The field widening of a sweep mirror field-widened Fourier transform imaging spectrometer is achieved by adding a sweep mirror before its optical system. However, the addition of the sweep mirror causes the cross distribution of rays from different fields and different channels, which presents a difficulty in the suppression of stray light, and thus the traditional stray light suppression design is no longer valid. To solve this problem, a new design of stray light suppression is proposed. Based on the stray light suppression design, the point source transmittance (PST) stimulation and analysis of the visible and near-infrared optical system and the shortwave infrared optical system is presented for the sweep mirror at different field of view locations of 0°, 15°, and -15°. The analysis results show that as for all sweep mirror locations, the PST along the spatial direction and spectral direction at 0.5° off-field angle can be reduced to 10 ^-3. Based on the observation mode, the signal source and stray light source of the on-orbit imaging spectrometer are analyzed. In addition, based on the PST stimulation results, the stray light irradiance at the focal plane is studied and the signal-to-stray ratio is calculated. It shows that the signal-to-stray ratio of the visible and near-infrared optical system is 0.1% and that of the shortwave infrared optical system is 0.6%. The results indicate that the proposed stray light suppression method is effective and can meet the requirement of the space-borne imaging spectrometer.