Abstract

An optical technique that is based on defocused digital speckle photography is proposed for the evaluation of phase objects. Phase objects are different kinds of transparent or semi-transparent media that allow light to be transmitted. A phase object inserted in a laser speckle field introduces speckle displacement, from which information about the object may be extracted. It is shown that one may use speckle displacements to determine both the phase gradients and the positions of phase objects. As an illustration the positions and focal lengths of two weak lenses have been derived from defocused laser speckle displacement.

© 2004 Optical Society of America

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References

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  1. W. Merzkirch, “Approaches in flow visualization,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 559–569.
  2. U. Wernekinck, W. Merzkirch, “Speckle photography of spatially extended refractive-index fields,” Appl. Opt. 26, 31–32 (1987).
    [CrossRef] [PubMed]
  3. N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements (Springer-Verlag, Berlin, 1998).
    [CrossRef]
  4. M. Sjödahl, “Digital speckle photography,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, Chichester, UK, 2001), pp. 289–336.
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  6. M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
    [CrossRef] [PubMed]
  7. G. E. P. Box, W. G. Hunter, J. S. Hunter, “Simple modeling with least squares (regression analysis),” in Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building (Wiley, New York, 1978), pp. 453–509.

1994

1987

Box, G. E. P.

G. E. P. Box, W. G. Hunter, J. S. Hunter, “Simple modeling with least squares (regression analysis),” in Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building (Wiley, New York, 1978), pp. 453–509.

Fomin, N. A.

N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements (Springer-Verlag, Berlin, 1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Hunter, J. S.

G. E. P. Box, W. G. Hunter, J. S. Hunter, “Simple modeling with least squares (regression analysis),” in Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building (Wiley, New York, 1978), pp. 453–509.

Hunter, W. G.

G. E. P. Box, W. G. Hunter, J. S. Hunter, “Simple modeling with least squares (regression analysis),” in Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building (Wiley, New York, 1978), pp. 453–509.

Merzkirch, W.

U. Wernekinck, W. Merzkirch, “Speckle photography of spatially extended refractive-index fields,” Appl. Opt. 26, 31–32 (1987).
[CrossRef] [PubMed]

W. Merzkirch, “Approaches in flow visualization,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 559–569.

Sjödahl, M.

M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
[CrossRef] [PubMed]

M. Sjödahl, “Digital speckle photography,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, Chichester, UK, 2001), pp. 289–336.

Wernekinck, U.

Appl. Opt.

Other

G. E. P. Box, W. G. Hunter, J. S. Hunter, “Simple modeling with least squares (regression analysis),” in Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building (Wiley, New York, 1978), pp. 453–509.

W. Merzkirch, “Approaches in flow visualization,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 559–569.

N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements (Springer-Verlag, Berlin, 1998).
[CrossRef]

M. Sjödahl, “Digital speckle photography,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, Chichester, UK, 2001), pp. 289–336.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

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Figures (6)

Fig. 1
Fig. 1

Optical setup used to derive the propagation of modified mutual intensity. The imaging plane is at a distance L′ in front of the phase object. Δ is the distance between the principal planes of the camera lens.

Fig. 2
Fig. 2

Experimental setup. Different imaging planes are obtained by focusing at different distances.

Fig. 3
Fig. 3

Speckle displacement field when a weak lens is used as a phase object and focusing is behind the lens.

Fig. 4
Fig. 4

(a) Example of displacements in the x direction versus x positions on the detector. A small departure from linearity can be seen at the edges of the detector positions, depending on aberrations in the thin lens used as the phase object. (b) Fitted line subtracted from the displacements in the x direction plotted versus the x positions on the detector. The displacements have a larger spread at the edges of the detector positions.

Fig. 5
Fig. 5

Position, in front of the detector, of imaging planes versus slope coefficient β of displacement fields. β = 0 gives the position of the principal planes of the thin lens, which is ∼380 mm.

Fig. 6
Fig. 6

Position, in front of the diffuser, of imaging planes versus slope coefficient β of the displacement fields when a telecentric imaging system is used. Focal length f of the weak lens becomes 9740 ± 40 mm, and its position is 602.5 ± 0.2 mm in front of the diffuser.

Equations (17)

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Jr, r=I expiϕrδr-r.
Js, s=C  expiϕrexpik|s|2-|s|22L+rLs-sd2r,
ϕr=ϕ0r0+rϕr,
Js, s=C expiϕ0expik |s|2-|s|22L×expik rLs-s+Lkϕrd2r.
s-s=-Lkϕr.
γ=Θ-sin Θπ2,
Θ=2 arccos|Ap|D.
p=-mLkϕr,
tlr=exp-i k2f|r|2=expiϕ,
ϕr=-kfr.
f=mLr/p.
p=βl,
L=Ld-a-b-Δ,
r=sL+aa=-lmL+aa,
f=-LL+aβa.
f=-Ld-Δ-fcmm+12Ld-Δ-fcmm+12+fcmm+1fcmm+1β.
f=-L/β.

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