Abstract

To overcome the phase shift error in phase shifting interferometry, a two-step random phase retrieval approach based on Gram-Schmidt (GS) orthonormalization and Lissajous ellipse fitting (LEF) method (GS&LEF) is proposed. It doesn’t need pre-filtering, and can obtain relatively accurate phase distribution with only two phase shifted interferograms and less computational time. It is suitable for different background intensity, modulation amplitude distributions and noises. Last but not least, this method is effective for circular, straight or complex fringes. The simulations and experiments verify the correctness and feasibility of GS&LEF.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (3)

2016 (5)

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

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

2015 (1)

2013 (1)

2012 (1)

2011 (3)

2009 (2)

2008 (1)

2004 (1)

1997 (1)

1995 (1)

1994 (1)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

1992 (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

1988 (1)

1974 (1)

Belenguer, T.

Brangaccio, D. J.

Bruning, J. H.

Carazo, J. M.

Chai, L.

Chen, Q.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Chen, W.

Chen, Y. C.

de Groot, P. J.

Deck, L. L.

Deng, J.

Estrada, J. C.

Farrant, D. I.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Gallagher, J. E.

Han, B.

Han, H.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Hao, J.

Herriott, D. R.

Hibino, K.

Hou, X.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Ishikawa, K.

Ji, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Jin, W.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

Kinnstaetter, K.

Larkin, K. G.

Lee, C. M.

Li, C.

Liang, C. W.

Liang, R.

Lin, P. C.

Liu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Liu, S.

Lohmann, A. W.

Lv, X.

Oikawa, Y.

Oreb, B. F.

Patorski, K.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Qin, J.

Quiroga, J. A.

Rosenfeld, D. P.

Schwider, J.

Servin, M.

Sorzano, C. O.

Streibl, N.

Tian, C.

Tian, X.

Trusiak, M.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Vargas, J.

Wan, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wang, J.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wang, K.

Wang, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Wang, Z.

White, A. D.

Wu, D.

Wu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Wu, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Xu, J.

Xu, Q.

Xu, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Yatabe, K.

Zhang, Y.

Appl. Opt. (5)

J. Opt. Soc. Am. A (3)

J. Optics-UK (2)

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert-Huang prefiltering,” J. Optics-UK 18(10), 105604 (2016).
[Crossref]

Meas. Sci. Technol. (2)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Opt. Express (8)

J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
[Crossref] [PubMed]

Y. Zhang, X. Tian, and R. Liang, “Fringe-print-through error analysis and correction in snapshot phase-shifting interference microscope,” Opt. Express 25(22), 26554–26566 (2017).
[Crossref] [PubMed]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Hyper ellipse fitting in subspace method for phase-shifting interferometry: Practical implementation with automatic pixel selection,” Opt. Express 25(23), 29401–29416 (2017).
[Crossref]

Y. Zhang, X. Tian, and R. Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15071 (2018).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[Crossref] [PubMed]

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

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

Opt. Lasers Eng. (1)

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Opt. Lett. (4)

Other (2)

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007), Chap. 1–7.

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007), Chap. 14.

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Figures (11)

Fig. 1
Fig. 1 Simulated phase distribution and two phase shifted interferograms. (a) The theoretical phase distribution, (b) and (c) the first interferogram and the second interferogram.
Fig. 2
Fig. 2 The calculated phase distributions and phase error distributions by GS and GS&LEF in different simulations.
Fig. 3
Fig. 3 The ellipses before and after using LEF method for GS&LEF in different simulations.
Fig. 4
Fig. 4 The RMS phase error curves of GS and GS&LEF for different phase shifts with different levels of noises. (a) – (l) the phase shifts are respectively 0.4 rad, 0.6 rad, 0.8 rad, 1.0 rad, 1.2 rad, 1.4 rad, 1.6 rad, 1.8 rad, 2.0 rad, 2.2 rad, 2.4 rad and 2.6 rad.
Fig. 5
Fig. 5 Simulated interferogram with straight fringes and plane phase distribution. (a) One of the simulated interferograms, (b) theoretical phase distribution (PV = 25.1327 rad, RMS = 7.2733 rad).
Fig. 6
Fig. 6 Simulated results of straight fringes using GS and GS&LEF. (a) and (b) the extracted phase distributions using GS and GS&LEF, (c) and (d) the phase error distributions after using GS and GS&LEF.
Fig. 7
Fig. 7 Simulated complex interferogram and phase distribution. (a) One of the simulated interferograms, (b) theoretical phase distribution (PV = 57.8814 rad, RMS = 11.0403 rad) .
Fig. 8
Fig. 8 Simulated results of complex fringes using GS and GS&LEF. (a) and (b) the extracted phase distributions using GS and GS&LEF, (c) and (d) the phase error distributions after using GS and GS&LEF.
Fig. 9
Fig. 9 Experimental results of the circular fringes. (a) One of the phase shifted interferograms, (b) the reference phase distribution obtained by 4-step PSA (PV = 38.4319 rad, RMS = 7.9988 rad), (c) and (d) the extracted phase distributions obtained by GS (PV = 37.7590 rad, RMS = 7.9190 rad) and GS&LEF (PV = 38.2341 rad, RMS = 7.9983 rad), (e) and (f) the differences between the reference and extracted phase distributions by GS and GS&LEF, (g) and (h) the ellipses before and after using LEF method.
Fig. 10
Fig. 10 Experimental results of the straight fringes. (a) One of the phase shifted interferograms, (b) the reference phase distribution obtained by 4-step PSA (PV = 21.6492 rad, RMS = 6.0171 rad), (c) and (d) the extracted phase distributions obtained by GS (PV = 24.0863 rad, RMS = 6.0483 rad) and GS&LEF (PV = 22.1756 rad, RMS = 6.0483 rad), (e) and (f) the differences between the reference and extracted phase distributions by GS and GS&LEF, (g) and (h) the ellipses before and after using LEF method.
Fig. 11
Fig. 11 Experimental results of the complex fringes. (a) One of the phase shifted interferograms, (b) the reference phase distribution obtained by 4-step PSA (PV = 135.5400 rad, RMS = 30.5750 rad), (c) and (d) the extracted phase distributions obtained by GS (PV = 135.1772 rad, RMS = 30.5159 rad) and GS&LEF (PV = 135.9246 rad, RMS = 30.5455 rad), (e) and (f) the differences between the reference and extracted phase distributions by GS and GS&LEF, (g) and (h) the ellipses before and after using LEF method.

Tables (1)

Tables Icon

Table 1 The RMS phase errors and computational time of GS and GS&LEF in different simulations.

Equations (20)

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

I m ( x,y )=a( x,y )+b( x,y )cos( φ( x,y )+ δ m ).
u ˜ 1 = u 1 / u 1 .
u ^ 2 = u 2 u 2 , u ˜ 1 u ˜ 1 .
u ˜ 2 = u ^ 2 / u ^ 2 .
I ˜ 1 = I 1 / I 1 = a+bcos( φ ) Θ 1
cos( φ )= I ˜ 1 a Θ 1 b Θ 1
I ^ 2 = I 2 I 2 , I ˜ 1 I ˜ 1 =a+bcos( φ+δ )( x=1 N x y=1 N y ( a+bcos( φ+δ ) )( a+bcos( φ ) Θ 1 ) )( a+bcos( φ ) Θ 1 ) = a( Θ 1 Θ ) Θ 1 + b( Θ 1 cos( δ )Θ ) Θ 1 cos( φ )bsin( δ )sin( φ )
I ˜ 2 = I ^ 2 / I ^ 2 == a( Θ 1 Θ ) Θ 1 Θ 2 + b( Θ 1 cos( δ )Θ ) Θ 1 Θ 2 cos( φ ) bsin( φ )sin( δ ) Θ 2 = a( Θ 1 Θ ) Θ 1 Θ 2 + Θ 1 cos( δ )Θ Θ 2 I ˜ 1 a( Θ 1 cos( δ )Θ ) Θ 1 Θ 2 bsin( φ )sin( δ ) Θ 2
cos( φ )= D a Θ 1 b Θ 1
sin( φ )= 1 bsin( δ ) ( Θ 2 N +( Θ 1 cos( δ )Θ ) D a( 1cos( δ ) ) )
cos( φ )=D D +E
sin( φ )=A N +B D +C
( A N +B D +C ) 2 + ( D D +E ) 2 =1
A 2 N 2 +2AB N D +( B 2 + D 2 ) D 2 +2AC N +2DE D + C 2 + E 2 1=0
F=a x 2 +bxy+c y 2 +dx+fy+g
a x = 2 a f 2 +c d 2 +g b 2 bdf4acg ( b 2 4ac )( ( ac ) 2 + b 2 ( a+c ) ) , a y = 2 a f 2 +c d 2 +g b 2 bdf4acg ( b 2 4ac )( ( ac ) 2 + b 2 ( a+c ) ) x 0 = 2cdbf b 2 4ac , y 0 = 2afbd b 2 4ac θ= 1 2 arctan b ac fora<c θ= π 2 + 1 2 arctan b ac fora>c
[ N c D c ]=T*[ N x 0 D y 0 ]
T=[ cos( θ ) sin( θ ) sin( θ ) cos( θ ) ]*[ 1 0 0 r ]*[ cos θ sin θ sin θ cos θ ]
φ=arctan( N c D c )
a 1 ( x,y )=exp[ 0.02( x 2 + y 2 ) ] a 2 ( x,y )=0.9exp[ 0.02( x 2 + y 2 ) ] b 1 ( x,y )=0.9exp[ 0.02( x 2 + y 2 ) ] b 2 ( x,y )=0.8exp[ 0.02( x 2 + y 2 ) ]

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