Super-resolution discrete-Fourier-transform spectroscopy using precisely periodic radiation beyond time window size limitation

Fourier transform spectroscopy (FTS) has been widely used in a variety of fields in research, industry, and medicine due to its high signal-to-noise ratio, simultaneous acquisition of signals in 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 limitation using discrete Fourier transform spectroscopy (dFTS) with precisely periodic pulsed terahertz (THz) radiation. Since infinitesimal resolution can be achieved at periodically discrete frequencies when the time window size is exactly matched to the repetition period T, a combination of THz-dFTS with a spectral interleaving technique achieves a spectral resolution only limited by the spectral interleaving interval. Linewidths narrower than 1/(50T) are fully resolved allowing the attribution of rotational-transition absorption lines of low-pressure molecular gases within a 1.25 MHz band. The proposed method represents a powerful tool to improve spectrometer performance and accelerate the practical use of various types of FTS.


Introduction
Fourier transform spectroscopy (FTS) is a spectroscopic technique where 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.
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 [1][2][3][4]. Therefore, as the time window is increased, 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 -4-size is restricted to a single repetition period to avoid the coexistence of multiple signals. However, if the repeating phenomenon is observed using precisely periodic radiation, the time window 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][6][7][8][9][10][11][12]. DCS acquires the temporal waveforms of repeating phenomena with a time window extending over many repetition periods and by FT achieves a spectral resolution finer than the repetition frequency. The spectral resolution is still limited by the time window, along with massive volume of datapoints required for rapid sampling over a long window.
Such a huge number of data results in stringent restrictions for the data acquisition and FT calculation.
If the spectral resolution in FTS can be improved beyond the repetition frequency without the need to acquire the huge number of datapoints, the scope of application of FTS will be further extended. In this article, we demonstrate a significant spectral resolution imptovement over the time window limitation by using discrete Fourier transform spectroscopy (dFTS) in THz region. We show that this is only possible when the time window size is exactly matched to the repetition period of precisely periodic THz pulse.

Principle of operation
-5-First we consider the measured temporal waveform h(t) of a phenomenon and its FT spectrum H(f) given as

This equation indicates that a spectral component H(f) is obtained by multiplying h(t)
by a frequency signal exp(-2πift) and then integrating the product for an infinite integration period. This process is illustrated in Fig. 1(a), where cos2πft is shown as the real part of exp(-2πift). Although the achieved spectral resolution is infinitesimal in Eq. (1), the practical resolution is limited by the achievable finite integration period due to SNR, the acquisition time, and/or the stage travel range.
Next we consider the case where h(t) is repeated by precisely periodic 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 marks of scale into the frequency gap between data points, namely, spectral interleaving [13][14][15]. If incremental sweeping of T is repeated to , 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 τ = T, the frequency interval between the spectral data points is universally constant, and hence the absolute frequency of the spectrum is -7-secured to the frequency standard.

Experimental setup
To observe a repeating phenomenon precisely by using a stabilized, mode-locked THz pulse train and acquiring its temporal waveform with τ = T, we applied the asynchronous optical sampling (ASOPS) method [16][17][18][19].   [20] and room temperature Doppler broadening linewidth [21] for this gas sample are also indicated as blue solid and broken lines in Fig. 3 where J and K are rotational quantum numbers. The first term in Eq. (3) indicates that many groups of absorption lines regularly spaced by 2B (= 18.40 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 both of these two features simultaneously with conventional THz spectroscopy [23,24]. Recently, gapless THz-DCS, based on a combination of DCS with spectral interleaving in the THz region, has been successfully used to observe them [15]. However, the huge number of the 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 CH 3 CN at a pressure of 30 Pa.  Fig. 5(a). As the resolution and spacing are no longer equal, a frequency gap is created between successive comb modes. Hence the remaining (N p -1)/N p frequencies, namely comb gaps, lack any significant information due to the absent of the radiation even though the mode-resolved comb spectrum is composed of a huge number of datapoints. Therefore, the data quantity ratio of the comb modes to the comb gaps, namely, the signal efficiency with respect to all spectral data points, -13-is relatively 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. Although a quantitative comparison is given later, the large reduction of the amount of data is a great advantage of dFTS over DCS.
We next discuss the achievable spectral resolution of the two methods.
When 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, as shown in the lower part of Fig. 5

A1. Theory
The FT spectrum H(f) for the measured temporal waveform h(t) of a phenomenon [see Fig. 1(a)] is given by where k is an integer. In this case, the spectral resolution is infinitesimal. On the other hand, when the phenomenon repeats at a time period T, as shown in Fig. 1(b), the repeating phenomenon g(t) is given by If we observe g(t) for a time window of 0<t<T, as shown in Fig. 1(c), its FT spectrum G(f) is given by Therefore, if f n = n/T, the corresponding FT spectrum G(f n ) is given by indicating that Eq. (7) is equivalent to Eq. (4). That is to say, if a phenomenon repeats with a time period T and we observe it for a finite time window of 0<t<T, its FT spectrum provides spectral information with an infinitesimal resolution at a frequency f n (= n/T). In other words, infinitesimal resolution can be achieved at a discrete frequency f n in FTS with a finite time window.
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 δ is the delta function. The FT of Eq. (8) is where t m = mT/N. Eq. (9) is the simple dFT of g s (t) with time window T. It is known that information is not lost by sampling when the sampling frequency is faster than two-times the maximum frequency of the data (sampling theorem) [27]. Therefore, discrete spectral data with an infinitesimal spectral resolution is given by taking the On the other hand, to perform the same experiment using a combination of THz-DCS with the spectral interleaving [15], one has to first acquire the temporal waveform of the THz pulse train with a time window size equal to 200 repetition periods (= 800 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 (data/waveform) * 20 (waveforms)], which is 200 times larger than that in THz-dFTS.