Abstract

Interferometric speckle techniques are plagued by the omnipresence of phase singularities, impairing the phase unwrapping process. To reduce the number of phase singularities by physical means, an incoherent averaging of multiple speckle fields may be applied. It turns out, however, that the results may strongly deviate from the expected N behavior. Using speckle-shearing interferometry as an example, we investigate the mechanism behind the reduction of phase singularities, both by calculations and by computer simulations. Key to an understanding of the reduction mechanism during incoherent averaging is the representation of the physical averaging process in terms of certain vector fields associated with each speckle field.

© 2014 Optical Society of America

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  1. J. Leendertz and J. Butters, J. Phys. E 6, 1107 (1973).
    [CrossRef]
  2. Y. T. Hung, Opt. Eng. 21, 213391 (1982).
    [CrossRef]
  3. D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010).
    [CrossRef]
  4. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
  5. J. Adachi and G.-O. Ishikawa, Nonlinearity 20, 1907 (2007).
    [CrossRef]
  6. I. Yamaguchi, Opt. Acta 28, 1359 (1981).
    [CrossRef]
  7. M. Sjödahl, Appl. Opt. 34, 7998 (1995).
    [CrossRef]
  8. P. Šmíd, P. Horváth, and M. Hrabovský, Appl. Opt. 46, 3709 (2007).
    [CrossRef]
  9. S. Liu and L. Yang, Opt. Eng. 46, 051012 (2007).
    [CrossRef]
  10. J. M. Huntley, Appl. Opt. 40, 3901 (2001).
    [CrossRef]
  11. D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  12. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).
    [CrossRef]
  13. R. Dändliker and F. M. Mottier, ZAMP 22, 369 (1971).
    [CrossRef]
  14. S. Equis and P. Jacquot, Proc. SPIE 6341, 634138 (2006).
    [CrossRef]
  15. J. W. Goodman, Statistical Optics (Wiley, 1985).

2010 (1)

D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010).
[CrossRef]

2007 (3)

J. Adachi and G.-O. Ishikawa, Nonlinearity 20, 1907 (2007).
[CrossRef]

S. Liu and L. Yang, Opt. Eng. 46, 051012 (2007).
[CrossRef]

P. Šmíd, P. Horváth, and M. Hrabovský, Appl. Opt. 46, 3709 (2007).
[CrossRef]

2006 (1)

S. Equis and P. Jacquot, Proc. SPIE 6341, 634138 (2006).
[CrossRef]

2001 (1)

1995 (1)

1982 (1)

Y. T. Hung, Opt. Eng. 21, 213391 (1982).
[CrossRef]

1981 (1)

I. Yamaguchi, Opt. Acta 28, 1359 (1981).
[CrossRef]

1976 (1)

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).

1973 (1)

J. Leendertz and J. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

1971 (1)

R. Dändliker and F. M. Mottier, ZAMP 22, 369 (1971).
[CrossRef]

Adachi, J.

J. Adachi and G.-O. Ishikawa, Nonlinearity 20, 1907 (2007).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).

Butters, J.

J. Leendertz and J. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

Dändliker, R.

R. Dändliker and F. M. Mottier, ZAMP 22, 369 (1971).
[CrossRef]

Equis, S.

S. Equis and P. Jacquot, Proc. SPIE 6341, 634138 (2006).
[CrossRef]

Francis, D.

D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Goodman, J. W.

J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 1985).

Groves, R. M.

D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010).
[CrossRef]

Horváth, P.

Hrabovský, M.

Hung, Y. T.

Y. T. Hung, Opt. Eng. 21, 213391 (1982).
[CrossRef]

Huntley, J. M.

Ishikawa, G.-O.

J. Adachi and G.-O. Ishikawa, Nonlinearity 20, 1907 (2007).
[CrossRef]

Jacquot, P.

S. Equis and P. Jacquot, Proc. SPIE 6341, 634138 (2006).
[CrossRef]

Leendertz, J.

J. Leendertz and J. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

Liu, S.

S. Liu and L. Yang, Opt. Eng. 46, 051012 (2007).
[CrossRef]

Mottier, F. M.

R. Dändliker and F. M. Mottier, ZAMP 22, 369 (1971).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Sjödahl, M.

Šmíd, P.

Tatam, R. P.

D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, Opt. Acta 28, 1359 (1981).
[CrossRef]

Yang, L.

S. Liu and L. Yang, Opt. Eng. 46, 051012 (2007).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Phys. E (1)

J. Leendertz and J. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

Meas. Sci. Technol. (1)

D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010).
[CrossRef]

Nonlinearity (1)

J. Adachi and G.-O. Ishikawa, Nonlinearity 20, 1907 (2007).
[CrossRef]

Opt. Acta (1)

I. Yamaguchi, Opt. Acta 28, 1359 (1981).
[CrossRef]

Opt. Eng. (2)

S. Liu and L. Yang, Opt. Eng. 46, 051012 (2007).
[CrossRef]

Y. T. Hung, Opt. Eng. 21, 213391 (1982).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).

Proc. SPIE (1)

S. Equis and P. Jacquot, Proc. SPIE 6341, 634138 (2006).
[CrossRef]

ZAMP (1)

R. Dändliker and F. M. Mottier, ZAMP 22, 369 (1971).
[CrossRef]

Other (2)

J. W. Goodman, Statistical Optics (Wiley, 1985).

D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

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Figures (9)

Fig. 1.
Fig. 1.

Model of the speckle-shearing interferometer used for the computer simulations.

Fig. 2.
Fig. 2.

Phase-singularity reduction for a light source consisting of nine mutually incoherent points. Vertical, reduction of the number of singularities compared to a single point source. Horizontal: a measure of inverse speckle size, varying from large to small speckles (the circles indicate the x-values referred to in Fig. 3). Small image, reduction curve averaged over 64 runs.

Fig. 3.
Fig. 3.

Different simulated speckle-shearing fields (b/w-inverted) of 1000×1000 pixels with a shear of 50 pixels, corresponding to the different areas visible in Fig. 2. Left: x=1, complete elimination of phase singularities. Middle left, x=71, no reduction. Middle right, x=91, reduction factor of 0.8. Right, x=100, no reduction.

Fig. 4.
Fig. 4.

Phase field and phase statistic for a standard speckle field (left) at x=40 and the speckle-shearing field derived from it (right). In black, phase edges (lines), phase singularities (illustrated as circles), and shear s. Both phase (upper part) and phase statistic (lower part) are shown. The phase statistic for the standard speckle field deviates randomly from the uniform distribution because of the low number of speckle grains in the field, while the statistic for the speckle-shearing field is systematically peaked at a phase value of zero.

Fig. 5.
Fig. 5.

Recombination of phase edges in the vector field representation. The basic complex amplitudes up and un*, which differ by a horizontal shift and a complex conjugation, are multiplied to form the associated vector field up·un*. The corresponding addition of phase angles explains the realignment of the phase edges due to shearing and the doubling of the initial singularities. The corresponding phase maps are also illustrated.

Fig. 6.
Fig. 6.

Phase field and phase statistic for medium speckle sizes. Left, x=69. Right, x=76. The phase statistic changed to uniform and, for even smaller speckle sizes, to concave.

Fig. 7.
Fig. 7.

Incoherent averaging represented by vector fields from the beginning (top) to the end of the averaging process (bottom). The vector field gradually changes direction according to the phase distribution of the individual speckle fields. The initial phase singularities and edges (sketched by dots) have disappeared, and the dominating direction is now along the positive horizontal axis.

Fig. 8.
Fig. 8.

Number of remaining phase singularities versus number of averaging speckle fields for x=40 (left), x=69 (middle), and x=76 (right); n denotes the averaging step after which the number of phase singularities fell to zero for the first time.

Fig. 9.
Fig. 9.

Deformation phase for a shear in horizontal direction. Left, point source. The fringes show a high visibility. Right, 20×60 light source. The phase values are restricted to a much smaller interval, and the fringes can only barely be seen.

Equations (3)

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Φ=arctan(I(4)I(2)I(1)I(3)),
v=ivi,
Φ=arg(v).

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