This paper presents the modified achromatic Savart Polariscope for constructing broadband spatially modulated snapshot imaging polarimeter. The modified achromatic Savart Polariscope can be made from two different birefringent crystals. The achievements and performances of the modified achromatic Savart Polariscope are demonstrated with numerical simulations. The chromatic variations in spatial carrier frequencies can be reduced by an order of magnitude. The spatially modulated snapshot imaging polarimeter constructed by the modified achromatic Savart Polariscopes can achieve the root mean square errors with a magnitude of 10 - 4 across the spectral range 0.48-0.96 ism as it does in monochromatic light conditions.