**2. Materials and Methods**

The DCVS apparatus used for the present measurements is described in detail elsewhere [29,35]. Basically, an OFC is spectrally filtered by means of a Fabry–Perot (FP) cavity acting as an interaction cell containing the absorbing gas. The Vernier ratio (*V*) between the FP and the OFC (i.e., the ratio between the FP's free spectral range Δ*FSR* and the OFC's repetition rate, *fr*) is established to be enough to resolve the FP-transmitted OFC modes with a high-resolution dispersion spectrometer (SOPRA, resolution 2 GHz @ 2 μm). Indeed, after FP-filtering, spectral fractions of the OFC of about 5 cm−<sup>1</sup> are dispersed at the output of the SOPRA instrument and detected with an InGaAs linear array detector. Different from the experimental setup described in [29,35], the OFC is a thulium-doped fiber-based mode-locked laser from IMRA America, Inc. It has a spectral coverage of about 40 nm to around 1970 nm with a repetition rate of about 400 MHz, with a maximum averaged power of about 4 W after the final amplification stage. The OFC is a self-referenced system, employing the first amplification stage (about 1 W of avg. power) to generate OFC emission around 1 μm through to the non-linear optical processes in optical fibers. The carrier-offset-frequency parameter of the OFC, *fo*, is consequently detected by beating the

teeth frequencies of the duplicated fraction of the fundamental comb emission with those of the 1 μm harmonic, and controlled by means of the phase-lock-loop (PLL) against the stable RF clock. The repetition frequency, *fr*, is controlled by detecting the beat of the comb modes with a fast InSb photodetector and by mixing it with RF synthesized frequency (*fRF*) to obtain the RF note at 150 MHz locked with a second PLL circuit. The reference clock for both *fr* and *fo* locks is a 10 MHz quartz-Rb-GPS system with a relative frequency stability of 6 × <sup>10</sup>−<sup>13</sup> in 1 s and an accuracy of 10−12, at worst. Hence, the frequency of the OFC modes is directly traceable to the primary frequency standard with these precision and accuracy figures when locked.

The FP is a linear cavity with silver-coated mirrors. The finesse is about *F* = 200 @ 2 μm with a transmission coefficient around 0.2%. The cavity length is variable and coarsely controllable by means of step-motors mounted in the kinematic mount of one of the two cavity mirrors. It's value is established by the chosen Vernier ratio *V* and by the requirement to be long enough to obtain an efficient enhancement of the absorption path. For the present measurements, where a simultaneous detection of the comb modes resonant with transitions of different isotopologues of the targeted molecule is needed, *V* should be fixed to a value which gives a Δ*FSR* as close as possible to a multiple of the isotope shift between the probed molecular transitions. Transitions of the 12C and 13C isotopes of the carbon dioxide molecule around 5005 cm−<sup>1</sup> are used to test the spectroscopic performances of the technique. In particular, the (2001–0000) R(18) of 13C16O2 @ 5004.84 cm−<sup>1</sup> and the (2001–0000) P(45) of 12C16O2 @ 5005.27 cm−<sup>1</sup> ro-vibrational transitions are selected in order to match the condition of simultaneous recording of both absorptions in the detected range of 5 cm−<sup>1</sup> of the cavity-transmitted and dispersed fraction of the interacting OFC for our spectrometer . The frequency shift between these two transitions is around Δ*νIS* = 12,863.55 MHz. A Δ*FSR* of the order of half of Δ*νIS* allows one to obtain an enhanced absorption path of about 2 m (i.e., *LFP* = 2.4 cm), while keeping the FP mode separation more than three times larger than the SOPRA spectral resolution. In addition, a noninteger *V* ratio is chosen to obtain rarefied resonance between OFC and FP while keeping the condition that the two FP transmissions are in resonance with the two selected CO2 transitions. Indeed, choosing *V* = 15.5 (i.e., 31 comb modes each 2 FP modes) tailors this condition, further helping in better FP-filtering of the OFC and a better resolved image of the transmitted modes by the SOPRA diffraction spectrometer. In Figure 1 is shown a scheme of the measurements: in one case the comb/FP configuration shown is resonant with the center of 13C16O2 transition. In such a case as shown in Figure 1a, both transitions are simultaneously probed by different comb teeth, while in Figure 1b, only the 12C16O2 transition shows an heavily saturated absorption.

The DCVS is performed under the condition of perfect resonance between the FP and the transmitted OFC modes for each value of their optical frequency. Consequently, the FP length is actively locked to set it on resonance with the maximum transmission of the Vernier resonant modes. To this aim, one of the FP mirrors is mounted in a PZT to obtain fine control of the cavity length by detecting a small part of the FP-transmitted light in a InGaAS detector. A 3 kHz modulation is applied to the PZT, and the first derivative of the FP output detected light is obtained by means of locking detection and used as an error signal to control the cavity length to the maximum transmission condition.

**Figure 1.** The two roto-vibrational transitions are probed around the two comb/FP configurations shown in figure, where the FP transmission as a function of frequency is shown in two cases. In case (**a**), one comb tooth and one FP mode are resonant with the 13C16O2 transition, while in case (**b**), they are resonant with the 12C16O2 transition. Far from resonances, the FP mode is an Airy function; around a resonance frequency, the mode is modified by the resonant refractive index of the FP medium. Here, the simulation is made by taking Voigt profiles for the two investigated transitions (Hitran parameters), and with f*<sup>r</sup>* = 400 MHz, Δ*FSR* = 6.2 GHz (i.e., 2Δ*FSR*/ *f*<sup>r</sup> = 31), and a FP finesse *F* = 200. In each inset immediately above the FP transmission, each FP mode is zoomed (broken axis plot). We observe that one FP mode every two transmits a comb tooth, and that between adjacent zoomed FP modes, there are 15 comb modes not transmitted and not shown in the inset. We observe that in the (**a**) case, both transitions are simultaneously probed by different comb teeth, the 13C16O2 transition giving an effect not visible in the graph. The (**b**) case shows a heavily saturated 12C16O2 transition.

The majority of the FP-transmitted light is sent to the SOPRA input slit, and the diffracted light at the output slit, for a given position of the diffraction grating, is detected by a liquid N2-cooled InGaAs array detector (PyloN-IR:1024, Princeton Instr.). The arrangement of the optical components before and after SOPRA is devised to obtain the largest spectral fraction in a single array's image, while keeping the maximum SOPRA resolution. In Figure 2d, an example of such an image is shown for *fr* = 398.99 MHz. For each image, 13 transmitted modes are detected, which is a spectral portion of the OFC of about 5 cm−<sup>1</sup> taking into account the 31 comb modes for each transmitted interval. The vertically integrated intensity profile of the dispersed image (Figure 2e) is used to calculate the transmitted contribution of each detected mode. As thoroughly described in [29], a knowledge of the FP-SOPRA system resolution function, i.e., its response to a monochromatic input,

is necessary in order to obtain successively the OFC modes transmissions. In this paper, in order to work with an analytical resolution function, we adopt a kind of continuous wavelets [43] approximation variant of the approach followed in [29]: we fit a single mode diode laser response by a superposition of Gaussian functions, treating their centers and widths as free parameters. In practice, a superposition made of seven of such wavelets is sufficient to reproduce the experimental diode laser response within the measurement noise. Once the resolution is determined in this way, the data analysis proceeds as in [29]: the resolution function, which is now an analytical function, is replicated on the set of *N* = 13 transmitted OFC modes, giving a total fitting function for the integrated image with free parameters given by the position of one of the peaks, the peak's separation, and by the *N* peaks intensities, which we write as I*M*(*fr*, *P*, *T*), where the arguments *fr*, and the pressure *P* and temperature *T* of the gas sample identify the experimental conditions, and where the subscript *M* identifies each one of the N recorded FP modes per image. Figure 2e shows an example of the fit.

The absorption spectrum of the tested molecular transitions is determined by recording the set of array images for a scan of the OFC frequencies around each transition frequency. The synthesized scan of *fr* is accomplished with a change of the *fRF* frequency in the *fr*-lock chain. Custom software was used to obtain an automated acquisition of such spectral images as a function of *fr*. In Figure 2b, the behaviors of the detected images as a function of the Δ*fr* for the FP-transmitted orders involved in the determination of the spectra of the 12CO2 and 13CO2 transitions are shown. For the given grating position, we label the relevant FP orders as the *M* = 0 order, resonant with the (2001-0000) R(18) of 13C16O2 transition, the *M* = +2 order, resonant with the (2001-0000) P(45) of 12C16O2 transition, and the *M* = −2 and *M* = +4 orders, used to calculate the transmittance spectra not-resonant with the CO2 transitions. In addition, the scan behaviors for the *M* = −12 order, partially resonant with the (2001–0000) R(14) transition of 13C16O2, as well as those of the not CO2 resonant *M* = −14 and *M* = −10 modes, are shown. Figure 2c shows details of the scan of these modes as a function of the transmitted comb mode frequency, while the corresponding integrated intensities I*<sup>M</sup>* are shown in Figure 2a.

The ratio between the integrated intensity for the molecule absorbed modes (M = 0 and +2) with the averaged intensity of the not-absorbed modes (M = −2 and +4) is used to calculate the transmittance spectrum of the 13CO2 and 12CO2 transitions, respectively, as shown in Figure 3 and in the inside graph of Figure 2a:

$$\mathcal{T}\_{M}^{\text{(s)}}(\Delta\nu, P, T) \quad = \quad 2\frac{\mathcal{T}\_{M}(f\_{r}, P, T)}{\mathcal{T}\_{-2}(f\_{r}, P, T) + \mathcal{T}\_{+4}(f\_{r}, P, T)} \tag{1} \tag{1}$$

On the right hand-side of Equation (1) we leave as arguments the parameters *fr*, *P* and *T* that set the experimental conditions of the acquisition, while for the task of the successive analysis procedure where optical frequencies are required to be compared with other results, the optical detuning Δ*ν* is used instead of *fr*. Δ*ν* is calculated as the optical detuning of the *M* mode with respect to the frequency of the *M* = 0 mode at the center of the *fr* scan:

Δ*ν* = *NM fr* − *N*<sup>0</sup> *frc* (*M* = 0, 2) (2)

with the *frc* repetition rate frequency at the center of the scan and with *N*<sup>2</sup> = *N*<sup>0</sup> + 31. *N*<sup>0</sup> is calculated from the integer ratio between the reported frequency [44] of the 13C16O2 transition and *frc* .

**Figure 2.** Spectra of the OFC modes filtered by the FP as a function of the OFC's Δ*fr* = *fr* − *frc* , with *frc* repetition rate frequency at the center of the scan. The spectra are the result of the analysis of the recorded images of the SOPRA-dispersed fraction of the FP-filtered OFC vs *fr*. Three of these modes are resonant with the (2001-0000) R(14) of 13C16O2 @ 5002.21 cm<sup>−</sup>1, (2001-0000) R(18) of 13C16O2 @ 5004.84 cm−1, and (2001-0000) P(45) of 12C16O2 @ 5005.27 cm−<sup>1</sup> ro-vibrational transitions; the other modes are not absorbed by the CO2 molecule. The modes are identified by the FP order scale of panel (**c**): the <sup>−</sup>12, 0 and +2 orders are the modes absorbed by 13C16O2 and 12C16O2 transitions, respectively; the <sup>−</sup>14, <sup>−</sup>10, <sup>−</sup>2 and +4 orders are the not-absorbed ones. (**a**) Integrated transmission spectrum. (**b**) Spectral behavior of the array images for each mode. (**c**) Spectral behavior of the SOPRA-resolution wavelet for each mode. (**d**) Detector array image of the portion of the FP-transmitted OFC modes for a given *fr*. (**e**) Vertically integrated intensity of the transmitted modes of image in panel (**d**) and the fit to a function resulting form the addition of 13 SOPRA-resolution functions; the intensity of the fitted resolution provides the transmission spectra values for each *fr* of panel (**a**). The inside graph of panel (**a**) shows the transmittance spectrum of the R(14) transition of the 13C16O2 from the *<sup>M</sup>* <sup>=</sup> <sup>−</sup>12 mode normalized by the average intensity of the <sup>−</sup><sup>14</sup> and −10 modes.

Due to the limited continuous scan of the *fr* in the OFC lock condition, the high frequency wings of the P(45) transition of 12C16O2 and of the R(14) transition of 13C16O2 are not recorded. Such a limitation should be overcome by using a better combination of the frequency parameters of OFC and FP. Uncertainties of the measured spectral parameters for such transitions are expected to be affected by this issue, as discussed in the following. The situation is critical for the R(14) transition of 13C16O2 because it is recorded by less than one half, as shown in the inside graph of Figure 2a. Consequently, it is not considered in the present isotopic ratio measurements.

**Figure 3.** FP transmittance of the absorbed comb modes resonant with the(2001-0000) R(18) of 13C16O2 (left side graphs) and (2001-0000) P(45) of 12C16O2 (right side graphs) transitions and the fit to the T (Δ*ν*, P, T) (Equation (4)) of all recorded spectra by using the global fit procedure. P = 55.0 mbar and T = 296 K for the CO2 gas sample for all acquisitions. The reduced *χ*<sup>2</sup> of the global fit is also shown.
