## 3 THz Multi-Heterodyne Detection
Figure 3: Observation on the spectrum analyzer of a single intemodal beatnote (IBN) for a 580-mA driving current and 23.3 \(\Omega\) device resistance: (**a**) Span = 20 MHz, RBW = 100 Hz; (**b**) Span = 20 MHz, RBW = 50 Hz. RBW: Resolution bandwidth. These heterodyne beatnotes (HBNs) are particularly useful when the ratio between the mode spacings of the two parenting combs are close to an integer value, as illustrated in Figure 5. The HBNs are acquired on a spectrum analyzer (Tektronix, RSA5106A), and two sample spectra are presented in Figure 2. In particular, Figure 6a shows an acquisition with \(\mathrm{f_{rep}}\) chosen as exact submultiple of \(\mathrm{f_{IBN}}\). In this configuration, assuming the DFG-QCL behaves as a comb, the frequency differences between each QCL-comb mode and its closest neighboring OR-comb mode are exactly the same. As a consequence, in the down-converted RF spectrum, all the HBN have to collapse at the same frequency, which is exactly the case of Figure 6a, confirming the comb-like nature of the DFG device. In Figure 6b, \(\mathrm{f_{rep}}\) is slightly detuned, and we can visualize five HBNs corresponding to five THz modes, equally spaced in the frequency domain, with a 40 MHz span and a 10 Hz resolution bandwidth (RBW). These HBNs correspond to the most intense THz modes emitted by the QCL device around 2.4 THz, as shown in Figure 1b. The HBNs signal levels are between 5 and 10 dBm, and these signal-to-noise ratios do not allow a characterization of the level of coherence of the emitted comb, i.e., application of the Fourier analysis of comb emission (FACE) technique. Yet, the realized experimental setup and the retrieval of the HBNs allow to measure frequencies of all the QCL modes with a very high precision. In fact, if we take into account the frequency of the most intense QCL mode N (\(\mathrm{f_{N}}\)), corresponding to the most intense HBN shown in Figure 6, we can write down the Equation (1)  
\[\mathrm{f_{N}}=\mathrm{M}\cdot\mathrm{f_{rep}}\pm\mathrm{f_{HBN}} \tag{1}\]  
where \(\mathrm{f_{HBN}}\) is the frequency of the HBN signal between the Nth QCL mode and the Mth OR-comb mode. Figure 5: Illustration of the multi-heterodyne down-conversion process. Schematic representation of the quantum cascade laser (QCL-) (red) and optically rectified (OR-) frequency comb (FC) (blue), whose modes are respectively spaced by \(\mathrm{f_{IBN}}\) and \(\mathrm{f_{rep}}\). These two repetition frequencies are tuned close to an integer ratio, allowing an ordered and distinguishable down-conversion to radio frequencies (RF). In fact, the down-converted RF-FC modes (green) are equally spaced by \(\Delta\), and their easily measurable RF frequencies are used to calibrate the absolute frequency scale of the QCL-FC, as described in the main text. Reprint with permission from [36] Copyright Communications Physics. By fixing constant values for I\({}_{\text{QCL}}\) and R\({}_{\text{QCL}}\), the QCL mode frequency f\({}_{\text{M}}\) remains constant. Then, by modifying the OR-comb repetition rate f\({}_{\text{rep}}\) and tracking the f\({}_{\text{HBN}}\) frequency, from a simple linear regression we can extrapolate the order M of the OR-comb mode, as shown in Figure 7. The linear regression of this dataset (green line) results in a precise estimation of the order M. In fact, since the retrieved value is 9705.01 with a 0.33 standard deviation, and since M is integer, we can round to 9705 as mode number M of the OR-comb. Therefore, the exact M order is used for the f\({}_{\text{N}}\) QCL mode frequency determination. Indeed, in Equation (1), where the QCL mode frequency is calculated, the only remaining sources of uncertainties are on the values of f\({}_{\text{rep}}\) and f\({}_{\text{HBN}}\). The latter, with the 1.0 MHz linewidth observed in Figure 6b, being predominant. As a consequence, the QCL frequency is determined as 2416068.1(1.0) MHz.