Abstract

In previous papers we proposed a digital method of correcting both amplitude and phase distortions caused by arbitrary phase-shift errors in standard four-frame phase-shifting interferometry (PSI), then extended it to the most generalized PSI, and showed the validity of this technique by computer simulations. Here some new simulations and a series of optical experiments with a plane wave, a spherical wave, and a piece of glass as objects are reported. The experimental results have further proved the correctness of our theoretical analysis and confirmed that our method is able to suppress double-frequency fringes in the retrieved amplitude map and the distortions in the phase map that are introduced by phase-shift errors such as to effectively eliminate the wave ripples and wall-like structures that are present in the unwrapped phase map owing to these errors. In addition, our technique can reduce the density of invalid pixels, which are barriers in phase unwrapping. Therefore the accuracy of both amplitude and phase measurements can be considerably improved.

© 2006 Optical Society of America

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References

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  1. K. Creath, "Phase measurement interferometry techniques," in Progress in Optics, E.Wolf, ed. (Elsevier, 1988), Vol. 26, pp. 349-393.
    [CrossRef]
  2. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  3. R. Schödel, A. Nicolaus, and G. Bönsch, "Phase-stepping interferometry: methods for reducing errors caused by camera nonlinearities," Appl. Opt. 41, 55-63 (2002).
    [CrossRef] [PubMed]
  4. Q. Liu, L. Z. Cai, and M. Z. He, "Digital correction of wave-front errors caused by detector nonlinearity of second order in phase-shifting interferometry," Opt. Commun. 239, 223-228 (2004).
    [CrossRef]
  5. P. de Groot, "Vibration in phase-shifting interferometry," J. Opt. Soc. Am. A 12, 354-365 (1995).
    [CrossRef]
  6. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital wave-front measuring interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983).
    [CrossRef] [PubMed]
  7. P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (1987).
    [CrossRef] [PubMed]
  8. K. G. Larkin and B. F. Oreb, "Design and assessment of symmetrical phase-shifting algorithms," J. Opt. Soc. Am. A 9, 1740-1748 (1992).
    [CrossRef]
  9. J. Schmit and K. Creath, "Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry," Appl. Opt. 34, 3610-3619 (1995).
    [CrossRef] [PubMed]
  10. B. Gutman and H. Weber, "Phase-shifter calibration and error detection in phase-shifting applications: a new method," Appl. Opt. 37, 7624-7631 (1998).
    [CrossRef]
  11. J. M. Huntley, "Suppression of phase errors from vibration in phase-shifting interferometry," J. Opt. Soc. Am. A 15, 2233-2241 (1998).
    [CrossRef]
  12. C. S. Guo, L. Zhang, H. T. Wang, J. Liao, and Y. Y. Zhu, "Phase-shifting error and its elimination in phase-shifting digital holography," Opt. Lett. 27, 1687-1689 (2002).
    [CrossRef]
  13. H. Kadono, Y. Bitoh, and S. Toyooka, "Statistical interferometry based on a fully developed speckle field: an experimental demonstration with noise analysis," J. Opt. Soc. Am. A 18, 1267-1274 (2001).
    [CrossRef]
  14. G. Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822-827 (1991).
    [CrossRef]
  15. L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Phase shift extraction and wavefront reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
    [CrossRef] [PubMed]
  16. L. Z. Cai, Q. Liu, and X. L. Yang, "Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects," Opt. Lett. 29, 183-185 (2004).
    [CrossRef] [PubMed]
  17. G. D. Lassahn, J. K. Lassahn, P. L. Taylor, and V. A. Deason, "Multiple fringe analysis with unknown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
    [CrossRef]
  18. X. Chen, M. Gramaglia, and J. A. Yeazell, "Phase-shifting interferometry with uncalibrated phase shifts," Appl. Opt. 39, 585-591 (2000).
    [CrossRef]
  19. L. Z. Cai, Q. Liu, and X. L. Yang, "Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors," Opt. Commun. 233, 21-26 (2004).
    [CrossRef]
  20. L. Z. Cai, Q. Liu, M. Z. He, and X. F. Meng, "Common form of the complex wave error caused by arbitrary phase shift errors in generalized phase-shifting interferometry and its digital correction," Opt. Commun. 249, 95-100 (2005).
    [CrossRef]

2005

L. Z. Cai, Q. Liu, M. Z. He, and X. F. Meng, "Common form of the complex wave error caused by arbitrary phase shift errors in generalized phase-shifting interferometry and its digital correction," Opt. Commun. 249, 95-100 (2005).
[CrossRef]

2004

Q. Liu, L. Z. Cai, and M. Z. He, "Digital correction of wave-front errors caused by detector nonlinearity of second order in phase-shifting interferometry," Opt. Commun. 239, 223-228 (2004).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, "Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors," Opt. Commun. 233, 21-26 (2004).
[CrossRef]

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

2003

2002

2001

2000

1998

1997

1995

1994

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, and V. A. Deason, "Multiple fringe analysis with unknown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
[CrossRef]

1992

1991

1987

1983

Bitoh, Y.

Bönsch, G.

Burow, R.

Cai, L. Z.

L. Z. Cai, Q. Liu, M. Z. He, and X. F. Meng, "Common form of the complex wave error caused by arbitrary phase shift errors in generalized phase-shifting interferometry and its digital correction," Opt. Commun. 249, 95-100 (2005).
[CrossRef]

Q. Liu, L. Z. Cai, and M. Z. He, "Digital correction of wave-front errors caused by detector nonlinearity of second order in phase-shifting interferometry," Opt. Commun. 239, 223-228 (2004).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, "Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors," Opt. Commun. 233, 21-26 (2004).
[CrossRef]

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

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Phase shift extraction and wavefront reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
[CrossRef] [PubMed]

Chen, X.

Creath, K.

de Groot, P.

Deason, V. A.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, and V. A. Deason, "Multiple fringe analysis with unknown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
[CrossRef]

Eiju, T.

Elssner, K.-E.

Gramaglia, M.

Grzanna, J.

Guo, C. S.

Gutman, B.

Hariharan, P.

He, M. Z.

L. Z. Cai, Q. Liu, M. Z. He, and X. F. Meng, "Common form of the complex wave error caused by arbitrary phase shift errors in generalized phase-shifting interferometry and its digital correction," Opt. Commun. 249, 95-100 (2005).
[CrossRef]

Q. Liu, L. Z. Cai, and M. Z. He, "Digital correction of wave-front errors caused by detector nonlinearity of second order in phase-shifting interferometry," Opt. Commun. 239, 223-228 (2004).
[CrossRef]

Huntley, J. M.

Kadono, H.

Lai, G.

Larkin, K. G.

Lassahn, G. D.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, and V. A. Deason, "Multiple fringe analysis with unknown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
[CrossRef]

Lassahn, J. K.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, and V. A. Deason, "Multiple fringe analysis with unknown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
[CrossRef]

Liao, J.

Liu, Q.

L. Z. Cai, Q. Liu, M. Z. He, and X. F. Meng, "Common form of the complex wave error caused by arbitrary phase shift errors in generalized phase-shifting interferometry and its digital correction," Opt. Commun. 249, 95-100 (2005).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, "Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors," Opt. Commun. 233, 21-26 (2004).
[CrossRef]

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

Q. Liu, L. Z. Cai, and M. Z. He, "Digital correction of wave-front errors caused by detector nonlinearity of second order in phase-shifting interferometry," Opt. Commun. 239, 223-228 (2004).
[CrossRef]

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Phase shift extraction and wavefront reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
[CrossRef] [PubMed]

Meng, X. F.

L. Z. Cai, Q. Liu, M. Z. He, and X. F. Meng, "Common form of the complex wave error caused by arbitrary phase shift errors in generalized phase-shifting interferometry and its digital correction," Opt. Commun. 249, 95-100 (2005).
[CrossRef]

Merkel, K.

Nicolaus, A.

Oreb, B. F.

Schmit, J.

Schödel, R.

Schwider, J.

Spolaczyk, R.

Taylor, P. L.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, and V. A. Deason, "Multiple fringe analysis with unknown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
[CrossRef]

Toyooka, S.

Wang, H. T.

Wang, Y. R.

Weber, H.

Yamaguchi, I.

Yang, X. L.

Yatagai, T.

Yeazell, J. A.

Zhang, L.

Zhang, T.

Zhu, Y. Y.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

Q. Liu, L. Z. Cai, and M. Z. He, "Digital correction of wave-front errors caused by detector nonlinearity of second order in phase-shifting interferometry," Opt. Commun. 239, 223-228 (2004).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, "Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors," Opt. Commun. 233, 21-26 (2004).
[CrossRef]

L. Z. Cai, Q. Liu, M. Z. He, and X. F. Meng, "Common form of the complex wave error caused by arbitrary phase shift errors in generalized phase-shifting interferometry and its digital correction," Opt. Commun. 249, 95-100 (2005).
[CrossRef]

Opt. Eng.

G. D. Lassahn, J. K. Lassahn, P. L. Taylor, and V. A. Deason, "Multiple fringe analysis with unknown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
[CrossRef]

Opt. Lett.

Other

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

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Figures (9)

Fig. 1
Fig. 1

Simulation results in the recording plane with the standard five-frame PSI algorithm and a slightly tilted plane object wave: (a), (b) amplitude and phase distributions, respectively, of the test object field; (c) profile extracted along the central horizontal line of (b); (d), (e) amplitude and phase, respectively, before correction; (f) profile extracted along the central line of (e); (g), (h) amplitude and phase, respectively, after correction; (i) profile extracted along the central line of (h).

Fig. 2
Fig. 2

Simulation results in the recording plane with the general three-frame PSI algorithm and a spherical object wave. The figures are arranged in the same order as in Fig. 1, and each has the same meaning as the corresponding one there.

Fig. 3
Fig. 3

Optical setup for our experiments.

Fig. 4
Fig. 4

Experimental results with a slightly tilted plane object wave: (a) amplitude distribution of the uncorrected wave field in the CCD plane, (c) phase distribution of the uncorrected wave field in the CCD plane, (e) amplitude distribution of the uncorrected wave field in the original object plane, (g) phase distribution of the uncorrected wave field in the original object plane; (b), (d), (f), (h) the corresponding corrected counterparts of the figures at the left.

Fig. 5
Fig. 5

3-D pictures of phase distributions in the object plane extracted from Fig. 4: (a), (c), (e) uncorrected results for the wrapped phase profile along the x axis and the wrapped phase map and the unwrapped phase map of Fig. 4(g), respectively; (b), (d), (f) their corrected counterparts obtained from Fig. 4(h).

Fig. 6
Fig. 6

Experimental results with a spherical object wave. The figures are arranged in the same order as in Fig. 4, and each has the same meaning as the corresponding one there.

Fig. 7
Fig. 7

3-D pictures of the phase distributions in the object plane extracted from Fig. 6. The figures are arranged in the same order as in Fig. 5, and each has the same meaning as the corresponding one there, except that now (a), (c), and (e) were obtained from Fig. 6(g) and (b), (d), and (f) were obtained from Fig. 6(h).

Fig. 8
Fig. 8

Experimental results with a piece of glass inserted into the plane object wave. The figures are arranged in the same order as in Fig. 4, and each has the same meaning as the corresponding one there.

Fig. 9
Fig. 9

3-D pictures of phase distributions in the object plane extracted from Fig. 8: (a), (c) uncorrected wrapped and unwrapped phase maps of Fig. 8(g), respectively; (b), (d) the corresponding corrected counterparts obtained from Fig. 8(h).

Tables (1)

Tables Icon

Table 1 Calculated Residual Errors in the Experimental Results for the Plane Wave in Fig. 4

Equations (102)

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

π / 2
P H
P o
{ δ n }
P H , O ( r )
O ( r ) = A r     1 a n I n ( r ) ,
r = ( x , y ) , A r
a n
I n
{ δ n } ( n = 1 , 2 , , N ) ,
P H , Δ O ( r )
Δ O ( r ) = C 1 O ( r ) + C 2 O * ( r ) ,
O ( r ) = A ( r ) exp [ ( r ) ]
P H , *
C 1
C 2
C 1 = i a n exp ( i δ n ) Δ δ n ,
C 2 = i a n exp ( i δ n ) Δ δ n .
C 1
C 2
( { a n } , { δ n }
{ Δ δ n } )
cos 2 φ
sin 2 φ
( Δ A / A ) v = | C 2 | cos ( 2 φ φ 0 ) ,
( Δφ ) v = | C 2 | sin ( 2 φ φ 0 ) ,
φ 0
O 1 ( r )
P H
{ I n }
O 1 ( r ) , O 1 r ( r )
O 1 i ( r ) ,   and   K = | O 1 r | / | O 1 i | .
θ = 2 arcsin [ | O 1 r K O 1 i | / ( 2 | O 1 r | ) ] .
O 2
O 1
O 2 = O 1 r + i [ ( K O 1 i O 1 r cos θ ) / sin θ ] .
O = [ I 1 + I 5 2 I 3 + 2 i ( I 2 I 4 ) ] / ( 8 A r ) ,
O = { 2 I 1 ( 2 2 ) I 2 2 ( 2 1 ) I 3 + i ( 2 2 ) I 1 + 2 I 2 2 ( 2 1 ) I 3 } / ( 4 A r ) ,
λ = 5 3 2   nm
P o
P H   is   z = 216.5   mm
512 × 512   pixels
15 μ m
Δ α 1 = 0.26   rad
Δ α 2 = 0.34   rad
Δ α 3 = 0.46   rad
Δ α 4 = 0.40   rad
O 1
O 2
P H
O 2
O 1
π to π )
O 1
P H
Δ α 1 = 0.20
Δ α 2 = 0.45   rad
R = 2.5   m
( B S 1 )
M 2
L 01
S F 1
L 1
P o
L 1
L 02 , S F 2
L 2
P H
632.8   nm
P H
P o
65   mm
1536 × 1024   pixels
9 μ m
512 × 512   pixels
Δ α 1 = 15 °
Δ α 2 = 2 0 °
Δ α 3 = 12 °
Δ α 4 = 2 5 °
P H
P o
P H
P o
P H
P o
P o
L 1
S F 1
L 1
P o
P H
P o
P H
P o
P o
σ A     2 = | ( Δ A / A ) v | 2 , σ φ     2 = | ( Δ φ ) v | 2 ,
σ A
σ φ
P H   and   P o
cos 2 φ
sin 2 φ

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