Abstract

A theoretical analysis of fringe patterns and its experimental corroboration obtained by multiplication of two speckled images with electronic speckle-pattern interferometry (ESPI) are reported. A specifically designed digital filter is used to enhance the contrast and visibility of the inherently noisy multiplication fringes. Phase retrieval is achieved by a phase-stepping technique. Experimental results are presented for the in-plane-sensitive optical ESPI setup; however, out-of-plane and shearing setups may be used as well. The method represents an alternative to the subtraction and addition techniques in ESPI.

© 2000 Optical Society of America

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References

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  1. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989), Chap. 4.
    [CrossRef]
  2. J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilising speckle effect,” J. Phys. E 3, 214–218 (1970).
    [CrossRef]
  3. A. J. Moore, J. R. Tyrer, F. Mendoza Santoyo, “Phase extraction from ESPI addition fringes,” Appl. Opt. 33, 7312–7320 (1994).
    [CrossRef] [PubMed]
  4. N. Alcalá Ochoa and J. M. Huntley, “A convenient method to calibrate nonlinear phase modulators for use in phase shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
    [CrossRef]
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2, p. 17.
  6. N. Alcalá Ochoa, J. L. Marroquín, A. Dávila, “Phase recovery using a twin-pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
    [CrossRef]
  7. D. Kerr, G. H. Kauffmann, G. E. Galizzi, “Unwrapping of interferometric phase-fringe maps by the discrete cosine transform,” Appl. Opt. 35, 810–816 (1996).
    [CrossRef] [PubMed]

1999 (1)

N. Alcalá Ochoa, J. L. Marroquín, A. Dávila, “Phase recovery using a twin-pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

1998 (1)

N. Alcalá Ochoa and J. M. Huntley, “A convenient method to calibrate nonlinear phase modulators for use in phase shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[CrossRef]

1996 (1)

1994 (1)

1970 (1)

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilising speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Alcalá Ochoa, N.

N. Alcalá Ochoa, J. L. Marroquín, A. Dávila, “Phase recovery using a twin-pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

Alcalá Ochoa and J. M. Huntley, N.

N. Alcalá Ochoa and J. M. Huntley, “A convenient method to calibrate nonlinear phase modulators for use in phase shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[CrossRef]

Dávila, A.

N. Alcalá Ochoa, J. L. Marroquín, A. Dávila, “Phase recovery using a twin-pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

Galizzi, G. E.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2, p. 17.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989), Chap. 4.
[CrossRef]

Kauffmann, G. H.

Kerr, D.

Leendertz, J. A.

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilising speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Marroquín, J. L.

N. Alcalá Ochoa, J. L. Marroquín, A. Dávila, “Phase recovery using a twin-pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

Mendoza Santoyo, F.

Moore, A. J.

Tyrer, J. R.

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989), Chap. 4.
[CrossRef]

Appl. Opt. (2)

J. Phys. E (1)

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilising speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Opt. Commun. (1)

N. Alcalá Ochoa, J. L. Marroquín, A. Dávila, “Phase recovery using a twin-pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

Opt. Eng. (1)

N. Alcalá Ochoa and J. M. Huntley, “A convenient method to calibrate nonlinear phase modulators for use in phase shifting interferometry,” Opt. Eng. 37, 2501–2505 (1998).
[CrossRef]

Other (2)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), Chap. 2, p. 17.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989), Chap. 4.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Optical setup for in-plane ESPI with a 10-mW laser: M.O., optical magnification.

Fig. 2
Fig. 2

Multiplication of I 1 times I 2.

Fig. 3
Fig. 3

Multiplication fringes obtained from Fig. 2 after the digitally designed filter is applied.

Fig. 4
Fig. 4

Phase map from multiplication fringes obtained from Eq. (9).

Fig. 5
Fig. 5

Unwrapped phase from Fig. 4.

Equations (12)

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I1=Io+Ir+2IoIr cosψ,
I2=Io+Ir+2IoIr cosψ+φ,
Im=a2+b2/2cos2ψ+φ+2ab×cosψ+φ/2cosφ/2+b2/2cosφ,
σ2=Im2-Im2,
σn2=σ2/Im2.
σ2=a4-a22+a2b2+7/32b4+2a2b2-a2b2cosφ+3/32b4cos2φ.
Qk=4/35k2+7k+5/k,
a4-a22+a2b2+7/32b4=5a22-2k2Ir4,
2a2b2-a2b25a22-20k4+1Ir4.
σ25a221+αα+βcosφ,
σn2φ=51+αα+βcosφ.
φ=tan-1σn2φ-3π/2-σn2φ-π/2σn2φ-σn2φ-π.

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