To achieve higher resolution extreme ultraviolet lithography (EUVL) targeted toward sub-10 nm, reflective projection scanner image numerical apertures (NAi) are being increased beyond the current value of 0.33 to 0.55 and upward of 0.75 as a desirable target. Bragg reflectors using alternating silicon and molybdenum that have heretofore been coated as periodic multilayers cannot achieve desired reflected amplitudes as corresponding 0.25× mask numerical apertures (NAm) are accordingly increased. In addition, transverse magnetic-polarized image modulation decreases with NA, which becomes significant at 0.55 and above. We present here the optimization of non-regular alternating, or aperiodic, silicon-molybdenum multilayer reflective coatings that can achieve improved amplitude and polarization performance through angle as higher-NA EUVL lithography is pursued. Through the use of rigorous EM computation paired with a genetic optimization method, we show that amplitude apodization can be recovered to 60% peak reflectance for NAm values up to 0.2 (corresponding to NAi values of 0.8) while at the same time achieving a transverse electric degree of polarization exceeding 40%. In addition, aperiodic multilayers optimized for spectral bandwidth using refractory metals also show improvement over periodic designs.