1. Introduction

Causality is a fundamental concept in physics and is known to regulate the relationship between the real and imaginary parts (or between the amplitude and phase parts) of optical spectra through the Kramers-Kronig (K-K) transformation [1–3]. It has been reported that causality provides a practical method of compensating for a partial lack of spectral information in terahertz (THz) time-domain spectroscopy, where K-K relations can be used together with the values of both amplitude and phase known at some reference frequencies (called anchor points) within the measurement frequency range. Since the singly subtractive K-K relation was successfully exploited by Lucarini et al. to remove the artificial phase shift in THz reflection spectroscopy due to a spatial misplacement between the sample and reference materials [4], it has furthermore been applied to THz emission spectroscopy [5] and THz transmission spectroscopy [6,7].

In a previous paper [5], we proposed a causality-based method for determining the time origin (t = 0) in THz emission spectroscopy and demonstrated that a misplacement of the time origin can be detected with an accuracy that is an order of magnitude higher than the given temporal resolution of THz electric field. It has been shown that such an accurate time origin in THz emission allows us to obtain an in-depth physical understanding of ultrafast charge transport in semiconductors [8,9]. However, the core theoretical framework of the causality-based method was limited to situations where THz emission does not include the contribution of a delta-function-like instantaneous polarization, which is widely observed in nonlinear optical processes such as optical rectification [10].

In this work, we developed a generalized causality-based framework that can determine the time origin for all major mechanisms of THz emission including the occurrence of a delta-function-like instantaneous polarization. A multiply subtractive K-K relation [11,12] was used to derive a function for evaluating how well the causality in THz emission is satisfied versus a possible time-origin misplacement δt. We found that the function reveals the exact positions of t = 0 for trial THz waveforms and furthermore works for a recent example of THz waveforms emitted from biased conjugated polymers.

2. Generalized formulation of causality in THz emission spectroscopy

We provide the formulation of causality in transient THz emission phenomena and derive the function for determining the time origin, i.e., the moment at which real/virtual electron-hole pairs created by the peak of an ultrashort optical pulse start emitting a THz electromagnetic wave. Let us consider a general situation where the emitted THz electric field E(t) has conceivable retarded and instantaneous components and can be expressed as

By using the theory of complex functions in a way similar to that of [5], we obtain

Now, we deal with the case where a time-origin misplacement δt may accompany the measurement of THz electric field E(t). Because it gives rise to the artificial shift by –ωδt in the measured phase spectrum θ(ω) = argẼ(ω) [5,8], ReẼ(ω) and ImẼ(ω) in the formulas derived above should be replaced by ρ(ω)cosθ_cr(ω) and ρ(ω)sinθ_cr(ω), respectively, where ρ(ω) = |Ẽ(ω)| is the amplitude spectrum and θ_cr(ω) = θ(ω) + ωδt is the corrected phase spectrum. The use of Eq. (5) at another anchor point ω = ω₁ leads to the condition for δt that K(δt) = 0, with K(δt) defined by

3. Results and discussion

3.1 Verification with simulated THz waveforms

To examine whether the generalized framework described above works properly, we computed the causality-based function K(δt) for trial THz waveforms simulated with no time-origin misplacement. Figure 1(a) shows a trial THz waveform E(t) simulated for a transient current of J(t) = J₀Θ(t)exp(–γt)cosΩt, to which the previous version of K(δt) was applicable [5]. Here, J₀ is the magnitude of current, Θ(t) is the unit step function, Ω is the resonance angular frequency, and γ is the damping rate; the numerical data on E(t) was prepared with parameters of Ω/2π = 1.5 THz and γ = 1.1 THz and was furthermore convolved with a system response function that gives a temporal resolution of τ_res = 0.30 ps. This type of E(t) has been observed for Bloch oscillations in semiconductor superlattices [8,13] and gives an example of A ≠ 0 and B = C = 0 in Eq. (1). Figure 1(b) shows the amplitude spectrum ρ(ω) (solid curve) and the phase spectrum θ(ω) (dashed curve) calculated by performing the fast Fourier transform of E(t) in Fig. 1(a); ρ(ω) has the resonance peak at ω/2π = 1.5 THz, while θ(ω) has a change by ~π across this frequency.

 

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 τ_res = 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.

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We fed the numerical spectral data ρ(ω) and θ(ω) into Eq. (6) together with anchor points ω₀, ω₁, and ω₂. The causality-based functions K(δt) computed with (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 in an integration frequency range of ω'/2π = 0.0–4.0 THz are shown in Fig. 1(c) by curves 1–3, respectively. As seen in the figure, the present generalized version of K(δt) indeed has properties similar to those of the previous version [5]: K(δt) fluctuates greatly for δt < 0 (because causality is violated if the time origin is shifted forward from the current position) but is nearly equal to zero for δt > 0 with only a small fluctuation (because causality still holds if the time origin is shifted backward). We find that the small fluctuation, which arises from a finite temporal length of E(t) [5], is damped the most at δt ~0.30 ps and the deviation from the baseline (i.e., the K = 0 line) increases significantly as δt decreases from there, particularly in the magnified view of curves 1 and 3 in Fig. 1(d) (curve 2 exhibits it less clearly but is consistent with curves 1 and 3). This behavior reflects the fact that the THz signal in Fig. 1(a) penetrates down to t ~–0.30 ps owing to the given finite temporal resolution of τ_res = 0.30 ps. Thus, we can obtain the actual value of time-origin misplacement δt by following the same procedure as reported in [5]: we first search for the point δt' at which K(δt) starts deviating from the baseline versus possible misplacement δt, and we then estimate the actual misplacement to be δt = δt' – τ_res.

Here, we provide a practical criterion for specifying the value of δt' more quantitatively. When we consider the largest neighboring peak of K(δt) on the lower side of δt ~0.30 ps, i.e., the negative peak of curve 3 at δt = –0.23 ps in Fig. 1(d), we find that curves 1 and 3 at δt ~0.30 ps are as small as 1.5% of that peak height in terms of absolute values and we can determine that K(δt) starts deviating from the baseline at δt' = 0.31 ± 0.03 ps. Thus, the actual misplacement is given by δt = δt' – τ_res = 0.01 ± 0.03 ps, in which range the correct solution (i.e., δt = 0 ps for this simulation) is included properly. The same criterion applies to the other examples shown below. Because the clarity in the behavior of K(δt) around δt' depends on the choice of anchor points, we recommend preparing at least three sets of anchor points to obtain the value of δt together with its uncertainty.

Figure 2(a) shows another trial THz waveform E(t) simulated for a delta-function-like instantaneous polarization with a temporal resolution of τ_res = 0.030 ps. This type of E(t) gives an example of the important additional term with C ≠ 0 in Eq. (1). As seen in the figure, E(t) has a symmetric shape with respect to the largest negative peak at t = 0. Figure 2(b) shows the amplitude spectrum ρ(ω) and the phase spectrum θ(ω), which were obtained from E(t) and fed into Eq. (6). The causality-based functions K(δt) computed with (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 in an integration frequency range of ω'/2π = 0.0–50.0 THz are shown in Fig. 2(c) by curves 1–3, respectively. The deviation in K(δt) from the baseline starts at δt' = 0.029 ± 0.003 ps [see also the magnified view in Fig. 2(d)], indicating that the actual misplacement can be estimated properly to be δt = δt' – τ_res = –0.001 ± 0.003 ps.

 

Fig. 2 Analysis of THz emission data simulated for a delta-function-like instantaneous polarization with a resolution of τ_res = 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.

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The same analysis as made in Fig. 2 also works for an instantaneous polarization with an order-of-magnitude lower temporal resolution of τ_res = 0.30 ps treated in Fig. 3. The causality-based functions K(δt) computed with (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 in an integration frequency range of ω'/2π = 0.0–5.0 THz are shown in Fig. 3(c) by curves 1–3, respectively. The deviation in K(δt) from the baseline starts at δt' = 0.31 ± 0.04 ps [see also the magnified view in Fig. 3(d)], revealing the actual misplacement properly to be δt = δt' – τ_res = 0.01 ± 0.04 ps. Putting together the three different examples provided in Figs. 1–3, we find that the uncertainty in determining the time origin from the behavior of K(δt) is typically ± 10% of the given temporal resolution τ_res.

 

Fig. 3 Analysis of THz emission data simulated for a delta-function-like instantaneous polarization with a resolution of τ_res = 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.

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Now, let us make a few remarks on the present method. First, it excels the maximum entropy method (MEM) [14] at treating the additional term with C ≠ 0 in Eq. (1); the MEM made wrong predictions about the time origin in Figs. 2(a) and 3(a) because the MEM is not good at deriving flat phase spectra [see Figs. 2(b) and 3(b)] from amplitude spectra. Second, the present method is expected to work properly even for the cases where E(t) in Eq. (1) has several nonzero terms simultaneously; we treated separate cases for simplicity because Eq. (5) consists of linear operations and includes the principle of superposition for E(t).

3.2 Application to measured THz waveforms

Here, we demonstrate how the present method can be applied to a recent example of actual THz waveforms emitted from biased conjugated polymers. Figure 4(a) shows the THz waveform E(t) observed for a polythiophene film under an in-plane bias electric field of 12 kV/cm [15], with the time origin set to an arbitrary position. The THz electric field was detected using a 0.2-mm-thick (110) ZnTe electro-optic sensor, whose response function calculated within a theoretical model [16,17] had a nearly flat spectral shape up to 2 THz and thus led to a temporal resolution of τ_res = 0.50 ps. As seen in the figure, E(t) has the largest negative peak at t = 2.0 ps and smaller positive peaks. The amplitude spectrum ρ(ω) and the phase spectrum θ(ω) obtained from E(t) with the use of a window function and a zero-filling technique are shown in Fig. 4(b); ρ(ω) has a broad shape peaking at ω/2π ~1.5 THz, while θ(ω) has an almost linear component presumably due to a time-origin misplacement.

 

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 τ_res = 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).

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The causality-based functions K(δt) computed with (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 in an integration frequency range of ω'/2π = 0.2–5.0 THz are shown in Fig. 4(c) by curves 1–3, respectively. The deviation in K(δt) from the baseline is found to start at δt' = –1.46 ± 0.06 ps [see also the magnified view in Fig. 4(d)]. The actual misplacement is thus estimated to be δt = δt' – τ_res = –1.96 ± 0.06 ps, which requires us to correct the time origin to the point indicated by the arrow in Fig. 4(a). Note that the time origin determined here is very close to the position of the largest negative peak and is clearly way from those of the small positive peaks. This result strongly supports an interpretation of the THz emission described in [15]: the THz waveform shown in Fig. 4(a) should have appeared when it was emitted by the creation of a delta-function-like polarization and measured with slight distortion through the electro-optic sensor.

4. Summary

We presented a generalized causality-based function with respect to a possible misplacement δt of the time origin in THz emission spectroscopy, by deriving a multiply subtractive K-K relation for complex Fourier spectra of emitted THz electric fields. The function was found to have good properties that allow us to determine the time origin in trial THz waveforms simulated for a damped oscillation current and for delta-function-like instantaneous polarizations (created typically in nonlinear optical processes). Furthermore, we demonstrated how the time-origin misplacement can indeed be detected for a recent example of THz waveforms that were emitted from biased conjugated polymers and measured with slight distortion. The function will serve as a useful tool for inferring the time evolution of true current and polarization on the (sub)picosecond scale and allow for a better understanding of ultrafast phenomena in a variety of optoelectronic materials, such as charge separation in photovoltaic devices.