Abstract

Measurements of optical constants at terahertz—or T-ray—frequencies have been performed extensively using terahertz time-domain spectroscopy (THz-TDS). Spectrometers, together with physical models explaining the interaction between a sample and T-ray radiation, are progressively being developed. Nevertheless, measurement errors in the optical constants, so far, have not been systematically analyzed. This situation calls for a comprehensive analysis of measurement uncertainty in THz-TDS systems. The sources of error existing in a terahertz spectrometer and throughout the parameter estimation process are identified. The analysis herein quantifies the impact of each source on the output optical constants. The resulting analytical model is evaluated against experimental THz-TDS data.

© 2008 Optical Society of America

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  30. A. Gürtler, C. Winnewisser, H. Helm, and P. U. Jepsen, “Terahertz pulse propagation in the near field and the far field,” J. Opt. Soc. Am. A 17, 74-83 (2000).
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    [CrossRef]
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    [CrossRef]

2008 (2)

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

M. Franz, B. M. Fischer, and M. Walther, “The Christiansen effect in terahertz time-domain spectra of coarse-grained powders,” Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

2007 (2)

I. Pupeza, R. Wilk, and M. Koch, “Highly accurate optical material parameter determination with THz time domain spectroscopy,” Opt. Express 15, 4335-4350 (2007).
[CrossRef] [PubMed]

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 (2)

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]

W. Withayachumnankul, B. Ferguson, T. Rainsford, S. P. Mickan, and D. Abbott, “Direct Fabry-Pérot effect removal,” Fluct. Noise Lett. 6, L227-L239 (2006).
[CrossRef]

2005 (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]

2004 (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 Semiclassical Opt. 6, S786-S795 (2004).
[CrossRef]

2003 (1)

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

2001 (2)

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

M. Grabe, “Estimation of measurement uncertainties—an alternative to the ISO guide,” Metrologia 38, 97-106 (2001).
[CrossRef]

2000 (2)

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]

1997 (2)

I. H. Lira and W. Wöger, “The evaluation of standard uncertainty in the presence of limited resolution of indicating devices,” Meas. Sci. Technol. 8, 441-443 (1997).
[CrossRef]

J. M. Forniés-Marquina, J. Letosa, M. García-Gracia, and J. M. Artacho, “Error propagation for the transformation of time domain into frequency domain,” IEEE Trans. Magn. 33, 1456-1459 (1997).
[CrossRef]

1996 (3)

W. Bich, “Simple formula for the propagation of variances and covariances,” Metrologia 33, 181-183 (1996).
[CrossRef]

J. Letosa, M. García-Gracia, J. M. Forniés-Marquina, and J. M. Artacho, “Performance limits in TDR technique by Monte Carlo simulation,” IEEE Trans. Magn. 32, 958-961 (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)

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beams,” Appl. Phys. Lett. 67, 3523-3525 (1995).
[CrossRef]

1993 (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

1992 (2)

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

J. Son, J. V. Rudd, and J. F. Whitaker, “Noise characterization of a self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 733-735 (1992).
[CrossRef] [PubMed]

1990 (1)

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

1988 (1)

P. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
[CrossRef]

1965 (1)

J. E. Chamberlain, F. D. Findlay, and H. A. Gebbie, “Refractive index of air at 0.337-mm wave-length,” Nature 206, 886-887 (1965).
[CrossRef]

Abbott, D.

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

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. Ferguson, T. Rainsford, S. P. Mickan, and D. Abbott, “Direct Fabry-Pérot effect removal,” Fluct. Noise Lett. 6, L227-L239 (2006).
[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 Semiclassical Opt. 6, S786-S795 (2004).
[CrossRef]

Artacho, J. M.

J. M. Forniés-Marquina, J. Letosa, M. García-Gracia, and J. M. Artacho, “Error propagation for the transformation of time domain into frequency domain,” IEEE Trans. Magn. 33, 1456-1459 (1997).
[CrossRef]

J. Letosa, M. García-Gracia, J. M. Forniés-Marquina, and J. M. Artacho, “Performance limits in TDR technique by Monte Carlo simulation,” IEEE Trans. Magn. 32, 958-961 (1996).
[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]

Auston, D. H.

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

P. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
[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]

Baraniuk, R.

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]

Bich, W.

W. Bich, “Simple formula for the propagation of variances and covariances,” Metrologia 33, 181-183 (1996).
[CrossRef]

Bolivar, P.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[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]

Brucherseifer, M.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

Chamberlain, J. E.

J. E. Chamberlain, F. D. Findlay, and H. A. Gebbie, “Refractive index of air at 0.337-mm wave-length,” Nature 206, 886-887 (1965).
[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.

de Maagt, P.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

Dorney, T.

Duvillaret, L.

Ederra, I.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[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]

W. Withayachumnankul, B. Ferguson, T. Rainsford, S. P. Mickan, and D. Abbott, “Direct Fabry-Pérot effect removal,” Fluct. Noise Lett. 6, L227-L239 (2006).
[CrossRef]

Findlay, F. D.

J. E. Chamberlain, F. D. Findlay, and H. A. Gebbie, “Refractive index of air at 0.337-mm wave-length,” Nature 206, 886-887 (1965).
[CrossRef]

Fischer, B. M.

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

M. Franz, B. M. Fischer, and M. Walther, “The Christiansen effect in terahertz time-domain spectra of coarse-grained powders,” Appl. Phys. Lett. 92, 021107 (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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528-1558 (2007).
[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]

B. M. Fischer, M. Hoffmann, and P. U. Jepsen, “Dynamic range and numerical error propagation in terahertz time-domain spectroscopy, in Optical Terahertz Science and Technology, Technical Digest (CD) (Optical Society of America, 2005), paper TuD1.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1992).

Forniés-Marquina, J. M.

J. M. Forniés-Marquina, J. Letosa, M. García-Gracia, and J. M. Artacho, “Error propagation for the transformation of time domain into frequency domain,” IEEE Trans. Magn. 33, 1456-1459 (1997).
[CrossRef]

J. Letosa, M. García-Gracia, J. M. Forniés-Marquina, and J. M. Artacho, “Performance limits in TDR technique by Monte Carlo simulation,” IEEE Trans. Magn. 32, 958-961 (1996).
[CrossRef]

Franz, M.

M. Franz, B. M. Fischer, and M. Walther, “The Christiansen effect in terahertz time-domain spectra of coarse-grained powders,” Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

García-Gracia, M.

J. M. Forniés-Marquina, J. Letosa, M. García-Gracia, and J. M. Artacho, “Error propagation for the transformation of time domain into frequency domain,” IEEE Trans. Magn. 33, 1456-1459 (1997).
[CrossRef]

J. Letosa, M. García-Gracia, J. M. Forniés-Marquina, and J. M. Artacho, “Performance limits in TDR technique by Monte Carlo simulation,” IEEE Trans. Magn. 32, 958-961 (1996).
[CrossRef]

Garet, F.

Gebbie, H. A.

J. E. Chamberlain, F. D. Findlay, and H. A. Gebbie, “Refractive index of air at 0.337-mm wave-length,” Nature 206, 886-887 (1965).
[CrossRef]

Gonzalo, R.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

Grabe, M.

M. Grabe, “Estimation of measurement uncertainties—an alternative to the ISO guide,” Metrologia 38, 97-106 (2001).
[CrossRef]

M. Grabe, Measurement Uncertainties in Science and Technology (Springer, 2005).

Grischkowsky, D. R.

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

Gürtler, A.

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]

Haus, H. A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

Helm, H.

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]

A. Gürtler, C. Winnewisser, H. Helm, and P. U. Jepsen, “Terahertz pulse propagation in the near field and the far field,” J. Opt. Soc. Am. A 17, 74-83 (2000).
[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]

B. M. Fischer, M. Hoffmann, and P. U. Jepsen, “Dynamic range and numerical error propagation in terahertz time-domain spectroscopy, in Optical Terahertz Science and Technology, Technical Digest (CD) (Optical Society of America, 2005), paper TuD1.

Holker, M.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

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]

A. Gürtler, C. Winnewisser, H. Helm, and P. U. Jepsen, “Terahertz pulse propagation in the near field and the far field,” J. Opt. Soc. Am. A 17, 74-83 (2000).
[CrossRef]

B. M. Fischer, M. Hoffmann, and P. U. Jepsen, “Dynamic range and numerical error propagation in terahertz time-domain spectroscopy, in Optical Terahertz Science and Technology, Technical Digest (CD) (Optical Society of America, 2005), paper TuD1.

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]

Koch, M.

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]

Letosa, J.

J. M. Forniés-Marquina, J. Letosa, M. García-Gracia, and J. M. Artacho, “Error propagation for the transformation of time domain into frequency domain,” IEEE Trans. Magn. 33, 1456-1459 (1997).
[CrossRef]

J. Letosa, M. García-Gracia, J. M. Forniés-Marquina, and J. M. Artacho, “Performance limits in TDR technique by Monte Carlo simulation,” IEEE Trans. Magn. 32, 958-961 (1996).
[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]

Lira, I.

I. Lira, Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance, Series in Measurement and Technology (Institute of Physics, 2002).
[CrossRef]

Lira, I. H.

I. H. Lira and W. Wöger, “The evaluation of standard uncertainty in the presence of limited resolution of indicating devices,” Meas. Sci. Technol. 8, 441-443 (1997).
[CrossRef]

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[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]

W. Withayachumnankul, B. Ferguson, T. Rainsford, S. P. Mickan, and D. Abbott, “Direct Fabry-Pérot effect removal,” Fluct. Noise Lett. 6, L227-L239 (2006).
[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 Semiclassical Opt. 6, S786-S795 (2004).
[CrossRef]

Mittleman, D.

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 Semiclassical Opt. 6, S786-S795 (2004).
[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, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95, 1528-1558 (2007).
[CrossRef]

Nuss, M. C.

P. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
[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]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1992).

Pupeza, I.

Rainsford, T.

W. Withayachumnankul, B. Ferguson, T. Rainsford, S. P. Mickan, and D. Abbott, “Direct Fabry-Pérot effect removal,” Fluct. Noise Lett. 6, L227-L239 (2006).
[CrossRef]

Reynolds, A.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

Rivas, J.

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

Rudd, J. V.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, “Beam optics,” in Fundamentals of Photonics, Wiley Series in Pure and Applied Optics (Wiley, 1991), pp. 80-107.
[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 Semiclassical Opt. 6, S786-S795 (2004).
[CrossRef]

Smith, P.

P. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
[CrossRef]

Son, J.

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]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, “Beam optics,” in Fundamentals of Photonics, Wiley Series in Pure and Applied Optics (Wiley, 1991), pp. 80-107.
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1992).

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. R. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684-1691 (1990).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1992).

Walther, M.

M. Franz, B. M. Fischer, and M. Walther, “The Christiansen effect in terahertz time-domain spectra of coarse-grained powders,” Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

Whitaker, J. F.

Wilk, R.

Winnewisser, C.

Withayachumnankul, W.

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

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. Ferguson, T. Rainsford, S. P. Mickan, and D. Abbott, “Direct Fabry-Pérot effect removal,” Fluct. Noise Lett. 6, L227-L239 (2006).
[CrossRef]

Wöger, W.

I. H. Lira and W. Wöger, “The evaluation of standard uncertainty in the presence of limited resolution of indicating devices,” Meas. Sci. Technol. 8, 441-443 (1997).
[CrossRef]

Wu, Q.

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beams,” Appl. Phys. Lett. 67, 3523-3525 (1995).
[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]

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784-1786 (1992).
[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]

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 Semiclassical Opt. 6, S786-S795 (2004).
[CrossRef]

Zhang, X.-C.

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beams,” Appl. Phys. Lett. 67, 3523-3525 (1995).
[CrossRef]

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

M. Franz, B. M. Fischer, and M. Walther, “The Christiansen effect in terahertz time-domain spectra of coarse-grained powders,” Appl. Phys. Lett. 92, 021107 (2008).
[CrossRef]

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

Q. Wu and X.-C. Zhang, “Free-space electro-optic sampling of terahertz beams,” Appl. Phys. Lett. 67, 3523-3525 (1995).
[CrossRef]

Fluct. Noise Lett. (2)

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]

W. Withayachumnankul, B. Ferguson, T. Rainsford, S. P. Mickan, and D. Abbott, “Direct Fabry-Pérot effect removal,” Fluct. Noise Lett. 6, L227-L239 (2006).
[CrossRef]

IEEE J. Quantum Electron. (2)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983-996 (1993).
[CrossRef]

P. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

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]

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. Magn. (2)

J. Letosa, M. García-Gracia, J. M. Forniés-Marquina, and J. M. Artacho, “Performance limits in TDR technique by Monte Carlo simulation,” IEEE Trans. Magn. 32, 958-961 (1996).
[CrossRef]

J. M. Forniés-Marquina, J. Letosa, M. García-Gracia, and J. M. Artacho, “Error propagation for the transformation of time domain into frequency domain,” IEEE Trans. Magn. 33, 1456-1459 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

P. Bolivar, M. Brucherseifer, J. Rivas, R. Gonzalo, I. Ederra, A. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51, 1062-1066 (2003).
[CrossRef]

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

J. Opt. B: Quantum Semiclassical Opt. (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 Semiclassical Opt. 6, S786-S795 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Meas. Sci. Technol. (1)

I. H. Lira and W. Wöger, “The evaluation of standard uncertainty in the presence of limited resolution of indicating devices,” Meas. Sci. Technol. 8, 441-443 (1997).
[CrossRef]

Metrologia (2)

M. Grabe, “Estimation of measurement uncertainties—an alternative to the ISO guide,” Metrologia 38, 97-106 (2001).
[CrossRef]

W. Bich, “Simple formula for the propagation of variances and covariances,” Metrologia 33, 181-183 (1996).
[CrossRef]

Nature (1)

J. E. Chamberlain, F. D. Findlay, and H. A. Gebbie, “Refractive index of air at 0.337-mm wave-length,” Nature 206, 886-887 (1965).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. IEEE (1)

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 (8)

I. Lira, Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance, Series in Measurement and Technology (Institute of Physics, 2002).
[CrossRef]

Joint Committee for Guides in Metrology, Evaluation of Measurement Data—Supplement 1 to the Guide to the Expression of Uncertainty in Measurement—Propagation of Distributions Using a Monte Carlo Method (Joint Committee for Guides in Metrology, 2006).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1992).

B. E. A. Saleh and M. C. Teich, “Beam optics,” in Fundamentals of Photonics, Wiley Series in Pure and Applied Optics (Wiley, 1991), pp. 80-107.
[CrossRef]

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

ISO, International Vocabulary of Basic and General Terms in Metrology (VIM) (International Organization for Standardization, 2004).

M. Grabe, Measurement Uncertainties in Science and Technology (Springer, 2005).

B. M. Fischer, M. Hoffmann, and P. U. Jepsen, “Dynamic range and numerical error propagation in terahertz time-domain spectroscopy, in Optical Terahertz Science and Technology, Technical Digest (CD) (Optical Society of America, 2005), paper TuD1.

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Figures (7)

Fig. 1
Fig. 1

Parameter estimation process using a THz-TDS measuring system. The process is mainly composed of a T-ray spectrometer, a physical T-ray propagation model, and a random process. The solid boxes represent well-reported parts of the THz-TDS measurement process, whereas the dotted boxes have not been fully analyzed and are now addressed in this paper. Modified from [5].

Fig. 2
Fig. 2

THz-TDS system configured in transmission mode. The system consists of an ultrafast optical laser, T-ray emitter–receiver, an optical delay line, a set of mirrors, and a material sample. The emitter and receiver shown are photoconductive antennas. The optical beam path is indicated by small arrowheads and the T-ray beam path by large arrowheads.

Fig. 3
Fig. 3

Sources of error in a THz-TDS measurement. The sources of error in the dashed boxes occur in both the THz-TDS measurement and the parameter extraction process. The errors produced by these sources are classified as either random or systematic. They cause the variances and deviations, which propagate down the process, and eventually contribute to the uncertainty in the extracted optical constants.

Fig. 4
Fig. 4

Amplitude variance of the time-domain signal. The amplitude variance is plotted against the time and the number of measurements. Any two succeeding measurements are separated by approximately 40 s . As the number of measurements increases, the variance increases drastically. The inset shows the arithmetic mean of the 60 measurements. Interestingly, the two peaks in the variance occur at 11.8 and 12.7 ps , whereas the negative and positive peaks in the mean signal are at 11 and 12.4 ps , respectively. The result is most probably dominated by delay-line registration and mechanical drift.

Fig. 5
Fig. 5

Tilted sample in a T-ray beam path. This exaggerated figure illustrates a small tilt angle from the normal, which might occur due to manual misalignment of the sample. The T-ray path inside the sample, l θ , is longer than the sample thickness, l, by the factor of 1 / cos θ t . The refraction angle, θ t , is related to the incident angle (and the tilting angle), θ i , through Snell’s law, n sin θ t = n 0 sin θ i , but for a small tilting angle, θ t θ i .

Fig. 6
Fig. 6

Average signals and standard deviations for reference and lactose. The reference and lactose signals are each averaged over 10 measurements. The signals have a temporal resolution of 0.0167 ps , and a total duration of 34.16 ps . The inset shows the spectra of the reference and sample.

Fig. 7
Fig. 7

Uncertainty for the lactose measurement. The combined uncertainties in the optical constants are plotted in comparison to the mean values of the optical constants and the standard deviations introduced by various sources of error. The combined uncertainty is calculated with the coverage factor k P = 1 . Both subfigures share the same vertical scale. In (a) the refractive index of the lactose–HDPE pellet is approximately 1.46, compared to its combined uncertainty of 10 3 . The major sources contributing to the combined uncertainty are signal noise and thickness uncertainty. In (b) the extinction coefficient is of the order of 10 3 , compared to its combined uncertainty of the order of 10 4 . The major source contributing to the combined uncertainty is signal noise. The arrowheads indicate the low-frequency resonances of α lactose at 0.53 and 1.37 THz .

Equations (73)

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E sam ( ω ) = η 4 n ̂ ( ω ) n 0 [ n ̂ ( ω ) + n 0 ] 2 exp { j n ̂ ( ω ) ω l θ c } E ( ω ) ,
E ref ( ω ) = η exp { j n 0 ω l c } E ( ω ) ,
H 0 ( ω ) = 4 n ̂ ( ω ) n 0 [ n ̂ ( ω ) + n 0 ] 2 exp { κ ( ω ) ω l c } exp { j [ n ( ω ) n 0 ] ω l c } .
H ( ω ) = 4 n ( ω ) n 0 [ n ( ω ) + n 0 ] 2 exp { κ ( ω ) ω l c } exp { j [ n ( ω ) n 0 ] ω l c } .
H ( ω ) = [ n ( ω ) n 0 ] ω l c ,
ln H ( ω ) = ln [ 4 n ( ω ) n 0 ( n ( ω ) + n 0 ) 2 ] κ ( ω ) ω l c .
n ( ω ) = n 0 c ω l H ( ω ) ,
κ ( ω ) = c ω l { ln [ 4 n ( ω ) n 0 ( n ( ω ) + n 0 ) 2 ] ln H ( ω ) } .
s n , E 2 ( ω ) = ( c ω l ) 2 { A sam ( ω ) E sam ( ω ) 4 + A ref ( ω ) E ref ( ω ) 4 } ,
s κ , E 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 , E 2 ( ω ) n ( ω ) 2 } ,
A sam ( ω ) = k I 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) ,
A ref ( ω ) = k I 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) ,
B sam ( ω ) = k R 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) ,
B ref ( ω ) = k R 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) .
s n , E sam 2 ( ω ) = ( c ω l ) 2 A sam ( ω ) E sam ( ω ) 4 ,
s κ , E sam 2 ( ω ) = ( c ω l ) 2 { B sam ( ω ) E sam ( ω ) 4 + ( n ( ω ) n 0 n ( ω ) + n 0 ) 2 s n , E sam 2 ( ω ) n ( ω ) 2 } .
s n , E ref 2 ( ω ) = ( c ω l ) 2 A ref ( ω ) E ref ( ω ) 4 ,
s κ , E ref 2 ( ω ) = ( c ω l ) 2 { B ref ( ω ) E ref ( ω ) 4 + ( n ( ω ) n 0 n ( ω ) + n 0 ) 2 s n , E ref 2 ( ω ) n ( ω ) 2 } .
s n , l 2 ( ω ) = [ n ( ω ) n 0 l ] 2 s l 2 ,
s κ , l 2 ( ω ) = [ κ ( ω ) l ] 2 s l 2 + [ c n ( ω ) ω l ( n ( ω ) n 0 n ( ω ) + n 0 ) ] 2 s n , l 2 ( ω ) ,
s n , δ 2 ( ω ) = [ n ( ω ) n 0 l ] 2 δ l 2 12 ,
s κ , δ 2 ( ω ) = [ κ ( ω ) l ] 2 δ l 2 12 + [ c n ( ω ) ω l ( n ( ω ) n 0 n ( ω ) + n 0 ) ] 2 s n , δ 2 ( ω ) .
l θ = l cos θ t .
f l = l ( 1 cos f θ 1 ) .
f n , θ ( ω ) = n ( ω ) n 0 l f l ,
f κ , θ ( ω ) = κ ( ω ) l f l + c n ( ω ) ω l n ( ω ) n 0 n ( ω ) + n 0 f n , θ ( ω ) .
f n , θ ( ω ) = [ n ( ω ) n 0 ] ( 1 cos f θ 1 ) ,
f κ , θ ( ω ) = κ ( ω ) ( 1 cos f θ 1 ) + c n ( ω ) ω l n ( ω ) n 0 n ( ω ) + n 0 f n , θ ( ω ) .
f H ( ω ) = H appx ( ω ) H exact ( ω ) , = arg { 4 n ̂ ( ω ) n 0 [ n ̂ ( ω ) + n 0 ] 2 } .
f ln H ( ω ) = ln H appx ( ω ) ln H exact ( ω ) , = ln 4 n ( ω ) n 0 [ n ( ω ) + n 0 ] 2 ln 4 n ̂ ( ω ) n 0 [ n ̂ ( ω ) + n 0 ] 2 , = ln n ( ω ) n ̂ ( ω ) [ n ̂ ( ω ) + n 0 n ( ω ) + n 0 ] 2 .
f n , H ( ω ) = c ω l f H ( ω ) .
f κ , H ( ω ) = c ω l [ f ln H + 1 n ( ω ) n ( ω ) n 0 n ( ω ) + n 0 f n , H ( ω ) ] .
H FP ( ω ) = FP ( ω ) H ( ω ) ,
FP ( ω ) = { 1 [ n ̂ ( ω ) n 0 n ̂ ( ω ) + n 0 ] 2 exp [ 2 j n ̂ ( ω ) ω l c ] } 1 .
f FP ( ω ) = H ( ω ) H FP ( ω ) , = arg { FP ( ω ) } .
f ln FP ( ω ) = ln H ( ω ) ln H FP ( ω ) , = ln FP ( ω ) .
f n , FP ( ω ) = c ω l f FP ( ω ) .
f κ , FP ( ω ) = c ω l [ f ln FP + 1 n ( ω ) n ( ω ) n 0 n ( ω ) + n 0 f n , FP ( ω ) ] .
n 0 , exact = 1 + 86.26 ( 5748 + T ) p T 2 10 6 ,
f n , n 0 ( ω ) = f n 0 ,
f κ , n 0 ( ω ) = c ω l n ( ω ) n 0 n ( ω ) n 0 f n 0 .
u n ¯ ( ω ) = k P s n , E sam 2 N E sam + s n , E ref 2 N E ref + s n , l 2 N l + s n , δ 2 + f n , θ + f n , H + f n , FP + f n 0 ,
u κ ¯ ( ω ) = k P s κ , E sam 2 N E sam + s κ , E ref 2 N E ref + s κ , l 2 N l + s κ , δ 2 + f κ , θ + f κ , H + f κ , FP + f κ , n 0 ,
E ( ω ) = k E ( k ) exp ( j ω k τ ) ,
E r ( ω ) = k E ( k ) cos ( ω k τ ) ,
E i ( ω ) = k E ( k ) sin ( ω k τ ) .
s E r 2 ( ω ) = k cos 2 ( ω k τ ) s E 2 ( k ) ,
s E i 2 ( ω ) = k sin 2 ( ω k τ ) s E 2 ( k ) ,
s E r E i ( ω ) = k sin ( ω k τ ) cos ( ω k τ ) s E 2 ( k ) = 1 2 k sin ( 2 ω k τ ) s E 2 ( k ) .
E ( ω ) = E r ( ω ) 2 + E i ( ω ) 2 ,
E ( ω ) = arctan ( E i ( ω ) E r ( ω ) ) .
s E 2 ( ω ) = 1 E ( ω ) 2 [ E r ( ω ) 2 s E r 2 ( ω ) + E i 2 ( ω ) s E i 2 ( ω ) + 2 E r ( ω ) E i ( ω ) s E r E i ( ω ) ] ,
s E 2 ( ω ) = 1 E ( ω ) 4 [ E i ( ω ) 2 s E r 2 ( ω ) + E r ( ω ) 2 s E i 2 ( ω ) 2 E r ( ω ) E i ( ω ) s E r E i ( ω ) ] .
s E 2 ( ω ) = 1 E ( ω ) 2 k [ E r ( ω ) cos ( ω k τ ) E i ( ω ) sin ( ω k τ ) ] 2 s E 2 ( k ) ,
s E 2 ( ω ) = 1 E ( ω ) 4 k [ E i ( ω ) cos ( ω k τ ) + E r ( ω ) sin ( ω k τ ) ] 2 s E 2 ( k ) .
s E 2 ( ω ) = 1 E ( ω ) 2 k R 2 [ E ( ω ) exp ( j ω k τ ) ] s E 2 ( k ) ,
s E 2 ( ω ) = 1 E ( ω ) 4 k I 2 [ E ( ω ) exp ( j ω k τ ) ] s E 2 ( k ) ,
s E sam 2 ( ω ) = 1 E sam ( ω ) 2 k R 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) ,
s E ref 2 ( ω ) = 1 E ref ( ω ) 2 k R 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) ,
s E sam 2 ( ω ) = 1 E sam ( ω ) 4 k I 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) ,
s E ref 2 ( ω ) = 1 E ref ( ω ) 4 k I 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) .
H ( ω ) = E sam ( ω ) E ref ( ω ) ,
H ( ω ) = E sam ( ω ) E ref ( ω ) .
s H 2 ( ω ) = 1 E ref ( ω ) 2 s E sam 2 ( ω ) + E sam ( ω ) 2 E ref ( ω ) 4 s E ref 2 ( ω ) ,
s H 2 ( ω ) = s E sam 2 ( ω ) + s E ref 2 ( ω ) .
s H 2 ( ω ) = 1 E ref ( ω ) E sam ( ω ) 2 k R 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) + E sam ( ω ) 2 E ref ( ω ) 6 k R 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) ,
s H 2 ( ω ) = 1 E sam ( ω ) 4 k I 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) + 1 E ref ( ω ) 4 k I 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) .
n ( ω ) = n 0 c ω l H ( ω ) ,
κ ( ω ) = c ω l { ln [ 4 n ( ω ) n 0 ( n ( ω ) + n 0 ) 2 ] ln H ( ω ) } .
s n 2 ( ω ) = ( c ω l ) 2 s H 2 ( ω ) ,
s κ 2 ( ω ) = [ c ω l H ( ω ) ] 2 s H 2 ( ω ) + [ c ω l ( n ( ω ) n 0 n ( ω ) + n 0 ) ] 2 s n 2 ( ω ) n ( ω ) 2 .
s n 2 ( ω ) = ( c ω l ) 2 { 1 E sam ( ω ) 4 k I 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) + 1 E ref ( ω ) 4 k I 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) } ,
s κ 2 ( ω ) = ( c ω l ) 2 { 1 E sam ( ω ) 4 k R 2 [ E sam ( ω ) exp ( j ω k τ ) ] s E sam 2 ( k ) + 1 E ref ( ω ) 4 k R 2 [ E ref ( ω ) exp ( j ω k τ ) ] s E ref 2 ( k ) + ( n ( ω ) n 0 n ( ω ) + n 0 ) 2 s n , E 2 ( ω ) n ( ω ) 2 } .

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