Abstract

It has been suggested recently that the Transfer Function of instruments such as Coherence Scanning Interferometers could be measured via a single measurement of a large spherical artefact [Appl. Opt. 53(8), 1554–1563 (2014)]. In the current paper we present analytical solutions for the Fourier transform of the ’foil’ model used in this technique, which thus avoids the artefacts resulting from the numerical approach used earlier. The Fourier transform of a partial spherical shell is found to contain points of zero amplitude for spatial frequencies that lie within the Transfer Function. This implies that the Transfer Function is unmeasurable at these points when a single spherical artefact is used, in situations where the foil model is a valid representation of the physical system. We propose extensions to the method to address this issue.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [PubMed]
  3. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  15. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 2nd Revised edition (Academic Press, 1980).
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    [Crossref]
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2014 (1)

2013 (2)

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, “Determination of the transfer function for optical surface topography measuring instruments - a review,” Meas. Sci. Technol. 24(5), 052001 (2013).
[Crossref]

J. Coupland, R. Mandal, K. Palodhi, and R. Leach, “Coherence scanning interferometry: linear theory of surface measurement,” Appl. Opt. 52(16), 3662–3670 (2013).
[Crossref] [PubMed]

2012 (1)

R. Mandal, K. Palodhi, J. Coupland, R. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).

2008 (1)

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

2006 (2)

L. Onural, “Impulse functions over curves and surfaces and their applications to diffraction,” J. Math. Anal. Appl. 322, 18–27 (2006).
[Crossref]

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

2002 (1)

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

2001 (1)

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A working person’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

1999 (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

1995 (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]

1990 (2)

Balasubramanian, N.

N. Balasubramanian, “Optical system for surface topography measurement,” U.S. Patent 4340306 (1982).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Beverage, J. L.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

Chim, S. S. C.

Conchello, J. A.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Coupland, J.

Coupland, J. M.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, “Determination of the transfer function for optical surface topography measuring instruments - a review,” Meas. Sci. Technol. 24(5), 052001 (2013).
[Crossref]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

J. M. Coupland, School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire, LE11 3TU. (personal communication2014)

de Groot, P.

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]

Descour, M. R.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

Foreman, M. R.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, “Determination of the transfer function for optical surface topography measuring instruments - a review,” Meas. Sci. Technol. 24(5), 052001 (2013).
[Crossref]

Giusca, C. L.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, “Determination of the transfer function for optical surface topography measuring instruments - a review,” Meas. Sci. Technol. 24(5), 052001 (2013).
[Crossref]

Gradshteyn, I.S.

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 2nd Revised edition (Academic Press, 1980).

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Kino, G. S.

Leach, R.

Leach, R. K.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, “Determination of the transfer function for optical surface topography measuring instruments - a review,” Meas. Sci. Technol. 24(5), 052001 (2013).
[Crossref]

Lee, B. S.

Lobera, J.

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

Mandal, R.

Mansfield, D.

R. Mandal, J. Coupland, R. Leach, and D. Mansfield, “Coherence scanning interferometry: measurement and correction of three-dimensional transfer and point-spread characteristics,” Appl. Opt. 53(8), 1554–1563 (2014).
[Crossref] [PubMed]

R. Mandal, K. Palodhi, J. Coupland, R. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Nehorai, A.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

Onural, L.

L. Onural, “Impulse functions over curves and surfaces and their applications to diffraction,” J. Math. Anal. Appl. 322, 18–27 (2006).
[Crossref]

Palodhi, K.

J. Coupland, R. Mandal, K. Palodhi, and R. Leach, “Coherence scanning interferometry: linear theory of surface measurement,” Appl. Opt. 52(16), 3662–3670 (2013).
[Crossref] [PubMed]

R. Mandal, K. Palodhi, J. Coupland, R. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).

Ryzhik, I.M.

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 2nd Revised edition (Academic Press, 1980).

Sarder, P.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

Schaefer, L. H.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A working person’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

Shack, R. V.

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[Crossref] [PubMed]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Strand, T. C.

Swedlow, J. R.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A working person’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

Török, P.

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, “Determination of the transfer function for optical surface topography measuring instruments - a review,” Meas. Sci. Technol. 24(5), 052001 (2013).
[Crossref]

Wallace, W.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A working person’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

Appl. Opt. (4)

Biotechniques (1)

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A working person’s guide to deconvolution in light microscopy,” Biotechniques 31(5), 1076–1097 (2001).
[PubMed]

IEEE Signal Process. Mag. (1)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23(3), 32–45 (2006).
[Crossref]

J. Math. Anal. Appl. (1)

L. Onural, “Impulse functions over curves and surfaces and their applications to diffraction,” J. Math. Anal. Appl. 322, 18–27 (2006).
[Crossref]

J. Microsc. (1)

J. L. Beverage, R. V. Shack, and M. R. Descour, “Measurement of the three-dimensional microscope point spread function using a Shack-Hartmann wavefront sensor,” J. Microsc. 205(1), 61–75 (2002).
[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]

Meas. Sci. Technol. (2)

M. R. Foreman, C. L. Giusca, J. M. Coupland, P. Török, and R. K. Leach, “Determination of the transfer function for optical surface topography measuring instruments - a review,” Meas. Sci. Technol. 24(5), 052001 (2013).
[Crossref]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
[Crossref]

Methods (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19(3), 373–385 (1999).
[Crossref] [PubMed]

Proc. SPIE (1)

R. Mandal, K. Palodhi, J. Coupland, R. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).

Other (4)

N. Balasubramanian, “Optical system for surface topography measurement,” U.S. Patent 4340306 (1982).

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

J. M. Coupland, School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire, LE11 3TU. (personal communication2014)

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 2nd Revised edition (Academic Press, 1980).

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Figures (5)

Fig. 1
Fig. 1 Slice through the 3D Fourier transform | F * ( k x * , k Z * ) | of a full spherical shell (a) and partial spherical shell ((b)–(d)), respectively. The logarithm of the absolute values is shown to allow visualisation of the full dynamic range.
Fig. 2
Fig. 2 The construction that demonstrates the region in which the TF will take non-zero values, shown in red. It is symmetric throughout a rotation about the kz axis. The TF will be taken to be zero in the blue region.
Fig. 3
Fig. 3 The absolute value of the FT of a hemispherical shell in a region where non-zero values of the TF would typically be found. This function has zeros periodically along the z axis.
Fig. 4
Fig. 4 (a) the absolute value of the FT of the cap of a sphere that is given by the section of the surface of a 53 μm ball within an intersecting cone with a half angle of 33.37 degrees starting at its centre. A region around the point where the TF is non-zero is shown. (b) A slice through the section of the FT that coincides with the TF according to the construction given in the text above.
Fig. 5
Fig. 5 Part (a) The cap of an oblate spheroid that is imaged after a rotation of -3 degrees about the y- axis and, part (b) the section of its FT that lies within the region defined above as containing non-zero values of the TF. The breaking of the symmetry of the object, combined with the rotation leads to the location of the zero valued points in the FT being shifted from the kz - axis

Equations (15)

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O ˜ ( k ) = Δ ˜ ( k ) H ˜ ( k )
F ( k ) = f ( r ) exp ( i r . k ) d 3 r
r . k = r k [ sin ( α ) cos ( β ) sin ( θ ) cos ( ϕ ) + sin ( α ) sin ( β ) sin ( θ ) sin ( ϕ ) + cos ( α ) cos ( θ ) ]
F ( k ) = 0 f ( r ) d r 0 π r 2 sin ( θ ) d θ 0 2 π d ϕ exp [ i r k sin ( α ) sin ( θ ) cos ( β ϕ ) ] × exp [ i r k cos ( α ) cos ( θ ) ] = 2 π 0 f ( r ) d r 0 π r 2 sin ( θ ) d θ J 0 ( r k sin ( α ) sin ( θ ) ) exp [ i r k cos ( α ) cos ( θ ) ] = 2 π 0 f ( r ) r 2 d r 1 1 d cos ( θ ) J 0 ( r k sin ( α ) ( 1 cos 2 ( θ ) ) 1 / 2 ) exp [ i r k cos ( α ) cos ( θ ) ] = 4 π 0 f ( r ) r 2 d r 0 1 d y J 0 ( r k sin ( α ) ( 1 y 2 ) 1 / 2 ) cos [ r k cos ( α ) y ]
F ( k ) = 4 π 0 f ( r ) r 2 d r sin ( ( r 2 k 2 sin 2 ( α ) + r 2 k 2 cos 2 ( α ) ) 1 / 2 ) ( r 2 k 2 sin 2 ( α ) + r 2 k 2 cos 2 ( α ) ) 1 / 2 = 4 π 0 f ( r ) r d r sin ( r k ) k
f ( r ) = δ ( r r 0 )
F ( k ) = 4 π r 0 2 sin ( r 0 k ) r 0 k
F ( k x , k y ) = 2 π 0 f ( r ) r 2 d r y 0 1 d y J 0 ( r k x ( 1 y 2 ) 1 / 2 ) ) exp ( i r k z y )
F ( k x , k y ) = 2 π r 0 2 y 0 1 d y J 0 ( r 0 k x ( 1 y 2 ) 1 / 2 ) exp ( i r 0 k z y )
F * ( k x * , k z * ) = 1 2 y 0 1 d y J 0 ( k x * ( 1 y 2 ) 1 / 2 ) exp ( i k z * y )
F . T . { A ( r ) . B ( r ) } = F . T . { A ( r ) } F . T { B ( r ) }
F . T . { B ( z ) } = 2 sin ( r 0 . k z ) k z exp ( i r 0 . k z )
F . T . { A . B } = ( 4 π r 0 sin ( r 0 k ) k ) ( 2 sin ( r 0 . k z ) k z exp ( i r 0 . k z ) ) | k x , k y = 0
F ( k z ) = 8 π r 0 3 ( sinc ( r 0 κ ) ) ( sinc ( r 0 . ( k z κ ) ) exp ( i r 0 . ( k z κ ) ) ) d κ
F . T . { A . B } = ( 4 π r 0 2 sinc ( r 0 k ) ) ( 2 r 0 sinc ( r 0 k z ) exp { i r 0 [ 1 + cos ( θ ) ] k z } ) | k x , k y = 0

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