Abstract

White light interferometry is a widely used tool to extend the unambiguous measurement range of a monochromatic interferometer. In this work, we discuss Hilbert transformation analysis of a single white light interferogram acquired with a single-chip color CCD camera for step height measurement which lies beyond the unambiguous range of the monochromatic interferometry. The color interferogram is decomposed and phase maps for red, green, and blue components are calculated independently using Hilbert transformation. This procedure makes the measurement faster, simpler, and cost-effective. The usefulness of the technique is demonstrated on micro-sample.

© 2014 Optical Society of America

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References

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  2. D. Malacara, Optical Shop Testing (John Wiley & Sons Inc, 1992), vol. II, pp. 501–598.
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  11. U. Paul Kumar, B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “Two wavelength interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
    [Crossref]
  12. U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of discontinuous surfaces using multiple-wavelength interferometry,” Opt. Eng. 48(7), 073603 (2009).
    [Crossref]
  13. U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Multiple wavelength interferometry for surface profiling,” Proc. SPIE 7063, 70630W (2008).
    [Crossref]
  14. 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]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2012 (2)

U. Paul Kumar, W. Haifeng, N. Krishna Mohan, and M. P. Kothiyal, “White light Interferometry for surface profiling with 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 (1)

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Characterization of micro-lenses based on single interferogram analysis using Hilbert transformation,” Opt. Commun. 284(21), 5084–5092 (2011).
[Crossref]

2010 (1)

2009 (4)

S. K. Debnath, M. P. Kothiyal, and S. W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: Comparative study of single-frame techniques,” Opt. Lasers Eng. 47(11), 1125–1130 (2009).
[Crossref]

U. Paul Kumar, B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “Two wavelength interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of discontinuous surfaces using multiple-wavelength interferometry,” Opt. Eng. 48(7), 073603 (2009).
[Crossref]

N. Krishna Mohan and P. K. Rastogi, “Recent developments in Interferometry for Microsystems Metrology,” Opt. Lasers Eng. 47(2), 199–202 (2009).
[Crossref]

2008 (1)

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

2007 (1)

2006 (1)

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

2003 (3)

2001 (2)

1996 (1)

P. Sandoz, G. Tribillon, and H. Perrin, “High resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white light interferograms,” J. Mod. Opt. 43(4), 701–708 (1996).
[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(2), 389–401 (1995).
[Crossref]

1993 (1)

1992 (1)

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

1987 (1)

1985 (1)

1973 (1)

Bhaduri, B.

U. Paul Kumar, B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “Two wavelength interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

Bone, D. J.

Cheng, Y. Y.

Colbourne, P.

L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt. Eng. 42(5), 1417–1426 (2003).
[Crossref]

Creath, K.

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

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

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]

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]

Debnath, S. K.

S. K. Debnath, M. P. Kothiyal, and S. W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: Comparative study of single-frame techniques,” Opt. Lasers Eng. 47(11), 1125–1130 (2009).
[Crossref]

S. K. Debnath and M. P. Kothiyal, “Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer,” Appl. Opt. 46(22), 5103–5109 (2007).
[Crossref] [PubMed]

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]

Haifeng, W.

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

Hariharan, P.

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

Kadono, H.

Khomenko, A. V.

Kim, S. W.

S. K. Debnath, M. P. Kothiyal, and S. W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: Comparative study of single-frame techniques,” Opt. Lasers Eng. 47(11), 1125–1130 (2009).
[Crossref]

Kothiyal, M. P.

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

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Characterization of micro-lenses based on single interferogram analysis using Hilbert transformation,” Opt. Commun. 284(21), 5084–5092 (2011).
[Crossref]

U. P. Kumar, N. K. Mohan, and M. P. Kothiyal, “Time average vibration fringe analysis using Hilbert transformation,” Appl. Opt. 49(30), 5777–5786 (2010).
[Crossref] [PubMed]

S. K. Debnath, M. P. Kothiyal, and S. W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: Comparative study of single-frame techniques,” Opt. Lasers Eng. 47(11), 1125–1130 (2009).
[Crossref]

U. Paul Kumar, B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “Two wavelength interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of discontinuous surfaces using multiple-wavelength interferometry,” Opt. Eng. 48(7), 073603 (2009).
[Crossref]

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

S. K. Debnath and M. P. Kothiyal, “Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer,” Appl. Opt. 46(22), 5103–5109 (2007).
[Crossref] [PubMed]

Krishna Mohan, N.

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

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Characterization of micro-lenses based on single interferogram analysis using Hilbert transformation,” Opt. Commun. 284(21), 5084–5092 (2011).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of discontinuous surfaces using multiple-wavelength interferometry,” Opt. Eng. 48(7), 073603 (2009).
[Crossref]

U. Paul Kumar, B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “Two wavelength interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

N. Krishna Mohan and P. K. Rastogi, “Recent developments in Interferometry for Microsystems Metrology,” Opt. Lasers Eng. 47(2), 199–202 (2009).
[Crossref]

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

Kumar, U. P.

Larkin, K. G.

Madjarova, V. D.

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]

Mercado, J. T.

Mohan, N. K.

Oldfield, M. A.

Paul Kumar, U.

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

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Characterization of micro-lenses based on single interferogram analysis using Hilbert transformation,” Opt. Commun. 284(21), 5084–5092 (2011).
[Crossref]

U. Paul Kumar, B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “Two wavelength interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of discontinuous surfaces using multiple-wavelength interferometry,” Opt. Eng. 48(7), 073603 (2009).
[Crossref]

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

Perrin, H.

P. Sandoz, G. Tribillon, and H. Perrin, “High resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white light interferograms,” J. Mod. Opt. 43(4), 701–708 (1996).
[Crossref]

Pförtner, A.

Polhemus, C.

Rastogi, P. K.

N. Krishna Mohan and P. K. Rastogi, “Recent developments in Interferometry for Microsystems Metrology,” Opt. Lasers Eng. 47(2), 199–202 (2009).
[Crossref]

Sandoz, P.

P. Sandoz, G. Tribillon, and H. Perrin, “High resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white light interferograms,” J. Mod. Opt. 43(4), 701–708 (1996).
[Crossref]

Schmit, J.

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

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]

Toyooka, S.

Tribillon, G.

P. Sandoz, G. Tribillon, and H. Perrin, “High resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white light interferograms,” J. Mod. Opt. 43(4), 701–708 (1996).
[Crossref]

Wyant, J. C.

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

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

Yang, L.

L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt. Eng. 42(5), 1417–1426 (2003).
[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. Tools Manuf. (1)

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

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

P. Sandoz, G. Tribillon, and H. Perrin, “High resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white light interferograms,” J. Mod. Opt. 43(4), 701–708 (1996).
[Crossref]

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)

Opt. Commun. (1)

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Characterization of micro-lenses based on single interferogram analysis using Hilbert transformation,” Opt. Commun. 284(21), 5084–5092 (2011).
[Crossref]

Opt. Eng. (3)

U. Paul Kumar, N. Krishna Mohan, and M. P. Kothiyal, “Measurement of discontinuous surfaces using multiple-wavelength interferometry,” Opt. Eng. 48(7), 073603 (2009).
[Crossref]

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

L. Yang and P. Colbourne, “Digital laser micro-interferometer and its applications,” Opt. Eng. 42(5), 1417–1426 (2003).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (4)

S. K. Debnath, M. P. Kothiyal, and S. W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: Comparative study of single-frame techniques,” Opt. Lasers Eng. 47(11), 1125–1130 (2009).
[Crossref]

N. Krishna Mohan and P. K. Rastogi, “Recent developments in Interferometry for Microsystems Metrology,” Opt. Lasers Eng. 47(2), 199–202 (2009).
[Crossref]

U. Paul Kumar, B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “Two wavelength interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

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

Opt. Lett. (1)

Proc. SPIE (1)

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

Other (4)

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

E. Wolf, Progress in Optics (Elsevier, 1988), vol. XXVI, pp. 349–393.

D. Malacara, Optical Shop Testing (John Wiley & Sons Inc, 1992), vol. II, pp. 501–598.

L. H. Stefan, Hilbert Transforms in Signal Processing (Artech House Inc, 1996).

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Figures (4)

Fig. 1
Fig. 1 Eliminating the non-uniform signal bias using min-max method: (a) The original signal-p, envelopes of maxima and minimas and the mean ‘s’, (b) R, G, B components after removing non uniform bias, and (c) HT generated signals corresponding to the signals in Fig. 1(b).
Fig. 2
Fig. 2 Schematic of the microscopic white light interferometric system: CL, collimating lens; BS, Beam splitter; MO, Microscopic objective; RM, Reference mirror; PBS, Plate beam splitter.
Fig. 3
Fig. 3 Fringe analysis using Hilbert transformation: (a) white light interferogram, (b) interferogram at λ1, and (c) wrapped phase calculated.
Fig. 4
Fig. 4 Surface profile analysis using the fringe order method: (a) The linear fitting of the adjusted phase values measured at three wavelengths at a given pixel, (b) 2π ambiguity corrected single wavelength 3-D surface profile of an etched silicon sample using the phase at λ1, and (c) line scan profile showing the step height 645 nm.

Equations (12)

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

h 0 (x,y)= h b (x,y)+ h b (x,y)γ(x,y)cosδ(x,y)
h 1 =Im[ hilbert{ h 0 } ]= h b γsin(δ)
h 2 =Im[ hilbert{ h 1 } ]= h b γcos(δ)
δ= arctan( h 1 h 2 )
cos(δ)= 2 h 0 h 0max h 0min h 0max h 0min =C
S=Im[ hilbert{ C } ]=sin(δ)
δ= arctan( S C )
δ ' =δ+ξ
δ i =4π k i z
z a = 1 4π ( Δδ Δk )
n i =round( δ i 4π k i z a 2π )
z= λ i 4π ( δ i 2π n i )

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