Monolithic two-section InGaAs/GaAs double quantum well (DQW) passively mode-locked lasers (MLLs) with asymmetric waveguide, consisting of the layers of p-doped AlGaAs waveguide and no-doped InGaAsP waveguide, emitting at ~ 1.06 μm, with a fundamental repetition rate at ~ 19.56 GHz have been demonstrated. Modal gain characteristics, such as a gain bandwidth and a gain peak wavelength of the MLL, as a function of the saturable absorber (SA) bias voltage (Va) as well as the injection current of gain section (Ig), were investigated by the Hakki-Paoli method. With the increase of Va, the lasing wavelength and net modal gain peak of the MLL both exhibited red-shifts to longer wavelength significantly, while the modal gain bandwidth was narrowed. Both the net modal gain bandwidth and gain peak of the MLL followed a polynomial distribution versus the reverse bias at the absorber section. In addition, for the first time, it was found that Va had an obvious effect on the modal gain characteristics of the MLL.