Abstract

It is well known that an interferogram can be demodulated to find the wave-front shape if a linear carrier is introduced. We show that it can also be demodulated if it has many closed fringes or a circular carrier appears. A basic assumption is that the carrier fringes are of a bandwidth adequate to contain the wave-front distortion. This phase determination, called here demodulation, is made in the space domain, as opposed to demodulation in Fourier space, but the low-pass filter characteristics must be properly chosen. For academic purposes a holographic analogy of this demodulation process is also presented, which shows that the common technique of multiplying by a sine function and a cosine function is equivalent to holographically reconstructing with a tilted-flat wave front. Alternatively, a defocused (spherical) wave front can be used as a reference to perform the reconstruction or demodulation of some closed-fringe interferograms.

© 1998 Optical Society of America

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References

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  1. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  2. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  3. T. Kreis, “Digital holographic interference-phase measurement using the Fourier transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  4. T. Kreis, “Fourier transform evaluation of holographic interference patterns,” in International Conference on Photo Mechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 365–371 (1987).
    [CrossRef]
  5. A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Laser Eng. 23, 319–330 (1995).
    [CrossRef]
  6. M. Servín, J. L. Marroquín, F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
    [CrossRef] [PubMed]
  7. X. Peng, S. M. Zhou, Z. Gao, “An automatic demodulation technique for a non-linear carrier fringe pattern,” Optik 100, 11–14 (1995).
  8. V. I. Vlad, D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” in Progress in Optics, Vol. XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), pp. 261–317.
  9. D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

1997

1995

A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Laser Eng. 23, 319–330 (1995).
[CrossRef]

X. Peng, S. M. Zhou, Z. Gao, “An automatic demodulation technique for a non-linear carrier fringe pattern,” Optik 100, 11–14 (1995).

1986

1984

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1982

Cuevas, F. J.

Gao, Z.

X. Peng, S. M. Zhou, Z. Gao, “An automatic demodulation technique for a non-linear carrier fringe pattern,” Optik 100, 11–14 (1995).

Ina, H.

Kobayashi, S.

Kreis, T.

T. Kreis, “Digital holographic interference-phase measurement using the Fourier transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
[CrossRef]

T. Kreis, “Fourier transform evaluation of holographic interference patterns,” in International Conference on Photo Mechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 365–371 (1987).
[CrossRef]

Malacara, D.

D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

V. I. Vlad, D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” in Progress in Optics, Vol. XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), pp. 261–317.

Malacara, Z.

D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

Marroquín, J. L.

Mendoza-Santoyo, F.

A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Laser Eng. 23, 319–330 (1995).
[CrossRef]

Moore, A. J.

A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Laser Eng. 23, 319–330 (1995).
[CrossRef]

Peng, X.

X. Peng, S. M. Zhou, Z. Gao, “An automatic demodulation technique for a non-linear carrier fringe pattern,” Optik 100, 11–14 (1995).

Servín, M.

Takeda, M.

Vlad, V. I.

V. I. Vlad, D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” in Progress in Optics, Vol. XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), pp. 261–317.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Zhou, S. M.

X. Peng, S. M. Zhou, Z. Gao, “An automatic demodulation technique for a non-linear carrier fringe pattern,” Optik 100, 11–14 (1995).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Opt. Laser Eng.

A. J. Moore, F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Laser Eng. 23, 319–330 (1995).
[CrossRef]

Optik

X. Peng, S. M. Zhou, Z. Gao, “An automatic demodulation technique for a non-linear carrier fringe pattern,” Optik 100, 11–14 (1995).

Other

V. I. Vlad, D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” in Progress in Optics, Vol. XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), pp. 261–317.

D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).

T. Kreis, “Fourier transform evaluation of holographic interference patterns,” in International Conference on Photo Mechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 365–371 (1987).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Computer-simulated interferogram with a circular carrier in a 128 × 128 pixel grid, (b) spectrum.

Fig. 2
Fig. 2

Phase demodulation in an interferogram with a circular carrier with a spherical reference wave front.

Fig. 3
Fig. 3

Results from demodulation of the interferogram in Fig. 1 with a 3 × 3 averaging filter with 40 passes. (a) Circular carrier or defocused reconstructing wave front, (b) phase map, (c) unwrapped phase.

Fig. 4
Fig. 4

Results from demodulation of the same interferogram as in Fig. 3 but with a 3 × 3 averaging filter with 400 passes. (a) Circular carrier or defocused reconstructing wave front, (b) phase map, (c) unwrapped phase.

Fig. 5
Fig. 5

Interfering wave front, (a) a flat wave front and a spherical wave front, (b) a flat wave front, and a discontinuous wave front with two spherical portions, (c) signal for both cases.

Fig. 6
Fig. 6

Phase demodulation in an interferogram with a circular carrier with a tilted-plane reference wave front. (a) Minimum tilt, (b) greater than minimum tilt.

Fig. 7
Fig. 7

Fourier spectrum produced by an interferogram with a circular carrier (Gabor hologram) when illuminated with a tilted-flat reference wave front. The sign of the phase is reversed for negative values of y.

Fig. 8
Fig. 8

Demodulation of interferogram in Fig. 1 with a 3 × 1 convolution filter with 30 passes. (a) Reference reconstructing frequency, (b) phase map.

Equations (17)

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s x ,   y = a + b   cos   k D x 2 + y 2 - W x ,   y = a + b   cos   k DS 2 - W x ,   y = a + b 2 exp + ik DS 2 - W x ,   y + b 2 exp - ik DS 2 - W x ,   y ,
f x ,   y = 2 DS / λ .
DS 2 - W x ,   y S > 0 ,
D > 1 2 S W x ,   y S
r x ,   y = exp   i kD r x 2 + y 2 = exp   i kD r S 2 ,
r x ,   y s x ,   y = a   exp   i kD r S 2 + b 2 exp   ik D + D r S 2 - W x ,   y + b 2 exp - ik D - D r S 2 - W x ,   y .
f r x ,   y = 2 D r S / λ .
f - 1 x ,   y = 2 D + D r S λ - 1 λ W x ,   y S .
f + 1 x ,   y = 2 D - D r S λ - 1 λ W x ,   y S .
s x ,   y r x ,   y = z C x ,   y + iz S x ,   y = s x ,   y cos kD r S 2 + is x ,   y sin kD r S 2 .
z ¯ C x ,   y + i z ¯ S x ,   y = b 2 exp - ik D - D r S 2 - W x ,   y = b 2 cos   k D - D r S 2 - W x ,   y - i   b 2 sin   k D - D r S 2 - W x ,   y .
k D - D r S 2 - W x ,   y = - tan - 1 z ¯ S x ,   y z ¯ C x ,   y .
r x ,   y = exp   i 2 π f r x = cos 2 π f r x + i   sin 2 π f r x ,
s x ,   y r x ,   y = a   exp   i 2 π f r x + b 2 exp   i 2 π f r x + k DS 2 - W x ,   y + b 2 exp - i - 2 π f r x + k DS 2 - W x ,   y
s x ,   y r x ,   y = z C x ,   y + iz S x ,   y = s x ,   y cos 2 π f r x + is x ,   y sin 2 π f r x .
z ¯ C x ,   y + i z ¯ S x ,   y = b 2 exp - i - 2 π f r x + k DS 2 - W x ,   y = b 2 cos - 2 π f r x + k DS 2 - W x ,   y - i   b 2 sin - 2 π f r x + k DS 2 - W x ,   y .
- 2 π f r x + k DS 2 - W x ,   y = - tan - 1 z ¯ S x ,   y z ¯ C x ,   y ,

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