Abstract

The thickness of a sample material for a transmission-mode terahertz time-domain spectroscopy (THz-TDS) measurement is the subject of interest in this paper. A sample that is too thick or too thin can raise the problem of measurement uncertainty. Although greater thickness allows the terahertz radiation—or T-rays—to interact more with bulk material, the SNR rolls off with thickness due to signal attenuation. A sample that is too thin renders itself nearly invisible to T-rays, in such a way that the system can hardly sense the difference between the sample and a free space path. The optimal trade-off is analyzed and revealed in this paper, where our approach is to find the optimal thickness that results in the minimal uncertainty of measured optical constants. The derived model for optimal thickness is supported by the results from experiments performed with polyvinyl chloride (PVC), high-density polyethylene (HDPE), and lactose samples.

©2008 Optical Society of America

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  1. D. Abbott and X.-C. Zhang, “Scanning the issue: T-ray imaging, sensing, and retection,” Proc. IEEE 95, 1509–1513 (2007).
    [Crossref]
  2. R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. 76, 3191–3193 (2000).
    [Crossref]
  3. K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85, 863–865 (2004).
    [Crossref]
  4. Z. Jiang and X.-C. Zhang, “Measurement of spatio-temporal terahertz field distribution by using chirped pulse technology,” IEEE J. Quantum Electron. 36, 1214–1222 (2000).
    [Crossref]
  5. A. Bartels, A. Thoma, C. Janke, T. Dekorsy, A. Dreyhaupt, S. Winnerl, and M. Helm, “High-resolution THz spectrometer with kHz scan rates,” Opt. Express 14, 430–437 (2006). URL http://www.opticsexpr“ess.org/abstract.cfm?URI=oe-14-1-430.
    [Crossref] [PubMed]
  6. T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
    [Crossref]
  7. R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
    [Crossref] [PubMed]
  8. C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
    [Crossref] [PubMed]
  9. S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
    [Crossref]
  10. M. van Exter, C. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. 14, 1128–1130 (1989).
    [Crossref]
  11. 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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
    [Crossref]
  12. M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microw. Theory Tech. 38, 1684–1691 (1990).
    [Crossref]
  13. A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. 4, 179–184 (1998).
    [Crossref]
  14. N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
    [Crossref]
  15. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
    [Crossref]
  16. P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29–31 (2005).
    [Crossref] [PubMed]
  17. W. Feller, “The fundamental limit theorems in probability,” Bull. Amer. Math. Soc. 51, 800–832 (1945).
    [Crossref]
  18. H. F. Trotter, “An elementary proof of the central limit theorem,” Arch. Math. 10, 226–234 (1959).
    [Crossref]
  19. W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press).
    [Crossref]
  20. L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
    [Crossref]
  21. B. M. Fischer, “Broadband THz time-domain spectroscopy of biomolecules — A comprehensive study of the dielectric properties of biomaterials in the far-infrared,” Ph.D. thesis, Department of Molecular and Optical Physics, Freiburg Materials Research Center, Universität Freiburg (2005).
  22. Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J Korean Phys. Soc. 49, 513–517 (2006).
  23. S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
    [Crossref]
  24. R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
    [Crossref]
  25. B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
    [Crossref]
  26. A. G. Markelz, “Terahertz dielectric sensitivity to biomolecular structure and function,” IEEE J. Sel. Top. Quantum Electron. 14, 180–190 (2008).
    [Crossref]
  27. L. Duvillaret, F. Garet, and 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]
  28. L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
    [Crossref]

2008 (1)

A. G. Markelz, “Terahertz dielectric sensitivity to biomolecular structure and function,” IEEE J. Sel. Top. Quantum Electron. 14, 180–190 (2008).
[Crossref]

2007 (3)

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
[Crossref]

D. Abbott and X.-C. Zhang, “Scanning the issue: T-ray imaging, sensing, and retection,” Proc. IEEE 95, 1509–1513 (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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

2006 (4)

N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
[Crossref]

A. Bartels, A. Thoma, C. Janke, T. Dekorsy, A. Dreyhaupt, S. Winnerl, and M. Helm, “High-resolution THz spectrometer with kHz scan rates,” Opt. Express 14, 430–437 (2006). URL http://www.opticsexpr“ess.org/abstract.cfm?URI=oe-14-1-430.
[Crossref] [PubMed]

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J Korean Phys. Soc. 49, 513–517 (2006).

2005 (4)

B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
[Crossref]

T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29–31 (2005).
[Crossref] [PubMed]

2004 (2)

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85, 863–865 (2004).
[Crossref]

S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
[Crossref]

2000 (2)

Z. Jiang and X.-C. Zhang, “Measurement of spatio-temporal terahertz field distribution by using chirped pulse technology,” IEEE J. Quantum Electron. 36, 1214–1222 (2000).
[Crossref]

R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. 76, 3191–3193 (2000).
[Crossref]

1999 (1)

1998 (1)

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. 4, 179–184 (1998).
[Crossref]

1996 (3)

R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[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]

L. Duvillaret, F. Garet, and 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]

1995 (1)

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
[Crossref]

1990 (1)

M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microw. Theory Tech. 38, 1684–1691 (1990).
[Crossref]

1989 (1)

1959 (1)

H. F. Trotter, “An elementary proof of the central limit theorem,” Arch. Math. 10, 226–234 (1959).
[Crossref]

1945 (1)

W. Feller, “The fundamental limit theorems in probability,” Bull. Amer. Math. Soc. 51, 800–832 (1945).
[Crossref]

Abbott, D.

D. Abbott and X.-C. Zhang, “Scanning the issue: T-ray imaging, sensing, and retection,” Proc. IEEE 95, 1509–1513 (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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
[Crossref]

W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press).
[Crossref]

Araki, T.

T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

Bartels, A.

Berry, E.

N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
[Crossref]

Boyle, R. D.

N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
[Crossref]

Braunstein, S. L.

N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
[Crossref]

Brodschelm, A.

R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. 76, 3191–3193 (2000).
[Crossref]

Buchner, R.

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

Cohen, N.

N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
[Crossref]

Coutaz, J.-L.

L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
[Crossref]

L. Duvillaret, F. Garet, and 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]

Dekorsy, T.

Dreyhaupt, A.

Duvillaret, L.

L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
[Crossref]

L. Duvillaret, F. Garet, and 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]

Fattinger, C.

Feller, W.

W. Feller, “The fundamental limit theorems in probability,” Bull. Amer. Math. Soc. 51, 800–832 (1945).
[Crossref]

Ferguson, B.

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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

Fischer, B. 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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29–31 (2005).
[Crossref] [PubMed]

B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
[Crossref]

B. M. Fischer, “Broadband THz time-domain spectroscopy of biomolecules — A comprehensive study of the dielectric properties of biomaterials in the far-infrared,” Ph.D. thesis, Department of Molecular and Optical Physics, Freiburg Materials Research Center, Universität Freiburg (2005).

W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press).
[Crossref]

Garet, F.

L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
[Crossref]

L. Duvillaret, F. Garet, and 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]

Gordon, K. C.

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

Gorenflo, S.

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

Grischkowsky, D.

M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microw. Theory Tech. 38, 1684–1691 (1990).
[Crossref]

M. van Exter, C. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. 14, 1128–1130 (1989).
[Crossref]

Handley, J. W.

N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
[Crossref]

Helm, H.

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
[Crossref]

Helm, M.

Hinkov, I.

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

Hoffmann, M.

B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
[Crossref]

Huber, R.

R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. 76, 3191–3193 (2000).
[Crossref]

Jacobsen, R. H.

R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[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]

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
[Crossref]

Janke, C.

Jansen, C.

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
[Crossref]

Jeon, S.-G.

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J Korean Phys. Soc. 49, 513–517 (2006).

Jepsen, P. U.

B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
[Crossref]

P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29–31 (2005).
[Crossref] [PubMed]

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
[Crossref]

Jiang, Z.

Z. Jiang and X.-C. Zhang, “Measurement of spatio-temporal terahertz field distribution by using chirped pulse technology,” IEEE J. Quantum Electron. 36, 1214–1222 (2000).
[Crossref]

Jin, Y.-S.

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J Korean Phys. Soc. 49, 513–517 (2006).

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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

Keiding, S. R.

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
[Crossref]

Kim, G.-J.

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J Korean Phys. Soc. 49, 513–517 (2006).

Koch, M.

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
[Crossref]

Krausz, F.

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. 4, 179–184 (1998).
[Crossref]

Kürner, T.

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
[Crossref]

Lambrecht, A.

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

Leitenstorfer, A.

R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. 76, 3191–3193 (2000).
[Crossref]

Lin, 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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press).
[Crossref]

Liu, K.

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85, 863–865 (2004).
[Crossref]

Markelz, A. G.

A. G. Markelz, “Terahertz dielectric sensitivity to biomolecular structure and function,” IEEE J. Sel. Top. Quantum Electron. 14, 180–190 (2008).
[Crossref]

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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
[Crossref]

Mittleman, D.

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
[Crossref]

Mittleman, D. M.

R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[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]

Modjesch, G.

B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
[Crossref]

Munch, J.

S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
[Crossref]

Newnham, D. A.

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

Nuss, M. C.

R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[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.

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

Piesiewicz, R.

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

Poppe, A.

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. 4, 179–184 (1998).
[Crossref]

Rades, T.

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

Saneyoshi, E.

T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

Shvartsman, R.

S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
[Crossref]

Spielmann, C.

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. 4, 179–184 (1998).
[Crossref]

Strachan, C. J.

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

Taday, P. F.

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

Tauer, U.

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

Tauser, F.

R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. 76, 3191–3193 (2000).
[Crossref]

Thoma, A.

Thrane, L.

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
[Crossref]

Trotter, H. F.

H. F. Trotter, “An elementary proof of the central limit theorem,” Arch. Math. 10, 226–234 (1959).
[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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

van Exter, M.

M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microw. Theory Tech. 38, 1684–1691 (1990).
[Crossref]

M. van Exter, C. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. 14, 1128–1130 (1989).
[Crossref]

Wietzke, S.

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
[Crossref]

Winnerl, S.

Withayachumnankul, W.

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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press).
[Crossref]

Xu, J.

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85, 863–865 (2004).
[Crossref]

Xu, L.

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. 4, 179–184 (1998).
[Crossref]

Yasui, T.

T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

Zeitler, J. A.

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

Zhang, X. C.

S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
[Crossref]

Zhang, X.-C.

D. Abbott and X.-C. Zhang, “Scanning the issue: T-ray imaging, sensing, and retection,” Proc. IEEE 95, 1509–1513 (2007).
[Crossref]

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85, 863–865 (2004).
[Crossref]

Z. Jiang and X.-C. Zhang, “Measurement of spatio-temporal terahertz field distribution by using chirped pulse technology,” IEEE J. Quantum Electron. 36, 1214–1222 (2000).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. 76, 3191–3193 (2000).
[Crossref]

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. 85, 863–865 (2004).
[Crossref]

T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

Arch. Math. (1)

H. F. Trotter, “An elementary proof of the central limit theorem,” Arch. Math. 10, 226–234 (1959).
[Crossref]

Bull. Amer. Math. Soc. (1)

W. Feller, “The fundamental limit theorems in probability,” Bull. Amer. Math. Soc. 51, 800–832 (1945).
[Crossref]

Chem. Phys. Lett. (2)

S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. 421, 494–498 (2006).
[Crossref]

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
[Crossref]

Fluct. Noise Lett. (1)

N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. 6, L77–L84 (2006).
[Crossref]

IEEE J. Quantum Electron. (1)

Z. Jiang and X.-C. Zhang, “Measurement of spatio-temporal terahertz field distribution by using chirped pulse technology,” IEEE J. Quantum Electron. 36, 1214–1222 (2000).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (4)

A. G. Markelz, “Terahertz dielectric sensitivity to biomolecular structure and function,” IEEE J. Sel. Top. Quantum Electron. 14, 180–190 (2008).
[Crossref]

L. Duvillaret, F. Garet, and 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]

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

A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. 4, 179–184 (1998).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microw. Theory Tech. 38, 1684–1691 (1990).
[Crossref]

Int. J Infrared Milli. (1)

R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. 28, 363–371 (2007).
[Crossref]

J Korean Phys. Soc. (1)

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J Korean Phys. Soc. 49, 513–517 (2006).

J Opt. B-Quantum S. O. (1)

S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. 6, S786–S795 (2004).
[Crossref]

J. Pharm. Sci. (1)

C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. 94, 837–846 (2005).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Proc. IEEE (2)

D. Abbott and X.-C. Zhang, “Scanning the issue: T-ray imaging, sensing, and retection,” Proc. IEEE 95, 1509–1513 (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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528–1558 (2007).
[Crossref]

Semicond. Sci. Technol. (1)

B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. 20, S246–S253 (2005).
[Crossref]

Other (2)

B. M. Fischer, “Broadband THz time-domain spectroscopy of biomolecules — A comprehensive study of the dielectric properties of biomaterials in the far-infrared,” Ph.D. thesis, Department of Molecular and Optical Physics, Freiburg Materials Research Center, Universität Freiburg (2005).

W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. Standard deviation in the refractive index against the sample thickness and frequency. The contours represent the simulated magnitude of sn, which is in fact comparable to that of sκ . The optical constants are set to n-jκ=3.0–0.1 j at all frequencies (κ=0.1 equals α=41.9 cm-1 at 1 THz). Additive white Gaussian noise limits the maximum dynamic range of the reference spectrum to 40 dB. The position of the global minimum relative to the frequency corresponds to the peak position of the reference spectrum. The profile of the optimum thickness determined numerically appears jagged due to the discrete nature of the simulation.

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Fig. 2.
Fig. 2. Standard deviation in the optical constants against the sample thickness. The plots are cross-sectional profiles of Fig. 1, at selected frequencies 0.5, 1.0, and 1.5 THz. The optimum thicknesses, where the standard deviation is minimal, are indicated by arrowheads. By moving towards a thicker sample, the standard deviation rapidly increases to the point comparable to the optical constants’ values. The standard deviation at high frequency is more sensitive to the thickness increment, as the T-ray magnitude at high frequencies is relatively low.

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Fig. 3.
Fig. 3. Optical constants of PVC. The constants are calculated from the average signal probing a 10-mm-thick sample. The available bandwidth of the measurement is from 0.05 to 0.55 THz. The ripples in the curves are due to Fabry-Pérot reflections.

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Fig. 4.
Fig. 4. Optimum thickness for PVC. The optimum thickness is determined from the measured absorption coefficient. The solid line is a second-order polynomial fit to the absorption coefficient between 0.05 and 0.55 THz. The horizontal dotted lines indicate the sample thicknesses of 1, 10, 20, and 50 mm, available for the measurements.

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Fig. 5.
Fig. 5. Standard deviations in the optical constants of PVC. Each standard deviation is determined from ten reference and ten sample signals. The dotted lines represent the raw profiles obtained from Eq. (3), whereas the solid lines are from an analytical function, exp(a 1 x 6+a 2 x 5+…+a 6 x+a 7), fitted to the dotted lines. The missing part of the profiles in the high frequency range corresponds to the low SNR portion of the measured spectra.

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Fig. 6.
Fig. 6. Optical constants of HDPE. The constants are calculated from the average signal probing the 50-mm-thick sample. The noise in the refractive index at low frequencies is due to the low SNR of the probing signal.

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Fig. 7.
Fig. 7. Optimum thickness for HDPE. The optimum thickness is determined from the measured absorption coefficient. The solid line is a second-order polynomial fit to the absorption coefficient between 0.05 and 1 THz. The horizontal dotted lines indicate the sample thicknesses of 1, 10, 20, 50, 100, and 200 mm, available for the measurements.

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Fig. 8.
Fig. 8. Standard deviations in the optical constants of HDPE. Each standard deviation is determined from ten reference and ten sample signals. The dotted lines represent the raw profiles obtained from Eq. (3), whereas the solid lines are from an analytical function, exp(a 1 x 5+a 2 x 4+…+a 5 x+a 6), fitted to the dotted lines. The missing part of the profiles in the high frequency range corresponds to the low SNR portion of the measured spectra.

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Fig. 9.
Fig. 9. Optical constants of the lactose mix. The constants are calculated from the average signal probing the 1.6-mm sample. The absorption peaks are pronounced at 0.53, 1.2, and 1.37 THz, as indicated by the arrowheads, before the spectrum is overwhelmed by noise beyond 1.6 THz.

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Fig. 10.
Fig. 10. Optimum thickness for the lactose mix. The optimum thickness is determined from the measured absorption coefficient. The horizontal dotted lines indicate the sample thicknesses of 0.4, 0.8, 1.6, 2.4, 3.2, to 4.0 mm, available for the measurements.

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Fig. 11.
Fig. 11. Standard deviation in the extinction coefficient of the lactose mix. The values at each thickness, indicated by the crosses, are determined from eight reference and eight sample signals, using the Monte Carlo method. The frequencies 0.53 and 1.37 THz correspond to the two first significant absorption peaks of the lactose spectrum. The missing part of the 1.37-THz profile is due to the low SNR portion of the measured spectra. The standard deviation in the index of refraction has a comparable value, and hence is not shown here. The dotted lines are fitted to the measurements only for visualization.

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Tables (1)

Tables Icon

Table 1. Optimum sample thickness for some common materials determined by the proposed method. The optimum thickness for THz-TDS measurement is determined using Eq. (11). The absorption coefficients, measured at room temperature, are taken from various sources: water [20]; PMMA, TPX [21]; HDPE [22]. Note that the absorption coefficient can widely vary from sample to sample, in particular for plastics.

View Table

Equations (16)

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H ( ω ) = E sam ( ω ) E ref ( ω ) = τ τ ′ · exp { − κ ( ω ) ω l c } · exp { − j [ n ( ω ) − n 0 ] ω l c } ,
n ( ω ) = n 0 − c ω l ∠ H ( ω ) ,
κ ( ω ) = c ω l { ln ∣ τ τ ′ ∣ − ln ∣ H ( ω ) ∣ } .
s n 2 ( ω ) = ( c ω l ) 2 { A sam ( ω ) ∣ E sam ( ω ) ∣ 4 + A ref ( ω ) ∣ E ref ( ω ) ∣ 4 } ,
s κ 2 ( ω ) = ( c ω l ) 2 { B sam ( ω ) ∣ E sam ( ω ) ∣ 4 + B ref ( ω ) ∣ E ref ( ω ) ∣ 4 + ( n ( ω ) − n 0 n ( ω ) + n 0 ) 2 s n 2 ( ω ) n ( ω ) 2 } ,
A sam ( ω ) = ∑ k ℑ 2 [ E sam ( ω ) exp ( j ω k Δ ) ] s E sam 2 ( k ) ,
A ref ( ω ) = ∑ k ℑ 2 [ E ref ( ω ) exp ( j ω k Δ ) ] s E ref 2 ( k ) ,
B sam ( ω ) = ∑ k ℜ 2 [ E sam ( ω ) exp ( j ω k Δ ) ] s E sam 2 ( k ) ,
B ref ( ω ) = ∑ k ℜ 2 [ E ref ( ω ) exp ( j ω k Δ ) ] s E ref 2 ( k ) .
∣ E sam ( ω ) ∣ 4 = ( τ τ ′ ) 4 exp { − 4 κ ( ω ) ω l c } ∣ E ref ( ω ) ∣ 4 .
S n 2 ( ω ) = C l 2 { A sam ′ ( ω ) exp ( − 4 κ ω l c ) + A ref ( ω ) } ,
C = c 2 ω 2 ∣ E ref ( ω ) ∣ 4 and A sam ′ ( ω ) = A sam ( ω ) ( τ τ ′ ) 4 .
∂ s n 2 ( ω ) ∂ l = ( 4 C l 2 κ ω c − 2 C l 3 ) A sam ′ ( ω ) exp ( − 4 κ ω l c ) − 2 C l 3 A ref ( ω ) .
A sam ( ω ) ≈ ( τ τ ′ ) 4 exp { − 4 κ ( ω ) ω l c } A ref ( ω ) .
( 4 C l 2 κ ω c − 2 C l 3 ) A ref ( ω ) − 2 C l 3 A ref ( ω ) = 0 ,
l opt = c ω κ ( ω ) = 2 α ( ω ) .

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