Harmonic instability in a quantum cascade laser with Fabry–Perot cavity

Abstract

A new state of instability called harmonic instability, which is characterized by the appearance of side modes separated by multiple of free spectral range from each other, was discovered in quantum cascade lasers (QCLs) a few years ago. However, a detailed analysis using a model beyond the two-level density-matrix (DM) equations as well as incorporating phenomena such as the detuning of the primary-mode frequency from the line-center frequency and the frequency dependence of the background refractive index, resulting in group velocity dispersion (GVD), has not been performed yet. In this paper, we present a comprehensive analysis of harmonic instability in a QCL with a Fabry–Perot (FP) cavity. Starting from three-level DM equations, which include the resonant tunneling phenomenon and scattering rates between all three states, and then by using Maxwell's equation, we derive a closed-form expression for the gain of the side modes, from which quantities pertinent to instability can be determined. We also take the aforementioned phenomena into account in our theory. By using our theory, we show the way of determining primary-mode detuning from the line center. Furthermore, we study the effects of GVD on instability in detail, showing that the output from an FP QCL demonstrates the characteristic of a frequency-modulated wave up to a certain value of dispersion. Above this value, because of the significant deviation of the side-mode amplitude ratio from unity, the output shows neither frequency-modulated-like nor amplitude-modulated-like behavior.