## Introduction

A Fourier-transform spectrometer1 measures the time-domain interference between two waves while their relative phase-shift is varied. The Fourier transform of the interference pattern provides the spectrum. Advanced implementations use lasers as light sources. Their increased brightness compared to incoherent sources leads to a shorter measurement time or an increased signal-to-noise ratio (SNR)2. Even further refined systems incorporate optical frequency combs (OFC) as light sources. This enhances the spectral resolution, which is ultimately limited by the OFCs individual comb linewidth; improves sensitivity3; and reduces the effect of instrument line shape on the measurement2. Experiments comprising two OFCs, known as dual-comb spectroscopy (DCS), emulate the mechanically scanned delay in Fourier-transform spectroscopy by two separate pulse trains with slightly different repetition frequencies (asynchronous pulse trains). DCS is rapidly advancing since it was first proposed4 and demonstrated in a Ti:Sapphire-driven mid-infrared system5. The key reason is the simple and fast electronic detection of the interferogram generated by two heterodyned asynchronous pulse trains on a photodetector or by means of electro-optic sampling6. Advantages arise in terms of acquisition speed, spectral resolution, accuracy, robustness, and SNR over traditional scanning Fourier-transform spectroscopy. The principle is extendable to nearly all related spectroscopy methods, and several fundamental experimental demonstrations have been published. They included Doppler-free spectroscopy7, anti-Stokes Raman-spectroscopy8, and two-photon spectroscopy9, as well as scattering-type scanning near-field optical microscopy10,11. DCS applications include trace gas detection, lidars, and kinetics of chemical reactions, as reviewed in 2 and 12.

One major challenge with DCS is implementing an interferometer with two combs. They must be mutually coherent and operate at different repetition rates12, whilst spanning the necessary bandwidth in the examined spectral region. One promising way of realizing such a light source is by electronically modulating a continuous wave laser with two intensity modulators at different frequencies13. More commonly though, the two OFCs are generated by separate mode-locked femtosecond oscillators running at slightly offset repetition rates. This involves the complex active stabilization of repetition rates and carrier-envelope offset frequencies or their active tracking and a posteriori correction. To date, erbium fiber lasers14 show the best performance, based on active stabilization or a posteriori correction schemes.

An emerging way is to generate a dual-comb output from a single laser cavity. This could simplify the technique, as many noise sources are expected to be shared by both pulse trains. This alleviates the need for active stabilization over short time periods. So far, there have been only a few demonstrations of such a scheme. They rely on different effects to generate a repetition frequency offset between the two outputs. For example, polarization multiplexing in MIXSELs15, in bulk oscillators16 and chromatic dispersion in a dual-wavelength fiber laser system17 have been demonstrated. In bulk solid-state lasers, the directional dependence of the Kerr nonlinearity18,19 was exploited. A selection of passive dual-comb lasers is given in Supplementary Table 1. It compares average and peak powers, repetition rates, differences in repetition rates, pulse durations, spectral spans, central wavelengths and types of laser technology. Average and peak powers are particularly important for spectral broadening, pulse compression and spectral extension into the THz, infrared (IR), ultraviolet (UV) and vacuum ultraviolet (VUV) spectral regions where most of the strong and spectroscopically relevant transitions are situated. The conversion efficiency, which is intrinsically low for these nonlinear processes, must be compensated by a high photon flux driving laser.

In this work, we present a thin-disk based dual-comb femtosecond oscillator. Thanks to the thin-disk platform20, the presented system shows more than one order of magnitude higher output average and peak powers compared to previous dual-comb demonstrations. No noise is added by external optically active media, since the thin-disk laser provides enough power without subsequent amplification. The scheme effectively rejects common noise without active stabilization. We report a broad tuning range of the difference of the repetition frequencies $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ during mode-locked laser operation by simply adjusting the relative longitudinal offset of the end mirrors. In contrast to optical systems based on a single cavity in Supplementary Table 1, this allows optimal and flexible choice of the important parameter $$\Delta {f}_{{{{{\rm{rep}}}}}}$$.

Given its output parameters, it represents a relatively simple and compact table-top source for DCS, particularly promising for further nonlinear conversion into the IR, UV and VUV spectral regions and realization of different nonlinear spectroscopy schemes.

## Results

The experimental setup (Fig. 1) is a high-power Kerr-lens mode-locked (KLM) Yb:YAG thin-disk oscillator pumped by a 500 W-class fiber-coupled diode laser. It features a thin-disk module consisting of the disk and the pump cavity made by TRUMPF and high-power optimized dispersive mirrors designed and manufactured by Dr V. Pervak and his team. The dual layout of this thin-disk laser contains two nearly identical laser resonators, which share all cavity mirrors, including the active area on the thin-disk (Fig. 1c, pump spot). An exception to this principle constitutes the end mirrors and output couplers. They are split horizontally to allow the alignment of the two cavities independently21. The spatial separation is roughly 5 mm in the vertical (sagittal) plane (Fig. 1a, separation drawn horizontally (tangentially) for simplified visualization). The spatial separation allows individual fine tuning of the repetition frequencies. Their difference $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ is a key parameter in any DCS systems. It determines the dead-time between consecutive interferograms and the line-spacing of the down-sampled RF comb22. In the presented arrangement, $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ can be easily tuned, up to the kilohertz range, by longitudinally translating one of the split output couplers. The tuning of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ up to 1 MHz was experimentally verified. This simple tunability of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ can allow the implementation of fast scanning methods, such as those that rely on the inversion of the sign of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$, demonstrated in23. This is a major advantage over DCS systems that rely on different methods to introduce $$\Delta {f}_{{{{{\rm{rep}}}}}}$$, i.e., via birefringence24 or chromatic dispersion17. Except for25,26, those systems are quite limited in tunability and their tunability range (see Supplementary Table 1 for more information), because changes involve rearranging or realigning the cavities as opposed to simply translating a mirror.

The configuration of an individual cavity, without the stacked dual-cavity feature, was inspired by Brons et al.27. A telescope comprising two concave mirrors with a radius of curvature of 300 mm focuses both cavity modes into two distinct foci within a Kerr medium composed of a 4 mm sapphire plate placed under Brewster’s angle. Radial restriction of the modes by hard apertures and soft aperture effects in the gain medium combined with the self-focusing in the sapphire plate enables KLM28. Despite the vertical displacement of the laser beams, it is possible to use a single sapphire plate to mode-lock both cavities. Similar to conventional KLM thin-disk oscillators, both cavities can be mode-locked at the same time by manually perturbing one of the telescope mirrors. To achieve soliton mode-locking in the net-negative dispersion regime, two highly dispersive mirrors, compensating a Group Delay Dispersion of approximately −3000 fs2 per reflection, resulting in −12,000 fs2 total intra-cavity dispersion per round trip, are placed inside the resonators.

The common-path cavity design ensures temporal stability of the dual-output system as well as a reasonable mutual coherence time between the two frequency combs in the absence of any active stabilization mechanisms. The two cavities share nearly the same intra-cavity optical components and the same pump source. Thus, they experience nearly equal perturbations originating from air flows, mechanical vibrations, and the intensity noise of the pump source. However, the system can be somewhat more sensitive to the drifts and vibrations associated with the displacement of the beams in a plane parallel to the optical table.

### Output characteristics of the dual-comb system

The spectra of both pulse trains, measured with an optical spectrum analyzer (OSA, Yokogawa), are compared in Fig. 2. They show almost identical spectral shapes and widths, indicating almost identical output pulses. Both spectra are centered at 1030 nm, and they have slightly different average output powers of 16 and 15 W, which corresponds to 265 and 250 nJ per pulse, respectively. This yields a combined optical-to-optical efficiency of 12% at the 270 W pump power. The repetition frequency $${f}_{{{{{\rm{rep}}}}}}$$ was set to 60.4 MHz. The sech2-shaped output pulses have 176 and 186 fs FWHM pulse durations, measured with a commercial autocorrelator (APE). The peak powers are calculated to be around 1.32 and 1.2 MW, respectively. To the best of our knowledge, the achieved peak power exceeds the currently reported highest peak power from a single laser dual-comb system of 0.03 MW18 by approximately 40 times. It is also more than 18 times larger than the currently reported highest peak power of 0.07 MW from a dual-comb CPA system29. The small differences between the two outputs can arise from the fact that the optical axes are not identical resulting in different angles of incidence on the telescope mirrors and Kerr medium. This can be mitigated easily by attenuating the more powerful beam to match the lower power one. The different beam positions on the thin-disk can also result in a slight gain difference between the two lasing modes. Unequal amounts of defects on the mirrors, as well as a mismatch in the size of the hard apertures, also can create differences in linear losses.

The time evolution of the two repetition frequencies and their difference is shown in Fig. 3. Although both repetition frequencies decrease monotonically with time, their difference $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuates only weakly with a standard deviation of 0.1 Hz on a minute time-scale. This may suggest that $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ is also fluctuating with a similar standard deviation on a millisecond time scale, the typical acquisition times for fast DCS experiments. Careful analysis of the DCS time trace (see Fig. 4a) shows that $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuations (see Supplementary information for details) are non-negligible, and it confirms the standard deviation of approximately 0.1 Hz. The $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuations are happening on the time scale 10–100 ms (see Supplementary Fig. 1). The presented stability level (in Fig. 3) is comparable to previous publications17,18,30,31, which is remarkable considering the at least one order of magnitude higher output average powers and intra-cavity power of approximately 150 W. Further active stabilization of the difference in repetition frequencies should not be required, but it could be implemented for more challenging experiments. A posteriori numerical correction of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuations is desirable, and it significantly improves the results as shown further in text. Furthermore, the pulse energy is over one order of magnitude higher than the 7.7 nJ used in the CPA system in29 and it far exceeds the energies delivered by state-of-the-art single laser systems. In future versions of the presented system, the pulse energy could be scaled up easily as described by Brons et al.20 by increasing the focal length of the cavities curved mirrors. For example, a set of −600 mm mirrors instead of −300 mm mirrors should yield a twofold increase.

### Dual-comb measurements

By combining two laser beams on a photodiode (see setup in Fig. 1b), interferograms are recorded in the time domain. Fig. 4a shows a 100 ms trace with 11 consecutive interferograms. $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ was adjusted to 128 Hz, which corresponds to 7.8 ms temporal separation between consecutive interferograms and a down-conversion factor of $$a=\frac{\varDelta f{{{{\rm{rep}}}}}}{f{{{{\rm{rep}}}}}}=2.12* {10}^{-6}$$22. The optical spectrum (Fig. 4c) can be obtained by taking a Fourier transform of an individual interferogram centered in a 20-µs time-window (Fig. 4b) and scaling it appropriately by the down-conversion factor. No sample was used in this measurement. The spectrum agrees well with a reference spectrum recorded by an OSA (Yokogawa). Characteristic Kelly sidebands of our oscillators at 1012 and 1052 nm are clearly visible in Fig. 4c, which confirms spectral accuracy of our DCS laser source for spectral spans as large as 40 nm.

To investigate fluctuations of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$, the arrival time of the interferograms and their jitter are evaluated in time domain (see Supplementary information). This analysis showed that the correcting the jitter of the repetition rate difference (or burst arrival time) is crucial. However, we still assume that $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ remains constant on the microsecond timescale and therefore cannot affect the width of the spectra during measurement. This may indicate that the dominating instabilities are due to fluctuations in the repetition rate difference ($$\Delta {f}_{{{{{\rm{rep}}}}}}$$). They can be compensated numerically using a simple post-processing technique (see Supplementary information for details).

To resolve the individual comb lines, the measurement time must be increased to yield enough data to enhance frequency resolution. Furthermore, the increase in the acquisition time will improve the SNR if the two frequency combs stay coherent. After correcting the interferograms, which only involves corrections of the difference in repetition frequencies, the SNR scales as $$\sim \sqrt{T}$$ where $$T$$ is the measurement time (see Supplementary information).

Figure 5a depicts optical spectra calculated from a trace containing 40 interferograms. A comparison between the spectra with apodization and the spectra without any correction shows that the $${{{{{{\rm{SNR}}}}}}}_{{{{{{\rm{apod}}}}}}.}$$ = 1417 for the apodized case is significantly improved compared to the spectra without any correction with $${{{{{\rm{SNR}}}}}}$$ = 25. The huge increase in SNR and cleaning up of the spectrum is due to the relatively narrow window (2 µs) applied during apodization.

Figure 5b depicts optical spectra with the $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ correction technique noted above applied, including apodization and the spectra without any correction. This case shows $${{{{{{\rm{SNR}}}}}}}_{{{{{{\rm{time\; corr}}}}}}.-{{{{{\rm{apod}}}}}}.}$$ = 3112 compared to $${{{{{\rm{SNR}}}}}}$$ = 25 of the non-corrected case while the good agreement with the reference measurement remains unchanged. The inset of Fig. 5b shows that the spectral resolution is already sufficiently high to resolve the comb structure of the optical spectrum.

The 312 ms time trace with 40 consecutive interferograms was analyzed after applying the $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ correction. The analysis showed that the FWHM of a single comb line (Fig. 6) of the spectrum is 3.1 Hz in the RF domain. Interleaving multiple measurements can bring the spectral resolution close to the width of the comb line in the optical domain32. Moreover, using the aforementioned $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ correction technique on arbitrarily long measurements containing a vast number of interferograms can improve the SNR even further. With minor corrections to the $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ only, a time-domain trace of 312 ms can be transformed without artifacts, increasing a posteriori the effective coherence time by simple numerical means33.

In addition, we performed a spectroscopic measurement of the transmission of a home-built Fabry–Pérot etalon with a free spectral range of 140 GHz and an FWHM-linewidth of 24 GHz. The reference spectrum (OSA) and the measured (retrieved) spectrum are shown in Fig. 7. The retrieved spectrum was acquired from a single 80 µs interferogram time-trace. The characteristic Fabry–Pérot transmission lines are clearly visible and both measurements agree well within a reasonable deviation. Thus, the single 80 µs interferogram time-trace leads to a spectral resolution of at least 24 GHz.

Finally, to verify the real-life applicability of our dual-comb spectrometer, we performed spectroscopic measurements on acetylene (C2H2). Acetylene is a well-studied molecule having overtone absorption around 1034 nm. It has been measured in similar DCS experiments multiple times34. Our initial laser spectrum was long-pass filtered to have optimal overlap with the acetylene absorption spectrum around 1034 nm. Since the spectrum was narrowed, a higher Δfrep = 510 Hz was realized without aliasing due to the higher beating orders. The measurement setup is shown in more detail in the Supplementary material. With this higher $$\varDelta {f}_{{{{{{\rm{rep}}}}}}}$$, 167 interferograms were recorded within 330 ms. In the 20 µs long time trace shown in Fig. 8a, free-induction decay of acetylene can clearly be seen. The Fourier transform of this interferogram and the corresponding measurement performed by our OSA are shown in Fig. 8b. However, the dual-comb measurement takes only 20 µs while the measurement with the state-of-the-art spectrum analyzer (Yokogawa AQ6370) requires 10 min in high-resolution (0.02 nm) mode. This result confirms the applicability of our high-power dual-comb spectrometer to real-life spectroscopic measurements.

## Discussion

The passive dual-comb oscillator relies on thin-disk technology with its intrinsic power scalability. Hence, this demonstration should be able to approach the parameters of a state-of-the-art single output system of around 62 MW peak and 270 W average powers20,27. The presented dual-comb cavity layout is independent of the mode-locking mechanism. Therefore, an application to powerful SESAM mode-locked systems35,36 is not out of scope. Moreover, the disk geometry, with its large, flat, and easily scalable active area, allows for a simple integration of multiple spatial modes. Thus, not only the dual-output systems but multiple-output lasers can be demonstrated. Multiple-output oscillators would open a way toward a very compact, simple, cost-effective multi-comb system (such as a tri-comb system) for 2D spectroscopy37, and generally nonlinear multi-dimensional spectroscopy and imaging.

Interestingly, the loss of coherence between interferograms due to $$\Delta {f}_{{{{{\rm{CEO}}}}}}$$ fluctuations is not evident during at least 312 ms measurement time (equivalent to 40 bursts). It was shown that a simple numerical technique to correct the interferogram arrival time can compensate for the $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuations sufficiently well for the presented measurements. In addition, apodization helps to increase the SNR ratio mainly by removing the side-interferograms associated with the side-pulse generated by the thin-disk dual-comb system (see Supplementary information for details). In practical dual-comb applications though, different sample types (gas, solid or liquid) and different spectral resolution can be of interest, which determine the duration of the apodization window. The ideal dual-comb system would provide delays between interferograms on the order of these response times, rendering this noise suppression scheme unnecessary. Ultimately, the spectral resolution is going to define the time window of interest. For example, in liquids (or even solids) the lines are broadened to around 100 GHz that makes it easy to record within approximately 5 µs duration of the interferogram. Gas at atmospheric pressure with about 1 GHz broadened linewidth can be sufficiently sampled during 0.5 ms that corresponds to 1/10th of the time window between the interferograms. Our Acetylene measurement uses a 20-µs apodization window (see Fig. 8), and reaches the 5.65 GHz resolution of Yokogawa OSA. In addition, it is necessary to state with regard to the correction of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuations, that it will be more challenging to deal with interferograms having much more complex temporal structures originating from complex real-world samples.

One very attractive approach for passive fCEO stabilization was been demonstrated with KLM thin-disk oscillators by means of intra-pulse difference frequency generation and simultaneous broadband conversion in mid-IR range at ~100 mW average power38. The required spectral broadening and temporal pulse compression also were demonstrated for a large span of peak and average powers in fibers or multipass cells39. Furthermore, a system driven by a mode-locked thin-disk Ho:YAG oscillator running at 2.1 μm wavelength40 showed excellent conversion efficiencies41 and ultra-broadband coverage (500–2250 cm−1) of the IR region. Therefore, combining the passive dual-comb approach and the Ho:YAG thin-disk technology should be possible in the near future. This would open a route toward ultra-broadband DCS of diverse chemical and biological samples42. Interestingly, the systems can be upgraded with an electro-optic sampling unit6 avoiding the use of liquid nitrogen detectors.

However, the extension of the spectral range toward UV and VUV might be even more interesting as no dual-comb lasers are available in this spectral range2. Efficient spectral broadening or nonlinear frequency conversion to UV (257 nm) is already possible at the current power level, and it will be even simpler for power-scaled systems. The overall passive stability of the system, in particular, low $$\Delta {f}_{{{{{\rm{CEO}}}}}}$$ fluctuations estimated to be 12 mrad (see Supplementary information), paves the way to UV and VUV dual-combs spectroscopy. Yet, this topic clearly requires further investigation, since the required $$\Delta {f}_{{{{{\rm{CEO}}}}}}$$ relative stability scales with frequency, and harmonic generation might add significant amount of noise. The results presented provide a good basis and motivation for further research, in particular, for detailed quantitative characterization of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ and $$\Delta {f}_{{{{{\rm{CEO}}}}}}$$ noise. Also, the upper limit of the mutual coherence time set by $$\Delta {f}_{{{{{\rm{CEO}}}}}}$$ fluctuations should be measured.

It is worth noting that relatively simple techniques can be applied to the dual-comb systems to stabilize $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ actively. This might be necessary once the number of interferograms becomes too large to be numerically corrected a posteriori. Relatively straightforward direct averaging of the interferograms in a field-programmable gate array can be implemented instead. Stabilizing $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ is expected to be simpler than stabilizing $${f}_{{{{{\rm{rep}}}}}}$$ due solely to its small residual jitter on the order of 0.1 Hz (0.1%). The origin of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuations requires further investigations. We believe that these fluctuations can be drastically reduced with acoustic isolation of the oscillator and interferometer and by optimizing the mechanical components and disk cooling flow.

A recently published dual-comb thin-disk oscillator based on polarization splitting43 delivers similar output parameters and additionally confirms our finding with regard to the importance of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ fluctuations.

We have developed a passive dual-frequency comb spectrometer based on a single free-running Yb:YAG KLM thin-disk laser. To the best of our knowledge, the peak (1.2 MW) and average power (15 W) are an order of magnitude higher than all other dual-comb systems. The two cavities emit nearly identical pulse trains, with a mutual $$\Delta {f}_{{{{{\rm{CEO}}}}}}$$ coherence time of at least 2.4 s. This experimentally achieved coherence is well manifested during two single interferograms showing comb lines in the spectrum after Fourier transformation without any active stabilization or numerical correction. The passive stability of the difference in repetition rates was measured to be around 0.1 Hz over 60 s. The evaluation of the interferogram trace (see Supplementary information) shows similar stability on the ms time scale. It was shown that, in combination with the simple correction of the fluctuations of $$\Delta {f}_{{{{{\rm{rep}}}}}}$$ and apodization, a narrow comb linewidth of 3.1 Hz in the RF domain has been achieved32. Finally, spectroscopic measurements with a Fabry–Pérot etalon and spectroscopy on acetylene (C2H2) have been demonstrated.

Ultrafast thin-disk oscillators have been demonstrated with average powers approaching 300 W and peak powers as high as 62 MW. Extension toward UV and VUV range44,45, seems especially promising taking into account the power scalability of our platform and excellent mutual coherence of the two combs.

## Methods

The experimental setup of the laser source is explained in detail in the results section. The Fabry–Pérot etalon was built with two output couplers with 98.4% and 95% reflectivity. Further details on the acetylene measurement setup and the evaluation method are explained in the Supplementary material.