Abstract

An advanced spatial carrier phase-shifting (SCPS) algorithm based on least-squares iteration is proposed to extract the phase distribution from a single spatial carrier interferogram. The proposed algorithm divides the spatial carrier interferogram into four phase-shifted interferograms. By compensating for the effects of the variations of phase shifts between pixels and the variations of background and contrast, the proposed algorithm determines the local phase shifts and phase distribution simultaneously and accurately. Numerical simulations show that the accuracy of the proposed algorithm is obviously improved by compensating for the effects of background and contrast variations. The peak to valley of the residual phase error remains less than 0.002rad when the magnitude of spatial carrier is in the range from π/5 to π/2 and the direction of the spatial carrier is in the range from 25° to 65°. Numerical simulations and experiments demonstrate that the proposed algorithm exhibits higher precision than the existing SCPS algorithms. The proposed algorithm is sensitive to random noise, but the error can be reduced by N times if N measurements are taken and averaged.

© 2008 Optical Society of America

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References

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  1. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. Robinson and G. Reid, eds. (IOP Publishing, 1993), pp. 95-140.
  2. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  3. M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61-67(1991).
    [CrossRef]
  4. P. H. Chan and P. J. Bryanston-Cross, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343-354 (1995).
    [CrossRef]
  5. M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459-2466 (1995).
    [CrossRef]
  6. M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853-1862(1995).
    [CrossRef]
  7. H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. 46, 1057-1065 (2007).
    [CrossRef] [PubMed]
  8. A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase-shifting applied to interferogram intensity contrast determination,” Appl. Opt. 46, 4613-4624 (2007).
    [CrossRef] [PubMed]
  9. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671-1673 (2004).
    [CrossRef] [PubMed]
  10. D. Malacara and S. L. DeVore, “Optical interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 1992).
  11. M. Pirga and M. Kujawinska, “Errors in two-dimensional spatial-carrier phase-shifting method for closed fringe pattern analysis,” Proc. SPIE 2860, 72-83 (1996).
    [CrossRef]

2007 (2)

2004 (1)

1996 (1)

M. Pirga and M. Kujawinska, “Errors in two-dimensional spatial-carrier phase-shifting method for closed fringe pattern analysis,” Proc. SPIE 2860, 72-83 (1996).
[CrossRef]

1995 (3)

P. H. Chan and P. J. Bryanston-Cross, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853-1862(1995).
[CrossRef]

1993 (1)

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. Robinson and G. Reid, eds. (IOP Publishing, 1993), pp. 95-140.

1992 (1)

D. Malacara and S. L. DeVore, “Optical interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 1992).

1991 (1)

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61-67(1991).
[CrossRef]

1982 (1)

Bryanston-Cross, P. J.

P. H. Chan and P. J. Bryanston-Cross, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

Chan, P. H.

P. H. Chan and P. J. Bryanston-Cross, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

Chen, M.

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. Robinson and G. Reid, eds. (IOP Publishing, 1993), pp. 95-140.

Cuevas, F. J.

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853-1862(1995).
[CrossRef]

DeVore, S. L.

D. Malacara and S. L. DeVore, “Optical interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 1992).

Guo, H.

Han, B.

Ina, H.

Kobayashi, S.

Kujawinska, M.

M. Pirga and M. Kujawinska, “Errors in two-dimensional spatial-carrier phase-shifting method for closed fringe pattern analysis,” Proc. SPIE 2860, 72-83 (1996).
[CrossRef]

M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61-67(1991).
[CrossRef]

Malacara, D.

D. Malacara and S. L. DeVore, “Optical interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 1992).

Patorski, K.

Pirga, M.

M. Pirga and M. Kujawinska, “Errors in two-dimensional spatial-carrier phase-shifting method for closed fringe pattern analysis,” Proc. SPIE 2860, 72-83 (1996).
[CrossRef]

M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

Servin, M.

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853-1862(1995).
[CrossRef]

Styk, A.

Takeda, M.

Wang, Z.

Wójciak, J.

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61-67(1991).
[CrossRef]

Yang, Q.

Appl. Opt. (2)

J. Mod. Opt. (1)

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853-1862(1995).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

Opt. Lasers Eng. (1)

P. H. Chan and P. J. Bryanston-Cross, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

M. Pirga and M. Kujawinska, “Errors in two-dimensional spatial-carrier phase-shifting method for closed fringe pattern analysis,” Proc. SPIE 2860, 72-83 (1996).
[CrossRef]

M. Kujawinska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61-67(1991).
[CrossRef]

Other (2)

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. Robinson and G. Reid, eds. (IOP Publishing, 1993), pp. 95-140.

D. Malacara and S. L. DeVore, “Optical interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 1992).

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Figures (5)

Fig. 1
Fig. 1

Residual phase error with and without compensation as a function of the parameter r b : (a) PV and (b) RMS.

Fig. 2
Fig. 2

Residual phase errors as a function of the (a) magnitude and (b) direction of spatial carrier frequency.

Fig. 3
Fig. 3

Residual phase errors as a function of the RMS of random noise.

Fig. 4
Fig. 4

Simulation results: (a) carrier fringe, (b) local phase shifts along the y axis, (d) phase distribution obtained by the proposed method, and (c), (e) residual phase errors corresponding to (b), (d), respectively. (f) Local phase shifts along the y axis, (h) phase distribution obtained by Guo’s method, and (g), (i) residual phase errors corresponding to (f), (h), respectively.

Fig. 5
Fig. 5

Experimental results: (a) interferogram with spatial carrier frequency, (b) phase map obtained by the proposed method, (c) phase map obtained by the five-point SCPS method, and (d) phase map obtained by Zygo’s PSI method.

Equations (19)

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I ( x , y ) = A ( x , y ) + B ( x , y ) cos [ φ ( x , y ) + u c x + v c y ] ,
u ( x , y ) = [ Φ ( x , y ) ] / x = φ ( x , y ) / x + u c ,
v ( x , y ) = [ Φ ( x , y ) ] / y = φ ( x , y ) / y + v c ,
I 1 ( x , y ) = I ( x , y ) = A 1 + B 1 cos [ Φ ( x , y ) ] ,
I 2 ( x , y ) = I ( x + 1 , y ) = A 2 + B 2 cos [ Φ ( x , y ) + u ( x , y ) ] ,
I 3 ( x , y ) = I ( x , y + 1 ) = A 3 + B 3 cos [ Φ ( x , y ) + v ( x , y ) ] ,
I 4 ( x , y ) = I ( x + 1 , y + 1 ) = A 4 + B 4 cos [ Φ ( x , y ) + u ( x , y ) + v ( x , y ) ] ,
I n ( x , y ) A n ( x , y ) + A 1 ( x , y ) = a 1 ( x , y ) + b 1 ( x , y ) B n ( x , y ) B 1 ( x , y ) cos [ θ n ( x , y ) ] + c 1 ( x , y ) B n ( x , y ) B 1 ( x , y ) sin [ θ n ( x , y ) ] .
S ( x , y ) = n = 1 N [ I n e ( x , y ) I n ( x , y ) ] 2 ,
S ( x , y ) a 1 ( x , y ) = 0 , S ( x , y ) b 1 ( x , y ) = 0 , S ( x , y ) c 1 ( x , y ) = 0.
[ a 1 b 1 c 1 ] = [ N n = 1 N B n B 1 cos δ n n = 1 N B n B 1 sin δ n n = 1 N B n B 1 cos δ n n = 1 N B n 2 B 1 2 cos 2 δ n n = 1 N B n 2 B 1 2 cos δ n sin δ n n = 1 N B n B 1 sin δ n n = 1 N B n 2 B 1 2 sin δ n cos δ n n = 1 N B n 2 B 1 2 sin 2 δ n ] 1 [ n = 1 N I n t A n + A 1 n = 1 N ( I n t A n + A 1 ) B n B 1 cos δ n n = 1 N ( I n t A n + A 1 ) B n B 1 sin δ n ] .
A ( x , y ) = a 1 ( x , y ) ,
B ( x , y ) = ( c 1 ( x , y ) ) 2 + ( b 1 ( x , y ) ) 2 ,
Φ ( x , y ) = φ ( x , y ) + u c x + v c y = tan 1 ( c 1 ( x , y ) / b 1 ( x , y ) ) ,
I n t ( x , y ) A n ( x , y ) = b n B n ( x , y ) cos [ Φ n ( x , y ) ] + c n B n ( x , y ) sin Φ n ( x , y ) ,
[ b n c n ] = [ x = 1 X y = 1 Y B n 2 cos 2 Φ n x = 1 X y = 1 Y B n 2 sin Φ n cos Φ n x = 1 X y = 1 Y B n 2 sin Φ cos Φ n x = 1 X y = 1 Y B n 2 sin 2 Φ n ] 1 [ x = 1 X y = 1 Y ( I n A n ) B n cos Φ n x = 1 X y = 1 Y ( I n A n ) B n sin Φ n ] .
θ n = tan 1 ( c n / b n ) .
| ( θ n i θ 1 i ) ( θ n i 1 θ 1 i 1 ) | < ε ,
I ( x , y ) = 130 exp [ r a ( x 2 + y 2 ) ] + 120 exp [ r b ( x 2 + y 2 ) ] cos [ 2 π ( x 2 + y 2 ) + u c x + v c y ] + n ( x , y ) ,

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