Abstract

A novel method is presented to extract phase distribution from phase-shifted interferograms with unknown tilt phase shifts. The proposed method can estimate the tilt phase shift between two temporal phase-shifted interferograms with high accuracy, by extending the regularized optical flow method with the spatial image processing and frequency estimation technology. With all the estimated tilt phase shifts, the phase component encoded in the interferograms can be extracted by the least-squares method. Both simulation and experimental results have fully proved the feasibility of the proposed method. Particularly, a flat-based diffractive optical element with quasi-continuous surface is tested by the proposed method with introduction of considerably large tilt phase shift amounts (i.e., the highest estimated tilt phase shift amount between two consecutive frame reaches 6.18λ). The phase extraction result is in good agreement with that of Zygo’s MetroPro software under steady-state testing conditions, and the residual difference between them is discussed. In comparison with the previous methods, the proposed method not only has relatively little restrictions on the amounts or orientations of the tilt phase shifts, but also works well with interferograms including open and closed fringes in any combination.

© 2013 OSA

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References

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2013

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013).
[CrossRef]

J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng.51(5), 637–641 (2013).
[CrossRef]

2011

2009

2008

2005

2004

2002

2000

Aboutanios, E.

S. Ye and E. Aboutanios, “Two dimensional frequency estimation by interpolation on Fourier coefficients,” in Proc. of IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 3353–3356 (2012).
[CrossRef]

Álvarez-Herrero, A.

Apostol, D.

Arévalo Aguilar, L. M.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013).
[CrossRef]

Belenguer, T.

Cai, L. Z.

Carazo, J. M.

Chai, L.

J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt.47(3), 480–485 (2008).
[CrossRef] [PubMed]

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt.10(7), 075011 (2008).
[CrossRef]

Chen, L.

Chen, M.

Cheng, X. C.

Damian, V.

Dobroiu, A.

Dong, G. Y.

Estrada, J. C.

Gao, P.

Geist, E.

Guerrero-Sánchez, F.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013).
[CrossRef]

Guo, H.

Han, B.

Harder, I.

Juarez-Salazar, R.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013).
[CrossRef]

Li, B.

Lindlein, N.

Ma, S.

Mantel, K.

Meneses-Fabian, C.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013).
[CrossRef]

Meng, X. F.

Nascov, V.

Quiroga, J. A.

Robledo-Sánchez, C.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013).
[CrossRef]

Shen, X. X.

Soloviev, O.

Sorzano, C. O. S.

Sun, W. J.

Tuya, W.

Vargas, J.

Vdovin, G.

Wang, Y. R.

Wang, Z.

Wei, C.

Xu, J.

J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt.47(3), 480–485 (2008).
[CrossRef] [PubMed]

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt.10(7), 075011 (2008).
[CrossRef]

Xu, Q.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt.10(7), 075011 (2008).
[CrossRef]

J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt.47(3), 480–485 (2008).
[CrossRef] [PubMed]

Xu, X. F.

Yao, B.

Ye, S.

S. Ye and E. Aboutanios, “Two dimensional frequency estimation by interpolation on Fourier coefficients,” in Proc. of IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 3353–3356 (2012).
[CrossRef]

Zhang, H.

Zhu, R.

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt.10(7), 075011 (2008).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013).
[CrossRef]

J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng.51(5), 637–641 (2013).
[CrossRef]

Opt. Lett.

Other

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007), Chap. 14, pp. 547–666.

S. Ye and E. Aboutanios, “Two dimensional frequency estimation by interpolation on Fourier coefficients,” in Proc. of IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 3353–3356 (2012).
[CrossRef]

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB (Prentice Hall, 2004).

L. G. Shapiro and G. C. Stockman, Computer Vision (Prentice Hall, Upper Saddle River, New Jersey, USA, 2001).

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Figures (6)

Fig. 1
Fig. 1

The flow chart of the proposed method.

Fig. 2
Fig. 2

(a), (b): The simulated background-suppressed inteferograms I ˜ 1 and I ˜ 2 ; (c) The simulated phase-shifting plane δ 2 (x,y) ; (d) The map δ ¯ est (x,y) ; (e),(f): The binary map BM1 via Eq. (11) before and after being processed with the connected-component labeling as well as the morphological operations; (g) The binary map BM2 ; (h) The map δ ˜ est (x,y) computed via Eq. (14); (i) The estimated phase-shifting plane δ ˜ 2 (x,y) ; (j) The residual error of the estimated phase-shifting plane, i.e. the wrapped difference between (c) and (i). The data shown in (c), (d), (h)-(j) are all in radians. In (e)-(g), the black and gray color represent the values of zero and one, respectively.

Fig. 3
Fig. 3

(a) The first interferogram used in this experiment; (b) The wrapped reference phase map by the AIA method using all the interferograms; (c) The wrapped phase extraction result by the AIA method with eight interferograms (i.e., one of the interferograms is excluded); (d) The wrapped phase extraction result by the proposed method using all the interferograms; (e) The wrapped phase difference between (b) and (c), after removing the bias; (f) The wrapped phase difference between (b) and (d), after removing the bias. The data shown in (b)-(f) are in radians.

Fig. 4
Fig. 4

(a),(b) The first two interferogram used in this experiment with a image size of 240×240 , corresponding to a part of the DOE area; (c) The wrapped phase extraction result by the proposed method; (d) The phase extraction result by the Zygo’s MetroPro software with the piston and tilt components removed, where the phase data related to the pixels in white color are missing; (e) The wrapped phase difference between (c) and (d), with the piston and tilt components removed; (f) The normalized estimated contrast parameters B ˜ (x,y) shown in the log10 scale (the definition of B ˜ (x,y) can be found in section 2.6). The data shown in (c)-(e) are in radians.

Fig. 5
Fig. 5

The estimated phase-shifting planes: (a) between the first and the second interferograms; (b) between the second and the third interferograms; (c) between the third and the fourth interferograms; (d) between the fourth and the fifth interferograms; (e) between the fifth and the sixth interferograms. All the data shown in these figures are in radians.

Fig. 6
Fig. 6

Difference between phase extraction results by the Zygo’s MetroPro software at different testing status, to demonstrate the relative retrace error. (a) The typical interferogram related to the testing status 1; (b) The typical interferogram related to the testing status 2; (c) The difference in phase extraction results by the Zygo’s MetroPro software, between the testing status 1 and the testing status 2 (the phase data related to the pixels in white color are missing); (d) the same as (c), but with the piston and tilt components removed. The data shown in (c), (d) are in radians.

Tables (1)

Tables Icon

Table 1 Residual Errors of the Estimated Phase-shifting Planes

Equations (21)

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I n ( x, y )=A( x, y )+B( x, y )cos[ φ( x, y )+ δ n ( x, y ) ] =A( x, y )+B( x, y )cos[ φ( x, y )+( k xn x+ k yn y+ d n ) ]
φ=arctan I 3 I 2 +( I 1 I 3 )cos δ 2 +( I 2 I 1 )cos δ 3 ( I 1 I 3 )sin δ 2 +( I 2 I 1 )sin δ 3
u k+1 = u ¯ k I x [ I x u ¯ k + I y v ¯ k + I t ] / ( ρ 2 + I x 2 + I y 2 ) v k+1 = v ¯ k I y [ I x u ¯ k + I y v ¯ k + I t ] / ( ρ 2 + I x 2 + I y 2 )
η=arctan( ν/u )
SPT{ }=F T 1 { ( ω x +i ω y ω x 2 + ω y 2 )FT{ } }
φ est =arctan( iexp( iη )SPT{ I ˜ 1 } I ˜ 1 )
φ est ( x,y ){ mod( φ( x,y ),2π ), 0<δ<π mod( φ( x,y ),2π ), π<δ<2π
I ˜ 1 '( x,y )= I ˜ 2 ( x,y )B(x,y)cos[ φ ( x,y ) ] I ˜ 2 '( x,y )= I ˜ 1 ( x,y )cos[ φ ( x,y )δ ]=cos[ φ ( x,y )+ δ ]
δ est ( x,y )=mod( φ est ( x,y )+ φ est ( x,y ),2π )2πδ
δ est ( x,y )=mod( φ est ( x,y )+ φ est ( x,y ),2π )δ
BM1( x,y )={ 1, | δ ¯ est ( x,y )π |π/2 0, | δ ¯ est ( x,y )π |>π/2
BM2( x,y )={ 1, p( x,y )su b 2m , m=1,2,...,[ ( n+1 ) /2 ] 0, p( x,y )su b 2m , m=1,2,...,[ ( n+1 ) /2 ]
BM2( x,y )={ 1, p( x,y )su b 2m , m=1,2,...,[ ( n+1 ) /2 ] 0, p( x,y )su b 2m , m=1,2,...,[ ( n+1 ) /2 ]
δ ˜ est ( x,y )={ δ ¯ est ( x,y ), BM2( x,y )=1 2π δ ¯ est ( x,y ), BM2( x,y )=0
d est =angle{ x,y exp[ j δ ˜ est ( x,y ) ]exp[ j( k x,est x+ k y,est y ) ] }
δ ˜ 2 (x, y)= k x,est x+ k y,est y+ d est
φ ˜ est,ij ( x,y )={ φ est,ij ( x,y ), BM 2 ij ( x,y )=1 2π φ est , ij ( x,y ), BM 2 ij ( x,y )=0
val1=| x,y { exp[ j φ ˜ est,ij ( x,y ) ]exp[ j φ ˜ est,ik ( x,y ) ] } | val2=| x,y { exp[ j φ ˜ est,ij ( x,y ) ]exp[ j φ ˜ est,ik ( x,y ) ] } |
I n ( x, y )=a( x, y )+b( x, y )cos[ δ n ( x, y ) ]+c( x, y )sin[ δ n ( x, y ) ]
[ a est ( x, y ) b est ( x, y ) c est ( x, y ) ]= [ N n=1 N cos[ δ est,n ( x, y ) ] n=1 N sin[ δ est,n ( x, y ) ] n=1 N cos[ δ est,n ( x, y ) ] n=1 N cos 2 [ δ est,n ( x, y ) ] n=1 N c s n ( x,y ) n=1 N sin[ δ est,n ( x, y ) ] n=1 N c s n ( x,y ) n=1 N sin 2 [ δ est,n ( x, y ) ] ] 1 ×[ n=1 N I n ( x, y ) n=1 N I n ( x, y )cos[ δ est,n ( x, y ) ] n=1 N I n ( x, y )sin[ δ est,n ( x, y ) ] ]
φ ˜ ( x, y )=arctan[ c est ( x,y ) / b est ( x,y ) ]

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