Abstract

The surface shape and optical thickness variation of a lithium niobate (LNB) wafer were measured simultaneously using a wavelength-tuning interferometer with a new phase-shifting algorithm. It is necessary to suppress the harmonic signals for testing a highly reflective sample such as a crystal wafer. The LNB wafer subjected to polishing, which is in optical contact with a fused-silica (FS) supporting plate, generates six different overlapping interference fringes. The reflectivity of the wafer is typically 15%, yielding significant harmonic signals. The new algorithm can flexibly select the phase-shift interval and effectively suppress the harmonic signals and crosstalk. Experimental results indicated that the optical thickness variation of the LNB wafer was measured with an accuracy of 2 nm.

© 2014 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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2012 (1)

M. Servin and J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory,” Opt. Lasers Eng. 50(8), 1009–1014 (2012).
[Crossref]

2009 (2)

2004 (3)

2003 (2)

2000 (2)

1999 (1)

1998 (2)

1997 (3)

1996 (1)

1995 (3)

1992 (1)

1991 (1)

1990 (2)

1987 (1)

1983 (1)

1981 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

1975 (1)

T. Tsuruta and Y. Ichihara, “Accurate measurement of lens thickness by using white-light fringes,” Jpn. J. Appl. Phys. 14(14) 369–372 (1975).

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Burke, J.

Burow, R.

Chen, J.

Creath, K.

de Groot, P.

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
[Crossref] [PubMed]

P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

Deck, L.

P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

Deck, L. L.

Dorrío, B. V.

Elssner, K. E.

Estrada, J. C.

Fairman, P. S.

Farrant, D. I.

Freischlad, K.

Fukano, T.

Gallagher, J. E.

Grzanna, J.

Hanayama, R.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Herriott, D. R.

Hibino, K.

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

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(6), 1241–1249 (2004).
[Crossref] [PubMed]

K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express 12(23), 5579–5594 (2004).
[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(19), 3888–3895 (2003).
[Crossref] [PubMed]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts: reply to comment,” J. Opt. Soc. Am. A 15(5), 1234–1235 (1998).
[Crossref]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
[Crossref]

K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36(10), 2084–2093 (1997).
[Crossref] [PubMed]

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(4), 761–768 (1995).
[Crossref]

Hopler, M. D.

Ichihara, Y.

T. Tsuruta and Y. Ichihara, “Accurate measurement of lens thickness by using white-light fringes,” Jpn. J. Appl. Phys. 14(14) 369–372 (1975).

Ishii, Y.

Koliopoulos, C. L.

Larkin, K. G.

Littman, M. G.

Liu, K.

Malacara-Doblado, D.

Malacara-Hernández, D.

Merkel, K.

Mitsuishi, M.

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

Murata, K.

Okada, K.

Oreb, B. F.

Ose, T.

Phillion, D. W.

Quiroga, J. A.

Rogers, J. R.

Rosenfeld, D. P.

Sakuta, H.

Schmit, J.

Schwider, J.

Servin, M.

Spolaczyk, R.

Surrel, Y.

Tsujiuchi, J.

Tsuruta, T.

T. Tsuruta and Y. Ichihara, “Accurate measurement of lens thickness by using white-light fringes,” Jpn. J. Appl. Phys. 14(14) 369–372 (1975).

Warisawa, S.

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

White, A. D.

Yamaguchi, I.

Appl. Opt. (13)

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(22), 3280–3285 (1990).
[Crossref] [PubMed]

T. Fukano and I. Yamaguchi, “Separation of measurement of the refractive index and the geometrical thickness by use of a wavelength-scanning interferometer with a confocal microscope,” Appl. Opt. 38(19), 4065–4073 (1999).
[Crossref] [PubMed]

L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 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(19), 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(6), 1241–1249 (2004).
[Crossref] [PubMed]

M. D. Hopler and J. R. Rogers, “Interferometric measurement of group and phase refractive index,” Appl. Opt. 30(7), 735–744 (1991).
[Crossref] [PubMed]

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
[Crossref] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref] [PubMed]

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
[Crossref] [PubMed]

D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997).
[Crossref] [PubMed]

K. Hibino, “Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36(10), 2084–2093 (1997).
[Crossref] [PubMed]

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(11), 2693–2703 (1974).
[Crossref] [PubMed]

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(21), 3421–3432 (1983).
[Crossref] [PubMed]

J. Mod. Opt. (1)

P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

J. Opt. Soc. Am. A (6)

Jpn. J. Appl. Phys. (1)

T. Tsuruta and Y. Ichihara, “Accurate measurement of lens thickness by using white-light fringes,” Jpn. J. Appl. Phys. 14(14) 369–372 (1975).

Opt. Express (3)

Opt. Lasers Eng. (1)

M. Servin and J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory,” Opt. Lasers Eng. 50(8), 1009–1014 (2012).
[Crossref]

Opt. Lett. (3)

Opt. Rev. (1)

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Other (1)

P. de Groot, “Method and system for profiling objects having multiple reflective surfaces using wavelength-tuning phase-shifting interferometry,” U.S. Patent 6, 359,692 (March 19, 2002).

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Figures (9)

Fig. 1
Fig. 1

Wavelength-tuning Fizeau interferometer with LNB wafer and FS supporting plate.

Fig. 2
Fig. 2

Geometric layout of the reference surface and measurement sample.

Fig. 3
Fig. 3

Sampling functions iF1 and F2 of Surrel’s 2N - 1 algorithm (N = 15).

Fig. 4
Fig. 4

(a) 4N - 3 algorithm (N = 24), (b) rotation of 4N - 3 algorithm.

Fig. 5
Fig. 5

Sampling functions of 4N - 3 algorithm: (a) m = 1, (b) m = 5 (N = 16).

Fig. 6
Fig. 6

PV phase error caused by (a) phase-shift miscalibration, (b) 1st-order nonlinearity of phase shift, and (c) 2nd-order nonlinearity of phase shift.

Fig. 7
Fig. 7

(a) Laboratory photo of LNB wafer in wavelength-tuning interferometer, (b) raw interferogram at 632.8 nm.

Fig. 8
Fig. 8

Measured (a) optical thickness variation of LNB wafer, (b) optical thickness variation of FS, (c) surface shape of LNB wafer, (d) surface shape of FS.

Fig. 9
Fig. 9

Phase of surface shape of LNB wafer before unwrapping obtained by (a) 4N3 algorithm with polynomial window function (N = 16), (b) Larkin-Oreb N + 1 algorithm (N = 16).

Tables (2)

Tables Icon

Table 1 Relative Frequencies of the Interference Fringes.

Tables Icon

Table 2 Performance of Several Window Functions for Harmonic Signal Suppression.

Equations (10)

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

I( x,y,t )= A 0 + m=1 A m cos[ ν m t φ m ( x,y ) ] ,
D=2( pL+q n 1 T 1 +r n 2 T 2 ),
ν m ( p,q,r )= 4πL λ 0 2 [ p+q n 1 T 1 L ( 1 λ 0 n 1 d n 1 dλ )+r n 2 T 2 L ( 1 λ 0 n 2 d n 2 dλ ) ]( dλ dt ).
φ m ( x,y )=arctan r=1 M b r I( x,y, α r ) r=1 M a r I( x,y, α r ) ,
a r = 2 N 2 ( N| Nr | )cos[ 2π N ( rN ) ], b r = 2 N 2 ( N| Nr | )sin[ 2π N ( rN ) ].
F 1 ( ν )= r=1 M b r exp( i α r ν ) , F 2 ( ν )= r=1 M a r exp( i α r ν ) ,
α r = α 0r [ 1+ ε 0 + ε 1 α 0r π + ε 2 ( α 0r π ) 2 ++ ε p ( α 0r π ) p ],
a r = 2 N w r cos 2mπ N [ r( 2N1 ) ], b r = 2 N w r sin 2mπ N [ r( 2N1 ) ],
w r = 1 N 3 [ 1 6 r( r+1 )( r+2 ) ]( 1rN ), w r = 1 N 3 [ 1 3 N( 2 N 2 +1 )+ 1 2 | X |( | X | 2 2N| X |1 ) ]( N+1r3N3 ), ( X=r2N+1 ) w r = 1 N 3 [ 1 2 ( 4Nr )( 4Nr1 )( 4Nr2 ) ]( 3N2r4N3 ).
δλ= λ 2 4π n 1 T 1 δφ0.06527nm.

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