Abstract

Thin-film sensing with a film thickness much less than a wavelength is an important challenge in conventional transmission-mode terahertz time-domain spectroscopy (THz-TDS). Since the interaction length between terahertz waves and a sample film is short, a small change in the transmitted signal compared with the reference is considerably obscured by system uncertainties. In this article, several possible thin-film measurement procedures are carefully investigated. It is suggested that an alternating sample and reference measurement approach is most robust for thin-film sensing. In addition, a closed-form criterion is developed to determine the critical thickness, i.e., the minimal thickness of a film unambiguously detectable by transmission-mode THz-TDS. The analysis considers influences from the Fresnel transmission at interfaces and the Fabry-Pérot reflections, in addition to the propagation across the film. The experimental results show that typical THz-TDS systems can detect polymer films with a thickness down to a few microns, two orders of magnitude less than the wavelength. For reasonably accurate characterization, it is recommended that the film thickness be at least ten times above this limit. The analysis is readily extended to biomolecular and semiconductor films. The criterion can be used to estimate the system-dependent performance in thin-film sensing applications, and can help to ascertain whether an alternative terahertz sensing modality is necessary.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2012

J. F. O’Hara, W. Withayachumnankul, I. Al-Naib, “A review on thin-film sensing with terahertz waves,” J. Infrared Millimeter Terahertz Waves 33, 245–291 (2012).
[CrossRef]

2009

M. Scheller, C. Jansen, M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282, 1304–1306 (2009).
[CrossRef]

2008

2007

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

2006

2005

2003

W. Zhang, A. K. Azad, D. Grischkowsky, “Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN,” Appl. Phys. Lett. 82, 2841–2843 (2003).
[CrossRef]

2002

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

K.-S. Lee, T.-M. Lu, X.-C. Zhang, “Tera tool,” IEEE Circuits Devices Mag. 18, 23–28 (2002).

2001

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

T. D. Dorney, R. G. Baraniuk, D. M. Mittleman, “Material parameter estimation with terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18, 1562–1571 (2001).
[CrossRef]

2000

Z. Jiang, M. Li, X.-C. Zhang, “Dielectric constant measurement of thin films by differential time-domain spectroscopy,” Appl. Phys. Lett. 76, 3221–3223 (2000).
[CrossRef]

1996

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2, 739–746 (1996).
[CrossRef]

1990

Abbott, D.

W. Withayachumnankul, B. M. Fischer, D. Abbott, “Material thickness optimization for transmission-mode terahertz time-domain spectroscopy,” Opt. Express 16, 7382–7396 (2008).
[CrossRef] [PubMed]

W. Withayachumnankul, B. Fischer, H. Lin, D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B 25, 1059–1072 (2008).
[CrossRef]

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Al-Naib, I.

J. F. O’Hara, W. Withayachumnankul, I. Al-Naib, “A review on thin-film sensing with terahertz waves,” J. Infrared Millimeter Terahertz Waves 33, 245–291 (2012).
[CrossRef]

Atakaramians, S.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Avitzour, Y.

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

Azad, A. K.

A. K. Azad, J. Dai, W. Zhang, “Transmission properties of terahertz pulses through subwavelength double split-ring resonators,” Opt. Lett. 31, 634–636 (2006).
[CrossRef] [PubMed]

W. Zhang, A. K. Azad, D. Grischkowsky, “Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN,” Appl. Phys. Lett. 82, 2841–2843 (2003).
[CrossRef]

Balakrishnan, J.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Baraniuk, R. G.

Chen, Y.

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

Cohen, G.

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

Coutaz, J.-L.

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2, 739–746 (1996).
[CrossRef]

Dai, J.

Dorney, T. D.

Duvillaret, L.

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2, 739–746 (1996).
[CrossRef]

Earley, S.

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

Fattinger, C.

Ferguson, B.

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Fischer, B.

Fischer, B. M.

Garet, F.

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2, 739–746 (1996).
[CrossRef]

Geltner, I.

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

Grischkowsky, D.

W. Zhang, A. K. Azad, D. Grischkowsky, “Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN,” Appl. Phys. Lett. 82, 2841–2843 (2003).
[CrossRef]

D. Grischkowsky, S. Keiding, M. van Exter, C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

Hashimshony, D.

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

Jansen, C.

M. Scheller, C. Jansen, M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282, 1304–1306 (2009).
[CrossRef]

Jepsen, P. U.

Jiang, Z.

Z. Jiang, M. Li, X.-C. Zhang, “Dielectric constant measurement of thin films by differential time-domain spectroscopy,” Appl. Phys. Lett. 76, 3221–3223 (2000).
[CrossRef]

Jones, I.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Keiding, S.

Koch, M.

M. Scheller, C. Jansen, M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282, 1304–1306 (2009).
[CrossRef]

Lee, K.-S.

K.-S. Lee, T.-M. Lu, X.-C. Zhang, “Tera tool,” IEEE Circuits Devices Mag. 18, 23–28 (2002).

Li, M.

Z. Jiang, M. Li, X.-C. Zhang, “Dielectric constant measurement of thin films by differential time-domain spectroscopy,” Appl. Phys. Lett. 76, 3221–3223 (2000).
[CrossRef]

Lin, H.

W. Withayachumnankul, B. Fischer, H. Lin, D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B 25, 1059–1072 (2008).
[CrossRef]

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Liu, H.

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Liu, H.-B.

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

Lu, T.-M.

K.-S. Lee, T.-M. Lu, X.-C. Zhang, “Tera tool,” IEEE Circuits Devices Mag. 18, 23–28 (2002).

MacColl, R.

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Mannella, C. A.

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Menikh, A.

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Mickan, S. P.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Mittleman, D. M.

Munch, J.

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Naftaly, M.

W. Withayachumnankul, M. Naftaly, “Fundamentals of measurement in terahertz time-domain spectroscopy,” J. Infrared Millimeter Terahertz Waves (2013), accepted for publication.
[CrossRef]

Ng, B. W.-H.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

O’Hara, J. F.

J. F. O’Hara, W. Withayachumnankul, I. Al-Naib, “A review on thin-film sensing with terahertz waves,” J. Infrared Millimeter Terahertz Waves 33, 245–291 (2012).
[CrossRef]

Plopper, G.

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

Png, G. M.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Scheller, M.

M. Scheller, C. Jansen, M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282, 1304–1306 (2009).
[CrossRef]

Smith, C.

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

Ung, B. S. Y.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

van Exter, M.

Withayachumnankul, W.

J. F. O’Hara, W. Withayachumnankul, I. Al-Naib, “A review on thin-film sensing with terahertz waves,” J. Infrared Millimeter Terahertz Waves 33, 245–291 (2012).
[CrossRef]

W. Withayachumnankul, B. M. Fischer, D. Abbott, “Material thickness optimization for transmission-mode terahertz time-domain spectroscopy,” Opt. Express 16, 7382–7396 (2008).
[CrossRef] [PubMed]

W. Withayachumnankul, B. Fischer, H. Lin, D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B 25, 1059–1072 (2008).
[CrossRef]

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

W. Withayachumnankul, M. Naftaly, “Fundamentals of measurement in terahertz time-domain spectroscopy,” J. Infrared Millimeter Terahertz Waves (2013), accepted for publication.
[CrossRef]

W. Withayachumnankul, “Limitation in thin-film detection with transmission-mode terahertz time-domain spectroscopy” (2011), ArXiv:1111.3498v1.

Yin, X. X.

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Zhang, W.

A. K. Azad, J. Dai, W. Zhang, “Transmission properties of terahertz pulses through subwavelength double split-ring resonators,” Opt. Lett. 31, 634–636 (2006).
[CrossRef] [PubMed]

W. Zhang, A. K. Azad, D. Grischkowsky, “Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN,” Appl. Phys. Lett. 82, 2841–2843 (2003).
[CrossRef]

Zhang, X.-C.

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

K.-S. Lee, T.-M. Lu, X.-C. Zhang, “Tera tool,” IEEE Circuits Devices Mag. 18, 23–28 (2002).

Z. Jiang, M. Li, X.-C. Zhang, “Dielectric constant measurement of thin films by differential time-domain spectroscopy,” Appl. Phys. Lett. 76, 3221–3223 (2000).
[CrossRef]

Zigler, A.

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

Appl. Phys. Lett.

Z. Jiang, M. Li, X.-C. Zhang, “Dielectric constant measurement of thin films by differential time-domain spectroscopy,” Appl. Phys. Lett. 76, 3221–3223 (2000).
[CrossRef]

W. Zhang, A. K. Azad, D. Grischkowsky, “Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN,” Appl. Phys. Lett. 82, 2841–2843 (2003).
[CrossRef]

Biosens. Bioelectron.

H.-B. Liu, G. Plopper, S. Earley, Y. Chen, B. Ferguson, X.-C. Zhang, “Sensing minute changes in biological cell monolayers with THz differential time-domain spectroscopy,” Biosens. Bioelectron. 22, 1075–1080 (2007).
[CrossRef]

IEEE Circuits Devices Mag.

K.-S. Lee, T.-M. Lu, X.-C. Zhang, “Tera tool,” IEEE Circuits Devices Mag. 18, 23–28 (2002).

IEEE J. Sel. Top. Quantum Electron.

L. Duvillaret, F. Garet, J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2, 739–746 (1996).
[CrossRef]

J. Appl. Phys.

D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, C. Smith, “Characterization of the electrical properties and thickness of thin epitaxial semiconductor layers by THz reflection spectroscopy,” J. Appl. Phys. 90, 5778–5781 (2001).
[CrossRef]

J. Infrared Millimeter Terahertz Waves

J. F. O’Hara, W. Withayachumnankul, I. Al-Naib, “A review on thin-film sensing with terahertz waves,” J. Infrared Millimeter Terahertz Waves 33, 245–291 (2012).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

M. Scheller, C. Jansen, M. Koch, “Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282, 1304–1306 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

S. P. Mickan, A. Menikh, H. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, X.-C. Zhang, “Label-free bioaffinity detection using terahertz technology,” Phys. Med. Biol. 47, 3789–3795 (2002).
[CrossRef] [PubMed]

Proc. IEEE

W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[CrossRef]

Other

W. Withayachumnankul, M. Naftaly, “Fundamentals of measurement in terahertz time-domain spectroscopy,” J. Infrared Millimeter Terahertz Waves (2013), accepted for publication.
[CrossRef]

ISO, Guide to the Expression of Uncertainty in Measurement (GUM), 1st ed. (ISO, 1993).

W. Withayachumnankul, “Limitation in thin-film detection with transmission-mode terahertz time-domain spectroscopy” (2011), ArXiv:1111.3498v1.

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Figures (7)

Fig. 1
Fig. 1

Typical measurement procedures. (a) Case 1: measuring the same target consecutively and then averaging. (b) Case 2: measuring the reference and sample alternately and then normalizing each adjacent pair before averaging. (c) Case 3: measuring the reference and sample alternately and then averaging each group. The dotted arrows denote the time progress, during which a number of measurements are conducted. The letters ‘R’ and ‘S’ represent reference and sample measurements, respectively. The amplitude values are for an explanatory purpose only. The shaded areas denote the confidence intervals, which are proportional to the standard deviations.

Fig. 2
Fig. 2

Confidence intervals in the detection scheme. (a) The confidence intervals of the sample and reference measurements are well separated. The presence of a sample is confirmed. (b) The two confidence intervals partially overlap. The presence of a sample is undecided.

Fig. 3
Fig. 3

Different geometries for thin-film measurement. (a) Sample on top of the substrate, typically obtained from spin coating, chemical vapor deposition, or molecular binding; 0 = 1. (b) Sample embedded in the substrate; 0 = s. (c) Freestanding film; 0 = s = 1. In the first two cases, the substrate is considered thick enough to allow time-gating of the internal reflections originating therein.

Fig. 4
Fig. 4

Schematic diagram of modified THz-TDS setup with an 8-F confocal geometry. The notation M1,..., M4 denote the off-axis parabolic mirrors. The terahertz beam with a frequency-independent beam waist is obtained between mirrors M2 and M3.

Fig. 5
Fig. 5

Critical film thicknesses for the THz-TDS system under consideration. (a) Critical thicknesses, lc,p, derived from the phase component of four different datasets by using Eq. (13), and (b) Critical thicknesses, lc,m, derived from the amplitude component by using Eq. (18). The indices annotate the number designator of the dataset in use.

Fig. 6
Fig. 6

Phase difference compared with confidence interval. The phase difference is obtained from the averaged sample and reference phase spectra, |arg(Esam) − arg(Eref)|, and the confidence interval is from a geometrical combination of their corresponding confidence intervals, k P s arg ( E sam ) 2 / N sam + s arg ( E ref ) 2 / N ref, where kP = 1. The numbers indicate different film thicknesses.

Fig. 7
Fig. 7

Optical parameters of the photoresist and silicon. (a) Refractive index, n, and (b) extinction coefficient, κ. The shaded areas indicate unreliable parts of the data. At 1 THz, the complex refractive index for the photoresist equals 1.6 − 0.08 j, and the refractive index for the silicon is 3.44 with negligible loss.

Tables (1)

Tables Icon

Table 1 Minimum detectable film thicknesses for some common materials. The phase-based and amplitude-based minimum thicknesses lc,p and lc,m are determined by using Eqs. (13) and (18), respectively. For a substrate-backed film, the reference is assumed to be a bare substrate, as illustrated in Fig. 3(a). For a freestanding film, the reference is assumed to be free space, as in Fig. 3(c). For all the calculations, sarg(Eref) = 3.2 × 10−2 rad, sln|Eref| = 3.8 × 10−3, Nref = 20, ω = 2π × 1012 rad, and kP = 1.

Equations (32)

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x ¯ k P N s x μ x x ¯ + k P N s x ,
Δ n ( ω ) > k P s n , E sam 2 N sam + s n , E ref 2 N ref + s n , l 2 N l , or
Δ κ ( ω ) > k P s κ , E sam 2 N sam + s κ , E ref 2 N ref + s κ , l 2 N l ,
Δ n ( ω ) > k P N s n ( ω ) , and
Δ κ ( ω ) > k P N s κ ( ω ) ,
Δ n ( ω ) > Δ κ ( ω ) , and
s n ( ω ) s κ ( ω ) .
E ref ( ω ) = τ a s τ s a exp ( j n ^ 0 ω l c ) exp ( j n ^ s ω l s c ) ,
E sam ( ω ) = τ a f τ f s τ s a exp ( j n ^ f ω l c ) exp ( j n ^ s ω l s c ) FP f ,
FP f = [ 1 ρ f a ρ f s exp ( j 2 n ^ f ω l c ) ] 1 ,
H ( ω ) = E sam ( ω ) / E ref ( ω ) = τ a f τ f s τ a s exp [ j ( n ^ f n ^ 0 ) ω l c ] FP f .
arg ( H ( ω ) ) = arg ( E sam ) arg ( E ref ) = arg ( τ total ) Δ n ω l c + arg ( FP f ) ,
| arg ( E sam ) arg ( E ref ) | > k P s arg ( E sam ) 2 N sam + s arg ( E ref ) 2 N ref .
| arg ( H ) | > k P 2 N ref s arg ( E ref ) .
| Δ n ω l c arg ( τ total ) arg ( FP f ) | > k P 2 N ref s arg ( E ref ) .
| Δ n 2 n f ( ρ total ) ( ρ total ) 1 | ω l c > k P 2 N ref s arg ( E ref ) ,
l c , p > k P | Δ n 2 n f ( ρ total ) / [ ( ρ total ) 1 ] | c ω 2 N ref s arg ( E ref ) ,
| ln | E sam | ln | E ref | | > k P s ln | E sam | 2 N sam + s ln | E ref | 2 N ref .
| ln | H ( ω ) | | > k P 2 N ref s ln | E ref | .
ln | H ( ω ) | = ln | τ total | Δ κ ω l c + ln | FP f | ,
| ln | τ total | Δ κ ω l c + ln | FP f | | > k P 2 N ref s ln | E ref | .
l c , m > k P | Δ κ | c ω 2 N ref s ln | E ref | .
arg ( FP f ) = arg [ 1 ρ total exp ( j k l ) ] 1 ,
arg ( FP f ) 0 = arg ( 1 ρ total ) = arg ( τ total ) ,
arg ( FP f ) = arctan [ ( FP f ) / ( FP f ) ] arctan [ ( ρ total ) sin k l ( ρ total ) cos k l 1 ] .
arg ( FP f ) 1 l d d l arctan [ ( ρ total ) sin k l ( ρ total ) cos k l 1 ] l = 0 2 n f ω l c ( ρ total ) ( ρ total 1 ) .
arg ( FP f ) arg ( τ total ) + 2 n f ω l c ( ρ total ) ( ρ total ) 1 .
ln | FP f | = ln | 1 ρ total exp ( j k l ) | 1 .
ln | FP f | 0 = ln | 1 ρ total | 1 = ln | τ total | .
ln | FP f | 1 2 ln [ ( 1 ( ρ total ) cos k l ) 2 + ( ( ρ total ) sin k l ) 2 ] .
ln | FP f | 1 l d d l { 1 2 ln [ ( 1 ( ρ total ) cos k l ) 2 + ( ( ρ total ) sin k l ) 2 ] } l = 0 0 .
ln | FP f | ln | τ total | .

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