Abstract

A new phase-shifting interferometry analysis technique has been developed to overcome the errors introduced by nonlinear, irregular, or unknown phase-step increments. In the presence of a spatial carrier frequency, by observation of the phase of the first-order maximum in the Fourier domain, the global phase-step positions can be measured, phase-shifting elements can be calibrated, and the accuracy of phase-shifting analysis can be improved. Furthermore, reliance on the calibration accuracy of transducers used in phase-shifting interferometry can be reduced; and phase-retrieval errors (e.g., fringe print-through) introduced by uncalibrated fluctuations in the phase-shifting phase increments can be alleviated. The method operates deterministically and does not rely on iterative global error minimization. Relative to other techniques, the number of recorded interferograms required for analysis can be reduced.

© 2001 Optical Society of America

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    [CrossRef]
  2. R. Crane, “Interference phase measurement,” Appl. Opt. 8, 538–542 (1969).
  3. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  4. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront corrections systems (for telescopes),” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  5. J. Schwider, R. Burrow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef]
  6. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase-calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  7. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  8. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibel, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1995).
    [CrossRef]
  9. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. A. Dobroiu, P. C. Lagofatu, D. Apostol, V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8, 738–745 (1997).
    [CrossRef]
  23. A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  31. P. P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
    [CrossRef]
  32. D. A. Tichenor, G. D. Kubiak, M. E. Malinowski, R. H. Stulen, S. J. Haney, K. W. Berger, L. A. Brown, W. C. Sweatt, J. E. Bjorkholm, R. R. Freeman, M. D. Himel, A. A. MacDowell, D. M. Tennant, O. R. Wood, J. Bokor, T. E. Jewell, W. M. Mansfield, W. K. Waskiewicz, D. L. White, D. L. Windt, “Soft-x-ray projection lithography experiments using Schwarzschild imaging optics,” Appl. Opt. 32, 7068–7071 (1993).
    [CrossRef] [PubMed]
  33. K. A. Goldberg, P. Naulleau, J. Bokor, “EUV interferometric measurements of diffraction-limited optics,” J. Vac. Sci. Technol. B 17, 2982–2986 (1999).
    [CrossRef]
  34. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  35. P. P. Naulleau, K. A. Goldberg, “Dual-domain point diffraction interferometer,” Appl. Opt. 38, 3523–3533 (1999).
    [CrossRef]
  36. N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
    [CrossRef]

2000

1999

1998

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

1997

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[CrossRef]

A. Dobroiu, P. C. Lagofatu, D. Apostol, V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8, 738–745 (1997).
[CrossRef]

K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, 2084–2093 (1997).
[CrossRef] [PubMed]

D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
[CrossRef]

1996

1995

1994

1993

1992

1991

1990

1988

1987

1986

1985

1984

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

1983

1982

1975

1974

1969

R. Crane, “Interference phase measurement,” Appl. Opt. 8, 538–542 (1969).

1966

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

A. Dobroiu, P. C. Lagofatu, D. Apostol, V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8, 738–745 (1997).
[CrossRef]

Attwood, D.

Bachor, H.-A.

Berger, K. W.

Bjorkholm, J. E.

Bokor, J.

Bone, D. J.

Brangaccio, D. J.

Brophy, C. P.

Brown, L. A.

Bruning, J. H.

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed. D. Malacara, ed. (Wiley, New York, 1992), pp. 522–524.

Burrow, R.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Chang, C.

Chen, M.

Cornejo-Rodriguez, A.

Crane, R.

R. Crane, “Interference phase measurement,” Appl. Opt. 8, 538–542 (1969).

Creath, K.

J. Schmidt, K. Creath, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

A. Dobroiu, P. C. Lagofatu, D. Apostol, V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8, 738–745 (1997).
[CrossRef]

de Groot, P. J.

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

A. Dobroiu, P. C. Lagofatu, D. Apostol, V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8, 738–745 (1997).
[CrossRef]

Eiju, T.

Elssner, K. E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibel, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1995).
[CrossRef]

Farrant, D. I.

Frankena, H. J.

Freeman, R. R.

Gallagher, J. E.

Goldberg, K. A.

P. P. Naulleau, K. A. Goldberg, “Dual-domain point diffraction interferometer,” Appl. Opt. 38, 3523–3533 (1999).
[CrossRef]

P. P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

K. A. Goldberg, P. Naulleau, J. Bokor, “EUV interferometric measurements of diffraction-limited optics,” J. Vac. Sci. Technol. B 17, 2982–2986 (1999).
[CrossRef]

K. A. Goldberg, “EUV interferometry,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1997).

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed. D. Malacara, ed. (Wiley, New York, 1992), pp. 522–524.

Grzanna, J.

Guo, H.

Han, G.-S.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[CrossRef]

G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
[CrossRef] [PubMed]

Haney, S. J.

Hariharan, P.

Herriott, D. R.

Hibino, K.

Himel, M. D.

Honda, T.

Ina, H.

Jewell, T. E.

Kang, M.-G.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[CrossRef]

Kim, S.-W.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[CrossRef]

G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
[CrossRef] [PubMed]

Kinnstaetter, K.

Kinoshita, S.

Kobayashi, S.

Kubiak, G. D.

Lagofatu, P. C.

A. Dobroiu, P. C. Lagofatu, D. Apostol, V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8, 738–745 (1997).
[CrossRef]

Larkin, K. G.

Lee, S. H.

Lohmann, A. W.

MacDowell, A. A.

Malinowski, M. E.

Mansfield, W. M.

Merkel, K.

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Naulleau, P.

K. A. Goldberg, P. Naulleau, J. Bokor, “EUV interferometric measurements of diffraction-limited optics,” J. Vac. Sci. Technol. B 17, 2982–2986 (1999).
[CrossRef]

Naulleau, P. P.

Nugent, K. A.

Ohyama, N.

Oreb, B. F.

Phillion, D. W.

Rathjen, C.

Rosenfeld, D. P.

Sandeman, R. J.

Schmidt, J.

J. Schmidt, K. Creath, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Schreiber, H.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibel, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1995).
[CrossRef]

Schwider, J.

Smorenburg, C.

Spolaczyk, R.

Streibel, N.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibel, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1995).
[CrossRef]

Streibl, N.

Stulen, R. H.

Surrel, Y.

Sweatt, W. C.

Takeda, M.

Tennant, D. M.

Tichenor, D. A.

Tsujiuchi, J.

Waskiewicz, W. K.

Wei, C.

White, A. D.

White, D. L.

Windt, D. L.

Wingerden, J.

Wood, O. R.

Wyant, J. C.

Zoller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibel, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1995).
[CrossRef]

Appl. Opt.

R. Crane, “Interference phase measurement,” Appl. Opt. 8, 538–542 (1969).

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront corrections systems (for telescopes),” Appl. Opt. 14, 2622–2626 (1975).
[CrossRef] [PubMed]

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

K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24, 3101–3105 (1985).
[CrossRef] [PubMed]

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
[CrossRef] [PubMed]

K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef] [PubMed]

J. Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
[CrossRef] [PubMed]

K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, 2084–2093 (1997).
[CrossRef] [PubMed]

D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
[CrossRef]

P. P. Naulleau, K. A. Goldberg, “Dual-domain point diffraction interferometer,” Appl. Opt. 38, 3523–3533 (1999).
[CrossRef]

P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[CrossRef]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

P. P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

M. Chen, H. Guo, C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39, 3894–3898 (2000).
[CrossRef]

D. A. Tichenor, G. D. Kubiak, M. E. Malinowski, R. H. Stulen, S. J. Haney, K. W. Berger, L. A. Brown, W. C. Sweatt, J. E. Bjorkholm, R. R. Freeman, M. D. Himel, A. A. MacDowell, D. M. Tennant, O. R. Wood, J. Bokor, T. E. Jewell, W. M. Mansfield, W. K. Waskiewicz, D. L. White, D. L. Windt, “Soft-x-ray projection lithography experiments using Schwarzschild imaging optics,” Appl. Opt. 32, 7068–7071 (1993).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase-calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

K. A. Goldberg, P. Naulleau, J. Bokor, “EUV interferometric measurements of diffraction-limited optics,” J. Vac. Sci. Technol. B 17, 2982–2986 (1999).
[CrossRef]

Meas. Sci. Technol.

A. Dobroiu, P. C. Lagofatu, D. Apostol, V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8, 738–745 (1997).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9, 1451–1455 (1998).
[CrossRef]

Metrologia

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng.

S.-W. Kim, M.-G. Kang, G.-S. Han, “Accelerated phase-measuring algorithm of least squares for phase-shifting interferometry,” Opt. Eng. 36, 3101–3106 (1997).
[CrossRef]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibel, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1995).
[CrossRef]

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Other

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed. D. Malacara, ed. (Wiley, New York, 1992), pp. 522–524.

K. A. Goldberg, “EUV interferometry,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1997).

J. Schmidt, K. Creath, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

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Figures (5)

Fig. 1
Fig. 1

Complex-plane phasor representation of the spatial carrier frequency peak of the Fourier transform of six hypothetical interferograms. The resultant, measured amplitude (indicated by the gray line segments with small filled circles) is the sum of three terms, as described by Eq. (7a): The largest term is the first-order peak. It is the phase of this term in each measurement of the series that reveals the phase-step values we seek.

Fig. 2
Fig. 2

Detail from a typical interferogram image from a phase-shifting series recorded with the EUV PSPDI. The test optic is a molybdenum silicon multilayer-coated Schwarzschild objective operating at a 13.4-nm wavelength. The detail subtends 60% of the full 0.088 N.A. Below the interferogram is an intensity cross section taken through the central portion of the interferogram, indicated by the white lines at the edges. The cross section shows the high fringe contrast.

Fig. 3
Fig. 3

Complex-plane phasor representation of the spatial carrier frequency Fourier-domain peak for a phase-shifting series of ten interferograms. Small amplitude fluctuations and a nonuniform step size can be seen. The phase-step values are calculated directly from the angle, and the uncertainty is estimated from the variation of the magnitude.

Fig. 4
Fig. 4

(Top) Phase-step values and (Bottom) the step increments (Δ n - Δ n-1) are shown for the phase-shifting series of ten interferograms. On both graphs the dashed lines indicate quarter-cycle phase steps.

Fig. 5
Fig. 5

Wave-front phase maps reveal aberrations in the system wave front. (a) First five interferograms analyzed with the Hariharan technique: ϕ1. (b) FTPSD method combined with the LSM, applied to the first four interferograms: ϕ2. Phase-map sections shown are taken from the center portion of the interferogram of Fig. 2; the gray scale is bounded on the range (-2.274 to 2.216 nm). (c) The difference ϕ1 - ϕ2 reveals strong fringe print-through at twice the fundamental fringe frequency, coming primarily from ϕ1. The difference is displayed on the gray-scale range (-0.180 to 0.139 nm). Below (c) are cross sections of ϕ1 and ϕ2 plotted with a small constant displacement for clarity. Horizontal cross sections are taken from the central region of the wave-front data, indicated by horizontal lines at the edges. The difference ϕ1 - ϕ2 is also shown. The rms magnitude of the difference wave front, attributable primarily to fringe print-through, is roughly 11% of the full-wave-front magnitudes.

Equations (24)

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Inr=Ar+Brcosϕr+Δn.
ϕrϕ0r+k0·r.
Inr=Ar+Brcosϕ0r+k0·r+Δn.
Inr=Ar+expiΔnCrexpik0·r+exp-iΔnC*rexp-ik0·r,
Cr12Brexpiϕ0r,
ink=ak+expiΔnck-k0+exp-iΔnc*k+k0,
ink0=ak0+expiΔnc0+exp-iΔnc*2k0
expiΔnc0,
Δntan-1ink0, or ΔnImlnink0.
ink0= Inrexpik0·rdr,
Δntan-1 Inrexpik0·rdr.
pexpiΔnc0,
qak0+exp-iΔnc*2k0,
ink0=p+q,
δΔn|q|/|p|.
|q|>12M1-M0.
δΔnM1-M0M1+M0.
Inx=Ax+Bxcosϕx+Δn=a0x+a1xcos Δn+a2xsin Δn.
a0xAx,a1xBxcos ϕx,a2x-Bxsin ϕx.
Ei2E2xin=1NInxi-a0xi-a1xicos Δn-a2xisin Δn2.
NΣ cos ΔnΣ sin ΔnΣ cos ΔnΣ cos2 ΔnΣ cos Δn sin ΔnΣ sin ΔnΣ cos Δn sin ΔnΣ sin2 Δn×a0xia1xia2xi=ΣInxiΣInxicos ΔnΣInxisin Δn,
AΔaxi=bxi, Δ.
axi=A-1Δbxi, Δ.
ϕx=tan-1-a2xa1x, or ϕx=tan-1-a2x, a1x.

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