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Single-shot measurement of a terahertz field pulse waveform by electro-optic sampling using a chirped optical pulse and a spectrometer was demonstrated by and Jiang and Zhang [Appl. Phys. Lett. 72, 1945 (1998)]. We have performed an experimental and theoretical investigation into the dependence of the waveform thus measured on the chirp rate and spectral resolution. It was found that the waveform exhibits multicyclic behavior at a chirp rate of −0.24  THz², which corresponds to a chirped-pulse width of over 10 ps, for the monocyclic original terahertz field, while it approaches the monocyclic behavior with decreasing pulse width. Further, broadening of the spectral resolution of the spectrometer gives rise to a monocyclic waveform in the chirp rate range where the waveform is expected to be multicyclic. In addition, we have derived an analytical expression for the terahertz field pulse waveform thus measured without using the method of stationary phase. The theoretical results were found to be consistent with measured ones. Finally, we examined the spectral bandwidth and resolution of terahertz spectroscopy using this method. ©2008 American Institute of Physics

Contents

• INTRODUCTION
• THEORETICAL BACKGROUND
• EXPERIMENTAL PROCEDURE
• EXPERIMENTAL RESULTS AND DISCUSSION
  • A.Dependence of the terahertz field waveform on the chirp rate
  • B.Comparison between the experimental and calculated results
  • C.Dependence on spectral resolution of the spectrometer
  • D.Spectral bandwidth and resolution of terahertz spectroscopy using EODCP
• CONCLUSION
• ACKNOWLEDGMENTS
• REFERENCES
• FIGURES
• FOOTNOTES

Optical constants such as complex permittivity have been measured for various materials since the development of terahertz time-domain spectroscopy (THz-TDS) using a femtosecond (fs) laser.^{1,2,3,4,5,6,7,8} This method allows simultaneous determination of both the real and imaginary parts of the optical constant across a spectral range from a few tens of gigahertz to several terahertz because it allows measurement of the change due to the optical constant in the amplitude and phase of a terahertz electric field.

THz-TDS is based on a pump-probe method. A fs optical pulse is first divided into two beams. One (pump) is used to generate a terahertz field pulse, which, for example, is emitted from a semiconductor such as InAs by irradiating its surface with the pump pulse. In measurements using the electro-optic (EO) effect, the other (probe) is overlapped with the terahertz field pulse within an EO crystal such as ZnTe, and the magnitude of the Pockels effect proportional to the amplitude of the terahertz field is obtained from measurement of the intensity of the probe pulse subject to the birefringence in the crossed-polarizer configuration. The temporal waveform of the terahertz field is measured by changing the time interval between the pump and probe pulses with an optical delay line. Thus, THz-TDS is not applicable to single-shot measurement of the temporal waveform of a terahertz field pulse.

A method that allows single-shot measurement of a terahertz field waveform was proposed by Jiang and Zhang.⁹ In the method, a fs probe pulse is linearly chirped and overlapped with a terahertz field pulse within the EO crystal. The chirped probe pulse is modulated by the terahertz field and dispersed onto a multichannel detector combined with a spectrometer. Since the wavelength axis can be converted to the time axis using the value of the chirp rate, the temporal waveform of the terahertz field is derived from the two spectra of the probe pulses with and without terahertz field modulation. They used a chirped pulse with a temporal width of approximately 30 ps and obtained a terahertz field pulse three times broader than the original pulse width measured by THz-TDS. Moreover, severely distorted terahertz field waveforms were observed even for chirp probe pulses with a time width of several picoseconds.^10,11,12 Sun et al.¹³ analyzed the dependence of the temporal resolution of electro-optic detection using a chirped probe pulse (EODCP) on the chirp rate by assuming that the temporal widths of the chirped pulse and the terahertz field are much longer than the oscillation period of the optical beam and that the stationary phase method is applicable. The EODCP-derived terahertz field pulse waveform in their analysis was monocyclic for a monocyclic original terahertz field. It has recently been demonstrated that the temporal resolution of EODCP can be significantly enhanced by combination with an interferometric retrieval algorithm; the original terahertz field waveform is reproduced well by the algorithm.^11,12

Single-shot measurement of the terahertz field pulse has been achieved by overlapping the terahertz radiation and the fs optical probe beam noncollinearly within an EO crystal, where the time window is determined by the spot size of the probe beam and the crossing angle between the two beams.¹⁴ Further, it has been demonstrated that for a time window of about 5 ps, the terahertz field waveform obtained from the single-shot measurement almost coincides with that from THz-TDS.

Complex systems such as liquids and proteins show low-frequency vibrations in the terahertz spectral range, and such vibrations are believed to play an important role in the dynamics of chemical reactions in solutions and proteins. Further, the origin of the boson peak observed in the low-frequency vibrational spectrum of the complex system is under debate.^15,16 Thus, the low-frequency dynamics of the complex system has been extensively investigated by means of a variety of experimental methods such as due to inelastic neutron^17,18,19 and light scatterings,²⁰ as well as by THz-TDS.^{2,3,4,5,6,7,8} Most of the measurements, however, have been made for systems in thermal equilibrium. It is considered necessary to study how such low-frequency vibrations change during a reaction in order to explore the mechanism of the reaction in solutions and proteins.

This can be achieved by optical pump-terahertz probe (OPTP) spectroscopy. In dye solutions, the solvation dynamics after optical excitation of the dye molecule have been probed by a terahertz field pulse, and it has been suggested that librational modes localized close to the solute occur owing to the abrupt change caused by the excitation in the dipole moment of the dye molecule.²¹ In liquid hexane, the transient photoconductivity and recombination dynamics of quasifree electrons generated by two-photon ionization induced by the irradiation with a fs UV pulse have been probed by the terahertz pulse.²²

THz-TDS is mostly adopted for the terahertz probe in OPTP spectroscopy. In such a case, a fs laser pulse is first divided into two beams, and one is used for exciting the sample. The other is used for THz-TDS; that is, it is further divided into two beams: one for the generation of the probe terahertz field pulse and the other for sampling the terahertz field. Hence, two optical delay lines are employed among the three fs pulses thus divided. OPTP spectroscopy with THz-TDS has several drawbacks. An intense fs laser such as that generated from a chirped-pulse-amplification system is often used in pump-probe spectroscopy to excite as many molecules as possible in the sample. The shot-to-shot fluctuation of such an intense laser deforms the terahertz field waveform obtained from THz-TDS. Even if the shot-to-shot fluctuation is small, the terahertz field pulse shape is deformed if the system under study does not relax within the period of pulse repetition or undergoes an irreversible process. Further, the use of two optical delay lines makes measurement overly time consuming. In fact, in the above mentioned OPTP studies, the amplitude of the terahertz field pulse was measured only at the specified detection delay time at which the waveform had its peak while the pump-probe delay time was scanned, but its temporal waveform was not measured.^21,22 These measurements do not provide the so-called time-resolved terahertz spectrum. If EODCP is employed for the terahertz probe in OPTP spectroscopy, these drawbacks can be overcome.^23,24 However, no experimental investigation has been performed to determine how the terahertz field waveform measured by EODCP depends on the chirp rate and spectral resolution of the spectrometer, although the chirp rate dependence of the terahertz field waveform has been analyzed under limited conditions using the stationary phase method.¹³ Such dependence determines the parameters such as the spectral resolution and bandwidth of terahertz spectroscopy using EODCP.

Thus, in this study, we measured the terahertz field waveform by EODCP as a function of the chirp rate and the spectral resolution of the spectrometer, and derived an expression for the temporal shape of the terahertz field pulse obtained from EODCP without using the stationary phase method. The experimental results were compared with the theoretical predictions. Further, in this paper, we examine the experimental conditions such as the chirp rate and the spectral resolution of the spectrometer in the application of EODCP to terahertz spectroscopy.

In this section, we derive an expression of the temporal shape of the terahertz field pulse obtained from EODCP. Denoting the electric field waveform of a chirped probe pulse by C(t), we give the temporal shape of the chirped pulse modulated by a terahertz field E_THz(t) as follows:

where k is a modulation coefficient and is the delay time between the terahertz field pulse and the chirped pulse. The first term on the right-hand side of Eq. (1) appears because the probe pulse passes partly through the two crossed polarizers, e.g., owing to the inherent residual birefringence of the EO crystal or the finite extinction ratio of the polarizers, where b is the transmission coefficient. By comparing the terahertz field waveform obtained from EODCP with numerical results, Yellampalle et al.¹⁰ showed that Eq. (1) is valid for an EO crystal with sufficient residual birefringence, as in the present study using a 1-mm-thick ZnTe crystal. However, the equation needs to be modified if analyzer detuning or an additional waveplate is used in the case where there is no residual birefringence.¹⁰

The waveform of the chirped pulse is defined as C(t)=exp(−t²T−iat²−i₀t), where T_c, 2a, and ₀ are the width, chirp rate, and central frequency of the pulse, respectively. We represent a monocyclic original terahertz field waveform by E_THz(t)=(t/T)exp(−t²/T²), which is a symmetrically bipolar function and represents a global feature of the terahertz field waveform generated by a fs laser. If the modulated probe pulse is measured with a spectrometer with a multichannel detector, the spectral intensity I()_on is expressed by

where g(−) is the spectral response function of the spectrometer; we set g(−)=(−) in this derivation. It is possible to solve Eq. (2) by using Gaussian integrals and dealing with complex numbers in the polar form. Further, we follow the procedure of Jiang and Zhang⁹ to derive the terahertz field waveform from EODCP, namely,

where I()_off is obtained by setting M(t)=bC(t) in Eq. (2). Finally we obtain an expression of the terahertz field waveform as

with

where =₀−, =T+T⁻², and =T+a². Here the quadratic term of k is neglected on the assumption that the modulation coefficient is very small and kE_THzb. We confirmed that the waveform obtained from Eq. (4) agrees with that from Eq. (3) through the numerical integration of I()_on and I()_off with the same parameter values.

A light source was a regeneratively amplified Ti:sapphire laser (Clark-MXR CPA-1000). It generated an optical pulse with a temporal width of less than 150 fs [full width at half maximum (FWHM)], a wavelength of 800 nm, a repetition rate of about 1 kHz, and a pulse energy of 600  µJ. The fs laser beam was divided into two beams: one was irradiated onto an InAs(100) wafer to generate the terahertz field pulse, and the other was chirped by passing it through a pair of gratings. We used two types of grating pairs, i.e., 1200 and 600 grooves/mm, to change the chirp rate.

A schematic diagram of the experimental setup for EODCP is illustrated in Fig. 1. We employed a double-beam configuration to obtain I()_on and I()_off simultaneously. This configuration reduces the effect of the shot-to-shot fluctuation of the laser on the derived terahertz waveform using EODCP. The chirped beam was first passed through a polarizer and then divided into two beams. One beam, the reference R(), was led into a fiber that transferred the light to a spectrometer (Acton Research 300i) with three automatically exchangeable gratings (150, 1200, and 1800 grooves/mm), and was detected with a charge-coupled device (CCD) image sensor. The other, the signal S(), was aligned to travel collinearly with the terahertz field pulse, modulated by the terahertz field within a 1-mm-thick ZnTe crystal, and was passed through an analyzer into the other fiber. The reference and signal beams were detected simultaneously on two different areas of the image sensor. S()_off and R()_off were measured simultaneously in the absence of the terahertz field, while R()_on and S()_on [=I()_on] were measured in the presence of it. I()_off was obtained by R()_onS()_off/R()_off. Thus, we could measure I()_off and I()_on simultaneously in the double-beam configuration and derive the terahertz field waveform using Eq. (3). In the present study, the data were acquired by averaging over a few hundred shots to achieve a good signal-to-noise ratio. Further, an optical delay line was employed in order to vary the time interval between the probe pulse and the terahertz field pulse within the EO crystal. This was necessary to obtain the value of the chirp rate experimentally. The terahertz field waveform thus obtained was shifted on the frequency axis by changing the time interval to derive the value of the chirp rate from the magnitude of the shift as a function of the time interval and convert the abscissa for the terahertz field waveform from frequency to time, as shown below.

Figure 1.

THz-TDS was performed by the conventional method; that is, a fs optical probe pulse was used instead of a chirped one, and the spectrometer with a CCD sensor was replaced with a photodiode. The time interval for sampling the terahertz field was about 100 fs.

Figure 2(a) depicts the terahertz waveform obtained by EODCP at a chirp rate of −0.24  THz². The waveform is shifted along the abscissa (wavelength) when the delay time between the chirped probe and terahertz field pulses is changed. Figure 2(b) was obtained by tracking corresponding peak positions [arrow in Fig. 2(a)] in the terahertz waveforms at different delay times. We derived the value of the chirp rate from the slope of the line fitted to the data points. It was found that this value agrees with that obtained by a second-harmonic-generation frequency-resolved optical grating measurement and the estimated value from the distance between the two gratings and the number of grooves ruled on the grating per unit length. The upper axis in Fig. 2(a) is the time axis obtained from the wavelength axis using the chirp rate value. The temporal width of the chirped pulse at this chirp rate was determined to be about 15 ps (FWHM) by cross-correlation measurement between the original fs pulse and the chirped one. The terahertz waveform (black line) is compared with that obtained by THz-TDS (thick gray one) in Fig. 2(c). The waveform due to THz-TDS is monocyclic, although small long-lived oscillation due to absorption by water vapor in the terahertz frequency range is involved; all the measurements in the present study were made in the atmosphere. The waveform measured by EODCP is multicyclic. Further, the amplitude of the oscillation in the waveform derived by EODCP increases gradually with time and then decreases, as seen in Fig. 2(c), except for the central part around 2.5 ps. Such multicyclic behavior was not observed in the terahertz field waveform obtained from EODCP by Jiang and Zhang,⁹ although the temporal width of the chirped probe pulse in their measurement is comparable to that in this case. However, there are a few experimental studies that report distorted waveforms similar to the multicyclic behavior obtained here from EODCP using a several picosecond wide chirped pulse, although the effect of the distortion is not explicitly mentioned.^10,11,12 Further, it should be noted that the temporal period of the multicycle in the terahertz waveform obtained by EODCP is roughly the same as that of the monocycle obtained by THz-TDS [Fig. 2(c)], while the former was three times longer than the latter in the measurement by Jiang and Zhang.⁹ This implies that the temporal resolution of EODCP is comparable to that of THz-TDS in our case. The experimental results at a chirp rate of −1.3  THz² for different delay times are shown in Fig. 3. It is obvious that the number of oscillations is small compared with that at −0.24  THz².

Figure 2.

Figure 3.

The terahertz field waveforms calculated from Eq. (4) for delay times of −10, 0, and 10 ps are illustrated in Fig. 4, together with the original terahertz field pulse (inset) used for the calculation, where T_c=22  ps, a=−0.12  THz², T=0.45  ps, and ₀=375  THz (=800  nm). The values of T and ₀ are fixed for the calculations below. These parameter values are about the same as those used in the measurement for Fig. 2(a). However, the original terahertz pulse is not symmetrically bipolar, with long-lived oscillations due to the absorption by water vapor, as seen in Fig. 2(c), and the temporal envelope of the chirped pulse deviates from a Gaussian curve in the measurement. The waveform thus calculated reproduces the characteristic features observed in Fig. 2(a). Multicyclic behavior is obviously seen in Fig. 4. In both the experimental and theoretical results, the waveform consists of two damping oscillations moving in opposite directions on the abscissa except for the central part. The chirped probe pulse is modulated by the terahertz field pulse around the peak of the spectrum of the probe pulse at a delay time of 0 ps. The two damping oscillations are inversion symmetric about a wavelength of 800 nm at 0 ps in Fig. 4, while they are asymmetric at ±10  ps. That is, the amplitude at the start point of the damping oscillation on the longer-wavelength side is larger than that on the shorter-wavelength side at 10 ps, while it is smaller at −10  ps. This feature is remarkably enhanced in the measured waveform shown in Fig. 2(a). The waveforms are calculated at [a(THz²),T_c(ps)]=(−0.12,22), (−0.65,4), and (−1.5,1.8), with =0  ps in Fig. 5. The waveform is found to change from monocyclic to multicyclic behavior with decreasing magnitude of chirp rate. This tendency agrees with that seen in the experimental results. It should be noted that the multicyclic behavior does not emerge in the waveform derived by Sun et al.¹⁰ on the assumption that the stationary phase method is valid. Hence, analysis based on the stationary phase method is not applicable to the case where the terahertz field waveform derived by EODCP is multicyclic.

Figure 4.

Figure 5.

Figure 6(a) shows the dependence of the terahertz field waveform obtained by EODCP on the spectral resolution of the spectrometer at a chirp rate of −1.9  THz². The waveform does not exhibit monocyclic behavior at the high spectral resolution (0.2 nm, FWHM), while it becomes monocyclic at the low resolution (3 nm). According to the fact that the temporal resolution is low at the low spectral resolution in Fig. 6(a), the bandwidth of the spectrum obtained from the Fourier transform of the temporal waveform is narrower at a spectral resolution of 3 nm than at 0.2 nm in Fig. 6(b). Further, we calculated the terahertz field waveform from Eq. (3) through numerical integration of Eq. (2) at a chirp rate of −0.24  THz² in Fig. 7, where the -function and Gaussian function with widths of 0.2, 0.6, and 3.4 nm were used for g(−) in Eq. (2). It is found that the multicyclic behavior observed at the high spectral resolutions disappears completely at a resolution of 3.4 nm. Thus, the spectral resolution of the spectrometer plays an important role in determining the temporal shape of the terahertz field obtained by EODCP.

Figure 6.

Figure 7.

The terahertz field waveform measured by Jiang and Zhang⁹ does not show multicyclic features despite the fact that the chirp rate and temporal width of the chirped probe pulse are comparable to those in our case at −0.24  THz². This may be because the spectral resolution of the spectrometer used by them was not high enough to resolve the multicyclic waveform, as seen in Fig. 6(b), although its value is not described in the literature.⁹

The spectral resolution and bandwidth of THz-TDS is determined by the whole time window of observation and the temporal profile of the terahertz field pulse, respectively, according to time-frequency relation based on the Fourier transform. On the other hand, the spectral bandwidth of terahertz spectroscopy using EODCP is determined not only by the time profile of the original terahertz field pulse but also by the temporal resolution of the measurement.

The temporal resolution due to the spectral resolution () of the spectrometer is given by /(2|a|) for EODCP; this value must ideally be smaller than the temporal width of the original terahertz waveform. For example, in the case that the spectral resolution and chirp rate are 0.2 nm and −0.24  THz², respectively, the temporal resolution is about 0.4 ps. The terahertz waveform calculated at a 0.2-nm resolution is found to coincide fairly well with that at the infinite resolution in Fig. 7. On the other hand, when the temporal resolution is about 1.2 ps at a 0.6-nm resolution, the terahertz waveform broadens compared with that at the infinite resolution in Fig. 7. Further, it is found from Fig. 6 that the terahertz field waveform measured broadens at a 3-nm resolution compared with that at 0.2 nm, and hence the bandwidth of the frequency spectrum is narrower than that at 0.2 nm.

The temporal period of the multicycle in the terahertz field waveform measured by EODCP at a chirp rate of −0.24  THz² and a spectral resolution of 0.2 nm is roughly the same as that of the monocycle obtained by THz-TDS [Fig. 2(c)]. The frequency spectrum obtained by EODCP (Fig. 8), accordingly, has a spectral bandwidth (about 2 THz) comparable to that obtained by THz-TDS [Fig. 6(b)], although the spectral shapes are not the same.

Figure 8.

As for the time window of the observation, a range of a few tens of picoseconds is required to obtain spectral resolution as small as 1  cm⁻¹ (30 GHz). EODCP with a small chirp rate leads to a narrow time window of observation under the limited spectral width of the probe pulse (~10  nm in the present study), and hence a high spectral resolution is not achieved. In Fig. 6(b), the frequency spectrum of the terahertz field pulse obtained by THz-TDS shows sharp gaps due to absorption by water vapor, e.g., around 1.2 THz. However that obtained by EODCP at −1.9  THz² (chirp rate) and 0.2 nm (spectral resolution) does not because of its narrow time window (~5  ps), i.e., low spectral resolution (~0.15  THz). On the other hand, there exist sharp gaps due to absorption by water vapor in the frequency spectrum obtained at −0.24  THz² (Fig. 8). Therefore, EODCP with a small chirp rate has a crucial drawback for use in terahertz spectroscopy, even if the terahertz field waveform approaches monocyclic behavior. Thus, we must use a chirped probe pulse with a chirp rate that assures a time window for observation of a few tens of picoseconds, although we will possibly encounter multicyclic behavior in the terahertz waveform in such a case.

The multicyclic behavior in the terahertz field waveform calculated from Eq. (4) at a chirp rate of −0.24  THz² leads to the oscillating behavior in the frequency spectrum shown in Fig. 8, although the envelope connecting the peaks in the spectrum coincides roughly with the spectrum obtained from the Fourier transform of the original terahertz field waveform (inset of Fig. 4). We consider the effect of the oscillating behavior on the derivation of the optical constants of materials by terahertz spectroscopy using EODCP. The quadratic term of the modulation coefficient is neglected in the derivation of Eq. (4), which implies that the EODCP instrument is assumed to exhibit a linear response to the terahertz field for the small modulations considered in this work. On the assumption of such a linear response, the waveform of the terahertz field obtained by EODCP is expressed by

where (t) and E₀(t) are the time profiles of the instrumental response and the original terahertz field, respectively. If the terahertz field is transmitted through the material under study and measured by EODCP, its waveform is given by

where the material is assumed to exhibit a linear response to the terahertz field and the response function is denoted as (t). Thus, the complex transmission coefficient () of the material is derived by

with

where the terms with superscripts, e.g.,_sig(), are the Fourier transforms of the corresponding terms without superscripts, e.g., E_sig(t). The convolution theorem is used in the derivation of Eq. (8). The Fourier-transformed spectra of the instrument response function and the original terahertz waveform are canceled out in Eq. (7), and so have no effect on derivation of the optical constant of the material. The complex refractive index is derived from the complex transmission coefficient.²⁵

The terahertz field waveform obtained by EODCP depends on the delay time between the terahertz field and chirped probe pulses, as shown in Figs. 2(a)3. This implies that the instrument response of EODCP varies with the delay time. If the refractive index of the material under study is not unity in the terahertz spectral range, the terahertz field pulse transmitted through the material is delayed from that passing through free space by an amount determined by the refractive index. Thus, the two terahertz field pulses temporally modulate different portions of the chirped probe pulse. Therefore, the instrument response is different for the two terahertz fields. A method for removing the effect of variable instrument response is to make the two terahertz field pulses overlap with the same (temporal) portion of the chirped probe pulse using an optical delay line, although the time difference between the two terahertz field pulses must be measured in advance. The chirped probe beam can be modulated by the terahertz field at any temporal portion using an optical delay line, as seen in Figs. 2(a)3. Further investigation on how to correct for the change in the instrument response will be reported elsewhere. Thus, one can determine the optical constant of a material by terahertz spectroscopy using EODCP with spectral resolution and bandwidth values which are comparable to those obtained by THz-TDS if appropriate values are adopted for the chirp rate and the spectral resolution of the spectrometer and if the instrument response () is canceled out through the calculation of the optical constant in EODCP.

We have measured the temporal waveform of a terahertz field pulse using EO detection with a chirped optical probe beam and a spectrometer as a function of the chirp rate and spectral resolution of the spectrometer. A double-beam configuration was employed in this measurement to simultaneously measure the probe pulses with and without terahertz field modulation. The waveform thus measured at a chirp rate of −0.24  THz² exhibited multicyclic behavior and approached monocyclic behavior with increasing magnitude of the chirp rate, although the original terahertz field waveform was monocyclic. Further, it was found that the broadening of the spectral resolution of the spectrometer leads to a change from a multicyclic waveform to a monocyclic waveform. Furthermore, an expression for the terahertz field waveform measured using the chirped probe pulse was derived without using the stationary phase method. The chirp rate and spectral resolution dependences of the terahertz field waveform measured were in agreement with those calculated. Moreover, it was demonstrated that this technique will be applicable to the terahertz spectroscopy if appropriate values are adopted for the chirp rate and the spectral resolution of the spectrometer and if the instrument response is canceled out through the calculation of the complex transmission coefficient.

We would like to thank Professor M. Hosoda, Dr. A. Yokoyama, and Mr. Y. Toyota for the fruitful discussions.

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Fig. 1. The schematic diagram of the experimental setup for the EODCP of the terahertz field pulse with a spectrometer equipped with a CCD image sensor. A double-beam configuration was employed to obtain the probe pulses with modulation due to the terahertz field and without it simultaneously. ITO is indium tin oxide. First citation in article

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Fig. 2. (a) The terahertz field waveforms obtained by EODCP at five delay times between the input terahertz field and chirped probe pulses, at a chirp rate of −0.24  THz². The ordinate of each waveform is shifted for the sake of clarity. (b) Corresponding peak positions [arrow in (a)] in the terahertz waveform, represented by frequency, plotted as a function of the delay time (open squares). The chirp rate is obtained from the slope of the solid line fitted to the data. The upper abscissa is derived from the lower one using the chirp rate value in (a). (c) The waveform obtained by EODCP (black line) at −0.24  THz² is compared with that obtained by THz-TDS (thick gray line). First citation in article

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Fig. 3. Terahertz field waveforms obtained by EODCP at five delay times, at a chirp rate of −1.3  THz². The ordinate of each waveform is shifted for the sake of clarity. The upper abscissa is derived from the lower one using the chirp rate value. First citation in article

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Fig. 4. The terahertz field waveforms calculated from Eq. (4) at three delay times and a chirp rate of −0.24  THz², where T_c=22  ps, T=0.45  ps, and ₀=375  THz. The ordinate of each waveform is shifted for the sake of clarity. The original terahertz field waveform, E_THz(t) in Eq. (1), is depicted in the inset. The values of T and ₀ are fixed for the calculations below. First citation in article

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Fig. 5. Chirp rate dependence of the terahertz field waveform calculated from Eq. (4), where [a(THz²),T_c(ps)]=(−0.12,22), (−0.65,4), and (−1.5,1.8), with =0  ps. The chirp rate value is given by 2a. The ordinate of each waveform is shifted for the sake of clarity. First citation in article

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Fig. 6. (a) Terahertz field waveforms measured by EODCP at two spectral resolutions of the spectrometer [0.2 nm (black line) and 3 nm (thick gray line)] at a chirp rate of −1.9  THz². (b) Frequency spectra obtained from the Fourier transforms of the two terahertz field waveforms, together with that obtained by THz-TDS. A logarithmic scale is used for the ordinate. First citation in article

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Fig. 7. Terahertz field waveform calculated from Eq. (3) through numerical integration of Eq. (2) at a chirp rate of −0.24  THz², where the -function and Gaussian functions with widths of 0.2, 0.6, and 3.4 nm are used for g(−). The ordinate of each waveform is shifted for the sake of clarity. First citation in article

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Fig. 8. Frequency spectra obtained from the Fourier transform of the terahertz field waveform calculated from Eq. (4) at a chirp rate of −0.24  THz² (black line) and the original waveform (thick gray line). The frequency spectrum obtained by EODCP (experiment) at −0.24  THz² is plotted, where the ordinate is shifted for the sake of clarity. A logarithmic scale is used for the ordinate. First citation in article

^*Electronic mail: murakami.hiroshi@jaea.go.jp.

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