Abstract: Frequency combs generated by coherent stimulated emission have revolutionized the precision at which we are able to measure time, frequency, and distance. Directly measuring the frequency of the radiation offers a much higher resolution, as time can be measured with a higher accuracy than distance, and conversion between frequency and wavelength can be done without much concern for accuracy degradation. FC generation in the visible and near-IR have enjoyed much progress in the last decade however there is still a lack of viable methods for producing FCs in the mid-IR and THz spectrum. Typically, combs are generated with mode-locked pulses or sending coherent light through a non-linear resonator, however there is a lack of materials available to achieve this in the longer wavelengths. Current approaches for FCs at longer wavelength typically include non-linear frequency conversion, which suffers from low conversion efficiency. Quantum Cascade lasers (QCLs) are a ubiquitous source of coherent radiation in the mid-infrared to THz regions of the spectrum. However, passive mode-locking is hard to achieve in QCLs because of their inherently low gain recovery time, due to intersubband operation, on the order of 1ps when compared to the round trip time on the order of 100ps. Despite the difficulty of actively or passively locking QCLs, experimental evidence has shown that frequency combs are indeed generated by free-running QCLs [1,2], i.e. no additional active or passive elements. Given proper dispersion compensation and a broadband gain medium the QCL generates coherent frequency combs with a frequency modulated phase relation.

In an attempt to explicate this behavior, a theoretical, frequency-domain model was developed via a perturbative solution of the Maxwell Bloch equation in the frequency domain[3], illustrating that frequency modulation is indeed a natural consequence of spatial hole burning in an inhomogeneously broadened gain medium (the origin of multi-mode lasing) and a short gain recovery time, which favors constant intensity. This combination of multi-mode operation with constant intensity is a signature of frequency modulation. The theoretical model predicts a pseudo-random frequency modulation of the laser with a general period of oscillations equal to the gain recovery time and an amplitude equal to the gain bandwidth. The work proposed here attempts to explain the necessity of a pseudo-random FM signal as well as take into account spectral hole burning which was excluded in the FD model.

This time domain model is developed using the Optical Bloch Equations (OBE). Initially the question of coherence must be investigated. It is our hypothesis that the dynamics of the laser, in fact, do not rely on coherent processes. In order to prove this, we compare the OBE under a full coherent interaction with a modified rate equation that is an approximation of the OBE. We operate under the assumption that the time rate of change is slower than the loss of coherence. This approximation greatly lightens the computational load, allowing for a more realistic model (more inhomogeneously broadened spectral bins, a longer cavity length, etc.). With the validation of this assumption we assert that the operation regime with the most stimulated emission will result in the lowest threshold, rationalizing the necessity of a pseudo random frequency modulation signal. In order to achieve this, we input various forms of an FM signal into a model utilizing real world specifications of both mid-infrared and THz QCLs. We show that indeed, with an FM signal similar to that produced in the FD model, the gain provided by the active medium peaks for a very random signal that fully spans the gain bandwidth, we trace the root cause of this to be spatial hole burning. Further experimental results have shown that under certain conditions the QCLs exhibit amplitude modulation as well as frequency modulation. In order to keep our model current, we develop analytical solutions for spatial hole burning in the cavity and investigate under what conditions of AM and FM is the spatial hole burning reduced.