1. Introduction Quantum cascade lasers (QCLs) promise to be efficient, cheap and compact sources of frequency combs in the mid- and far-infrared portions of the electromagnetic spectrum. Experimental results have shown comb generation in the mid-infrared as well as, more recently, the terahertz [1, 2] based on free running QCLs. The community appears to collectively agree that high order nonlinear optical processes, such as four wave mixing, are the main mode proliferation mechanisms that contribute to comb formation [3]. In contrast, it has been argued that group velocity dispersion (GVD) is the thorn in the design of frequency combs as it leads to pulse broadening and limits the full exploitation of the gain bandwidth of the material [4]. In the terahertz regime, the two widest comb generating devices demonstrated so far [1, 2], have shown a strong variation of the beatnote’s linewidth with changing injection current, thus indicating that they experience "comb" regimes of operation that cover only a fraction of the whole dynamic range of these lasers. Here we present results from a coupled ensemble Monte-Carlo (EMC) and Maxwell-Bloch (MB) equations simulations of the device in Ref. [1]. We show that the correct compensation of group velocity dispersion is essential for the frequency stability of the laser in question and we also investigate the effect of different model parameters onto the simulated beatnote’s linewidth. 2. Theoretical model The MB equations are a semi-classical model describing the light-matter interaction in microscopic systems, where the coherent coupling between the optical field and the gain medium is treated within a density matrix formalism, whereas the effect of the induced polarization onto the incident optical field is captured via the classical Maxwell’s equations. This model is a generalization of the rate equations approach, which allows us to include optical nonlinearities and coherence effects into electron transport simulations and thus incorporate the physics of four wave mixing and resonant tunnelling into the system dynamics. We numerically solve the Maxwell-Bloch equations for a three level system with a single resonant tunnelling and a single optical transition included, whereas the physics of incoherent scattering mechanisms is treated phenomenologically via a rate equations approach [5]. 3. Results After a stationary EMC simulation to extract the scattering rates of the QCL in [1], we propagated the Maxwell-Bloch equations system in time until steady state was reached. Without dispersion compensation mechanisms, the simulated laser failed to produce a beatnote with a narrow linewidth, but rather settled for a chaotic multimode regime of operation (Fig. 1b). Similar behavior was experimentally detected in Ref. [1, 2]. We then used the same model to perform numerical gain dispersion measurements, in order to evaluate the linear and higher order group delay induced by the optical transitions. With this information at hand, we managed to emulate dispersion compensation by applying a non-causal linear filter with fixed magnitude and phase response at every iteration of our simulation. Running again our simulations, however this time employing dispersion compensation, we observed narrowing of the beatnote linewidth as compared to the uncompensated case (Fig. 1c).We investigated this in detail and the results are plotted in Fig. 1. Due to the inherent limitation of computational resources, we could not calculate the physical linewidth limit, since this would require simulation durations in the order of tens of thousands of round trips. The narrowing of the FWHM of the beatnote signal with increasing number of simulated round trips, superposed to the FFT resolution limit, is depicted in Fig. 1a. From there it becomes evident that for linewidths in the order of several MHz, the Fourier transform resolution is the limiting factor in our calculation. To further prove that the simulated dispersion compensated device is indeed a frequency comb, we applied the SWIFT spectroscopic technique on our data to extract and compare the correlation and spectrum product. The results from this operation are plotted in Fig. 2 and show good agreement with experiment [6].     «

1. Introduction Quantum cascade lasers (QCLs) promise to be efficient, cheap and compact sources of frequency combs in the mid- and far-infrared portions of the electromagnetic spectrum. Experimental results have shown comb generation in the mid-infrared as well as, more recently, the terahertz [1, 2] based on free running QCLs. The community appears to collectively agree that high order nonlinear optical processes, such as four wave mixing, are the main mode proliferation mechanisms that contr...     »