Abstract

A self-calibrating algorithm for phase-shift interferometry is described that is able to cancel the effect of accidental relative tilts that may occur during phase stepping. The algorithm is able to retrieve both the phase steps and the tilts that accompany them. Only three phase-shifted interferograms are needed, and no other information about the intentional phase shifts or possible tilts has to be supplied. This purpose is achieved by division of the interferogram space into blocks on which a previously reported self-calibrating algorithm is applied and the actual values of the local phase shifts are calculated. The information thus obtained is used for extracting the global shift and tilt values. Further improvement in the results is achieved by means of a fitting routine.

© 2002 Optical Society of America

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References

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  1. D. W. Robinson, G. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics, Bristol, UK, 1993).
  2. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
    [CrossRef]
  3. M. Chen, H. Guo, C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39, 3894–3898 (2000).
    [CrossRef]
  4. A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9, 744–750 (1998).
    [CrossRef]
  5. A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
    [CrossRef]

2000 (1)

1998 (2)

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9, 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

1997 (1)

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9, 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Chen, M.

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9, 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9, 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Farrant, D. I.

Guo, H.

Hibino, K.

Larkin, K. G.

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9, 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Oreb, B. F.

Wei, C.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (2)

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9, 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Other (1)

D. W. Robinson, G. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics, Bristol, UK, 1993).

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Figures (3)

Fig. 1
Fig. 1

(a) One phase-shifted interferogram (of three) simulated with a relative tilt. (b) Corresponding contrast map calculated, taking into account uniform arbitrary phase shifts. The pseudofringes have variable visibility, depending on the difference between actual and supposed phase shifts. In a small region this visibility is zero.

Fig. 2
Fig. 2

Mean phase shifts found within individual blocks by applying a self-calibrating algorithm. The calculated contrast map, for which the phase shifts were assumed constant inside each block, is shown. Unlike Fig. 1(b) the pseudofringes are displayed here largely amplified to make them clearly visible. Pseudofringes have a visibility minimum in the center of the blocks. Block size was taken at 32 × 32 pixels.

Fig. 3
Fig. 3

Test on the real set of interferograms of an approximately spherical wave front. (a) Contrast map calculated with an arbitrary set of uniform phase shifts and showing strong pseudofringes. (b) Corresponding phase map affected by tilts, better seen in (c) the phase profile of a horizontal line of pixels, at mid-height, as shoulders on the slopes, especially on the right-hand side. The same information is in (d), (e), and (f) after the tilts were measured and compensated. The phase profile is visibly better. Ideally this profile is a set of parabola arcs.

Equations (7)

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Ikx, y=Imx, y1+Vx, ycosϕx, y-βk, k=0, 1, 2,
βkx, y=βk0+txkx+tyky, k=1, 2,
V=c2+s21/2I0-c,
c=I1-I0sin β2-I2-I0sin β1cos β1-1sin β2-cos β2-1sin β1,
s=cos β1-1I2-I0-cos β2-1I1-I0cos β1-1sin β2-cos β2-1sin β1.
W=NpixelsVx, y-V02Npixels.
tx1=-2.43×10-4×2π/pixel=-0.091×2π/x side,tx2=-5.94×10-4×2π/pixel=-0.219×2π/x side,ty1=+2.98×10-5×2π/pixel=+0.009×2π/y side,ty2=+7.15×10-5×2π/pixel=+0.021×2π/y side.

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