Soft gain aperture effect from pump beam size inside the laser crystal is crucial for obtaining stable Kerr lens mode-locking and generating ultrabroad-bandwidth pulses in Ti∶sapphire laser. Modelocking dynamics is experimentally investigated for a diode-pumped all-solid-state laser. At the pump power of 4 W,the femtosecond laser pulses tunable from 794 nm to 835 nm are generated with high output power of 570mW and extremely large bandwidths at the maximum of 135 nm,which can support the transform-limited pulse width in sub-10 fs time duration. Moreover,this laser is extended to blue pulses tunable from 418 nm to 429 nm by using a 260 μm-thick BBO crystal.The measured ultrabroad bandwidths are unprecedent in this spectral region,and their transformlimited pulse durations are sub-15 fs.