Abstract

Recently we proposed an alternative method of displacement analysis in vortex metrology, based on the application of the Fourier optics techniques, that is suitable for an intermediate range of displacement measurements ranging below the resolution of speckle photography and above that of the conventional vortex metrology. However, for smaller displacements, we introduce an approach to perform the Fourier analysis from vortex networks. In this work, we present an enhanced method for measuring uniform in-plane displacements, taking advantage of the capability of determining the subpixel locations of vortices and having the ability to track the homologous vortices onto a plane. It is shown that high-quality fringe systems can be synthesized and analyzed to accurately measure in an extended range of displacements and for highly decorrelated speckle patterns. Experimental results supporting the validity of the method are presented and discussed.

© 2013 Optical Society of America

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References

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    [CrossRef]
  4. L. Angel-Toro, M. Tebaldi, and N. Bolognini, “Multiple-aperture speckle method applied to local displacement measurements,” Opt. Commun. 274, 23–31 (2007).
    [CrossRef]
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    [CrossRef]
  6. L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Speckle decorrelation influence on measurements quality in vortex metrology,” Opt. Commun. 285, 4312–4316 (2012).
    [CrossRef]
  7. L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Vortex metrology by using Fourier analysis techniques: vortex networks correlation fringes,” Appl. Opt. 51, 7411–7419(2012).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (2)

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Speckle decorrelation influence on measurements quality in vortex metrology,” Opt. Commun. 285, 4312–4316 (2012).
[CrossRef]

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Vortex metrology by using Fourier analysis techniques: vortex networks correlation fringes,” Appl. Opt. 51, 7411–7419(2012).
[CrossRef]

2007 (2)

L. Angel-Toro, M. Tebaldi, and N. Bolognini, “Multiple-aperture speckle method applied to local displacement measurements,” Opt. Commun. 274, 23–31 (2007).
[CrossRef]

L. Angel, M. Tebaldi, and N. Bolognini, “Multiple rotation assessment through isothetic fringes in speckle photography,” Appl. Opt. 46, 2676–2682 (2007).
[CrossRef]

2006 (2)

2003 (1)

2001 (1)

L. Angel-Toro, M. Tebaldi, M. Trivi, and N. Bolognini, “Properties of speckle patterns generated through multiaperture pupils,” Opt. Commun. 192, 37–47 (2001).
[CrossRef]

2000 (1)

1979 (1)

1976 (1)

Angel, L.

Angel-Toro, L.

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Speckle decorrelation influence on measurements quality in vortex metrology,” Opt. Commun. 285, 4312–4316 (2012).
[CrossRef]

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Vortex metrology by using Fourier analysis techniques: vortex networks correlation fringes,” Appl. Opt. 51, 7411–7419(2012).
[CrossRef]

L. Angel-Toro, M. Tebaldi, and N. Bolognini, “Multiple-aperture speckle method applied to local displacement measurements,” Opt. Commun. 274, 23–31 (2007).
[CrossRef]

L. Angel-Toro, M. Tebaldi, M. Trivi, and N. Bolognini, “Properties of speckle patterns generated through multiaperture pupils,” Opt. Commun. 192, 37–47 (2001).
[CrossRef]

L. Angel-Toro, M. Tebaldi, N. Bolognini, and M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

Bolognini, N.

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Vortex metrology by using Fourier analysis techniques: vortex networks correlation fringes,” Appl. Opt. 51, 7411–7419(2012).
[CrossRef]

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Speckle decorrelation influence on measurements quality in vortex metrology,” Opt. Commun. 285, 4312–4316 (2012).
[CrossRef]

L. Angel-Toro, M. Tebaldi, and N. Bolognini, “Multiple-aperture speckle method applied to local displacement measurements,” Opt. Commun. 274, 23–31 (2007).
[CrossRef]

L. Angel, M. Tebaldi, and N. Bolognini, “Multiple rotation assessment through isothetic fringes in speckle photography,” Appl. Opt. 46, 2676–2682 (2007).
[CrossRef]

L. Angel-Toro, M. Tebaldi, M. Trivi, and N. Bolognini, “Properties of speckle patterns generated through multiaperture pupils,” Opt. Commun. 192, 37–47 (2001).
[CrossRef]

L. Angel-Toro, M. Tebaldi, N. Bolognini, and M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

Chiang, F. P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005), Chap. 4.

Hanson, S. G.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
[CrossRef]

W. Wang, S. G. Hanson, and M. Takeda, “Optical vortex metrology,” in Advances in Speckle Metrology and Related Techniques, G. H. Kaufmann, ed. (Wiley-VCH, 2011), Chap. 5, pp. 207–238.

Ishijima, R.

Khetan, R. P.

Miyamoto, Y.

Senthilkumaran, P.

Sierra-Sosa, D.

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Vortex metrology by using Fourier analysis techniques: vortex networks correlation fringes,” Appl. Opt. 51, 7411–7419(2012).
[CrossRef]

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Speckle decorrelation influence on measurements quality in vortex metrology,” Opt. Commun. 285, 4312–4316 (2012).
[CrossRef]

Takeda, M.

Tebaldi, M.

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Vortex metrology by using Fourier analysis techniques: vortex networks correlation fringes,” Appl. Opt. 51, 7411–7419(2012).
[CrossRef]

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Speckle decorrelation influence on measurements quality in vortex metrology,” Opt. Commun. 285, 4312–4316 (2012).
[CrossRef]

L. Angel-Toro, M. Tebaldi, and N. Bolognini, “Multiple-aperture speckle method applied to local displacement measurements,” Opt. Commun. 274, 23–31 (2007).
[CrossRef]

L. Angel, M. Tebaldi, and N. Bolognini, “Multiple rotation assessment through isothetic fringes in speckle photography,” Appl. Opt. 46, 2676–2682 (2007).
[CrossRef]

L. Angel-Toro, M. Tebaldi, M. Trivi, and N. Bolognini, “Properties of speckle patterns generated through multiaperture pupils,” Opt. Commun. 192, 37–47 (2001).
[CrossRef]

L. Angel-Toro, M. Tebaldi, N. Bolognini, and M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

Trivi, M.

L. Angel-Toro, M. Tebaldi, M. Trivi, and N. Bolognini, “Properties of speckle patterns generated through multiaperture pupils,” Opt. Commun. 192, 37–47 (2001).
[CrossRef]

L. Angel-Toro, M. Tebaldi, N. Bolognini, and M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

Wada, A.

Wang, W.

Yokozeki, T.

Appl. Opt. (5)

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

Opt. Commun. (3)

L. Angel-Toro, M. Tebaldi, and N. Bolognini, “Multiple-aperture speckle method applied to local displacement measurements,” Opt. Commun. 274, 23–31 (2007).
[CrossRef]

L. Angel-Toro, D. Sierra-Sosa, M. Tebaldi, and N. Bolognini, “Speckle decorrelation influence on measurements quality in vortex metrology,” Opt. Commun. 285, 4312–4316 (2012).
[CrossRef]

L. Angel-Toro, M. Tebaldi, M. Trivi, and N. Bolognini, “Properties of speckle patterns generated through multiaperture pupils,” Opt. Commun. 192, 37–47 (2001).
[CrossRef]

Opt. Express (2)

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005), Chap. 4.

W. Wang, S. G. Hanson, and M. Takeda, “Optical vortex metrology,” in Advances in Speckle Metrology and Related Techniques, G. H. Kaufmann, ed. (Wiley-VCH, 2011), Chap. 5, pp. 207–238.

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Figures (4)

Fig. 1.
Fig. 1.

Scheme representing the input and diffraction planes, including the different axes and coordinate systems introduced in the text.

Fig. 2.
Fig. 2.

Experimental arrangements for recording the speckled images before and after an in-plane displacement for (a) undecorrelated and (b) decorrelated speckle patterns. λ , wavelength; D, diffuser; M, mask; CMOS, camera; Δ x , in-plane displacement.

Fig. 3.
Fig. 3.

(a) Measured displacements obtained by using the conventional information processing methods from vortex metrology. (b) Measured average periods of synthetic correlation fringes by following the method presented in this paper, with λ z = 600 pixels 2 .

Fig. 4.
Fig. 4.

(a) Measured displacements by using both the synthetic vortex correlation fringes from [7] (crosses) and the method introduced by Wang et al. in [9,10] (triangles). (b), the results obtained by means of both the vortex correlation fringes method [7] (circles) and by using the synthetic vortex correlation fringes method (crosses) presented in this paper.

Tables (4)

Tables Icon

Table 1. Effect of Decorrelation on Synthetic Vortex Correlation Fringe Visibilitya

Tables Icon

Table 2. Undecorrelated and Decorrelated Data Synthesis

Tables Icon

Table 3. Enlarged Synthetic Vortex Correlation Fringes and Their Corresponding Intensity Profiles by Setting a Different Scale Factora

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Table 4. Synthetic Vortex Correlation Fringes and Their Intensity Fringe Profiles for Two Different Horizontal Displacementsa

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

U 2 ( x , y ) = 1 λ z U 1 ( ξ , η ) [ e i k z ( x ξ + y η ) ] d ξ d η ,
A ( x , y ) = i = 1 N A i ( x , y ) ,
I ( x , y ) = N I 0 + 2 I 0 i , j = 1 i < j N cos [ φ j ( x , y ; ξ j , η j ) φ i ( x , y ; ξ i , η i ) ] ,
φ i ( x , y ; ξ i , η i ) φ 0 + 2 π λ { z + 1 2 z [ ( x ξ i ) 2 + ( y η i ) 2 ] } ,
I ( x , y ) = N I 0 + 2 I 0 i , j = 1 i < j N cos { π λ z [ ( ξ j 2 + η j 2 ) ( ξ i 2 + η i 2 ) ] 2 π λ z [ ( ξ j ξ i ) x + ( η j η i ) y ] } .
I ( x , y ) = I 0 [ N + i , j = 1 i < j N cos { π λ z [ ( ξ j 2 + η j 2 ) ( ξ i 2 + η i 2 ) ] 2 π λ z [ ( ξ j ξ i ) x + ( η j η i ) y ] } ] ,
I ( x , y ) = I 0 [ N + i , j = 1 i < j N cos { 2 π λ z [ ( ξ ˜ j ξ ˜ i ) x + ( η ˜ j η ˜ i ) y ] } ] 1 2 + ( 1 2 N ) i , j = 1 i < j N cos { 2 π λ z [ ( ξ ˜ j ξ ˜ i ) x + ( η ˜ j η ˜ i ) y ] } .
I ( x , y ) = 1 2 + 1 2 N i , j = 1 i < j N cos { 2 π λ z [ ( ξ j ξ i ) ( x n 2 ) + ( η j η i ) ( y m 2 ) ] } .
Λ i j = λ z [ ( ξ j ξ i ) 2 + ( η j η i ) 2 ] 1 2 .
I ˜ ( x , y ) = V ( f x , f y ) F ( f x , f y ) e 2 π i ( f x x + f y y ) d f x f y ,
V ( f x , f y ) = e i arctan ( f y / f x ) .

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