## Abstract

Fourier transform spectroscopy (FTS) has been widely used in a variety of fields due to its high signal-to-noise ratio, simultaneous acquisition of a broad spectrum, and versatility for different radiation sources. Further improvement of the spectroscopic performance will widen its scope of applications. Here, we demonstrate improved spectral resolution by overcoming the time-window size limitation using a mode-locked terahertz (THz) pulse train as precisely periodic pulsed radiation in discrete Fourier transform spectroscopy (dFTS). Since infinitesimal resolution can be achieved at harmonic components of its repetition frequency $1/T$ when the time-window size is exactly matched to the repetition period $T$, a combination of dFTS with a spectral interleaving technique achieves a spectral resolution limited only by the spectral interleaving interval. Linewidths narrower than $1/(50T)$ are fully resolved by THz-dFTS, allowing rotational-transition absorption lines of low-pressure molecular gases to be attributed within a 1.25 MHz band.

© 2015 Optical Society of America

## 1. INTRODUCTION

Fourier transform spectroscopy (FTS) is a spectroscopic technique in which spectra are obtained by measuring a temporal waveform or interferogram of electromagnetic radiation, or other types of radiation, and calculating its Fourier transform (FT). FTS possesses inherent advantages over conventional dispersive spectrometers, such as a high signal-to-noise ratio (SNR), simultaneous acquisition of signals in a broad spectrum, and versatility for different radiation sources. Therefore, various types of FTS has been widely used in the fields of research, industry, and medicine, such as FT infrared (FT-IR) spectroscopy [1], terahertz (THz) time-domain spectroscopy (THz-TDS) [2], FT electrical/optical spectral analysis, FT nuclear magnetic resonance (FT-NMR) [3], and FT mass spectroscopy (FT-MS) [4].

When the temporal waveform of a phenomenon is measured, the spectral resolution is simply determined by the inverse of the measurement time-window size during which the temporal waveform is observed. Therefore, as the time-window size is extended, the spectral resolution is enhanced. However, when the majority of the signal components are temporally localized, excessive extension of the window size increases the noise contribution as well as the acquisition time. Furthermore, in the case of optical FTS, the travel range of a translation stage used for time-delay scanning practically limits the spectral resolution.

When the phenomenon repeats, it is generally accepted that the achievable spectral resolution is limited to its repetition frequency because the maximum window size is restricted to a single repetition period to avoid the coexistence of multiple signals. On the other hand, recent progress in ultrashort pulsed lasers and related technologies enables us to generate a mode-locked pulse train in the ultraviolet, visible, infrared, and THz regions, which can be considered precisely periodic pulsed radiation. If the phenomenon is made to repeat by using precisely periodic pulsed radiation, a sequence of phenomena are essentially copies of the same phenomenon separated by an interval equal to the inverse of the repetition frequency. In this case, the time-window size may be expanded beyond a single period without accumulation of timing errors. Recently, dual comb spectroscopy (DCS) has emerged in the ultraviolet, visible, infrared, and even THz regions [5 –13]. The mode-resolved DCS acquires the temporal waveforms of repeating phenomena with a time-window size extending over many repetition periods and, by FT, achieves a spectral resolution finer than the repetition frequency. However, the spectral resolution is still limited by the time window. Furthermore, there is a huge number of data points due to fine temporal sampling over a large time window, resulting in stringent restrictions for data acquisition and FT calculation.

If the spectral resolution of FTS can be improved beyond the repetition frequency without the need to acquire a huge number of data points, the scope of application of FTS will be further extended. Recently, two interesting techniques have been implemented in DCS: coherent averaging [10,11] and spectral interleaving [8,10,11,13]. The former enables not only co-adding sequential interferograms for high SNR, but also achieves spectral resolution equal to the intrinsic linewidth of the optical comb mode without the need to acquire the mode-resolved spectrum. Furthermore, it can greatly reduce the number of data points. On the other hand, in the latter, the spectral resolution is determined by the spectral interleaving interval rather than the frequency interval of the optical comb by filling the comb mode gap with additional frequency marks. Furthermore, the combination of coherent averaging with spectral interleaving has been reported [10,11]. However, the detailed mechanism of its spectral resolution improvement has not yet been clearly explained. Furthermore, its high spectroscopic performance is underutilized except the near-infrared region.

In this article, we demonstrate a significant spectral resolution improvement over the time-window size limitation by using discrete Fourier transform spectroscopy (dFTS) with a mode-locked THz pulse train. We explain the mechanism and limit of the spectral resolution improvement from the viewpoint of the temporal connection of many signal pieces at different timings in the temporal waveform, and verify its correctness theoretically and experimentally. The resolution improvement is possible when the time-window size is exactly matched to the repetition period of the mode-locked THz pulse train, which is central to coherent averaging. More interestingly, the resolution improvement can be maintained by introducing null data to restore the temporal connection even though the time-window size is not always equal to the repetition period. Finally, we perform high-resolution, high-accuracy, broadband spectroscopy of multiple absorption lines of low-pressure molecular gas in the broad THz region.

## 2. METHODS

#### A. Principle of Operation

First we consider the measured temporal waveform $h(t)$ of a phenomenon and its FT spectrum $H(f)$ given by

This equation indicates that a spectral component $H(f)$ is obtained by multiplying $h(t)$ by a frequency signal $\mathrm{exp}(-2\pi ift)$, and then integrating the product for an infinite integration period. This process is illustrated in Fig. 1(a), where $\mathrm{cos}\text{\hspace{0.17em}}2\pi ft$ is shown as the real part of $\mathrm{exp}(-2\pi ift)$. Although the achieved spectral resolution is infinitesimal in Eq. (1), the practical resolution is limited by the achievable finite integration period due to the SNR, the acquisition time, and/or the stage travel range.Next we consider the case where $h(t)$ is made to repeat by using precisely periodic pulsed radiation with a repetition period of $T$. When the relaxation time of $h(t)$ is longer than $T$, a series of signals $h(t)$ temporally overlap, each subsequent event being delayed by an integer multiple of $T$ [see Fig. 1(b)]. When the time series data is acquired with a finite time-window size $\tau $, the observed time window includes signal contributions from multiple periods, for example, signals (A), (B), (C), and (D), as depicted in Fig. 1(c). To obtain the FT spectrum of these, signals (A), (B), (C), and (D) are summed and multiplied by $\mathrm{exp}(-2\pi ift)$ before being integrated over $\tau $, as shown in Fig. 1(d). Here, if $T$ is sufficiently stable and is exactly matched to $\tau $, signals (A), (B), (C), and (D) can be temporally connected to form a single, temporally continuous signal alongside $\mathrm{exp}(-2\pi ift)$ [see Fig. 1(e)]. Despite the finite time-window size ($=\tau =T$), this connection is equivalent to acquiring the temporal waveform of $h(t)$ without the limitation of the time-window size. Because of the smooth connection of $\mathrm{exp}(-2\pi ift)$, an infinitesimal spectral resolution can be achieved at discrete frequencies:

where $n$ is the order of the spectral data points, and $N$ is the total number of sampling points in the temporal waveform [see the top of Fig. 1(f)]. The spectral data at the discrete frequencies are calculated simply by taking the discrete Fourier transform (dFT) of the signals in Fig. 1(c).Although these spectral data points provide infinitesimal spectral resolution, their discrete spectral distribution limits the spectral sampling interval to $1/T$. To harness the improved spectral resolution for broadband spectroscopy, we must interleave additional data points into the frequency gap between original data points, namely, spectral interleaving [8,10,11,13]. If incremental sweeping of $T$ is repeated to interleave the additional points (${f}_{n}^{\prime},{f}_{n}^{\prime \prime},\dots $) [see the middle of Fig. 1(f)], and all of the resulting dFT spectra are overlaid, the frequency gaps can be filled in [see the bottom of Fig. 1(f)], allowing us to obtain a more densely distributed, discrete spectrum, which has great potential for broadband high-resolution spectroscopy. Since this procedure is equivalent to obtaining the dFT spectral components while tuning $T$ with a fixed $N$ in Eq. (2), the spectral sampling density becomes higher due to the increase in the number of data points. Furthermore, if $T$ is phase-locked to a microwave frequency standard while maintaining $\tau =T$, the frequency interval between the spectral data points is universally constant, and, hence, the absolute frequency of the spectrum has the same accuracy as the frequency standard. The increased spectral sampling density made possible by the spectral interleaving enhances the spectral accuracy as well as the spectral resolution.

#### B. FT of a Periodically Repeating Phenomenon

Here, we consider the principle of operation mathematically. The FT spectrum $H(f)$ for the measured temporal waveform $h(t)$ of a phenomenon [see Fig. 1(a)] is given by

The temporal signal is sampled, and its FT transform is calculated on a computer. The sampled data ${g}_{s}(t)$ is expressed by

where $N$ is the total number of sampling points, and $\delta $ is the delta function. The FT of Eq. (7) is#### C. Experimental Setup

To repeat a phenomenon precisely by using a stabilized, mode-locked THz pulse train and acquire its temporal waveform with $\tau =T$, we applied the asynchronous optical sampling (ASOPS) method [15,16]. Since the ASOPS method enables us to expand the picosecond time scale of THz transient signals up to the microsecond scale, the resulting slowed temporal waveform can be directly and precisely measured by a data acquisition board without the need for mechanical time-delay scanning. This nonmechanical nature of the time-delay scanning enables us to acquire a temporal waveform where $\tau $ is accurately matched to $T$, ensuring a smooth temporal connection of the signal contributions.

Figure 2 is a schematic diagram of the THz-dFTS setup, which contains dual mode-locked femtosecond lasers and a THz optical setup for low-pressure gas spectroscopy. We used dual mode-locked Er-doped fiber lasers (center wavelength ${\lambda}_{c}=1550\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, pulse duration $\mathrm{\Delta}\tau =50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}$, mean power ${P}_{\text{mean}}=500\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mW}$) for THz-dFTS. Their repetition frequencies (${f}_{\text{rep}1}=\mathrm{250,000,000}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Hz}$, and ${f}_{\text{rep}2}={f}_{\text{rep}1}+{f}_{\text{offset}}=250,000,050\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Hz}$) and the frequency offset between them (${f}_{\text{offset}}={f}_{\text{rep}2}-{f}_{\text{rep}1}=50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Hz}$) were stabilized by using two independent laser control systems referenced to a rubidium frequency standard (Rb-FS, $\text{accuracy}=5\times {10}^{-11}$, $\text{instability}=2\times {10}^{-11}$ at 1 s). Furthermore, ${f}_{\text{rep}1}$ and ${f}_{\text{rep}2}$ could be separately tuned over a frequency range of $\pm 0.8\%$ by changing the cavity length with a stepper motor and a piezoelectric actuator. After wavelength conversion of the two laser beams by second-harmonic-generation (SHG) crystals, pulsed THz radiation was emitted from a dipole-shaped, low-temperature-grown (LTG), GaAs photoconductive antenna (PCA1) triggered by pump light (${\lambda}_{c}=775\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, $\mathrm{\Delta}\tau =80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}$, ${P}_{\text{mean}}=19\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mW}$), passed through a low-pressure gas cell ($\text{length}=500\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, $\text{diameter}=40\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$), and was then detected by another dipole-shaped LTG GaAs photoconductive antenna (PCA2) triggered by probe light (${\lambda}_{c}=775\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, $\mathrm{\Delta}\tau =80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}$, ${P}_{\text{mean}}=9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mW}$). The optical path in which the THz beam propagated was purged with dry nitrogen gas to avoid absorption by atmospheric moisture, except for the part in the gas cell. A small portion of the output light from the two lasers was fed into a sum-frequency-generation cross correlator (SFG-XC). The resulting SFG signal was used to generate a time origin signal for the ASOPS measurement. After amplification with a current preamplifier (AMP, $\text{bandwidth}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$, $\text{gain}=4\times {10}^{6}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{V}/\mathrm{A}$), the temporal waveform of the output current from PCA2 was acquired with a digitizer ($\text{sampling rate}=2\times {10}^{6}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{samples}/\mathrm{s}$, $\text{resolution}=20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{bit}$) by using the SFG-XC’s output as a trigger signal and the frequency standard’s output as a clock signal. Then, the time scale of the observed signal was magnified by a temporal magnification factor of ${f}_{\text{rep}1}/{f}_{\text{offset}}$ ($=250,000,000/50=5,000,000$) [16]. This sampling rate and this temporal magnification factor enabled us to measure the temporal waveform of the THz electric field signal at a sampling interval of 100 fs during a time-window size of 4 ns ($=T$), corresponding to 40,000 sampling data, at a scan rate of 50 Hz. When 1000 temporal waveforms of pulsed THz electric field were accumulated in the time domain, SNR of 300 was achieved without the gas sample. After signal accumulation in the time domain, a THz discrete amplitude spectrum with a spectral sampling interval of 250 MHz was obtained by taking its dFT. It is important to note that the pulse timing (${f}_{\text{rep}1}$ and ${f}_{\text{rep}2}$), the trigger timing (${f}_{\text{offset}}$), and the data acquisition timing in the digitizer were completely synchronized by use of Rb-FS for a common time base. Such excellent synchronization is a critical factor in correctly achieving the temporal connection of many signal pieces at different timings in the temporal waveform. In this setup, since $T$ ($=1/{f}_{\text{rep}1}$) is phase locked to Rb-FS while maintaining $\tau =T$, the frequency interval between the spectral data points is universally constant, and, hence, the absolute frequency of the spectrum is secured to Rb-FS.

The spectral interleaving in the dFTS spectrum was performed by incremental increases of $T$, or ${f}_{\text{rep}1}$, with the laser control systems. For example, in the rotational-transition absorption spectroscopy of water mentioned later, incremental increases of ${f}_{\text{rep}1}$ and ${f}_{\text{rep}2}$ by 11,220.8 Hz were repeated 10 times while keeping ${f}_{\text{offset}}$ at 50 Hz. In this case, a single shift of ${f}_{\text{rep}1}$ by 0.004488% resulted in sweeping of the spectral sampling data around the water absorption line at 0.557 THz by 10% of frequency interval in the dFTS spectrum ($=25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$) because the shift of ${f}_{\text{rep}1}$ is multiplied by $n$ ($=2,228$ at 0.557 THz) in Eq. (2).

## 3. RESULTS

#### A. Spectroscopy of Low-Pressure Water Vapor

To investigate the importance of consistency between $\tau $ and $T$ in the temporal connection of Fig. 1(e), we performed low-pressure spectroscopy of water vapor. Water vapor shows sharp absorption lines in the THz region due to rotational transitions of the asymmetric top molecule. Here, the rotational transition ${1}_{10}\leftarrow {1}_{01}$ at 0.557 THz was measured. To satisfy the condition that the phenomenon relaxation time be longer than $T$ [see Figs. 1(b) and 1(c)], a mixture of water vapor ($\text{partial pressure}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Pa}$) and nitrogen ($\text{partial pressure}=140\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Pa}$) was introduced into a low-pressure gas cell. The absorption line at 0.557 THz has an expected pressure-broadening linewidth of 10.6 MHz full width at half-maximum (FWHM) from the self-broadening of water vapor and the collision broadening induced by nitrogen [17]. Since the absorption relaxation time is determined by the inverse of its linewidth ($=94\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$), the absorption phenomenon relaxes over 23 repetition periods of the THz pulse train. Figure 3(a) shows the absorption spectrum obtained when $\tau =T$. In this experiment, we accumulated 5000 temporal waveforms for each dFT spectrum, and repeated the spectral interleaving 10 times at a spectral interleaving interval of 25 MHz. Baseline correction of the spectrum was not applied to the spectra in this article. We confirmed the absorption spectrum with a linewidth of 25 MHz at 0.557 THz. Although the observed spectral linewidth was limited by spectral interleaving interval rather than pressure broadening, it was 10 times better than the inverse of $\tau $ ($=250\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$). This result indicated that the temporal connection of different signal contributions enhanced the spectral resolution to greater than the limit determined by the time-window size. On the other hand, when $\tau =0.9995T$, the spectral shape was distorted [see Fig. 3(b); spectral interleaving $\mathrm{interval}=25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$]. This spectral distortion results from the partially discontinuous absorption transient due to the failure of the temporal connection. In this way, the exact consistency between $\tau $ and $T$ is important in the proposed method. More interestingly, the spectral shape was recovered by introducing null data corresponding to $0.0005T$, restoring the temporal connection [see Fig. 3(c)]. Although the spectral information may have been somewhat lost by padding null data, the zero padding is useful for the correct temporal connection when $\tau \ne T$.

For reference, we performed a similar comparison experiment when the absorption relaxation time was shorter than $T$. To this end, the spectral linewidth of the same water absorption line was increased to 504 MHz by setting partial pressures of water vapor and nitrogen at 1000 Pa and 3500 Pa, respectively. Under this condition, the relaxation time is 1.99 ns, which is around a half of $T$. Figures 3(d) and 3(e) show a comparison of the absorption spectra between $\tau =T$ and $\tau =0.9995T$ (spectral sampling $\text{interval}=25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$). Similar absorption spectra ($\text{linewidth}=504\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$) were confirmed for the two of them. This is because the absorption relaxation completes within a range of $T$ and, hence, the temporal connection of the noise signal has no meaning.

Next, we investigated the achievable spectral resolution limit of the proposed method by measuring the pressure broadening characteristic of the narrow water absorption line. To determine the spectral linewidth, we fitted a Lorentzian function to the measured spectral profile. The red circles in Fig. 3(f) show the FWHM of the observed absorption line with respect to the total pressure of this sample gas, which was varied within a range from 1 Pa to 10 kPa. The green triangles indicate the spectral interleaving interval, whereas the expected pressure broadening linewidth [17] and the room-temperature Doppler broadening linewidth [18] are indicated as blue solid and dashed lines, respectively. The expected pressure broadening linewidth does not follow a straight line due to the different mixture ratios used for each total pressure. The spectral linewidth determined by THz-dFTS was limited by the spectral interleaving interval or the expected pressure broadened linewidth, whichever is larger. In this way, since each data point has an infinitesimal spectral resolution, the spectral interleaving interval finally determines the practical spectral resolution achieved by the proposed method.

#### B. Spectroscopy of Low-Pressure Acetonitrile Gas

Finally, we applied THz-dFTS to gas-phase acetonitrile (${\mathrm{CH}}_{3}\mathrm{CN}$) to demonstrate its capacity to simultaneously probe multiple absorption lines. ${\mathrm{CH}}_{3}\mathrm{CN}$ is an important molecular gas in astronomic and atmospheric gas analyses because it is not only a very abundant species in the interstellar medium, but is also a volatile organic gas compound found in the atmosphere. Since ${\mathrm{CH}}_{3}\mathrm{CN}$ is a symmetric top molecule with a rotational constant, $B$, of 9.199 GHz and a centrifugal distortion constant, ${D}_{JK}$, of 17.74 MHz [19], the frequencies of its rotational transitions are given by,

where $J$ and $K$ are rotational quantum numbers. From this equation, the molecule displays two characteristic features in its THz spectrum. The first term in Eq. (9) indicates that many groups of absorption lines regularly spaced by 2B ($=18.40\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$) appear. The second term indicates that each group includes a series of closely spaced absorption lines of decreasing strength. It has been difficult to observe these two features simultaneously with conventional THz spectroscopy [20,21]. Recently, gapless THz-DCS, based on a combination of DCS with spectral interleaving in the THz region, has been successfully used to observe them [13]. However, the huge volume of sampling data produced by this method hinders its extensive use in practical gas analysis.To confirm the first spectral feature of the symmetric top molecule, we performed broadband THz-dFTS of gas-phase ${\mathrm{CH}}_{3}\mathrm{CN}$ at a pressure of 30 Pa. Figure 4(a) shows its absorption spectrum within a frequency range from 0 to 1 THz when the spectral interleaving interval around 0.65 THz was set at 12.5 MHz. Thirty-nine groups of absorption lines periodically appeared with a frequency separation of 18.40 GHz. These groups can be assigned to rotational quantum numbers from $J=15$ around 0.29 THz to $J=53$ around 0.98 THz. The observed dispersion of the peak absorbance is due to the mismatching between the absorption line positions and the spectral sampling positions rather than low SNR. Figure 4(b) shows the magnified spectrum for the 0.6 to 0.7 THz region, indicating that six individual groups ($J=32$ to 37) have fine spectral features related to the rotational transitions of ${\mathrm{CH}}_{3}\mathrm{CN}$ molecules in the ground state. We also observed the spectral features of the vibrationally excited molecules for those six groups [see red asterisks in Fig. 4(b)]. Since such fine spectral features were not clearly observed in the gapless THz-DCS [13], it is evidence of the enhanced spectroscopic performance of the THz-dFTS.

Next, to confirm the second feature of the symmetric top molecule, we spectrally expanded the absorption line group of $J=34$. Nine sharp absorption lines were clearly observed, as shown in Fig. 4(c). By performing multipeak fitting analysis based on a Lorentzian function, we correctly assigned them to $K=2$ to 10. However, two absorption peaks, for $K=0$ and 1, were not clear because the frequency separation between them ($=15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$) is comparable to the spectral interleaving interval ($=12.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$). To spectrally resolve these two absorption peaks, we further reduced the spectral interleaving interval to 1.25 MHz. This yielded a measured profile containing two fully resolved absorption lines for $K=0$ and $K=1$, as shown in Fig. 4(d). We determined their center frequencies to be 0.64326876 THz for $K=0$ and 0.64325750 THz for $K=1$. The discrepancy of them from values in the JPL database (see green dashed line) [22] was 1.10 MHz for $K=0$ and 0.12 MHz for $K=1$, which were both within the spectral interleaving interval of 1.25 MHz. In this way, we successfully observed the spectral signatures of ${\mathrm{CH}}_{3}\mathrm{CN}$ characterized by both $B$ and ${D}_{JK}$ using a single instrument. Regarding the result of $K=0$, the discrepancy of 1.10 MHz in the current work was twice larger than that of 0.578 MHz in gapless THz-DCS [13] even though the spectral interleaving interval ($=1.25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$) in the former was half of that ($=2.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$) in the latter. We consider that the lack of the spectral baseline correction under insufficient SNR limits the accuracy rather than the spectral interleaving interval.

## 4. DISCUSSION

To highlight dFTS, it is important to compare it with mode-resolved DCS because these two methods are analogous. The data volume requirements and achievable spectral resolution of the two techniques are key. We first discuss the total amount of data of the acquired temporal waveform. In mode-resolved DCS, the temporal waveform of multiple periodic signals of a phenomenon is acquired with a time window extending over many repetition periods. The mode-resolved comb spectrum is obtained by calculating its FT. It contains a series of frequency spikes with a frequency spacing equal to the repetition frequency and a linewidth equal to the reciprocal of the time window. For example, if the time window is extended up to ${N}_{p}$ times the repetition period, the linewidth is reduced to $1/{N}_{p}$ of the frequency spacing, as shown in the upper part of Fig. 5(a). As the linewidth and spacing are no longer equal, a frequency gap is created between successive comb modes. Hence, the remaining $({N}_{p}-1)/{N}_{p}$ of frequencies, namely, comb gaps, lack any significant information due to the absence of radiation. Therefore, even though the mode-resolved comb spectrum is composed of a huge number of data points, the data quantity ratio of the comb modes to the comb gaps, namely, the signal efficiency with respect to all spectral data points, is considerably low. On the other hand, the dFTS spectrum is equivalent to a spectrum in which only the peaks of each comb mode in the upper part of Fig. 5(a) are sampled with infinitesimal spectral resolution, as shown in the upper part of Fig. 5(b). Therefore, all spectral data points contribute to the signal components, and, hence, the signal efficiency is high.

We here compared the total amount of data of the acquired temporal waveform between THz-dFTS and mode-resolved THz-DCS. For example, in the experiment of Fig. 4(d), we repeated the spectral interleaving 20 times at an interval of 1.25 MHz to obtain the absorbance spectrum. To do so, we acquired 20 temporal waveforms at 20 different $T$ values ($\text{sampling interval}=100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}$, time-window $\text{size}=4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$), and, hence, the total number of sampling data was 800,000 [$=\mathrm{40,000}\text{\hspace{0.17em}}\text{\hspace{0.17em}}(\text{data}/\text{waveform})\times 20(\text{waveforms})$]. On the other hand, to perform the same experiment using a combination of mode-resoled THz-DCS with spectral interleaving [13], one has to first acquire the temporal waveform of the THz pulse train with a time-window size equal to 200 repetition periods ($=800\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$) at a sampling step of 100 fs to reduce the comb mode linewidth to 1.25 MHz. Then, one has to acquire 20 temporal waveforms at different $T$ values for 20 times spectral interleaving. In this case, the total number of sampling data is 160,000,000 [$=8,000,000(\text{data}/\text{waveform})\times 20(\text{waveforms})$], which is 200 times larger than that in THz-dFTS. In this way, the large reduction of the amount of data is a great advantage of dFTS over the mode-resolved DCS.

We next discuss the achievable spectral resolution of the two methods. When mode-resolved DCS is combined with the spectral interleaving technique, the spectral resolution will be limited by the comb mode linewidth or the spectral interleaving interval, whichever is larger, as shown in the lower part of Fig. 5(a). On the other hand, in the case of dFTS, the actual resolution is determined by only the spectral interleaving interval because spectral data points provide the infinitesimal spectral resolution, as shown in the lower part of Fig. 5(b). The spectral resolution determined by only the spectral interleaving interval is another advantage of dFTS over mode-resolved DCS. However, in reality, a few experimental factors may hinder the infinitesimal spectral resolution. The achievable spectral resolution in the upper part of Fig. 5(b) is limited by how long ago the oldest phenomenon was included in the observed time window. For example, if we set the waiting time for 20 ms before starting the data acquisition, the theoretical limit of the minimum spectral resolution is 50 Hz because 2,000,000 phenomena induced within the waiting time are included in the time window. Also, the spectral resolution will be limited by timing jitter between dual lasers because the timing jitter makes the temporal connection difficult due to fluctuation of the temporal magnification factor. However, even though these effects are considered, dFTS still has the advantage of the achievable spectral resolution over the mode-resolved DCS.

## 5. CONCLUSIONS

We demonstrated that time-window size did not limit the spectral resolution in dFTS using precisely periodic pulsed radiation when the time-window size is matched to the repetition period. The combination of dFTS with the spectral interleaving technique enables the narrow low-pressure absorption lines to be assigned within a spectral interleaving interval of 1.25 MHz. Further reduction of the spectral interleaving interval may improve the spectral accuracy as well as the spectral resolution. It should also be emphasized that THz-dFTS significantly reduces the required data volume while achieving spectroscopic performance equal to or greater than THz-DCS. Although THz-dFTS was demonstrated based on the asynchronous optical sampling method in this article, mechanical time-delay scanning in THz-TDS may be also used for THz-dFTS if the requirement $\tau =T$ can be precisely satisfied.

It will be possible to apply dFTS to a variety of FTS techniques using other radiation sources. For example, dFTS can be easily implemented in the experimental setup of DCS in other wavelength regions because of their similar setups. Also, FT spectral analysis, FT-NMR, or FT-MS may be combined with dFTS if the phenomenon is made to repeat by using precisely periodic pulsed radiation and measured precisely with a time-window size equal to one repetition period.

In addition to the improvement of spectral resolution and accuracy, the proposed method will be useful to enhance measurement sensitivity. For example, the confined acoustic modes in a free-standing semiconductor membrane was resonantly enhanced by the combination of the proposed method with optical pump–probe spectroscopy, and the subharmonic component of the mechanical resonance at 19 GHz was observed 20 times more sensitively than the nonresonant measurement [23].

The high flexibility and functionality of dFTS, allowing it to be applied to various types of FTS, will accelerate its practical adoption in the fields of science, industry, and medicine.

## FUNDING INFORMATION

Canon Foundation; Japan Science and Technology Corporation (JST); Ministry of Education, Culture, Sports, Science, and Technology (MEXT); Osaka University.

## ACKNOWLEGMENT

The authors are grateful to Prof. Kaoru Minoshima of the University of Electro-Communications and Prof. Tetsuo Iwata of Tokushima University for fruitful discussions.

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