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1 1 Supplementary Figures a) f rep,1 Δf f rep,2 = f rep,1 +Δf RF Domain Optical Domain b) Aliasing region Supplementary Figure 1. Multi-heterdoyne beat note of two slightly shifted frequency combs. a Case where the one-to-one mapping condition is verified b Case where the one-to-one mapping condition is not verified. The aliasing effect is highlighted on the RF domain.

2 2 a).1 V(t) (a.u.) Slice 1 * Slice 2 * Slice N Slice 3 * * b) d) Without Sample With Sample Transmission Voltage (db scale) c) Time (s) τ/e Frequency (MHz) Frequency (MHz) Supplementary Figure 2. Description of the data analysis performed on the time domain signal generated by a dual-comb spectrometer based on QCL combs. a Typical time domain interferogram. The time domain signal is divided in Nacq slices. An apodization is performed on each small time domain signal before performing a Fourier transformation. b Spectrogram of the Nacq multi-heterodyne beat spectra. Left: Case where the spectra are not aligned. Right: The Nacq are aligned to the first multi-heterodyne spectra. c Typical multi-heterodyne beat notes averaged over an entire acquisition after aligning all the spectra. This data corresponds to one step of the acquisition of the transmission of the low finesse GaAs etalon. d Transmission spectrum of the investigated sample (low-finesse GaAs sample) on the RF domain.

3 3 Low power peak high power peak time (s) Low power peak high power peak -2 σ y(τ) time (ms) Supplementary Figure 3. Amplitude oscillations of the amplitude of a multi-heterodyne beat note when only photon noise is taken in consideration. The simulations parameters are: P1 = P2 = 5 mw, N = 6 modes, η =.7, f = 7 MHz and R = 68.7 the ratio between the two beat note amplitude voltages. a Time domain evolution of both beat note amplitudes. b Allan deviation of the amplitude oscillations of both beat notes.

4 4 a) Software control BS SA Gas cell Temperature control ref. signal.1 V(t) (a.u.) Sample Comb Time (μs) Oscilloscope 5 LO Comb sub-step 1: sample transmission measurement LO comb Sample comb Detector bandwidth Δ Δ sub-step 2: Sample comb tunned c) Total detunning (GHz) b) sample signal Transmission sub-step 3: LO comb tunned Δ Transmission d) Δ 1 FSR Number of steps 1 12 Measured detuning of sample comb Measurement 1 Measurement 2 Reconstructed sample spectrum. Supplementary Figure 4. Method for interleaving several transmission spectra acquired by tunning the QCL combs in a dualcomb spectroscopy set up. a Schematic view of the set-up. A home-made software controls both the temperature tuning of both QCL combs as well as the acquisition made by the oscilloscope. SA: spectrum analyser. BS: beam splitter. LO: local oscillator. b Representation the 3 sub-steps in order to measure a single transmission spectrum Ti in the RF domain and the corresponding frequency shift i. c Total detuning of the sample comb measured with the oscilloscope. d Representation of the construction of the interleaved spectrum T based on the knowledge of Ti and i.

5 5 Supplementary Note 1 The beating of two combs creates a plethora of down-mixed beat notes in the multi-hetedoyne beat signal. The purpose of this appendix is to give as a reminder (see for example [1]) the conditions on the comb repetition frequencies and offsets frequencies in order to obtain a one-to-one mapping between the optical comb lines and the down-mixed comb lines, necessary to achieve the link between the optical and RF domain. Let s assume that the comb lines of the two combs are described by: f n,1 = f ceo,1 + nf rep f k,2 = f ceo,2 + k(f rep + f) (1) where n and k are integers, f rep,1 is the repetition frequency, f is the difference in repetition frequency and f ceo,1 (resp. f ceo,2 ) is the carier-enveloppe offset frequency of the comb 1 (resp. comb 2). We also assume that the comb 1 (resp. comb 2) has N 1 (resp. N 2 ) lines and that f ceo,1 f ceo,2. On the most general case, the line f n,1 will create N 2 down mixed lines when beating with the N 2 available lines of comb 2. These beat notes are at the frequency difference between the two lines, i.e.: f beat = f k,2 f n,1 = f ceo,2 + kf rep,2 (f ceo,1 + nf rep,1 ) = f ceo + (k n)f rep,1 + k f n {1...N 1 }, k {1...N 2 } (2) Expression (2) represents the most general expression of the multi-heterodyne beat. The one-to-one mapping is achieved when n = k and its represented schematically on Supplementary Figure 1a. However, an aliasing effect can occur between the lines k and n = k + 1, as represented schematically on Supplementary Figure 1b. In fact, we can retrieve the most general one-to-one map condition by using expression (2). In the most general case, the aliasing will be observed if the beating between k = N = min(n 1, N 2 ) and n = N + 1 is at lower frequencies that the beating between k = n = min(n 1, N 2 ). Mathematically, this corresponds to the condition: with f ceo = f ceo,2 f ceo,1, which gives finally: f ceo + N f f ceo + f rep N f (3) f ceo + N f f rep 2 (4) Expression (4) represents the most general expression for the one-to-one mapping condition between the multiheterdoyne beat of two combs and the combs optical frequencies. Supplementary Note 2 In this section we derive the expression of the signal-to-noise ratio (SNR) of the amplitude of the beat note generated by the beating of two combs. We suppose that the system is photon-noise limited and that the power of the combs is equally distributed among the N comb lines. The electrical power generated by one of the beatings of the multi-heterodyne beat is given by: ( eη ) 2Pline,1 P e = 2R P line,2 = 2 R ( eη ) 2Ptot,1 h N 2 P tot,2 (5) h where we supposed that the power of the combs is equally distributed among the N comb lines, P tot = NP line for each comb. We should note here that the signal is proportional to the product of the power of each comb line. However, the variance of the current induced by the photon shot noise is proportional to the sum of the total power on the detector, expressed by: ( e σ 2 2 η ) = 2R (P tot,1 + P tot,2 ) f (6) h

6 6 which gives the SNR of the multi-heterodyne beat: SNR combs = η P 1 P 2 N 2 (7) h f P 1 + P 2 Now we give more details about the amplitude oscillations of a multi-heterodyne beat note in the case where only photon noise of both lasers is taken into account. We generate a time domain signal with a normal distributed noise, where its mean value is estimated by using equation 5 and its variance by using equation 6. The simulated time domain signal is represented in Supplementary Figure 3. We assume that the power is equally distributed over the modes, as described in the previous section. However, as it is shown in the main text, the beat notes amplitudes can vary significantly. In order to take this into account, we measured the ratio between the amplitudes of the two peaks (number 11 and number 24), and we used this ratio in order to simulate the two relative amplitude oscillations. Finally, we computed the Allan deviation of this simulated time domain signal. Supplementary Methods This sections describes in detail the algorithm allowing the calculation of the multi-heterodyne beat spectrum as well as the transmission of the measured sample. The time domain signal corresponding to the multi-heterodyne beat of the two combs is acquired with the full sampling rate (2.5 GSample/s) over the entire memory of the oscilloscope (1 MSamples with the L memory option, 25 MSamples with the XL memory option). A typical time domain signal is showed in Supplementary Figure 2a, acquired with 5 MSamples (2 ms total acquisition time). The entire sample is splitted into N acq small time domain signals of time τ acq (called the single acquisition time in the main text), as shown in Supplementary Figure 2a. The value of τ acq = 3.2µs is set by knowing the frequency resolution needed to resolve all the multi-heterodyne beat notes and must also satisfy the condition for coherent accumulation of the signal explained in the main text. Each of this N acq time domain signals are then apodized. A Fast Fourier Transform (FFT) algorithm is then used to compute the N acq amplitude spectra corresponding to the non-averaged multi-heterodyne spectra. During the acquisition (tens of ms), the heterodyne beat signal drifts by less than 2 khz. We apply a small frequency shift to all the acquired spectra in order to align all the spectra in respect to the first one. Supplementary Figure 2b shows to spectrogram of the multi-heterodyne spectra when they are not aligned (left) and after alignment (right). After the alignment of the spectra, we average all the spectra to get a single multi-heterodyne spectrum averaged over the total acquisition time. A zoom of a typical spectrum is represented on Supplementary Figure 2c, where we show the averaged multiheterodyne spectrum when the sample comb is going through a sample (red) as well as when there is no sample (blue). Finally, the algorithm detects the amplitude value of each multi-heterodyne beat note. A threshold value can be set to detect the multi-heterodyne beat notes which amplitude values are higher then this threshold. The transmission of sample in the RF domain is then computed as it is related to the voltage generated by a heterodyne beat on the detector by: ( ) 2 V S ( k ) T ( k ) = (8) V NS ( k ) where T ( k ) is the transmission at the frequency ( k ), V S ( k ) (resp. V NS ( k )) is the voltage generated by a multiheterodyne beat at the frequency ( k ) measured with the sample signal (resp. the reference signal). Now we describe in detail the method used for tuning both QCL combs for interleaving several transmission spectra and therefore achieving high resolution spectroscopy. We remind that the spacing between the comb lines are usually between 4 and 2 GHz, determined by the cavity length of the devices. For achieving high resolution spectroscopy (typically 1 MHz), both QCL combs are swept by at least one FSR by changing their temperatures in N step steps. Supplementary Figure 4a shows the set-up used for achieving high resolution spectroscopy. A home-made software controls the temperature of the QCL combs by acting on the temperature controller of each device. The same software sets the beginning of the acquisition by triggering the high-bandwidth oscilloscope, to which the two detectors are connected. The oscilloscope measures the time domain signal created by the multi-heterodyne beat. A RF spectrum analyser is also used for controlling the position of the multi-heterodyne beat during the sweep. At each step i (i {1...N step }), we measure the transmission spectrum T i and the corresponding frequency shift i. For measuring these two quantities, each step i is divided into 3 sub-steps, as shown schematically in Supplementary Figure 4b. The first sub-step consists of acquiring the multi-heterodyne signal with the two detectors, as described in Supplementary Figure 4b. The multi-heterodyne signals are acquired synchronously as the two channels of the oscilloscope

7 are sharing the same trigger. These two time domain signals are then treated as described in the methods in order to compute the peak intensities. At the end of the first sub-step, the transmission on the RF domain T i is obtained. No absolute frequency reference is known at this point. The second sub-step consists of tuning the QCL comb going through the sample (sample comb). The temperature controller of the Peltier cooler is used to reduce the temperature of the sample comb, moving the sample comb to higher frequencies. The impact of this tuning on the RF domain can either be a shift of the multi-heterodyne beat spectrum towards the lower or higher frequencies. This effect, originating from the fact that we do not know the absolute position of the combs, can be solved by measuring the comb line spacings of both QCL combs at the beginning of the sweep. In fact, by knowing the relative values of the comb line spacings and by observing to which direction the multi-heterodyne beat spectrum shifts, we can know the relative position of each comb. In our case, we check that the multi-heterodyne beat spectrum shifts towards the higher frequencies in the RF-domain when the comb line spacing of the sample comb is higher than the LO comb line spacing. We stress that this verification is only done at the initial step and is valid for all the subsequent steps. This corresponds to the second schematic represented in 4b. At the end of the second sub-step, the multi-heterodyne signals are again acquired synchronously. By comparing the multi-heterodyne beat spectrum at the end of sub-steps 1 and 2, we measure the value of the frequency shift i for the step i. The frequency shift i is measured by determining the frequency position of the highest-amplitude heterodyne beat note for the sub-steps 1 and 2 with the oscilloscope, and computing their frequency difference. The accuracy of this measurement is given by the inverse of the acquisition time, typically hundreds of khz. At this point, the LO comb was still not tuned and remains at its original position. The third sub-step consists of tuning the LO comb by also reducing its temperature, as represented in Supplementary Figure 4b. This has also the effect of moving the LO comb towards the higher frequencies in the optical domain. As a result, the multi-heterodyne beat spectrum shifts towards the lower frequencies. We stress that the LO comb does not need to be shifted by the exact amount i as the sample comb was shifted. The important point is that the frequency shift i is measured only when the sample comb is tuned. This is possible because only one comb interrogates the sample. At the end of the third sub-step, we have acquired a transmission spectra T i in the RF domain and measured the value of the corresponding frequency shift i. These steps are then repeated in order to sweep the two combs over the value of one FSR or more. In our case, we have f rep 7.5 GHz and we set i to 8 MHz. At the end of the sweep, we obtain N step transmission spectra T i in the RF domain as well as N step values for frequency shifts i, as represented in 2d. At this point, all the transmission spectra T i are interleaved step by step, by applying the frequency shift i to the transmission spectrum T i+1, as explained schematically in Supplementary Figure 4c, resulting in a single interleaved spectrum T. Finally, a total frequency offset is added to the interleaved spectrum T in order to achieve the wavenumber scale calibration, as the absolute value of the combs is still not known. This can be done by knowing the frequency value of a reference absorption line. In our case, this absolute scale was determined by fitting HITRAN simulation to the interleaved transmission T, where the fitting parameter is the value of the offset frequency. We also used this value to perform the wavenumber scale calibration in the transmission measurement of the low-finesse GaAs etalon. As discussed on the main text, the QCL combs are not actively stabilised during the transmission measurement. As the software controlling the acquisition waits until the temperature of each QCL comb is stable before performing an acquisition, long term drifts are presented in the set-up. More precisely, the drifts occurring to one QCL comb while waiting for the temperature stabilisation of the other comb are typically around 1 MHz. As a consequence, the interleaved transmission spectrum is affected, and a significant technical noise is added to the noise measured on a single acquisition (described in the main text). Therefore, we apply a moving average filter to smooth the interleaved transmission spectrum to reduce the impact of these drifts. The expression of a moving average filter is given by: 7 T ( k ) = 1 M i= M 2 i= M 2 T ( k+i ) (9) where T ( k ) is the value of the interleaved transmission at the frequency k (k {1...N step N peaks }, where N peaks is the number of detected peaks) and M is the length of the filter. Typical values for an interleaved transmission spectrum are N step 1 and N peaks 5. The filter length was set to M = 11 during the water transmission measurement, and no moving average filter was used (M = 1) for the low-finesse etalon transmission measurement. As M << N step N peaks, the filter acts as low pass filter only removing the effect of the drifts previously described. At the same time, the frequency resolution is decreased. An estimation of the frequency resolution after the moving average filter is M v. The limitations created by this drift on the interleaved transmission spectrum would be significantly reduced if these drifts would be compensated, as it was done for the measurement of the equidistance of

8 8 the comb spacing. [1] S. Schiller, Spectrometry with frequency combs, Optics Letters, vol. 27, no. 9, (22).

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