Abstract

We develop a generalized causality-based framework for determining the time origin in terahertz emission spectroscopy. Our framework is formulated in terms of a multiply subtractive Kramers-Kronig relation and can treat all major mechanisms of terahertz emission, which include the occurrence of a delta-function-like instantaneous polarization observed typically in nonlinear optical processes. We show that a function derived within our framework properly determines the positions of t = 0 both for simulated terahertz waveforms and for a measured one obtained in biased conjugated polymers. This function will be useful for an in-depth understanding of ultrafast phenomena involving terahertz emission in various optoelectronic materials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Causality-based method for determining the time origin in terahertz emission spectroscopy

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Opt. Express 19(13) 12759-12765 (2011)

Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy

Takeya Unuma, Yusuke Ino, Makoto Kuwata-Gonokami, Erik M. Vartiainen, Kai-Erik Peiponen, and Kazuhiko Hirakawa
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Comparison of subtractive Kramers-Kronig analysis and maximum entropy model in resolving phase from finite spectral range reflectance data

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References

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  1. R. de L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. 12(6), 547–557 (1926).
    [Crossref]
  2. H. A. Kramers, “La diffusion de la lumière par les atomes,” in Atti del Congresso Internazionale dei Fisici, Como (Zanichelli, 1927), Vol. 2, pp. 545–557.
  3. V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).
  4. V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B Condens. Matter Mater. Phys. 72(12), 125107 (2005).
    [Crossref]
  5. T. Unuma, Y. Ino, K.-E. Peiponen, E. M. Vartiainen, M. Kuwata-Gonokami, and K. Hirakawa, “Causality-based method for determining the time origin in terahertz emission spectroscopy,” Opt. Express 19(13), 12759–12765 (2011).
    [Crossref] [PubMed]
  6. M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
    [Crossref]
  7. H. Son, D.-H. Choi, and G.-S. Park, “Improved thickness estimation of liquid water using Kramers-Kronig relations for determination of precise optical parameters in terahertz transmission spectroscopy,” Opt. Express 25(4), 4509–4518 (2017).
    [Crossref] [PubMed]
  8. T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
    [Crossref]
  9. A. Naka, K. Hirakawa, and T. Unuma, “Capacitive response and room-temperature terahertz gain of a Wannier-Stark ladder system in GaAs-based superlattices,” Appl. Phys. Express 9(11), 112101 (2016).
    [Crossref]
  10. K. Sakai and M. Tani, “Introduction to terahertz pulses,” in Terahertz Optoelectronics ed. K. Sakai (Springer, 2005).
  11. K. F. Palmer, M. Z. Williams, and B. A. Budde, “Multiply subtractive kramers-kronig analysis of optical data,” Appl. Opt. 37(13), 2660–2673 (1998).
    [Crossref] [PubMed]
  12. V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4), 409–414 (2003).
    [Crossref]
  13. T. Unuma and A. Matsuda, “Temperature-dependent spectral linewidths of terahertz Bloch oscillations in biased semiconductor superlattices,” Appl. Phys. Lett. 112(16), 162107 (2018).
    [Crossref]
  14. T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010).
    [Crossref] [PubMed]
  15. T. Unuma, N. Yamada, and H. Kishida, “Terahertz emission from biased conjugated polymers excited by femtosecond laser pulses,” Appl. Phys. Express 9(12), 121601 (2016).
    [Crossref]
  16. Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70(14), 1784–1786 (1997).
    [Crossref]
  17. G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
    [Crossref]

2018 (1)

T. Unuma and A. Matsuda, “Temperature-dependent spectral linewidths of terahertz Bloch oscillations in biased semiconductor superlattices,” Appl. Phys. Lett. 112(16), 162107 (2018).
[Crossref]

2017 (1)

2016 (3)

A. Naka, K. Hirakawa, and T. Unuma, “Capacitive response and room-temperature terahertz gain of a Wannier-Stark ladder system in GaAs-based superlattices,” Appl. Phys. Express 9(11), 112101 (2016).
[Crossref]

M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
[Crossref]

T. Unuma, N. Yamada, and H. Kishida, “Terahertz emission from biased conjugated polymers excited by femtosecond laser pulses,” Appl. Phys. Express 9(12), 121601 (2016).
[Crossref]

2011 (1)

2010 (2)

T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
[Crossref]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010).
[Crossref] [PubMed]

2005 (1)

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B Condens. Matter Mater. Phys. 72(12), 125107 (2005).
[Crossref]

2003 (1)

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4), 409–414 (2003).
[Crossref]

1999 (1)

G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
[Crossref]

1998 (1)

1997 (1)

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70(14), 1784–1786 (1997).
[Crossref]

1926 (1)

Bastard, G.

T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
[Crossref]

Bernier, M.

M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
[Crossref]

Budde, B. A.

Choi, D.-H.

Coutaz, J.-L.

M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
[Crossref]

de L. Kronig, R.

Gallot, G.

G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
[Crossref]

Garet, F.

M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
[Crossref]

Grischkowsky, D.

G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
[Crossref]

Hirakawa, K.

A. Naka, K. Hirakawa, and T. Unuma, “Capacitive response and room-temperature terahertz gain of a Wannier-Stark ladder system in GaAs-based superlattices,” Appl. Phys. Express 9(11), 112101 (2016).
[Crossref]

T. Unuma, Y. Ino, K.-E. Peiponen, E. M. Vartiainen, M. Kuwata-Gonokami, and K. Hirakawa, “Causality-based method for determining the time origin in terahertz emission spectroscopy,” Opt. Express 19(13), 12759–12765 (2011).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
[Crossref]

Ino, Y.

T. Unuma, Y. Ino, K.-E. Peiponen, E. M. Vartiainen, M. Kuwata-Gonokami, and K. Hirakawa, “Causality-based method for determining the time origin in terahertz emission spectroscopy,” Opt. Express 19(13), 12759–12765 (2011).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
[Crossref]

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B Condens. Matter Mater. Phys. 72(12), 125107 (2005).
[Crossref]

Jeon, T.-I.

G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
[Crossref]

Kishida, H.

T. Unuma, N. Yamada, and H. Kishida, “Terahertz emission from biased conjugated polymers excited by femtosecond laser pulses,” Appl. Phys. Express 9(12), 121601 (2016).
[Crossref]

Kuwata-Gonokami, M.

T. Unuma, Y. Ino, K.-E. Peiponen, E. M. Vartiainen, M. Kuwata-Gonokami, and K. Hirakawa, “Causality-based method for determining the time origin in terahertz emission spectroscopy,” Opt. Express 19(13), 12759–12765 (2011).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
[Crossref]

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B Condens. Matter Mater. Phys. 72(12), 125107 (2005).
[Crossref]

Lucarini, V.

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B Condens. Matter Mater. Phys. 72(12), 125107 (2005).
[Crossref]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4), 409–414 (2003).
[Crossref]

Matsuda, A.

T. Unuma and A. Matsuda, “Temperature-dependent spectral linewidths of terahertz Bloch oscillations in biased semiconductor superlattices,” Appl. Phys. Lett. 112(16), 162107 (2018).
[Crossref]

McGowan, R. W.

G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
[Crossref]

Minamide, H.

M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
[Crossref]

Naka, A.

A. Naka, K. Hirakawa, and T. Unuma, “Capacitive response and room-temperature terahertz gain of a Wannier-Stark ladder system in GaAs-based superlattices,” Appl. Phys. Express 9(11), 112101 (2016).
[Crossref]

Palmer, K. F.

Park, G.-S.

Peiponen, K.-E.

T. Unuma, Y. Ino, K.-E. Peiponen, E. M. Vartiainen, M. Kuwata-Gonokami, and K. Hirakawa, “Causality-based method for determining the time origin in terahertz emission spectroscopy,” Opt. Express 19(13), 12759–12765 (2011).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010).
[Crossref] [PubMed]

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B Condens. Matter Mater. Phys. 72(12), 125107 (2005).
[Crossref]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4), 409–414 (2003).
[Crossref]

Saarinen, J. J.

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4), 409–414 (2003).
[Crossref]

Sato, A.

M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
[Crossref]

Son, H.

Unuma, T.

T. Unuma and A. Matsuda, “Temperature-dependent spectral linewidths of terahertz Bloch oscillations in biased semiconductor superlattices,” Appl. Phys. Lett. 112(16), 162107 (2018).
[Crossref]

T. Unuma, N. Yamada, and H. Kishida, “Terahertz emission from biased conjugated polymers excited by femtosecond laser pulses,” Appl. Phys. Express 9(12), 121601 (2016).
[Crossref]

A. Naka, K. Hirakawa, and T. Unuma, “Capacitive response and room-temperature terahertz gain of a Wannier-Stark ladder system in GaAs-based superlattices,” Appl. Phys. Express 9(11), 112101 (2016).
[Crossref]

T. Unuma, Y. Ino, K.-E. Peiponen, E. M. Vartiainen, M. Kuwata-Gonokami, and K. Hirakawa, “Causality-based method for determining the time origin in terahertz emission spectroscopy,” Opt. Express 19(13), 12759–12765 (2011).
[Crossref] [PubMed]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
[Crossref]

T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010).
[Crossref] [PubMed]

Vartiainen, E. M.

Williams, M. Z.

Wu, Q.

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70(14), 1784–1786 (1997).
[Crossref]

Yamada, N.

T. Unuma, N. Yamada, and H. Kishida, “Terahertz emission from biased conjugated polymers excited by femtosecond laser pulses,” Appl. Phys. Express 9(12), 121601 (2016).
[Crossref]

Zhang, J.

G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
[Crossref]

Zhang, X.-C.

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70(14), 1784–1786 (1997).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Express (2)

T. Unuma, N. Yamada, and H. Kishida, “Terahertz emission from biased conjugated polymers excited by femtosecond laser pulses,” Appl. Phys. Express 9(12), 121601 (2016).
[Crossref]

A. Naka, K. Hirakawa, and T. Unuma, “Capacitive response and room-temperature terahertz gain of a Wannier-Stark ladder system in GaAs-based superlattices,” Appl. Phys. Express 9(11), 112101 (2016).
[Crossref]

Appl. Phys. Lett. (3)

Q. Wu and X.-C. Zhang, “7 terahertz broadband GaP electro-optic sensor,” Appl. Phys. Lett. 70(14), 1784–1786 (1997).
[Crossref]

G. Gallot, J. Zhang, R. W. McGowan, T.-I. Jeon, and D. Grischkowsky, “Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,” Appl. Phys. Lett. 74(23), 3450–3452 (1999).
[Crossref]

T. Unuma and A. Matsuda, “Temperature-dependent spectral linewidths of terahertz Bloch oscillations in biased semiconductor superlattices,” Appl. Phys. Lett. 112(16), 162107 (2018).
[Crossref]

IEEE Trans. Terahertz Sci. Technol. (1)

M. Bernier, F. Garet, J.-L. Coutaz, H. Minamide, and A. Sato, “Accurate characterization of resonant samples in the terahertz regime through a technique combining time-domain spectroscopy and Kramers-Kronig analysis,” IEEE Trans. Terahertz Sci. Technol. 6(3), 442–450 (2016).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4), 409–414 (2003).
[Crossref]

Opt. Express (3)

Phys. Rev. B Condens. Matter Mater. Phys. (2)

T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B Condens. Matter Mater. Phys. 81(12), 125329 (2010).
[Crossref]

V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B Condens. Matter Mater. Phys. 72(12), 125107 (2005).
[Crossref]

Other (3)

H. A. Kramers, “La diffusion de la lumière par les atomes,” in Atti del Congresso Internazionale dei Fisici, Como (Zanichelli, 1927), Vol. 2, pp. 545–557.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).

K. Sakai and M. Tani, “Introduction to terahertz pulses,” in Terahertz Optoelectronics ed. K. Sakai (Springer, 2005).

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Figures (4)

Fig. 1
Fig. 1 Analysis of THz emission data simulated for a damped cosine current. (a) Temporal waveform of THz electric field E(t) convolved with a resolution of τres = 0.30 ps. (b) Fourier spectra of amplitude ρ(ω) and phase θ(ω). (c) Causality-based functions K(δt) versus possible time-origin misplacement δt (curves 1–3) computed with the three different sets of anchor points (ω0/2π, ω1/2π, ω2/2π) = (0.98, 1.22, 2.06), (0.98, 1.50, 2.06), and (0.98, 1.79, 2.06) THz. (d) Magnified view of (c) around the point where causality is best satisfied.
Fig. 2
Fig. 2 Analysis of THz emission data simulated for a delta-function-like instantaneous polarization with a resolution of τres = 0.030 ps. (a) Temporal waveform of THz electric field E(t). (b) Fourier spectra of amplitude ρ(ω) and phase θ(ω). (c) Causality-based functions K(δt) versus possible time-origin misplacement δt (curves 1–3) computed with the three different sets of anchor points (ω0/2π, ω1/2π, ω2/2π) = (5.63, 8.50, 33.9), (5.63, 17.7, 33.9), and (5.63, 28.9, 33.9) THz. (d) Magnified view of (c) around the point where causality is best satisfied.
Fig. 3
Fig. 3 Analysis of THz emission data simulated for a delta-function-like instantaneous polarization with a resolution of τres = 0.30 ps. (a) Temporal waveform of THz electric field E(t). (b) Fourier spectra of amplitude ρ(ω) and phase θ(ω). (c) Causality-based functions K(δt) versus possible time-origin misplacement δt (curves 1–3) computed with the three different sets of anchor points (ω0/2π, ω1/2π, ω2/2π) = (0.56, 0.85, 3.40), (0.56, 1.77, 3.40), and (0.56, 2.89, 3.40) THz. (d) Magnified view of (c) around the point where causality is best satisfied.
Fig. 4
Fig. 4 Analysis of THz emission data measured for a biased polythiophene film excited by femtosecond laser pulses. (a) Temporal waveform of THz electric field E(t) detected experimentally with a resolution of τres = 0.50 ps and an arbitrary time origin [15]. (b) Fourier spectra of amplitude ρ(ω) and phase θ(ω). (c) Causality-based functions K(δt) versus possible time-origin misplacement δt (curves 1–3) computed with the three different sets of anchor points (ω0/2π, ω1/2π, ω2/2π) = (0.47,0.70, 3.16), (0.47, 1.55, 3.16), and (0.47, 2.80, 3.16) THz. (d) Magnified view of (c) around the point where causality is best satisfied. This analysis indicates that the time origin is corrected to the point denoted by the arrow in (a).

Equations (6)

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E(t)= E ret (t)+Aδ(t)+B δ ˙ (t)+C δ ¨ (t).
P E ˜ ( ω )A+iB ω +C ω 2 ω ω d ω iπ[ E ˜ (ω)A+iBω+C ω 2 ]=0,
E ˜ (ω)= E(t) exp(iωt)dt.
Im E ˜ (ω)+Bω= 1 π P Re E ˜ ( ω )+C ω 2 ω ω d ω .
2Im E ˜ (ω) ω Im E ˜ ( ω 0 ) ω 0 Im E ˜ ( ω 2 ) ω 2 = 2 π P 0 Re E ˜ ( ω )( 1 ω 2 ω 0 2 + 1 ω 2 ω 2 2 2 ω 2 ω 2 ) d ω ,
K(δt)= ρ( ω 0 )sin[θ( ω 0 )+ ω 0 δt] ω 0 + ρ( ω 2 )sin[θ( ω 2 )+ ω 2 δt] ω 2 2ρ( ω 1 )sin[θ( ω 1 )+ ω 1 δt] ω 1 + 2 π P 0 ρ( ω )cos[θ( ω )+ ω δt]( 1 ω 2 ω 0 2 + 1 ω 2 ω 2 2 2 ω 2 ω 1 2 ) d ω .