Stable numerical schemes for nonlinear dispersive equations with counter-propagation and gain dynamics

We develop a stable and efficient numerical scheme for modeling the optical field evolution in a nonlinear dispersive cavity with counter-propagating waves and complex, semiconductor physics gain dynamics that is expensive to evaluate. Our stability analysis is characterized by a von Neumann analysis that shows that many standard numerical schemes are unstable due to competing physical effects in the propagation equations. We show that the combination of a predictor–corrector scheme with an operator splitting not only results in a stable scheme, but also provides a highly efficient, single-stage evaluation of the gain dynamics. Given that the gain dynamics is the rate-limiting step of the algorithm, our method circumvents the numerical instability induced by the other cavity physics when evaluating the gain in an efficient manner. We demonstrate the stability and efficiency of the algorithm on a diode laser model that includes three waveguides and semiconductor gain dynamics. The laser is able to produce a repeating temporal waveform and stable optical comb lines, thus demonstrating that frequency comb generation may be possible in chip-scale, diode lasers.