Abstract

Confocal microscopy is one of the dominating measurement techniques in surface metrology, with an enhanced lateral resolution compared to alternative optical methods. However, the axial resolution in confocal microscopy is strongly dependent on the accuracy of signal evaluation algorithms, which are limited by random noise. Here, we discuss the influence of various noise sources on confocal intensity signal evaluating algorithms, including center-of-mass, parabolic least-square fit, and cross-correlation-based methods. We derive results in closed form for the uncertainty in height evaluation on surface microstructures, also accounting for the number of axially measured intensity values and a threshold that is commonly applied before signal evaluation. The validity of our results is verified by numerical Monte Carlo simulations. In addition, we implemented all three algorithms and analyzed their numerical efficiency. Our results can serve as guidance for a suitable choice of measurement parameters in confocal surface topography measurement, and thus lead to a shorter measurement time in practical applications.

© 2017 Optical Society of America

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References

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  1. A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
    [Crossref]
  2. A. Weidner, J. Seewig, and E. Reithmeier, “3D roughness evaluation of cylinder liner surfaces based on structure-oriented parameters,” Meas. Sci. Technol. 17, 477–482 (2006).
    [Crossref]
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    [Crossref]
  5. M. Rahlves, B. Roth, and E. Reithmeier, “Systematic errors on curved microstructures caused by aberrations in confocal surface metrology,” Opt. Express 23, 9640–9648 (2015).
    [Crossref]
  6. M. Fleischer, R. Windecker, and H. J. Tiziani, “Theoretical limits of scanning white-light interferometry signal evaluation algorithms,” Appl. Opt. 40, 2815–2820 (2001).
    [Crossref]
  7. A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Signal evaluation for high-speed confocal measurements,” Appl. Opt. 41, 7410–7415 (2002).
    [Crossref]
  8. F. Mauch, W. Lyda, M. Gronle, and W. Osten, “Improved signal model for confocal sensors accounting for object depending artifacts,” Opt. Express 20, 19936–19945 (2012).
    [Crossref]
  9. J. Aguilar and E. Méndez, “On the limitations of the confocal scanning optical microscope as a profilometer,” J. Mod. Opt. 42, 1785–1794 (1995).
    [Crossref]
  10. T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).
  11. N. Langholz, J. Seewig, and E. Reithmeier, “Robust surface fitting—using weights based on à priori knowledge about the measurement process,” Wear 266, 515–517 (2009).
    [Crossref]
  12. D. A. Hall, “Review nonlinearity in piezoelectric ceramics,” J. Mater. Sci. 36, 4575–4601 (2001).
    [Crossref]
  13. G. C. Holst, CCD Arrays Cameras and Displays (SPIE, 1996).
  14. Joint Committee for Guides in Metrology, “Evaluation of measurement data—guide to the expression of uncertainty in measurement,” (2008).
  15. J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27, 025006 (2016).
    [Crossref]
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).
  17. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).
  18. M. Rahlves, “Calibration of confocal microscopes,” Ph.D. dissertation (Leibniz University Hannover, 2011).

2016 (1)

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27, 025006 (2016).
[Crossref]

2015 (1)

2012 (1)

2009 (1)

N. Langholz, J. Seewig, and E. Reithmeier, “Robust surface fitting—using weights based on à priori knowledge about the measurement process,” Wear 266, 515–517 (2009).
[Crossref]

2006 (2)

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[Crossref]

A. Weidner, J. Seewig, and E. Reithmeier, “3D roughness evaluation of cylinder liner surfaces based on structure-oriented parameters,” Meas. Sci. Technol. 17, 477–482 (2006).
[Crossref]

2002 (1)

2001 (2)

1995 (1)

J. Aguilar and E. Méndez, “On the limitations of the confocal scanning optical microscope as a profilometer,” J. Mod. Opt. 42, 1785–1794 (1995).
[Crossref]

1994 (1)

Aguilar, J.

J. Aguilar and E. Méndez, “On the limitations of the confocal scanning optical microscope as a profilometer,” J. Mod. Opt. 42, 1785–1794 (1995).
[Crossref]

Brain, K.

Fleischer, M.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Gronle, M.

Gu, M.

Hall, D. A.

D. A. Hall, “Review nonlinearity in piezoelectric ceramics,” J. Mater. Sci. 36, 4575–4601 (2001).
[Crossref]

Hoffmann, J.

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[Crossref]

Holst, G. C.

G. C. Holst, CCD Arrays Cameras and Displays (SPIE, 1996).

Langholz, N.

N. Langholz, J. Seewig, and E. Reithmeier, “Robust surface fitting—using weights based on à priori knowledge about the measurement process,” Wear 266, 515–517 (2009).
[Crossref]

Liu, C.

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27, 025006 (2016).
[Crossref]

Liu, J.

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27, 025006 (2016).
[Crossref]

Lyda, W.

Mauch, F.

Méndez, E.

J. Aguilar and E. Méndez, “On the limitations of the confocal scanning optical microscope as a profilometer,” J. Mod. Opt. 42, 1785–1794 (1995).
[Crossref]

Osten, W.

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

Peggs, G.

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[Crossref]

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

Rahlves, M.

Reithmeier, E.

M. Rahlves, B. Roth, and E. Reithmeier, “Systematic errors on curved microstructures caused by aberrations in confocal surface metrology,” Opt. Express 23, 9640–9648 (2015).
[Crossref]

N. Langholz, J. Seewig, and E. Reithmeier, “Robust surface fitting—using weights based on à priori knowledge about the measurement process,” Wear 266, 515–517 (2009).
[Crossref]

A. Weidner, J. Seewig, and E. Reithmeier, “3D roughness evaluation of cylinder liner surfaces based on structure-oriented parameters,” Meas. Sci. Technol. 17, 477–482 (2006).
[Crossref]

Roth, B.

Ruprecht, A. K.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Seewig, J.

N. Langholz, J. Seewig, and E. Reithmeier, “Robust surface fitting—using weights based on à priori knowledge about the measurement process,” Wear 266, 515–517 (2009).
[Crossref]

A. Weidner, J. Seewig, and E. Reithmeier, “3D roughness evaluation of cylinder liner surfaces based on structure-oriented parameters,” Meas. Sci. Technol. 17, 477–482 (2006).
[Crossref]

Sheppard, C. J. R.

Tan, J.

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27, 025006 (2016).
[Crossref]

Tiziani, H. J.

Wang, H.

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27, 025006 (2016).
[Crossref]

Weckenmann, A.

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[Crossref]

Weidner, A.

A. Weidner, J. Seewig, and E. Reithmeier, “3D roughness evaluation of cylinder liner surfaces based on structure-oriented parameters,” Meas. Sci. Technol. 17, 477–482 (2006).
[Crossref]

Wiesendanger, T. F.

Wilson, T.

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

Windecker, R.

Zhou, H.

Appl. Opt. (3)

J. Mater. Sci. (1)

D. A. Hall, “Review nonlinearity in piezoelectric ceramics,” J. Mater. Sci. 36, 4575–4601 (2001).
[Crossref]

J. Mod. Opt. (1)

J. Aguilar and E. Méndez, “On the limitations of the confocal scanning optical microscope as a profilometer,” J. Mod. Opt. 42, 1785–1794 (1995).
[Crossref]

Meas. Sci. Technol. (3)

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27, 025006 (2016).
[Crossref]

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[Crossref]

A. Weidner, J. Seewig, and E. Reithmeier, “3D roughness evaluation of cylinder liner surfaces based on structure-oriented parameters,” Meas. Sci. Technol. 17, 477–482 (2006).
[Crossref]

Opt. Express (2)

Wear (1)

N. Langholz, J. Seewig, and E. Reithmeier, “Robust surface fitting—using weights based on à priori knowledge about the measurement process,” Wear 266, 515–517 (2009).
[Crossref]

Other (7)

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

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

M. Rahlves, “Calibration of confocal microscopes,” Ph.D. dissertation (Leibniz University Hannover, 2011).

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

G. C. Holst, CCD Arrays Cameras and Displays (SPIE, 1996).

Joint Committee for Guides in Metrology, “Evaluation of measurement data—guide to the expression of uncertainty in measurement,” (2008).

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Figures (6)

Fig. 1.
Fig. 1. Theoretical point spread function (solid line) and noisy intensity data I i (circles).
Fig. 2.
Fig. 2. Noisy intensity data of a measured confocal curve (left) and a sketch of intensity measurement points around the threshold (right). The measured intensity is assumed to be equally distributed in the interval Δ I .
Fig. 3.
Fig. 3. Theoretical results and MC simulations. (a) Influence of the number of intensity measurements N on the uncertainty σ h : σ z = 0 ; (b)  σ I = 0 .
Fig. 4.
Fig. 4. Dependence of the uncertainty σ h on the number of intensity measurement points N (only intensity noise is considered). The threshold was applied to the simulated confocal curve after adding random noise during the MC simulations.
Fig. 5.
Fig. 5. Theoretical results and MC simulations. (a) Influence of the threshold a on the uncertainty σ h : σ z = 0 ; (b)  σ I = 0 . Note that the number of sampling points N is fixed for each data point in the figure, which leads to a decrease of the sampling size Δ z and thus to a decrease in uncertainty when increasing γ in case of the LSQ algorithm.
Fig. 6.
Fig. 6. Elapsed time during signal evaluation of 10 6 confocal curves for different algorithms as a function of number of intensity values within the FWHM ( a = 0.5 ).

Equations (28)

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I ( z ) = I 0 ( sin ( u / 2 ) u / 2 ) 2 with u = 8 π z λ sin ( α 2 ) 2 ,
Δ z = γ · FWHM N + 1 .
h = z i | max { I i } .
χ 2 = i = N / 2 N / 2 ( I i f ( z ) ) 2 min ,
h = i = N / 2 N / 2 z i I i i = N / 2 N / 2 I i .
c ( z ) = i = N / 2 N / 2 I i · f ( z i z ) ,
h = z | max { c ( z ) } .
I [ z i ] I 0 · exp [ 1 2 · ( 8 ln ( 2 ) · ( z ^ i h ) FWHM ) 2 ] I 0 · [ 1 4 ln ( 2 ) ( z ^ i h FWHM ) 2 ] + O ( z i 4 ) ,
σ h 2 = i = N / 2 N / 2 { ( h I i ) 2 σ I 2 + ( h z ^ i ) 2 σ z 2 } .
f ( z ) = a 0 + a 1 z + a 2 z 2 ,
h = FWHM 2 8 ln ( 2 ) I 0 · a 1 σ h = h 2 = FWHM 2 8 ln ( 2 ) I 0 · a 1 2 ,
a = M 1 x with a = [ a 0 , a 1 , a 2 ] T
M = j = N / 2 N / 2 ( 1 z j z j 2 z j z j 2 z j 3 z j 2 z j 3 z j 4 ) , x = j = N / 2 N / 2 ( I j ( z ^ j ) z j I j ( z ^ j ) z j 2 I j ( z ^ j ) ) .
a 1 = 12 Δ z 2 N ( N 2 + 3 N + 2 ) j = N / 2 N / 2 z j I j ( z ^ j ) ,
h I i | i = j = 3 FWHM 2 z i 2 ln ( 2 ) I 0 Δ z 2 N ( N 2 + 3 N + 2 ) , h z ^ i | i = j = 12 z i z ^ i Δ z 2 N ( N 2 + 3 N + 2 ) .
σ h , Fit 3 16 N ( FWHM σ I ln ( 2 ) γ I 0 ) 2 + 9 σ z 2 5 N .
h I i | i = j = z i j = N / 2 N / 2 I j j = N / 2 N / 2 z j I j ( j = N / 2 N / 2 I j ) 2 = 3 γ FWHM i I 0 [ ( ln ( 2 ) γ 2 3 ) N 2 + ( 2 ln ( 2 ) γ 2 ) N 3 ]
h z ^ i | i = j = z i j = N / 2 N / 2 I j I j ( z ^ j ) z ^ j | i = j j = N / 2 N / 2 z j I j ( j = N / 2 N / 2 I j ) I j ( z ^ j ) z ^ j | i = j = 24 ln ( 2 ) γ 2 i 2 · ( N + 1 ) 1 [ ( ln ( 2 ) γ 2 3 ) N 2 + ( 2 ln ( 2 ) γ 2 ) 6 ) N 3 ] 2 .
σ h , 1 1 N ( 2 γ FWHM σ I 5 I 0 ) 2 + ( 36 γ 2 ln ( 2 ) ) 2 σ z 2 N ( 45 30 ln ( 2 ) γ 2 ) .
N ^ 2.52 σ I I 0 ( N + 1 ) .
h k = FWHM I 0 4 ( k N ^ ) · ( i = N / 2 N / 2 I i ) 1 ,
σ h , 2 = k = 0 2 N ^ P k ( h k h ¯ k ) 2 ,
P k = ( 2 N ^ k ) · 0.5 2 N ^ .
σ h , 2 = FWHM I 0 4 ( i = N / 2 N / 2 I i ) 1 k = 0 2 N ^ ( 2 N ^ k ) · 0.5 2 N ^ · k 2 .
σ h , 2 0.13 σ I I 0 FWHM 2 N .
σ h , CoM = σ h , 1 2 + σ h , 2 2 .
z c ( z ) = i = N / 2 N / 2 8 ln ( 2 ) I 0 I i · z i z FWHM 2 .
h = z = i = N / 2 N / 2 z i I i i = N / 2 N / 2 I i .

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