Abstract

It is well known that phase-shifting interferometry suffers from inaccuracy in the presence of phase-shifting errors. We have proved the limitation of using a 4-phase algorithm to reduce the phase-measurement error in the presence of the phase-shifting error. A class of 4 + 1-phase error compensating algorithms is formulated. It is shown that the proposed algorithms can effectively minimize the effects of the constant phase-shifting error and possess a superior performance than existing error-compensating algorithms. The effectiveness of the proposed algorithm is demonstrated by computer simulations and experiments.

© 2004 Optical Society of America

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References

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  1. P. Hariharan, D. Malacara, Interference, Interferometry, and Interferometric Metrology (SPIE Optical Engineering Press, Bellingham, Wa., 1995).
  2. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  3. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  4. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 13, 2504–2505 (1987).
    [CrossRef]
  5. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  6. D. C. Ghiglia, Mark D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  7. H. Zhao, M. J. Lalor, D. R. Burton, “Error-compensating algorithms in phase-shifting interferometry: a comparison by error analysis,” Opt. Lasers Eng. 31, 381–400 (1999).
    [CrossRef]

1999 (1)

H. Zhao, M. J. Lalor, D. R. Burton, “Error-compensating algorithms in phase-shifting interferometry: a comparison by error analysis,” Opt. Lasers Eng. 31, 381–400 (1999).
[CrossRef]

1993 (1)

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

1992 (1)

1987 (1)

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 13, 2504–2505 (1987).
[CrossRef]

1983 (1)

Burow, R.

Burton, D. R.

H. Zhao, M. J. Lalor, D. R. Burton, “Error-compensating algorithms in phase-shifting interferometry: a comparison by error analysis,” Opt. Lasers Eng. 31, 381–400 (1999).
[CrossRef]

Eiju, T.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 13, 2504–2505 (1987).
[CrossRef]

Elssner, K.-E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia, Mark D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

Grzanna, J.

Hariharan, P.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 13, 2504–2505 (1987).
[CrossRef]

P. Hariharan, D. Malacara, Interference, Interferometry, and Interferometric Metrology (SPIE Optical Engineering Press, Bellingham, Wa., 1995).

Lalor, M. J.

H. Zhao, M. J. Lalor, D. R. Burton, “Error-compensating algorithms in phase-shifting interferometry: a comparison by error analysis,” Opt. Lasers Eng. 31, 381–400 (1999).
[CrossRef]

Larkin, K. G.

Malacara, D.

P. Hariharan, D. Malacara, Interference, Interferometry, and Interferometric Metrology (SPIE Optical Engineering Press, Bellingham, Wa., 1995).

Merkel, K.

Oreb, B. F.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 13, 2504–2505 (1987).
[CrossRef]

Pritt, Mark D.

D. C. Ghiglia, Mark D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

Schreiber, H.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

Spolaczyk, R.

Streibl, N.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Zhao, H.

H. Zhao, M. J. Lalor, D. R. Burton, “Error-compensating algorithms in phase-shifting interferometry: a comparison by error analysis,” Opt. Lasers Eng. 31, 381–400 (1999).
[CrossRef]

Zoller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt. (2)

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 13, 2504–2505 (1987).
[CrossRef]

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

Opt. Eng. (1)

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Opt. Lasers Eng. (1)

H. Zhao, M. J. Lalor, D. R. Burton, “Error-compensating algorithms in phase-shifting interferometry: a comparison by error analysis,” Opt. Lasers Eng. 31, 381–400 (1999).
[CrossRef]

Other (2)

P. Hariharan, D. Malacara, Interference, Interferometry, and Interferometric Metrology (SPIE Optical Engineering Press, Bellingham, Wa., 1995).

D. C. Ghiglia, Mark D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

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Figures (15)

Fig. 1
Fig. 1

Phase-shifting interferometry by use of a parallel grating projection.

Fig. 2
Fig. 2

Semicylinder sitting on the reference.

Fig. 3
Fig. 3

Simulated fringes in the presence of linear phase-shifting errors: (a) without object; (b) with object.

Fig. 4
Fig. 4

Measured profile of a semicylinder via three PSI algorithms in Eqs. (10), (11), and (13).

Fig. 5
Fig. 5

Phase-measurement error against the phase-shifting error via three PSI algorithms in Eqs. (10), (11), and (13).

Fig. 6
Fig. 6

Simulated fringes in the presence of a quadratic phase-shifting error ε1 = 0.002: (a) without object; (b) with object.

Fig. 7
Fig. 7

Measurement phase errors in the presence of a quadratic phase-shifting error.

Fig. 8
Fig. 8

Simulated fringes in the presence of a quadratic phase-shifting error: (a) without object; (b) with object.

Fig. 9
Fig. 9

Measurement errors in the presence of uniform distributed random phase-shifting errors.

Fig. 10
Fig. 10

Actual 3-D surface of the step-shape object to be measured.

Fig. 11
Fig. 11

Experiment fringes without and with the step profile to be measured: (a) without the step profile; (b) with the step profile.

Fig. 12
Fig. 12

Measured profile of the object in Fig. 11 by use of the common 4 + 1-phase algorithm in Eq. (13).

Fig. 13
Fig. 13

Measured profile of the object in Fig. 11 by use of Schwider’s 4-phase algorithm.

Fig. 14
Fig. 14

Measured profile of the object in Fig. 11 by use of the proposed 4 + 1-phase algorithm in Eq. (10).

Fig. 15
Fig. 15

Measurement errors obtained by use of the algorithms in Eqs. (10), (11), (13), respectively: (a) common 4 + 1-phase algorithm; (b) Schwider’s 4-phase algorithm; (c) proposed 4 + 1-phase algorithm.

Equations (23)

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Iix, y=Ax, y+Bx, ycosφx, y+i π2i=0, 1, 2, 3if N=4i=0, 1, 2, 3, 4if N=4+1,
Iix, y=Ax, y+Bx, ycosφx, y+iπ2+εi=0, 1, 2, 3if N=4i=0, 1, 2, 3, 4if N=4+1,
φ˜x, y=arctana0I0+a1I1+a2I2+a3I3+a4I4b0I0+b1I1+b2I2+b3I3+b4I4,
φ˜x, y=arctanC1 sin φx, y+C2 cos φx, yD1 cos φx, y+D2 sin φx, y,
C1=-a1 cos ε+a2 sin 2ε+a3 cos 3ε-a4 sin 4ε,C2=a0-a1 sin ε-a2 cos 2ε+a3 sin 3ε+a4 cos 4ε,D1=b0-b1 sin ε-b2 cos 2ε+b3 sin 3ε+b4 cos 4ε,D2=-b1 cos ε+b2 sin 2ε+b3 cos 3ε-b4 sin 4ε,
cosnε=1, sinnε=nε for n=0, 1, 2, 3, 4.
φ˜x, y=arctan-a1+a3+2a2-4a4εb0-b2+b4+-b1+3b3εtan φx, y+a0-a2+a4+-a1+3a3εb0-b2+b4+-b1+3b3ε1-b1-b3+-2b2+4b4εb0-b2+b4+-b1+3b3εtan φx, y.
-a1+a3=b0-b1+b4a0+a4=a2a0+a1+a2+a3+a4=0b1=b3b0+b1+b2+b3+b4=02a2-4a4=-b1+3b3-a1+3a3=-2b2+4b4.
φ˜x, y=φx, y+arctan3b0-4a4+b4ε2b0-4a4+b0-2a4-b4ε.
Δφx, y+arctan3b0-4a4+b4ε2b0-4a4+b0-2a4-b4ε.
φx, y=arctan3I0-6I1+4I2-2I3+I42I0+2I1-4I2+2I3-2I4.
φx, y=arctan3I2-I1+I3+I4I1+I2+I4-3I3,
arctan3ε2+ε.
φx, y=arctan2I3-I1I0+I4-2I2.
Iix, y=Ax, y+Bx, y×cosφx, y+ti1+ε1+ε2ti2π for i=0, 1, 2, 3, 4,
Iix, y=Ax, y+Bx, y×cosφx, y+i π2+εi for i=0, 1, 2, 3, 4,
Iix, y=Ax, y+Bx, y×cosφx, y+i π2 i=0, 1, 2, 3, 4.
φ=arctani=04 aiAx, y+Bx, ycosφx, y+i π2i=04 biAx, y+Bx, ycosφx, y+i π2=arctanAx, yi=04 ai+Bx, ya0-a2+a4cos φx, y+-a1+a3sin φx, yAx, yi=04 bi+Bx, yb0-b2+b4cos φx, y+-b1+b3sin φx, y.
a0+a1+a2+a3+a4=0a0-a2+a4=0b0+b1+b2+b3+b4=0-b1+b3=0-a1+a3=b0-b2+b4.
φ˜x, y=arctan-a1+a3+2a2-4a4εb0-b2+b4+-b1+3b3εtan φx, y+a0-a2+a4+-a1+3a3εb0-b2+b4+-b1+3b3ε1-b1-b3+-2b2+4b4εb0-b2+b4+-b1+3b3εtan φx, y.
φ˜x, y=φx, y+Δφx, y=arctantan φx, y+tan Δφx, y1-tan φx, ytan Δφx, y.
-a1+a3+2a2-4a4εb0-b2+b4+-b1+3b3ε=1a0-a2+a4+-a1+3a3εb0-b2+b4+-b1+3b3ε=b1-b3+-2b2+4b4εb0-b2+b4+-b1+3b3ε.
-a1+a3=b0-b2+b42a2-4a4=-b1+3b3a0+a4-a2=b1-b3-a1+3a3=-2b2+4b4.

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