Abstract
Since the beginning of this millennium, frequency comb generators have reshaped frequency metrology and related areas. After more than two decades since their first realization, several other ways to generate frequency combs, in any spectral region, have been demonstrated, each way with its peculiar features. This trend has triggered the need to quantitatively assess how close the new comb realizations are to an ideal comb, a feature that will be called combness throughout this paper. We will briefly review the very dynamic area of novel frequency comb sources and we will describe the techniques that have been recently developed to quantitatively assess the key parameters of old and new frequency combs, in view of the specific applications. Finally, we will try to sketch future steps in this recently born research area.
1 Introduction
For many years, at the end of the previous century, the development of ultrafast laser sources did not cross frequency metrology that strived to reference optical frequencies to the Cesium clock transition in the microwaves, at 9.19 GHz, chosen as primary frequency standard. To this purpose, frequency chains including plenty of phase-referenced ultra-stable continuous-wave lasers had been developed. Then, across the year 2000, at the beginning of this millennium, the intrinsic correlation in between time and frequency domains was finally unveiled and exploited: the frequency comb (FC) generator was born. This new concept appeared to be groundbreaking not only for frequency metrology and plenty of related areas, but also for new concepts of laser sources, designed to generate “combs” in a number of different ways and in widely different spectral regions.
1.1 The evolution and diversification of optical frequency comb synthesizers
FCs are nowadays key tools in many fields of fundamental and applied research [1]. Visible/near-infrared FCs can be generated by means of controlled, frequency-stabilized mode-locked femtosecond lasers [2], [3], [4]. In recent years, FCs have been rapidly developed on fiber-based platforms, achieving a high operation power and stability, and broad spectral coverage in the visible and near-infrared spectral regions.
The miniaturization of the sources, together with the expansion of their operation towards other spectral regions (ultraviolet, mid- and far-infrared), is crucial for broadening their application range (telecommunications [5], [6], molecular spectroscopy and gas sensing [7]).
Specifically, the availability of broadband and compact radiation sources, spectrally covering the ultraviolet range while exhibiting optical phase coherence and high repetition rates opens the doors to precision frequency metrology, photoelectron spectroscopy and attosecond science [8], [9], [10], [11], [12], [13]. In the absence of convenient broadband laser gain media and optics suitable for building mode-locked oscillators in the ultraviolet range, frequency upconversion of the pulse train emitted by visible/near-infrared mode-locked lasers via coherent high-order harmonic generation is the only available approach at the moment [14].
On the other side of the electromagnetic spectrum, the generation of mid-to-far infrared FCs is possible via two different approaches. The first one consists in down-converting near-infrared FCs via difference-frequency generation (DFG) [15], [16], [17], [18], [19] or using synchronously-pumped optical parametric oscillators (OPOs) [20], [21]. Conceptually, a DFG-based mid-infrared FC (DFG-comb) is obtained by mixing a visible/near-infrared FC with a pump laser in a non-linear crystal, with matching wavelength requirement to reach the mid infrared [22], [23], [24], [25]. At even longer wavelengths, the generation of a THz FC via optical rectification has been demonstrated [26].
On the miniaturization front, the most interesting results have recently been achieved with the following technologies: microresonators, quantum cascade lasers and interband cascade lasers. All of them are characterized by a high third-order (Kerr) nonlinearity and by the same FC formation mechanism, that is degenerate and non-degenerate four-wave mixing (FWM).
Whispering gallery mode resonators (microresonators) are ring microcavities with large Q factors
On the other hand, quantum cascade lasers (QCLs) are current-driven semiconductor lasers based on intersubband transitions in quantum wells, emitting high-power coherent radiation in the mid and far infrared [35], [36], [37]. Due to the active region structure, in particular for high-performance room-temperature mid-infrared devices, the upper lasing state lifetime is very short compared to the cavity round-trip time (about two orders of magnitude). As a consequence, in continuous-wave operation, energy cannot be stored during the round trip, the sustenance of optical pulses is prevented and classical pulsed passive mode locking is generally not achievable [38], [39], [40]. In this regard, new results related to the implementation of graphene-based saturable absorbers were recently demonstrated [41].
Active mode locking emerged as an alternative to passive mode locking. Active pulsed mode locking has been successfully demonstrated both in the mid infrared [42], [43] and THz [44], [45] ranges. The limitation of this approach derives from the need of close-to-threshold operation in order to mitigate gain saturation, severely limiting the emitted power, and from the length of the pulses that cannot reach the inverse of the gain bandwidth.
However, by using broadband Fabry–Pérot QCLs [46], [47] designed to have low group velocity dispersion, FC generation has been demonstrated in free-running operation (QCL-combs) both in the mid-infrared and in the THz range [38], [48], [49]. Starting from the independent longitudinal modes generated by a Fabry–Pérot multimode laser, degenerate and non-degenerate FWM processes induce a proliferation of modes over the laser emission spectrum [50], [51]. The original modes are then injection-locked by the modes generated by FWM, ensuring correlation among all the longitudinal modes, giving birth to a FC [40] with a fixed phase relation, but a non-pulsed emission. In the THz range the full phase stabilization of both the FC degrees of freedom (offset and mode spacing) has been demonstrated [52], while FC operation over the entire available gain bandwidth has been achieved by conveniently increasing the mirror losses of the Fabry–Perot cavity through coating the back facet with an epitaxially-grown multilayer graphene film [53].
More recently, interband cascade lasers (ICLs) [54], [55], [56], [57], [58] also proved to be able to generate FCs [59], [60], [61]. Both QCLs and ICLs could be successfully exploited for dual-comb spectroscopy (DCS) [62], [63], [64] and free-space communication [65], [66].
Advanced techniques for FC characterization have been developed. Frequency-resolved optical gating (FROG) [33], [67], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [68], [69] and asynchronous upconversion sampling (ASUPS) [70] have been developed and applied to characterize FC emission in the time domain. Other techniques, suitable for studying quasi-continuous-wave mid- and far-infrared radiation, have been developed. The list comprises optical and RF spectrum monitoring, intermodal beatnote spectroscopy [38] and shifted wave interference Fourier transform spectroscopy (SWIFTS) [48], single-frequency counting and multi-heterodyne detection using a dual-comb setup for frequency equispacing estimation and frequency stability characterization [62], [71], the Vernier technique using a high-finesse optical cavity for technical and quantum frequency noise estimation [72]. All these techniques afford coherence estimation of the FC emission. Among them, only the SWIFTS technique can access the phase relation between the modes [73], [74], [75]. In particular, it allows for the retrieval of the phase of each mode compared to the ones of its first neighbors. As a consequence, this technique enables to obtain the phase relation inherent to continuous portions of the FC spectrum. The main limitation is that SWIFTS relies on a cumulative sum, therefore the result is particularly subject to noise. Moreover, relying on a scan (in particular the mechanical scan of the interferometer arm, usually lasting 5–15 min [48]) it does not allow for a synchronous retrieval of all the modal phases, preventing monitoring of the time evolution of the phase relation.
In 2019 our research group proposed an alternative characterization technique named Fourier-transform analysis of comb emission (FACE). The experimental procedure, firstly reported in ref. [76] and then deeply discussed in ref. [77], takes advantage of the multiheterodyne detection scheme, also used in dual-comb spectroscopy setups, the major difference being that the sample to be investigated is not a molecular species but the sample FC itself. Following this scheme, the sample FC is mixed with a second, fully controlled and stabilized, reference FC (LO), generating a down-converted FC in the radio frequencies (RF) domain. The phase information, encoded in the RF beat notes, is then retrieved by means of a subsequent Fourier-transform analysis. The main advantages of the FACE technique are its great generality, i.e. its applicability to any FC source, regardless of its wavelength or temporal waveform, and its remarkably simple experimental setup, in which only a fast mixer and a reference FC (spectrally overlapping with the sample one) are needed for the down-conversion. Another important aspect is that, unlike SWIFT, FROG, and SPIDER, the FACE technique (being based on DCS) does not require a mechanical scanning arm, and can therefore provide a simultaneous and real-time sampling of the investigated FC.
In this work, a mid-infrared QCL-comb is investigated as sample, its spectral coherence is evaluated showing how the coherence time of each FC mode can be retrieved. This parameter is proposed as estimator for the combness.
2 Discussion
2.1 Evaluating the coherence of frequency comb modes
We will illustrate the approach of the Fourier-transform analysis of comb emission (FACE) considering the steps necessary for deriving a coherence plot of a mid-infrared laser; we refer for the details to ref. [77]. In a nutshell, FACE is a dual-comb spectroscopy (DCS) without spectroscopic sample, where a local oscillator FC (LO-FC, repetition frequency
Beside the simplest case, when the repetition frequencies of the two FCs are just slightly detuned of a quantity Δν
rep,
where k is an integer number defining the harmonic mixing ratio. In this work, we will consider a case of semi-harmonic mixing, when
As a result of this analysis we have the amplitude and the phase of each oscillator in the heterodyne beats as a function of the stride time. The phase knowledge opens the possibility to investigate the time coherence of the oscillators. To this purpose we set three threshold values for the oscillator phase angle (Δθ coh = 0.05, 0.1, and 0.2 rad) and calculate the average time (τ choerence in figure), that the rewound phase ΘΔm − Θ0 needs in order to drift of the threshold value. The result is shown in Figure 2. The RBN, the phase reference, reaches the full acquisition duration without hitting the threshold, and is plotted as a gray dot. For the other oscillators the coherence time is found of the order or below the stride duration, and as expected, is decreasing for increasing |Δm|. For the three threshold values we found that a power law τ = A/|Δm| α is a good fit for the data (once the Δm = 0 point is excluded). The exponent fit parameter is α = 0.77 ± 0.07 for both Δθ coh = 0.05 rad and 0.2 rad, while α = 0.83 ± 0.08 for Δθ coh = 0.1 rad. As a further check, we can use the classical expression for the complex degree of coherence [78]
where
Finally, considering ΔΘΔm ≡ ΘΔm (t + τ) − ΘΔm (t) as a stationary (t–independent) stochastic Gaussian process, we obtain[4]
where averaging is now made on the process realizations, i.e. over t if the further assumption of ergodicity is made, and the last passage stems from a well-known identity of the characteristic function
We conclude observing that this kind of characterization concerns the FC interferometric coherence, as it measures the phase stability of each S-FC mode; the phase stability is also influenced by the phase stabilization electronic loop. Moreover, since it is related to slow phase drifts, it cannot be estimated by the traditional PSD analysis.
3 Conclusions
Considering the wide variety of possible frequency comb patterns that can be generated exploiting different physical phenomena, devices and materials, the need to perform quantitative measurements clearly arises in order to assess how close a specific comb is to an ideal frequency comb, a feature that we call combness. We have shown that a key parameter to assess the combness is the phase stability of each single mode, relative to all the others. After discussing different existing methodologies to make such a characterization, we focus the attention on what we consider the most rigorous technique to this goal, the FACE technique. In combination with a fully-fledged complex field data analysis, FACE allows to determine the actual coherence time of each and every comb mode, providing quantitative data for FC use in experiments/applications with specific targets and requirements.
This perspective work highlights also the need to summarize the many data that are usually collected, to build smart and easy–to–use quantitative combness indicators. In addition, smarter ways to get data, reducing the overall data load and measurement time, can help the ranking of existing FCs, in view of a wider and wider range of applications, more or less demanding of the actual FC properties. As always happens when accurate quantitative approaches are set-up, we believe that FC applications will mostly benefit from these new methodologies, as well as future FC generators.
Funding source: HORIZON EUROPE Framework Programme
Award Identifier / Grant number: Laserlab-Europe Project [G.A. n. 871124]
Award Identifier / Grant number: MUQUABIS Project [G.A. n. 101070546]
Funding source: QuantERA
Award Identifier / Grant number: [G.A. n. 101017733] – QATACOMB Project
Funding source: European Commission
Award Identifier / Grant number: I-PHOQS Infrastructure [IR0000016, ID D2B8D520]
Funding source: Horizon 2020 Framework Programme
Award Identifier / Grant number: Qombs Project [G.A. n. 820419]
Acknowledgments
The authors gratefully thank Prof. Dr. Jérome Faist (ETH Zurich) for having provided the quantum cascade laser and the company Menlo Systems for having provided the DFG-comb.
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Research funding: The authors acknowledge financial support by the European Union’s NextGenerationEU Programme (https://doi.org/10.13039/100018693) with the I-PHOQS Infrastructure [IR0000016, ID D2B8D520, CUP B53C22001750006] “Integrated infrastructure initiative in Photonic and Quantum Sciences”, by the European Union’s Research and Innovation Programmes Horizon 2020 (https://doi.org/10.13039/100010661) and Horizon Europe (https://doi.org/10.13039/100018693) with the Qombs Project [G.A. n. 820419] “Quantum simulation and entanglement engineering in quantum cascade laser frequency combs”, the Laserlab-Europe Project [G.A. n. 871124], and the MUQUABIS Project [G.A. n. 101070546] “Multiscale quantum bio-imaging and spectroscopy”, by the European Union’s QuantERA II (https://doi.org/10.13039/501100020314) [G.A. n. 101017733] – QATACOMB Project “Quantum correlations in terahertz QCL combs”, and by the Italian ESFRI Roadmap (Extreme Light Infrastructure – ELI Project).
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: Authors state no conflicts of interest.
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Informed consent: Informed consent was obtained from all individuals included in this study.
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Ethical approval: The conducted research is not related to either human or animals use.
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Data availability: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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