Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry

doi:10.1364/OPEX.12.005579

Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry

Kenichi Hibino, Ryohei Hanayama, Jan Burke, and Bozenko Oreb »View Author Affiliations

Optics Express, Vol. 12, Issue 23, pp. 5579-5594 (2004)
http://dx.doi.org/10.1364/OPEX.12.005579

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Abstract

The interferometric surface measurement of single or stacked parallel plates presents considerable technical difficulties due to multiple-beam interference. To apply phase-shifting methods, it is necessary to use a pathlength-dependent technique such as wavelength scanning, which separates interference signals from various surfaces in frequency space. The detection window for frequency analysis has to be optimized for maximum tolerance against frequency detuning due to material dispersion and scanning nonlinearities, as well as for suppression of noise from other frequencies. We introduce a new class of phase-shifting algorithms that fulfill these requirements and allow continuous tuning of phase detection to any frequency of interest. We show results for a four-surface stack of near-parallel plates, measured in a Fizeau interferometer.

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Research Papers

History
Original Manuscript: September 23, 2004
Revised Manuscript: October 27, 2004
Published: November 15, 2004

Citation
Kenichi Hibino, Ryohei Hanayama, Jan Burke, and Bozenko Oreb, "Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry," Opt. Express 12, 5579-5594 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-23-5579

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References

K. Creath, �??Phase-measurement interferometry techniques,�?? in Progress in Optics XXVI, E. Wolf, ed. (North Holland, Amsterdam, 1988). [CrossRef]
Y. Surrel, �??Design of algorithms for phase measurements by the use of phase stepping,�?? Appl. Opt. 35, 51-60 (1996). [CrossRef] [PubMed]
D.W. Phillion, �??General method for generating phase-shifting interferometry algorithms,�?? Appl. Opt. 36, 8098-8115 (1997). [CrossRef]
K. Hibino, B.F. Oreb, D.I. Farrant and K.G. Larkin, �??Phase shifting for nonsinusoidal waveforms with phase-shift errors,�?? J. Opt. Soc. Am. A 12, 761-768 (1995). [CrossRef]
M.V. Mantravadi, �??Testing Nearly Parallel Plates,�?? in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, New York, 1992), 22.
J. Schwider, R. Burow, K.E. El�?ner, J. Grzanna, R. Spolaczyk and K. Merkel, �??Digital wavefront measuring interferometry: some systematic error sources,�?? Appl. Opt. 22, 3421-3432 (1983). [CrossRef] [PubMed]
P. de Groot, L. Deck, �??Surface profiling by analysis of white-light interferograms in the spatial frequency domain,�?? J. Mod. Opt. 42.2, 389-401 (1995). [CrossRef]
S.-W. Kim and G.-H. Kim, �??Thickness-profile measurement of transparent thin-film layers by white-light scanning interferometry,�?? Appl. Opt. 38, 5968-5973 (1999). [CrossRef]
Y. Ishii, J. Chen and K. Murata, �??Digital phase-measuring interferometry with a tunable laser diode,�?? Opt. Lett. 12, 233-235 (1987). [CrossRef] [PubMed]
K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, �??Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,�?? Appl. Opt. 29, 3280-3285 (1990). [CrossRef] [PubMed]
M. Takeda and H. Yamamoto, �??Fourier-transform speckle profilometry: three dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,�?? Appl. Opt. 33, 7829-7837 (1994). [CrossRef] [PubMed]
H.J. Tiziani, B. Franze, and P. Haible, �??Wavelength shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,�?? J. Mod. Opt. 44.8, 1485-1496 (1997). [CrossRef]
P.J. de Groot, �??Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,�?? Appl. Opt. 39, 2658-2663 (2000). [CrossRef]
K. Hibino and T. Takatsuji, �??Suppression of multiple-beam interference noise in testing an optical-parallel plate by wavelength-scanning interferometry,�?? Opt. Rev. 9.2, 60-65 (2002). [CrossRef]
A. Yamamoto and I. Yamaguchi, �??Profilometry of sloped plane surfaces by wavelength scanning interferometry,�?? Opt. Rev. 9.3, 112-121 (2002). [CrossRef]
L.L. Deck, �??Fourier-transform phase-shifting interferometry,�?? Appl. Opt. 42, 2354-2365 (2003). [CrossRef] [PubMed]
K. Hibino, B.F. Oreb, and P.S. Fairman, �??Wavelength-scanning interferometry of a transparent parallel plate with refractive index dispersion,�?? Appl. Opt. 42, 3888-3895 (2003). [CrossRef] [PubMed]
K. Hibino, B.F. Oreb, P.S. Fairman, and J. Burke, �??Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,�?? Appl. Opt. 43, 1241-1249 (2004). [CrossRef] [PubMed]
F.J. Harris, �??On the use of windows for harmonic analysis with the discrete Fourier transform,�?? Proc. IEEE 66.1, 51-83 (1978). [CrossRef]
E.W. Weisstein, �??Apodization Function,�?? <a href= "http://mathworld.wolfram.com/ApodizationFunction.html">http://mathworld.wolfram.com/ApodizationFunction.html</a>
R. Bracewell, The Fourier transform and its applications (3rd ed., McGraw Hill, New York, 1999).
P. de Groot, �??Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,�?? Appl. Opt. 34, 4723-4730 (1995). [CrossRef]
J. Schmit and K. Creath, �??Window function influence on phase error in phase-shifting algorithms,�?? Appl. Opt. 35, 5642-5649 (1996). [CrossRef] [PubMed]
P.S. Fairman, B.K. Ward, B.F. Oreb, D.I. Farrant, Y. Gilliand, C.H. Freund, A.J. Leistner, J.A. Seckold, and C.J. Walsh, �??300-mm-aperture phase-shifting Fizeau interferometer,�?? Opt. Eng. 38.8, 1371-1380 (1999). [CrossRef]
B.F. Oreb, D.I. Farrant, C.J. Walsh, G. Forbes, and P.S. Fairman, �??Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,�?? Appl. Opt. 39, 5161-5171 (2000). [CrossRef]
J.H. Bruning, D.R. Herriott, J.E. Gallagher, D.P. Rosenfeld, A.D. White and D.J. Brangaccio, �??Digital wavefront measuring interferometer for testing optical surfaces and lenses,�?? Appl. Opt. 13, 2693-2703 (1974). [CrossRef] [PubMed]
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]
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]
D. Malacara-Doblado, B.V. Dorrío, D. Malacara-Hernández, �??Graphic tool to produce tailored symmetrical phase-shifting algorithms,�?? Opt. Lett 25, 64-66 (2000). [CrossRef]
K. Freischlad and C.L. Koliopoulos, �??Fourier description of digital phase-measuring interferometry,�?? J. Opt. Soc. Am. A 7, 542-551 (1990). [CrossRef]
K. Hibino, B.F. Oreb, D.I. Farrant and K.G. Larkin, �??Phase-shifting algorithms for nonlinear and spatially non-uniform phase-shifts,�?? J. Opt. Soc. Am. A 14, 918-930 (1997). [CrossRef]
Y. Surrel, �??Phase-shifting algorithms for nonlinear and spatially non-uniform phase-shifts: comment�?? J. Opt. Soc. Am. A 15, 1227-1233 (1998). [CrossRef]
K. Hibino, B.F. Oreb, D.I. Farrant and K.G. Larkin, �??Phase-shifting algorithms for nonlinear and spatially non-uniform phase-shifts: reply to comment,�?? J. Opt. Soc. Am. A 15, 1234-1235 (1998). [CrossRef]
K. Hibino, �??Susceptibility of error-compensating phase-shifting algorithms to random noise,�?? Appl. Opt. 36, 2084-2093 (1997). [CrossRef] [PubMed]
K. Liu and M.G. Littman, �??Novel geometry for single-mode scanning of tunable lasers,�?? Opt. Lett. 6, 117-118 (1981). [CrossRef] [PubMed]

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[1] K. Creath, �??Phase-measurement interferometry techniques,�?? in Progress in Optics XXVI, E. Wolf, ed. (North Holland, Amsterdam, 1988). [CrossRef]

[2] Y. Surrel, �??Design of algorithms for phase measurements by the use of phase stepping,�?? Appl. Opt. 35, 51-60 (1996). [CrossRef] [PubMed]

[3] D.W. Phillion, �??General method for generating phase-shifting interferometry algorithms,�?? Appl. Opt. 36, 8098-8115 (1997). [CrossRef]

[4] K. Hibino, B.F. Oreb, D.I. Farrant and K.G. Larkin, �??Phase shifting for nonsinusoidal waveforms with phase-shift errors,�?? J. Opt. Soc. Am. A 12, 761-768 (1995). [CrossRef]

[5] M.V. Mantravadi, �??Testing Nearly Parallel Plates,�?? in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, New York, 1992), 22.

[6] J. Schwider, R. Burow, K.E. El�?ner, J. Grzanna, R. Spolaczyk and K. Merkel, �??Digital wavefront measuring interferometry: some systematic error sources,�?? Appl. Opt. 22, 3421-3432 (1983). [CrossRef] [PubMed]

[7] P. de Groot, L. Deck, �??Surface profiling by analysis of white-light interferograms in the spatial frequency domain,�?? J. Mod. Opt. 42.2, 389-401 (1995). [CrossRef]

[8] S.-W. Kim and G.-H. Kim, �??Thickness-profile measurement of transparent thin-film layers by white-light scanning interferometry,�?? Appl. Opt. 38, 5968-5973 (1999). [CrossRef]

[9] Y. Ishii, J. Chen and K. Murata, �??Digital phase-measuring interferometry with a tunable laser diode,�?? Opt. Lett. 12, 233-235 (1987). [CrossRef] [PubMed]

[10] K. Okada, H. Sakuta, T. Ose, and J. Tsujiuchi, �??Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,�?? Appl. Opt. 29, 3280-3285 (1990). [CrossRef] [PubMed]

[11] M. Takeda and H. Yamamoto, �??Fourier-transform speckle profilometry: three dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,�?? Appl. Opt. 33, 7829-7837 (1994). [CrossRef] [PubMed]

[12] H.J. Tiziani, B. Franze, and P. Haible, �??Wavelength shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,�?? J. Mod. Opt. 44.8, 1485-1496 (1997). [CrossRef]

[13] P.J. de Groot, �??Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,�?? Appl. Opt. 39, 2658-2663 (2000). [CrossRef]

[14] K. Hibino and T. Takatsuji, �??Suppression of multiple-beam interference noise in testing an optical-parallel plate by wavelength-scanning interferometry,�?? Opt. Rev. 9.2, 60-65 (2002). [CrossRef]

[15] A. Yamamoto and I. Yamaguchi, �??Profilometry of sloped plane surfaces by wavelength scanning interferometry,�?? Opt. Rev. 9.3, 112-121 (2002). [CrossRef]

[16] L.L. Deck, �??Fourier-transform phase-shifting interferometry,�?? Appl. Opt. 42, 2354-2365 (2003). [CrossRef] [PubMed]

[17] K. Hibino, B.F. Oreb, and P.S. Fairman, �??Wavelength-scanning interferometry of a transparent parallel plate with refractive index dispersion,�?? Appl. Opt. 42, 3888-3895 (2003). [CrossRef] [PubMed]

[18] K. Hibino, B.F. Oreb, P.S. Fairman, and J. Burke, �??Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,�?? Appl. Opt. 43, 1241-1249 (2004). [CrossRef] [PubMed]

[19] F.J. Harris, �??On the use of windows for harmonic analysis with the discrete Fourier transform,�?? Proc. IEEE 66.1, 51-83 (1978). [CrossRef]

[20] E.W. Weisstein, �??Apodization Function,�?? <a href= "http://mathworld.wolfram.com/ApodizationFunction.html">http://mathworld.wolfram.com/ApodizationFunction.html</a>

[21] R. Bracewell, The Fourier transform and its applications (3rd ed., McGraw Hill, New York, 1999).

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

[23] J. Schmit and K. Creath, �??Window function influence on phase error in phase-shifting algorithms,�?? Appl. Opt. 35, 5642-5649 (1996). [CrossRef] [PubMed]

[24] P.S. Fairman, B.K. Ward, B.F. Oreb, D.I. Farrant, Y. Gilliand, C.H. Freund, A.J. Leistner, J.A. Seckold, and C.J. Walsh, �??300-mm-aperture phase-shifting Fizeau interferometer,�?? Opt. Eng. 38.8, 1371-1380 (1999). [CrossRef]

[25] B.F. Oreb, D.I. Farrant, C.J. Walsh, G. Forbes, and P.S. Fairman, �??Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,�?? Appl. Opt. 39, 5161-5171 (2000). [CrossRef]

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

[27] 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]

[28] 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]

[29] D. Malacara-Doblado, B.V. Dorrío, D. Malacara-Hernández, �??Graphic tool to produce tailored symmetrical phase-shifting algorithms,�?? Opt. Lett 25, 64-66 (2000). [CrossRef]

[30] K. Freischlad and C.L. Koliopoulos, �??Fourier description of digital phase-measuring interferometry,�?? J. Opt. Soc. Am. A 7, 542-551 (1990). [CrossRef]

[31] K. Hibino, B.F. Oreb, D.I. Farrant and K.G. Larkin, �??Phase-shifting algorithms for nonlinear and spatially non-uniform phase-shifts,�?? J. Opt. Soc. Am. A 14, 918-930 (1997). [CrossRef]

[32] Y. Surrel, �??Phase-shifting algorithms for nonlinear and spatially non-uniform phase-shifts: comment�?? J. Opt. Soc. Am. A 15, 1227-1233 (1998). [CrossRef]

[33] K. Hibino, B.F. Oreb, D.I. Farrant and K.G. Larkin, �??Phase-shifting algorithms for nonlinear and spatially non-uniform phase-shifts: reply to comment,�?? J. Opt. Soc. Am. A 15, 1234-1235 (1998). [CrossRef]

[34] K. Hibino, �??Susceptibility of error-compensating phase-shifting algorithms to random noise,�?? Appl. Opt. 36, 2084-2093 (1997). [CrossRef] [PubMed]

[35] K. Liu and M.G. Littman, �??Novel geometry for single-mode scanning of tunable lasers,�?? Opt. Lett. 6, 117-118 (1981). [CrossRef] [PubMed]

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Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry