Acceleration of protons by the radiation pressure of a circularly polarized laser pulse with the intensity up to 1021 W/cm^2 from a double-layer or multi-ion-mixed thin foil is investigated by two-dimensional particle-in-cell simulations. The double-layer foil is composed of a heavy ion layer and a proton layer. It is found that the radiation pressure acceleration can be classified into three regimes according to the laser intensity due to the different critical intensities for laser transparency with different ion species. When the laser intensity is moderately high, the laser pushes the electrons neither so slowly nor so quickly that the protons can catch up with the electrons, while the heavy ions cannot. Therefore, the protons can be accelerated efficiently. The proton beam generated from the double-layer foil is of better quality and higher energy than that from a pure proton foil with the same areal electron density. When the laser intensity is relatively low, both the protons and heavy ions are accelerated together, which is not favorable to the proton acceleration. When the laser intensity is relatively high, neither the heavy ions nor the protons can be accelerated efficiently due to the laser transparency through the target.
