  We present a complete numerical analysis and simulation of the full
spatio-temporal dynamics of Kerr-lens mode-locking (KLM) in a laser on all
time-scales. The KLM dynamics, which is the workhorse mechanism for generating
ultrashort pulses, relies on the intricate coupling between the spatial
nonlinear evolution due to self focusing and the temporal nonlinear compression
due to self-phase modulation (SPM) and dispersion. 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 one round-trip to the next. We employ a nonlinear ABCD
formalism that fully handles all relevant effects in the laser, namely - self
focusing and diffraction, dispersion and SPM, space-dependent loss and gain
saturation. We confirm the validity of our model by reproducing the
pulse-formation in KLM in all aspects: The evolution of the pulse energy,
duration, and gain is observed during the entire cavity buildup (from
spontaneous noise to steady state), 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 space-time evolution of the
pulse is a key enabler to analyse and optimize the KLM operation, as well as to
explore new nonlinear space-time phenomena.
