  It is shown that a multilayer comprised of alternating thin superconducting
and insulating layers on a thick substrate can fully screen the applied
magnetic field exceeding the superheating fields $H_s$ of both the
superconducting layers and the substrate, the maximum Meissner field is
achieved at an optimum multilayer thickness. For instance, a dirty layer of
thickness $\sim 0.1\; \mu$m at the Nb surface could increase $H_s\simeq 240$ mT
of a clean Nb up to $H_s\simeq 290$ mT. Optimized multilayers of Nb$_3$Sn, NbN,
some of the iron pnictides, or alloyed Nb deposited onto the surface of the Nb
resonator cavities could potentially double the rf breakdown field, pushing the
peak accelerating electric fields above 100 MV/m while protecting the cavity
from dendritic thermomagnetic avalanches caused by local penetration of
vortices.
