2.1.

Principle of Operation

THz-DCS can be performed in the frequency domain^11,12 or time domain.¹³ In the time-domain THz-DCS, the THz frequency comb spectrum is obtained by a combination of asynchronous optical sampling (ASOPS)^45–47 and Fourier transform (FT), as shown in Fig. 1(a). Using two mode-locked lasers with slightly mismatched repetition rates ($f_{\mathrm{rep}1}$ and $f_{\mathrm{rep}2}$, $\mathrm{\Delta }{f}_{\mathrm{rep}}=f_{\mathrm{rep}2}-f_{\mathrm{rep}1}$) for THz generation and detection, a THz pulse train (with a repetition rate of $f_{\mathrm{rep}1}$, consecutive THz pulses of $N$, and a time window size of $N/f_{\mathrm{rep}1}$) is slowed down to an RF pulse train (with a repetition rate of $\mathrm{\Delta }{f}_{\mathrm{rep}}$, consecutive RF pulses of $N$, and a time window size of $N/\mathrm{\Delta }{f}_{\mathrm{rep}}$) by the TMF ($f_{\mathrm{rep}1}/\mathrm{\Delta }{f}_{\mathrm{rep}}$) based on the temporal magnifying function of ASOPS; this enables a direct acquisition of the RF pulse train using a data acquisition board without the need for mechanical time-delay scanning. FT of the RF pulse train results in the mode-resolved RF comb spectrum with a frequency spacing of $\mathrm{\Delta }{f}_{\mathrm{rep}}$. Finally, the mode-resolved THz comb spectrum is obtained by calibrating the frequency scale of the RF comb spectrum with the inverse of the TMF. The mode-resolved THz comb spectrum contains a series of successive narrow lines with a frequency spacing equal to the repetition rate ($f_{\mathrm{rep}1}$) and a spectral sampling step equal to the inverse of the time window ($f_{\mathrm{rep}1}/N$). The linewidth of each comb mode is equal to the spectral sampling step.

Fig. 1

Principle of operation. (a) Flowchart of time-domain THz-DCS. (b) Acquisition of the temporal waveform using the adaptive sampling method.

Details of the adaptive sampling method have been discussed elsewhere,³⁹ so only its brief description is given here. If the TMF is fluctuated by the residual timing jitter of the free-running SCDCL, linearity of the time scale of the RF pulse train is distorted as shown in the upper row of Fig. 1(b). Such temporal nonlinearity is transferred to the frequency scale of the RF comb and THz comb, which will seriously degrade the spectral resolution. If the RF pulse train is acquired by a sampling clock synchronized with the residual timing jitter, namely an adaptive clock, 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. It is also possible to acquire accumulated data over a longer period of time to realize an improved SNR.

2.2.

Single-Cavity Dual-Comb Laser

The upper-left part of Fig. 2 shows a free-running SCDCL composed of an all-fiber ring-cavity oscillator, a band-pass filter, and erbium-doped fiber amplifiers (EDFAs). The dual-comb light beams with different center wavelengths and different $f_{\mathrm{rep}}$ were obtained from the fiber oscillator by multiplexing the mode-locking operation in the wavelength region.^29,32,34 The fiber oscillator has an all-fiber ring cavity in which the dual-comb light beams propagate along a common-path route. The cavity consists of a hybrid wavelength division multiplexer and isolator (WDM/ISO), a piece of 0.46-m-long erbium-doped fiber (EDF) pumped by a 980-nm laser diode, a polarization controller (PC), a single-wall carbon nanotube saturable absorber (SA), an in-line polarizer (ILP) with polarization-maintained fiber (PMF) pigtails, and a 40% fiber output coupler (OC). In addition, a 65-cm-long dispersion compensation fiber was installed to optimize the intracavity dispersion at $\sim 4.0\mathrm{fs}/\mathrm{nm}$ at 1560 nm. The total length of the cavity was $\sim 4.21\mathrm{m}$. Due to the considerable birefringence of the PMF and the use of the ILP, spectral filtering and polarization-dependent loss tuning enable dual-comb lasing oscillation by adjusting the PC.²⁹ Dual-comb light with different center wavelengths and different $f_{\mathrm{rep}}$ (${\lambda }_1$-comb with $f_{\mathrm{rep}1}$ and ${\lambda }_2$-comb with $f_{\mathrm{rep}2}$) from the oscillator was separated into two independent combs with different center wavelengths and different $f_{\mathrm{rep}}$ by the bandpass filter, and the resulting two combs were further amplified by EDFAs. The difference in their center wavelengths is negligible for generation and detection of THz comb.

Fig. 2

Configuration of comb-mode-resolved adaptive sampling THz-DCS. SFG-X, sum-frequency-generation cross-correlator; BBO, beta-barium borate crystal; PC-THz comb, photocarrier THz comb; M, double-balanced mixer; FM, frequency multiplier (frequency multiplication factor $N=40$); PCA, photoconductive antenna; and AMP, current preamplifier.

2.3.

Adaptive Clock Generator

The adaptive clock generator (ACG) provides the adaptive clock to suppress the residual long-term drift and timing jitter in the repetition rate difference $\mathrm{\Delta }{f}_{\mathrm{rep}}$ of the free-running SCDCL, enabling us to recover the temporal linearity of the acquired temporal waveform. To trace the timing fluctuation in real time, we used photoconductive heterodyning mixing between a reference CW-THz radiation and two photocarrier THz combs (PC-THz comb1 with $f_{\mathrm{rep}1}$ and PC-THz comb2 with $f_{\mathrm{rep}2}$) seeded by the SCDCL output,³⁹ as shown in the middle part of Fig. 2. Two bowtie-shaped, low-temperature-grown GaAs photoconductive antennas (BT-PCA1 and BT-PCA2) were optically excited by the second-harmonic-generation light of the optical dual combs (not shown in Fig. 2). When the CW-THz radiation ($f_{\mathrm{THz}}=0.1\mathrm{THz}$, $\text{linewidth}<0.6\mathrm{Hz}$, and $\text{average power}=2.5\mathrm{mW}$) from an active frequency multiplier (FM) chain sourced by a microwave frequency synthesizer was also incident on both BT-PCA1 and BT-PCA2, two RF beat signals were generated between the CW-THz and the nearest adjacent PC-THz comb modes. Their frequencies ($f_{\text{beat}1}$ and $f_{\text{beat}2}$) are given by $|f_{\mathrm{THz}}-m{f}_{\mathrm{rep}1}|$ and $|f_{\mathrm{THz}}-m{f}_{\mathrm{rep}2}|$, respectively, assuming that the same $m$-order comb lines are involved. The $f_{\text{beat}2}-f_{\text{beat}1}$ signal that carries the timing fluctuation information ($=m{f}_{\mathrm{rep}2}-m{f}_{\mathrm{rep}1}=m\mathrm{\Delta }{f}_{\mathrm{rep}}$) was obtained by electrical mixing in a double-balanced mixer (M). The resulting signal was further frequency-multiplied by an FM (frequency multiplication factor $N=40$), which serves as the adaptive sampling clock.

2.4.

THz Dual-Comb Spectroscopy

The right part of Fig. 2 shows the experimental setup of THz-DCS. THz comb was radiated from a fiber-coupled, strip-line-shaped LT-InGaAs/InAlAs photoconductive antenna (PCA1, TERA 15-TX-FC, Menlo Systems, $\text{bias voltage}=20\mathrm{V}$, optical power of 20 mW) excited by ${\lambda }_1$-comb pump light and then passed through a low-pressure gas cell ($\text{length}=38\mathrm{cm}$, $\text{diameter}=17\mathrm{mm}$). The THz comb was then detected by another fiber-coupled, dipole-shaped LT-InGaAs/InAlAs photoconductive antenna (PCA2, TERA 15-RX-FC, Menlo Systems, $\text{optical power}=20\mathrm{mW}$) that was pumped by the ${\lambda }_2$-comb light. The electrical output of PCA2 was amplified by a current preamplifier (AMP, $\text{bandwidth}=3.8\mathrm{MHz}$, $\text{gain}=1\times {10}^6\mathrm{V}/\mathrm{A}$).

A portion of the separated ${\lambda }_1$-comb and ${\lambda }_2$-comb lights was fed into a sum-frequency-generation cross-correlator (SFG-X), with a setup that was constructed based on a noncollinear configuration with a piece of $\beta \text{-}{\mathrm{BaB}}_2{\mathrm{O}}_4$ (BBO) crystal. The resulting SFG pulse that occurs every $1/\mathrm{\Delta }{f}_{\mathrm{rep}}$ served as the trigger signal for the data acquisition board.

The temporal waveform of the amplifier output was acquired with a data acquisition board. The mode-resolved THz comb spectrum was obtained by taking Fourier transform of the temporal waveform accumulated in the time domain. A rubidium frequency standard (Stanford Research FS725, $\text{accuracy}=5\times {10}^{-11}$ and $\text{instability}=2\times {10}^{-11}$ at 1 s) is used to provide the common time-base signal for the CW-THz source and the data acquisition board (not shown in Fig. 2).

2.5.

Sample

Acetonitrile (${\mathrm{CH}}_3\mathrm{CN}$) is a symmetric top molecule with a rotational constant $B$ of 9.194 GHz and a centrifugal distortion constant $D_{\mathit{JK}}$ of 17.74 MHz,⁴⁸ and its gas phase has relatively complicated spectral absorption features in the THz domain. Frequencies of its rotational transitions are given as

3.1.

Basic Performance of SCDCL

When the cavity EDF was pumped beyond its mode-locking threshold and the intracavity PC was properly adjusted, dual-comb mode-locking oscillation was achieved at two different center wavelengths with similar peak intensities, as shown in Fig. 3(a). Two spectral peaks at 1532.5 and 1557.7 nm have 3 dB bandwidths of 4.2 and 3.6 nm, respectively, corresponding to the ${\lambda }_1$-comb light and ${\lambda }_2$-comb light. Due to the finely adjusted, low intracavity anomalous dispersion, the ${\lambda }_1$-comb light and ${\lambda }_2$-comb light have slightly different repetition rates, $f_{\mathrm{rep}1}$ ($\sim 48.804486\mathrm{MHz}$) and $f_{\mathrm{rep}\mathrm{2}}$ ($\sim 48.804296\mathrm{MHz}$), and their difference ($\mathrm{\Delta }{f}_{\mathrm{rep}}$) is only $\sim 190\mathrm{Hz}$, as shown in Fig. 3(b). After separate amplification by EDFAs, the dual-comb lights are spectrally broadened and temporally compressed to $\sim 110\mathrm{fs}$ at full-width-at-half-maximum (FWHM) by propagating through the standard single-mode fiber. The achieved pulse duration is sufficient to drive the broadband THz comb spectrum.

Fig. 3

Performance of the dual-comb fiber laser. (a) Output spectrum of the laser. (b) RF spectrum of the dual-comb pulses. (c) Fluctuations of $f_{\mathrm{rep}1}$, $f_{\mathrm{rep}2}$, and $\mathrm{\Delta }{f}_{\mathrm{rep}}$. (d) Measured frequency instability of $f_{\mathrm{rep}1}$ and $\mathrm{\Delta }{f}_{\mathrm{rep}}$.

The temporal drifts of $f_{\mathrm{rep}1}$, $f_{\mathrm{rep}2}$, and $\mathrm{\Delta }{f}_{\mathrm{rep}}$ under free-running conditions were monitored in time because they alter the TMF between the THz comb and RF comb. As shown in Fig. 3(c), the drifts of both repetition rates are $\sim 2.5\mathrm{Hz}$ in 120 s and follow the same trend owing to the identical fluctuations in the completely shared path for both pulses. The resulting fluctuation of $\mathrm{\Delta }{f}_{\mathrm{rep}}$ is merely 5.1 mHz in terms of standard deviation without active stabilization. The frequency instability of $f_{\mathrm{rep}1}$ and $\mathrm{\Delta }{f}_{\mathrm{rep}}$ in the SCDCL was measured for different gate times as indicated by the red solid and hollow circles in Fig. 3(d). For comparison, we also evaluated the frequency instability of $f_{\mathrm{rep}1}$ and $\mathrm{\Delta }{f}_{\mathrm{rep}}$ for two independently stabilized fiber-comb lasers [see the blue solid and hollow circles in Fig. 3(d)] and two independently free-running fiber-comb lasers [see the green solid and hollow circles in Fig. 3(d)]. When comparing the SCDCL with two independently free-running fiber-comb lasers, $f_{\mathrm{rep}1}$ and $\mathrm{\Delta }{f}_{\mathrm{rep}}$ show similar trends. The fluctuation of $f_{\mathrm{rep}1}$ for the SCDCL is nearly one order larger than that of free-running lasers when the gate time is more than 1 s, which might be due to the better outer packaging of the fiber-comb laser for environmental immunity. However, the frequency stability of $\mathrm{\Delta }{f}_{\mathrm{rep}}$ in the SCDCL was significantly better than that in the two free-running fiber-comb lasers over a short time. Next, when comparing the SCDCL with two independently stabilized fiber-comb lasers, the SCDCL shows better short-term stability in $\mathrm{\Delta }{f}_{\mathrm{rep}}$ than the two independently stabilized fiber-comb lasers. By contrast, the two independently stabilized fiber-comb lasers show better long-term stability in $\mathrm{\Delta }{f}_{\mathrm{rep}}$ and $f_{\mathrm{rep}1}$ than the SCDCL. Importantly, even though the SCDCL is inferior to the two independently stabilized fiber-comb lasers in terms of the long-term stability of $\mathrm{\Delta }{f}_{\mathrm{rep}}$ and $f_{\mathrm{rep}1}$ 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.

3.2.

Performance of Adaptive Sampling THz-DCS

To investigate the effectiveness of the proposed adaptive sampling THz-DCS method, 100,000 temporal waveforms of 10-consecutive THz pulse train were acquired and accumulated by the adaptive sampling method. For a comparison, similar temporal waveforms were acquired based on a constant sampling method, which is widely used for data acquisition of the previous DCS with SCDCL.^31,33 In the case of the constant sampling method as shown in the upper part of Fig. 4(a), 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, leading to a low efficiency in the signal accumulation. Obviously, when the number of signal accumulation is largely increased, the constant sampling method with the SCDCL is not suitable to extend the temporal window of the accumulated temporal waveform up to multiple pulse periods. Therefore, previous DCS with SCDCL was demonstrated in a short acquisition time of the temporal waveform (typically, $<1\mathrm{s}$).^31,33 When the adaptive sampling method was used for THz-DCS with SCDCL, each THz pulse was clearly observed in the accumulated temporal waveform, as shown in the lower part of Fig. 4(a). Moreover, each THz pulse was the complete duplicate of each other, coincident with the behavior of THz comb in the time domain. As shown in the inset of Fig. 4(a), the asymmetric bipolar monocycle pulse shape is due to the dipole radiation from the photoconductive antenna, with its temporal waveform given by a time derivative of transient photocurrent induced by the pump light. These results imply that the adaptive sampling method has the capability to significantly suppress the instability of TMF over a long data acquisition time.

Fig. 4

(a) Comparison of the temporal waveforms averaged 100,000 times obtained using different sampling clocks. Inset: a zoomed-in plot of the main THz pulse. (b) Comb-mode-resolved THz spectrum through air at room pressure. Inset: a zoomed-in plot around 0.5672 THz.

Next, the mode-resolved THz comb spectrum was obtained by calculating the FT of the temporal waveform of the adaptive sampling RF pulse train [see the lower part of Fig. 4(a)], as shown in Fig. 4(b). A $\sim 50\text{-}\mathrm{dB}$ power dynamic range was achieved in the 0.1- to 1.5-THz frequency band. An expanded view of the spectral region around 0.567 THz is shown in the inset of Fig. 4(b), indicating multiple discrete lines with a constant frequency spacing of 48.8 MHz exactly equal to $f_{\mathrm{rep}1}$. The spectral width of each comb line was 4.88 MHz, which is exactly equal to the theoretical FT spectral resolution given by an inverse of the time window size ($f_{\mathrm{rep}1}/10$ in this case). Because we later investigate the pressure-broadening characteristics of the sample gas within the range of absorption linewidth from a few tens MHz to 100 MHz, the comb-mode linewidth was set to be $f_{\mathrm{rep}1}/10$ (= 4.88 MHz) to acquire the spectrum with a minimum required resolution and data size. In the previous study, the constant sampling THz-DCS with dual stabilized lasers could be demonstrated with the time window size of $100/f_{\mathrm{rep}1}$,^49,50 whereas the adaptive sampling THz-DCS with dual free-running lasers was more powerful than the constant sampling THz-DCS with dual stabilized lasers.³⁹ We consider that there is a space to further decrease the comb-mode linewidth in the present system. In this way, the combination of the adaptive sampling method with the SCDCL has the potential to reach spectral resolution of MHz level or below without the influence of the residual timing jitter in SCDCL. In Fig. 4(b), two spectral dips appeared at 0.557 and 0.752 THz with an FWHM linewidth of 7 GHz in the mode-resolved THz comb spectrum, which was caused by atmospheric water vapor existing in the THz optical path outside the gas cell. This is a simple demonstration of pressure-broadening gas spectroscopy in the THz region.

3.3.

Near-Doppler-Limited Spectroscopy of Low-Pressure Acetonitrile/Air Gas

To demonstrate the high-resolution spectroscopic capability of the proposed system, THz spectroscopy of mixed gas of acetonitrile (${\mathrm{CH}}_3\mathrm{CN}$) and air was performed at low pressures. ${\mathrm{CH}}_3\mathrm{CN}$ has a rotational constant $B$ of 9.194 GHz and a centrifugal distortion constant $D_{JK}$ of 17.74 MHz⁴⁸ and is a good candidate for the demonstration of Doppler-limited gas spectroscopy in the THz region because of the closely spaced rotational transitions on the order of tens of MHz in each manifold. The gas cell was filled with a mixture of ${\mathrm{CH}}_3\mathrm{CN}$ and air with a total pressure of 360 Pa. The absorption spectrum of this mixed gas was obtained by normalizing the mode-resolved THz comb spectrum measured in the ${\mathrm{CH}}_3\mathrm{CN}/\text{air}$-filled gas cell with that measured in the vacuum gas cell. Figure 5(a) shows the broadband spectrum of absorbance coefficient within the frequency range of 0.2 to 0.7 THz. 29 manifolds were observed with a frequency separation of 18.388 GHz equal to $2B$ and were correctly assigned to $J=10$ to 38 by comparison with the Jet Propulsion Laboratory (JPL) spectral database.⁵¹ Figure 5(b) shows a zoomed-in plot of Fig. 5(a) within the frequency range of 0.31 to 0.37 THz. In addition to $J=16$ to 19 at the ground state, the manifolds of the rotational transitions at the vibrationally excited states⁴⁹ were clearly observed as marked by the red asterisks. These spectral features could be obtained by the enhanced spectral resolving power and sensitivity in the low-pressure gas condition, which constitutes a significant enhancement in the spectroscopic performance compared with the previous demonstration.³⁴

Fig. 5

Comb-mode-resolved THz spectroscopy of a mixture gas sample of ${\mathrm{CH}}_3\mathrm{CN}$ and air with a total pressure of 360 Pa. Absorption spectra of ${\mathrm{CH}}_3\mathrm{CN}$ within the frequency range of (a) 0.2 to 0.72 THz and (b) 0.31 to 0.37 THz. The red stars indicate the manifolds of the rotational transitions for the vibrationally excited states. (c) Comparison of the absorption spectra between the database fitting and the experimental data and their residual and (d) the corresponding zoomed-in plot around 0.3310 and 0.3677 THz. (e) Absorption spectra around 0.3310 THz and (f) 0.3677 THz.

Each manifold is composed of a number of closely spaced rotational transitions, assigned by $K$, as tabulated in the JPL spectral database.⁵¹ Multipeak fitting analysis was performed to determine a ${\mathrm{CH}}_3\mathrm{CN}$ partial pressure and the corresponding linewidth of the rotational transitions in which a Lorentzian lineshape was used for each rotational line and a global linewidth parameter was applied to all of the lines. The line positions and intensities were fixed parameters, while the acetonitrile partial pressure and linewidth were left as free parameters. An example is presented in the red and blue plots of Fig. 5(c), yielding a ${\mathrm{CH}}_3\mathrm{CN}$ partial pressure of 127 ($\pm 1$) Pa and an FWHM linewidth of 77 ($\pm 2$) MHz. The database fitting curve (red line) was in good agreement with the experimental data (blue line) for all peaks in the frequency range of 0.2 to 0.4 THz with a slight residual (black line). The reason for the periodic modulation of the residual is that the fitting model does not include the rotational transitions at the vibrationally excited states.⁵¹ This is better illustrated in an expanded view in Fig. 5(d) around 0.3310 and 0.3677 THz, which shows that the large residual is mostly around the vibrationally excited states. Figures 5(e) and 5(f) show zoomed-in spectra for $J=17$ at 0.3310 THz and $J=19$ at 0.3677 THz, in which the blue plots show the experimental data. The frequency spacing of the experimental data points, corresponding to the spectral sampling interval, was 48.8 MHz, equal to $f_{\mathrm{rep}1}$. The curve fitting results based on multipeak fitting analysis are shown by the red line while literature values of JPL spectral database are shown by the green line in Figs. 5(e) and 5(f). Most of the absorption lines have frequency separation between each other equal to or less than the spectral sampling interval (see green lines and blue plots). Such undersampling acquisition of fine spectral features leads to disagreement between experimental plots and absorption line positions. However, it is important to note that even such an undersampling spectral acquisition is effective for near-Doppler-limited gas spectroscopy with the help of curve fitting analysis, which is discussed later.

The validity of the adaptive sampling THz-DCS scheme was more precisely evaluated by measuring the pressure broadening characteristics of ${\mathrm{CH}}_3\mathrm{CN}$ rotation transition in the ${\mathrm{CH}}_3\mathrm{CN}/\text{air}$-mixed gas. Similar to the procedure for 360 Pa as shown in Fig. 5, a series of absorption spectra of rotational transitions in $J=17$ at 0.331 THz was measured when reducing the total pressure of the ${\mathrm{CH}}_3\mathrm{CN}/\text{air}$-mixed gas from 430, 330, 280, 256, 149, to 115 Pa with an uncertainty of $\pm 2\mathrm{Pa}$, as shown by blue plots in Figs. 6(a)–6(f), respectively. To perform a quantitative analysis, we again performed multipeak fitting analysis based on a Lorentzian lineshape under the assumption that all $K$ lines have the same linewidth. The total pressure was measured by a pressure gauge to help achieve a sample with the desired approximate concentration and was not used in the fitting procedure. As shown by the red lines in Fig. 6(a)–6(f), the curve fitting spectra match the experimental spectra very well. Both the experimental data and the curve fitting results clearly indicate the pressure-dependent change in the shape of the absorption spectrum. Table 1 shows the result of quantitative analysis for the ${\mathrm{CH}}_3\mathrm{CN}$ partial pressure and the corresponding linewidth of rotational transition with respect to different total pressures of this mixed gas sample. The uncertainties provided in Table 1 are the confidence intervals calculated as part of the fitting procedure. The partial pressure and the linewidth were determined with good uncertainty under all total pressures, indicating the validity of the adaptive sampling THz-DCS and the following curve fitting analysis. In particular, the linewidth was determined to be 25 MHz at a total pressure of 115 Pa. This quantitative analysis indicates the capability of the proposed system to realize THz spectroscopy of MHz-order absorption features with the reduced system complexity.

Fig. 6

Mode-resolved absorption characterization of ${\mathrm{CH}}_3\mathrm{CN}$ around 0.331 THz at (a) 430 Pa, (b) 330 Pa, (c) 280 Pa, (d) 256 Pa, (e) 149 Pa, and (f) 115 Pa.

Table 1

Quantitative analysis of CH3CN/air-mixed gas.