Abstract

An advanced iterative algorithm is presented to extract phase distribution from randomly and spatially nonuniform phase-shifted interferograms. The proposed algorithm divides the interferograms into small blocks and retrieves local phase shifts accurately by iterations. Therefore, the phase distribution can be calculated with high precision by eliminating the effect of tilts occurring during phase shifting. Simulated results and experiments demonstrate that the proposed algorithm exhibits high precision and converges faster than previous algorithms even when the tilt errors are up to 27.6% of the normal phase step.

© 2008 Optical Society of America

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References

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  1. K. Creath, "Phase-shifting interferometry techniques," in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), Vol. 26, pp. 349-393.
    [CrossRef]
  2. K. Creath, "Temporal phase measurement methods," in Interferogram Analysis, D. Robinson and G. Reid, eds. (IOP Publishing, 1993), pp. 95-140.
  3. K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
    [CrossRef]
  4. Z. Wang and B. Han, "Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms," Opt. Lett. 29, 1671-1673 (2004).
    [CrossRef] [PubMed]
  5. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts," J. Opt. Soc. Am. A 14, 918-930 (1997).
    [CrossRef]
  6. M. Chen, H. Guo, and C. Wei, "Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000).
    [CrossRef]
  7. A. Dobroiu, A. Apostol, V. Nascov, and V. Damian, "Tilt-compensating algorithm for phase-shift interferometry," Appl. Opt. 41, 2435-2439 (2002).
    [CrossRef] [PubMed]
  8. J. C. Wyant, "Inteferometric optical metrology: basic system and principles," Laser Focus 18, 65-71 (1982).

2004 (1)

2002 (1)

2000 (1)

1997 (1)

1991 (1)

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

1982 (1)

J. C. Wyant, "Inteferometric optical metrology: basic system and principles," Laser Focus 18, 65-71 (1982).

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

Laser Focus (1)

J. C. Wyant, "Inteferometric optical metrology: basic system and principles," Laser Focus 18, 65-71 (1982).

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, 118-124 (1991).
[CrossRef]

Opt. Lett. (1)

Other (2)

K. Creath, "Phase-shifting interferometry techniques," in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), Vol. 26, pp. 349-393.
[CrossRef]

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

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Figures (4)

Fig. 1
Fig. 1

Interferograms with different phase-shift values and tilts: (a) d 1 = 0 , k x 1 = 0 , k y 1 = 0 ; (b) d 2 = 1.5 , k x 2 = 0.167 , k y 2 = 0.67 ; (c) d 3 = 3.1 , k x 3 = 0.167 , k y 3 = 1.33 ; (d) d 4 = 4.8 , k x 4 = 0.167 , k y 4 = 1.33 .

Fig. 2
Fig. 2

(Color online) Simulation results: (a) Phase extracted by the iterative algorithm of Wang and Han, (b) residual error of (a), (c) phase extracted by the proposed iterative algorithm, (d) residual error of (c).

Fig. 3
Fig. 3

(Color online) Relationship between residual errors and number of iterations: (a) PV of the residual errors, (b) rms of the residual errors.

Fig. 4
Fig. 4

(Color online) Comparison among the phases extracted by Zygo's software, the conventional four-frame algorithm, the algorithm of Wang and Han, and the proposed algorithm. (a)–(d) Four randomly and spatially nonuniform phase-shifted interferograms. (e)–(h) Phase map extracted by Zygo's software, the conventional four-frame algorithm, the algorithm of Wang and Han, and the proposed algorithm, respectively. (i)–(k) Residual errors of the conventional four-frame algorithm, the algorithm of Wang and Han, and the proposed algorithm compared with Zygo's software. (1) The difference in result between the algorithm of Wang and Han and the proposed algorithm.

Tables (3)

Tables Icon

Table 1 Iterative Results of the Phase-Shift Planes for Four Different Sets and Corresponding Residual Errors a

Tables Icon

Table 2 PV and rms of the Phase Extracted by All Four Algorithms and Relative Errors between Them a

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Table 3 Translation and Tilt Errors of the Phase-Shift Plane a

Equations (10)

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I n t ( x , y ) = A ( x , y ) + B ( x , y ) cos [ Φ ( x , y ) + k x n x + k y n y + d n ] ,
I n t ( x , y ) = a ( x , y ) + b ( x , y ) cos ( k x n x + k y n y + d n ) + c ( x , y ) sin ( k x n x + k y n y + d n ) .
S ( x , y ) = n = 1 N [ I n e ( x , y ) I n t ( x , y ) ] 2 ,
S ( x , y ) a ( x , y ) = 0 ,   S ( x , y ) b ( x , y ) = 0 ,   S ( x , y ) c ( x , y ) = 0 .
[ a ( x , y ) b ( x , y ) c ( x , y ) ] = [ N n = 1 N cos   δ n n = 1 N sin   δ n n = 1 N cos   δ n n = 1 N cos 2 δ n n = 1 N cos   δ n   sin   δ n n = 1 N sin   δ n n = 1 N sin   δ n   cos   δ n n = 1 N sin 2 δ n ] 1 × [ n = 1 N I n e ( x , y ) n = 1 N I n e ( x , y ) cos   δ n n = 1 N I n e ( x , y ) sin   δ n ] .
Φ ( x , y ) = tan 1 [ c ( x , y ) / b ( x , y ) ] .
I n k t ( x , y ) = a n ( k ) + b n ( k ) cos   Φ ( x , y ) + c n ( k ) sin   Φ ( x , y ) .
[ a n ( k ) b n ( k ) c n ( k ) ] = [ X Y / K k cos   Φ k sin   Φ k cos   Φ k cos 2 Φ k sin   Φ   cos   Φ k sin   Φ k sin   Φ   cos   Φ k sin 2 Φ ] 1 × [ k I k I   cos   Φ k I   sin   Φ ] ,
d n k = tan 1 [ c n ( k ) / b n ( k ) ] .
{ | ( d n i d 1 i ) ( d n i 1 d 1 i 1 ) | < ε 1 | ( k x n i k x 1 i ) ( k x n i 1 k x 1 i 1 ) | + | ( k y n i k y 1 i ) ( k y n i 1 k y 1 i 1 ) | < ε 2 ,

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