Cleaning of contamination of optical surfaces by amorphous carbon (a-C) is highly relevant for extreme ultraviolet (EUV) lithography. We have studied the mechanisms for a-C removal from a Si surface. By comparing a-C removal in a surface wave discharge (SWD) plasma and an EUV-induced plasma, the cleaning mechanisms for hydrogen and helium gas environments were determined. The C-atom removal per incident ion was estimated for different sample bias voltages and ion fluxes. It was found that H2 plasmas generally had higher cleaning rates than He plasmas: up to seven times higher for more negatively biased samples in EUV induced plasma. Moreover, for H2, EUV induced plasma was found to be 2-3 times more efficient at removing carbon than the SWD plasma. It was observed carbon removal during exposure to He is due to physical sputtering by He+ ions. In H2, on the other hand, the increase in carbon removal rates is due to chemical sputtering. This is a new C cleaning mechanism for EUV-induced plasma, which we call "EUV-reactive ion sputtering".