We describe short-cavity In0.35Ga0.65As/GaAs multiple quantum well (MQW) lasers with undoped and p-doped active regions. The epilayer structure consists of four 5.7 nm QWs separated by 20.1 nm barriers in a GaAs core. The cladding layers consist of Al0.8Ga0.2As. In the case of p-doped devices a 4.5 nm carbon (C)-doped region (2.5 multiplied by 1019 cm-3) was inserted above each QW, separated by a 3.1 nm GaAs spacer, resulting in a modulation-doped core region. Using a CAIBE process, short-cavity ridge-waveguide lasers are fabricated in a triple-mesa geometry suitable for on-wafer probing. The best device (6 multiplied by 130 micrometers squared) with an undoped active region attained a damping-limited direct modulation bandwidth exceeding 40 GHz at a cw bias current of 160 mA. In contrast, the p-doped devices, demonstrating a maximum bandwidth of 37 GHz, are still limited by power dissipation. (alpha) -factors as low as 1.4 and 1.5 for undoped and p-doped devices, respectively, are extracted from measurements of the sub- threshold gain spectra. In addition, we demonstrate eye diagrams at 25 Gbit/s (limited by the pulse pattern generator) for these laser diodes. A complete characterization of dc and rf properties of these lasers is presented.