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We have developed a scheme for determining the time origin by the maximum entropy method (MEM) in time-domain terahertz (THz) emission spectroscopy. By applying the MEM to trial damped sinusoidal waveforms, we confirmed that the MEM gives true phase shifts across the resonance features and that its inherent uncertainty in determining the time origin is ±15 fs for 100-fs-class excitation/sampling optical pulses. Furthermore, when the MEM was applied to a THz waveform recorded experimentally with a finite sampling interval for the Bloch oscillation in a semiconductor superlattice, a misplacement of the time origin was indeed detected with an accuracy limited by the worse of the MEM inherent uncertainty and the sampling interval.
©2010 Optical Society of America
1. Introduction
Time-domain terahertz (THz) emission spectroscopy with the electro-optic sampling technique [1,2] is a useful experimental tool for investigating various kinds of carrier dynamics and elementary excitations, such as transient carrier velocities in inorganic [3,4] and organic [5] semiconductors, dynamical conductivities in semiconductor superlattices [6], and ultrafast spin changes in ferromagnets [7]. The uniqueness of this spectroscopy lies in the phase-sensitive and flat-band detection of emitted THz electric fields, which allows us to directly obtain both the real and imaginary parts of the radiation signals without resorting to the Kramers-Kronig relation.
However, it is not easy to determine the time origin in emitted THz waveforms E THz(t), i.e., the moment when carriers excited by the peak of femtosecond laser pulses start emitting THz radiation. A possible misplacement δt of the time origin in E THz(t) gives rise to an artificial phase shift by −ωδt in its complex Fourier transform defined by
and deforms both the real and imaginary parts of the spectrum. This is indeed very troublesome particularly when complex ac conductivities, Reσ(ω) and Imσ(ω), are derived from the spectral shapes of emitted THz electric fields, ReẼ THz(ω) and ImẼ THz(ω), respectively [6,8]. Thus, the determination of the time origin in E THz(t) is crucial in making the most of its unique information. The conventional way of removing δt is based on an empirical search for the −ωδt component in the phase spectrum argẼ THz(ω) [6]. However, when −ωδt is comparable to or smaller than true phase shifts across resonance features, the empirical method does not work very accurately for such a spectral shape of argẼ THz(ω).In this work, we have demonstrated a systematic determination of the time origin in time-domain THz emission spectroscopy. For this purpose, the maximum entropy method (MEM) was employed to calculate a true phase shift from the power spectrum |Ẽ THz(ω)|2 and to separate it from the −ωδt component in the phase spectrum argẼ THz(ω). By testing the MEM on THz waveforms simulated for two types of damped sinusoidal oscillation current, we confirmed that the MEM properly derives true phase shifts across the resonance features and that its inherent uncertainty in determining the time origin is ±15 fs for 100-fs-class excitation/sampling optical pulses. Furthermore, when the MEM was applied to a THz waveform recorded experimentally with an arbitrary time origin and a finite sampling interval for the Bloch oscillation in a GaAs-based superlattice, a misplacement of the time origin was detected with a practically sufficient accuracy that is comparable to the sampling interval of the data points.
2. Scheme for determining the time origin
The principle of determining the time origin in THz emission spectroscopy is as follows: Given that E THz(t) consists of signal and small random noise, the fluctuation of E THz(t) does not have any particular polarity. In terms of the information theory, this situation can be formulated into the condition that the entropy for E THz(t) should be maximized [9]. The entropy for E THz(t) is known to be a functional of the power spectrum |Ẽ THz(ω)|2 [9]. By maximizing the entropy, we obtain an analytical expression for |Ẽ THz(ω)|2:
Here, Δt is the sampling interval of the data points in E THz(t), and 2M is the number of the data points in |Ẽ THz(ω)|2 in a frequency range used for the MEM analysis. am (m = 1, 2, …, M) and b are parameters called the MEM coefficients, which can be determined in such a way that Eq. (2) fits the measured power spectrum [9–11].Equation (2) allows us to restrict the form of Ẽ THz(ω) to
with a phase factor exp[iφ err(ω)]. We obtain an important message from Eq. (3) that the phase spectrum argẼ THz(ω) can be separated into two parts:withHere, ψM(ω) is the true phase shift, called the MEM phase, that is derived from the spectral feature of the power spectrum |Ẽ THz(ω)|2. φ err(ω) is the remaining part, called the error phase [10,11], that consists of random noise and a systematic shift by −ωδt due to a misplacement δt of the time origin. It should be noted that, since |Ẽ THz(ω)|2 does not depend on the position of t = 0, the misplacement of the time origin does not affect ψM(ω) and appears only in φ err(ω) as the −ωδt component.A major benefit of the MEM is that φ err(ω) has a simpler spectral shape than the original phase spectrum argẼ THz(ω), which consists of both ψM(ω) and φ err(ω). Thus, the MEM will allow us to detect δt in the slope of φ err(ω) more accurately than the empirical method does in the slope of argẼ THz(ω). The scheme described above for removing δt in THz emission spectroscopy is similar to that developed for removing the spatial misplacement between sample and reference in THz reflection spectroscopy [12,13].
3. Results and discussion
3.1. Application to trial THz waveforms
To confirm how properly the MEM derives the true phase shift from the power spectrum |Ẽ THz(ω)|2, we first applied the MEM to trial THz waveforms E THz(t) simulated for two types of damped sinusoidal oscillation current J(t) = J 0Θ(t)exp(−γt)sinω 0 t and J 0Θ(t)exp(−γt)cosω 0 t. Here, J 0 is the magnitude of the current, Θ(t) the unit step function, γ the dephasing rate, and ω 0 the eigenfrequency. In this simulation, where E THz(t) has the exact time origin (i.e., δt = 0) and does not include random noise, the difference between the exact phase spectrum argẼ THz(ω) and the MEM phase spectrum ψM(ω) is attributed to numerical errors in the MEM algorithm itself. We can thus estimate its inherent uncertainty in detecting δt from the slope of the error phase spectrum φ err(ω) = argẼ THz(ω)−ψM(ω). Numerical data on trial THz waveforms were prepared by convolving ∂J(t)/∂t with a system response function that characterizes a temporal resolution τ res.
Figures 1(a) and 2(a) show the trial THz waveforms E THz(t) simulated for the damped sine and cosine currents, respectively, with parameters ω 0/2π = 1.5 THz, γ = 1.1 THz, and τ res = 300 fs. Here, vertical dashed lines denote the exact position of t = 0 used for the simulations. As seen in the figures, the two THz waveforms have rather different features: First, peak A is at t = 0 in Fig. 1(a), whereas peak A' is at t < 0 in Fig. 2(a). Second, peak A is of similar height to peak C and much higher than peak D in Fig. 1(a), whereas peak A' is much lower than peak C' and of similar height to peak D' in Fig. 2(a). Third, peak B is significantly higher than peak C in Fig. 1(a), whereas peak B' is of similar height to peak C' in Fig. 2(a).
Fig. 1 MEM analysis of a trial THz waveform simulated for a damped sine current J(t) = J 0Θ(t)exp(−γt)sinω 0 t. (a) THz waveform E THz(t) simulated with ω 0/2π = 1.5 THz, γ = 1.1 THz, and a temporal resolution of τ res = 300 fs. (b) Spectra of amplitude |Ẽ THz(ω)| (solid curve) and phase argẼ THz(ω) (dash-dotted curve). (c) MEM phase spectra ψM(ω) computed in three different frequency ranges. (d) Error phase spectra φ err(ω) (solid curves). Dashed lines are the linear fits to φ err(ω).
Fig. 2 MEM analysis of a trial THz waveform simulated for a damped cosine current J(t) = J 0Θ(t)exp(−γt)cosω 0 t. (a) THz waveform E THz(t) simulated with ω 0/2π = 1.5 THz, γ = 1.1 THz, and a temporal resolution of τ res = 300 fs. (b) Spectra of amplitude |Ẽ THz(ω)| (solid curve) and phase argẼ THz(ω) (dash-dotted curve). (c) MEM phase spectra ψM(ω) computed in three different frequency ranges. (d) Error phase spectra φ err(ω) (solid curves). Dashed lines are the linear fits to φ err(ω).
Figure 1(b) shows the amplitude spectrum |Ẽ THz(ω)| (solid curve) and the phase spectrum argẼ THz(ω) (dash-dotted curve) calculated from the THz waveform E THz(t) of Fig. 1(a). As seen in the figure, |Ẽ THz(ω)| has a resonance peak at ω/2π = 1.48 THz and argẼ THz(ω) displays a true phase shift from −π/2 to π/2 across the resonance feature. We then fitted the power spectrum |Ẽ THz(ω)|2 with Eq. (2) to determine MEM coefficients am (m = 1, 2, …, M) and b, which can be numerically done in a Toeplitz matrix equation [12]. Figure 1(c) shows the MEM phase spectrum ψM(ω) computed using Eq. (5) with the obtained set of am in three different frequency ranges: 0.14–2.99 THz (blue), 0.20–2.94 THz (red), and 0.29–2.90 THz (green) [14]. As seen in the figure, ψM(ω) changes across the resonance feature and curls up near the edges of the respective frequency ranges. The difference between argẼ THz(ω) and ψM(ω), i.e., the error phase spectrum φ err(ω), is shown in Fig. 1(d) by solid curves in the corresponding colors. As seen in the figure, φ err(ω) exhibits small but finite values, indicating numerical errors inherent to the MEM algorithm. We performed a least-squares fitting of φ err(ω) with a straight line in the respective frequency ranges (dashed lines). From different slopes of these three linear fits to φ err(ω), we find that a typical error in detecting δt by the MEM is δt err = ±15 fs for the given temporal resolution of τ res = 300 fs.
Similarly, the spectral data on |Ẽ THz(ω)| (solid curve) and argẼ THz(ω) (dash-dotted curve), ψM(ω), and φ err(ω) for the THz waveform E THz(t) of Fig. 2(a) are shown in Figs. 2(b)–(d), respectively. As seen in the figures, the same analysis works for the THz waveform of Fig. 2(a), except that the obtained error phase spectrum φ err(ω) has an offset by −π/2 due to the initial phase of the oscillatory waveform. It should be noted that, since ω 0 δt err is calculated to be ±0.14 and an order of magnitude smaller than π/2, the MEM can definitely distinguish the two THz waveforms shown in Figs. 1(a) and 2(a).
3.2. Application to actual THz waveforms
Below, we give an example of MEM application to actual THz waveforms observed for the Bloch oscillation in semiconductor superlattices. The THz emission from superlattice samples excited by femtosecond laser pulses (pulse duration: ~100 fs) under dc bias electric fields F was measured in the time domain using a ZnTe electro-optic sensor that has flat sensitivity up to 3.5 THz. More experimental details are presented in Refs [6]. and [15].
Figure 3(a) shows the THz waveform E THz(t) observed for a GaAs(7.5 nm)/AlAs(0.8 nm) superlattice biased at F = 8.2 kV/cm. Here, the time origin is placed at a tentative position; a possible misplacement δt of the time origin exists in E THz(t). The sampling interval of the data points in E THz(t) is Δt = 30 fs. Figure 3(b) shows the amplitude spectrum |Ẽ THz(ω)| (solid curve) and the phase spectrum argẼ THz(ω) (dash-dotted curve) calculated from the THz waveform E THz(t) of Fig. 3(a). As seen in the figure, |Ẽ THz(ω)| has a resonance peak at ω/2π = 1.47 THz, nearly equal to the expected Bloch frequency eFd/h (d: superlattice period) [15]. argẼ THz(ω) exhibits no clear linear phase shift at this stage because the −ωδt component was already partially minimized by the empirical method.
Fig. 3 MEM analysis of a THz waveform recorded experimentally for the Bloch oscillation in a GaAs/AlAs superlattice at 10 K. (a) THz waveform E THz(t) recorded with a tentative time origin (t = 0). (b) Spectra of amplitude |Ẽ THz(ω)| (solid curve) and phase argẼ THz(ω) (dash-dotted curve). (c) MEM phase spectra ψM(ω) computed in three different frequency ranges. (d) Error phase spectra φ err(ω) (solid curves). Dashed lines are the linear fits to φ err(ω), the slopes of which show that the time origin should be shifted to the position indicated by a red vertical line in (a).
We then calculated MEM coefficients am (m = 1, 2, …, M) and b for the measured power spectrum |Ẽ THz(ω)|2, following the same procedure described in the previous subsection. Figure 3(c) shows the MEM phase spectrum ψM(ω) computed using Eq. (5) with the obtained set of am in three different frequency ranges: 0.22–2.88 THz (blue), 0.28–2.97 THz (red), and 0.38–2.58 THz (green) [14]. The error phase spectrum φ err(ω) is shown in Fig. 3(d) by solid curves in the corresponding colors. As seen in the figures, ψM(ω) displays a phase shift by ~π across the resonance feature in |Ẽ THz(ω)|, and φ err(ω) exhibits a nearly linear phase shift due to misplacement of the time origin. By fitting the obtained spectral shapes of φ err(ω) with straight lines (dashed lines) and estimating their slopes, we determined δt to be 180 ± 20 fs, the uncertainty of which is comparable to the sampling interval of the original data points (Δt = 30 fs) shown in Fig. 3(a). This means that δt has been detected with a practically sufficient accuracy and that the time origin should be shifted to the position indicated by a red vertical line in Fig. 3(a).
Now, let us discuss the advantage that the MEM has over the empirical method in THz emission spectroscopy. As described above, the MEM reveals the reason why the −ωδt component is not easy to empirically find in the original phase spectrum argẼ THz(ω) of Fig. 3(b): the −ωδt phase shift is indeed obscured by the true phase shift [see Figs. 3(c) and 3(d)] in argẼ THz(ω). This gives the implication that, when the MEM is applied to different spectral shapes [16], the MEM will excel the empirical method more greatly at detecting δt in THz emission with larger or more complicated true phase shifts.
3.3. Note
Finally, we make a comment on the THz waveforms shown in Figs. 1(a) and 2(a). So far, the THz waveforms of the type of Fig. 1(a) have been predicted for Bloch oscillations [17]. As seen in Fig. 3(a), however, actual THz waveforms observed for the Bloch oscillation in semiconductor superlattices are of the type of Fig. 2(a), indicating that the transient current J(t) starts from its maximum as damped cosω 0 t at t = 0. Here, we would like to emphasize that the MEM can make a clear distinction between the two. Very recently, it has been reported that the observed phase of the Bloch oscillation is ascribed to a capacitive response of electrons distributed on the Wannier-Stark ladder with translational symmetry [18].
4. Summary
In summary, we have systematically determined the time origin by the MEM in time-domain THz emission spectroscopy. By testing the MEM on THz waveforms simulated for damped sinusoidal oscillation currents, we confirmed that the MEM can properly derive true phase shifts across the resonance features and that its inherent uncertainty in determining the time origin is ±15 fs, i.e., much smaller than the typical pulse duration of readily available femtosecond lasers. Furthermore, when the MEM was applied to a THz waveform recorded experimentally with a finite sampling interval for the Bloch oscillation in a semiconductor superlattice, a misplacement of the time origin was detected with a practically sufficient accuracy that is comparable to the sampling interval of the data points.
Acknowledgments
This work was partly supported by Grants-in-Aid from JSPS (Nos. 22241036 and 17-10614) and a Special Coordination Fund for Promoting Science and Technology from MEXT (NanoQuine).
References and links
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14. We chose these frequency ranges by considering a tradeoff: the resonance feature is more accurately captured in wider frequency ranges, while the numerical error in solving the Toeplitz matrix equation becomes less in narrower frequency ranges.
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