We demonstrate the generation of an intense femtosecond optical vortex (OV) pulse by employing an OV converter set between two laser amplifiers in a chirped-pulse amplification (CPA) system of a Ti:sapphire laser. The OV converter is composed of a liquid-crystal spatial light modulator (LC-SLM) exhibiting a blazed-phase computer-generated hologram, a concave mirror, and a flat mirror in the 4f setup. Owing to the intrinsic nature of the 4f setup, the OV converter is free from chromatic and topological-charge dispersions, which are always induced in a spiral phase plate conventionally used to convert an intense Gaussian laser pulse to an OV pulse, while we can avoid damage to the LC-SLM by the irradiation of a low-energy pulse before the second amplifier. We have increased the throughput of the OV converter to 42% by systematically investigating the diffraction efficiency of the blazed-phase hologram on the LC-SLM, which relaxes the gain condition required for the second amplifier. The combination of the high-throughput OV converter and the two-stage amplification enables us to generate OV pulses with an energy of 1.63 mJ and a pulse duration of 60 fs at a wavelength of ~720 nm, at which the gain of the Ti:sapphire laser is only 60% of the peak gain around 800 nm.