Abstract

Coherence scanning interferometry (CSI) offers three-dimensional (3D) measurement of surface topography with high precision and accuracy. Defocus within the interferometric objective lens, however, is commonly present in CSI measurements and reduces both the resolving power of the imaging system and the ability to measure tilted surfaces. This Letter extends the linear theory of CSI to consider the effects of defocus on the 3D transfer function and the point spread function in an otherwise ideal CSI instrument. The results are compared with measurements of these functions in a real instrument. This work provides further evidence for the validity of the linear systems theory of CSI.

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References

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2017 (1)

2016 (1)

2015 (1)

2014 (1)

2013 (2)

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, Meas. Sci. Technol. 24, 052001 (2013).
[Crossref]

J. M. Coupland, R. Mandal, K. Palodhi, and R. K. Leach, Appl. Opt. 52, 3662 (2013).
[Crossref]

2011 (1)

2008 (1)

J. M. Coupland and J. Lobera, Meas. Sci. Technol. 19, 07010 (2008).

1997 (1)

R. M. von Bunau, H. Fukuda, and T. Terasawa, Jpn. J. Appl. Phys. 36, 7494 (1997).
[Crossref]

1992 (1)

1990 (1)

C. J. R. Sheppard and C. J. Cogswell, J. Microsc. 159, 179 (1990).
[Crossref]

1984 (1)

C. J. R. Sheppard and D. K. Hamilton, Opt. Acta 31, 723 (1984).
[Crossref]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Cogswell, C. J.

C. J. R. Sheppard and C. J. Cogswell, J. Microsc. 159, 179 (1990).
[Crossref]

Coupland, J. M.

R. Su, Y. Wang, J. M. Coupland, and R. K. Leach, Opt. Express 25, 3297 (2017).
[Crossref]

N. Nikolaev, J. Petzing, and J. M. Coupland, Appl. Opt. 55, 3555 (2016).
[Crossref]

R. Mandal, J. M. Coupland, R. K. Leach, and D. Mansfield, Appl. Opt. 53, 1554 (2014).
[Crossref]

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, Meas. Sci. Technol. 24, 052001 (2013).
[Crossref]

J. M. Coupland, R. Mandal, K. Palodhi, and R. K. Leach, Appl. Opt. 52, 3662 (2013).
[Crossref]

J. M. Coupland and J. Lobera, Meas. Sci. Technol. 19, 07010 (2008).

J. Petzing, J. M. Coupland, and R. K. Leach, in NPL Measurement Good Practice Guide No.  116 (2010), p. 59.

Creath, K.

J. Schmit, K. Creath, and J. C. Wyant, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 667.

de Groot, P.

P. de Groot, Adv. Opt. Photon. 7, 1 (2015).
[Crossref]

P. de Groot, in Optical Measurement of Surface Topography, R. K. Leach, ed. (Springer, 2011), p. 187.

Dong, J.

Foreman, M. R.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, Meas. Sci. Technol. 24, 052001 (2013).
[Crossref]

Fukuda, H.

R. M. von Bunau, H. Fukuda, and T. Terasawa, Jpn. J. Appl. Phys. 36, 7494 (1997).
[Crossref]

Giusca, C. L.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, Meas. Sci. Technol. 24, 052001 (2013).
[Crossref]

Gu, M.

Hamilton, D. K.

C. J. R. Sheppard and D. K. Hamilton, Opt. Acta 31, 723 (1984).
[Crossref]

Leach, R. K.

R. Su, Y. Wang, J. M. Coupland, and R. K. Leach, Opt. Express 25, 3297 (2017).
[Crossref]

R. Mandal, J. M. Coupland, R. K. Leach, and D. Mansfield, Appl. Opt. 53, 1554 (2014).
[Crossref]

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, Meas. Sci. Technol. 24, 052001 (2013).
[Crossref]

J. M. Coupland, R. Mandal, K. Palodhi, and R. K. Leach, Appl. Opt. 52, 3662 (2013).
[Crossref]

J. Petzing, J. M. Coupland, and R. K. Leach, in NPL Measurement Good Practice Guide No.  116 (2010), p. 59.

Li, Y.

Lobera, J.

J. M. Coupland and J. Lobera, Meas. Sci. Technol. 19, 07010 (2008).

Lu, R.

Mandal, R.

Mansfield, D.

Nikolaev, N.

Palodhi, K.

Petzing, J.

N. Nikolaev, J. Petzing, and J. M. Coupland, Appl. Opt. 55, 3555 (2016).
[Crossref]

J. Petzing, J. M. Coupland, and R. K. Leach, in NPL Measurement Good Practice Guide No.  116 (2010), p. 59.

Schmit, J.

J. Schmit, K. Creath, and J. C. Wyant, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 667.

Sheppard, C. J. R.

M. Gu and C. J. R. Sheppard, Appl. Opt. 31, 2541 (1992).
[Crossref]

C. J. R. Sheppard and C. J. Cogswell, J. Microsc. 159, 179 (1990).
[Crossref]

C. J. R. Sheppard and D. K. Hamilton, Opt. Acta 31, 723 (1984).
[Crossref]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Su, R.

Terasawa, T.

R. M. von Bunau, H. Fukuda, and T. Terasawa, Jpn. J. Appl. Phys. 36, 7494 (1997).
[Crossref]

Török, P.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, Meas. Sci. Technol. 24, 052001 (2013).
[Crossref]

von Bunau, R. M.

R. M. von Bunau, H. Fukuda, and T. Terasawa, Jpn. J. Appl. Phys. 36, 7494 (1997).
[Crossref]

Wang, Y.

Wu, K.

Wyant, J. C.

J. Schmit, K. Creath, and J. C. Wyant, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 667.

Adv. Opt. Photon. (1)

Appl. Opt. (5)

J. Microsc. (1)

C. J. R. Sheppard and C. J. Cogswell, J. Microsc. 159, 179 (1990).
[Crossref]

Jpn. J. Appl. Phys. (1)

R. M. von Bunau, H. Fukuda, and T. Terasawa, Jpn. J. Appl. Phys. 36, 7494 (1997).
[Crossref]

Meas. Sci. Technol. (2)

J. M. Coupland and J. Lobera, Meas. Sci. Technol. 19, 07010 (2008).

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, Meas. Sci. Technol. 24, 052001 (2013).
[Crossref]

Opt. Acta (1)

C. J. R. Sheppard and D. K. Hamilton, Opt. Acta 31, 723 (1984).
[Crossref]

Opt. Express (1)

Other (4)

J. Petzing, J. M. Coupland, and R. K. Leach, in NPL Measurement Good Practice Guide No.  116 (2010), p. 59.

P. de Groot, in Optical Measurement of Surface Topography, R. K. Leach, ed. (Springer, 2011), p. 187.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

J. Schmit, K. Creath, and J. C. Wyant, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 667.

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Figures (6)

Fig. 1.
Fig. 1. Common interferometers in CSI: (a) Michelson and (b) Mirau type.
Fig. 2.
Fig. 2. (a) Intensity image of the roughened optical flat in the xy plane and cross-sectional CSI images of the surface taken at different focus conditions. The estimated axial offsets are (b) 2.4  μm, (c) 1.3  μm, (d) 0.1  μm, (e) 1.5 μm, and (f) 3.3 μm. (Direction is defined in Fig. 1.)
Fig. 3.
Fig. 3. Cross-sectional views of (a) the TF of an optical system with a finite NA and (b) the TF of CSI (monochromatic illumination is assumed).
Fig. 4.
Fig. 4. Cross-sectional views of the theoretical TFs and PSFs of CSI at different focus conditions: (a) 2.4  μm, (b) 1.3  μm, (c) 0.1  μm, (d) 1.5 μm, and (e) 3.3 μm. Note the change in z-axis of the PSF plot.
Fig. 5.
Fig. 5. Top-view CSI images of the spherical cap at different focus conditions (top row) and the corresponding height maps (bottom row): (a) 2.4  μm, (b) 1.3  μm, (c) 0.1  μm, (d) 1.5 μm, and (e) 3.3 μm.
Fig. 6.
Fig. 6. Measured TFs and derived PSFs at different focus conditions: (a) 2.4  μm, (b) 1.3  μm, (c) 0.1  μm, (d) 1.5 μm, and (e) 3.3 μm.

Equations (5)

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

I(k)=O(k)H(k),
o(x,y,z)=4πjRw(x,y)δ[zs(x,y)],
G(k,k0)=j4πk02δ(|k|k0)step(k·z^|k|1NA2).
H(k)=(|k|22k·z^)×G(kr,k0)G(kkr,k0)d3krS(k0)dk0,
Hd(k)=(|k|22k·z^)G(kr,k0)exp(j4πkrδz·z^)×G(kkr,k0)d3krS(k0)dk0.

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