Harmonic frequency combs in quantum cascade lasers: Time-domain and frequency-domain theory

Harmonic frequency combs, in which the lasing modes are separated by a period of tens of free spectral ranges from each other, have been recently discovered in quantum cascade lasers (QCLs). There is an ongoing debate how the harmonic combs can be formed and stable under continuous pumping. Here we reproduce the harmonic state of lasing in QCLs by space-time-domain numerical simulations and show that the corresponding optical wave has a frequency-modulated nature, with the fundamental beat note in the electron population being suppressed. To understand the physics behind the formation and stability of the harmonic state, we develop a frequency-domain linearized analytic theory which analyzes the stability of single-mode lasing with respect to the harmonic state formation and the instabilities of the resulting harmonic state. Our analysis shows how the instability of a single lasing mode can result in the growth of harmonic side modes. The coupling between the two side modes leads to enhancement of the gain for frequency-modulated waves as compared to the gain for amplitude-modulated waves. The dependence of the sideband gain spectrum on group-velocity dispersion is also studied. While the instability of a single lasing mode with respect to the growth of side bands separated by many free spectral ranges does not prove that the harmonic state can be self-starting, we analyze the instability of three strong harmonic modes and demonstrate that the harmonic state can be stable and self-supported.