## 3 Results and Discussion
The presented algorithm has a total number of ten free parameters, representing the coefficients of frequency calibration from step (iii) (5 parameters), the offset in absorption spectrum (1), CO\({}_{2}\) gas concentration \(c_{\text{CO}_{2}}\) (1), FWHMs of the fundamental and hot band transitions (2), and gas temperature \(T_{\text{QCL}}\) (1). The results of the fitting algorithm applied to the exemplary spectra from Figure 3a are presented by the red solid line in Figure 3c. Figure 4 compares the results of our QCL-based temperature sensing technique with the Pt1000 sensor data (\(T_{\text{Pt1000}}\)) obtained on a representative \"static\" heating circle with 5% CO\({}_{2}\) sample gas. Figure 4: Temperature obtained from the QCL technique and the Pt1000 sensors during a heating cycle of the gas cell. Inset: Correlation between the temperature measured by the QCL technique and the Pt1000 sensors and quadratic fit (black line). Evidently, the QCLAS temperature values deviate significantly from the values measured with the Pt1000 device. The inset in Figure 4 shows the correlation between \(T_{\text{QCL}}\) and \(T_{\text{Pt1000}}\). As can be seen, the two signals are almost linearly related with slight contributions of quadratic order (best fit: black solid line). Some aspects have to be considered in order to explain the discrepancy between \(T_{\text{QCL}}\) and \(T_{\text{Pt1000}}\): firstly, intrapulse QCL spectra are generally disturbed from their ideal behavior as a result of the instantaneous QCL laser line width as well as the limited electronic bandwidth of the detection system [17,18]. The magnitude of this effect is directly correlated with the down chirp rate of the QCL. Since the chirp rate decreases toward the end of the laser pulse, absorption spectra are less disturbed in the low frequency range. For our measurements, we indeed find that spectra are affected by the detector system response and thus \(T_{\text{QCL}}\) strongly depends on the choice of pulse section, where the fitting algorithm should be applied. Limiting the fitting section to a small range at the end of the pulse will generally result in a higher accuracy of \(T_{\text{QCL}}\), however, at the cost of statistical certainty. Secondly, uncertainties in spectroscopic parameters used to calculate the temperature from Equation (1) may lead to systematic errors in \(T_{\text{QCL}}\). The line-strength parameter \(S_{j}\) have a stated minimum uncertainty of 20%, whereas the uncertainty in the lower-state energies \(E_{j}\) is not given [15]. The correlation between \(T_{\text{QCL}}\) and \(T_{\text{Pt1000}}\) given in Figure 4 is, within the accuracy of our measurement, independent of the sample and day-to-day variances and thus can be used as an inherent calibration in order to correct for the above-mentioned systematic errors in \(T_{\text{QCL}}\). A further issue to be considered is non-uniform temperature fields in the gas cell. Given that the QCL spectra are line-of-sight measurements, \(T_{\text{QCL}}\) inherently represents an averaged temperature with contributions from the center and the boundary area of the gas cell, while \(T_{\text{Pt1000}}\) probes the temperature at a rather localized position close to its center with a slow response to temperature variations. The discrepancy between \(T_{\text{QCL}}\) and \(T_{\text{Pt1000}}\) is particularly strong for strongly non-static conditions, as seen for instance in the heat-up and cool-down processes from Figure 4. This finding supports the conclusion that temperature gradients in the gas cell contribute to offsets between \(T_{\text{QCL}}\) and \(T_{\text{Pt1000}}\). The \(T_{\text{QCL}}\) data are also subjected to low-frequency drift variations of \(<\)1 K (see, e.g., the time range 3800-4200 s in Figure 4).