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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 13 — Jun. 20, 2011
  • pp: 12759–12765
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Causality-based method for determining the time origin in terahertz emission spectroscopy

Takeya Unuma, Yusuke Ino, Kai-Erik Peiponen, Erik M. Vartiainen, Makoto Kuwata-Gonokami, and Kazuhiko Hirakawa  »View Author Affiliations


Optics Express, Vol. 19, Issue 13, pp. 12759-12765 (2011)
http://dx.doi.org/10.1364/OE.19.012759


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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.

© 2011 OSA

1. Introduction

The Kramers-Kronig (K-K) relation is a fundamental pair of dispersion formulas derived from causality in physics and optoelectronics. It has been widely used for the transformation between the real and imaginary parts (or between the amplitude and phase parts) of complex optical spectra, depending on which part can be obtained more easily in practical situations [1

1. R. de L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. 12(6), 547–557 (1926). [CrossRef]

3

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

]. Dispersion integrals appearing with (semi)infinite frequency ranges in conventional expressions for the K-K relation converge rather slowly, while experimental data are recorded only in finite frequency ranges. It has been reported that the convergence of dispersion integrals can be greatly improved in alternative expressions for the K-K relation, i.e., the singly and multiply subtractive K-K relations, when values for both the real and imaginary parts of complex optical spectra are known at a few points, called anchor points, within a measurement frequency range [4

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

7

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]

].

In the present work, we have determined the time origin on the basis of causality in THz emission spectroscopy. For this purpose, the singly subtractive K-K relation was derived from the causality principle to provide a function for detecting misplacement δt. By investigating the properties of the function in trial THz waveforms simulated for two types of transient current, we obtained a practical procedure for evaluating δt in general THz waveforms. When the procedure was applied to a THz waveform recorded experimentally for the Bloch oscillation in a GaAs/AlAs superlattice, a significant misplacement was indeed detected in good agreement with the value obtained previously by the maximum entropy method. We found that the singly subtractive K-K relation works for the detection of δt with an accuracy that is an order of magnitude higher than the given temporal resolutions.

2. Causality principle in transient THz emission

We describe the causality principle in transient THz emission and derive an equation for determining the time origin (t = 0), i.e., the moment when carriers excited by the peak of an ultrashort optical pulse start emitting THz radiation. Let us suppose that the emitted THz electric field E THz(t) consists of instantaneous and retarded carrier responses, expressed as
ETHz(t)=Aδ(t)+Eret(t).
(1)
Here, A is a constant and causality requires the absence of any advanced carrier response:
Eret(t)=0    for    t<0.
(2)
The Fourier transform of Eq. (1) is given by

E˜THz(ω)A=0Eret(t)exp(iωt)dt.
(3)

When ω is extended to a complex number, the right hand side of Eq. (3) is an analytic function in the upper half of the complex ω-plane. Using the residue theorem, we obtain
PE˜THz(ω)Aωωdωiπ[E˜THz(ω)A]=0,
(4)
where P denotes the Cauchy principal value. The imaginary part of Eq. (4) provides a conventional form of the K-K relation that we exploit in this study for THz(ω):
ReE˜THz(ω)=A+1πPImE˜THz(ω)ωωdω.
(5)
In practice, this type of integral over an infinite frequency range is known to be sensitive to the way of extrapolating input spectral data into immeasurable frequency ranges. To construct an integral that has faster convergence [7

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]

], we set an anchor point ωa within a measurement frequency range and calculate the subtraction Re THz(ω) – Re THz(ωa) using Eq. (5). We thus obtain the singly subtractive K-K relation:
ReE˜THz(ω)=ReE˜THz(ωa)+1πP(1ωω1ωωa)ImE˜THz(ω)dω=ReE˜THz(ωa)+2πP0(ωω2ω2ωω2ωa2)ImE˜THz(ω)dω.
(6)
For the latter equality in Eq. (6), we have used the fact that E THz(t) is a real-valued function and hence Im THz(–ω) = –Im THz(ω).

3. Results and discussion

3.1. Application to trial THz waveforms

To clarify how the causality-based function K(δt) behaves with a possible misplacement δt, we performed numerical calculations of K(δt) from trial THz waveforms E THz(t) simulated for two types of transient current J(t) = J 0Θ(t)exp(–γt)cosω 0 t and J 0Θ(t)γtexp(–γt), where J 0 is the magnitude of the current, Θ(t) the unit step function, γ the damping rate, and ω 0 the resonance frequency. The former and latter types of J(t) correspond to the Bloch oscillation [10

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]

] and Drude-Smith transport [11

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]

] of carriers under dc bias electric fields, respectively. Numerical data on E THz(t) ∝ ∂J(t)/∂t were prepared both before and after it was convolved with a system response function that gives a temporal resolution of τ res = 0.30 ps.

The trial THz waveforms E THz(t) simulated for J(t) = J 0Θ(t)exp(–γt)cosω 0 t before and after the convolution with τ res = 0.30 ps are shown in Fig. 1(a)
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).
by red and black curves, respectively. Here, ω 0/2π = 1.5 THz, γ = 1.1 THz, and the vertical dashed line denotes the exact position of t = 0 used for the simulation. The red curve provides an example of THz emission that has an instantaneous signal at t = 0, i.e., the (t) term in Eq. (1), as well as a clear oscillatory signal with the resonance frequency. Due to the finite temporal resolution, the black curve exhibits a slight penetration of the THz signals into the region where t < 0. Following an algorithm of the fast Fourier transform (FFT), we computed the amplitude spectra ρ(ω) and phase spectra θ exp(ω) to feed them into Eq. (7). The spectral data are displayed in Fig. 1(b) for τ res = 0.30 ps.

Figure 1(c) shows plots of K(δt) for the red curve in Fig. 1(a), calculated using Eq. (7) in an integration frequency range of ω'/2π = 0.0–500.0 THz [12

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

] with three different pairs of anchor points (ω 1/2π, ω 2/2π) = (1.10, 4.12), (1.10, 5.11), and (1.10, 7.74) THz (orange, green, and purple curves, respectively). As seen in the figure, K(δt) fluctuates greatly for δt < 0 but is nearly equal to zero for δt > 0 with only a small fluctuation. This is because Eq. (2) is violated if the time origin is shifted forward from the current position (i.e., δt < 0), while Eq. (2) still holds even if the time origin is shifted backward (i.e., δt > 0). It should be noted that, in practice, K(δt) = 0 for δt > 0 is not completely realized by numerical data with a finite temporal length on E THz(t), which is treated extensionally as a periodic repetition of its original waveform in the FFT algorithm [13

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

]. The inset in Fig. 1(c) provides a magnified view around δt = 0: the small fluctuations for δt > 0 in the three curves are damped the most and exhibit similar behaviors in the limit of δt → +0. This indicates that, if THz signals are recorded without temporal broadening, causality is best satisfied at the actual value for δt (i.e., δt = 0 in this simulation).

Figure 1(d) shows plots of K(δt) for the black curve in Fig. 1(a), calculated using Eq. (7) in an integration frequency range of ω'/2π = 0.0–3.5 THz [12

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

] with three different pairs of anchor points (ω 1/2π, ω 2/2π) = (0.96, 2.03), (0.96, 2.05), and (0.96, 2.07) THz (orange, green, and purple curves, respectively). The three curves have small fluctuations for δt > 0, which are damped the most at δt ~ 0.30 ps [see the inset in Fig. 1(d)] and connected to large fluctuations for δt < 0 through an intermediate region of δt = 0.00–0.30 ps. Thus, K(δt) properly reflects the slight penetration of the THz signals down to t ~ –0.30 ps (= –τ res) shown in Fig. 1(a).

Similar numerical data on E THz(t), ρ(ω), θ exp(ω), and K(δt) are shown in 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).
for J(t) = J 0Θ(t)γtexp(–γt) with γ = 5.0 THz. For this type of J(t), E THz(t) starts from a finite value at t = 0 and does not exhibit a clear resonance, as plotted by the red curve in Fig. 2(a). When the THz signals are traced with τ res = 0.30 ps, they have a nearly monocycle waveform plotted by the black curve in Fig. 2(a) and contain many low-frequency components with the amplitude spectrum peaked at 0.62 THz in Fig. 2(b). K(δt) was calculated for the red curve in Fig. 2(a) with (ω 1/2π, ω 2/2π) = (0.10, 5.94), (0.10, 6.27), and (0.10, 6.60) THz [see Fig. 2(c)] and also for the black curve in Fig. 2(a) with (ω 1/2π, ω 2/2π) = (0.08, 1.88), (0.08, 1.91), and (0.08, 1.94) THz [see Fig. 2(d)]. We found that the same analysis as described above also works for J(t) = J 0Θ(t)γtexp(–γt).

3.2. Application to actual THz waveforms

We calculated the causality-based function K(δt) by substituting the experimental data on ρ(ω) and θ exp(ω) into Eq. (7) with an integration frequency range of ω'/2π = 0.2–3.0 THz [12

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

]. Figure 3(c) shows plots of K(δt) computed with (ω 1/2π, ω 2/2π) = (0.69, 2.09), (0.69, 2.15), and (0.69, 2.22) THz (orange, green, and purple curves, respectively). The three curves have small fluctuations for δt > 0, which appear along the K = 0 line and are damped the most at δt' = 0.47 ± 0.03 ps [see the inset in Fig. 3(c)]. The actual misplacement is thus estimated to be δt = δt'τ res = 0.19 ± 0.03 ps, the uncertainty of which is an order of magnitude smaller than the given temporal resolution τ res = 0.28 ps. As a result, the time origin is corrected to the position indicated by the blue vertical line in Fig. 3(a) and the phase spectrum changes accordingly from the dash-dotted curve to the blue curve in Fig. 3(b). This agrees well with the result obtained previously by the maximum entropy method (δt = 0.18 ± 0.02 ps) [9

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]

].

3.3. Remarks

Finally, we would like to mention that the causality-based method is complementary to the maximum entropy method, which calculates the true phase shift from the power spectrum ρ(ω)2 and thereby reveals the –ωδt phase shift [9

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]

]. As described above, the causality-based method allows us to detect misplacement δt itself rather than the –ωδt phase shift. Therefore, the causality-based method will work more effectively than the maximum entropy method in transient THz emission that consists of relatively low-frequency components.

4. Summary

We derived the singly subtractive K-K relation from the causality principle and introduced a function for detecting the misplacement δt of the time origin in THz emission spectroscopy. The causality-based function was tested on simulated THz waveforms, which led to a practical procedure for evaluating δt in general THz waveforms. Indeed, the misplacement was properly detected when this method was applied to a THz waveform recorded experimentally for the Bloch oscillation in a semiconductor superlattice. We found that the singly subtractive K-K relation works for the detection of δt with an accuracy that is an order of magnitude higher than the given temporal resolutions.

Acknowledgments

This work was partly supported by a KAKENHI (No. 23104716) and a Special Coordination Fund for Promoting Science and Technology (NanoQuine) from MEXT.

References and links

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.

OCIS Codes
(000.3860) General : Mathematical methods in physics
(300.2140) Spectroscopy : Emission
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Spectroscopy

History
Original Manuscript: April 26, 2011
Revised Manuscript: June 5, 2011
Manuscript Accepted: June 6, 2011
Published: June 16, 2011

Citation
Takeya Unuma, Yusuke Ino, Kai-Erik Peiponen, Erik M. Vartiainen, Makoto Kuwata-Gonokami, and Kazuhiko Hirakawa, "Causality-based method for determining the time origin in terahertz emission spectroscopy," Opt. Express 19, 12759-12765 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12759


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References

  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.

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