## Abstract

Dual-comb spectroscopy (DCS) is useful for gas spectroscopy due to the high potential of optical frequency comb (OFC). However, fast Fourier transform (FFT) calculation of a huge amount of temporal data spends significantly longer time than the acquisition time of an interferogram. In this article, we demonstrate frequency-domain DCS by a combination of DCS with lock-in detection, namely LID-DCS. LID-DCS directly extracts an arbitrary OFC mode from a vast number of OFC modes without the need for FFT calculation. Usefulness of LID-DCS is demonstrated in the rapid monitoring of transient signal change and spectroscopy of hydrogen cyanide gas.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Recent advances in optical frequency comb (OFC) [1–3] enable us to benefit from a group of a vast number of phase-locked narrow-linewidth continuous-wave (CW) lights with a constant frequency spacing *f _{rep}* (typically, 50 to 100 MHz) over a broad spectral range. The inherent mode-locking nature and active laser control make it possible to use the OFC as an optical frequency ruler traceable to a microwave or radio-frequency (RF) frequency standard. To fully utilize both its narrow spectral linewidth and broadband spectral coverage for broadband spectroscopy, it is essential to acquire the mode-resolved OFC spectrum. Fourier transform spectroscopy [4] can be used for this purpose by using a long mechanical scanning of a reference arm (typically, sub-meter to a few meters length) [5]; however, such mechanical scanning hampers rapid data acquisition. Virtually-imaged-phased-array (VIPA) spectroscopy [6,7,8] is a promising method for rapid acquisition of mode-resolved OFC spectrum. By spatially developing OFC spectrum with a combination of VIPA and a diffraction grating [9,10], the mode-resolved OFC spectrum can be acquired all at once as a two-dimensional (2D) spectrograph by a camera without the need for mechanical scanning. However, VIPA spectroscopy is limited for OFCs with

*f*larger than a few GHz due to its spectral resolving power.

_{rep}Recently, dual-comb spectroscopy (DCS) [11–14] has appeared as a technique for acquiring the mode-resolved OFC spectrum via its replica in RF regions, namely RF combs, by using dual OFCs with slightly mismatched frequency spacing (= *f _{rep1}* and

*f*). Due to its rapid, precise, and accurate acquisition of the spectrum, DCS has found many applications in optical frequency metrology; examples include gas spectroscopy [15], gas thermometry [16], solid spectroscopy [17], spectroscopic ellipsometry [18], hyper-spectral imaging [19], and coherent Raman imaging [20]. Among them, gas spectroscopy is one interesting application because DCS-based gas analysis has several advantages over conventional gas analysis including gas chromatography: real-time data acquisition, simultaneous analysis of multiple gasses, and no need for sample preparation. For example, the broad-band DCS covering from 158 to 300 THz, corresponding to 1.0 to 1.9 µm, has been effectively applied for simultaneous analysis of acetylene, methane, and water vapor [21]. Also, DCS has been used for monitoring of atmospheric gas [22,23] and gas turbine exhaust [24]. Furthermore, such DCS has been extended to the mid-infrared region [25] and even the terahertz (THz) region [26,27].

_{rep2}In usual DCS, after a temporal waveform of a single interferogram or consecutive interferograms was acquired in time domain, mode-resolved OFC spectrum is obtained by fast Fourier transform (FFT) calculation of the acquired temporal waveform. However, FFT calculation consumes time due to a huge amount of temporal data for the mode-resolved OFC spectra. Due to this FFT calculation, the actual measurement rate significantly decreases even though the acquisition rate of temporal waveform can be increased up to a difference of *f _{rep}* between dual OFCs (= Δ

*f*=

_{rep}*f*-

_{rep2}*f*); it will hamper monitoring of transient signal change. Low duty factor in the ultra-discrete tooth-like spectrum of OFC is another practical limitation of DCS. For example, when the mode-resolved OFC spectrum is measured by a spectral resolution of

_{rep1}*f*/100, the mode linewidth of the measured OFC spectrum is decreased down to

_{rep}*f*/100. However, spectral data points except mode peaks fall in gap regions between OFC modes; only 1% of the spectral data points gives the information on the signal of mode peaks, and the remaining 99% of them gives no information due to noise region without OFC modes. Furthermore, in the case of gas spectroscopy, the absorption lines of the gas molecule are localized at specific spectral region [21]; it is not always necessary to acquire the whole spectral range of OFC, and only the spectral information at the absorption lines is sufficient for simple analysis. Therefore, the spectral analysis with minimum required spectral information is greatly desired for efficient and fast DCS.

_{rep}One possible method of overcoming these limitations is the frequency-domain acquisition in DCS by use of lock-in detection (LID). Since the RF comb has a highly stable, discrete spectrum in frequency domain, one can extract only a specific RF comb mode by selection of a LID reference frequency; simultaneously, other unnecessary RF comb modes and gap data points can be rejected. This leads to the great reduction of the data size. More importantly, since the LID is based on the frequency-domain measurement, it needs no FFT calculation to obtain the spectral information, enabling fast processing in DCS. Its acquisition time is dependent on a LID time constant independently of Δ*f _{rep}*. Combination of LID with DCS, namely LID-DCS, has been successfully demonstrated in DCS-based distance measurement, in which the optical phase of the specific RF comb mode was measured by LID [28,29]. However, there are no attempts to apply LID-DCS for gas spectroscopy requiring the optical amplitude measurement of specific RF comb mode.

In this paper, we evaluate the basic performance of LID-DCS by comparing with usual DCS from viewpoint of net measurement time and signal-to-nose ratio (SNR). We further demonstrate use of LID-DCS for spectroscopy of hydrogen cyanide gas.

## 2. Experimental setup

Figure 1(a) shows a principle of operation in LID-DCS. In the frequency-domain description of DCS, two OFCs with a slightly different repetition frequency (signal OFC, mode spacing = *f _{rep1}*; local OFC, mode spacing =

*f*=

_{rep2}*f*+ Δ

_{rep1}*f*) generates a secondary frequency comb in RF region, namely RF comb (mode spacing = Δ

_{rep}*f*), via the multi-frequency heterodyning interference between them. In usual DCS, the RF comb is acquired as an RF interferogram in time domain and then is obtained as the mode-resolved spectrum by FFT calculation of the RF interferogram. In LID-DCS, a lock-in amplifier (LIA) enables us to acquire both amplitude and phase of a frequency signal synchronized with a LID reference-frequency signal. Therefore, one can select an arbitrary mode from the mode-resolved RF comb spectrum without the need for FFT calculation by tuning the LID reference frequency to coincide with a target RF-comb-mode frequency.

_{rep}Figure 1(b) shows an experimental setup of LID-DCS, which is based on the free-space optical setup except light sources. The reason for use of the free-space optical setup is in the flexibility of the setup construction. Two mode-locked erbium-doped fiber combs (OCLS-HSC-D100-TKSM, Neoark Co., Japan; center wavelength = 1560 nm, spectral bandwidth = 50 nm; signal OFC, carrier-envelope-offset frequency = *f _{ceo1}* = 10.5 MHz,

*f*= 100.000188 MHz ; local OFC,

_{rep1}*f*= 10.5 MHz,

_{ceo2}*f*= 99.999976 MHz; Δ

_{rep2}*f*=

_{rep}*f*–

_{rep2}*f*= 212 Hz) were used for light sources in LID-DCS. We used a rubidium frequency standard (Rb-FS, Stanford Research Systems, Inc., FS725; frequency = 10 MHz, accuracy = 5×10

_{rep1}^{−11}; instability = 2×10

^{−11}at 1 s) for a frequency reference in these dual OFCs. The local OFC, equipped with an intra-cavity electro-optical modulator for laser control, was tightly and coherently locked to the signal OFC with a frequency offset using a narrow-linewidth continuous-wave (CW) laser (CWL, Redfern Integrated Optics, Inc., Santa Clara, California, USA, PLANEX; center wavelength = 1550 nm; FWHM < 2.0 kHz) for an intermediate laser [17,18]. Polarization of the signal OFC light and the local OFC light was aligned at the vertical direction by use pairs of a quarter waveplate (λ/4) and a half waveplate (λ/2). After spatially overlapping of them for optical interference by a beam splitter (BS), the dual OFC lights passed through a band-pass filter (BPF, pass band = 1550 ± 10 nm) for bandwidth reduction and another λ/2 for polarization rotation by 45°. Then, the dual OFC lights were split for a signal light and a reference light by a polarization beam splitter (PBS). A sample was placed into the optical path of the signal light. The RF combs of the signal light and the reference light, namely signal RF comb and reference one, were respectively detected by a pair of photodetectors (PDs, Thorlabs, PDA10CF-EC; wavelength = 800–1700nm; bandwidth < 150 MHz). We extract an arbitrary comb mode from the signal RF comb by a radio-frequency LIA (RF-LIA1, Stanford Research Systems, SR844; frequency range = 25kHz ∼ 200 MHz, time constant = no or 100 µs to 30 ks). We further extracted the same-order comb mode of the reference RF comb for a reference to compensate the common-mode fluctuation in amplitude, arising from dual OFC and/or environmental disturbance, by use of another RF-LIA (RF-LIA2, Stanford Research Systems, SR844) in real-time. Then, we calculated amplitude ratio between them as a normalized amplitude spectrum. LID reference-frequency signals for RF-LIA1 and RF-LIA2 were generated from a RF waveform generator (RF-WG, Keysight Technologies, 33510B, frequency range < 20 MHz). Since dual OFCs and the RF-WG share the same Rb-FS for the common frequency reference, the LID reference-frequency signal can be synchronized with the arbitrary RF comb mode.

For comparison with LID-DCS, we performed usual DCS using the same optical setup except the reference light in Fig. 1(b). Detail of its experimental setup is given elsewhere [18,19]. The detected electrical signal was acquired using a digitizer (National Instruments Corp., NI PXIe-5122; resolution = 14 bit). The sampling clock signal was synchronized with *f _{rep2}*. We made an FFT calculation program to obtain full spectrum of amplitude and phase in OFC with LabView2017 (National Instruments Corp., 64 bit) and performed it in a computer (National Instruments Corp., PXIe-8840, Intel Core i7, Processor base frequency = 2.60 GHz, Cache = 6 MB smart cache, RAM = 8GB).

## 3. Results

#### 3.1 Performance evaluation of LID-DCS and DCS

Before performing the performance evaluation, we defined the net measurement time and optical spectral resolution in the following experiments. The net measurement time of LID-DCS was defined as an acquisition time of accumulated signal, and was given in the unit of “ms/spectral_point” because LID-DCS gives a single point in the spectrum; that of DCS was defined as a sum of acquisition time of interferogram and its FFT calculation time, and was given in the unit of “ms/spectrum” because DCS gives the whole spectrum. On the other hand, the optical spectral resolution of LID-DCS was calculated by the product of LID spectral resolution { = 1/[2π*(LID time constant)]} and the frequency-scale conversion factor (= *f _{rep1}*/Δ

*f*); that of DCS was given by an inverse of time window size in the interferogram. We first measured a fluctuation of spectral amplitude at a certain optical frequency in LID-DCS and DCS when the net measurement time of a single spectral point in LID-DCS ( = 42 ms) was set to be equal to the acquisition time of 9 consecutive interferogram in DCS ( = 42 ms) while maintaining the same optical spectral resolution in them. Figures 2(a) and 2(b) compare a fluctuation of spectral amplitude at 193.554964 THz between LID-DCS (optical spectral resolution = 10.8 MHz, LID time constant = 3 ms, number of signal accumulation = 14, net measurement time = 42 ms/spectral_point) and DCS (optical spectral resolution = 11.1 MHz, number of consecutive interferograms = 9, time window size = 9/

_{rep}*f*= 90 ns, sampling time interval = 1/

_{rep1}*f*- 1/

_{rep2}*f*= 21.2 fs, number of temporal data = 4,716,980, acquisition time of interferogram = 9/Δ

_{rep1}*f*= 42 ms/data, number of interferogram accumulation = 1, FFT calculation time = 3.08 s/data, net measurement time = 3.12 s/spectrum) with respect to the number of measured data. Similar fluctuation of spectral amplitude was observed in both. When we defined SNR as a ratio of the mean to the standard deviation in spectral amplitude, SNR in them is significantly similar to each other: 30.6 for LID-DCS and 31.9 for DCS. Therefore, use of LID in DCS does not contribute to negative effect in SNR. Although the total number of data plots was equal to each other in Figs. 2(a) and 2(b), there is a large difference of the net measurement time between them if the total number of measured data is converted into the total net measurement time: 2.8 s for LID-DCS and 206 s for DCS. The net measurement time of LID-DCS is determined by the LID time constant and the number of signal accumulation; in DCS, most of the net measurement time is occupied by the FFT calculation time rather than the acquisition time of interferogram.

_{rep}Considering the results in Figs. 2(a) and 2(b), the LID-DCS can reduce the net measurement time while maintaining the SNR and the spectral resolution similar to DCS. Figure 2(c) compares SNR of spectral amplitude at 193.559964 THz with respect to the net measurement time between LID-DCS (see red circle) and DCS (see blue circle). We here adjusted the optical spectral resolution of DCS while maintaining the constant optical resolution of LID-DCS to match the net measurement time of LID-DCS and that of DCS (see red and blue triangles). The linear relation was confirmed between SNR and net measurement time in both methods. Starting point of the slope in LID-DCS (see red line) was significantly higher SNR and shorter net measurement time than that in DCS because of no FFT calculation. The slope coefficient was determined to be 0.37 for LID-DCS and 0.57 for DCS, respectively. In LID-DCS, while the residual timing jitter between dual OFCs somewhat fluctuates frequency of RF comb modes, the LID reference frequency is always fixed at a constant value. Lock-in detection of such frequency-fluctuated signal at a fixed frequency makes the LID-DCS sensitive to the timing jitter and hence limits the slope coefficient. In the case of DCS, we applied the phase compensation for the RF interferogram, in which the phase of the RF interferogram was preset to null for every interferogram by the self-triggering of the RF interferogram in the acquisition of temporal waveform, making DCS robust to the residual timing jitter. Difference of slope coefficient between LID-DCS and DCS is mainly due to these effects.

#### 3.2 Temporal response of LID-DCS and DCS

We next evaluated the temporal response of LID-DCS and DCS when the intensity of the measured signal light was transiently fluctuated. To this end, we chopped the optical beam with a glass plate (BK7, thickness = 1 mm), leading to a transient change in the optical intensity. Figure 3 shows the temporal response of the spectral amplitude at 193.554964 THz for (a) LID-DCS (optical spectral resolution = 1.1 MHz, LID time constant = 30 ms, number of signal accumulation = 1, net acquisition time = 30 ms/spectral_point) and (b) DCS (optical spectral resolution = 100 MHz, number of consecutive interferograms = 1, time window size = 1/*f _{rep1}* = 10 ns, sampling time interval = 21.2 fs, number of temporal data = 471,698, number of signal accumulation = 6, acquisition time of interferogram = 1/Δ

*f*= 4.7 ms/data, FFT calculation time = 0.373 s/data, net acquisition time = 0.401 s/spectrum) with respect to the elapsed time. In this experiment, optical resolution of DCS was 100-times worse than that of LID-DCS to maintain the real-time capability of FFT calculation in DCS by reducing the data size of temporal waveform of the RF interferogram. Nevertheless, the DCS has less discrete sampling points and could not respond to such transient fluctuation sensitively; in contrast, the LID-DCS well responds to the fluctuation by sufficient number of sampling points. Therefore, the LID-DCS will be more powerful than DCS for monitoring of transient signal change, such as gas concentration measurement under air turbulence.

_{rep}#### 3.3 Spectroscopy of cyanide gas (H^{13}C^{14}N)

Finally, we demonstrated spectroscopy of cyanide gas (H^{13}C^{14}N) by LID-DCS and DCS. H^{13}C^{14}N gas, contained in a gas cell (cell length = 15 cm , gas pressure = 25 Torr), was placed into an optical path of the signal light. We selected P(9) absorption line of H^{13}C^{14}N gas (center frequency = 193.544907 THz, expected pressure-broadening linewidth = 2.25 GHz) for measurement. Red lines in Fig. 4(a) show a mode-resolved amplitude spectrum of OFC within the frequency range from 193.535 THz to 195.555 THz, measured by LID-DCS (optical spectral resolution = 3.24 MHz, LID time constant = 10 ms, number of signal accumulation = 4700, net acquisition time = 47 s/spectral_point). We here extracted 21 RF comb modes at a frequency interval of 1 GHz across the P(9) absorption line by scanning the LID reference-frequency at a frequency interval of 2.12 kHz, leading to the discretely spectral sampling (optical spectral resolution = 3.24 MHz, optical spectral sampling interval = 1 GHz). Since the switching speed of LID frequency is sufficiently fast compared with the LID time constant, the acquisition time for the spectrum is limited by the LID time constant. The total acquisition time was 987 s. Although the net acquisition time was set to be 47 s/spectral_point in this demonstration for high SNR spectrum, there is enough space for further reduction of the acquisition time by reducing the number of signal accumulation. For comparison, we measured the same absorption line by DCS (optical spectral resolution = 10.0 MHz, number of consecutive interferograms = 10, time window size = 100 ns, sampling time interval = 21.2 fs, number of temporal data = 4,716,980, number of signal accumulation = 1000, acquisition time of interferogram = 47 ms/data, FFT calculation time = 3.5 s/data, net acquisition time = 50.5 s/spectrum), as shown in concentrated blue lines of Fig. 4(a). We performed a single FFT calculation of the interferogram after the time-domain averaging of interferograms to reduce the number of FFT calculations. Envelopes of amplitude spectrum measured by LID-DCS and DCS were almost overlapped to each other as shown in the difference plot of them. The total acquisition time in LID-DCS was longer that in DCS when monitoring the spectrum itself; however, when monitoring only the single spectral point on the absorption spectral dip for simple gas analysis, the total acquisition time in LID-DCS is 10-times shorter than that in DCS. To determine the center frequency and the linewidth of the measured P(9) absorption line measured by LID-DCS, we performed the curve-fitting analysis using the Voigt function. Red plots and red line in Fig. 4(b) show the experimental data of the amplitude spectrum and the corresponding fitting result, respectively. For comparison, literature value of this absorption line position [30] is indicated as a green line in Fig. 4(b). The center frequency and the linewidth were determined to be 193.545100 THz and 2.26 GHz. The deviation of them from the expected values might be due to the instability or calibration error of the system used in the DCS, such as frequency counters, sinusoidal function generators, feedback controllers, and so on. However, these results indicate the correct frequency scale and high applicability for gas spectroscopy of LID-DCS.

## 4. Discussion

One may have doubts about the advantage of LID-DCS compared with usual CW spectroscopy because only a single OFC mode was extracted in this article. CW spectroscopy equipped with tunable CW laser diode [30] benefits from rapid wavelength scanning; however, the spectral resolution and accuracy are respectively limited by optical frequency fluctuation of an unstabilized tunable laser and performance of a wavelength meter. On the other hand, LID-DCS can rapidly select an arbitrary OFC mode within the broadband OFC spectrum, and spectral resolution and accuracy of the selected OFC mode are well secured by phase-locking control of OFC to a frequency standard. Also, one may consider the similarity between LID-DCS and optical vector network analyzer (OVNA) [31], which detects spectral feature of a sample. The spectral resolution and accuracy of OVNA are limited by factors similar to CW spectroscopy. In this way, LID-DCS has the advantage of spectral resolution and accuracy over CW spectroscopy and OVNA although its acquisition speed of spectrum might be sometimes less than that in CW spectroscopy and OVNA.

Simultaneous acquisition of optical amplitude and phase will be another advantage although either the amplitude or phase signal was acquired in this article or the previous article [28,29]. Furthermore, a low frequency electrical noise can be suppressed in the LID-DCS owing to an inherent heterodyne detection mechanism of DCS, while the CW spectroscopy requires an additional intensity or frequency modulator to suppress the low frequency electrical noise. Multi-channel detection is an interesting option for the further extension of LID-DCS although a single-channel LID-DCS was used for gas spectroscopy in this article. The state-of-art multi-channel lock-in detection [29,32,33] will bring interesting options to LID-DCS. For example, multi-channel LID-DCS enables simultaneous monitoring of different gas samples without the time delay of FFT calculation in DCS. Also, simultaneous monitoring of different absorption lines in the same gas sample makes it possible to determine the gas temperature together with gas concentration [34]. These will be a powerful tool for analysis of combustion process in industry. Work is in progress to investigate multi-channel LID-DCS.

One may consider a possibility to further reduce the FFT calculation time in DCS by devising the data processing. FFT calculation time depends on the FFT calculation algorithm and PC speed. Although full spectrum of amplitude and phase in OFC was obtained by usual FFT calculation algorithm in this article, the specific FFT calculation algorithm suitable to obtain narrow-band spectral data acquisition may enables the further reduction of FFT calculation time in DCS. In this case, one have to compare the speed of electronics in LID-DCS and the speed of PC in DCS. This comparison will be our future work.

## 5. Conclusion

We demonstrated use of LID in DCS. This combination, LID-DCS, has potential to largely reduce the time spent for FFT calculation of a huge amount of temporal data because it depends on the frequency-domain measurement without the need for FFT calculation. The advantage of LID-DCS over DCS was highlighted in the rapid data acquisition under the condition of monochromatic spectral analysis due to no FFT calculation. Although the large amount of spectral data points available in the usual DCS are sometimes useful because of such as the application of spectral fitting to improve the accuracy of gas spectroscopy, LID-DCS benefits from the faster temporal response than usual DCS while maintaining the high resolution and accuracy comparable to usual DCS. Such characteristics of LID-DCS will be a powerful tool for monitoring of transient signal change, such as gas concentration measurement under air turbulence. Furthermore, options for multi-channel detection or imaging will expand the application fields of LID-DCS.

## Funding

Exploratory Research for Advanced Technology (ERATO) (JPMJER1304); Institute of Post-LED Photonics; Grant-in-Aid for Exploratory Research, Japan Society for the Promotion of Science (JSPS) (19H00987).

## Acknowledgments

The authors wish to acknowledge Ms. Shoko Lewis of Tokushima Univ., Japan, for her help in preparation of the manuscript.

## References

**1. **T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Accurate measurement of large optical frequency differences with a mode-locked laser,” Opt. Lett. **24**(13), 881–883 (1999). [CrossRef]

**2. **M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, T. Udem, M. Weitz, T. W. Hänsch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon, and A. Clairon, “Measurement of the hydrogen 1S-2S transition frequency by phase coherent comparison with a microwave cesium fountain clock,” Phys. Rev. Lett. **84**(24), 5496–5499 (2000). [CrossRef]

**3. **T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature **416**(6877), 233–237 (2002). [CrossRef]

**4. **P. R. Griffiths, *Chemical Infrared Fourier Transform Spectroscopy* (Wiley, 1975).

**5. **P. Maslowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A **93**(2), 021802 (2016). [CrossRef]

**6. **S. A. Diddams, L. Holloberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature **445**(7128), 627–630 (2007). [CrossRef]

**7. **M. J. Thorpe, D. Balslev-Clausen, M. S. Kirchner, and J. Ye, “Cavity-enhanced optical frequency comb spectroscopy: application to human breath analysis,” Opt. Express **16**(4), 2387–2397 (2008). [CrossRef]

**8. **A. J. Fleisher, B. J. Bjork, T. Q. Bui, K. C. Cossel, M. Okumura, and J. Ye, “Mid-infrared time-resolved frequency comb spectroscopy of transient free radicals,” J. Phys. Chem. Lett. **5**(13), 2241–2246 (2014). [CrossRef]

**9. **M. Shirasaki, “Large angular dispersion by a virtually imaged phased array and its application to a wavelength demultiplexer,” Opt. Lett. **21**(5), 366–368 (1996). [CrossRef]

**10. **S. Xiao and A. M. Weiner, “2-D wavelength demultiplexer with potential for ≥1000 channels in the C-band,” Opt. Express **12**(13), 2895–2902 (2004). [CrossRef]

**11. **S. Schiller, “Spectrometry with frequency combs,” Opt. Lett. **27**(9), 766–768 (2002). [CrossRef]

**12. **F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. **29**(13), 1542–1544 (2004). [CrossRef]

**13. **T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. **88**(24), 241104 (2006). [CrossRef]

**14. **I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica **3**(4), 414–426 (2016). [CrossRef]

**15. **A. M. Zolot, F. Giorgetta, E. Baumann, W. Swann, I. Coddington, and N. Newbury, “Broad-band frequency references in the near-infrared: accurate dual comb spectroscopy of methane and acetylene,” J. Quant. Spectrosc. Radiat. Transfer **118**, 26–39 (2013). [CrossRef]

**16. **Y. Shimizu, S. Okubo, A. Onae, K. M. T. Yamada, and H. Inaba, “Molecular gas thermometry on acetylene using dual-comb spectroscopy: analysis of rotational energy distribution,” Appl. Phys. B: Lasers Opt. **124**(4), 71 (2018). [CrossRef]

**17. **A. Asahara, A. Nishiyama, S. Yoshida, K. Kondo, Y. Nakajima, and K. Minoshima, “Dual-comb spectroscopy for rapid characterization of complex optical properties of solids,” Opt. Lett. **41**(21), 4971–4974 (2016). [CrossRef]

**18. **T. Minamikawa, Y. Hsieh, K. Shibuya, E. Hase, Y. Kaneoka, S. Okubo, H. Inaba, Y. Mizutani, H. Yamamoto, T. Iwata, and T. Yasui, “Dual-comb spectroscopic ellipsometry,” Nat. Commun. **8**(1), 610 (2017). [CrossRef]

**19. **K. Shibuya, T. Minamikawa, Y. Mizutani, H. Yamamoto, K. Minoshima, T. Yasui, and T. Iwata, “Scan-less hyperspectral dual-comb single-pixel-imaging in both amplitude and phase,” Opt. Express **25**(18), 21947–21957 (2017). [CrossRef]

**20. **T. Ideguchi, S. Holzner, B. Bernhardt, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Coherent Raman spectro-imaging with laser frequency combs,” Nature **502**(7471), 355–358 (2013). [CrossRef]

**21. **S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F. L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9µm,” Appl. Phys. Express **8**(8), 082402 (2015). [CrossRef]

**22. **G. B. Rieker, F. R. Giorgetta, W. C. Swann, J. Kofler, A. M. Zolot, L. C. Sinclair, E. Baumann, C. Cromer, G. Petron, C. Sweeney, P. P. Tans, I. Coddington, and N. R. Newbury, “Frequency-comb-based remote sensing of green gases over kilometer air paths,” Optica **1**(5), 290–298 (2014). [CrossRef]

**23. **K. C. Cossel, E. M. Waxman, F. R. Giorgetta, M. Cermak, I. R. Coddington, D. Hesselius, S. Ruben, W. C. Swann, G.-W. Truong, G. B. Rieker, and N. R. Newbury, “Open-path dual-comb spectroscopy to an airborne retroreflector,” Optica **4**(7), 724–728 (2017). [CrossRef]

**24. **P. J. Schroeder, R. J. Wright, S. Coburn, B. Sodergren, K. C. Cossel, S. Droste, G. W. Truong, E. Baumann, F. R. Giorgetta, I. Coddington, N. R. Newbury, and G. B. Rieker, “Dual frequency comb laser absorption spectroscopy in a 16 MW gas turbine exhaust,” Proc. Combust. Inst. **36**(3), 4565–4573 (2017). [CrossRef]

**25. **B. Berinhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picque, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 µm Cr^{2+}:ZnSe femtosecond lasers,” Appl. Phys. B: Lasers Opt. **100**(1), 3–8 (2010). [CrossRef]

**26. **T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, “Terahertz frequency comb by multi-frequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. **88**(24), 241104 (2006). [CrossRef]

**27. **Y. D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. **4**(1), 3816 (2015). [CrossRef]

**28. **S. Yokoyama, T. Yokoyama, Y. Hagihara, T. Araki, and T. Yasui, “A distance meter using a terahertz intermode beat in an optical frequency comb,” Opt. Express **17**(20), 17324–17333 (2009). [CrossRef]

**29. **R. Yang, F. Pollinge, K. Meiners-Hagen, M. Krystek, J. Tan, and H. Bosse, “Absolute distance measurement by dual-comb interferometry with multi-channel digital lock-in phase detection,” Meas. Sci. Technol. **26**(8), 084001 (2015). [CrossRef]

**30. **W. C. Swann and S. L. Gilbert, “Line centers, pressure shift, and pressure broadening of 1530-1560 nm hydrogen cyanide wavelength calibration lines,” J. Opt. Soc. Am. B **22**(8), 1749–1756 (2005). [CrossRef]

**31. **Z. Tang, S. Pan, and J. Yao, “A high resolution optical vector network analyzer based on a wideband and wavelength-tunable optical single-sideband modulator,” Opt. Express **20**(6), 6555–6560 (2012). [CrossRef]

**32. **N. Ishii, E. Tokunaga, S. Adachi, T. Kimura, H. Matsuda, and T. Kobayashi, “Optical frequency- and vibrational time-resolved two-dimensional spectroscopy by real-time impulsive resonant coherent Raman scattering in polydiacetylene,” Phys. Rev. A **70**(2), 023811 (2004). [CrossRef]

**33. **P. Mao, Z. Wang, W. Dang, and Y. Wenga, “Multi-channel lock-in amplifier assisted femtosecond time-resolved fluorescence non-collinear optical parametric amplification spectroscopy with efficient rejection of superfluorescence background,” Rev. Sci. Instrum. **86**(12), 123113 (2015). [CrossRef]

**34. **Y. Deguchi, M. Noda, Y. Fukuda, Y. Ichinose, Y. Endo, M. Inada, Y. Abe, and S. Iwasaki, “Industrial applications of temperature and species concentration monitoring using laser diagnostics,” Meas. Sci. Technol. **13**(10), R103–R115 (2002). [CrossRef]