Abstract

To accurately obtain the phase distribution of an optical surface under test, the accurate phase extraction algorithm is essential. To overcome the phase shift error, a random two-step phase shifting algorithm, which can be used in the fluctuating and non-uniform background intensity and modulation amplitude, Lissajous ellipse fitting, and least squares iterative phase shifting algorithm (LEF&LSI PSA), is proposed; pre-filtering interferograms are not necessary, but they can get relatively accurate phase distribution and unknown phase shift value. The simulation and experiment verify the correctness and feasibility of the LEF & LSI PSA.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (3)

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

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. Opt. 18(10), 105604 (2016).
[Crossref]

2014 (1)

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

2013 (1)

2010 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

2009 (2)

2008 (2)

2007 (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

2006 (2)

2004 (2)

2003 (2)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

1997 (2)

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)

Brangaccio, D. J.

Bruning, J. H.

Cai, L. Z.

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

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. Opt. 18(10), 105604 (2016).
[Crossref]

Chen, Y. C.

Cheng, X. C.

de Groot, P. J.

Deck, L. L.

Dong, G. Y.

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]

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. Opt. 18(10), 105604 (2016).
[Crossref]

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]

Kinnstaetter, K.

Larkin, K. G.

Lee, C. M.

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. Opt. 18(10), 105604 (2016).
[Crossref]

Liu, Q.

Liu, S.

Lohmann, A. W.

Meng, X. F.

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. Opt. 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]

Quiroga, J. A.

Rosenfeld, D. P.

Schwider, J.

Servin, M.

Shen, X. X.

Streibl, N.

Sun, W. J.

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. Opt. 18(10), 105604 (2016).
[Crossref]

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. Opt. 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. Opt. 18(10), 105604 (2016).
[Crossref]

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, Y. R.

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Wang, Z.

White, A. 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. Opt. 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. Opt. 18(10), 105604 (2016).
[Crossref]

Xu, J.

Xu, Q.

Xu, X. F.

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]

Yan, R. S.

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[Crossref]

Yang, X. L.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003).
[Crossref] [PubMed]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Zhang, H.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

Zhang, Y.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, “Generalized phase-shifting interferometry with arbitrary unknown phase shifts: Direct wave-front reconstruction by blind phase shift extraction and its experimental verification,” Appl. Phys. Lett. 90(12), 121124 (2007).
[Crossref]

J. Opt. (2)

X. F. Xu, L. Z. Cai, Y. R. Wang, and R. S. Yan, “Direct phase shift ectraction and wavefront reconstruction in two-step generalized phase-shifting interferometry,” J. Opt. 12(1), 015301 (2010).
[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. Opt. 18(10), 105604 (2016).
[Crossref]

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

Meas. Sci. Technol. (2)

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]

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

Opt. Commun. (1)

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, “Sensitivity adjustable contouring by digital holography and a virtual reference wavefront,” Opt. Commun. 221(1-3), 49–54 (2003).
[Crossref]

Opt. Express (3)

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. (6)

Optik (Stuttg.) (1)

R. S. Yan, L. Z. Cai, and X. F. Meng, “Correction of wave-front retrieval errors caused by the imperfect collimation of reference beam in phase-shifting interferometry,” Optik (Stuttg.) 125(2), 601–605 (2014).
[Crossref]

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 (7)

Fig. 1
Fig. 1 Flow chart of LEF & LSI-PSA.
Fig. 2
Fig. 2 Simulated tested surface and two phase shifting interferograms. (a) The tested surface, (b) the first interferogram, and (c) the second interferogram.
Fig. 3
Fig. 3 The phase errors of LEF and LEF & LSI PSAs for different phase shift values.
Fig. 4
Fig. 4 (a) The phase error (RMS) and (b) phase shift error of LEF and LEF & LSI PSAs with different noises.
Fig. 5
Fig. 5 Four phase shifted interferograms with π/2 phase shift.
Fig. 6
Fig. 6 The 2D maps of the calculated phase distributions by (a) 4 step PSA (PV = 452.7nm, RMS = 44.3 nm), (b) LEF PSA (PV = 703.4 nm, RMS = 54.3 nm) and (c) LEF & LSI PSA (PV = 493.3 nm, RMS = 48.4 nm).
Fig. 7
Fig. 7 The calculated result by LEF & LSI PSA. (a) the iterative curve, (b) and (c) the calculated phase and phase shift with different iterative times.

Tables (4)

Tables Icon

Table 1 The calculated phase distribution, phase error and iterative curve LEF & LSI method in different simulations.

Tables Icon

Table 2 The phase shift error, RMS phase error and processing time of LEF and LEF & LSI PSAs in 4 different simulations

Tables Icon

Table 3 The background intensity and modulation amplitude in 8 different situations

Tables Icon

Table 4 The phase shift errors and RMS phase errors of LEF and LEF & LSI PSAs with different fluctuations in the background intensity and modulation amplitude distribution

Equations (22)

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

I 1 ( x,y )= A 1 ( x,y )+ B 1 ( x,y )cos( φ( x,y ) ) I 2 ( x,y )= A 2 ( x,y )+ B 2 ( x,y )cos( φ( x,y )+α ).
I 1 ( x,y )=a+bcos( φ( x,y ) ) I 2 ( x,y )=a+bcos( φ( x,y )+α ).
I dif = I 1 I 2 =2bsin( φ+ α 2 )sin( α 2 ) I sum = I 1 + I 2 =2a+2bcos( φ+ α 2 )cos( α 2 ).
sin( φ+ α 2 )= I dif 2bsin( α 2 ) cos( φ+ α 2 )= I sum 2a 2bcos( α 2 ) .
( I dif x 0 a x ) 2 + ( I sum y 0 a y ) 2 =1.
a x =2bsin( α 2 ), a y =2bcos( α 2 ), x 0 =0, y 0 =2a.
I 2 dif a x 2 + I 2 sum a y 2 2 x 0 I dif a x 2 2 y 0 I sum a y 2 + x 0 2 a x 2 + y 0 2 a y 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 .
α=2 tan 1 ( a x a y ).
φ= tan 1 ( I dif x 0 I sum y 0 a y a x ) tan 1 ( a x a y ).
φ= tan 1 ( I dif I sum y 0 a y a x )
I i,j = A i,j + B i,j cos( φ j + α i ).
I i,j = a i + b i cos φ j + c i sin φ j .
S i = j=1 N ( I i,j I i,j ) 2 = j=1 N ( a i + b i cos φ j + c i sin φ j I i,j ) 2 .
X i = S i -1 R i .
S i =[ N j=1 N cos φ j j=1 N sin φ j j=1 N cos φ j j=1 N cos 2 φ j j=1 N sin φ j cos φ j j=1 N sin φ j j=1 M sin φ j cos φ j j=1 N sin 2 φ j ].
X i = [ a i b i c i ] T .
R i = [ j=1 N I i,j j=1 N I i,j cos φ j j=1 N I i,j sin φ j ] T .
α i = tan 1 ( c i b i ).
cosφ= I 1 A 1 B 1 sinφ= ( B 2 I 1 B 1 cosΔα I 2 )( B 2 B 1 A 1 cosΔα A 2 ) B 2 sinΔα .
φ= tan 1 ( ( B 2 I 1 cosΔα B 1 I 2 )( B 2 A 1 cosΔα B 1 A 2 ) ( I 1 A 1 ) B 2 sinΔα ).

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