## 3 Results and Discussion  
Figure 3a shows exemplary intensity spectra subsequently recorded on sample gas (\(I_{s}\), blue) and reference gas (\(I_{r}\), black). From these two time domain spectra, the sample gas temperature and CO\({}_{2}\) concentration is obtained in a multi-step fitting algorithm, which is also illustrated in Figure 3:  
(i) Linear baseline correction of the measured sample gas and reference gas spectrum--see dotted green lines in section (a). (ii) Calculation of absorption from the corrected sample gas and reference gas spectra and Beer-Lambert's law, \(\alpha=ln(I_{r}^{*}/I_{s}^{*})/d\)--see section (b). (iii) Conversion of the absorption spectrum from time-domain to frequency-domain by a fourth-order polynomial--see blue curve in section (c). Note that the abscissae of time-domain and frequency-domain spectra are inverted as the laser experiences a frequency down chirp with pulse time. The coefficients of this polynomial are included as free parameters in the fitting routine. Initial guess values were obtained from comparison of CO\({}_{2}\) peak positions in the absorption spectrum to the HITRAN database. (iv) Fitting of the absorption spectrum using a Levenberg-Marquardt least-squares algorithm--see section (c) and the explanations below. The gas temperature and CO\({}_{2}\) concentration are included in the fitting procedure as a free parameter. The best-fit values  
Figure 3: Exemplary data and illustration of the algorithm for temperature measurements: (**a**) Time domain intensity spectra recorded on 5% CO\({}_{2}\) gas (blue) and a reference gas (N\({}_{2}\) gas, black), obtained at \(T\sim\) 293 K, (**b**) Time-domain absorption spectrum derived from the spectra in section (**a**), and (**c**) Frequency-domain absorption spectrum (blue) and best-fit to the HITRAN-based absorption model (red, see text). Residuals (yellow) are shifted and multiplied by a factor of ten. of these parameters represent the final value of the sensing technique, hereafter labeled as \(T_{\text{QCL}}\) (\"QCLAS temperature\") and \(c_{\text{CO}_{2}}\). The simulation of temperature-dependent absorption spectra in the fitting algorithm of step (iv) is based on Boltzmann statistics. The ratio \(R\) of the integrated absorption coefficients of two single absorption lines is then given by,  
\[R(T)=(\frac{S_{1}}{S_{2}})\text{exp}[-\frac{(E_{2}-E_{1})}{k}(\frac{1}{T_{0}}- \frac{1}{T})], \tag{1}\]  
where \(S_{j}\) and \(E_{j}\) are the spectral line intensity at a given temperature \(T_{0}\) and the lower-state energy of the transition \(j\), respectively, and have been taken from the HITRAN database [15]. Terms from stimulated emission have been neglected here, which for our purposes is well justified for temperatures below 400 K. For the fitting procedure applied to the experimental absorption spectra in step (iv), several simplifying assumptions were made: (a) the shape of single absorption lines is fully Lorentzian, (b) peak positions of CO\({}_{2}\) absorption lines are fixed to its HITRAN values at standard conditions for temperature and pressure [15], (c) the full width at half maximum (FWHM) of all CO\({}_{2}\) absorption lines is equal within the same class of transitions (fundamental or hot band transitions) and, thus, independent of the rotational states, and (d) only transitions with spectral line intensities \(S_{j}>10^{-21}\) cm/molecule and line positions in the range 2347-2353 cm\({}^{-1}\) are included in the fitting procedure. While these simplifications make the simulation and fitting process significantly more stable and efficient, induced systematic deviations are generally less than 0.2 K for all measurements discussed in the present work. Further, a parameter for an offset in the absorption spectrum was included in the fitting procedure in order to account for baseline variations in the experimental data.