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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 Optical Society of America
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 [1–3]. 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 [4–7].
Recently, the singly subtractive K-K relation has been applied to another type of problem in terahertz (THz) reflection spectroscopy [8]. Here, both the amplitude and phase spectra of THz electric fields are obtained in principle but an artificial phase shift usually arises from the spatial misplacement between the sample and reference materials. A similar artifact also appears in THz emission spectroscopy, where the exact position of the time origin (t = 0) is not given experimentally. In a previous paper, we demonstrated that the misplacement δt of the time origin in emitted THz waveforms can be detected by the maximum entropy method [9]. However, it is an elaborate algorithm based on the information theory and works in a less intuitive way for physics and optoelectronics.
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
Here, A is a constant and causality requires the absence of any advanced carrier response:The Fourier transform of Eq. (1) is given byWhen 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
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 Ẽ THz(ω):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], we set an anchor point ωa within a measurement frequency range and calculate the subtraction ReẼ THz(ω) – ReẼ THz(ωa) using Eq. (5). We thus obtain the singly subtractive K-K relation: For the latter equality in Eq. (6), we have used the fact that E THz(t) is a real-valued function and hence ImẼ THz(–ω) = –ImẼ THz(ω).Now, we consider the case where E THz(t) has a time-origin misplacement δt. In this case, since the phase spectrum θ exp(ω) = argẼ THz(ω) includes an artificial linear component of –ωδt and the amplitude spectrum ρ(ω) = |Ẽ THz(ω)| remains unchanged, ReẼ THz(ω) and ImẼ THz(ω) are replaced by ρ(ω)cos[θ exp(ω)+ωδt] and ρ(ω)sin[θ exp(ω)+ωδt], respectively. With two anchor points ωa = ω 1 and ω 2, Eq. (6) gives the condition for δt that K(δt) = 0, where K(δt) is defined by
K(δt) has some properties that can be used to find the actual solution for δt numerically, as described in the next section. For the computation of K(δt), we choose a pair of anchor points around the quarter-maximum points of ρ(ω) considering a trade-off: the –ωδt phase shift is reflected more accurately in K(δt) by more distant anchor points, while the spectral distortion due to noise and system response functions is reflected less in K(δt) by anchor points nearer to the maximum point of ρ(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] and Drude-Smith transport [11] 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) 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 Aδ(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.
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).
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] 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]. 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] 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 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).
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).
From the simulation results, we obtain a practical procedure for determining the time origin in E THz(t): First, we calculate the causality-based function K(δt) from numerical data on E THz(t) and search for the point δt' at which the fluctuation in K(δt) is damped the most versus possible misplacement δt. Then, with the given temporal resolution τ res of E THz(t), we estimate the actual misplacement to be δt = δt' – τ res and remove it from E THz(t).
3.2. Application to actual THz waveforms
Below, we perform this procedure in an example of actual THz waveforms recorded experimentally with an arbitrary time origin. Figure 3(a) shows a THz waveform E THz(t) observed for the Bloch oscillation in a GaAs(7.5 nm)/AlAs(0.8 nm) superlattice under a dc bias electric field of F = 8.2 kV/cm [9]. Here, the THz signals were measured using a ZnTe electro-optic sensor with nearly flat sensitivity up to 3.5 THz [14], which corresponds to a temporal resolution of τ res = 0.28 ps. More details of the experiment are given in Ref [10]. The amplitude spectrum ρ(ω) and phase spectrum θ exp(ω) obtained from the THz waveform in Fig. 3(a) after zero filling of the waveform edges are shown in Fig. 3(b) by black solid and dash-dotted curves, respectively. ρ(ω) exhibits a resonance feature at ω/2π = 1.47 THz, nearly equal to the expected Bloch frequency eFd/h (d: superlattice period) [10]. It should be noted that, though θ exp(ω) exhibits no clear linear component in Fig. 3(b), E THz(t) has a possible misplacement δt beyond the empirical search for the –ωδt phase shift.
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).
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]. 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].
Note that the measurement frequency range is wide enough and the signal-to-noise ratio is high enough to find proper anchor points in Fig. 3(b); otherwise, the computation of K(δt) would be unstable with integration frequency ranges and lead to an inaccurate estimation of the actual misplacement δt. When K(δt) is calculated with proper anchor points, noise in ρ(ω) and θ exp(ω) is usually the main factor in the uncertainty of δt detection, partly depending on zero filling of the E THz(t) edges related to the frequency interval of spectral data points.
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]. 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
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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.
