Abstract

Imaging systems based on terahertz (THz) time-domain spectroscopy offer a range of unique modalities owing to the broad bandwidth, subpicosecond duration, and phase-sensitive detection of the THz pulses. Furthermore, the possibility exists for combining spectroscopic characterization or identification with imaging because the radiation is broadband in nature. To achieve this, we require novel methods for real-time analysis of THz waveforms. This paper describes a robust algorithm for extracting material parameters from measured THz waveforms. Our algorithm simultaneously obtains both the thickness and the complex refractive index of an unknown sample under certain conditions. In contrast, most spectroscopic transmission measurements require knowledge of the sample’s thickness for an accurate determination of its optical parameters. Our approach relies on a model-based estimation, a gradient descent search, and the total variation measure. We explore the limits of this technique and compare the results with literature data for optical parameters of several different materials.

© 2001 Optical Society of America

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References

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  1. P. R. Smith, D. H. Auston, M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255–260 (1988).
    [CrossRef]
  2. M. van Exter, D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
    [CrossRef]
  3. M. C. Nuss, J. Orenstein, “Terahertz time-domain spectroscopy (THz-TDS),” in Millimeter and Submillimeter Wave Spectroscopy of Solids, G. Grüner, ed. (Springer–Verlag, Heidelberg, Germany, 1998), and references therein.
  4. D. M. Mittleman, R. H. Jacobsen, M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
    [CrossRef]
  5. B. B. Hu, M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20, 1716–1718 (1995).
    [CrossRef] [PubMed]
  6. D. M. Mittleman, S. Hunsche, L. Boivin, M. C. Nuss, “T-ray tomography,” Opt. Lett. 22, 904–906 (1997).
    [CrossRef] [PubMed]
  7. D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
    [CrossRef]
  8. Q. Wu, F. G. Sun, P. Campbell, X.-C. Zhang, “Dynamic range of an electro-optic field sensor and its imaging applications,” Appl. Phys. Lett. 68, 3224–3226 (1996).
    [CrossRef]
  9. Z. G. Lu, P. Campbell, X.-C. Zhang, “Free-space electro-optic sampling with a high-repetition-rate regenerative amplified laser,” Appl. Phys. Lett. 71, 593–595 (1997).
    [CrossRef]
  10. D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
    [CrossRef]
  11. L. Hui, F. Zhongyu, Y. Jianqi, “Infrared imaging solar spectrograph at Purple Mountain Observatory,” Sol. Phys. 185, 69–76 (1999).
    [CrossRef]
  12. K. Sato, T. Sato, H. Sone, T. Takagi, “Development of a high-speed time-resolved spectroscope and its application to analysis of time-varying optical spectra,” IEEE Trans. Instrum. Meas. IM-36, 1045–1049 (1987).
    [CrossRef]
  13. R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
    [CrossRef]
  14. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer–Verlag, Heidelberg, Germany, 1998), pp. 2–7.
  15. L. Duvillaret, F. Garet, J. 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]
  16. L. Duvillaret, F. Garet, J. Coutaz, “Highly precise determination of both optical constants and sample thickness in teraherz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
    [CrossRef]
  17. E. Hecht, Optics, 2nd ed. (Addison–Wesley, Reading, Mass., 1987).
  18. M. Dobrin, C. Savit, Introduction to Geophysical Prospecting, 4th ed. (McGraw-Hill, New York, 1988).
  19. M. van Exter, C. Fattinger, D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. 14, 1128 (1989).
    [CrossRef]
  20. P. E. Ciddar, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566–1573 (1996).
    [CrossRef]
  21. S. Haykin, Adaptive Filter Theory (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  22. J. E. Odegard, C. S. Burrus, “Discrete finite variation: a new measure of smoothness for the design of wavelet basis,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 1467–1470.
  23. F. Jones, Lebesgue Integration on Euclidean Space (Jones and Bartlett, Boston, Mass., 1993).
  24. 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]
  25. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

1999 (3)

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

L. Hui, F. Zhongyu, Y. Jianqi, “Infrared imaging solar spectrograph at Purple Mountain Observatory,” Sol. Phys. 185, 69–76 (1999).
[CrossRef]

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

1998 (1)

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

1997 (3)

R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
[CrossRef]

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

Z. G. Lu, P. Campbell, X.-C. Zhang, “Free-space electro-optic sampling with a high-repetition-rate regenerative amplified laser,” Appl. Phys. Lett. 71, 593–595 (1997).
[CrossRef]

1996 (4)

Q. Wu, F. G. Sun, P. Campbell, X.-C. Zhang, “Dynamic range of an electro-optic field sensor and its imaging applications,” Appl. Phys. Lett. 68, 3224–3226 (1996).
[CrossRef]

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

L. Duvillaret, F. Garet, J. 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]

P. E. Ciddar, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566–1573 (1996).
[CrossRef]

1995 (1)

1990 (2)

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

1989 (1)

1988 (1)

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

1987 (1)

K. Sato, T. Sato, H. Sone, T. Takagi, “Development of a high-speed time-resolved spectroscope and its application to analysis of time-varying optical spectra,” IEEE Trans. Instrum. Meas. IM-36, 1045–1049 (1987).
[CrossRef]

Auston, D. H.

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

Baraniuk, R. G.

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

Boivin, L.

Browne, M. T.

R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
[CrossRef]

Burge, R. E.

R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
[CrossRef]

Burrus, C. S.

J. E. Odegard, C. S. Burrus, “Discrete finite variation: a new measure of smoothness for the design of wavelet basis,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 1467–1470.

Campbell, P.

Z. G. Lu, P. Campbell, X.-C. Zhang, “Free-space electro-optic sampling with a high-repetition-rate regenerative amplified laser,” Appl. Phys. Lett. 71, 593–595 (1997).
[CrossRef]

Q. Wu, F. G. Sun, P. Campbell, X.-C. Zhang, “Dynamic range of an electro-optic field sensor and its imaging applications,” Appl. Phys. Lett. 68, 3224–3226 (1996).
[CrossRef]

Charalambous, P.

R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
[CrossRef]

Ciddar, P. E.

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer–Verlag, Heidelberg, Germany, 1998), pp. 2–7.

Coutaz, J.

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

L. Duvillaret, F. Garet, J. 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]

Dobrin, M.

M. Dobrin, C. Savit, Introduction to Geophysical Prospecting, 4th ed. (McGraw-Hill, New York, 1988).

Duvillaret, L.

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

L. Duvillaret, F. Garet, J. 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.

Garet, F.

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

L. Duvillaret, F. Garet, J. 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]

Grischkowsky, D.

Gupta, M.

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

Haykin, S.

S. Haykin, Adaptive Filter Theory (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison–Wesley, Reading, Mass., 1987).

Hu, B. B.

Hui, L.

L. Hui, F. Zhongyu, Y. Jianqi, “Infrared imaging solar spectrograph at Purple Mountain Observatory,” Sol. Phys. 185, 69–76 (1999).
[CrossRef]

Hunsche, S.

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

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

Jianqi, Y.

L. Hui, F. Zhongyu, Y. Jianqi, “Infrared imaging solar spectrograph at Purple Mountain Observatory,” Sol. Phys. 185, 69–76 (1999).
[CrossRef]

Jones, F.

F. Jones, Lebesgue Integration on Euclidean Space (Jones and Bartlett, Boston, Mass., 1993).

Keiding, S.

Knauer, J. N.

R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
[CrossRef]

Koch, M.

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer–Verlag, Heidelberg, Germany, 1998), pp. 2–7.

Lu, Z. G.

Z. G. Lu, P. Campbell, X.-C. Zhang, “Free-space electro-optic sampling with a high-repetition-rate regenerative amplified laser,” Appl. Phys. Lett. 71, 593–595 (1997).
[CrossRef]

Mittleman, D. M.

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

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

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

Neelamani, R.

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

Nuss, M. C.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

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

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

B. B. Hu, M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20, 1716–1718 (1995).
[CrossRef] [PubMed]

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

M. C. Nuss, J. Orenstein, “Terahertz time-domain spectroscopy (THz-TDS),” in Millimeter and Submillimeter Wave Spectroscopy of Solids, G. Grüner, ed. (Springer–Verlag, Heidelberg, Germany, 1998), and references therein.

Odegard, J. E.

J. E. Odegard, C. S. Burrus, “Discrete finite variation: a new measure of smoothness for the design of wavelet basis,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 1467–1470.

Orenstein, J.

M. C. Nuss, J. Orenstein, “Terahertz time-domain spectroscopy (THz-TDS),” in Millimeter and Submillimeter Wave Spectroscopy of Solids, G. Grüner, ed. (Springer–Verlag, Heidelberg, Germany, 1998), and references therein.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

Rudd, J. V.

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

Sato, K.

K. Sato, T. Sato, H. Sone, T. Takagi, “Development of a high-speed time-resolved spectroscope and its application to analysis of time-varying optical spectra,” IEEE Trans. Instrum. Meas. IM-36, 1045–1049 (1987).
[CrossRef]

Sato, T.

K. Sato, T. Sato, H. Sone, T. Takagi, “Development of a high-speed time-resolved spectroscope and its application to analysis of time-varying optical spectra,” IEEE Trans. Instrum. Meas. IM-36, 1045–1049 (1987).
[CrossRef]

Savit, C.

M. Dobrin, C. Savit, Introduction to Geophysical Prospecting, 4th ed. (McGraw-Hill, New York, 1988).

Smith, P. R.

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

Sone, H.

K. Sato, T. Sato, H. Sone, T. Takagi, “Development of a high-speed time-resolved spectroscope and its application to analysis of time-varying optical spectra,” IEEE Trans. Instrum. Meas. IM-36, 1045–1049 (1987).
[CrossRef]

Sun, F. G.

Q. Wu, F. G. Sun, P. Campbell, X.-C. Zhang, “Dynamic range of an electro-optic field sensor and its imaging applications,” Appl. Phys. Lett. 68, 3224–3226 (1996).
[CrossRef]

Takagi, T.

K. Sato, T. Sato, H. Sone, T. Takagi, “Development of a high-speed time-resolved spectroscope and its application to analysis of time-varying optical spectra,” IEEE Trans. Instrum. Meas. IM-36, 1045–1049 (1987).
[CrossRef]

van Exter, M.

Wu, Q.

Q. Wu, F. G. Sun, P. Campbell, X.-C. Zhang, “Dynamic range of an electro-optic field sensor and its imaging applications,” Appl. Phys. Lett. 68, 3224–3226 (1996).
[CrossRef]

Yuan, X.-C.

R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
[CrossRef]

Zhang, X.-C.

Z. G. Lu, P. Campbell, X.-C. Zhang, “Free-space electro-optic sampling with a high-repetition-rate regenerative amplified laser,” Appl. Phys. Lett. 71, 593–595 (1997).
[CrossRef]

Q. Wu, F. G. Sun, P. Campbell, X.-C. Zhang, “Dynamic range of an electro-optic field sensor and its imaging applications,” Appl. Phys. Lett. 68, 3224–3226 (1996).
[CrossRef]

Zhongyu, F.

L. Hui, F. Zhongyu, Y. Jianqi, “Infrared imaging solar spectrograph at Purple Mountain Observatory,” Sol. Phys. 185, 69–76 (1999).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (2)

D. M. Mittleman, M. Gupta, R. Neelamani, R. G. Baraniuk, J. V. Rudd, M. Koch, “Recent advances in terahertz imaging,” Appl. Phys. B 68, 1085–1094 (1999).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

Appl. Phys. Lett. (2)

Q. Wu, F. G. Sun, P. Campbell, X.-C. Zhang, “Dynamic range of an electro-optic field sensor and its imaging applications,” Appl. Phys. Lett. 68, 3224–3226 (1996).
[CrossRef]

Z. G. Lu, P. Campbell, X.-C. Zhang, “Free-space electro-optic sampling with a high-repetition-rate regenerative amplified laser,” Appl. Phys. Lett. 71, 593–595 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

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

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

L. Duvillaret, F. Garet, J. 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]

IEEE Trans. Instrum. Meas. (1)

K. Sato, T. Sato, H. Sone, T. Takagi, “Development of a high-speed time-resolved spectroscope and its application to analysis of time-varying optical spectra,” IEEE Trans. Instrum. Meas. IM-36, 1045–1049 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

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

Opt. Lett. (3)

Sol. Phys. (1)

L. Hui, F. Zhongyu, Y. Jianqi, “Infrared imaging solar spectrograph at Purple Mountain Observatory,” Sol. Phys. 185, 69–76 (1999).
[CrossRef]

Ultramicroscopy (1)

R. E. Burge, X.-C. Yuan, J. N. Knauer, M. T. Browne, P. Charalambous, “Scanning soft X-ray imaging at 10 nm resolution,” Ultramicroscopy 69, 259–278 (1997).
[CrossRef]

Other (8)

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer–Verlag, Heidelberg, Germany, 1998), pp. 2–7.

E. Hecht, Optics, 2nd ed. (Addison–Wesley, Reading, Mass., 1987).

M. Dobrin, C. Savit, Introduction to Geophysical Prospecting, 4th ed. (McGraw-Hill, New York, 1988).

S. Haykin, Adaptive Filter Theory (Prentice-Hall, Englewood Cliffs, N.J., 1996).

J. E. Odegard, C. S. Burrus, “Discrete finite variation: a new measure of smoothness for the design of wavelet basis,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 1467–1470.

F. Jones, Lebesgue Integration on Euclidean Space (Jones and Bartlett, Boston, Mass., 1993).

M. C. Nuss, J. Orenstein, “Terahertz time-domain spectroscopy (THz-TDS),” in Millimeter and Submillimeter Wave Spectroscopy of Solids, G. Grüner, ed. (Springer–Verlag, Heidelberg, Germany, 1998), and references therein.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

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Figures (9)

Fig. 1
Fig. 1

(a) Time-domain signals from the THz system without a sample (thick curves) and with a sample (thin curves) of a 0.51±0.02-mm-thick silicon wafer obtained through transmission. The parameter-estimation algorithm described in the text requires the primary transmission and two multiples. Three multiples are evident in the output. The signals are offset for clarity. (b) Corresponding discrete Fourier transform magnitudes of the signals from (a). (c) Fourier deconvolution of the sample versus no sample obtained using frequency values between 250 GHz and 1.4 THz.

Fig. 2
Fig. 2

Transmission and reflection pathways for a THz wave through a planar, homogenous material. We rotate the sample in the drawing to exaggerate nonnormal incidence and clarify the multiple-reflection pathways. Our model describes the interactions at each transmission or reflection interface and the propagation through the material. The resulting signal includes the first transmission through the sample and two multiples caused by internal reflections.

Fig. 3
Fig. 3

Final real index of refraction obtained from our algorithm at various guessed thicknesses for a sample of GaAs. As we approach the appropriate thickness, the oscillations in the complex index of refraction decrease. The general trend as the guessed thickness increases, however, is the decrease in amplitude for the complex refractive index. This leads us to use the deepest local minimum and the total variation of degree of one metric in Eqs. (19)–(21).

Fig. 4
Fig. 4

(a) Total error between the model and measured signals, according to Eq. (15), for a 0.51±0.02-mm-thick sample of silicon plotted over a wide range of thicknesses. The box indicates the boundaries for the next graph. (b) We compare the total error curve (circles) against the total variation metric (triangles) for the intermediate stepping distance 0.01 mm. The deepest local minimum for total variation provides a better indicator of the actual thickness than total error since it has a wider concave region and is more noise tolerant. The measured thickness with error region is shown. (c) Corresponding real and imaginary indices of refraction for the thickness identified by the modified total-variation deepest local minimum of the final pass (solid lines). Literature data puts the complex index of refraction of silicon at 3.42-j0 (dashed lines).

Fig. 5
Fig. 5

(a) Total-variation measure for GaAs shown with a 0.1-mm stepping distance. (b) Complex refractive index for the estimated thickness (solid line) compared with the index from the literature (dashed line). We note the slight dispersion of the material.

Fig. 6
Fig. 6

(a) Total error and total-variation measure for InP shown with a 0.1-mm stepping distance. We note that the initial stepping distance did not produce a local minimum for total error, although one does exist for total variation. (b) Real refractive index for the estimated thickness compared with the literature data. Literature data is not available for the imaginary refractive index over the frequency range shown, but an imaginary value of 8.7×10-3 at 6 THz is reported in the literature.25

Fig. 7
Fig. 7

Complex index of refraction for LiNbO3 (ordinary axis) displayed at the estimated thickness of 0.49 mm. The real refractive index tracks the literature data well; however, some low-frequency noise exists in the raw-data signals. We note the increasing absorption at higher frequencies captured by our estimate, as expected.

Fig. 8
Fig. 8

(a) Total error (circles) and total variation (triangles) for a simulated material with a low real index of refraction. The deepest local minimum for total error does not occur at the proper thickness with a 0.1-mm-thickness stepping size. (b) Total error and total variation as in (a) with a 0.001-mm-thickness stepping size. A global minimum is possible for both measures, but real data often provide only a deepest local minimum, owing to noise. Total variation is a more robust and computationally efficient metric to indicate the correct material thickness, owing to the larger thickness step size allowed.

Fig. 9
Fig. 9

Limits of our method displayed for a simulated material at a signal-to-noise ratio of 1500:1. The circles indicate the test cases that passed for the thickness indicated by the vertical dashed lines. At each point, the real refractive index was constant across frequency, while the imaginary component was zero. The darker area represents the passing region. The area below the passing region is signal-to-noise limited. The lighter area at the top indicates a region that should pass; however, the initial stepping distance for thickness needs to be smaller and/or the data record needs to be longer. At larger thicknesses, the multiple reflections might exceed the data record length.

Equations (24)

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tab(ω)=2n˜a(ω)cos θn˜a(ω)cos β+n˜b(ω)cos θ ,
rab(ω)=n˜b(ω)cos θ-n˜a(ω)cos βn˜a(ω)cos β+n˜b(ω)cos θ,
β=arcsinnasin θnb.
pb(ω, d)=exp-jn˜b(ω)ωdc,
d=lcos β.
Eref(ω)=Einitial(ω)pair(ω, x),
n˜air(ω)=1.00027-j0,
m=d cos(θ-β),whereθβ.
Eprimary(ω)=Einitial(ω)pair[ω,(x-m)]×t01 psample(ω, d)t10,
Efirstmultiple(ω)=Einitial(ω)pair[ω,(x-m)]×t01 psample(ω, d)r10 psample(ω, d)×r10 psample(ω, d)t10=Einitial(ω)pair[ω,(x-m)]×t01 psample(ω, d)t10r102 psample2(ω, d),
Esecondmultiple(ω)=Einitial(ω)pair[ω,(x-m)]×t01 psample(ω, d)t10r104 psample4(ω, d).
Ecomplete(ω)=Einitial(ω)pair[ω,(x-m)]t01psample(ω,d)×t10{1+k=12[r 102p sample2(ω,d)]h}FP(ω).
Hˆ(ω)=Ecomplete(ω)Eref (ω)=4n˜air(ω)n˜sample(ω)cos θ cos β[n˜air(ω)cos β+n˜sample(ω)cos θ]2×exp-j[dn˜sample(ω)-mn˜air(ω)]ωcFP(ω).
H(ω)=Esample(ω)Eref (ω).
mER(ω)=|H(ω)|-|Hˆ(ω)|,
pER(ω)=H(ω)-Hˆ(ω).
ER=ω|mER(ω)|+|pER(ω)|.
lupper=Δt cn1-nair,ltower=Δt cn2-nair,
n=argmax[|h(t)|]cl+1.00027,
nnew(ω)=nold(ω)+ pER(ω),
κnew(ω)=κold(ω)+ mER(ω),
D[m]=|n[m-1]-n[m]|+|κ[m-1]-κ[m]|,
TV=D[m],
TV2=|D[m]-D[m+1]|.

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