Abstract

We propose a novel phase-shift calibration algorithm. With this technique we determine the unknown phase shift between two interferograms by examining the sums and differences of the intensities on each interferogram at the same spatial location, i.e., I1(x, y)±I2(x, y). These intensities are normalized so that they become sinusoidal in form. A uniformly illuminated region of the interferograms that contains at least a 2π variation in phase is examined. The extrema of these sums and differences are found in this region and are used to find the unknown phase shift. An error analysis of the algorithm is provided. In addition, an error-correction algorithm is implemented. The method is tested by numerical simulation and implemented experimentally. The numerical tests, including digitization error, indicate that the phase step has a root-mean-square (RMS) phase error of less than 10-6 deg. Even in the presence of added intensity noise (5% amplitude) the RMS error does not exceed 1 deg. The accuracy of the technique is not sensitive to nonlinearity in the interferogram.

© 2000 Optical Society of America

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References

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  1. P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).
  2. D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker Inc., New York, 1998).
  3. D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).
  4. Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  5. P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  6. K. Jambunathan, L. S. Wang, B. N. Dobbins, S. P. He, “Semiautomatic phase shifting calibration using digital speckle pattern interferometry,” Opt. Laser Technol. 27, 145–151 (1995).
    [CrossRef]
  7. N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. (Bellingham) 37, 2501–2505 (1998).
    [CrossRef]
  8. Hedser van Brug, “Phase-step calibration for phase-stepped interferometry,” Appl. Opt. 38, 3549–3555 (1999).
    [CrossRef]

1999 (1)

1998 (1)

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. (Bellingham) 37, 2501–2505 (1998).
[CrossRef]

1995 (1)

K. Jambunathan, L. S. Wang, B. N. Dobbins, S. P. He, “Semiautomatic phase shifting calibration using digital speckle pattern interferometry,” Opt. Laser Technol. 27, 145–151 (1995).
[CrossRef]

1985 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Cheng, Y. Y.

Dobbins, B. N.

K. Jambunathan, L. S. Wang, B. N. Dobbins, S. P. He, “Semiautomatic phase shifting calibration using digital speckle pattern interferometry,” Opt. Laser Technol. 27, 145–151 (1995).
[CrossRef]

He, S. P.

K. Jambunathan, L. S. Wang, B. N. Dobbins, S. P. He, “Semiautomatic phase shifting calibration using digital speckle pattern interferometry,” Opt. Laser Technol. 27, 145–151 (1995).
[CrossRef]

Huntley, J. M.

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. (Bellingham) 37, 2501–2505 (1998).
[CrossRef]

Jambunathan, K.

K. Jambunathan, L. S. Wang, B. N. Dobbins, S. P. He, “Semiautomatic phase shifting calibration using digital speckle pattern interferometry,” Opt. Laser Technol. 27, 145–151 (1995).
[CrossRef]

Malacara, D.

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker Inc., New York, 1998).

Malacara, Zacarias

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker Inc., New York, 1998).

Ochoa, N. A.

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. (Bellingham) 37, 2501–2505 (1998).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).

Servin, M.

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker Inc., New York, 1998).

van Brug, Hedser

Wang, L. S.

K. Jambunathan, L. S. Wang, B. N. Dobbins, S. P. He, “Semiautomatic phase shifting calibration using digital speckle pattern interferometry,” Opt. Laser Technol. 27, 145–151 (1995).
[CrossRef]

Wyant, J. C.

Appl. Opt. (2)

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng. (Bellingham) (1)

N. A. Ochoa, J. M. Huntley, “Convenient method for calibrating nonlinear phase modulators for use in phase-shifting interferometry,” Opt. Eng. (Bellingham) 37, 2501–2505 (1998).
[CrossRef]

Opt. Laser Technol. (1)

K. Jambunathan, L. S. Wang, B. N. Dobbins, S. P. He, “Semiautomatic phase shifting calibration using digital speckle pattern interferometry,” Opt. Laser Technol. 27, 145–151 (1995).
[CrossRef]

Other (3)

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, Berlin, 1994).

D. Malacara, M. Servin, Zacarias Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker Inc., New York, 1998).

D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).

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Figures (4)

Fig. 1
Fig. 1

RMS phase error as a function of the phase step in the presence of noise. (a) Intensity noise with an amplitude of 5% of the modulation intensity, Im, is added to the simulated interferograms (the simulated interferograms have digitization noise). (b) Intensity noise of 10%. Solid curves, RMS error for the phase calibration at this level of noise; dashed curves, reduced error achieved by implementing the error correction scheme.

Fig. 2
Fig. 2

RMS phase error as a function of added intensity noise. The error in the calibration of a phase step of 15 deg is measured. Solid curve, error for the calibration algorithm; dashed curve, error when the error-correction algorithm is implemented in conjunction with the calibration algorithm.

Fig. 3
Fig. 3

Raw data from the region of interest on two interferograms. The two sets of data (solid and dashed curves) are from interferograms that have a phase separation of five cycles as determined by the phase calibration. They are nearly identical, as expected.

Fig. 4
Fig. 4

Calibration curve for the phase shifter (PZT). The symbols indicate the voltages at which interferograms were recorded and the corresponding phase shifts. The solid curve is a third-order polynomial fit to these data points.

Equations (19)

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I1(x, y) = Ib(x, y) + Im(x, y)cos[δ(x, y)],
I2(x, y) = Ib(x, y) + Im(x, y)cos[δ(x, y) + ϕ],
In1(x, y) = cos[δ(x, y)],
In2(x, y) = cos[δ(x, y) + ϕ].
Q12+ = (1 + cos ϕ)cos[δ(x, y)] - sin ϕ sin[δ(x, y)].
R12+ = max[Q12+] - min[Q12+] = 22 + 2 cos ϕ.
Q12- = (1 - cos ϕ)cos[δ(x, y)]+sin ϕ sin[δ(x, y)].
R12- = max[Q12-] - min[Q12-] = 22-2 cos ϕ.
cos ϕ = (R12+)2 - (R12-)216.
(R12+)2+(R12-)2 = 16.
δϕ=-δC/sin ϕ.
cos ϕ2(δϕ)2+sin ϕδϕ + δ C = 0.
δϕ =-tan ϕ1 + 1 - 2δCcos ϕ tan2 ϕ1/2.
 = 16 - [(R12+)2 + (R12-)2].
(R12+ + ΔR12+)2 + (R12- + ΔR12-)2 = 16,
2R12+ΔR12+ + 2R12-ΔR12-  
= 16 - [(R12+)2 + (R12-)2] = .
ΔR12+ = R12+2[(R12+)2 + (R12-)2],
ΔR12- = R12-2[(R12+)2 + (R12-)2],

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