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A regime of multi-cascade proton acceleration in the interaction of$10^{21}-10^{22}$ W/cm$^2$ laser pulse with a structured target is proposed.The regime is based on the electron charge displacement under the action oflaser ponderomotive force and on the effect of relativistically induced slabtransparency which allows to realize idea of multi-cascade acceleration. It isshown that a target comprising several properly spaced apart thin foils canoptimize the acceleration process and give at the output quasi-monoenergeticbeams of protons with energies up to hundreds of MeV with energy spread of justfew percent.

Multi-Cascade Proton Acceleration by Superintense Laser Pulse in the Regime of Relativistically Induced Slab Transparency