Comb-mode-resolved adaptive sampling terahertz dual-comb spectroscopy with a free-running single-cavity fiber laser

Mode-resolved dual-comb spectroscopy (DCS) is an emerging spectroscopic tool with the potential to simultaneously achieve a broad spectral coverage and ultrahigh spectral resolution in terahertz (THz) spectroscopy. However, the need for two independently stabilized ultrafast lasers significantly hampers the potential application of DCS techniques. In this article, we demonstrate mode-resolved DCS in the THz region based on a free-running single-cavity dual-comb fiber laser with adaptive sampling. Low-pressure spectroscopy of acetonitrile gas with absorption features approaching the Doppler limit is demonstrated by comb-mode-resolved measurements with a spectral sampling spacing of 48.8 MHz, a spectral resolution of less than 5 MHz and a signal-to-noise ratio of ~50 dB. The successful demonstration of the proposed method clearly indicates the great potential for the realization of low-complexity, MHz-resolution THz spectroscopy instrumentation.


INTRODUCTION
Coherent spectroscopic techniques in the terahertz (THz) or far-infrared region (frequencies of 0.1 ~ 10 THz and wavelengths of 30 μm ~ 3 mm) are the enabling technology for a wide variety of important applications, ranging from material characterization for intriguing and highly complex interactions [1,2] to submillimeterwave electronic device/system characterization [3]. Among these techniques, photonic-based THz time-domain spectroscopy (THz-TDS) has developed into arguably the most widely used technology in the past three decades [4,5]. THz-TDS takes advantage of the broadband radiation of ultrafast lasers and optically pumped THz emitters/receivers relative to its narrowband electronic-based counterparts to realize measurements over a large THz spectral range. THz-TDS has been applied to the study of a diverse range of samples, such as the rotational transitions of polar gas molecules [6], the hydrogen bonding signature in aqueous systems [7], and the dynamics and self-assembly of proteins [8]. This well-established THz-TDS scheme uses ultrashort pulses from a mode-locked laser for THz radiation and delayed probe pulses for THz detection; however, it has a limited spectral resolution and accuracy due to the constraints on the travel range, repeatability, and speed of the mechanical delay lines. To overcome this limitation, asynchronous optical sampling (ASOPS) THz-TDS [9][10][11] has been developed, using two mode-locked femtosecond lasers with a small repetition rate offset between them. By avoiding the physical delay lines, the equivalent temporal delay range can be largely increased; thus, the spectral resolution -4-can be significantly enhanced to ~1 GHz. The emergence of optical frequency combs, which provide unprecedented freedom and accuracy in manipulating optical frequencies, also benefits the development of their counterparts in the THz region.
THz dual-comb spectroscopy (THz-DCS) [12][13][14] has become a promising pathway towards ultrahigh-resolution broadband THz spectroscopy. While directly generating two THz combs is possible using a pair of THz quantum cascade lasers (QCLs) [15][16][17] for example, such THz sources have a relatively large comb tooth spacing and often poor mutual coherence between the combs in place of compact device size. This poses limitations to their spectral resolution and sampling spacing. If two stabilized optical combs with a high mutual stability are used for the generation and detection of a THz comb, the mixing of dual THz combs with excellent mutual coherence can be mapped to an RF comb with the temporal magnification factor (TMF), which is given by the ratio of the repetition rate to the repetition rate offset (frep1/∆frep). An RF comb is easily accessible with low-bandwidth electronics and further processed to yield highbandwidth, high-resolution THz spectroscopic information. It has been demonstrated that the spectral resolution can reach the THz comb tooth linewidth, which is on the order of MHz or better [18,19]. By sweeping the lasers' comb tooth spacing via slight adjustments in the lasers' cavity lengths, gapless THz spectral sampling was achieved [20,21]. However, the practical use of mode-resolved THz-DCS is still hampered by the need for two frequency-stabilized optical frequency comb sources, which are expensive and complex.
-5-Various schemes have been investigated to further reduce the complexity of THz-DCS systems and optical DCS systems. Recently, advances in the endeavor to generate a pair of frequency combs from a free-running single-cavity laser [22][23][24][25][26][27][28][29][30][31] have shown great potential towards this goal. By propagating through the same cavity, the dual optical combs experience almost the same disturbances, and the commonmode fluctuations thereby prevent the decline of the mutual coherence between the dual combs. Such a single-cavity dual-comb laser has been applied to THz and microwave frequency measurements [32,33] and optical spectroscopy [28,[34][35][36].
Furthermore, a single-cavity dual-comb laser has been successfully demonstrated to realize low-complexity THz-DCS [37,38]; however, the spectral resolution remained at approximately 1 GHz and was insufficient for low-pressure, Doppler-limited gas spectroscopy. In this case, the long-term instability in the TMF caused by the residual timing jitter of the laser eventually limits the spectral resolution [39,40]. On the other hand, adaptive sampling has been shown to be able to compensate for the timing jitter in DCS in the optical and THz regions [41,42]. Adaptive sampling can reconstruct the relative coherence for relatively long time scales between the two combs. Furthermore, from the viewpoint of the suppressing the timing jitter, it has been demonstrated that a combination of adaptive sampling with two free-running lasers is more powerful than a combination of constant sampling with two stabilized lasers [42]. However, two similarly constructed ultrafast lasers are still required. A combination of adaptive sampling with a free-running single-cavity dual-comb laser will be the ultimate form of -6-THz-DCS for high spectroscopic performance and reduced system complexity.
In this paper, we demonstrate a comb-mode-resolved THz-DCS scheme based on a simple free-running, single-cavity dual-comb fiber laser and adaptive sampling. By extending the temporal sampling window to over 200 ns while maintaining the data fidelity over a long acquisition period, a mode-resolved THz comb spectrum with a frequency sampling spacing of 48.8 MHz and a spectral resolution of less than 5 MHz is obtained with a signal-to-noise ratio (SNR) of ~ 50 dB.

METHODS
THz-DCS can be performed in the frequency domain [12,13] or time domain [14]. In time-domain THz-DCS, the THz frequency comb spectrum is obtained by a combination of ASOPS and a Fourier transformation (FT), as shown in Fig. 1 However, the RF pulse train is subject to the residual timing jitter of the laser source due to variations in the TMF, leading to a nonlinearity in the RF time scale. The fluctuating RF pulse train is shown in the upper row of Fig. 1(b) [42]. Such temporal fluctuations will be transferred to the frequency scale of the RF comb and THz comb, which will seriously degrade the spectral resolution and accuracy. If the RF pulse train is acquired by a sampling clock synchronized with such fluctuations as shown in the middle row of Fig. 1(b), the time-scale linearity of the sampled signal can be recovered (see the lower row of Fig. 1(b)). In this case, the time window size can be effectively extended without an accumulation of timing errors, and it is also possible to acquire accumulated data over a longer period of time to realize an improved SNR.
As shown in Fig. 2, the comb-mode-resolved adaptive sampling THz-DCS setup is seeded by a free-running single-cavity dual-comb fiber laser, which has an all-fiber ring cavity in which the dual-comb light beams propagate along a commonpath route. The dual-comb light beams with different center wavelengths were obtained by multiplexing the mode-locking operation in the wavelength region [32,34,37]. The cavity consists of a hybrid wavelength division multiplexer and isolator -9-A portion of the separated λ1-comb and λ2-comb light is fed into a sumfrequency-generation cross-correlator (SFG-X), whose setup is constructed based on a noncollinear configuration with a piece of a β-BaB2O4 (BBO) crystal. The resulting SFG pulse that occurs every 1/∆frep serves as the trigger signal for the data acquisition board.
The other important part of the setup compared to our previous THz-DCS system [37] is the adaptive clock generator, which suppresses the long-term drift and timing jitter in the repetition rate difference ∆frep of the free-running, dual-comb laser.
As shown in Fig. 2, the adaptive clock, which can trace the timing fluctuation in real time, is generated by heterodyning photoconductive mixing of reference CW-THz radiation and two PC-THz combs (PC-THz comb1 and PC-THz comb2) seeded by the optical dual combs [42]. In this scheme, two bowtie-shaped, low-temperature-grown

A. Performance of the dual-comb fiber laser
When the EDF is pumped above its mode-locking threshold and the intracavity PC is properly adjusted, dual-comb lasing is achieved with similar peak spectral intensities at two different center wavelengths, as shown in Fig. 3(a). The spectra at 1532.5 nm and 1557.7 nm have 3 dB bandwidths of 4.2 nm and 3.6 nm, respectively (namely, the λ1-comb light and λ2-comb light). The interval between their wavelengths is 25.2 nm, consistent with the birefringence introduced by the PMF fiber. is only ~190 Hz, as shown in Fig. 3(b). After separate amplification by EDFAs, the -11-dual-comb lights are spectrally broadened and temporally compressed to ~110 fs at full-width-at-half-maximum (FWHM) by propagating through standard single-mode fiber (SMF), which are sufficient to drive the broadband THz comb spectrum.
The temporal drifts of frep1, frep2 and ∆frep under free-running conditions are monitored in time, since they affect the TMF between the THz comb and RF comb (see Fig. 1). As shown in Fig. 3(c) However, the frequency stability of ∆frep in the single-cavity dual-comb laser is significantly better than that in the two free-running fiber-comb lasers over a short time.
Next, when comparing the single-cavity dual-comb laser with two independently stabilized fiber-comb lasers, the single-cavity dual-comb laser shows better short-term stability in ∆frep than the two independently stabilized fiber-comb lasers. On the other -12-hand, the two independently stabilized fiber-comb lasers show better long-term stability in ∆frep and frep1 than the single-cavity dual-comb laser. Importantly, even though the single-cavity dual-comb laser is inferior to the two independently stabilized fiber-comb lasers in terms of the long-term stability of ∆frep and frep1 due to the lack of active laser stabilization, the adaptive sampling method allows us to overcome such inferiority significantly as demonstrated later, leading to a reduced system complexity and excellent spectroscopic performance.

B. Performance of the adaptive sampling scheme
To investigate the effectiveness of the proposed adaptive sampling THz-DCS method, 100,000 temporal waveforms of a train of 10 consecutive THz pulses were acquired and accumulated with the adaptive sampling clock method. For a comparison, similar temporal waveforms were acquired based on the conventional method with a constant sampling clock. As shown in the upper part of Fig. 4(a), when using the constant sampling method, the THz pulses almost disappeared except for the first pulse, because the residual timing jitter causes random walk-off of the temporal sampling positions in each time-delay scan and hence quite a low efficiency in the signal accumulation. Obviously, the constant sampling method is not suitable to extend the temporal window of the accumulated temporal waveform to multiple pulse periods in the case of a free-running single-cavity dual-comb laser. However, by using the adaptive sampling method, each THz pulse can be clearly observed in the accumulated temporal waveform, as shown in the lower part of Fig. 4(a). Moreover, -13-the peak amplitude of the pulsed THz electric field remains constant for each THz pulse after many averages. These results imply that the adaptive sampling method has the capability to suppress the instability of the TMF over a long data acquisition time. Next, the comb-mode-resolved THz comb spectra were obtained by calculating the FT of the adaptive sampling temporal waveform [see the lower part of Fig. 4(a)], as shown in Fig. 4(b).
from the atmospheric water vapor. Figure 5(b) shows a zoomed-in plot of the linear power spectra for Fig. 5(a) within the spectral range of 0.3305 to 0.3315 THz for the vacuum gas cell (blue plot) and the CH3CN/air gas cell (red plot). Although a periodic modulation is observed in the spectral envelope of the THz comb mode due to internal reflections in the THz optical setup, it can be canceled by calculating the absorption spectrum from both spectra. Because this frequency range has no absorption lines from water vapor, the difference in the spectra reflects the absorption features of the low-pressure CH3CN gas.
The spectrum of the absorption coefficient for CH3CN gas was obtained by normalizing the mode-resolved THz comb spectrum measured in the 360 Pa CH3CN/air gas cell with that measured in the vacuum gas cell. Figure 6(a) shows the broadband spectrum of the absorbance coefficient, indicating multiple manifolds with a peak absorption coefficient of approximately 0.1 cm -1 . The manifolds were separated by ~18.388 GHz, exactly equal to 2B, and a series of manifolds within this frequency range were correctly assigned to J = 10 to 38. Figure 6(b) shows a zoomed-in plot of the assumption that all K lines have the same linewidth. As shown by the red line in Fig. 7(a) -(f), the database fitting spectra match the experimental data very well. These analyses suggest that our THz-DCS scheme with a reduced system complexity has the capability to realize low-pressure gas spectroscopy with a MHz-order spectral resolution.

DISSUSSION
We first discuss the potential of the mode-resolved adaptive sampling THz-DCS scheme for low-pressure, Doppler-limited gas spectroscopy. We confirmed clear differences in the absorption linewidth as the pressure was reduced. By using multipeak curve fitting analysis based on the JPL spectral database, the results -18-indicated that the observed pressure broadening coefficient lies between the values for self-broadening (912 kHz/Pa) and broadening by nitrogen (91.2 kHz/Pa) [45,46]. We next discuss the possibility of employing the proposed method for THz spectroscopy with improved precision. Although we demonstrated the potential of the proposed method for Doppler-limited, low-pressure gas spectroscopy, the frequency comb spacing of frep1 is still coarse for a full analysis of each rotational transition with a MHz-order structure. A promising approach for THz spectroscopy with improved precision is to use the gapless technique in the THz comb, in which the frequency spacing of two stabilized THz combs is precisely swept to interleave additional comb lines into the original comb lines [20,21]. In this case, the THz comb spacing can be

CONCLUSIONS
We demonstrated the capability of realizing mode-resolved THz comb spectroscopy with a free-running single-cavity dual-comb fiber laser. By using adaptive sampling with the free-running laser, the long-term instability of the TMF was effectively suppressed, facilitating the long-term acquisition and temporal accumulation of THz temporal waveforms with a time window extending to multiple laser pulse periods. This results in a broadband, mode-resolved THz comb spectrum with a frequency sampling spacing of 48.804 MHz and a spectral resolution of less than 5 MHz. Low-pressure CH3CN/air gas with absorption spectral features on the order of GHz and MHz was measured, and good agreement with the theoretical predictions was achieved. The resolving capability of MHz-level spectral -20-characteristics using a simple fiber laser could greatly expand the applicability of precise THz spectroscopic techniques to much broader areas.