Abstract

A new technique for directly extracting phase gradients from two-dimensional (2-D) interferometer fringe data is presented. One finds the gradients by tracking the maximum modulus of the continuous wavelet transform of the fringe data and the phase distribution that is obtained, with a small error, by integration. Problems associated with phase unwrapping are thereby avoided. The technique is compared with standard methods, and excellent agreement is found. In common with Fourier-transform methods, the technique is capable of extracting the full 2-D phase distribution from a single image.

© 1999 Optical Society of America

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References

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  1. K. Creath, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, The Netherlands, 1988), Vol. 26, p. 384.
  2. M. Takeda, Indust. Metrol. 1, 79 (1990).
    [CrossRef]
  3. H. Singh and J. Sirkis, Appl. Opt. 33, 5016 (1994).
    [CrossRef] [PubMed]
  4. J. Marroquin, M. Servin, and R. Vera, Opt. Lett. 23, 238 (1998).
    [CrossRef]
  5. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992), Chap. 2.
    [CrossRef]
  6. D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).
  7. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

1998 (1)

1994 (1)

1990 (1)

M. Takeda, Indust. Metrol. 1, 79 (1990).
[CrossRef]

1946 (1)

D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).

Creath, K.

K. Creath, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, The Netherlands, 1988), Vol. 26, p. 384.

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992), Chap. 2.
[CrossRef]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Gabor, D.

D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).

Marroquin, J.

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Servin, M.

Singh, H.

Sirkis, J.

Takeda, M.

M. Takeda, Indust. Metrol. 1, 79 (1990).
[CrossRef]

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Vera, R.

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Appl. Opt. (1)

Indust. Metrol. (1)

M. Takeda, Indust. Metrol. 1, 79 (1990).
[CrossRef]

J. Inst. Electr. Eng. (1)

D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).

Opt. Lett. (1)

Other (3)

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992), Chap. 2.
[CrossRef]

K. Creath, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, The Netherlands, 1988), Vol. 26, p. 384.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

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Figures (4)

Fig. 1
Fig. 1

Fringe pattern for the bare Michelson interferometer.

Fig. 2
Fig. 2

Contour plot of the modulus of the CWT Φfa,b for the interference fringes shown in the inset at the top, which correspond to a typical row of Fig. 1. The white dashed curve gives the phase gradient.

Fig. 3
Fig. 3

Phase distribution for the bare interferome-ter: (a) wavelet-based method, (b) phase-stepping method.

Fig. 4
Fig. 4

Contour plot of the difference in phase for the two distributions shown in Fig. 3.

Equations (3)

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Φfa,b=-fxha,b*xdx,
ha,bx=a-1/2hx-ba,
hx=expiω0xexp-x22,

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