Evidence of linear chirp in mid-infrared quantum cascade lasers: supplementary material

A 7.3 GHz RF tone (Rohde and Schwarz SMF 100A) is used as the local oscillator (LO) to downmix the amplified beatnote derived from the QCL, and the signal from the QWIP; these are then further downmixed and integrated on a fast (50 MS/s) lock-in amplifier (Zurich Instruments HF2LI), with an integration time corresponding to τ ≈ 0. 5λ0/v (v the mirror velocity and λ0 the central wavelength). A stabilised helium neon (HeNe) laser is also aligned to the interferometer (Brucker IFS 66/S), allowing the mirror displacement to be measured. This HeNe signal, the DC autocorrelation taken from the MCT, and lock-in signals are acquired on a 4-channel fast oscilloscope (LeCroy HDO6104). We generally oversample as far as the memory allows us, and then subsequently digitally filter the traces for a processing gain in SNR of a factor √ N for an oversampling factor N. Care is taken to keep the coaxial cable lengths for the X and Y channels the same, as well as to use the identical filters, so as not to induce any artificial phase shifts between the components. Synchronisation between the DC autocorrelation and RF autocorrelation traces is achieved by comparing the positions of the centreburst (zero path difference) in the DC autocorrelation and the smallest peak in |x(t) + iy(t)|2. As there is an AM component to the field oscillating at the repetition rate, as made clear by the existence of a measurable beatnote when light is shined directly onto the detector, such a peak must exist. Zero crossings of the HeNe signal are identified, and linear interpolation is used to improve the precision in their estimate. The 3 single-sided interferograms are then resampled accordingly using a cubic interpolator. We thus move from units of the scope’s internal LO t to the relative path delay τ. Being considered narrowband signals (fractional bandwidth < 100cm−1/1200cm−1 ≈ 8.4%), phase correction steps are deemed unnecessary. The interferograms are then apodised using the Mertz window [1], zero-padded, left circular shifted to the centreburst, and Fourier transformed, as is conventionally done for single-sided interferograms. A first estimate of the comb repetition rate is then found either by adding the lock-in demodulation frequency to the LO frequency used for the downconversion, which is usually selected to be below the fundamental beat tone of the QCL, or by measuring the delay in the interferogram from the centreburst to the first satellite. The estimate is then improved iteratively on the DC autocorrelation spectrum, which generally has a higher SNR, by choosing the values of ω̂0 and ω̂r which maximise ∑n I(ω̂0 + nω̂r). For stability, we made use of a low-noise current driver (Wavelength Electronics QCL2000) and thermoelectric cooler (Wavelength Electronics PTC 10 K-CH, with a standard Peltier element), in combination with an RF self-referencing scheme to mitigate drift. To verify the stability, we measured the deviation of the signal from the lock-in at a fixed path difference at rapid intervals over the course of more than 20 minutes. As shown from the Allan deviation[2] in Fig. S1 (a), there is no obvious drift in the signal over 100 seconds. The stability is confirmed in Fig. S1 (b), where superimposed are 13 amplitude and phase difference traces, for measurements taken sequentially at the same operating point with no power cycling in-between. As can be seen, they are very consistent and one can legitimately average.