Theory of mode locking with a slow saturable absorber

A closed-form solution is presented for the pulses of a homogeneously broadened laser mode locked by a saturable absorber of relaxation time much longer than the pulsewidth. With suitable approximations, the pulse shape is a secant hyperbolic. The system is described in terms of the linewidth of the system, the saturation energies of the saturable absorber and the laser medium, the ratio of saturable-absorber loss to resonator loss, and the ratio of cavity round-trip time and laser-medium relaxation time. The energy and width of the pulse are obtained as functions of these parameters. Graphs are presented for the operating regimes as bounded by the choice of the preceding parameters. The theoretical predictions are compared with second-harmonic-generation (SHG) autocorrelation traces of mode-locked dye-laser pulses.