Three operating modes of femtosecond and picosecond pulses,which are independent self mode-locking,cross mode-locking and multi-pulse operating,are presented in a two-beam-pumped Ti:sapphire laser.Group velocity dispersion,self-phase modulation,cross-phase modulation,gain competition and self-amplitude modulation in the femtosecond and picosecond cavities are analyzed.The results show that,in the independent self mode-locking mode,group velocity dispersion and selfphase modulation largely determine the prop...