Abstract

Frequency Domain Analysis of measurement results from coherence scanning interferometers provides an estimate of the topography of the surface scattering the light, with claims of nm levels of accuracy being achieved. In the following work we use simulations of the measurement result to show that the limited set of spatial frequencies passed by the instrument can lead to errors in excess of 200 nm for surfaces with curvature. In addition we present a method that takes the uncertainty in the amplitude and phase of each element of the transfer function and provides the upper and lower limit on the location of the surface provided by the Frequency Domain Analysis method. This provides an idea of the level of accuracy that the spatial frequency components must be known to in order to reproduce curved surfaces well.

© 2015 Optical Society of America

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References

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  1. N. Balasubramanian, “Optical system for surface topography measurement,” United States Patent4340306 (July20, 1982).
  2. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32(19), 3438–3441 (1993).
    [Crossref] [PubMed]
  3. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29(26), 3775–3783 (1990).
    [Crossref] [PubMed]
  4. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13(4), 832–843 (1996).
    [Crossref]
  5. C. Ai and E. L. Novak, “Centroid approach for estimating modulation peak in broad-bandwidth interferometry,” United States Patent5633715A (May27, 1997).
  6. P. Sandoz, “Wavelet transform as a processing tool in white-light interferometry,” Opt. Lett. 22(14), 1065–1067 (1997).
    [Crossref] [PubMed]
  7. I. Lee-Bennett, “Advances in non-contacting surface metrology,” in Frontiers in Optics 2004/Laser Science XXII/Diffractive Optics and Micro-Optics/Optical Fabrication and Testing, OSA Technical Digest (CD) (OSA, 2004), paper OTuC1.
  8. 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]
  9. P. de Groot and L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18(17), 1462–1464 (1993).
    [Crossref] [PubMed]
  10. P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” Appl. Opt. 41(22), 4571–4578 (2002).
    [Crossref] [PubMed]
  11. 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]
  12. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light(Cambridge University, 1999).
    [Crossref]
  13. J. M. Coupland and J. Lobera, “Holography, tomography and 3-D microscopy as linear filtering operations,” Meas. Sci. Technol. 19(7), 074012 (2008).
    [Crossref]
  14. J. W. Goodman, Introduction to Fourier Optics, Vol. 2 (McGraw-Hill, 1968).
  15. A. J. Henning, J. M. Huntley, and C. L. Giusca, “Obtaining the Transfer Function of optical instruments using large calibrated reference objects,” Opt. Express 23(13), 16617–16627 (2015).
    [Crossref] [PubMed]

2015 (1)

2013 (1)

2008 (1)

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

2002 (1)

1997 (1)

1996 (1)

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]

1993 (2)

1990 (1)

Ai, C.

C. Ai and E. L. Novak, “Centroid approach for estimating modulation peak in broad-bandwidth interferometry,” United States Patent5633715A (May27, 1997).

Balasubramanian, N.

N. Balasubramanian, “Optical system for surface topography measurement,” United States Patent4340306 (July20, 1982).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light(Cambridge University, 1999).
[Crossref]

Caber, P. J.

Chim, S. S. C.

Colonna de Lega, X.

Coupland, J.

Coupland, J. M.

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

de Groot, P.

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]

P. de Groot and L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18(17), 1462–1464 (1993).
[Crossref] [PubMed]

Giusca, C. L.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, Vol. 2 (McGraw-Hill, 1968).

Henning, A. J.

Huntley, J. M.

Kino, G. S.

Kramer, J.

Larkin, K. G.

Leach, R.

Lee-Bennett, I.

I. Lee-Bennett, “Advances in non-contacting surface metrology,” in Frontiers in Optics 2004/Laser Science XXII/Diffractive Optics and Micro-Optics/Optical Fabrication and Testing, OSA Technical Digest (CD) (OSA, 2004), paper OTuC1.

Lobera, J.

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

Mandal, R.

Novak, E. L.

C. Ai and E. L. Novak, “Centroid approach for estimating modulation peak in broad-bandwidth interferometry,” United States Patent5633715A (May27, 1997).

Palodhi, K.

Sandoz, P.

Turzhitsky, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light(Cambridge University, 1999).
[Crossref]

Appl. Opt. (4)

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]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

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

Opt. Express (1)

Opt. Lett. (2)

Other (5)

I. Lee-Bennett, “Advances in non-contacting surface metrology,” in Frontiers in Optics 2004/Laser Science XXII/Diffractive Optics and Micro-Optics/Optical Fabrication and Testing, OSA Technical Digest (CD) (OSA, 2004), paper OTuC1.

C. Ai and E. L. Novak, “Centroid approach for estimating modulation peak in broad-bandwidth interferometry,” United States Patent5633715A (May27, 1997).

N. Balasubramanian, “Optical system for surface topography measurement,” United States Patent4340306 (July20, 1982).

J. W. Goodman, Introduction to Fourier Optics, Vol. 2 (McGraw-Hill, 1968).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light(Cambridge University, 1999).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 The non-zero components of the TF of a CSI in k- space. This is a slice through the 3 dimensional volume, with the spatial frequencies that are passed by a CSI corresponding to those shown in green. The TF is symmetric throughout a rotation about the kz axis. Here kmax and kmin correspond to the wavenumbers of the shortest and longest wavelengths of the illuminating light respectively, and Θ corresponds to the half angle of the objective lens.
Fig. 2
Fig. 2 Part (a) shows a slice passing through the kz axis of the log plot of the 3-D FT of the cap of an infinitesimally thin spherical shell that is measured when using a lens of NA = 0.7. The red region in part (b) is the non-zero components of the TF on a slice through the 3-D TF, once again the slice passes through the kz axis. Part (c) shows the log |FT| within the non-zero region of the TF. Part (d) shows the real part of the IFT of the data represented in part (c), twice the intensity of which are the fringes that would be recorded by such a CSI.
Fig. 3
Fig. 3 Part (a) shows a slice through the product of the TF of an instrument with an NA of 0.7, using illuminating light from 400 nm to 600 nm, and the |FT| of the relevant cap of a sphere. In part (b) the fringes that would be measured are shown, obtained by taking twice the real part of the IFT of the data illustrated in (a). Part (c) shows the data as recorded by a single pixel, data taken along the dashed line in (b), while the red line in part (d) shows the unwrapped phase of the terms in the FT, the blue line is the magnitude of the terms scaled to fit the phase axis, and is just included to show the region where the terms have significant magnitude.
Fig. 4
Fig. 4 The surface recovered and errors for a lens with NA 0.5. Part (a) shows a slice through the section of log|FT| of the surface of the ball, while part (b) shows a slice through the log|(FT)(TF)|. Part (c) shows the surface recovered (red) and the ’ideal’ surface used to generate the FT (blue), part (e) shows the difference between these two surfaces. Parts (d) and (f) show the same, product of the FT and the TF, illustrated in (b), which has the effect of cropping the FT to the passband of the instrument. The red arrows show the location of the central spatial frequency used to calculate the location of the surface.
Fig. 5
Fig. 5 The surface recovered and errors for a lens with NA 0.7. Part (a) shows a slice through the section of log|FT| of the surface of the ball, while part (b) shows a slice through the log|(FT)(TF)|. Part (c) shows the surface recovered (red) and the ’ideal’ surface used to generate the FT (blue), part (e) shows the difference between these two surfaces. Parts (d) and (f) show the same, product of the FT and the TF, illustrated in (b), which has the effect of cropping the FT to the passband of the instrument. The red arrows show the location of the central spatial frequency used to calculate the location of the surface.
Fig. 6
Fig. 6 Part (a) shows a vector with an uncertainty in the angle and magnitude, the green box is the region that it can end in taking the uncertainty into account, and the circle of radius U0 is the smallest that completely encloses the green box. In (b) the location of the end point of a vector, again with uncertainty in its magnitude and angle, starting anywhere in the circle around the end of the first vector. Its end point cannot lie outside of the red circle of radius |U0 +U1|. In part (c) an example of the summation each of the vectors corresponding to elements with a common kz is shown (blue), and the region the vector must end within, the green circle, taking into account the uncertainty in length and phase. Part (d) shows schematically the unwrapped phase found for successive wavenumbers (blue crosses), and the upper and lower limits to the phase taking into account the uncertainties (green crosses).
Fig. 7
Fig. 7 An example of the upper and lower limits to the surface when uncertainty is included, shown in magenta and green. The surface without uncertainty is shown in blue. The results are for NA = 0.7

Equations (12)

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O ˜ ( k ) = 2 Δ ˜ ( k ) H ˜ ( k )
F ( k ) = 8 π r 0 3 sinc ( r 0 κ 2 + k x 2 + k y 2 ) sinc ( r 0 ( k z κ ) ) exp ( i r 0 [ 1 + cos Θ ] ( k z κ ) ) d κ
f ( r ) = F ( k ) exp ( i k . r ) d k x d k y d k z
F T p l a n e = exp ( i k . r 0 ) | θ = θ 0 , ϕ = ϕ 0
S i g n a l k , r = A k exp ( i [ k x x + k y y ] ) exp ( i k z z )
B k , x , y exp ( i k z z ) = A k exp ( i [ k x x + k y y ] ) exp ( i k z z )
F z ( k z ) | ( x , y ) = k x k y B k , x , y | k z
U 0 = | B k , x , y | 2 + ( | B k , x , y | + | Δ B k , x , y | ) 2 2 | B k , x , y | ( | B k , x , y | + | Δ B k , x , y | ) cos ( Δ θ k , x , y ) | k z
U max = k x k y | B k , x , y | 2 + ( | B k , x , y | + | Δ B k , x , y | ) 2 2 | B k , x , y | ( | B k , x , y | + | Δ B k , x , y | ) cos ( Δ θ k , x , y ) | k z
Δ θ = a s i n ( U max x 0 2 + y 0 2 )
β 1 = m n = 0 m 1 p n w n + m n = 0 m 1 w n ε n n = 0 m 1 w n n = 0 m 1 p n n = 0 m 1 w n n = 0 m 1 ε n m n = 0 m 1 w n 2 ( n = 0 m 1 w n ) 2
β 1 ε n = m w n r = 0 m 1 w r m r = 0 m 1 w r 2 ( r = 0 m 1 w r ) 2

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