Abstract

White-light scanning interferometry is an effective and widely used technology for measuring the microscopic three-dimensional morphology of an object. However, it is easily affected by external disturbances and appears to have a non-uniform sampling problem, which reduces the measurement accuracy. In this study, an effective correction algorithm is presented, in which a Hilbert transform and a correlation analysis of the white light interference envelope curves, as well as the simulated ideal interference signal envelope, are employed for a robust and high precision signal correction. In addition, the proposed method is at least 4 times as accurate as a traditional method and achieves a high repeatability, which is analyzed through a simulation and contrast experiments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
    [Crossref]
  2. H. Schreiber and J. Bruning, “Phase Shifting Interferometry,” pp. 547–666,(2006).
  3. U. Paul Kumar, N. Krishna Mohan, and M. Kothiyal, “Multiple wavelength interferometry for surface profiling,” Proc. SPIE 7063, 70630W (2008).
    [Crossref]
  4. J. C. Wyant and K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tool Manu. 32(1-2), 5–10 (1992).
    [Crossref]
  5. W. Osten, Optical inspection of Microsystems, (CRC Press, 2007.
  6. U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of static and vibrating microsystems using microscopicTV holography,” Optik 122(1), 49–54 (2011).
    [Crossref]
  7. P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (2000).
  8. U. Paul Kumar, M. P. Kothiyal, and N. K. Mohan, “Microscopic TV shearography for characterization of microsystems,” Opt. Lett. 34(10), 1612–1614 (2009).
    [Crossref]
  9. W. Steinchen and L. Yang, [Digital shearography: Theory and Application of digital speckle pattern shearing interferometry], PM100, SPIE Press, Washington DC, (2003).
  10. U. P. Kumar, Y. Kalyani, N. K. Mohan, and M. P. Kothiyal, “Time-average TV holography for vibration fringe analysis,” Appl. Opt. 48(16), 3094–3101 (2009).
    [Crossref]
  11. J. Schmit and A. Pakuła, “White Light Interferometry,” in: N. Ida and N. Meyendorf, eds., Handbook of Advanced Non-Destructive Evaluation, Springer International Publishing, Cham, pp. 1–47 (2019).
  12. J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
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  14. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26(14), 2810–2816 (1987).
    [Crossref]
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    [Crossref]
  16. D. G. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50(19), 3360–3368 (2011).
    [Crossref]
  17. U. P. Kumar, N. Krishna Mohan, M. Kothiyal, and A. Asundi, “Deformation and shape measurement using multiple wavelength microscopic TV holography,” Opt. Eng. 48(2), 023601 (2009).
    [Crossref]
  18. L. Fei, X. Lu, H. Wang, W. Zhang, J. Tian, and L. Zhong, “Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms,” Opt. Express 22(25), 30910–30923 (2014).
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    [Crossref]
  21. D. S. Mehta and V. Srivastava, “Quantitative phase imaging of human red blood cells using phase-shifting white light interference microscopy with colour fringe analysis,” Appl. Phys. Lett. 101(20), 203701 (2012).
    [Crossref]
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    [Crossref]
  24. Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
  27. D. Zweig and R. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” 1333, (1990).
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    [Crossref]
  29. R. Bracewell, The Fourier Transform and its Application, (McGrawHill Book Company, 2000).

2016 (1)

Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
[Crossref]

2015 (2)

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

2014 (2)

L. Fei, X. Lu, H. Wang, W. Zhang, J. Tian, and L. Zhong, “Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms,” Opt. Express 22(25), 30910–30923 (2014).
[Crossref]

A. Saad, Y. Jiang, Y. Liu, and Z. Wang, “The measurement of the diameter change of a piezoelectric transducer cylinder with the white-light interferometry,” Opt. Lasers Eng. 56, 169–172 (2014).
[Crossref]

2012 (2)

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50(8), 1084–1088 (2012).
[Crossref]

D. S. Mehta and V. Srivastava, “Quantitative phase imaging of human red blood cells using phase-shifting white light interference microscopy with colour fringe analysis,” Appl. Phys. Lett. 101(20), 203701 (2012).
[Crossref]

2011 (2)

D. G. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50(19), 3360–3368 (2011).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of static and vibrating microsystems using microscopicTV holography,” Optik 122(1), 49–54 (2011).
[Crossref]

2009 (3)

2008 (1)

U. Paul Kumar, N. Krishna Mohan, and M. Kothiyal, “Multiple wavelength interferometry for surface profiling,” Proc. SPIE 7063, 70630W (2008).
[Crossref]

2006 (1)

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

2003 (2)

2000 (1)

M. Born and E. Wolf, “Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light,” Phys. Today 53(10), 77–78 (2000).
[Crossref]

1992 (1)

J. C. Wyant and K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tool Manu. 32(1-2), 5–10 (1992).
[Crossref]

1988 (1)

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1987 (1)

1985 (1)

1973 (1)

Abdelsalam, D. G.

Asundi, A.

U. P. Kumar, N. Krishna Mohan, M. Kothiyal, and A. Asundi, “Deformation and shape measurement using multiple wavelength microscopic TV holography,” Opt. Eng. 48(2), 023601 (2009).
[Crossref]

Born, M.

M. Born and E. Wolf, “Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light,” Phys. Today 53(10), 77–78 (2000).
[Crossref]

Bracewell, R.

R. Bracewell, The Fourier Transform and its Application, (McGrawHill Book Company, 2000).

Bruning, J.

H. Schreiber and J. Bruning, “Phase Shifting Interferometry,” pp. 547–666,(2006).

Chang, S.

Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
[Crossref]

Chen, L.

Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
[Crossref]

Cheng, J.

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

Cheng, Y.

Creath, K.

J. C. Wyant and K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tool Manu. 32(1-2), 5–10 (1992).
[Crossref]

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26(14), 2810–2816 (1987).
[Crossref]

de Groot, P.

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

Fei, L.

Gao, Z.

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

Haifeng, W.

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50(8), 1084–1088 (2012).
[Crossref]

Hariharan, P.

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

Hufnagel, R.

D. Zweig and R. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” 1333, (1990).

Jiang, Y.

A. Saad, Y. Jiang, Y. Liu, and Z. Wang, “The measurement of the diameter change of a piezoelectric transducer cylinder with the white-light interferometry,” Opt. Lasers Eng. 56, 169–172 (2014).
[Crossref]

Kadono, H.

Kalyani, Y.

Kim, D.

Kothiyal, M.

U. P. Kumar, N. Krishna Mohan, M. Kothiyal, and A. Asundi, “Deformation and shape measurement using multiple wavelength microscopic TV holography,” Opt. Eng. 48(2), 023601 (2009).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. Kothiyal, “Multiple wavelength interferometry for surface profiling,” Proc. SPIE 7063, 70630W (2008).
[Crossref]

Kothiyal, M. P.

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50(8), 1084–1088 (2012).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of static and vibrating microsystems using microscopicTV holography,” Optik 122(1), 49–54 (2011).
[Crossref]

U. P. Kumar, Y. Kalyani, N. K. Mohan, and M. P. Kothiyal, “Time-average TV holography for vibration fringe analysis,” Appl. Opt. 48(16), 3094–3101 (2009).
[Crossref]

U. Paul Kumar, M. P. Kothiyal, and N. K. Mohan, “Microscopic TV shearography for characterization of microsystems,” Opt. Lett. 34(10), 1612–1614 (2009).
[Crossref]

Krishna Mohan, N.

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of static and vibrating microsystems using microscopicTV holography,” Optik 122(1), 49–54 (2011).
[Crossref]

U. P. Kumar, N. Krishna Mohan, M. Kothiyal, and A. Asundi, “Deformation and shape measurement using multiple wavelength microscopic TV holography,” Opt. Eng. 48(2), 023601 (2009).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. Kothiyal, “Multiple wavelength interferometry for surface profiling,” Proc. SPIE 7063, 70630W (2008).
[Crossref]

Kumar, U. P.

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50(8), 1084–1088 (2012).
[Crossref]

U. P. Kumar, N. Krishna Mohan, M. Kothiyal, and A. Asundi, “Deformation and shape measurement using multiple wavelength microscopic TV holography,” Opt. Eng. 48(2), 023601 (2009).
[Crossref]

U. P. Kumar, Y. Kalyani, N. K. Mohan, and M. P. Kothiyal, “Time-average TV holography for vibration fringe analysis,” Appl. Opt. 48(16), 3094–3101 (2009).
[Crossref]

Lei, Z.

Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
[Crossref]

Li, M.

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

Liu, X.

Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
[Crossref]

Liu, Y.

A. Saad, Y. Jiang, Y. Liu, and Z. Wang, “The measurement of the diameter change of a piezoelectric transducer cylinder with the white-light interferometry,” Opt. Lasers Eng. 56, 169–172 (2014).
[Crossref]

Lu, W.

Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
[Crossref]

Lu, X.

Madjarova, V. D.

Magnusson, R.

Mehta, D. S.

D. S. Mehta and V. Srivastava, “Quantitative phase imaging of human red blood cells using phase-shifting white light interference microscopy with colour fringe analysis,” Appl. Phys. Lett. 101(20), 203701 (2012).
[Crossref]

Mohan, N. K.

Osten, W.

W. Osten, Optical inspection of Microsystems, (CRC Press, 2007.

Pakula, A.

J. Schmit and A. Pakuła, “White Light Interferometry,” in: N. Ida and N. Meyendorf, eds., Handbook of Advanced Non-Destructive Evaluation, Springer International Publishing, Cham, pp. 1–47 (2019).

Paul Kumar, U.

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of static and vibrating microsystems using microscopicTV holography,” Optik 122(1), 49–54 (2011).
[Crossref]

U. Paul Kumar, M. P. Kothiyal, and N. K. Mohan, “Microscopic TV shearography for characterization of microsystems,” Opt. Lett. 34(10), 1612–1614 (2009).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. Kothiyal, “Multiple wavelength interferometry for surface profiling,” Proc. SPIE 7063, 70630W (2008).
[Crossref]

Pförtner, A.

Polhemus, C.

Rastogi, P. K.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (2000).

Saad, A.

A. Saad, Y. Jiang, Y. Liu, and Z. Wang, “The measurement of the diameter change of a piezoelectric transducer cylinder with the white-light interferometry,” Opt. Lasers Eng. 56, 169–172 (2014).
[Crossref]

Schmit, J.

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

J. Schmit and A. Pakuła, “White Light Interferometry,” in: N. Ida and N. Meyendorf, eds., Handbook of Advanced Non-Destructive Evaluation, Springer International Publishing, Cham, pp. 1–47 (2019).

Schreiber, H.

H. Schreiber and J. Bruning, “Phase Shifting Interferometry,” pp. 547–666,(2006).

Schwider, J.

Srivastava, V.

D. S. Mehta and V. Srivastava, “Quantitative phase imaging of human red blood cells using phase-shifting white light interference microscopy with colour fringe analysis,” Appl. Phys. Lett. 101(20), 203701 (2012).
[Crossref]

Steinchen, W.

W. Steinchen and L. Yang, [Digital shearography: Theory and Application of digital speckle pattern shearing interferometry], PM100, SPIE Press, Washington DC, (2003).

Tian, J.

Toyooka, S.

Wang, H.

Wang, S.

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

Wang, Z.

A. Saad, Y. Jiang, Y. Liu, and Z. Wang, “The measurement of the diameter change of a piezoelectric transducer cylinder with the white-light interferometry,” Opt. Lasers Eng. 56, 169–172 (2014).
[Crossref]

Wolf, E.

M. Born and E. Wolf, “Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light,” Phys. Today 53(10), 77–78 (2000).
[Crossref]

Wyant, J. C.

J. C. Wyant and K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tool Manu. 32(1-2), 5–10 (1992).
[Crossref]

Y. Cheng and J. C. Wyant, “Multiple-wavelength phase-shifting interferometry,” Appl. Opt. 24(6), 804–807 (1985).
[Crossref]

Yang, L.

W. Steinchen and L. Yang, [Digital shearography: Theory and Application of digital speckle pattern shearing interferometry], PM100, SPIE Press, Washington DC, (2003).

Yang, Z.

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

Ye, J.

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

Yuan, Q.

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

Zhang, W.

Zhong, L.

Zweig, D.

D. Zweig and R. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” 1333, (1990).

Adv. Opt. Photonics (1)

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

Appl. Opt. (6)

Appl. Phys. Lett. (1)

D. S. Mehta and V. Srivastava, “Quantitative phase imaging of human red blood cells using phase-shifting white light interference microscopy with colour fringe analysis,” Appl. Phys. Lett. 101(20), 203701 (2012).
[Crossref]

Int. J. Mach. Tool Manu. (1)

J. C. Wyant and K. Creath, “Advances in interferometric optical profiling,” Int. J. Mach. Tool Manu. 32(1-2), 5–10 (1992).
[Crossref]

Measurement (1)

Z. Lei, X. Liu, L. Chen, W. Lu, and S. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016).
[Crossref]

Opt. Eng. (2)

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

U. P. Kumar, N. Krishna Mohan, M. Kothiyal, and A. Asundi, “Deformation and shape measurement using multiple wavelength microscopic TV holography,” Opt. Eng. 48(2), 023601 (2009).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (2)

U. P. Kumar, W. Haifeng, N. K. Mohan, and M. P. Kothiyal, “White light interferometry for surface profiling with a colour CCD,” Opt. Lasers Eng. 50(8), 1084–1088 (2012).
[Crossref]

A. Saad, Y. Jiang, Y. Liu, and Z. Wang, “The measurement of the diameter change of a piezoelectric transducer cylinder with the white-light interferometry,” Opt. Lasers Eng. 56, 169–172 (2014).
[Crossref]

Opt. Lett. (1)

Optik (1)

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of static and vibrating microsystems using microscopicTV holography,” Optik 122(1), 49–54 (2011).
[Crossref]

Phys. Today (1)

M. Born and E. Wolf, “Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light,” Phys. Today 53(10), 77–78 (2000).
[Crossref]

Proc. SPIE (2)

S. Wang, Z. Gao, M. Li, J. Ye, J. Cheng, Z. Yang, and Q. Yuan, “Design, assembly and calibration of white-light microscopy interferometer,” Proc. SPIE 9677, 96771Q (2015).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. Kothiyal, “Multiple wavelength interferometry for surface profiling,” Proc. SPIE 7063, 70630W (2008).
[Crossref]

Prog. Opt. (1)

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Other (7)

H. Schreiber and J. Bruning, “Phase Shifting Interferometry,” pp. 547–666,(2006).

W. Osten, Optical inspection of Microsystems, (CRC Press, 2007.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (2000).

W. Steinchen and L. Yang, [Digital shearography: Theory and Application of digital speckle pattern shearing interferometry], PM100, SPIE Press, Washington DC, (2003).

J. Schmit and A. Pakuła, “White Light Interferometry,” in: N. Ida and N. Meyendorf, eds., Handbook of Advanced Non-Destructive Evaluation, Springer International Publishing, Cham, pp. 1–47 (2019).

D. Zweig and R. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” 1333, (1990).

R. Bracewell, The Fourier Transform and its Application, (McGrawHill Book Company, 2000).

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Figures (8)

Fig. 1.
Fig. 1. Hilbert transform envelope curve of interference signal
Fig. 2.
Fig. 2. Envelope curve correlation analysis: (a) principle of envelope correlation calculation and (b) correlation coefficients of reference window sequences and series of matching window sequences.
Fig. 3.
Fig. 3. Schematic flow chart of the correction method.
Fig. 4.
Fig. 4. Signal correction process: (a) Interference signal with NUSPs, (b) Ideal interference signal, (c) Correlation coefficients between the envelope curve from (a) and (b), and (d) corrected interference signal.
Fig. 5.
Fig. 5. Comparison of shape recovery result: (a) PLM, (b) FTM, (c) HT-ESM, and (d) cross-section profile at Y = 150.
Fig. 6.
Fig. 6. Experimental measurement of step surface: (a) layout of the Mirau-type WLSI measurement system and (b) interferograms of a standard step surface.
Fig. 7.
Fig. 7. Process of experimental calculation: (a) collected interference signal from CCD, (b) ideal simulated interference signal, (c) correlation coefficients, and (d) corrected interference signal.
Fig. 8.
Fig. 8. Measurement results from different methods: (a) PLM, (b) FTM, (c) HT-ESM, and (d) cross-section profile at Y = 400.

Tables (2)

Tables Icon

Table 1. Comparison of height errors in the simulation results.

Tables Icon

Table 2. Measurement results of standard step surface.

Equations (12)

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

I ( x , y , z ) = I 0 { 1 + e x p [ ( z h ( x , y ) z 0 l c ) 2 ] c o s ( 4 π z h ( x , y ) z 0 λ c + ϕ ( x , y ) ) }
I ( z ) = I 0 ( 1 + e x p [ ( z h z 0 l c ) 2 ] c o s ( 4 π z h z 0 λ c + ϕ ) )
l c = λ c 2 Δ λ
I ( z ) = + I ( τ ) π ( z τ ) d τ
I ( z ) = H [ I ( z ) ]
I ^ ( z ) = I 2 ( z ) + I 2 ( z )
H [ I ( z ) c o s ( ω c z ) ] I ( z ) s i n ( ω c z )
H [ I ( z ) s i n ( ω c z ) ] I ( z ) c o s ( ω c z )
H [ I ( z ) ] e x p [ ( z h z 0 l c ) 2 ] s i n ( 4 π z h z 0 λ c + ϕ )
H [ H [ I ( z ) ] ] e x p [ ( z h z 0 l c ) 2 ] c o s ( 4 π z h z 0 λ c + ϕ )
I ^ ( z ) = e x p [ ( z h z 0 l c ) 2 ]
r s = i = 1 n [ ( I r i I ¯ r ) ( I m i I ¯ m ) ] i = 1 n ( I r i I ¯ r ) 2 i = 1 n ( I m i I ¯ m ) 2