Abstract

Wavelength-tuning interferometry has been widely used for measuring the thickness variation of optical devices used in the semiconductor industry. However, in wavelength-tuning interferometry, the nonlinearity of phase shift causes a spatially uniform error in the calculated phase distribution. In this study, the spatially uniform error is formulated using Taylor series. A new 9-sample phase-shifting algorithm is proposed, with which the uniform spatial phase error can be eliminated. The characteristics of 9-sample algorithm is discussed using Fourier representation and RMS error analysis. Finally, optical-thickness variation of transparent plate is measured using the proposed algorithm and wavelength-tuning Fizeau interferometer and the error is compared with 7-sample algorithm.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (2)

C. J. Evans, E. C. Browy, T. H. C. Childs, and E. Paul, “Interferometric measurements of single crystal diamond tool wear,” CIRP Ann. Manuf. Technol. 64(1), 125–128 (2015).

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Absolute optical thickness measurement of transparent plate using excess fraction method and wavelength-tuning Fizeau interferometer,” Opt. Express 23(4), 4065–4073 (2015).
[PubMed]

2013 (1)

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurements of freeform optics,” CIRP Ann. Manuf. Technol. 62(2), 823–846 (2013).

2009 (1)

2008 (1)

C. J. Evans, “Uncertainty evolution for measurements of peak-to-valley surface form errors,” CIRP Ann. Manuf. Technol. 57(1), 509–512 (2008).

2006 (1)

R. Schmitt and D. Doerner, “Measurement technology for the machine integrated determination of form deviations in optical surfaces,” CIRP Ann. Manuf. Technol. 55(1), 559–562 (2006).

1997 (1)

1996 (1)

1995 (4)

1992 (1)

1990 (1)

1983 (1)

1981 (1)

1978 (1)

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

1974 (1)

Brangaccio, D. J.

Browy, E. C.

C. J. Evans, E. C. Browy, T. H. C. Childs, and E. Paul, “Interferometric measurements of single crystal diamond tool wear,” CIRP Ann. Manuf. Technol. 64(1), 125–128 (2015).

Bruning, J. H.

Burow, R.

Childs, T. H. C.

C. J. Evans, E. C. Browy, T. H. C. Childs, and E. Paul, “Interferometric measurements of single crystal diamond tool wear,” CIRP Ann. Manuf. Technol. 64(1), 125–128 (2015).

Creath, K.

de Groot, P. J.

Doerner, D.

R. Schmitt and D. Doerner, “Measurement technology for the machine integrated determination of form deviations in optical surfaces,” CIRP Ann. Manuf. Technol. 55(1), 559–562 (2006).

Elssner, K. E.

Estrada, J. C.

Evans, C.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurements of freeform optics,” CIRP Ann. Manuf. Technol. 62(2), 823–846 (2013).

Evans, C. J.

C. J. Evans, E. C. Browy, T. H. C. Childs, and E. Paul, “Interferometric measurements of single crystal diamond tool wear,” CIRP Ann. Manuf. Technol. 64(1), 125–128 (2015).

C. J. Evans, “Uncertainty evolution for measurements of peak-to-valley surface form errors,” CIRP Ann. Manuf. Technol. 57(1), 509–512 (2008).

Fang, F. Z.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurements of freeform optics,” CIRP Ann. Manuf. Technol. 62(2), 823–846 (2013).

Farrant, D. I.

Freischlad, K.

Gallagher, J. E.

Groot, P.

Grzanna, J.

Harris, F. J.

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

Herriott, D. R.

Hibino, K.

Kim, Y.

Koliopoulos, C. L.

Larkin, K. G.

Littman, M. G.

Liu, K.

Merkel, K.

Mitsuishi, M.

Oreb, B. F.

Paul, E.

C. J. Evans, E. C. Browy, T. H. C. Childs, and E. Paul, “Interferometric measurements of single crystal diamond tool wear,” CIRP Ann. Manuf. Technol. 64(1), 125–128 (2015).

Quiroga, J. A.

Rosenfeld, D. P.

Schmit, J.

Schmitt, R.

R. Schmitt and D. Doerner, “Measurement technology for the machine integrated determination of form deviations in optical surfaces,” CIRP Ann. Manuf. Technol. 55(1), 559–562 (2006).

Schwider, J.

Servin, M.

Spolaczyk, R.

Sugita, N.

Surrel, Y.

Weckenmann, A.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurements of freeform optics,” CIRP Ann. Manuf. Technol. 62(2), 823–846 (2013).

White, A. D.

Zhang, G. X.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurements of freeform optics,” CIRP Ann. Manuf. Technol. 62(2), 823–846 (2013).

Zhang, X. D.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurements of freeform optics,” CIRP Ann. Manuf. Technol. 62(2), 823–846 (2013).

Appl. Opt. (5)

CIRP Ann. Manuf. Technol. (4)

C. J. Evans, “Uncertainty evolution for measurements of peak-to-valley surface form errors,” CIRP Ann. Manuf. Technol. 57(1), 509–512 (2008).

R. Schmitt and D. Doerner, “Measurement technology for the machine integrated determination of form deviations in optical surfaces,” CIRP Ann. Manuf. Technol. 55(1), 559–562 (2006).

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurements of freeform optics,” CIRP Ann. Manuf. Technol. 62(2), 823–846 (2013).

C. J. Evans, E. C. Browy, T. H. C. Childs, and E. Paul, “Interferometric measurements of single crystal diamond tool wear,” CIRP Ann. Manuf. Technol. 64(1), 125–128 (2015).

J. Opt. Soc. Am. A (5)

Opt. Express (1)

Opt. Lett. (2)

Proc. IEEE (1)

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

Other (2)

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series (Marcel Dekker, 1998), pp. 169–245.

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Figures (5)

Fig. 1
Fig. 1 Two-surface interferometer.
Fig. 2
Fig. 2 Sampling functions of (a) 9-sample algorithm, (b) de Groot 7-sample algorithm [14] (c) Hibino 7-sample algorithm [9], and (d) Surrel 2N – 1 algorithm (N = 6) [11].
Fig. 3
Fig. 3 Wavelength-tuning Fizeau interferometer. PBS denotes polarization beam splitter; QWP is quarter-wave plate; HWP is half-wave plate.
Fig. 4
Fig. 4 (a) Experimental photo and (b) raw fringe pattern of the fused silica plate at 632.8 nm.
Fig. 5
Fig. 5 Calculated optical-thickness variation of a fused silica transparent plate at 632.8 nm.

Tables (1)

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Table 1 RMS error at λs = 43 mm.

Equations (22)

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I( α r )= I 0 [ 1+γcos( α r φ ) ],
φ=Arctan r=1 M b r I( α r ) r=1 M a r I( α r ) .
α r = α 0r [ 1+ ε 0 + ε 1 ( α 0r π )+ ε 2 ( α 0r π ) 2 + ],
Δφ=Arctan r=1 M b r I( α r ) r=1 M a r I( α r ) φ=π( X ε 0 sin2φ+Y ε 1 +Z ε 1 cos2φ+ ),
X= 1 2 r=1 M ( α 0r π )( a r sin α 0r + b r cos α 0r ) =0,
Y= 1 2 r=1 M ( α 0r π ) 2 ( a r cos α 0r + b r sin α 0r ) =0,
Z= 1 2 r=1 M ( α 0r π ) 2 ( a r cos α 0r b r sin α 0r ) =0,
r=1 M a r sin( m α 0r ) = r=1 M b r cos( m α 0r ) =0,
r=1 M a r cos α 0r = r=1 M b r sin α 0r =δ( m,1 ).
Y= 1 2 r=1 M ( α 0r π ) 2 w r .
α 0r = π 2 ( r M+1 2 ).
w r =[ w 1 , w 2 , w 3 , w 4 , w 5 , w 4 , w 3 , w 2 , w 1 ].
r=1 9 α 0r w r sin( 2 α 0r ) =0,
r=1 9 α 0r 2 w r =0,
r=1 9 α 0r 2 w r cos( 2 α 0r )=0,
r=1 9 w r =2,
r=1 9 w r cos( m α 0r )=0( m=1,2,3 ),
w r =[ 1 16 , 1 16 , 1 4 , 9 16 , 5 8 , 9 16 , 1 4 , 1 16 , 1 16 ].
φ=Arctan I 2 +9 I 4 9 I 6 I 8 I 1 +4 I 3 10 I 5 +4 I 7 + I 9 .
F 1 ( ν )= r=1 M b r exp( i α 0r ν ) =i r=1 M b r sin( α 0r ν ) ,
F 2 ( ν )= r=1 M a r exp( i α 0r ν ) = r=1 M a r cos( α 0r ν ) .
δλ= λ 2 4πnT δφ0.01716nm.

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