Dual-comb thin-disk oscillator

Dual-comb spectroscopy (DCS) normally operates with two independent, relatively low power and actively synchronized laser sources. This hinders the wide adoption for practical implementations and frequency conversion into deep UV and VUV spectral ranges. Here, we report a fully passive, high power dual-comb laser based on thin-disk technology and its application to direct frequency comb spectroscopy. The peak power (1.2 MW) and the average power (15 W) of our Yb:YAG thin-disk dual-comb system are more than one-order-of-magnitude higher than in any previous systems. The scheme allows easy adjustment of the repetition frequency difference during operation. Both combs share all cavity components which leads to an excellent mutual stability. A time-domain signal recorded over 10 ms without any active stabilization was sufficient to resolve individual comb lines after Fourier transformation.


Data processing
In the following description, the dataset is a one-dimensional array containing sampling time t and signal values .
The 40 measured interferograms (bursts) are numbered by . The 312 ms long time trace consists of 1.95×10 8 data points separated by an interval of 1.6 ns. This corresponds to a sampling rate of 625 MSa/s. The trace is recorded with 14-bit vertical resolution (oscilloscope Keysight DSOS254A).

Burst arrival time correction
The optical spectrum was reconstructed from the temporal signal (the set of interferograms) with the help of the discrete Fourier transformation DFT (pyFFTW). DFT was applied to the whole dataset. However, additional numerical data processing (phase correction) is necessary to compensate for the fluctuations of the Δ rep or in other words, a timing jitter between the interferograms. This jitter was corrected numerically with the following method.
(Supplementary Fig. 1 shows burst arrival time correction results for 40 bursts.) Step 1: Assessment of the deviation. By finding the array index of the maximum of each burst idx peak ( ) (with SciPy's peak finding algorithm scipy.signal.find_peaks) in the time signal, the interferograms' arrival time can be measured with measured ( ) = (idx peak ( )). This value could be fitted linearly with the function fit . The slope  should be the inverse of the difference in repetition rate. Evaluating the fit function at the burst index yields the expected interferogram arrival time fit ( ) = fit (idx peak ( )).
Step 2: Calculating a correction factor. The ratio of expected arrival time (taken from the linear fit) to the measured arrival time serves as a multiplicative correction factor: However, this correction factor is only well defined at the maximum of each burst, because in between bursts no arrival time can be measured. Nevertheless, it must be applied to the entire time vector of the measurement, hence it is to be interpolated to the continuous correction factor . In this case, a simple univariate spline Step 3: Applying the correction. The time vector may now be corrected by multiplying it by the correction factor: corrected = ⋅ . However, the corrected time vector does not correspond to the signal values anymore. This must be accounted for by interpolating the signal values onto the new temporal vector with SciPy's interpolate function (scipy.interpolate.interp1d). Here, the linear interpolation algorithm was used. Note that the corrected time vector is not evenly spaced anymore. To find the frequency vector after the DFT, the temporal spacing of the uncorrected time vector must be used. This is correct, since it corresponds to the sampling rate of the DAQ system.

Evaluation of the time correction.
To assess the performance of this correction algorithm, one may compare the deviation between the linear fit and the actual burst arrival time before and after the correction in Supplementary   Fig 1. Initially, the absolute deviations between expected and measured burst are 11.1 μs peak to peak. After the correction algorithm was applied, these deviations are suppressed by a factor of 4270 down to 2.6 ns peak to peak.
Note that the data were sampled in 1.6 ns intervals, so the remaining fluctuations correspond to two sampling intervals. The 11.1 μs peak-to-peak fluctuation corresponds to a relative fluctuation of 1.4×10 -3 , taking into account the 7.8 ms separation between interferograms. This, in turn, corresponds to approximately 0.18 Hz peak to peak or 59 mHz standard deviation when the Δ rep is tuned to 128 Hz.

Apodization
Significant satellite interferograms corresponding to side-pulses in the mode-locked trace can be observed, see Supplementary. Fig 11. In addition, there is significant noise between bursts. It has a purely electronic origin and may be filtered out. Apodization is a well-known technique used in such situations. For apodization, a supergaussian local filter function of order and width at position n with amplitude one was used: In the presented measurements, the filter was parameterized with = 6, = 2 . The location of each burst n was found numerically by SciPy's peak finding algorithm. If more than one interferogram is in the data set, the sum of all local filter functions is used: The corrected signal is simply apodized = total ( ) ⋅ raw . The multiplication here is carried out element-wise. Supplementary Fig. 4 Interferogram apodization. Example of the apodization process showing the original data (blue), filter data (orange), and the local filter function (green). Note that the filter amplitude was scaled up to 6000 to be visible in the graph.

Acetylene spectroscopy measurement setup
We have chosen acetylene gas for a "real-life" test of our thin-disk dual-comb spectrometer. The laser spectrum was shaped with the help of a bandpass filter (central wavelength 1030 nm, FWHM 10 nm) and a slightly tilted longpass filter (Thorlabs FELH1050). In our setup, the tilt was set to spectrally select the laser output around 1034nm.

Performance of the data correction
This section summarizes how the data evaluation is influenced by both correction methods, arrival time correction and apodization. All temporal signals are zero-padded to double their length to interpolate the spectrum. applied. Note that the signal-to-noise ratio (SNR) of 25.9 is relatively low, and the spectral shape is substantially buried in the noise floor. SNR is defined as a ratio of the spectral peak magnitude to the background signal mean value. The mean value of the background signal is calculated in 1-3.5 MHz range in RF domain or 238.8-285 THz in optical domain. In Supplementary Fig. 7, the arrival time correction algorithm was applied but without the apodization. A two-fold improvement in SNR time corr. to 55 and a clear agreement of the reference spectrum and the dual-comb spectrum can be observed.
Finally, applying both apodization and arrival time correction yields Supplementary Fig. 8. Here, the background noise is practically eliminated from the measurement, and the SNR time corr.−apod. is improved by a factor of almost 56 to 3112. The spectral shapes of reference measurement and dual-comb measurement clearly agree. Even the weak spectral Kelly sidebands at around 285.5 and 296.5 THz, initially buried in the noise, appear after applying the time correction and apodization. Supplementary Fig. 9 displays SNR versus the number of bursts for timing jittercorrected and non-corrected measurements. Signals have been apodized in both cases. SNR scales with √n bursts, which is expected for uncorrelated noise.

Central frequency drift.
A drift in the center frequency of a spectrum corresponding to an individual interferogram is directly related to a drift in the difference in the carrier-envelope-offset frequencies between the two frequency combs. To assess this quantity, the DFTs of 4 sets of 10 individual bursts were analyzed. The sets are disjointed, i.e. the first set contains the interferograms 1 through 10, the second contains the interferograms 11 through 20, and so on. The raw data containing eleven interferograms is 80 µs long. To distinguish the location of the comb lines, the data were symmetrically zero-padded to a length of 0.93 s. A total of 40 interferograms were considered in the evaluation.
Arrival time correction and apodization were applied. Supplementary Fig. 10 a) shows the raw data of the first set of 10 bursts. Supplementary Fig. 10 b) shows the DFT spectra in the RF domain. Supplementary Fig. 10 c) is a zoomed-in view of the comb line centered at 16.686 MHz and its neighbors. In b) and c), the orange markers highlight the location of the comb line's peak. Supplementary Fig. 10 we can calculate the relative CEP noise Δ 0 to be12 mrad.

Tuning the repetition rate difference
Initially, we were interested in the detuning range 100 Hz-1 kHz particularly relevant for the dual-comb spectroscopy. For very narrow spectra, higher Δ rep values could be of interest, so we experimentally verified that our dual-comb laser source can easily be tuned within this range and can be detuned even up to Δ rep = 1 MHz at rep = 60 MHz. Practically there is no limit for the lower detuning value. However, there can be a limit for the upper detuning value. There are two possible reasons for it. 1. The detuning of one of the cavities changes the roundtrip time and, thus the pulse train repetition rate. This, in turn, influences the pulse energy p = rep ⋅ avg.
Since all optical elements are shared in the laser cavities (Kerr medium, separation between the focusing curved mirrors), Kerr-lens mode-locking is optimized for only a certain pulse energies range. Considering our previous experiments, we can estimate that a 5 % energy decrease (or increase) will lead to instabilities in one of the arms.
This 5 % change in the cavity length corresponds to approximately 3 MHz maximal acceptable detuning. 2. The detuning of one of the cavity arms changes the stability zone of this cavity. As Kerr-lens mode-locking is quite sensitive to the cavity position in the stability zone, this will manifest itself in a decreased pulse energy of the detuned cavity up to the point where this cavity will cease mode-locking. These limits depend strongly on the specific cavity configuration.

Analysis of temporal side pulses
Satellite pulses are a well-known phenomenon in mode-locked lasers. In DCS they can lead to satellite interferograms which might disturb spectroscopic measurements. In our thin disk oscillator satellite pulses were present and measured with a a autocorrelator (APE pulseCheck). In both, the top and bottom cavity, satellite pulses appear with around 4 picoseconds temporal distance to the main pulse which is displayed in Supplementary Fig. 11a). The main pulse measured to be of 178fs duration (FHWM, sech² fit) for our top cavitythe bottom cavity shows similar behaviour. These pulses lead to satellite interferograms as displayed in Fig. 11b), taken from a measurement set with Δ rep = 188 Hz and Δ rep = 61.1 MHz.
The temporal position sat of in the downconverted satellite interferogram with respect to the main interferogram can be estimated with the down-conversion factor a and the real side-pulse difference in timing 4 ps: Fortunately, with careful adjustment of oscillator parameters such as mirror dispersion and pump power, it was possible to suppress the side-pulses for the presented measurement set. In addition to the two satellite interferograms shown in Supplementary Fig. 11b), the side pulses beat with themselves and create weak satellite interferograms 3 and 4 which in our case were probably burried in noise.