Abstract

We propose a numerical parameter fitting method to determine the time delay between two temporally overlapped echo pulses in terahertz (THz) tomography measurements. The method is based on multiple-regression analysis with the least-squares method and is applied to decrease the minimum paint film thickness for THz paint meters. Applying multiple-regression analysis to paint thickness measurements is five times more sensitive with regard to the minimum thickness than numerical Fourier deconvolution. We apply the proposed method to determine the optical thickness, geometrical thickness, and group refractive index of dry paint film and wet paint film. The proposed method is useful for decreasing the minimum thickness for a THz paint meter and other THz tomography measurements.

© 2007 Optical Society of America

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  1. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, "T-ray imaging," IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996).
    [CrossRef]
  2. D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, "T-ray tomography," Opt. Lett. 22, 904-906 (1997).
    [CrossRef] [PubMed]
  3. A. J. Fitzgerald, B. E. Cole, and P. F. Taday, "Nondestructive analysis of tablet coating thicknesses using terahertz pulsed imaging," J. Pharm. Sci. 94, 177-183 (2005).
    [CrossRef] [PubMed]
  4. R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
    [CrossRef] [PubMed]
  5. D. E. Bray and D. McBride, Nondestructive Testing Techniques (Wiley, 1992).
  6. T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, "A terahertz paint meter for noncontact monitoring of thickness and drying progress in paint film," Appl. Opt. 44, 6849-6856 (2005).
    [CrossRef] [PubMed]
  7. G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
    [CrossRef]
  8. T. Yasuda, T. Yasui, T. Araki, and E. Abraham, "Real-time two-dimensional terahertz tomography of moving objects," Opt. Commun. 267, 128-136 (2006).
    [CrossRef]
  9. Q. Wu and X.-C. Zhang, "Free-space electro-optics sampling of mid-infrared pulses," Appl. Phys. Lett. 71, 1285-1286 (1997).
    [CrossRef]
  10. J. A. Blackburn, "Computer program for multicomponent spectrum analysis using least-squares method," Anal. Chem. 37, 1000-1003 (1965).
    [CrossRef]
  11. D. J. Leggett, "Numerical analysis of multicomponent spectra," Anal. Chem. 49, 276-281 (1971).
    [CrossRef]
  12. T. Yasui and T. Araki, "Dependence of terahertz electric field on electric bias and modulation frequency in pulsed terahertz emission from electrically modulated photoconductive antenna detected with free-space electro-optic sampling," Jpn. J. Appl. Phys. , Part 1 44, 1777-1780 (2005).
    [CrossRef]

2006 (1)

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, "Real-time two-dimensional terahertz tomography of moving objects," Opt. Commun. 267, 128-136 (2006).
[CrossRef]

2005 (3)

T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, "A terahertz paint meter for noncontact monitoring of thickness and drying progress in paint film," Appl. Opt. 44, 6849-6856 (2005).
[CrossRef] [PubMed]

A. J. Fitzgerald, B. E. Cole, and P. F. Taday, "Nondestructive analysis of tablet coating thicknesses using terahertz pulsed imaging," J. Pharm. Sci. 94, 177-183 (2005).
[CrossRef] [PubMed]

T. Yasui and T. Araki, "Dependence of terahertz electric field on electric bias and modulation frequency in pulsed terahertz emission from electrically modulated photoconductive antenna detected with free-space electro-optic sampling," Jpn. J. Appl. Phys. , Part 1 44, 1777-1780 (2005).
[CrossRef]

2002 (1)

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

1997 (2)

D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, "T-ray tomography," Opt. Lett. 22, 904-906 (1997).
[CrossRef] [PubMed]

Q. Wu and X.-C. Zhang, "Free-space electro-optics sampling of mid-infrared pulses," Appl. Phys. Lett. 71, 1285-1286 (1997).
[CrossRef]

1996 (1)

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, "T-ray imaging," IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996).
[CrossRef]

1992 (1)

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

1971 (1)

D. J. Leggett, "Numerical analysis of multicomponent spectra," Anal. Chem. 49, 276-281 (1971).
[CrossRef]

1965 (1)

J. A. Blackburn, "Computer program for multicomponent spectrum analysis using least-squares method," Anal. Chem. 37, 1000-1003 (1965).
[CrossRef]

Abraham, E.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, "Real-time two-dimensional terahertz tomography of moving objects," Opt. Commun. 267, 128-136 (2006).
[CrossRef]

Araki, T.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, "Real-time two-dimensional terahertz tomography of moving objects," Opt. Commun. 267, 128-136 (2006).
[CrossRef]

T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, "A terahertz paint meter for noncontact monitoring of thickness and drying progress in paint film," Appl. Opt. 44, 6849-6856 (2005).
[CrossRef] [PubMed]

T. Yasui and T. Araki, "Dependence of terahertz electric field on electric bias and modulation frequency in pulsed terahertz emission from electrically modulated photoconductive antenna detected with free-space electro-optic sampling," Jpn. J. Appl. Phys. , Part 1 44, 1777-1780 (2005).
[CrossRef]

Arnone, D. D.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

Blackburn, J. A.

J. A. Blackburn, "Computer program for multicomponent spectrum analysis using least-squares method," Anal. Chem. 37, 1000-1003 (1965).
[CrossRef]

Boivin, L.

Bray, D. E.

D. E. Bray and D. McBride, Nondestructive Testing Techniques (Wiley, 1992).

Busse, G.

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

Cole, B. E.

A. J. Fitzgerald, B. E. Cole, and P. F. Taday, "Nondestructive analysis of tablet coating thicknesses using terahertz pulsed imaging," J. Pharm. Sci. 94, 177-183 (2005).
[CrossRef] [PubMed]

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

Fitzgerald, A. J.

A. J. Fitzgerald, B. E. Cole, and P. F. Taday, "Nondestructive analysis of tablet coating thicknesses using terahertz pulsed imaging," J. Pharm. Sci. 94, 177-183 (2005).
[CrossRef] [PubMed]

Hunsche, S.

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, "T-ray imaging," IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996).
[CrossRef]

Karpen, W.

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

Leggett, D. J.

D. J. Leggett, "Numerical analysis of multicomponent spectra," Anal. Chem. 49, 276-281 (1971).
[CrossRef]

Linfield, E. H.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

McBride, D.

D. E. Bray and D. McBride, Nondestructive Testing Techniques (Wiley, 1992).

Mittleman, D. M.

D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, "T-ray tomography," Opt. Lett. 22, 904-906 (1997).
[CrossRef] [PubMed]

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, "T-ray imaging," IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996).
[CrossRef]

Nuss, M. C.

D. M. Mittleman, S. Hunsche, L. Boivin, and M. C. Nuss, "T-ray tomography," Opt. Lett. 22, 904-906 (1997).
[CrossRef] [PubMed]

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, "T-ray imaging," IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996).
[CrossRef]

Pepper, M.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

Pye, R. J.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

Sawanaka, K.

Taday, P. F.

A. J. Fitzgerald, B. E. Cole, and P. F. Taday, "Nondestructive analysis of tablet coating thicknesses using terahertz pulsed imaging," J. Pharm. Sci. 94, 177-183 (2005).
[CrossRef] [PubMed]

Wallace, V. P.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

Woodward, R. M.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

Wu, D.

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

Wu, Q.

Q. Wu and X.-C. Zhang, "Free-space electro-optics sampling of mid-infrared pulses," Appl. Phys. Lett. 71, 1285-1286 (1997).
[CrossRef]

Yasuda, T.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, "Real-time two-dimensional terahertz tomography of moving objects," Opt. Commun. 267, 128-136 (2006).
[CrossRef]

T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, "A terahertz paint meter for noncontact monitoring of thickness and drying progress in paint film," Appl. Opt. 44, 6849-6856 (2005).
[CrossRef] [PubMed]

Yasui, T.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, "Real-time two-dimensional terahertz tomography of moving objects," Opt. Commun. 267, 128-136 (2006).
[CrossRef]

T. Yasui and T. Araki, "Dependence of terahertz electric field on electric bias and modulation frequency in pulsed terahertz emission from electrically modulated photoconductive antenna detected with free-space electro-optic sampling," Jpn. J. Appl. Phys. , Part 1 44, 1777-1780 (2005).
[CrossRef]

T. Yasui, T. Yasuda, K. Sawanaka, and T. Araki, "A terahertz paint meter for noncontact monitoring of thickness and drying progress in paint film," Appl. Opt. 44, 6849-6856 (2005).
[CrossRef] [PubMed]

Zhang, X.-C.

Q. Wu and X.-C. Zhang, "Free-space electro-optics sampling of mid-infrared pulses," Appl. Phys. Lett. 71, 1285-1286 (1997).
[CrossRef]

Anal. Chem. (2)

J. A. Blackburn, "Computer program for multicomponent spectrum analysis using least-squares method," Anal. Chem. 37, 1000-1003 (1965).
[CrossRef]

D. J. Leggett, "Numerical analysis of multicomponent spectra," Anal. Chem. 49, 276-281 (1971).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Q. Wu and X.-C. Zhang, "Free-space electro-optics sampling of mid-infrared pulses," Appl. Phys. Lett. 71, 1285-1286 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, "T-ray imaging," IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996).
[CrossRef]

J. Appl. Phys. (1)

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

J. Pharm. Sci. (1)

A. J. Fitzgerald, B. E. Cole, and P. F. Taday, "Nondestructive analysis of tablet coating thicknesses using terahertz pulsed imaging," J. Pharm. Sci. 94, 177-183 (2005).
[CrossRef] [PubMed]

Jpn. J. Appl. Phys. (1)

T. Yasui and T. Araki, "Dependence of terahertz electric field on electric bias and modulation frequency in pulsed terahertz emission from electrically modulated photoconductive antenna detected with free-space electro-optic sampling," Jpn. J. Appl. Phys. , Part 1 44, 1777-1780 (2005).
[CrossRef]

Opt. Commun. (1)

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, "Real-time two-dimensional terahertz tomography of moving objects," Opt. Commun. 267, 128-136 (2006).
[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (1)

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47, 3853-3863 (2002).
[CrossRef] [PubMed]

Other (1)

D. E. Bray and D. McBride, Nondestructive Testing Techniques (Wiley, 1992).

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Figures (9)

Fig. 1
Fig. 1

Principle of the THz paint meter. (a) Single-layer paint film and (b) temporal waveforms of THz echo pulse.

Fig. 2
Fig. 2

Experimental setup. ML Ti:S laser: mode-locked Ti:sapphire laser; BS: beam splitter; L: objective lens; THz-EM: photoconductive antenna for THz generation; Si-L: hemispherical silicon lens; OAP1, OAP2, OAP3: off-axis parabolic mirrors; Si-BS: hemispherical silicon beam splitter; THz-DT: photoconductive antenna for THz detection.

Fig. 3
Fig. 3

Temporal waveform of an incident THz pulse [ E in ( t ) ] obtained when setting an aluminum substrate at the sample position.

Fig. 4
Fig. 4

Temporal waveform of the THz echo pulse of a single-layer dry paint film with 60 μm thickness. (a) Signal measured with THz paint meter, (b) signal simulated by multiple-regression analysis, and (c) residual between the measured signal and simulated signal.

Fig. 5
Fig. 5

Relationship between actual paint thickness measured with a contact-type thickness meter and simulated painting thickness by multiple regression analysis for five dry paint film samples with different thicknesses. Solid line represents d = d sim .

Fig. 6
Fig. 6

Change of simulated paint thickness for the 20   μm paint film when changing the S∕N ratio.

Fig. 7
Fig. 7

Parameters k 0 and k 2 for five dry paint film samples with different thickness.

Fig. 8
Fig. 8

Temporal evolution of THz echo pulse signal during the wet-to-dry transformation of a paint film. Temporal waveforms of the THz echo pulses were measured every 1 min after applying the paint.

Fig. 9
Fig. 9

Temporal change of each parameter for wet paint film with respect to elapsed time. (a) Deviations of Δ t and k 1 from the initial values at 0 min and (b) average group refractive index.

Tables (2)

Tables Icon

Table 1 Result of Multiple Regression Analysis for Dry Paint Films

Tables Icon

Table 2 Result of Multiple Regression Analysis for a Wet Paint Film

Equations (10)

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

d min = ( c Δ T ) / ( 2 n g ) ,
d = ( c Δ t ) / ( 2 n g ) .
E 1 st ( t ) = R × E in ( t + Δ t 1 ) ,
E 2nd ( t ) = ( 1 R ) 2 × exp ( 2 d × A ) × E in ( t + Δ t 2 ) α × E i n ( t + Δ t 2 ) ,
E all ( t ) = E 1 st ( t ) + E 2nd ( t ) R × E i n ( t + Δ t 1 ) + α × E i n ( t + Δ t 2 ) .
Y = k 0 × E i n ( t k 1 ) + k 2 × E i n ( t k 3 ) ,
k 0 , k 2 > 0
k 1 < k 3
k 0 + k 2 = 1 .
Δ d opt ( t ) = N g ( t ) Δ d geo ( t ) ,

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