Analytical and computational study of self-induced transparency mode locking in quantum cascade lasers

Abstract

The possibility of using the self-induced transparency effect to achieve laser mode locking has been discussed since the late 1960s but has never been observed. In prior work, we proposed that quantum cascade lasers are the ideal tool to realize self-induced transparency mode locking due to their rapid gain recovery times and relatively long coherence times, and because it is possible to interleave gain and absorbing periods. Here, we present designs of quantum cascade lasers that satisfy the requirements for self-induced transparency mode locking at both 8 and $12\text{ }\ensuremath{\mu}\text{m}$, indicating that it is possible to satisfy these requirements over a wide wavelength range. The coupled Maxwell-Bloch equations that define the dynamics in quantum cascade lasers that have both gain and absorbing periods have been solved both analytically and computationally. Analytical mode-locked solutions have previously been found under the conditions that there is no frequency detuning, the absorbing periods have a dipole moment twice that of the gain periods, the input pulse is a $\ensuremath{\pi}$ pulse in the gain medium, and the gain recovery times in the gain and absorbing periods are much longer than the coherence time ${T}_{2}$ and are short compared to the round-trip time. It was shown that the mode-locked pulse durations are on the order of ${T}_{2}$, which is typically about 100 fs in quantum cascade lasers. In this work, these analytical results are reviewed and extended to include the effects of partial inversion in the gain and absorbing periods and of frequency detuning. An energy theorem in the limit of long coherence times is derived. The Maxwell-Bloch equations have been solved computationally to determine the robustness of the mode-locked solutions when frequency detuning is present, the dipole moment of the absorbing periods differs from twice that of the gain periods, the gain relaxation time is on the order of 1--10 ps, as is typically obtained in quantum cascade lasers, and the initial pulse is not a $\ensuremath{\pi}$ pulse in the gain medium. We find that mode-locked solutions exist over a broad parameter range. We have also investigated the evolution of initial pulses that are initially much broader than the final mode-locked pulses. As long as the initial pulse duration is on the order of ${T}_{1}$ or shorter and has enough energy to create a $\ensuremath{\pi}$ pulse in the medium, a mode-locked pulse with a duration on the order of ${T}_{2}$ will ultimately form.