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.

© 2004 Optical Society of America

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References

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Appl. Opt. (16)

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]

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]

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]

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]

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

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]

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]

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

J. Mod. Opt. (2)

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. 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]

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

Opt. Eng. (1)

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]

Opt. Lett. (3)

Opt. Rev. (2)

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]

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]

Progress in Optics (1)

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

Other (3)

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).

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

Supplementary Material (1)

» Media 1: AVI (2188 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Wavelength-scanning Fizeau interferometer, with test object of stacked glass plates.

Fig. 2.
Fig. 2.

Anticipated frequency spectrum of fringe modulation from the configuration shown in Fig. 1.

Fig. 3.
Fig. 3.

Logarithmic plot of |Fa (ν)| and |Fb (ν)| for a two-period triangular window with N=32 and k=4.

Fig. 4.
Fig. 4.

Position of zeroes (roots) of the characteristic polynomials for a triangular window with N=32 and k=4. All roots are double roots, white dots and black rings denote one root each.

Fig. 5.
Fig. 5.

Position of zeroes of the characteristic polynomials for the auxiliary window correction function for with N=32 and k=4. White dots: single roots; white dots with black rings: double roots.

Fig. 6.
Fig. 6.

Filter functions Fa(ν) and iFb(ν) around the detection frequency for N=32 and k=4: (a) Triangular window; (b) Correction function.

Fig. 7.
Fig. 7.

Logarithmic plot of the sampling functions |Fa (ν)| and |Fb (ν)| of the new formula for N=32 and k=4.

Fig. 8.
Fig. 8.

PV phase errors caused by signal frequency detuning, for the fundamental frequency detection by three different window envelopes.

Fig. 9.
Fig. 9.

(a) Laboratory photo of BK7 stack under wavelength-tuning interferometer; the tray is pushed under the cylindrical tube for measurement. (b) Movie of five-beam interferogram (top) and two-beam control fringe pattern under wavelength shifting (2.13 MB). Total phase shift in external air gap, to be seen on control plate, is almost four fringes instead of two (see Section 4.1).

Fig. 10.
Fig. 10.

Measured phase distributions from a multiple-beam interferogram as in Fig. 9 (b). Top row: comparisons for thickness variation of internal air gap d; (a) direct measurement (φ8 ), (b) indirect measurement (φ3 -φ2 ). Middle row: indirect measurements; (c) p 1=(φ 2-φ 1); (d) p 2=(φ 4-φ 3). Bottom row: comparisons for thickness variations of entire glass stack, p 1+d+p 2 ; (e) direct measurement (φ7); (f) indirect measurement (φ 4-φ 1); (g) indirect measurement {(φ 2-φ 1)+φ8 +(φ4 -φ3 )}.

Tables (4)

Tables Icon

Table 1. Data sampling windows and their performance for suppression of higher harmonic signals and compensation for phase shift miscalibration.

Tables Icon

Table 2. Relative frequencies of interference signals for external air gap a=10.1 mm.

Tables Icon

Table 3. Performance of several algorithms for higher harmonic suppression, compensation for signal detuning and sensitivity to random noise. Signal detuning merits are repeated from Table 1 for better overview.

Tables Icon

Table 4. Error coefficients caused by a quadratic wavelength-scanning nonlinearity for several windows with N=50.

Equations (29)

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

ν 1 = 4 π a λ 0 2 d λ dt ,
φ k ( x , y ) = arctan r = 1 2 N 1 b r I ( x , y , α r ) r = 1 2 N 1 a r I ( x , y , α r ) ,
w ( r ) = r N for 1 r N ,
= 2 r N for N + 1 r 2 N 1 .
a r = w ( r ) cos k α r ,
b r = w ( r ) sin k α r .
F a ( ν ) = r = 1 2 N 1 a r exp ( i α r ν ν 1 )
F b ( ν ) = r = 1 2 N 1 b r exp ( i α r ν ν 1 ) ,
P ( z ) = c r = 1 2 N 1 ( a r + ib r ) z r 1 ,
P ( exp ( i δ ν ν 1 ) ) = F a ( ν ) + iF b ( ν ) ,
P ( z ) = 2 ζ 2 k N 2 ( z N 1 ) 2 ( z ζ k ) 2 ,
P ( exp ( ik δ ) ) = F a ( k ν 1 ) + iF b ( k ν 1 )
= 2 ,
z d dz P ( exp ( ik δ ) ) d d ν { F a ( k ν 1 ) iF b ( k ν 1 ) }
= 0 ,
P c ( z ) = r = 1 2 N 1 ( a r c + ib r c ) z r 1 = 2 ζ 2 k sin δ N π ( z N 1 ) 2 ( z ζ k 1 ) ( z ζ k + 1 ) ,
P opt ( z ) = P ( z ) + ε P c ( z )
w ( r ) = r N 1 14 sin ( 2 π r N ) for 1 r N ,
= 2 r N + 1 14 sin ( 2 π r N ) for N + 1 r 2 N 1 ,
S = r = 1 2 N 1 ( a r 2 + b r 2 ) .
w Hann ( r ) = 2 N cos 2 π 2 N ( r N ) for r = 1 , 2 , , 2 N 1
w Hammin g ( r ) = 2 N { 0.08 + 0.92 cos 2 π 2 N ( r N ) } for r = 1 , 2 , , 2 N .
ν k = k ν 1 ( 1 + ε 2 ν 1 t π ) ,
Δ φ k
= k ε 2 2 π m { ( iF b ( 2 ) ( m ν 1 ) + F a ( 2 ) ( m ν 1 ) ) cos ( φ k φ m ) + ( iF b ( 2 ) ( m ν 1 ) F a ( 2 ) ( m ν 1 ) ) cos ( φ k + φ m ) }
= π k ε 2 2 m { f ( k , m ) cos ( φ k φ m ) + g ( k , m ) cos ( φ k + φ m ) } ,
f ( k , m ) = 1 N 2 r = 1 2 N 1 ( r N ) 2 w ( r ) cos 2 π r N ( k m )
g ( k , m ) = 1 N 2 r = 1 2 N 1 ( r N ) 2 w ( r ) cos 2 π r N ( k + m ) ,
Δ λ = λ 0 2 2 a ( 2 1 N ) λ 0 2 a .

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