Lock-in-detection dual-comb spectroscopy

Dual-comb spectroscopy (DCS) is useful for gas spectroscopy due to 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 rapid monitoring of transient signal change and spectroscopy of hydrogen cyanide gas.


Introduction
Recent advances in optical frequency comb (OFC) [1][2][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 frep (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 frep larger than a few GHz due to its spectral resolving power.
Recently, dual-comb spectroscopy (DCS) [11][12][13][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 (= frep1 and frep2). Due to its rapid, precise, and accurate -5-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].
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 frep between dual OFCs (= ∆frep = frep2 -frep1); 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 -6-is measured by a spectral resolution of frep/100, the mode linewidth of the measured OFC spectrum is decreased down to frep/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.
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. Also, 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 ∆frep. 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, -7-29]. 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. two OFCs with a slightly different repetition frequency (signal OFC, mode spacing = frep1; local OFC, mode spacing = frep2 = frep1 + ∆frep) generates a secondary frequency comb in RF region, namely RF comb (mode spacing = ∆frep), via the multi-frequency heterodyning interference between them. In usual DCS, the RF comb are acquired as an RF interferogram in time domain and then are 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 by tuning the LID reference frequency to coincide with a target RF-comb-mode frequency.

Experimental setup
-8- Inc., FS725; frequency = 10 MHz, accuracy = 5×10 -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 [16][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 45º. 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 = 800-1700 nm; bandwidth < -9-150MHz). 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 ~ 200MHz, 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 OFCs such by the fluctuation of temperature, air flow and so on, 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 < 20MHz). 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 frep2. 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).

Performance evaluation of LID-DCS and DCS
We first investigated a relation between the number of measured spectra and the net measurement

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 = 1mm), leading to a transient change in the optical intensity. Figure 3  Therefore, the LID-DCS will be more powerful than DCS for monitoring of transient signal change, such as gas concentration measurement under air turbulence.

Discussion
One may have doubts about the advantage of LID-DCS compared with usual CW spectroscopy [30] because only a single OFC mode was extracted in this article. However, LID-DCS can rapidly select an arbitrary OFC mode within the broadband OFC spectrum, and frequency uncertainty of the selected OFC mode is always secured by a frequency standard. To realize the high spectral accuracy in CW spectroscopy as same accuracy as the LID-DCS, a CW laser must be stabilized to an OFC, indicating the fast wavelength tuning of the CW laser source would be difficult due to the requirement of a complicated method for the determination of absolute frequency, e.g., the tuning of the comb parameters such as frep, continuous wavelength sweeping of the CW laser source, and so on.
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 [29]. Furthermore, a low frequency electrical noise can be suppressed in the LID-DCS owing to an inherent heterodyne -20-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.
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.
A combination of LID-DCS with imaging is another interesting extension. Single-pixel imaging (SPI) has attracted attention as a scan-less imaging scheme [35,36] because SPI is particularly useful when -21-the appropriate image detector is not readily available; for example, the RF interferogram measured in DCS is too fast for camera acquisition. In SPI, while the sample object is coded sequentially by a series of mask patterns generated by a spatial light modulator or a digital mirror device, the corresponding total light intensity passing through the sample object is measured as time-series data by a single-channel photodetector; then, the original two-dimensional image is reconstructed mathematically by the correlation calculation between the known coded masks and the measured time-series data of total light intensity. Recently, SPI was effectively combined with DCS for ultrahighresolution and ultrahigh-dense hyper-spectral imaging [19]. While this combination enables us to perform the scan-less imaging without the need for the image detector, FFT calculation time of interferogram on every coding pattern hampers the scan-less advantage of SPI and is a bottleneck for rapid imaging. If LID-DCS is combined with SPI in place of DCS, the image acquisition time will be largely reduced.

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 frequencydomain measurement without the need for FFT calculation. Although the large amount of spectral data -22-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) MINOSHIMA Intelligent Optical Synthesizer Project (JPMJER1304), Japan Science and Technology Agency (JST), Japan, and Institute of Post-LED Photonics, Tokushima University, Japan.