Abstract

Phase-shifting is one of the most useful methods of phase recovery in digital interferometry in the estimation of small displacements, but miscalibration errors of the phase shifters are very common. In practice, the main problem associated with such errors is related to the response of the phase shifter devices, since they are dependent on mechanical and/or electrical parts. In this work, a novel technique to detect and measure calibration errors in phase-shifting interferometry, when an unexpected phase shift arises, is proposed. The described method uses the Radon transform, first as an automatic-calibrating technique, and then as a profile measuring procedure when analyzing a specific zone of an interferogram. After, once maximum and minimum value parameters have been registered, these can be used to measure calibration errors. Synthetic and real interferograms are included in the testing, which has thrown good approximations for both cases, notwithstanding the interferogram fringe distribution or its phase-shifting steps. Tests have shown that this algorithm is able to measure the deviations of the steps in phase-shifting interferometry. The developed algorithm can also be used as an alternative in the calibration of phase shifter devices.

© 2017 Optical Society of America

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References

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2016 (3)

2014 (2)

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

G. A. Ayubi, C. D. Perciante, J. L. Flores, J. M. Di Martino, and J. A. Ferrari, “Generation of phase-shifting algorithms with N arbitrarily spaced phase-steps,” Appl. Opt. 53(30), 7168–7176 (2014).
[Crossref] [PubMed]

2013 (2)

I. Elouedi, R. Fournier, A. Nait-Ali, and A. Hamouda, “Generalized multidirectional discrete Radon transform,” Signal Process. 93(1), 345–355 (2013).
[Crossref]

F. Zhang, L. Zhong, S. Liu, and X. Lu, “Spatiotemporal hybrid based non-iterative phase-shifting amount extraction method,” Opt. Commun. 300, 137–141 (2013).
[Crossref]

2011 (2)

2010 (1)

2009 (2)

J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express 17(18), 15766–15771 (2009).
[Crossref] [PubMed]

Z. Li, Y. Shi, C. Wang, D. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

2007 (1)

R. Langoju, A. Patil, and P. Rastogi, “A novel approach for characterizing the nonlinear phase steps of the pzt in interferometry,” Opt. Lasers Eng. 45(2), 258–264 (2007).
[Crossref]

2004 (1)

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1–3), 21–26 (2004).
[Crossref]

2001 (1)

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surfaces,” Opt. Commun. 190(1–6), 109–116 (2001).
[Crossref]

1998 (1)

1997 (1)

1996 (1)

1993 (1)

Ayubi, G. A.

Bo, T.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surfaces,” Opt. Commun. 190(1–6), 109–116 (2001).
[Crossref]

Cai, L. Z.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1–3), 21–26 (2004).
[Crossref]

Castillo, F.

Chai, L.

Di Martino, J. M.

Doblado, D. M.

Elouedi, I.

I. Elouedi, R. Fournier, A. Nait-Ali, and A. Hamouda, “Generalized multidirectional discrete Radon transform,” Signal Process. 93(1), 345–355 (2013).
[Crossref]

Farrant, D. I.

Ferrari, J. A.

Flores, J. L.

Fournier, R.

I. Elouedi, R. Fournier, A. Nait-Ali, and A. Hamouda, “Generalized multidirectional discrete Radon transform,” Signal Process. 93(1), 345–355 (2013).
[Crossref]

García-González, M. A.

Gutiérrez-García, J. C.

Gutiérrez-García, T. A.

Gutiérrez-García, V. A.

Hamouda, A.

I. Elouedi, R. Fournier, A. Nait-Ali, and A. Hamouda, “Generalized multidirectional discrete Radon transform,” Signal Process. 93(1), 345–355 (2013).
[Crossref]

Hernández, D. M.

Hibino, K.

Huang, K.

Z. Li, Y. Shi, C. Wang, D. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

Jin, W.

Langoju, R.

R. Langoju, A. Patil, and P. Rastogi, “A novel approach for characterizing the nonlinear phase steps of the pzt in interferometry,” Opt. Lasers Eng. 45(2), 258–264 (2007).
[Crossref]

Larkin, K. G.

Li, Y.

Li, Z.

Z. Li, Y. Shi, C. Wang, D. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

Liu, Q.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1–3), 21–26 (2004).
[Crossref]

Liu, S.

C. Tian and S. Liu, “Two-frame phase-shifting interferometry for testing optical surfaces,” Opt. Express 24(16), 18695–18708 (2016).
[Crossref] [PubMed]

F. Zhang, L. Zhong, S. Liu, and X. Lu, “Spatiotemporal hybrid based non-iterative phase-shifting amount extraction method,” Opt. Commun. 300, 137–141 (2013).
[Crossref]

Lu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

F. Zhang, L. Zhong, S. Liu, and X. Lu, “Spatiotemporal hybrid based non-iterative phase-shifting amount extraction method,” Opt. Commun. 300, 137–141 (2013).
[Crossref]

Ma, J.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Mosiño, J. F.

Nait-Ali, A.

I. Elouedi, R. Fournier, A. Nait-Ali, and A. Hamouda, “Generalized multidirectional discrete Radon transform,” Signal Process. 93(1), 345–355 (2013).
[Crossref]

Ngoi, B. K. A.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surfaces,” Opt. Commun. 190(1–6), 109–116 (2001).
[Crossref]

Oreb, B. F.

Pan, T.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Patil, A.

R. Langoju, A. Patil, and P. Rastogi, “A novel approach for characterizing the nonlinear phase steps of the pzt in interferometry,” Opt. Lasers Eng. 45(2), 258–264 (2007).
[Crossref]

Perciante, C. D.

Qin, D.

Z. Li, Y. Shi, C. Wang, D. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

Rastogi, P.

R. Langoju, A. Patil, and P. Rastogi, “A novel approach for characterizing the nonlinear phase steps of the pzt in interferometry,” Opt. Lasers Eng. 45(2), 258–264 (2007).
[Crossref]

Shi, Y.

Z. Li, Y. Shi, C. Wang, D. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

Shou, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Sivakumar, N. R.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surfaces,” Opt. Commun. 190(1–6), 109–116 (2001).
[Crossref]

Surrel, Y.

Tian, C.

Tian, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Venkatakrishnan, K.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surfaces,” Opt. Commun. 190(1–6), 109–116 (2001).
[Crossref]

Wang, C.

Z. Li, Y. Shi, C. Wang, D. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

Wang, H.

Wang, Z.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Xu, J.

Xu, Q.

Xu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Yang, X. L.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1–3), 21–26 (2004).
[Crossref]

Zhang, F.

F. Zhang, L. Zhong, S. Liu, and X. Lu, “Spatiotemporal hybrid based non-iterative phase-shifting amount extraction method,” Opt. Commun. 300, 137–141 (2013).
[Crossref]

Zheng, D.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Zhong, L.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

F. Zhang, L. Zhong, S. Liu, and X. Lu, “Spatiotemporal hybrid based non-iterative phase-shifting amount extraction method,” Opt. Commun. 300, 137–141 (2013).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Am. A (2)

Opt. Commun. (5)

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Z. Li, Y. Shi, C. Wang, D. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

F. Zhang, L. Zhong, S. Liu, and X. Lu, “Spatiotemporal hybrid based non-iterative phase-shifting amount extraction method,” Opt. Commun. 300, 137–141 (2013).
[Crossref]

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, and T. Bo, “Instantaneous phase shifting arrangement for microsurface profiling of flat surfaces,” Opt. Commun. 190(1–6), 109–116 (2001).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1–3), 21–26 (2004).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (2)

R. Langoju, A. Patil, and P. Rastogi, “A novel approach for characterizing the nonlinear phase steps of the pzt in interferometry,” Opt. Lasers Eng. 45(2), 258–264 (2007).
[Crossref]

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Signal Process. (1)

I. Elouedi, R. Fournier, A. Nait-Ali, and A. Hamouda, “Generalized multidirectional discrete Radon transform,” Signal Process. 93(1), 345–355 (2013).
[Crossref]

Other (7)

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw Hill, 2000), pp. 356–358.

S. Deans, “Radon and Abel Transforms,” in The Transforms and Applications Handbook, 2nd ed., A. D. Poularikas, ed. (CRC LLC, 2000), chapter 8.

D. Malacara, K. Creath, J. Schmit, and J. C. Wyant, “Testing of Aspheric Wavefronts and Surfaces,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, Inc., 2007), pp. 435–497.

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology. Theory, Algorithms, and Applications. (Wiley-VCH, 2014), pp. 241–270.

K. J. Gasvik, Optical Metrology, 3rd ed. (John Wiley & Sons, Inc., 2002).

H. Schreiber and J. H. Bruning, “Phase Shifting Interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (John Wiley & Sons, Inc., 2007), pp. 547–666.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005), pp. 359–398.

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Figures (9)

Fig. 1
Fig. 1

Radon transform applied to an image distribution.

Fig. 2
Fig. 2

Searching process of the adequate θA. (a) Flow diagram, and (b) Example applied to an interferogram.

Fig. 3
Fig. 3

Sinusoidal profile of the Radon transform at θA for four interferograms: (a) ε k = 0, and (b) considering a deviation in ε 3 .

Fig. 4
Fig. 4

Flow diagram of phase-shifting error detection process.

Fig. 5
Fig. 5

Synthetic interferograms: (a) SI1, (b) SI2 and (c) SI3.

Fig. 6
Fig. 6

The three θA search process for each SI. (a) [0°, 180°] Radon transforms for the three SI, and (b) Thresholding of the Radon transforms for the three SI.

Fig. 7
Fig. 7

Waveform of one cycle for the Radon transform of each SI. (a) of SI1, (b) of SI2, and (c) of SI3.

Fig. 8
Fig. 8

The θA search process for RI. (a) Real interferogram of a flat profile, (b) Radon transform of the real interferogram, and (c) Thresholding of the Radon transform.

Fig. 9
Fig. 9

Waveform of one cycle for the Radon transform of the real interferogram.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

I k ( x , y ) = A ( x ,   y ) + B ( x , y ) cos ( φ ( x ,   y ) + α k ) ,
I k ( x , y ) = A ( x ,   y ) + B ( x , y ) cos ( φ ( x ,   y ) + α k + ε k ) .
t a n ( φ ) = k b k I k k c k I k ,
φ w ( x , y ) = a t a n [ I 4 ( x , y ) I 2 ( x , y ) I 1 ( x , y ) I 3 ( x , y ) ]
f = R ( ρ , θ ) = f ( x , y ) δ ( ρ x c o s θ y s i n θ ) d x d y ,
I k 0.5 + 0.5 cos ( ρ + ( k 1 ) π 2 + ε k ) ,
ε 1 = | π P min I 1 | = | 0 P max I 1 | , ε 2 = | π 2 P min I 2 | = | π 2 P max I 2 | , ε 3 = | 0 P min I 3 | = | π P max I 3 | , ε 4 = | π 2 P min I 4 | = | π 2 P max I 4 | .

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