy=expjθsδrx-spz, ry.
Rwxyrx, ry= Wkx, kyexp2πjspzλ-2-kx2-ky21/2exp2πjkxrx+kyrydkxdky=δrx-spx, ry-spy.

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