Abstract

An iterative method based on least-squares fittings is proposed to retrieve wavefront phase from tilt phase-shift interferograms. In each iteration cycle, proposed method calculates wavefront phase and tilt phase shifts in x- and y-directions in three individual least-squares fitting steps. In tilt phase shifts extracting steps, phase shifts of interferograms columns or rows are calculated with least-squares method, and then tilt phase shifts of interferograms in x- or y-direction are determined by linear regressions. At least three interferograms of three by three pixels are required with proposed method. The performance of proposed method is demonstrated by simulations and experiments. Tilt gradients and translational phase shifts could be extracted with high accuracy and large wavefront tilts could be well handled with proposed method. The method could be applied to temporal phase-shift interferometers with uncalibrated transducers or that in vibrating environment.

© 2013 Optical Society of America

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References

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2013 (1)

2010 (1)

2009 (1)

2008 (2)

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]

2006 (1)

2005 (1)

2004 (1)

2002 (1)

2000 (1)

1999 (1)

S. Han and E. Novak, “Retrace error for the measurement of a long-radius optic,” Proc. SPIE3749, 597–598 (1999).
[CrossRef]

1994 (1)

1991 (2)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun.84(3-4), 118–124 (1991).
[CrossRef]

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE1400, 24–32 (1991).
[CrossRef]

Apostol, D.

Bruno, L.

Chai, L.

Chen, M.

Chen, Y. C.

Damian, V.

Dobroiu, A.

Guo, H.

Han, B.

Han, G. S.

Han, S.

S. Han and E. Novak, “Retrace error for the measurement of a long-radius optic,” Proc. SPIE3749, 597–598 (1999).
[CrossRef]

Hao, Q.

Hu, Y.

Kim, S. W.

Lee, C. M.

Li, Y.

Liang, C. W.

Lin, P. C.

Nascov, V.

Novak, E.

S. Han and E. Novak, “Retrace error for the measurement of a long-radius optic,” Proc. SPIE3749, 597–598 (1999).
[CrossRef]

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(3-4), 118–124 (1991).
[CrossRef]

Patorski, K.

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(3-4), 118–124 (1991).
[CrossRef]

Selberg, L. A.

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE1400, 24–32 (1991).
[CrossRef]

Soloviev, O.

Styk, A.

Szwaykowski, P.

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(3-4), 118–124 (1991).
[CrossRef]

Vdovin, G.

Wang, H.

Wang, Z.

Wei, C.

Xu, J.

Xu, Q.

Zhu, Q.

Appl. Opt. (5)

J. Opt. A, Pure Appl. Opt. (1)

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. 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(3-4), 118–124 (1991).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (2)

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE1400, 24–32 (1991).
[CrossRef]

S. Han and E. Novak, “Retrace error for the measurement of a long-radius optic,” Proc. SPIE3749, 597–598 (1999).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Schematic of tilt phase-shift interferograms. (b) Three steps in each iteration cycle.

Fig. 2
Fig. 2

Simulation results. The test wavefront is a spherical surface and its PV value is 0.075λ. Retrieved phases with Wang’s method (a), Xu’s method (b) and proposed method (c).

Fig. 3
Fig. 3

(a)-(f) Recorded interferograms in the presence of vibrations. The preset phase shift value is -π/2 between adjacent interferograms.

Fig. 4
Fig. 4

Phase retrieved from interferograms without vibration. This phase value is regarded as exact value for comparison.

Fig. 5
Fig. 5

(a) Phase retrieved from interferograms in Fig. 3 with Wang’s method, and (b) phase retrieval errors compared with accuracy phase values in Fig. 4.

Fig. 6
Fig. 6

(a) Phase retrieved from interferograms in Fig. 3 with Xu’s method, and (b) phase retrieval errors compared with accuracy phase values in Fig. 4.

Fig. 7
Fig. 7

(a) Phase retrieved from interferograms in Fig. 3 with proposed method, and (b) phase retrieval errors compared with accuracy phase values in Fig. 4.

Fig. 8
Fig. 8

Calculated wavefront tilts of interferograms in Fig. 3 in x- (a) and y-directions (b).

Fig. 9
Fig. 9

Comparison of convergence speeds. Residual iteration errors in translational phase shifts calculations (a) and in tilt gradients calculations (b).

Tables (1)

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Table 1 Extracted Results of Tilt Phase Shiftsa

Equations (23)

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I m t (x,y)=A(x,y)+B(x,y)cos[φ(x,y)+ k xm x+ k ym y+ δ m ] =A(x,y)+B(x,y)cos[φ(x,y)+ Δ xm (x)+ Δ ym (y)],
{ Δ xm (x)= k xm x+ δ xm Δ ym (y)= k ym y+ δ ym . δ m = δ xm + δ ym
I m t (x,y)=a(x,y)+b(x,y)cos[ Δ m (x,y)]+c(x,y)sin[ Δ m (x,y)],
S(x,y)= m=1 M [ I m t (x,y) I m (x,y)] 2 ,
S(x,y) a(x,y) =0, S(x,y) b(x,y) =0, S(x,y) c(x,y) =0.
α= M 1 β,
α= [a,b,c] T ,
β= [ m=1 M I m , m=1 M I m cos Δ m , m=1 M I m sin Δ m ] T ,
M=[ M m=1 M cos Δ m m=1 M sin Δ m m=1 M cos Δ m m=1 M cos 2 Δ m m=1 M cos Δ m sin Δ m m=1 M sin Δ m m=1 M sin Δ m cos Δ m m=1 M sin 2 Δ m ].
φ(x,y)= tan 1 [c(x,y)/b(x,y)].
I m t (x,y)=a ' m (x)+b ' m (x)cos[φ ' m (x,y)]+c ' m (x)sin[φ ' m (x,y)],
S ' m (x)= y=1 Y [ I m t (x,y) I m (x,y)] 2 .
α ' m (x)=M ' m 1 (x)β ' m (x),
α ' m (x)= [a ' m (x),b ' m (x),c ' m (x)] T ,
β ' m (x)= [ y=1 Y I m , y=1 Y I m cosφ ' m , y=1 Y I m sinφ ' m ] T ,
M ' m (x)=[ Y y=1 Y cosφ ' m y=1 Y sinφ ' m y=1 Y cosφ ' m y=1 Y cos 2 φ ' m y=1 Y cosφ ' m sinφ ' m y=1 Y sinφ ' m y=1 Y sinφ ' m cosφ ' m y=1 Y sin 2 φ ' m ].
Δ xm (x)= tan 1 [c ' m (x)/b ' m (x)].
α " m (y)=M " m 1 (y)β " m (y),
α " m (y)= [a " m (y),b " m (y),c " m (y)] T ,
β " m (y)= [ x=1 X I m , x=1 X I m cosφ " m , x=1 X I m sinφ " m ] T ,
M " m (y)=[ X x=1 X cosφ " m x=1 X sinφ " m x=1 X cosφ " m x=1 X cos 2 φ " m x=1 X cosφ " m sinφ " m x=1 X sinφ " m x=1 X sinφ " m cosφ " m x=1 X sin 2 φ " m ].
Δ ym (y)= tan 1 [c " m (y)/b " m (y)].
{ |[( δ xm i + δ ym i )( δ x1 i + δ y1 i )][( δ xm i1 + δ ym i1 )( δ x1 i1 + δ y1 i1 )]|< ε 1 |( k xm i k x1 i )( k xm i1 k x1 i1 )|+|( k ym i k y1 i )( k ym i1 k y1 i1 )|< ε 2 .

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