Description of Kerr lens mode-locked lasers with Poincaré maps in the complex plane

Abstract

We study the Kerr lens mode-locking (KLM) laser operation from the point of view of the spontaneous appearance of a new stable solution in a perturbed non-linear system. A description of KLM is possible in terms of a five variables iterative map. For usual values of the laser parameters, the complete map can be simplified to two maps with complex variables and one map with a real variable, which become uncoupled after a transient has evolved. After appropriate scaling, the two complex maps have the same form. This simplifies the calculation of the fixed points and their stability. It is found that, for appropriate parameters' values, KLM arises even in the absence of spatial apertures or bandwidth limitations. Hence, the Kerr perturbation modifies the system from non-dissipative to dissipative, this meaning a contraction of the phase space. It is also found that the phase space can expand for other parameters' values or initial conditions. If apertures are included in the model, the convergence to the mode-locked solution is faster and the size of its basin of attraction enlarges, but the dynamics remain essentially the same.