Abstract

A Michelson-type digital speckle photographic system has been proposed in which one light beam produces a Fourier transform and another beam produces an image at a recording plane, without interfering between themselves. Because the optical Fourier transform is insensitive to translation and the imaging technique is insensitive to tilt, the proposed system is able to simultaneously and independently determine both surface tilt and translation by two separate recordings, one before and another after the surface motion, without the need to obtain solutions for simultaneous equations. Experimental results are presented to verify the theoretical analysis.

© 2010 Optical Society of America

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References

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  1. H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–274 (1972).
    [CrossRef]
  2. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975).
    [CrossRef]
  3. A. Rosenberg and J. Politch, “Fringe parameters in speckle shearing interferometry,” Opt. Commun. 26, 301–304(1978).
    [CrossRef]
  4. P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R.S.Sirohi, ed. (Marcel Dekker, 1993).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

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2005

2003

1995

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1978

A. Rosenberg and J. Politch, “Fringe parameters in speckle shearing interferometry,” Opt. Commun. 26, 301–304(1978).
[CrossRef]

1972

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–274 (1972).
[CrossRef]

Benckert, L. R.

Bhaduri, B.

Diazdelacruz, J. M.

Fricke-Begemann, T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975).
[CrossRef]

Gopinathan, U.

Hennelly, B. M.

Kelly, D. P.

Liu, Y.

O’Neill, F. T.

Patten, R. F.

Politch, J.

A. Rosenberg and J. Politch, “Fringe parameters in speckle shearing interferometry,” Opt. Commun. 26, 301–304(1978).
[CrossRef]

Quan, C.

Rastogi, P. K.

P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R.S.Sirohi, ed. (Marcel Dekker, 1993).

Rosenberg, A.

A. Rosenberg and J. Politch, “Fringe parameters in speckle shearing interferometry,” Opt. Commun. 26, 301–304(1978).
[CrossRef]

Sheppard, C. J. R.

Sheridan, J. T.

Sjodahl, M.

Tay, C. J.

Tiziani, H.

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–274 (1972).
[CrossRef]

Ward, J. E.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–274 (1972).
[CrossRef]

A. Rosenberg and J. Politch, “Fringe parameters in speckle shearing interferometry,” Opt. Commun. 26, 301–304(1978).
[CrossRef]

Opt. Express

Opt. Lett.

Other

P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R.S.Sirohi, ed. (Marcel Dekker, 1993).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic of the Michelson-type optical arrangement for simultaneous measurement of tilt and translation: BS 1 , BS 2 , beam splitters; L, imaging lens; S, aperture stop; M 1 , M 2 , mirrors; P 1 , P 2 , polarizers; CCD, charge-coupled device.

Fig. 2
Fig. 2

Two arms of the Michelson-type setup: (a) OFT system with 2 f as the object-to-lens distance (Arm 1) and (b) imaging system with 2 f object and image distance (Arm 2).

Fig. 3
Fig. 3

Tilt measurement: (a) 2D plot of the cross-correlation coefficient with counterclockwise tilt by 0.7 mrad , (b) corresponding 3D plot, and (c) 2D plot of the cross-correlation coefficient with clockwise tilt by 2.44 mrad .

Fig. 4
Fig. 4

Translation measurement: (a) 2D plot of the cross- correlation coefficient with translation along the negative X direction by 500 μm , and (b) same with translation along the positive X direction by 1 mm .

Fig. 5
Fig. 5

Simultaneous tilt and translation measurement: 2D plot of the cross-correlation coefficient with counterclockwise tilt by 1.745 mrad and translation along the positive X direction by 1 mm .

Fig. 6
Fig. 6

Unambiguous tilt and translation measurement with additional Y translation: 2D plot of the cross-correlation coefficient with tilt counterclockwise by 1.0472 mrad and translation by 400 μm and 1 mm along the negative Y and positive X directions, respectively.

Fig. 7
Fig. 7

Cross-correlation peak detection with reference images: (a) when CC is performed between the reference image (obtained by closing Arm 2) and the image after the surface motion, (b) when CC is performed between the reference image (obtained by closing Arm 1) and the image after the surface motion, (c) 2D plot of the cross-correlation coefficient when CC is performed between the images recorded before and after the object motion with both arms open.

Equations (7)

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U 1 ( ξ ) = C F { u ( x ) } ,
u 2 ( x ) = u ( x ) .
I = | U 1 ( ξ ) | 2 + | u 2 ( x ) | 2 = | U 1 ( ξ ) | 2 + | u ( x ) | 2 = I 1 ( ξ ) + I 2 ( x ) ,
u ( x ) = u ( x δ ) exp [ j ( 2 π / λ ) θ x ] .
U 1 ( ξ ) = C F { u ( x δ ) exp [ j ( 2 π / λ ) θ x ] } = U 1 ( ξ θ f ) exp [ j 2 π ( ξ θ f ) δ / λ f ] .
u 2 ( x ) = u ( x + δ ) exp [ j ( 2 π / λ ) θ x ] .
I = | U 1 ( ξ ) | 2 + | u 2 ( x ) | 2 = | U 1 ( ξ θ f ) | 2 + | u ( x + δ ) | 2 = I 1 ( ξ θ f ) + I 2 ( x + δ ) ,

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