{'fraction_non_alphanumeric': 0.02723463687150838, 'fraction_numerical': 0.0, 'mean_word_length': 5.1239316239316235, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Kerr-Lens Mode-Locking: Numerical Simulation of the Spatio-Temporal Dynamics on All Time Scales

We present a complete numerical analysis and simulation of the full spatio-temporal dynamics of Kerr-lens mode-locking in a laser. This dynamic, which is the workhorse mechanism for generating ultrashort pulses, relies on the intricate coupling between the spatial nonlinear propagation and the temporal nonlinear compression. Our numerical tool emulates the dynamical evolution of the optical field in the cavity on all time-scales: the fast time scale of the pulse envelope within a single round trip, and the slow time-scale between round-trips. We employ a nonlinear ABCD formalism that fully handles all relevant effects in the laser, namely—self focusing and diffraction, dispersion and self-phase modulation, and space-dependent loss and gain saturation. We confirm the validity of our model by reproducing the pulse-formation in all aspects: The evolution of the pulse energy, duration, and gain during the entire cavity buildup, demonstrating the nonlinear mode competition in full, as well as the dependence of the final pulse in steady state on the interplay between gain bandwidth, dispersion, and self-phase modulation. The direct observation of the nonlinear evolution of the pulse in space-time is a key enabler to analyze and optimize the Kerr-lens mode-locking operation, as well as to explore new nonlinear phenomena.