Dual-comb interferometers, which can be used as high-precision spectrometers, have been widely developed for atmospheric sensing [1–3]. For instance, greenhouse gases such as methane have been extensively studied in controlled environments [4–7]. However, comb-based gas sensing outside the laboratory brings a new set of challenges [8]. Recently, there have been field demonstrations of methane detection addressing those challenges [3,9], but these dual-comb systems were based on fully locked combs and self-referenced combs to perform spectroscopy. As the need for portable instrument requires the system to be compact and cheap, further simplifications would be beneficial.

Self-correction algorithms have been proven to be effective with free-running dual-comb lasers originating from the same laser source [4,10] or in the case of highly mutually coherent lasers [11,12]. The range of operation of our self-correction algorithm is now extended to independent lasers with only minimal active stabilization.

In this paper, we present simplifications of a system used for methane sensing [9,13]. Reduction of the system’s complexity is achieved through the introduction of a second continuous-wave (CW) laser to the experimental setup and the use of a self-correcting algorithm [14]. The $ f - 2f $ interferometer originally used to stabilize the carrier-envelope offset (CEO) frequency becomes unnecessary as phase drifts are corrected in post-processing. As an additional simplification, the 100 kHz bandwidth piezo-electric transducer (PZT) used to stabilize rapid fluctuations of the laser’s repetition rate is dismissed, leaving only the 1 kHz bandwidth PZT to minimally stabilize the comb. The 100 kHz and 1 kHz bandwidth PZT controllers are thereafter referred to as “fast” and “slow” controllers, respectively. To demonstrate the performance of the simplified spectrometer, the lasers are amplified and broadened in highly nonlinear fiber (HNLF) up to 1700 nm to interrogate methane’s $ 2{\nu _3} $ overtone. The absorption spectrum is measured and compared for the actively locked and the minimally stabilized cases. A comparison of a spectral line to HITRAN database is also provided to assess the validity of the measurement.

The comb sources used are self-referenced fiber lasers based on the design fully described in [13]. Figure 1(a) shows one such source. Here, two frequency combs with a repetition rate $ {f_r} \approx 160 \;{\rm MHz} $ and repetition rate difference $ \Delta {f_r} = 828 \;{\rm Hz} $ are used. In the figure, the self-referencing section has been shaded to illustrate that it is no longer needed to produce coherent interferograms when the processing steps proposed in this paper are followed. This significantly reduces the hardware complexity of both sources since the PPLNs are no longer needed, and the requirements are greatly relaxed for the HNLF as coverage for an octave is no longer needed. The spectral broadening can thus be optimized for spectroscopy alone. Considering the two PPLN crystals alone are replaced by a single-frequency 1530 nm CW laser, the simplification represents a cost reduction of nearly one order of magnitude at the time of writing this manuscript.

 

Fig. 1. (a) Schematic of a single-frequency comb reproduced from [13] where components substituted by a second CW laser and a self-correction algorithm are indicated by the red-lined areas. SESAM, semiconductor saturable absorber mirror; PZT, piezo-electric transducer; OC, output coupler; ISO, isolator; WDM, wavelength division multiplexer; PM-HNLF, polarization-maintaining highly nonlinear fiber; PPLN, periodically poled lithium niobate; BPF, bandpass filter; PBS, polarization beam splitter. (b) Experimental setup for the dual-comb experiment where two frequency combs are used; CW, continuous-wave laser.

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To perform spectroscopy, the frequency combs are spectrally broadened up to 1700 nm using amplifiers and highly nonlinear fiber (OFS HNLF-PM). This second pair of amplifiers would not be required if the 20% TAP/WDMs were replaced by simple WDMs to give access to the full power generated by the first pair of amplifiers. Each comb is optically filtered with a bandpass filter centered on 1650 nm (Advance Fiber Resources). Spectroscopy with a greater optical bandwidth would be possible as long as the compression ratio $ ({f_r}/\Delta {f_r}) $ is adjusted so that the RF comb’s bandwidth matches the aliasing-free electrical bandwidth $ ({f_r}/2) $. The compression ratio can be increased by lowering the repetition rate difference of the lasers $ \Delta {f_r} $. There is however a limit to the maximum achievable ratio since the comb’s phase fluctuations are sampled at a rate equal to $ \Delta {f_r} $. Therefore, a higher compression ratio implies a lower bandwidth phase correction.

The dual-comb experimental setup is shown in Fig. 1(b). One of the comb interrogates a 5.5-cm-long cell filled with methane at the atmospheric pressure $ 740 \pm 10\%$ Torr (wavelength references). The two combs are recombined with a 50/50 coupler and mixed to produce an RF comb. The experimental setup is based on a single interferometer to measure the transmission spectrum of methane. A reference signal provided by a second interferometer [5] or provided through a second measurement without the gas cell [10] is usually required for intensity normalization. However, it has been chosen here to not calibrate the gas spectrum since it is out of the scope of this demonstration. The data are acquired on a GaGe acquisition board (CSE8389) with a sampling rate of 125 MS/s.

The amplified and broadened combs are split twice and mixed with two CW lasers. The 1564 nm beat signal is used to provide minimal active stabilization. A longitudinal mode of each comb at 1564 nm is mixed with the laser (RIO Planex RIO0095-3-15-1), and the beat note is fed to an FPGA running our open-source phase-locked loop software [15] to actively correct phase drifts of the comb at 1564 nm with a “slow” PZT controller (or actuator). The fast PZT controller that was in the initial design [13] has been removed since it introduces fast noise around its resonance frequency that degrades the quality of the dual-comb interference. Only the slow PZT controller is used to stabilize the laser’s repetition rate, while the CEO frequency is stabilized through the laser’s pump power. The idea behind this repetition rate stabilization loop is to avoid aliasing (by crossing DC or $ {f_r}/2 $) of the CW-comb beat signal by correcting its frequency variations (or its average frequency) with minimal hardware. Therefore, to compensate the additional fluctuations not corrected by the PZT controller, the beat note fed to the FPGA is also sent to the acquisition board. This information is used to perform an interferogram phase correction that replaces the effect of the fast PZT controller. This step is performed before using the self-correction algorithm.

The first idea was that the stabilization and correction of the mutual phase between the combs at 1564 nm would allow using the interferogram self-correction algorithm for the 1630–1670 nm region. In principle, with phase variations corrected at 1564 nm, only fluctuations of the repetition rate difference remain. These can be corrected with a resampling operation without phase wrapping issues, which interpolates the interferograms on a new linearly increasing time axis. However, significant fluctuations of the repetitions rates with a bandwidth larger than $ \Delta {f_r}/2 $ prohibit success with self-sufficient algorithms [16]. For that reason, the mutual coherence between the combs could not be retrieved, teeth were smeared, and a resolution loss was observed on the methane lines. This is shown in Fig. 2, which compares the case of the phase correction with one CW laser at 1564 nm followed by the self-correction with the fully stabilized case. The line R(3) appears broadened because of an inadequate correction of the interferograms’ repetition rate, which leaves an uncompensated stretching of the comb around the mode at 1564 nm. Since there are over 55,000 modes between 1564 nm and 1640 nm, the noise on $ \Delta {f_r} $ is amplified by a factor 55,000 at 1640 nm. To validate the measurement, the data for both scenarios have been compared to HITRAN to yield the residuals shown in the bottom panels of Fig. 2 where the broadening of R(3) is emphasized. The fitting conditions are discussed later in the manuscript.

 

Fig. 2. (top panel) Transmission spectrum in the $ 2{\nu _3}$ R(3) methane manifold region for the case of a phase correction with one CW laser, the fully stabilized case, and an HITRAN fit. (bottom panels) Residuals between the transmission spectra and the HITRAN fit.

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To solve this problem of uncorrected fluctuations, a CW laser operating at 1530 nm (RIO Planex RIO0095-3-59-1) is introduced to measure the combs’ relative phase at a second spectra location. The idea behind the use of two lasers is to provide an estimate of the CEO phase $ {\phi _{{\rm ceo}}} $ and repetition rate difference phase $ {\phi _{df_r}} $ of the RF comb [16]. Knowing these two quantities allows to calculate the phase at a desired frequency. Here, the phase at 1640 nm is extrapolated to pre-correct the interferograms such that self-correction can afterwards successfully retrieve the mutual comb coherence. One shall note that the pre-correction does not need to be perfect. It only needs to meet the self-correctability criterion [17], which requires the phase difference between two consecutive interferograms to be smaller than $ \pi $. The phase extrapolation at 1640 nm can be estimated with a linear combination of the phases at 1530 nm and 1564 nm. The following equations, where $ k $, $ l $, and $ m $ represent the mode number at 1530 nm, 1564 nm, and 1640 nm, respectively, are used [Eq. (1)]:

After the pre-correction, the self-correction algorithm previously reported in Ref. [14] extracts information from the interferograms themselves to perform phase correction for the CEO frequency and a resampling according to the repetition rate difference. Since the phase fluctuations occurring in the interferometer are not taken into account as they are not measured by the CW lasers, a self-correction is still required to properly phase-correct the interferograms.

The acquired interferograms, although phase corrected, result from quasi free-running combs, and thus their absolute optical frequency fluctuates during the measurement. However, theses fluctuations can be estimated to be below the laser’s mode spacing (160 MHz), which is nearly two orders of magnitude inferior to the few GHz linewidth of the observed methane features at atmospheric pressure. Therefore, at the considered scales, the noise on the comb’s absolute frequency is not significant. Since the optical position of a comb’s tooth is locked with CW lasers, the CW laser’s linewidth is required to be smaller than half the combs’ repetition rate to remain locked to the same tooth throughout the measurement. Alternately, frequency-stabilized CW lasers could provide an absolute frequency reference to lock the combs and guarantee frequency accuracy.

The transmission spectrum of methane is presented in Fig. 3. The R and Q branches of the $ 2{\nu _3} $ overtone are shown for the case of a self-referenced system (full active stabilization) and for the case of the simplified system with two CW lasers (minimal stabilization). Reasonable correspondence is observed as no notable difference is visible between the two cases. The residuals confirm that the two curves are the same within 1%, which is the standard deviation of the residuals limited by the measurement noise. The increasing variance of the residuals toward longer wavelengths is explained by the lower optical signal-to-noise ratio as the wavelength increases.

 

Fig. 3. Transmission spectrum of methane (R and Q branches) for the cases of a fully locked (red) and minimally stabilized frequency combs (black) where no distinction is visible within 1%.

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Fig. 4. Transmission spectrum (top) in the $ 2{\nu _3} $ R(3) methane manifold region for the experimental data without the CEO servo-loop (black) and as modeled by Voigt line shapes computed using parameters from the HITRAN 2016 database (red) and the corresponding residuals (bottom). Standard deviation of residuals $ \sigma $ is given on the bottom panel.

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Sixth-order piece-wise polynomial functions, where each function spans approximately 4 nm around each absorption line, have been removed from the baseline of the data to produce a flat-looking spectrum, thus allowing better comparison between presented scenarios. The frequency axis of Fig. 3 has been determined from an HITRAN fit shown in Fig. 4 and detailed further in text. As the dual-comb experiment performed here was one with two quasi free-running lasers, the frequency combs produced were not referenced to an optical standard, such as a stabilized laser or a well-known gas absorption feature. Consequently, the absolute frequency of the gas spectrum measured in the RF domain was determined in post-processing through the identification of the probed gas’ absorption lines.

To assess the validity of the measurement, the R(3) manifold of methane in the $ 2{\nu _3} $ band has been fitted to a modeled spectrum using a complex Voigt profile. Data from HITRAN 2016 have been used. Excellent correspondence is observed between the experimental data and the theoretical model, which confirms that the performance of the spectrometer is unaltered by the suggested simplifications.

The fit was performed with an optimization function that minimized the least-square error on the residuals where the pressure of the cell, its length, and its temperature were free parameters. The absolute frequency of the acquired data, the optical point spacing ($ {f_r} $), and the coefficients of a global fourth-order polynomial function were also parameters of the optimization. A pressure of 92 kPa, a cell length of 5.3 cm, and a temperature of 21°C were found, matching the uncertainty bounds given by the manufacturer.

In conclusion, simplifications to an existing dual-comb interferometer have been demonstrated. They greatly reduce the complexity of the instrument and its cost, thus improving its quality as a field-deployable instrument. Nonlinear broadening of the laser’s spectrum has been realized to perform near-infrared broadband spectroscopy of methane. Branches R and Q of methane in the $ 2{\nu _3} $ band were measured with and without the self-referencing interferometer to assess the performance of the simplified system.