Abstract

We propose a method for determining the time origin on the basis of causality in terahertz (THz) emission spectroscopy. The method is formulated in terms of the singly subtractive Kramers-Kronig relation, which is useful for the situation where not only the amplitude spectrum but also partial phase information is available within the measurement frequency range. Numerical analysis of several simulated and observed THz emission data shows that the misplacement of the time origin in THz waveforms can be detected by the method with an accuracy that is an order of magnitude higher than the given temporal resolutions.

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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. R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970).
    [CrossRef]
  5. R. K. Ahrenkiel, “Modified Kramers-Kronig analysis of optical spectra,” J. Opt. Soc. Am. 61(12), 1651–1655 (1971).
    [CrossRef]
  6. 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]
  7. V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4–6), 409–414 (2003).
    [CrossRef]
  8. 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 72(12), 125107 (2005).
    [CrossRef]
  9. 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]
  10. 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 81(12), 125329 (2010).
    [CrossRef]
  11. N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). For simplicity, the parameter c of the Drude-Smith model is set to –1 in the present paper.
    [CrossRef]
  12. Numerical results for K(δt) were insensitive to the way of cutting off the integration frequency range in Eq. (7).
  13. K(δt) becomes a periodic function of δt with a period equal to the original temporal length of ETHz(t).
  14. Even if ETHz(t) is recorded with non-flat sensitivity, causality should hold and thus our approach will also work. However, non-flat sensitivity will require a more complicated interpretation of THz signals.

2010

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 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

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 72(12), 125107 (2005).
[CrossRef]

2003

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4–6), 409–414 (2003).
[CrossRef]

2001

N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). For simplicity, the parameter c of the Drude-Smith model is set to –1 in the present paper.
[CrossRef]

1998

1971

1970

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970).
[CrossRef]

1926

Ahrenkiel, R. K.

Bachrach, R. Z.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970).
[CrossRef]

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 81(12), 125329 (2010).
[CrossRef]

Brown, F. C.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970).
[CrossRef]

Budde, B. A.

Hirakawa, K.

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 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]

Ino, Y.

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 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 72(12), 125107 (2005).
[CrossRef]

Kronig, R. de L.

Kuwata-Gonokami, M.

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 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 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 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(4–6), 409–414 (2003).
[CrossRef]

Palmer, K. F.

Peiponen, K.-E.

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 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(4–6), 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(4–6), 409–414 (2003).
[CrossRef]

Smith, N. V.

N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). For simplicity, the parameter c of the Drude-Smith model is set to –1 in the present paper.
[CrossRef]

Unuma, T.

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 81(12), 125329 (2010).
[CrossRef]

Vartiainen, E. M.

Williams, M. Z.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Commun.

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4–6), 409–414 (2003).
[CrossRef]

Opt. Express

Phys. Rev. B

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 72(12), 125107 (2005).
[CrossRef]

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 81(12), 125329 (2010).
[CrossRef]

N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). For simplicity, the parameter c of the Drude-Smith model is set to –1 in the present paper.
[CrossRef]

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970).
[CrossRef]

Other

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).

Numerical results for K(δt) were insensitive to the way of cutting off the integration frequency range in Eq. (7).

K(δt) becomes a periodic function of δt with a period equal to the original temporal length of ETHz(t).

Even if ETHz(t) is recorded with non-flat sensitivity, causality should hold and thus our approach will also work. However, non-flat sensitivity will require a more complicated interpretation of THz signals.

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Figures (3)

Fig. 1
Fig. 1

Analysis of trial THz waveforms simulated for current J(t) = J 0Θ(t)exp(–γt)cosω 0 t with ω 0/2π = 1.5 THz and γ = 1.1 THz. (a) THz waveforms E THz(t) before and after convolution with a temporal resolution of τ res = 0.30 ps (red and black curves, respectively). (b) Spectra of amplitude ρ(ω) and phase θ exp(ω). (c), (d) Causality-based function K(δt) versus possible time-origin misplacement δt computed with three different pairs of anchor points before and after the waveform convolution, respectively (insets: magnified views).

Fig. 2
Fig. 2

Analysis of trial THz waveforms simulated for current J(t) = J 0Θ(t)γtexp(–γt) with γ = 5.0 THz. (a) THz waveforms E THz(t) before and after convolution with a temporal resolution of τ res = 0.30 ps (red and black curves, respectively). (b) Spectra of amplitude ρ(ω) and phase θ exp(ω). (c), (d) Causality-based function K(δt) versus possible time-origin misplacement δt computed with three different pairs of anchor points before and after the waveform convolution, respectively (insets: magnified views).

Fig. 3
Fig. 3

Analysis of a THz waveform observed for the Bloch oscillation in a GaAs/AlAs superlattice. (a) THz waveform E THz(t) recorded experimentally with a temporal resolution of τ res = 0.28 ps and a tentative position of the time origin (t = 0). (b) Spectra of amplitude ρ(ω) and phase θ exp(ω). (c) Causality-based function K(δt) versus possible time-origin misplacement δt computed with three different pairs of anchor points (inset: a magnified view). As a result, the time origin is corrected to the position indicated by the blue vertical line in (a), giving the phase spectrum shown by the blue curve in (b).

Equations (7)

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E THz ( t ) = A δ ( t ) + E ret ( t ) .
E ret ( t ) = 0     for     t < 0.
E ˜ THz ( ω ) A = 0 E ret ( t ) exp ( i ω t ) d t .
P E ˜ THz ( ω ) A ω ω d ω i π [ E ˜ THz ( ω ) A ] = 0 ,
Re E ˜ THz ( ω ) = A + 1 π P Im E ˜ THz ( ω ) ω ω d ω .
Re E ˜ THz ( ω ) = Re E ˜ THz ( ω a ) + 1 π P ( 1 ω ω 1 ω ω a ) Im E ˜ THz ( ω ) d ω = Re E ˜ THz ( ω a ) + 2 π P 0 ( ω ω 2 ω 2 ω ω 2 ω a 2 ) Im E ˜ THz ( ω ) d ω .
K ( δ t ) = ρ ( ω 1 ) cos [ θ exp ( ω 1 ) + ω 1 δ t ] ρ ( ω 2 ) cos [ θ exp ( ω 2 ) + ω 2 δ t ]         2 π P 0 ( ω ω 2 ω 1 2 ω ω 2 ω 2 2 ) ρ ( ω ) sin [ θ exp ( ω ) + ω δ t ] d ω .

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