A lightsail accelerated via directed energy is a candidate technology to send a probe into the deep space in a time period compatible with human life. The light emitted by a ground-based large-aperture phased laser array is directed onto the lightsail to produce a thrust by transferring the momentum of the incident photons. Here we demonstrate that optimized multilayer structures allow ultralight spacecraft being accelerated by laser radiation pressure up to 20% of the light velocity, and eventually even above, as long as a compromise between efficiency and weight is achieved. Layer materials are selected to provide high reflectance in the Doppler-shifted laser wavelength range as well as high emissivity in the infrared, this last characteristic being required to survive to the temperature increase during the acceleration phase.