Abstract

When applied to the measurement of smooth surfaces, coherence scanning interferometry can be described by a three-dimensional linear filtering operation that is characterized either by the point-spread function in the space domain or equivalently by the transfer function (TF) in the spatial frequency domain. For an ideal, aberration-free instrument, these characteristics are defined uniquely by the numerical aperture of the objective lens and the bandwidth of the illumination source. In practice, however, physical imperfections such as those in lens aberrations, reference focus, and source alignment mean that the instrument performance is not ideal. Currently, these imperfections often go unnoticed as the instrument performance is typically only verified using rectilinear artifacts such as step heights and lateral grids. If an object of varying slope is measured, however, significant errors are often observed as the surface gradient increases. In this paper, a new method of calibration and adjustment using a silica micro-sphere as a calibration artifact is introduced. The silica microsphere was used to compute the point-spread and TF characteristics of the instrument, and the effect of these characteristics on instrument performance is discussed. Finally, a straightforward method to correct for phase and amplitude imperfections in the TF is described using a modified inverse filter.

© 2014 Optical Society of America

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References

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  1. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
    [CrossRef]
  2. L. Deck and P. De Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
    [CrossRef]
  3. R. K. Leach, Characterisation of Areal Surface Texture (Springer-Verlag, 2013).
  4. F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
    [CrossRef]
  5. 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]
  6. R. K. Leach, C. L. Giusca, and K. Naoi, “Development and characterisation of a new instrument for traceable measurement of areal surface texture,” Meas. Sci. Technol. 20, 125102 (2009).
    [CrossRef]
  7. W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white-light interferometers,” Appl. Opt. 51, 1795–1803 (2012).
    [CrossRef]
  8. P. De Groot and X. C. Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2006), pp. 30–37.
  9. G. Häusler and S. Ettl, “Limitations of optical 3D-sensors,” in Optical Measurement of Surface Topography, R. Leach, ed. (Springer-Verlag, 2011), pp. 23–48.
  10. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 456–520.
  11. J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 074012 (2008).
    [CrossRef]
  12. 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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
    [CrossRef]
  13. J. M. Coupland, R. Mandal, K. Palodhi, and R. K. Leach, “Coherence scanning interferometry: linear theory of surface measurement,” Appl. Opt. 52, 3662–3670 (2013).
    [CrossRef]
  14. K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the American Society of Precision Engineering Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, June 23–25, 2010.
  15. C. L. Giusca, R. K. Leach, and F. Helery, “Calibration of the scales of surface topography measuring instruments: part 2. Amplification, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
    [CrossRef]
  16. A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
    [CrossRef]
  17. J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
    [CrossRef]
  18. R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
    [CrossRef]
  19. Y. Zhou, Y. Ghim, A. Fard, and A. Davies, “Application of the random ball test for calibrating slope-dependent errors in profilometry measurements,” Appl. Opt. 52, 5925–5931 (2013).
    [CrossRef]
  20. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, 2001), pp. 351–354.
  21. P. De Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
    [CrossRef]

2013 (4)

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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
[CrossRef]

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

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

Y. Zhou, Y. Ghim, A. Fard, and A. Davies, “Application of the random ball test for calibrating slope-dependent errors in profilometry measurements,” Appl. Opt. 52, 5925–5931 (2013).
[CrossRef]

2012 (3)

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

W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white-light interferometers,” Appl. Opt. 51, 1795–1803 (2012).
[CrossRef]

C. L. Giusca, R. K. Leach, and F. Helery, “Calibration of the scales of surface topography measuring instruments: part 2. Amplification, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

2009 (1)

R. K. Leach, C. L. Giusca, and K. Naoi, “Development and characterisation of a new instrument for traceable measurement of areal surface texture,” Meas. Sci. Technol. 20, 125102 (2009).
[CrossRef]

2008 (2)

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

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement 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]

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, 389–401 (1995).
[CrossRef]

1994 (1)

1990 (1)

1956 (1)

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

Coupland, J. M.

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

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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
[CrossRef]

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

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

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

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

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the American Society of Precision Engineering Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, June 23–25, 2010.

Davies, A.

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, 389–401 (1995).
[CrossRef]

L. Deck and P. De Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
[CrossRef]

P. De Groot and X. C. Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2006), pp. 30–37.

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, 389–401 (1995).
[CrossRef]

L. Deck and P. De Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
[CrossRef]

Ettl, S.

G. Häusler and S. Ettl, “Limitations of optical 3D-sensors,” in Optical Measurement of Surface Topography, R. Leach, ed. (Springer-Verlag, 2011), pp. 23–48.

Fard, A.

Forbes, A.

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
[CrossRef]

Gao, F.

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

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 456–520.

Gates, J. W.

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

Ghim, Y.

Giusca, C.

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
[CrossRef]

C. L. Giusca, R. K. Leach, and F. Helery, “Calibration of the scales of surface topography measuring instruments: part 2. Amplification, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

R. K. Leach, C. L. Giusca, and K. Naoi, “Development and characterisation of a new instrument for traceable measurement of areal surface texture,” Meas. Sci. Technol. 20, 125102 (2009).
[CrossRef]

Häusler, G.

G. Häusler and S. Ettl, “Limitations of optical 3D-sensors,” in Optical Measurement of Surface Topography, R. Leach, ed. (Springer-Verlag, 2011), pp. 23–48.

Helery, F.

C. L. Giusca, R. K. Leach, and F. Helery, “Calibration of the scales of surface topography measuring instruments: part 2. Amplification, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

Henning, A.

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

Leach, R. K.

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
[CrossRef]

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

C. L. Giusca, R. K. Leach, and F. Helery, “Calibration of the scales of surface topography measuring instruments: part 2. Amplification, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

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

R. K. Leach, C. L. Giusca, and K. Naoi, “Development and characterisation of a new instrument for traceable measurement of areal surface texture,” Meas. Sci. Technol. 20, 125102 (2009).
[CrossRef]

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

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the American Society of Precision Engineering Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, June 23–25, 2010.

R. K. Leach, Characterisation of Areal Surface Texture (Springer-Verlag, 2013).

Lee, B. S.

Lega, X. C.

P. De Groot and X. C. Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2006), pp. 30–37.

Lehmann, P.

Lobera, J.

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

Mandal, R.

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

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

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

Mansfield, D.

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

Naoi, K.

R. K. Leach, C. L. Giusca, and K. Naoi, “Development and characterisation of a new instrument for traceable measurement of areal surface texture,” Meas. Sci. Technol. 20, 125102 (2009).
[CrossRef]

Niehues, J.

Palodhi, K.

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

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

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the American Society of Precision Engineering Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, June 23–25, 2010.

Petzing, J.

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

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, 2001), pp. 351–354.

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]

Smith, I.

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

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]

Strand, T. C.

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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
[CrossRef]

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]

Xie, W.

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]

Zhou, Y.

Appl. Opt. (5)

CIRP Ann. (1)

A. Henning, C. Giusca, A. Forbes, I. Smith, R. K. Leach, J. M. Coupland, and R. Mandal, “Correction for lateral distortion in coherence scanning interferometry,” CIRP Ann. 62, 547–550 (2013).
[CrossRef]

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]

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, 389–401 (1995).
[CrossRef]

J. Sci. Instrum. (1)

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

Meas. Sci. Technol. (5)

R. K. Leach, C. L. Giusca, and K. Naoi, “Development and characterisation of a new instrument for traceable measurement of areal surface texture,” Meas. Sci. Technol. 20, 125102 (2009).
[CrossRef]

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

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

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 systems—a review,” Meas. Sci. Technol. 24, 052001 (2013).
[CrossRef]

C. L. Giusca, R. K. Leach, and F. Helery, “Calibration of the scales of surface topography measuring instruments: part 2. Amplification, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

Proc. SPIE (1)

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

Other (6)

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, 2001), pp. 351–354.

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the American Society of Precision Engineering Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, June 23–25, 2010.

R. K. Leach, Characterisation of Areal Surface Texture (Springer-Verlag, 2013).

P. De Groot and X. C. Lega, “Interpreting interferometric height measurements using the instrument transfer function,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2006), pp. 30–37.

G. Häusler and S. Ettl, “Limitations of optical 3D-sensors,” in Optical Measurement of Surface Topography, R. Leach, ed. (Springer-Verlag, 2011), pp. 23–48.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), pp. 456–520.

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Figures (14)

Fig. 1.
Fig. 1.

Measurement of sinusoidal profile using CSI.

Fig. 2.
Fig. 2.

(a) Microsphere and (b) foil model.

Fig. 3.
Fig. 3.

Sections through (a) TF (absolute value) and (b) PSF (real part).

Fig. 4.
Fig. 4.

Sections through (a) interferogram and (b) its spectrum (real part).

Fig. 5.
Fig. 5.

Sections through the (a) filter and (b) fringe pattern after filtering (real part).

Fig. 6.
Fig. 6.

Sections through the foil model of the surface.

Fig. 7.
Fig. 7.

Sections through the measured TF (a) real part and (b) imaginary part.

Fig. 8.
Fig. 8.

Sections through the measured PSF (real part).

Fig. 9.
Fig. 9.

Gain of the weighting function.

Fig. 10.
Fig. 10.

Sections through the (a) corrected spectrum (real part) and (b) corresponding fringes (real part).

Fig. 11.
Fig. 11.

Error surface for the surface extracted from (a) instrument measurements and (b) corrected fringes.

Fig. 12.
Fig. 12.

Position of the droplets in total field of view.

Fig. 13.
Fig. 13.

Error surfaces extracted from instrument measurements at positions CBDE (clockwise from top left).

Fig. 14.
Fig. 14.

Error surfaces using the modified inverse filter at positions CBDE (clockwise from top left).

Equations (11)

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

O(r)=H(rr)Δ(r)d3r,
O˜(k)=Δ˜(k)H˜(k),
Δ(r)=4πjRW(rx,ry)δ(rzs(rx,ry)),
Δ(r)4πj(1n1+n)W(rx,ry)δ(rzs(rx,ry)).
H˜(k)=(|k|22k·o^)G˜NA(kr,k0)G˜NA(kkr,k0)d3krS(k0)dk0,
G˜NA(k,k0)=j4πk0δ(|k|k0)step(k·o^k01NA2),
θmax=sin1(ky/kz)max,
H˜inst(k)=O˜cal(k)Δ˜cal(k),
H˜inv(k)=1A(k)exp(jφ(k)).
H˜mod(k)=exp(jφ(k))W(k),
W(k)=(An(k)+0.027)(1+exp(An(k)×58+3.9)/1.5

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