Abstract

I develop and test several new phase-shifting formulas for a Fizeau interferometer, using a spherical cavity of high NA. The associated phase-shift miscalibrations provide a useful range for testing phase-shift error compensation experimentally. After removing the “double-frequency” cyclic phase variation from the error-fringe pattern by well-known methods, the residual phase oscillations occur mainly at 1 and 3 times the fringe frequency. The latter can be addressed by the theory of characteristic polynomials; the former cannot, but is removed by a variation on known averaging methods, and allows empirical optimization of phase-reconstruction performance, which can then be analyzed with characteristic polynomials again.

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  1. K. Creath and P. Hariharan, Appl. Opt. 33, 24 (1994).
    [CrossRef] [PubMed]
  2. P. de Groot, Appl. Opt. 34, 2856 (1995).
    [CrossRef]
  3. J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
    [CrossRef]
  4. K. Larkin and B. Oreb, Proc. SPIE 1755, 219 (1993).
    [CrossRef]
  5. J. Schwider, R. Burow, K. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, Appl. Opt. 22, 3421 (1983).
    [CrossRef] [PubMed]
  6. P. Hariharan, B. Oreb, and T. Eiju, Appl. Opt. 26, 2504 (1987).
    [CrossRef] [PubMed]
  7. Y. Surrel, Appl. Opt. 35, 51 (1996).
    [CrossRef] [PubMed]
  8. M. Küchel, “Method and apparatus for phase evaluation of pattern images used in optical measurement,” U.S. patent 5,361,312 (November 1, 1994).
  9. K. Hibino, R. Hanayama, J. Burke, and B. Oreb, Opt. Express 12, 5579 (2004).
    [CrossRef] [PubMed]
  10. Y. Surrel, J. Opt. Soc. Am. A 15, 1227 (1998).
    [CrossRef]
  11. T. Dresel, N. Lindlein, and J. Schwider, Optik (Jena) 112, 304 (2001).
  12. J. Młokosiewicz, W. Para, and W. Śmietański, Opt. Appl. XXV, 5 (1995).
  13. J. Schmit and K. Creath, Appl. Opt. 34, 3610 (1995).
    [CrossRef] [PubMed]
  14. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
    [CrossRef]
  15. P. Hariharan, Opt. Eng. 39, 967 (2000).
    [CrossRef]
  16. J. Huntley, Opt. Lasers Eng. 26, 131 (1997).
    [CrossRef]

2008

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

2004

2001

T. Dresel, N. Lindlein, and J. Schwider, Optik (Jena) 112, 304 (2001).

2000

P. Hariharan, Opt. Eng. 39, 967 (2000).
[CrossRef]

1998

1997

J. Huntley, Opt. Lasers Eng. 26, 131 (1997).
[CrossRef]

1996

1995

1994

1993

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
[CrossRef]

K. Larkin and B. Oreb, Proc. SPIE 1755, 219 (1993).
[CrossRef]

1987

1983

Burke, J.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

K. Hibino, R. Hanayama, J. Burke, and B. Oreb, Opt. Express 12, 5579 (2004).
[CrossRef] [PubMed]

Burow, R.

Creath, K.

de Groot, P.

Dresel, T.

T. Dresel, N. Lindlein, and J. Schwider, Optik (Jena) 112, 304 (2001).

Eiju, T.

Elssner, K.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
[CrossRef]

Green, K.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Grzanna, J.

Hanayama, R.

Hariharan, P.

Hibino, K.

Huntley, J.

J. Huntley, Opt. Lasers Eng. 26, 131 (1997).
[CrossRef]

Küchel, M.

M. Küchel, “Method and apparatus for phase evaluation of pattern images used in optical measurement,” U.S. patent 5,361,312 (November 1, 1994).

Larkin, K.

K. Larkin and B. Oreb, Proc. SPIE 1755, 219 (1993).
[CrossRef]

Leistner, A.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Lindlein, N.

T. Dresel, N. Lindlein, and J. Schwider, Optik (Jena) 112, 304 (2001).

Merkel, K.

Mlokosiewicz, J.

J. Młokosiewicz, W. Para, and W. Śmietański, Opt. Appl. XXV, 5 (1995).

Oreb, B.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

K. Hibino, R. Hanayama, J. Burke, and B. Oreb, Opt. Express 12, 5579 (2004).
[CrossRef] [PubMed]

K. Larkin and B. Oreb, Proc. SPIE 1755, 219 (1993).
[CrossRef]

P. Hariharan, B. Oreb, and T. Eiju, Appl. Opt. 26, 2504 (1987).
[CrossRef] [PubMed]

Para, W.

J. Młokosiewicz, W. Para, and W. Śmietański, Opt. Appl. XXV, 5 (1995).

Puhanic, E.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Schmit, J.

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
[CrossRef]

Schwider, J.

T. Dresel, N. Lindlein, and J. Schwider, Optik (Jena) 112, 304 (2001).

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
[CrossRef]

J. Schwider, R. Burow, K. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, Appl. Opt. 22, 3421 (1983).
[CrossRef] [PubMed]

Smietanski, W.

J. Młokosiewicz, W. Para, and W. Śmietański, Opt. Appl. XXV, 5 (1995).

Spolaczyk, R.

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
[CrossRef]

Stuart, W.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Surrel, Y.

Zöller, A.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Appl.

J. Młokosiewicz, W. Para, and W. Śmietański, Opt. Appl. XXV, 5 (1995).

Opt. Eng.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, Opt. Eng. 32, 1883 (1993).
[CrossRef]

P. Hariharan, Opt. Eng. 39, 967 (2000).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

J. Huntley, Opt. Lasers Eng. 26, 131 (1997).
[CrossRef]

Optik (Jena)

T. Dresel, N. Lindlein, and J. Schwider, Optik (Jena) 112, 304 (2001).

Proc. SPIE

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008).
[CrossRef]

K. Larkin and B. Oreb, Proc. SPIE 1755, 219 (1993).
[CrossRef]

Other

M. Küchel, “Method and apparatus for phase evaluation of pattern images used in optical measurement,” U.S. patent 5,361,312 (November 1, 1994).

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Figures (4)

Fig. 1
Fig. 1

(a) Interferogram with a few fringes to test for periodic phase errors. (b) “Double-frequency” fringe error; scale range: 5 to + 5 nm , rms = 1.95 nm , peak-to-valley amplitude 9 nm .

Fig. 2
Fig. 2

Comparison of errors from different phase-shifting formulas. Scale is 5 to + 5 nm to maintain comparison with Fig. 1. (a) Using Eq. (2); rms = 1.08 nm , PV error 3.0 nm . (b) Using Eq. (3); rms = 1.04 nm , PV error 2.8 nm . (c) Using Eq. (4); rms = 1.03 nm , PV error 3.4 nm . (d) Using Eq. (5); rms = 1.08 nm , PV error 3.2 nm .

Fig. 3
Fig. 3

Error fringes from using Eq. (2); scale as above. (a) Reference phase offset 0 ° . (b) Reference phase offset 180 ° [repeated from Fig. 2a for easier comparison].

Fig. 4
Fig. 4

Comparison of errors from averaging results with 180 ° offset reference phase. (a) Based on Eq. (2); rms = 0.65 nm , PV error 1.8 nm . (b) Based on Eq. (3); rms = 0.65 nm , PV error 2.0 nm . (c) Based on Eq. (4); rms = 0.85 nm , PV error 2.4 nm . (d) Based on Eq. (5); rms = 0.65 nm , PV error 2.0 nm .

Equations (7)

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

φ = arctan I 3 I 1 I 0 I 2
φ = arctan I 0 + 4 I 1 4 I 3 + I 4 I 0 2 I 1 + 6 I 2 2 I 3 I 4 .
φ = arctan 4 I 0 + 15 I 1 15 I 3 + 4 I 4 4 I 0 7 I 1 + 22 I 2 7 I 3 4 I 4 .
φ = arctan I 0 + 3 I 1 3 I 3 + I 4 I 0 I 1 + 4 I 2 I 3 I 4 ,
φ = arctan 0.71 I 0 + 2.81 I 1 2.81 I 3 + 0.71 I 4 0.71 I 0 1.39 I 1 + 4.20 I 2 1.39 I 3 0.71 I 4 .
φ = arctan I 2 4 I 3 + 4 I 5 I 6 I 2 + 2 I 3 6 I 4 + 2 I 5 + I 6 ,
φ = arctan I 0 + 4 I 1 + I 2 8 I 3 + I 4 + 4 I 5 I 6 I 0 2 I 1 + 7 I 2 7 I 4 + 2 I 5 + I 6 ,

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