We report the observation of a clear single-mode instability threshold in continuous-wave Fabry-Perot quantum cascade lasers (QCLs). The instability is characterized by the appearance of sidebands separated by tens of free spectral ranges (FSR) from the first lasing mode, at a pump current not much higher than the lasing threshold. As the current is increased, higher-order sidebands appear that preserve the initial spacing, and the spectra are suggestive of harmonically phase-locked waveforms. We present a theory of the instability that applies to all homogeneously broadened standing-wave lasers. The low instability threshold and the large sideband spacing can be explained by the combination of an unclamped, incoherent Lorentzian gain due to the population grating, and a coherent parametric gain caused by temporal population pulsations that changes the spectral gain line shape. The parametric term suppresses the gain of sidebands whose separation is much smaller than the reciprocal gain recovery time, while enhancing the gain of more distant sidebands. The large gain recovery frequency of the QCL compared to the FSR is essential to observe this parametric effect, which is responsible for the multiple-FSR sideband separation. We predict that by tuning the strength of the incoherent gain contribution, for example by engineering the modal overlap factors and the carrier diffusion, both amplitude-modulated (AM) or frequency-modulated emission can be achieved from QCLs. We provide initial evidence of an AM waveform emitted by a QCL with highly asymmetric facet reflectivities, thereby opening a promising route to ultrashort pulse generation in the mid-infrared. Together, the experiments and theory clarify a deep connection between parametric oscillation in optically pumped microresonators and the single-mode instability of lasers, tying together literature from the last 60 years.

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 applications 3 . Frequency combs with large intermodal spacing are employed in the field of microwave photonics for radiofrequency arbitrary waveform synthesis 4,5 and for generation of THz tones of high spectral purity in the future wireless communication networks 6,7 . We demonstrate for the first time self-starting harmonic frequency comb generation with a THz 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 the population inversion in the laser 8,9 . The mode spacing of the harmonic comb is shown to be uniform to within 5 × 10 −12 parts of the central frequency using multiheterodyne self-detection. This new harmonic comb state extends the range of applications of quantum cascade laser frequency combs 10-13 . Several techniques to generate optical frequency combs (OFCs) have been demonstratedin the last decades based on different nonlinear mechanisms that fulfill the modelocking condition. Originally, passively modelocked lasers based on saturable absorption and Kerr lensing were used to create short light pulses, and were subsequently shown to also constitute frequency combs. This type of modelocking is an example of amplitude-modulated modelocking, so-named for the temporal behavior of the electric field of the emitted light. However, these techniques usually result in elaborate optical systems. More recently, new routes promising chip-scale comb generators have been investigated based on optically-pumped ultra-high-quality-factor crystalline microresonators 14-16 and on broadband quantum cascade lasers (QCLs) with specially designed multistage active regions 10,17 . In both cases the essential underlying mechanism responsible for the generation of OFCs is cascaded four-wave mixing (FWM) enabled by a third-order χ (3) Kerr nonlinearity. The temporal behavior of these OFCs is not restricted to ultrashort pulses but can represent rather sophisticated waveforms due to a non-trivial relationship among the spectral phases of the comb teeth. In fact, the output of a QCL-based frequency comb resembles that of a frequency-modulated laser with nearly constant output intensity 10,18 .A novel mechanism of OFC generation in QCLs was suggested by the recent discovery of a new laser state 19 , which comprises many modes separated by higher harmonics of the 2 cavity free spectral range (FSR) (Figure 1a). This spectrum radically differs from that of fundamentally modelocked QCL combs where adjacent cavity modes are populated ( Figure 1b THz repetition rates in other semiconductor lasers 20 . Rather, the modes are locked passively due to the behavior of the QCL gain medium itself.In this work we employed two Fabry-Perot (FP) QCLs fabricated from the same growth process with 6 mm-long cavit...

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