Abstract

An iterative phase-shifting algorithm based on the least-squares principle is developed to overcome the random piston and tilt wavefront errors generated from the phase shifter. The algorithm iteratively calculates the phase distribution and the phase-shifting map to minimize the sum of squared errors in the interferograms. The performance of the algorithm is evaluated via computer simulations and validated by the Fizeau interferometer measurements. The results show that the proposed algorithm has a fast convergence rate and satisfactory phase-estimation accuracy, improving the measurement precision of the phase-shifting interferometers with significant phase-shifter errors.

© 2013 Optical Society of America

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References

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  1. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).
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  15. H. S. Chang, C. W. Liang, and C. C. Lee, “Using Hariharan algorithm to calibrate the piezoelectric transducers in Fizeau interferometer,” presented at Optics & Photonics Taiwan 2010, Tainan, Taiwan, 3–4 December2010.

2011 (1)

2008 (1)

2006 (2)

2004 (2)

2003 (1)

2002 (1)

2000 (1)

1994 (1)

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

1987 (1)

Apostol, A.

Basaran, C.

Bizuet, R.

Bruning, J. H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).

Cai, L.

Chai, L.

Chang, H. S.

H. S. Chang, C. W. Liang, and C. C. Lee, “Using Hariharan algorithm to calibrate the piezoelectric transducers in Fizeau interferometer,” presented at Optics & Photonics Taiwan 2010, Tainan, Taiwan, 3–4 December2010.

Chen, B.

Chen, M.

Damian, V.

Dobroiu, A.

Dong, G.

Eiju, T.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

Guo, H.

Han, B.

Han, G.-S.

Hariharan, P.

Kim, S.-W.

Lee, C. C.

H. S. Chang, C. W. Liang, and C. C. Lee, “Using Hariharan algorithm to calibrate the piezoelectric transducers in Fizeau interferometer,” presented at Optics & Photonics Taiwan 2010, Tainan, Taiwan, 3–4 December2010.

Liang, C. W.

H. S. Chang, C. W. Liang, and C. C. Lee, “Using Hariharan algorithm to calibrate the piezoelectric transducers in Fizeau interferometer,” presented at Optics & Photonics Taiwan 2010, Tainan, Taiwan, 3–4 December2010.

Liu, Q.

Martinez, A.

Meng, X.

Nascov, V.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Oreb, B. F.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

Rayas, J. A.

Rivera, M.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Schreiber, H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).

Shen, X.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

Wang, Z.

Wei, C.

Xu, J.

Xu, Q.

Xu, X.

Yang, X.

Appl. Opt. (6)

Opt. Commun. (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Other (3)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University, 2007).

H. S. Chang, C. W. Liang, and C. C. Lee, “Using Hariharan algorithm to calibrate the piezoelectric transducers in Fizeau interferometer,” presented at Optics & Photonics Taiwan 2010, Tainan, Taiwan, 3–4 December2010.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley-Interscience, 2007).

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Figures (4)

Fig. 1.
Fig. 1.

Simulation results with random pistons and tilts in the phase shifts: (a) log-scaled intensity RMSE as a function of iterations, (b) phase estimate, (c) phase residual, and (d) tilt-free phase residual.

Fig. 2.
Fig. 2.

Phase shifter configuration of the Fizeau interferometer.

Fig. 3.
Fig. 3.

Experimental results: (a) Hariharan phase distribution, (b) the phase distribution by the proposed algorithm, (c) Hariharan intensity-error map, (d) the intensity-error map by the proposed algorithm, (e) Hariharan fringe-modulation map, and (f) the fringe-modulation map by the proposed algorithm.

Fig. 4.
Fig. 4.

Comparisons of the simulation results with and without the uniformity assumption of the background and modulation intensity in the nonlinear least-squares fitting step: (a), (b) are background intensity estimates without and with the assumption; (c), (d) are modulation intensity estimates without and with the assumption; (e), (f) are tilt-free phase residual maps without and with the assumption; (g), (h) are log-scaled intensity RMSE as a function of iterations without and with the assumption, respectively.

Tables (1)

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Table 1. Statistics of the Simulation Results for 200 Sets of Random Samples

Equations (12)

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Ii(x,y)=Idc(x,y)+Iac(x,y)cos[ϕ(x,y)+pi+xTix+yTiy],fori=1,2,,M,
δi(x,y)=pi+xTix+yTiy=(i2)×α(x,y),fori=1,2,3,4,5.
Ii(x,y)=ui+vicos[ϕ(x,y)]+wisin[ϕ(x,y)],
Si=x,y[Iiest(x,y)Iimeas(x,y)]2,
Ii(x,y)=a(x,y)+b(x,y)cos[δi(x,y)]+c(x,y)sin[δi(x,y)],
S(x,y)=i[Iiest(x,y)Iimeas(x,y)]2.
Si(pi,Tix,Tiy)=x,y{Idc(x,y)+Iac(x,y)cos[ϕ(x,y)+pi+xTix+yTiy]Iimeas(x,y)}2.
(p^i,T^ix,T^iy)=argminpi,Tix,Tiyx,y{Idc+Iaccos[ϕ(x,y)+pi+xTix+yTiy]Iimeas(x,y)}2.
RMSE=1MNix,y[Idc(x,y)+Iac(x,y)cos[ϕ(x,y)+δi+xTix+yTiy]Iimeas(x,y)]2,
|RMSE(k)RMSE(k1)|<ε·max[Iimeas(x,y)],
Ierror(x,y)=1Mi=1M[Iiest(x,y)Iimeas(x,y)]2,
γ(x,y)=Iac(x,y)/Idc(x,y).

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