Abstract

We analyze the contributions of various error sources to uncertainty in the far-infrared optical constants (refractive index and absorption coefficient) measured by terahertz (THz) time-domain spectroscopy. We focus our study on the influence of noise. This noise study is made with a thick slab of transparent material for which the THz transmitted signal exhibits temporal echoes owing to reflections in the sample. Extracting data from each of these time-windowed echoes allows us to characterize the noise sources. In THz time-domain spectroscopy experiments in which photoswitches are used as antennae, the transmitting antenna constitutes the principal noise source. The uncertainty in the far-infrared optical constants can be strongly reduced when the extraction is performed with THz echoes that have encountered many reflections in the sample.

© 2000 Optical Society of America

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References

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  1. P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255–260 (1988); Ch. Fattinger and D. Grischkowsky, “Point source terahertz optics,” Appl. Phys. Lett. 53, 1480–1483 (1988).
    [CrossRef]
  2. J. Houghton and S. D. Smith, Infrared Physics (Oxford U. Press, London, 1966).
  3. R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).
  4. M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
    [CrossRef]
  5. D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]
  6. M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
    [CrossRef] [PubMed]
  7. S. D. Brorson, R. Buhleier, I. E. Trofimov, J. O. White, Ch. Ludwig, F. F. Balakirev, H.-U. Habermeier, and J. Kuhl, “Electrodynamics of high-temperature superconductors investigated with coherent terahertz spectroscopy,” J. Opt. Soc. Am. B 13, 1979–1993 (1996).
    [CrossRef]
  8. J. F. Whitaker, F. Gao, and Y. Liu, “Terahertz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, R. C. Eckardt, H. Everitt, D. D. Lowenthal, and N. Peyghambarian, eds., Proc. SPIE 2145, 168–177 (1994).
    [CrossRef]
  9. J. E. Pedersen and S. R. Keiding, “THz time-domain spectroscopy of nonpolar liquids,” IEEE J. Quantum Electron. 28, 2518–2522 (1992); S. R. Keiding, “Dipole correlation functions in liquid benzenes measured with THz time domain spectroscopy,” J. Chem. Phys. A 101, 5250–5254 (1997).
    [CrossRef]
  10. Y. Pastol, G. Arjavalingam, G. V. Kopcsay, and J.-M. Halbout, “Dielectric properties of uniaxial crystals measured with optoelectronically generated microwave transient radiation,” Appl. Phys. Lett. 55, 2277–2279 (1989).
    [CrossRef]
  11. Y. Pastol, G. Arjavalingam, J.-M. Halbout, and G. V. Kopcsay, “Absorption and dispersion of low-loss dielectrics measured with microwave transient radiation,” Electron. Lett. 25, 523–524 (1989).
    [CrossRef]
  12. R. A. Cheville and D. Grischkowsky, “Far-infrared terahertz time-domain spectroscopy of flames,” Opt. Lett. 20, 1646–1648 (1995).
    [CrossRef] [PubMed]
  13. H. Harde, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz coherent transients from methyl chloride vapor,” J. Opt. Soc. Am. B 11, 1018–1030 (1994).
    [CrossRef]
  14. S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
    [CrossRef]
  15. B. I. Greene, J. F. Federici, D. R. Dykaar, A. F. J. Levi, and L. Pfeiffer, “Picosecond pump and probe spectroscopy utilizing freely propagating terahertz radiation,” Opt. Lett. 16, 48–49 (1991).
    [CrossRef] [PubMed]
  16. S. S. Prabhu, S. E. Ralph, M. R. Melloch, and E. S. Harmon, “Carrier dynamics of low temperature grown GaAs observed via terahertz spectroscopy,” in Digest of Topical Meeting on Ultrafast Phenomena (Optical Society of America, Washington, D.C., 1996), paper UTuB5-1, pp. 146–148.
  17. P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. 20, 807–809 (1995).
    [CrossRef] [PubMed]
  18. P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13, 2424–2436 (1996).
    [CrossRef]
  19. 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]
  20. H. Sakai, G. A. Vanasse, and M. L. Forman, “Spectral recovery in Fourier spectroscopy,” J. Opt. Soc. Am. 58, 84 (1968).
    [CrossRef]
  21. P. U. Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, R3052–R3054 (1996).
    [CrossRef]
  22. L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in THz time domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 2, 739–746 (1996).
    [CrossRef]
  23. See, for example, E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985). Nevertheless, this is not the case for highly absorbing materials, such as superconducting thin films, for which another simplification holds.8 The general case does not lead to analytical expressions but can be numerically treated by a reliable and fast method.22
  24. Indeed, the Johnson noise is linked to the photoswitch resistance, which varies under laser illumination. Nevertheless, because of the slight decrease of the mean photoswitch resistance induced by the laser illumination, the Johnson noise is rather independent of the signal. Moreover, the shot noise is proportional to the square root of the signal amplitude and not to its amplitude.
  25. Both noise analysis and uncertainty reduction procedures have been applied with time-domain reflectometry equipment in our laboratory for the characterization of guided structures in the microwave domain. First results confirm the possibility of extending these two methods to another area of time-domain characterization.
  26. M. L. Boas, Mathematical Methods in the Physical Sciences (Wiley, New York, 1983).

1999 (1)

1998 (1)

S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

1996 (4)

P. U. Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, R3052–R3054 (1996).
[CrossRef]

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

S. D. Brorson, R. Buhleier, I. E. Trofimov, J. O. White, Ch. Ludwig, F. F. Balakirev, H.-U. Habermeier, and J. Kuhl, “Electrodynamics of high-temperature superconductors investigated with coherent terahertz spectroscopy,” J. Opt. Soc. Am. B 13, 1979–1993 (1996).
[CrossRef]

P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13, 2424–2436 (1996).
[CrossRef]

1995 (2)

1994 (2)

H. Harde, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz coherent transients from methyl chloride vapor,” J. Opt. Soc. Am. B 11, 1018–1030 (1994).
[CrossRef]

J. F. Whitaker, F. Gao, and Y. Liu, “Terahertz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, R. C. Eckardt, H. Everitt, D. D. Lowenthal, and N. Peyghambarian, eds., Proc. SPIE 2145, 168–177 (1994).
[CrossRef]

1991 (2)

M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
[CrossRef] [PubMed]

B. I. Greene, J. F. Federici, D. R. Dykaar, A. F. J. Levi, and L. Pfeiffer, “Picosecond pump and probe spectroscopy utilizing freely propagating terahertz radiation,” Opt. Lett. 16, 48–49 (1991).
[CrossRef] [PubMed]

1990 (2)

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

1989 (2)

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, and J.-M. Halbout, “Dielectric properties of uniaxial crystals measured with optoelectronically generated microwave transient radiation,” Appl. Phys. Lett. 55, 2277–2279 (1989).
[CrossRef]

Y. Pastol, G. Arjavalingam, J.-M. Halbout, and G. V. Kopcsay, “Absorption and dispersion of low-loss dielectrics measured with microwave transient radiation,” Electron. Lett. 25, 523–524 (1989).
[CrossRef]

1968 (1)

Arjavalingam, G.

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, and J.-M. Halbout, “Dielectric properties of uniaxial crystals measured with optoelectronically generated microwave transient radiation,” Appl. Phys. Lett. 55, 2277–2279 (1989).
[CrossRef]

Y. Pastol, G. Arjavalingam, J.-M. Halbout, and G. V. Kopcsay, “Absorption and dispersion of low-loss dielectrics measured with microwave transient radiation,” Electron. Lett. 25, 523–524 (1989).
[CrossRef]

Balakirev, F. F.

Barret, S.

S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

Brorson, S. D.

Buhleier, R.

Cheville, R. A.

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]

S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

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

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]

S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

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

Dykaar, D. R.

Fattinger, Ch.

Federici, J. F.

Forman, M. L.

Gao, F.

J. F. Whitaker, F. Gao, and Y. Liu, “Terahertz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, R. C. Eckardt, H. Everitt, D. D. Lowenthal, and N. Peyghambarian, eds., Proc. SPIE 2145, 168–177 (1994).
[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]

S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

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

Greene, B. I.

Grischkowsky, D.

Habermeier, H.-U.

Halbout, J.-M.

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, and J.-M. Halbout, “Dielectric properties of uniaxial crystals measured with optoelectronically generated microwave transient radiation,” Appl. Phys. Lett. 55, 2277–2279 (1989).
[CrossRef]

Y. Pastol, G. Arjavalingam, J.-M. Halbout, and G. V. Kopcsay, “Absorption and dispersion of low-loss dielectrics measured with microwave transient radiation,” Electron. Lett. 25, 523–524 (1989).
[CrossRef]

Harde, H.

Helm, H.

P. U. Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, R3052–R3054 (1996).
[CrossRef]

Jacobsen, R. H.

Jepsen, P. U.

Katzenellenbogen, N.

Keiding, S.

Keiding, S. R.

Kopcsay, G. V.

Y. Pastol, G. Arjavalingam, J.-M. Halbout, and G. V. Kopcsay, “Absorption and dispersion of low-loss dielectrics measured with microwave transient radiation,” Electron. Lett. 25, 523–524 (1989).
[CrossRef]

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, and J.-M. Halbout, “Dielectric properties of uniaxial crystals measured with optoelectronically generated microwave transient radiation,” Appl. Phys. Lett. 55, 2277–2279 (1989).
[CrossRef]

Kuhl, J.

Labbe-Lavigne, S.

S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

Levi, A. F. J.

Littlewood, P. B.

M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
[CrossRef] [PubMed]

Liu, Y.

J. F. Whitaker, F. Gao, and Y. Liu, “Terahertz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, R. C. Eckardt, H. Everitt, D. D. Lowenthal, and N. Peyghambarian, eds., Proc. SPIE 2145, 168–177 (1994).
[CrossRef]

Ludwig, Ch.

Mankiewich, P. M.

M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
[CrossRef] [PubMed]

Nuss, M. C.

M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
[CrossRef] [PubMed]

O’Malley, M. L.

M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
[CrossRef] [PubMed]

Pastol, Y.

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, and J.-M. Halbout, “Dielectric properties of uniaxial crystals measured with optoelectronically generated microwave transient radiation,” Appl. Phys. Lett. 55, 2277–2279 (1989).
[CrossRef]

Y. Pastol, G. Arjavalingam, J.-M. Halbout, and G. V. Kopcsay, “Absorption and dispersion of low-loss dielectrics measured with microwave transient radiation,” Electron. Lett. 25, 523–524 (1989).
[CrossRef]

Pfeiffer, L.

Sakai, H.

Schall, M.

P. U. Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, R3052–R3054 (1996).
[CrossRef]

Schyja, V.

P. U. Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, R3052–R3054 (1996).
[CrossRef]

Trofimov, I. E.

van Exter, M.

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

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

Vanasse, G. A.

Westerwick, E. H.

M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
[CrossRef] [PubMed]

Whitaker, J. F.

J. F. Whitaker, F. Gao, and Y. Liu, “Terahertz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, R. C. Eckardt, H. Everitt, D. D. Lowenthal, and N. Peyghambarian, eds., Proc. SPIE 2145, 168–177 (1994).
[CrossRef]

White, J. O.

Winnewisser, C.

P. U. Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, R3052–R3054 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, and J.-M. Halbout, “Dielectric properties of uniaxial crystals measured with optoelectronically generated microwave transient radiation,” Appl. Phys. Lett. 55, 2277–2279 (1989).
[CrossRef]

Electron. Lett. (1)

Y. Pastol, G. Arjavalingam, J.-M. Halbout, and G. V. Kopcsay, “Absorption and dispersion of low-loss dielectrics measured with microwave transient radiation,” Electron. Lett. 25, 523–524 (1989).
[CrossRef]

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

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

IEEE Trans. Microwave Theory Tech. (1)

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

J. Appl. Phys. (1)

S. Labbe-Lavigne, S. Barret, F. Garet, L. Duvillaret, and J.-L. Coutaz, “Far-infrared dielectric constant of porous sili-con layers measured by terahertz time-domain spectroscopy,” J. Appl. Phys. 83, 6007–6010 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

Opt. Lett. (3)

Phys. Rev. E (1)

P. U. Jepsen, C. Winnewisser, M. Schall, V. Schyja, S. R. Keiding, and H. Helm, “Detection of THz pulses by phase retardation in lithium tantalate,” Phys. Rev. E 53, R3052–R3054 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

M. C. Nuss, P. M. Mankiewich, M. L. O’Malley, E. H. Westerwick, and P. B. Littlewood, “Dynamic conductivity and ‘coherence peak’ in YBa2Cu3O7 superconductors,” Phys. Rev. Lett. 66, 3305–3308 (1991).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. F. Whitaker, F. Gao, and Y. Liu, “Terahertz-bandwidth pulses for coherent time-domain spectroscopy,” in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, R. C. Eckardt, H. Everitt, D. D. Lowenthal, and N. Peyghambarian, eds., Proc. SPIE 2145, 168–177 (1994).
[CrossRef]

Other (9)

J. E. Pedersen and S. R. Keiding, “THz time-domain spectroscopy of nonpolar liquids,” IEEE J. Quantum Electron. 28, 2518–2522 (1992); S. R. Keiding, “Dipole correlation functions in liquid benzenes measured with THz time domain spectroscopy,” J. Chem. Phys. A 101, 5250–5254 (1997).
[CrossRef]

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255–260 (1988); Ch. Fattinger and D. Grischkowsky, “Point source terahertz optics,” Appl. Phys. Lett. 53, 1480–1483 (1988).
[CrossRef]

J. Houghton and S. D. Smith, Infrared Physics (Oxford U. Press, London, 1966).

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

S. S. Prabhu, S. E. Ralph, M. R. Melloch, and E. S. Harmon, “Carrier dynamics of low temperature grown GaAs observed via terahertz spectroscopy,” in Digest of Topical Meeting on Ultrafast Phenomena (Optical Society of America, Washington, D.C., 1996), paper UTuB5-1, pp. 146–148.

See, for example, E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985). Nevertheless, this is not the case for highly absorbing materials, such as superconducting thin films, for which another simplification holds.8 The general case does not lead to analytical expressions but can be numerically treated by a reliable and fast method.22

Indeed, the Johnson noise is linked to the photoswitch resistance, which varies under laser illumination. Nevertheless, because of the slight decrease of the mean photoswitch resistance induced by the laser illumination, the Johnson noise is rather independent of the signal. Moreover, the shot noise is proportional to the square root of the signal amplitude and not to its amplitude.

Both noise analysis and uncertainty reduction procedures have been applied with time-domain reflectometry equipment in our laboratory for the characterization of guided structures in the microwave domain. First results confirm the possibility of extending these two methods to another area of time-domain characterization.

M. L. Boas, Mathematical Methods in the Physical Sciences (Wiley, New York, 1983).

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Figures (11)

Fig. 1
Fig. 1

Diagram of the procedure for extracting optical constants, calculating the uncertainties in optical constants, and determining noise parameters. FFT, fast Fourier transformation. Numbers in parentheses indicate the equations that are involved in the present procedure.

Fig. 2
Fig. 2

Temporal records of the THz signal without (reference) and with a 480-µm-thick silicon wafer.

Fig. 3
Fig. 3

Modulus of the complex transmission coefficient of a 480-µm-thick silicon wafer for several observable echoes, together with their standard deviations represented by the error bars.

Fig. 4
Fig. 4

Optical constants (circles) of an n-doped, 30-Ω·cm silicon wafer, calculated from relations (4) and (5), together with their standard deviations (error bars).

Fig. 5
Fig. 5

Uncertainties in both (a) the refractive index and (b) the absorption coefficient of silicon measured from a set of 12 measurements (circles) and calculated (solid curves) from relations (14) and (15) for echoes 0 and 2.

Fig. 6
Fig. 6

Standard deviation σρ of modulus ρ of the complex transmission coefficient of silicon versus ρ measured for five different echoes (circles) and fitted by relation (21) (solid curve) at a frequency of 250 GHz.

Fig. 7
Fig. 7

Dispersion of the noise parameters A(ω), B(ω), and C(ω) extracted from Fig. 3 by use of relation (21) (dots) together with the calculated value of noise parameter B(ω) (solid curve) and the shape of 1/|R(ω)|2 (dashed line).

Fig. 8
Fig. 8

Influence of noise parameters A(ω), B(ω), and C(ω) on the characterization of absorbing or transparent materials and on the dynamic of measurement.

Fig. 9
Fig. 9

(a) Optimal echo number p for the extraction of the optical constants n and α and (b) achieved reduction of their uncertainties versus n and αl. An example of an n-doped (30-Ω·cm) silicon wafer is represented by shaded squares.

Fig. 10
Fig. 10

Uncertainties in both (a) the refractive index and (b) the absorption coefficient of silicon, deduced from a set of 12 measurements (circles) for echoes 0–4 and calculated (solid curves) with relations (2), (14), (15), and (21).

Fig. 11
Fig. 11

Diagram of the various time windows involved in the noise analysis.

Equations (36)

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

σρ2(ω)=A(ω)ρ2(ω)+B(ω)ρ(ω)+C(ω),
Et(ω)=Ei(ω)τ(ω)P(ω, l)p=0+[r(ω)P2(ω, l)]p=PF(ω),
ρ(p, ω)|T(p, ω)|=4n2+κ2[(n-1)2+κ2]p[(n+1)2+κ2]p+1×exp-κ(2p+1) ωlc,
φ(p, ω)arg[T(p, ω)]=-[(2p+1)n-1] ωlc-2p×arctan2κn2+κ2-1-arctanκn(n+1)+κ2.
n=1-cωlφ(p, ω)(2p+1),
α=-2(2p+1)lln(n+1)2p+24n(n-1)2pρ(p, ω).
Δn=c(2p+1)ωlΔφ(p, ω),
Δα=2(2p+1)l(n-1)2-4pnn(n2-1)Δn+Δρ(p, ω)ρ(p, ω).
Texp(p, ω)=T(p, ω)+TN(p, ω)ρ(p, ω)exp[jφ(p, ω)]+ρN(p, ω)×exp[jφN(p, ω)],
ρexp(p, ω)ρ(p, ω)+ρN(p, ω)cos[φ(p, ω)-φN(p, ω)],
φexp(p, ω)=arctanρ(p, ω)sin[φ(p, ω)]+ρN(p, ω)sin[φN(p,ω)]ρ(p, ω)cos[φ(p, ω)]+ρN(p, ω)cos[φN(p, ω)].
σρ2(p, ω)=1M1M[ρexp(p, ω)-ρ(p, ω)]2=1M1MρN2(p, ω)cos2[φ(p, ω)-φN(p, ω)].
σρ2(p, ω)=ρN2(p, ω)cos2[φ(p, ω)-φN(p, ω)]=ρN2(p, ω)/2
σρ(p, ω)Δρ(p, ω)=[ρN2(p, ω)/2]1/2,
σφ2(p, ω)=1M1M arctan2sin[φ(p, ω)-φN(p, ω)]ρ(p, ω)/ρN(p, ω)+cos[φ(p, ω)-φN(p, ω)]sin2[φ(p, ω)-φN(p, ω)][ρ(p, ω)/ρN(p, ω)]2
σφ(p, ω)Δφ(p, ω)=[ρN2(p, ω)]1/22ρ(p, ω)=σρ(p, ω)ρ(p, ω).
Δn=c(2p+1)ωlσρ(p, ω)ρ(p, ω),
Δα=2Δnωc+1(2p+1)l(n-1)2-4pnn(n2-1).
Texp(p, ω)=S(p, ω)+SN(p, ω)R(ω)+RN(ω).
Texp(p, ω)T(p, ω)+SN(p, ω)-T(p, ω)RN(ω)R(ω),
σρ2(p, ω)=σS2(p, ω)|R(ω)|2+ρ2(p, ω)|R(ω)|2σR2(ω),
σsignal2(ω)=ρ2(ω)σe2(ω)+σsh2(ω)+σd2(ω),
σR2(ω)=σe2(ω)+2eR(ω)Δf+σd2(ω)
σS2(p, ω)=ρ2(p, ω)σe2(ω)+2eS(p, ω)Δf+σd2(ω),
σρ2(p, ω)=A(ω)ρ2(p, ω)+B(ω)ρ(p, ω)+C(ω),
A(ω)=2σe2(ω)+σd2(ω)+2eΔf|R(ω)||R(ω)|2,
B(ω)=2eΔf|R(ω)|,C(ω)=σd2(ω)|R(ω)|2.
A(ω)=2σe2(ω)/|R(ω)|2+B(ω)+C(ω).
σJ=4kBTΔfRswitch1/2,
T0 m|Rm(ω=2πm/T0)|2
=0T0R2(t)dt0TeffR2(t)dt=Teff n|R˜n(ω=2πn/Teff)|2,
|R(ω=2πn/Teff)||R˜n|(Teff/T0).
σd2(t)=1T00T0σd2(t)dt=mσd2(ω=2πm/T0).
σd2(t)=m|Rm(ω=2πm/T0)|2×T0TsignalC(ω=2πm/T0).
σd2(t)TeffTsignaln|R˜n|2C(ω=2πn/Teff).
σd2(t)TeffTsignalN|R˜(ω)|2C(ω),

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