**3. Results**

Three DCVS spectra of the (2001–0000) P(45) of 12C16O2 and (2001–0000) R(18) of 13C16O2 transitions are recorded with CO2 gas at a pressure of P = 55 mbar and at room temperature (T = 296 K). The total transmittance of the two absorbed modes for each recording is given by the following:

$$\mathcal{T}^{\text{(s)}}(\Delta\nu, \mathbf{P}, \mathbf{T}) \quad = \quad \mathcal{T}^{\text{(s)}}\_{M=0}(\Delta\nu, \mathbf{P}, \mathbf{T}) + \mathcal{T}^{\text{(s)}}\_{M=+2}(\Delta\nu, \mathbf{P}, \mathbf{T}) \tag{3}$$

with <sup>T</sup> (s) *<sup>M</sup>* calculated as described above (Equation (1)).

In order to determine isotope dependent relevant spectroscopic parameters between transitions, namely, frequency shift <sup>Δ</sup>*νIS* = *<sup>ν</sup>o*(12C) − *<sup>ν</sup>o*(13C) and natural isotopic concentration ratio R<sup>13</sup>*C*/12*<sup>C</sup>* = *a* 13C/*a* 12C with *aIS* isotopic abundance of the *IS* isotope in the gas sample, the T (s) data are fitted to a function that describes the absorption modified FP Airy transmission at maximal optical resonance [29,35,45]:

$$\mathcal{T}(\Delta\nu, \mathbf{P}, \mathbf{T}) \quad = \quad K \sum\_{M=0, +2} \mathcal{T}\_M(\Delta\nu, \mathbf{P}, \mathbf{T}) + (A + B\Delta\nu) \tag{4}$$

where a linear spectral background, with *A* and *B* as the frequency independent and slope parameters, respectively, and a scale factor *K* are considered to take into account possible not-compensated instrumental effects, due to the transmittance normalization. In Equation (4), T*<sup>M</sup>* is given by the following:

$$\begin{array}{rcl} \mathcal{T}\_{M}(\Delta\nu,\mathcal{P},\mathcal{T}) &=& \frac{\mathcal{T}\_{\max}(\mathfrak{a}\_{M})}{1 + F(\mathfrak{a}\_{M})\sin^{2}\left(\frac{\pi}{\Delta\_{PSK}}(\Delta\nu\Delta\mathfrak{a}\_{M} + \Delta\_{lock})\right)}\\\\ \mathcal{T}\_{\max}(\mathfrak{a}\_{M}) &=& \frac{T\_{m}^{2}e^{-\mathfrak{a}\_{M}L}}{\left(1 - R\_{m}e^{-\mathfrak{a}\_{ML}L}\right)^{2}}\\\\ F(\mathfrak{a}\_{M}) &=& 4R\_{m}\frac{e^{-\mathfrak{a}\_{ML}L}}{(1 - R\_{m}e^{-\mathfrak{a}\_{ML}L})^{2}} \end{array} \tag{5}$$

where *Tm* and *Rm* are the transmission and reflectivity coefficients of the FP mirrors, and *L* = *c*/2Δ*FSR* is the cavity length. The argument of sin<sup>2</sup> in the Airy function is the FP round trip phase shift, which is written as the sum of the empty cavity contribute, which is *π*Δ*lock*/Δ*FSR* for the locked cavity, and *π*Δ*ν*Δ*nM*/Δ*FSR*, representing the contribution due to the gas dispersion for the *M* mode. Δ*nM* and *α<sup>M</sup>* are the dispersion and absorption coefficients, respectively, induced on the *M*-mode by resonant absorption transitions of the gas sample, which are calculated as a function of the laser detuning Δ*ν* and the thermodynamic conditions, *P* and *T*, from the real and imaginary part of FP's refraction index variation: (*n* − 1)*<sup>M</sup>* = Δ*nM* + *iαM*/2*kM*, with *kM* being the wave vector module of the absorbed *M* comb mode. Because Δ*νIS* between the transitions is very large when compared to their linewidth in our thermodynamic conditions, any spectral interference effects between the two CO2 transitions can be safely neglected. Consequently, (*n* − 1)*<sup>M</sup>* contributions for the *M* = 0 and *M* = 2 modes can be considered to be only induced separately by the 13CO2 and 12CO2 transitions, respectively, simplifying the analysis. If we label as *t* = *a* and *t* = *b* these 13CO2 and 12CO2 transitions, and considering a Voigt profile for the CO2 absorptions, we have the following:

$$
\Delta n\_M + i \frac{a\_M}{2k} \quad = \quad - \frac{c N a^{IS} S\_t}{2 \pi^{3/2} k\_M \Delta \nu\_t^D} \int\_{-\infty}^{\infty} du \, \frac{e^{-u^2}}{\frac{\Delta \nu - \Delta \nu\_t^o}{\Delta \nu\_t^D} - u + i \frac{\Gamma\_t/2}{\Delta \nu\_t^D}} \tag{6}
$$

with the correspondence *M* = 0, 2 ⇔ *t* = *a*, *b*. In Equation (6), *c* is the light speed, *N* is the numeric density of the gas at the given thermodynamic conditions, *aIS* is the abundance of the isotopologue *IS* in the gas mixture, S*t* is the linestrength of the transition *t* per molecule [*St* = *SH I*/*aIS*, with *SH I* the linestrength value of the HITRAN database [44]. In this way, the isotopic abundances and their ratios can be measured independently of the transition's levels], Δ*ν<sup>D</sup> <sup>t</sup>* is the FWHM Doppler linewidth of the transition *t* of the isotopologue *IS* at *T*, Δ*ν<sup>o</sup> <sup>t</sup>* is the transition frequency detuning, <sup>Γ</sup>*<sup>t</sup>* <sup>=</sup> *<sup>δ</sup>*Γ*<sup>t</sup> <sup>δ</sup>*<sup>P</sup> P is the FWHM collisional linewidth contribution at *P* and *T* and *u* represents the Doppler shift of each molecular class velocity.

The least-squared fit procedure of T (Equation (4)) to T (s) determines the best values of the molecular parameters, Δ*ν<sup>o</sup> <sup>t</sup>* , *St*, *aIS*, Δ*ν<sup>D</sup> <sup>t</sup>* and Γ*<sup>t</sup>* for both transitions as well as of the instrumental-related parameters Δ*FSR*, *F*, Δ*lock*, *K*, *A*, and *B*. As in previous measurements [29,35,45], two different fit strategies are performed. In the first approach, the final values of the relevant parameters are calculated from the weighted average of those values resulting from the fits of each individual scan at pressure *P*. In the other approach, all acquired spectra are considered in a single global fit, where some fit parameters (i.e., molecular-related parameters, Δ*FSR*, and *F*) are considered shared between all the scans, while Δ*lock*, *K*, *A*, and *B* are local parameters, considered to be different for each scan. In addition, parameters that can be evaluated independently, such as Δ*FSR*, *St*, and Δ*ν<sup>D</sup> <sup>t</sup>* for both transitions are kept fixed during the fit procedure. The result of this global fit is graphically shown in Figure 3. A summary of the *ν<sup>o</sup> <sup>t</sup>* , *aIS* and *δ*Γ*t*/*δ*P parameters for both transitions and the determined values <sup>Δ</sup>*νIS* and R<sup>13</sup>*C*/12*<sup>C</sup>* from them are reported in Table 1. Their differences against the values reported in the HITRAN database [44] are also tabulated.

**Table 1.** <sup>Δ</sup>*νIS* and <sup>R</sup><sup>13</sup>*C*/12*<sup>C</sup>* determinations from the measured spectral parameters of the (2001-0000) R(18) of 13C16O2 and (2001-0000) P(45) of 12C16O2 transitions from DCVS measurements @ 5005 cm−<sup>1</sup> and comparison with HITRAN database values [44]. Absolute frequencies are calculated by *νo <sup>t</sup>* = N0f*rc* + <sup>f</sup>*<sup>o</sup>* + <sup>Δ</sup>*ν<sup>o</sup> <sup>t</sup>* with N0 order number of the OFC tooth transmitted by the M=0 FP mode and resonant with the 13C16O2 transition and f*rc* and *fo* repetition rate frequency at the center of the scan and offset frequency of the OFC, respectively. (Errors reported in parentheses). <sup>Δ</sup>(*DCVS*−*H I*) are the differences between HITRAN database values and the present measured values for each tabulated parameter. The HITRAN database values are *νo* = 150041403.1 (2) MHz, *a* <sup>13</sup>*<sup>C</sup>* = 0.01106 (1), *δ*Γ/*δP* = 5.92 (5) MHz/mbar for the (2001-0000) R(18) transition of 13C16O2 and *ν<sup>o</sup>* = 150,054,266.5 (2) MHz, *a* <sup>12</sup>*<sup>C</sup>* = 0.9842 (9), *<sup>δ</sup>*Γ/*δ<sup>P</sup>* = 4.56 (4) MHz/mbar for the (2001-0000) P(45) transition of 12C16O2. The frequency shift between them is Δ*νIS* = 12,863.5 (3) and the natural isotopic concentration ratio R<sup>13</sup>*C*/12*<sup>C</sup>* = 0.01123 (2). Errors of this differences take into account the error of the HITRAN values, which are added in quadrature to the measured ones.

