Abstract

In this work, we introduce an alternative method of analysis in vortex metrology based on the application of the Fourier optics techniques. The first part of the procedure is conducted as is usual in vortex metrology for uniform in-plane displacement determination. On the basis of two recorded intensity speckled distributions, corresponding to two states of a diffuser coherently illuminated, we numerically generate an analytical signal from each recorded intensity pattern by using a version of the Riesz integral transform. Then, from each analytical signal, a two-dimensional pseudophase map is generated in which the vortices are located and characterized in terms of their topological charges and their core’s structural properties. The second part of the procedure allows obtaining Young’s interference fringes when Fourier transforming the light passing through a diffracting mask with multiple apertures at the locations of the homologous vortices. In fact, we use the Fourier transform as a mathematical operation to compute the far-field diffraction intensity pattern corresponding to the multiaperture set. Each aperture from the set is associated with a rectangular hole that coincides both in shape and size with a pixel from recorded images. We show that the fringe analysis can be conducted as in speckle photography in an extended range of displacement measurements. Effects related with speckled decorrelation are also considered. Our experimental results agree with those of speckle photography in the range in which both techniques are applicable.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (2)

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt. 2012, 517591 (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]

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

2006 (2)

2005 (1)

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

2003 (1)

2000 (1)

1995 (1)

1979 (1)

1976 (1)

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

Buckland, J. R.

Chiang, F. P.

Gasvik, K. J.

K. J. Gasvik, Optical Metrology (Wiley, 2002).

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, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[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), pp. 207–238.

Huntley, J. M.

Ishii, N.

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Ishijima, R.

Khetan, R. P.

Masajada, J.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt. 2012, 517591 (2012).
[CrossRef]

Miyamoto, Y.

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, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Sato, S.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt. 2012, 517591 (2012).
[CrossRef]

Senthilkumaran, P.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt. 2012, 517591 (2012).
[CrossRef]

P. Senthilkumaran, “Optical phase singularities in detection of laser beam collimation,” Appl. Opt. 42, 6314–6320 (2003).
[CrossRef]

Sierra-Sosa, D.

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.

W. Wang, T. Yokozeki, R. Ishijima, and M. Takeda, “Optical vortex metrology based on the core structures of phase singularities in Laguerre-Gauss transform of a speckle pattern,” Opt. Express 14, 10195–10206 (2006).
[CrossRef]

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, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[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), pp. 207–238.

Tebaldi, M.

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 rotation assessment through isothetic fringes in speckle photography,” Appl. Opt. 46, 2676–2682 (2007).
[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, 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.

Wada, A.

Wang, W.

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, T. Yokozeki, R. Ishijima, and M. Takeda, “Optical vortex metrology based on the core structures of phase singularities in Laguerre-Gauss transform of a speckle pattern,” Opt. Express 14, 10195–10206 (2006).
[CrossRef]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[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), pp. 207–238.

Yokozeki, T.

Appl. Opt. (4)

Int. J. Opt. (1)

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt. 2012, 517591 (2012).
[CrossRef]

J. Opt. Soc. Am. A (2)

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]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[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]

Opt. Express (2)

Other (2)

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), pp. 207–238.

K. J. Gasvik, Optical Metrology (Wiley, 2002).

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Figures (3)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

Histograms for relative coordinate displacements between homologous vortices for (a)  Δ x and (b)  Δ y .

Fig. 3.
Fig. 3.

Relation between diffuser displacements (μm) and their respective measurements obtained from images (μm).

Tables (4)

Tables Icon

Table 1. Vortex-Network Correlation Fringes Obtained by Fourier Transforming the Light Passing through the Vortices One Pixel in Size, and Their Corresponding Intensity Profiles Through the Center of the Respective Image

Tables Icon

Table 2. Young’s Fringes Obtained by Multiplexing Speckles and Pixels Representing Vortices for Two Horizontal Displacements for Δ x = 180 μm (first column) and Δ x = 90 μm (right column)

Tables Icon

Table 3. Young’s Fringes Obtained by Multiplexing Speckles (left image) and Pixels Representing Vortices (central image) for a Horizontal 48 μm Displacement

Tables Icon

Table 4. Young’s Fringes Obtained by Multiplexing Decorrelated Speckle Distributions for Δ x = 180 μm (left column) and Δ x = 90 μm (right column)

Equations (2)

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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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