A simplest saturable absorber, in the form of an unpumped section, is introduced into a Fabry-Perot semiconductor laser with a strongly asymmetric broadened waveguide structure incorporating a relatively thick (80 nm) active layer. This allows for suppression of trailing oscillations and a decrease in the optical pulse width compared to the uniformly biased structure. Single optical pulses of ~80 ps full width at half maximum (FWHM) and ~35 W peak power (~3 nJ pulse energy, E(opt)), practically without trailing edge oscillations, were experimentally achieved under room temperature conditions by absorber-assisted gain-switching, using pumping current pulses of ~1.3 ns FWHM and ~17 A amplitude. The laser emission has a narrow (13 degrees FWHM in the transverse direction) far field.