Abstract

White-light interferometers are widely used for high-accuracy topography measurement in industrial and scientific applications. A common way to characterize a white-light interferometer is to assume small surface amplitudes resulting in linear transfer characteristics described by the instrument transfer function (ITF). However, the well-known batwing effect gives rise to systematic errors, causing extra nonlinearity to the ITF. In this paper a model to simulate an interference pattern in the image plane as it is obtained by a vertical scanning white-light interferometer is introduced in order to overcome the limitation of small surface amplitudes. Repeating the simulation procedure for different height positions of the object results in an image stack that can be analyzed by the same algorithms as real measurement data. The simulation results agree with experimental observations: the batwing effect occurs in certain situations and the correct amplitude of a rectangular grating structure can be obtained as long as the structure is optically resolved. Both simulation, as well as experimental results, provide transfer characteristics of broader bandwidth than predicted by theoretical approaches based on linear system behavior.

© 2012 Optical Society of America

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References

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    [CrossRef]
  7. F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurment errors using commercial scanning white-light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Co., 2005).
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    [CrossRef]

2011 (1)

J. Niehues and P. Lehmann, “Improvement of lateral resolution and reduction of batwings in vertical scanning white-light interferometry,” Proc. SPIE 8082, 80820W (2011).
[CrossRef]

2010 (3)

R. K. Leach and H. Haitjema, “Bandwidth characteristics and comparisons of surface texture measuring instruments,” Meas. Sci. Technol. 21, 032001 (2010).
[CrossRef]

J. M. Coupland and J. Lobera, “Measurement of steep surfaces using white light interferometry,” Strain 46, 69–78 (2010).
[CrossRef]

P. Lehmann, “Vertical scanning white-light interference microscopy on curved microstructures,” Opt. Lett. 35, 1768–1770 (2010).
[CrossRef]

2008 (1)

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurment errors using commercial scanning white-light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

2007 (1)

T. V. Vorburger, H.-G. Rhee, T. B. Renegar, J.-F. Song, and A. Zheng, “Comparison of optical and stylus methods for measurement of surface texture,” Int. J. Adv. Manuf. Technol. 33, 110–118 (2007).
[CrossRef]

2006 (2)

P. Lehmann, “Systematic effects in coherence peak and phase evaluation of signals obtained with a vertical scanning white-light Mirau interferometer,” Proc. SPIE 6188, 618811 (2006).
[CrossRef]

C. Kohler, U. Droste, K. Körner, and W. Osten, “Reduction of overshooting in 3-D fringe projection measurements by inverse filtering,” Tech. Mess. 73, 595–602 (2006).
[CrossRef]

2004 (1)

2002 (1)

2001 (1)

2000 (2)

1990 (1)

Chim, S. S. C.

Colonna de Lega, X.

P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” Appl. Opt. 41, 4571–4578 (2002).
[CrossRef]

P. de Groot and X. Colonna de Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Proceedings of Fringe 2005 (2005), pp. 30–37.

Coupland, J. M.

J. M. Coupland and J. Lobera, “Measurement of steep surfaces using white light interferometry,” Strain 46, 69–78 (2010).
[CrossRef]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurment errors using commercial scanning white-light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

de Groot, P.

P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” Appl. Opt. 41, 4571–4578 (2002).
[CrossRef]

P. de Groot and X. Colonna de Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Proceedings of Fringe 2005 (2005), pp. 30–37.

Droste, U.

C. Kohler, U. Droste, K. Körner, and W. Osten, “Reduction of overshooting in 3-D fringe projection measurements by inverse filtering,” Tech. Mess. 73, 595–602 (2006).
[CrossRef]

Fleischer, M.

Gao, F.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurment errors using commercial scanning white-light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Co., 2005).

Haitjema, H.

R. K. Leach and H. Haitjema, “Bandwidth characteristics and comparisons of surface texture measuring instruments,” Meas. Sci. Technol. 21, 032001 (2010).
[CrossRef]

Harasaki, A.

Kino, G. S.

Kohler, C.

C. Kohler, U. Droste, K. Körner, and W. Osten, “Reduction of overshooting in 3-D fringe projection measurements by inverse filtering,” Tech. Mess. 73, 595–602 (2006).
[CrossRef]

Körner, K.

C. Kohler, U. Droste, K. Körner, and W. Osten, “Reduction of overshooting in 3-D fringe projection measurements by inverse filtering,” Tech. Mess. 73, 595–602 (2006).
[CrossRef]

Kramer, J.

Leach, R. K.

R. K. Leach and H. Haitjema, “Bandwidth characteristics and comparisons of surface texture measuring instruments,” Meas. Sci. Technol. 21, 032001 (2010).
[CrossRef]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurment errors using commercial scanning white-light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Lehmann, P.

J. Niehues and P. Lehmann, “Improvement of lateral resolution and reduction of batwings in vertical scanning white-light interferometry,” Proc. SPIE 8082, 80820W (2011).
[CrossRef]

P. Lehmann, “Vertical scanning white-light interference microscopy on curved microstructures,” Opt. Lett. 35, 1768–1770 (2010).
[CrossRef]

P. Lehmann, “Systematic effects in coherence peak and phase evaluation of signals obtained with a vertical scanning white-light Mirau interferometer,” Proc. SPIE 6188, 618811 (2006).
[CrossRef]

Lobera, J.

J. M. Coupland and J. Lobera, “Measurement of steep surfaces using white light interferometry,” Strain 46, 69–78 (2010).
[CrossRef]

Niehues, J.

J. Niehues and P. Lehmann, “Improvement of lateral resolution and reduction of batwings in vertical scanning white-light interferometry,” Proc. SPIE 8082, 80820W (2011).
[CrossRef]

Osten, W.

C. Kohler, U. Droste, K. Körner, and W. Osten, “Reduction of overshooting in 3-D fringe projection measurements by inverse filtering,” Tech. Mess. 73, 595–602 (2006).
[CrossRef]

Pavlicek, P.

Petzing, J.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurment errors using commercial scanning white-light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Pförtner, A.

Renegar, T. B.

T. V. Vorburger, H.-G. Rhee, T. B. Renegar, J.-F. Song, and A. Zheng, “Comparison of optical and stylus methods for measurement of surface texture,” Int. J. Adv. Manuf. Technol. 33, 110–118 (2007).
[CrossRef]

Rhee, H.-G.

T. V. Vorburger, H.-G. Rhee, T. B. Renegar, J.-F. Song, and A. Zheng, “Comparison of optical and stylus methods for measurement of surface texture,” Int. J. Adv. Manuf. Technol. 33, 110–118 (2007).
[CrossRef]

Schwider, J.

Song, J.-F.

T. V. Vorburger, H.-G. Rhee, T. B. Renegar, J.-F. Song, and A. Zheng, “Comparison of optical and stylus methods for measurement of surface texture,” Int. J. Adv. Manuf. Technol. 33, 110–118 (2007).
[CrossRef]

Soubusta, J.

Tiziani, H. J.

Turzhitsky, M.

Vorburger, T. V.

T. V. Vorburger, H.-G. Rhee, T. B. Renegar, J.-F. Song, and A. Zheng, “Comparison of optical and stylus methods for measurement of surface texture,” Int. J. Adv. Manuf. Technol. 33, 110–118 (2007).
[CrossRef]

Wilson, T.

T. Wilson, Confocal Microscopy (Academic, 1990).

Windecker, R.

Wyant, J. C.

Zheng, A.

T. V. Vorburger, H.-G. Rhee, T. B. Renegar, J.-F. Song, and A. Zheng, “Comparison of optical and stylus methods for measurement of surface texture,” Int. J. Adv. Manuf. Technol. 33, 110–118 (2007).
[CrossRef]

Appl. Opt. (6)

Int. J. Adv. Manuf. Technol. (1)

T. V. Vorburger, H.-G. Rhee, T. B. Renegar, J.-F. Song, and A. Zheng, “Comparison of optical and stylus methods for measurement of surface texture,” Int. J. Adv. Manuf. Technol. 33, 110–118 (2007).
[CrossRef]

Meas. Sci. Technol. (2)

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurment errors using commercial scanning white-light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

R. K. Leach and H. Haitjema, “Bandwidth characteristics and comparisons of surface texture measuring instruments,” Meas. Sci. Technol. 21, 032001 (2010).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

J. Niehues and P. Lehmann, “Improvement of lateral resolution and reduction of batwings in vertical scanning white-light interferometry,” Proc. SPIE 8082, 80820W (2011).
[CrossRef]

P. Lehmann, “Systematic effects in coherence peak and phase evaluation of signals obtained with a vertical scanning white-light Mirau interferometer,” Proc. SPIE 6188, 618811 (2006).
[CrossRef]

Strain (1)

J. M. Coupland and J. Lobera, “Measurement of steep surfaces using white light interferometry,” Strain 46, 69–78 (2010).
[CrossRef]

Tech. Mess. (1)

C. Kohler, U. Droste, K. Körner, and W. Osten, “Reduction of overshooting in 3-D fringe projection measurements by inverse filtering,” Tech. Mess. 73, 595–602 (2006).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Co., 2005).

VDI/VDE guideline 2655, Part 1.1, “Calibration of interference microscopes and depth measurement standards for roughness measurement,” 2008.

P. de Groot and X. Colonna de Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Proceedings of Fringe 2005 (2005), pp. 30–37.

T. Wilson, Confocal Microscopy (Academic, 1990).

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Figures (13)

Fig. 1.
Fig. 1.

Graphic illustration of the MTF with the Rayleigh and the Sparrow frequency assuming NA=0.55, λ=0.6μm.

Fig. 2.
Fig. 2.

Rectangular grating with height h0=0.4λ and period T=2λ

Fig. 3.
Fig. 3.

Intensity signal corresponding to a single CCD pixel depending on the scanner position z.

Fig. 4.
Fig. 4.

Simulated intensity patterns for a rectangular grating with h0=λ/4 and T=2λ, scaled in gray levels (dark areas correspond to low intensity and bright areas correspond to high intensity). (a) Raw intensity without consideration of diffraction. (b) Image intensity pattern considering diffraction.

Fig. 5.
Fig. 5.

(a) Results of envelope and phase evaluation assuming h0=λ/4, T=2λ. (b) Enlarged section of phase evaluation result.

Fig. 6.
Fig. 6.

Correlograms at edges assuming h0=λ/4, T=2λ. (a) Raw and image intensity signals at upper and bottom edge. (b) Upper diagram: image correlograms according to (a), lower diagram: envelopes corresponding to the two correlograms.

Fig. 7.
Fig. 7.

Results of envelope and phase evaluation for a rectangular grating with h0=λ/2 and T=2λ.

Fig. 8.
Fig. 8.

(a) ITF functions (3-D view) calculated by envelope evaluation. (b) ITF for different height values.

Fig. 9.
Fig. 9.

ITF function based on the same intensity data as Fig. 8, but obtained by phase evaluation.

Fig. 10.
Fig. 10.

Comparison of the analytic ITF with simulated ITFs (h0=0.5λ, h0=0.25λ, h0=0.05λ).

Fig. 11.
Fig. 11.

Comparison of profiles resulting from simulated and measured correlograms (T=2μm, h0=192nm, λ=620nm).

Fig. 12.
Fig. 12.

Comparison of profiles resulting from simulated and measured correlograms (T=4μm, h0=192nm, λ=480nm).

Fig. 13.
Fig. 13.

Comparison of profiles resulting from simulated and measured correlograms (T=0.6μm, h0=160nm, λ=400nm).

Equations (12)

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OTF(Ω)=2π(arccos(Ω2Ω0)Ω2Ω01(Ω2Ω0)2),
ΩRayleigh=1δRayleigh,ΩSparrow=1δSparrow.
psf(ρ)=[2J1(u(ρ))u(ρ)]2
h(x,y)=h(x,y=0)={h0nTx<nT+T20nT+T2x<nT+Tn=0,1,2.
S(k)=12πΔkexp[(kk0)22Δk2],
ΔIn(x,y)0S(k)cos[2k(h(x,y)+n·Δzz0)+ϕ0]dk.
In(x,y)=I(1+exp{4[h(x,y)+n·Δzz0lc]2}cos{4πλ[h(x,y)+n·Δzz0]+ϕ0})
In,image(x,y)=In(x,y)*psf(x,y).
In,image(x,y)=F1{F{In(x,y)}·F{psf(x,y)}},
In,image(x,y=0)=F1{F{In(x,y=0)}·PSF(ξ,η=0)},
ITFenvelope(h0=λ/4,T=2λ)=0.44λ0.25λ=1.76,ITFphase(h0=λ/4,T=2λ)=0.25λ0.25λ=1.
ITFenvelope(h0=λ/2,T=2λ)=0.28λ0.5λ=0.56,ITFphase(h0=λ/2,T=2λ)=0.5λ0.5λ=1.

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