Abstract

Digital speckle photography is a useful tool for measuring the motion of optically rough surfaces from the speckle shift that takes place at the recording plane. A simple correlation based digital speckle photographic system has been proposed that implements two simultaneous optical extended fractional Fourier transforms (EFRTs) of different orders using only a single lens and detector to simultaneously detect both the magnitude and direction of translation and tilt by capturing only two frames: one before and another after the object motion. The dynamic range and sensitivity of the measurement can be varied readily by altering the position of the mirror/s used in the optical setup. Theoretical analysis and experiment results are presented.

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References

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    [CrossRef]
  2. 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]
  8. R. E. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31(1), 32–34 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2006 (2)

2005 (2)

2003 (1)

1997 (1)

1995 (1)

1993 (2)

1972 (1)

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5(4), 271–276 (1972).
[CrossRef]

1937 (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. U.S.A. 23(3), 158–164 (1937).
[CrossRef] [PubMed]

Benckert, L. R.

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. U.S.A. 23(3), 158–164 (1937).
[CrossRef] [PubMed]

Diazdelacruz, J. M.

Fricke-Begemann, T.

Gopinathan, U.

Hennelly, B. M.

Hua, J.

Kelly, D. P.

Li, G.

Liu, L.

Liu, Y.

Lohmann, A. W.

O’Neill, F. T.

Patten, R. E.

Sheridan, J. T.

Sjodahl, M.

Sjödahl, M.

Tiziani, H.

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5(4), 271–276 (1972).
[CrossRef]

Ward, J. E.

Appl. Opt. (5)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5(4), 271–276 (1972).
[CrossRef]

Opt. Lett. (1)

Proc. Natl. Acad. Sci. U.S.A. (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. U.S.A. 23(3), 158–164 (1937).
[CrossRef] [PubMed]

Other (2)

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

K. J. Gåsvik, Optical Metrology, 3rd ed., (John Wiley & Sons Ltd, Chichester, 2002).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the Michelson type optical arrangement for obtaining two simultaneous extended factional Fourier transforms: BS1, BS2, Beam splitters; L, lens; S, Aperture stop; M1, M2, Mirrors; P1, P2, Polarizers; CCD, charge coupled device.

Fig. 2
Fig. 2

Plot of the theoretical sensitivity for (a) translation and (b) tilt as a function of the ratio of the output distance to the focal length ( l / f ).

Fig. 3
Fig. 3

Plot of the experimental sensitivity for (a) translation and (b) tilt as a function of the ratio of the output distance to the focal length ( l / f ).

Fig. 4
Fig. 4

Translation measurement: (a) 2D plot of cross correlation coefficient, (b) 1st CC peak, and (c) 2nd CC peak.

Fig. 5
Fig. 5

Tilt measurement: 2D plot of cross correlation coefficient. The white arrows show the CC peaks due to EFRTs.

Fig. 6
Fig. 6

Simultaneous translation and tilt measurement: 2D plot of cross correlation coefficient. The white arrows show the CC peaks due to EFRTs.

Equations (17)

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U a , θ , b ( q ) = E F R T { f ( p ) } = C ( q ) f ( p ) exp ( i π a 2 p 2 tan θ i 2 π a b p q sin θ ) d p ,
a 2 = 1 λ f l f l 1 [ f 2 ( f l ) ( f l ) ] 1 / 2 ,
θ = arc cos ( f l f l f ) ,
b 2 = 1 λ f l f l 1 [ f 2 ( f l ) ( f l ) ] 1 / 2 ,
C ( q ) = exp [ i ( π / 4 θ / 2 ) ] ( 2 π sin θ ) 1 / 2 exp ( i π b 2 q 2 tan θ ) = exp [ i ϕ ( q ) ] ,
f ( p ) = f ( p ξ ) exp ( i κ p ) .
U a , θ , b ( q ) = E F R T { f ( p ) } = C ( q ) f ( p ξ ) exp ( i κ p ) exp ( i π a 2 p 2 tan θ i 2 π a b p q sin θ ) d p = C ( q ) f ( p ξ ) exp ( i κ p ) exp { i π a 2 ( p ξ ) 2 tan θ i π a 2 ξ 2 tan θ + i 2 π a 2 p ξ tan θ i 2 π a b p q sin θ } d p = C ( q ) exp ( i π a 2 ξ 2 tan θ ) f ( p ξ ) exp { i π a 2 ( p ξ ) 2 tan θ i ( p ξ ) [ 2 π a b q sin θ 2 π a 2 ξ tan θ κ ] i ξ [ 2 π a b q sin θ 2 π a 2 ξ tan θ κ ] } d p .
U a , θ , b ( q ) = C ( q ) f ( t ) exp { i π a 2 t 2 tan θ i 2 π a b t sin θ ( q a ξ cos θ b κ sin θ 2 π a b ) } d t = C ( q ) f ( t ) exp ( i π a 2 t 2 tan θ i 2 π a b t ( q Q ) sin θ ) d t ,
C ( q ) = C ( q ) exp ( i π a 2 ξ 2 tan θ ) exp { i ξ [ i 2 π a b q sin θ i 2 π a 2 ξ tan θ κ ] } = exp [ i ( π / 4 θ / 2 ) ] ( 2 π sin θ ) 1 / 2 exp { i π tan θ ( b 2 q 2 + a 2 ξ 2 2 a ξ b q cos θ + κ ξ tan θ / π ) } = exp [ i ϕ ( q ) ] ,
Q = a ξ cos θ b + κ sin θ 2 π a b = s ξ + t κ = s ξ + t α ,
I = | U a 1 , θ 1 , b 1 ( q 1 ) | 2 + | U a 2 , θ 2 , b 2 ( q 2 ) | 2 = I 1 ( q 1 ) + I 2 ( q 2 ) ,
I = | U a 1 , θ 1 , b 1 ( q 1 ) | 2 + | U a 2 , θ 2 , b 2 ( q 2 ) | 2 = | U a 1 , θ 1 , b 1 ( q 1 Q 1 ) | 2 + | U a 2 , θ 2 , b 2 ( q 2 Q 2 ) | 2         = I 1 ( q 1 Q 1 ) + I 2 ( q 2 Q 2 ) ,
Q 1 = s 1 ξ + t 1 α ,
Q 2 = s 2 ξ + t 2 α ,
Q = s ξ + t α = ( 1 l / f ) ξ + λ π ( l + l l l / f ) α .
s = ( 1 l / f ) ,
t = λ π ( l + l l l / f ) .

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