Self-starting harmonic frequency comb generation in a quantum cascade laser

Optical frequency combs1,2 establish a rigid phase-coherent link between microwave and optical domains and are emerging as high-precision tools in an increasing number of applications3. Frequency combs with large intermodal spacing are employed in the field of microwave photonics for radiofrequency arbitrary waveform synthesis4,5 and for the generation of terahertz tones of high spectral purity in future wireless communication networks6,7. Here, we demonstrate self-starting harmonic frequency comb generation with a terahertz repetition rate in a quantum cascade laser. The large intermodal spacing caused by the suppression of tens of adjacent cavity modes originates from a parametric contribution to the gain due to temporal modulations of population inversion in the laser8,9. Using multiheterodyne self-detection, the mode spacing of the harmonic comb is shown to be uniform to within 5 × 10−12 parts of the central frequency. This new harmonic comb state extends the range of applications of quantum cascade laser frequency combs10–13. Self-starting harmonic frequency comb generation with a THz repetition rate in a q﻿uantum cascade l﻿aser is demonstrated﻿. T﻿he mode spacing uniformity is verified to within 5 × 10−12 parts of the central frequency. The findings extend the range of applications of quantum cascade laser frequency combs.

cavity free spectral range (FSR) (Fig. 1a). Its spectrum differs radically from that of QCLs in fundamental comb operation, where adjacent cavity modes are populated (Fig. 1b). This new state is achieved by controlling the current so that the QCL first reaches a state of high single-mode intracavity intensity. When this intensity is large enough, an instability threshold is reached caused by the χ (3) population pulsation nonlinearity, favouring the appearance of modes separated by tens of FSRs from the first lasing mode. In this Letter, we verify the equidistance of these modes, thereby proving that QCLs are capable of harmonic comb operation and concomitant high-repetitionrate OFC generation. OFCs with repetition frequencies ranging between 10 and 1,000 GHz have already been demonstrated in optically pumped microresonators 14 , but the generation of highrepetition-rate OFCs based on QCLs operating in the harmonic regime presents the advantage of a truly monolithic, electrically driven source. It is notable that this type of comb operation does not require additional intracavity nonlinear elements such as saturable absorbers, or mode-selection elements such as Bragg reflectors, which were used to achieve passive harmonic mode locking at terahertz repetition rates in other semiconductor lasers 20 . Instead, the modes are locked passively due to the behaviour of the QCL gain medium itself.
Here, we use two Fabry-Pérot (FP) QCLs fabricated from the same growth process with 6-mm-long cavities and emitting at 4.5 μ m (see Methods for details). These devices exhibit four distinct laser states as a function of the injected current (Fig. 2a). Starting from single-mode operation and slowly increasing the bias, one can observe the harmonic state appearing at a pump current only fractionally higher than the lasing threshold. No beatnote at the cavity roundtrip frequency (f rt ) is observed in this regime (Fig. 2b), confirming the absence of interleaving FP modes.
At higher values of injected current, the laser transitions to a single-FSR-spaced state producing a single narrow intermodal beatnote (full-width at half-maximum, FWHM < 1 kHz) at f rt , indicating the occurrence of fundamental comb operation 10 . At even higher current, the single-FSR comb acquires a high-phasenoise pedestal-a typical signature of comb destabilization 21 . To verify the spacing uniformity of the modes of the harmonic state, techniques developed to investigate combs with an intermodal spacing in the lower gigahertz range (such as intermode beat spectroscopy 10 and shifted wave interference Fourier transform spectroscopy 12 ) cannot be applied because the terahertzscale beatnote frequency of the harmonic state is beyond the

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bandwidth of conventional mid-infrared (mid-IR) detectors and radiofrequency (RF) electronics. Instead, we use a technique that was first developed to characterize high-repetition-rate microresonator combs, in which the sample comb spectrum is downconverted from the optical to the RF domain by means of multiheterodyne beating with the modes of a finely spaced reference comb 22 . In this scheme, if the harmonic state constitutes a frequency comb, the downconverted spectrum will form an RF comb whose equidistant spacing can be accurately verified using electronic frequency counters.
For the multiheterodyne experiment we used two QCLs, one operating in the harmonic comb regime (QCL 1 ) and the other in a fundamental comb regime (QCL 2 ) acting as a reference comb with an intermodal spacing of ∼ 7.7 GHz. We employed a self-detection scheme in which the light emitted from QCL 1 is injected, after passing through an optical isolator, into the cavity of QCL 2 (Fig. 3f). The latter acts at the same time as a reference comb and a fast photomixer 23 , from which we can extract electrically the multiheterodyne signal generated by the intracavity beating of the optical fields of the two lasers. The attractive feature of this scheme is that it provides better signal stability than a standard approach utilizing an external fast photodiode. A description of this method is provided in the Supplementary Information, and its results are shown for comparison in Fig. 3g,h.
The emission spectra of the harmonic and reference comb measured using the self-detection scheme are presented in Fig. 3a,b. QCL 1 is operating in the harmonic comb regime, while QCL 2 is operating in a fundamental comb regime exhibiting adjacent cavity modes that constitute an equidistant grid with a spacing defined by f rt (Fig. 3c,d). Interestingly, several prominent peaks not lying on this grid can be identified in the spectrum of QCL 2 (marked by green arrowheads in Fig. 3b-d), corresponding to modes injected from QCL 1 into QCL 2 . The pairwise beating of the modes of the harmonic state with the nearest modes of the reference comb produces the multiheterodyne spectrum shown in Fig. 3e.
To assess the locking of the harmonic modes we further downconverted the multiheterodyne signal with an RF mixer and selected three beatnotes of the spectrum using bandpass filters whose outputs were fed into three synchronized frequency counters. From the measured frequencies, a histogram showing the statistics of the deviation from equidistant spacing of the RF comb can be constructed (Fig. 3g). The fractional frequency stability of the dual-comb system exhibits an inverse square-root dependence on the averaging time, indicating the dominance of white-noise frequency modulations in the system giving rise to random and uncorrelated fluctuations following a normal distribution (Fig. 3h). This allows the histogram to be fit with a Gaussian function, which yields a mean value of μ = − 27 Hz and a standard deviation of σ = 329 Hz. This result verifies the equidistant spacing of the harmonic comb with a relative accuracy of σ/f c = 5 × 10 −12 , as normalized to the optical carrier frequency of the laser (f c = 66.7 THz), being an order of magnitude smaller than for the measurement based on external detection. This net improvement is due to the higher stability of the multiheterodyne signal in the self-detection scheme (Fig. 3h), as discussed in Supplementary Section IIC.
To explain the occurrence of harmonic comb operation we resort to a perturbation theory of comb formation in QCLs considering the interaction of a two-level gain medium with a field comprising a central mode and two weak equally detuned sidebands in the laser cavity. The nature of the parametric gain responsible for adjacent mode skipping in the laser was studied in ref. 19 . Here, we apply a more general approach that includes, in a systematic way, the effects of nonequal sideband amplitudes, diffusion of the population grating and group velocity dispersion (GVD), which may hamper comb operation 21 . The complete derivation of our theory is provided in Supplementary Section III, while here we outline the main implications given by the solutions of our model for the real device parameters. The subthreshold GVD of QCL 1 , measured by a standard technique 24 , is displayed in Fig. 4a. The net parametric gain calculated as a function of sideband detuning is shown in Fig. 4b. It peaks at a frequency of 200 GHz (26 FSR of a 6-mm-long cavity)    The self-detected system shows more than an order of magnitude improvement in frequency stability. Error bars are defined as the confidence interval corresponding to one standard deviation. with respect to the central mode, indicating that the modes at this frequency are the first to oscillate, while the modes lying closer to the central pump are parametrically suppressed. Furthermore, we calculate that FWM can compensate for the non-zero dispersion of the real device up to a GVD of 6,000 fs 2 mm −1 , above which the onset of harmonic comb operation is hampered (Fig. 4c). These results are consistent with our experimental findings proving the occurrence of harmonic comb operation in a QCL with a GVD below 1,000 fs 2 mm −1 . The ability to generate and passively lock harmonic modes while skipping adjacent cavity resonances relies on a coherent instability enabled by the QCL gain medium, unveiling the compelling dynamics of the new laser state in QCLs. Locking of comb teeth with a spacing comparable to the gain recovery frequency represents a major step towards the demonstration of coherent mid-IR amplitude-modulated waveform emission from QCLs, long thought to be prevented by the underlying physical principles, and paving the way towards applications requiring short pulses of mid-IR light.
In this respect, an important point to address in future studies is measurement of the relative phases of the harmonic modes, which may be achieved by means of a multiheterodyne experiment involving a reference comb with known spectral phase or using spectrum analyser extension modules in the terahertz range. On the other hand, QCL harmonic comb generators may find applications in future wireless terahertz communication networks 25 , as they combine the functionality of a high-bandwidth photomixer and a comb source, promising the intracavity generation of powerful terahertz carrier signals, whose frequency can be designed by engineering the facet coatings of the device 19 and where the phase noise is inherently low due to a high degree of correlation among the optical modes that produce the beatnote. Merging this capability with the fact that QCLs can be optimized to have a flat frequency response over a large modulation bandwidth 26 may allow them to operate as compact unibody modems to transmit and receive digital data in the terahertz communication band.

Methods
Methods, including statements of data availability and any associated accession codes and references, are available at https://doi. org/10.1038/s41566-017-0026-y.