  We propose and substantiate experimentally the cascaded rotational Doppler
effect for interactions of spinning objects with light carrying angular
momentum. Based on the law of parity conservation for electromagnetic
interactions, we reveal that the frequency shift can be doubled through
cascading two rotational Doppler processes which are mirror-imaged to each
other. This effect is further experimentally verified with a rotating half-wave
plate, and the mirror-imaging process is achieved by reflecting the
frequency-shifted circularly polarized wave upon a mirror with a quarter-wave
plate in front of it. The mirror symmetry and thus parity conservation
guarantees that this doubled frequency shift can be further multiplied with
more successive mirror-imaging conjugations, with photons carrying spin and/or
orbital angular momentum, which could be widely applied for detection of
rotating systems ranging from molecules to celestial bodies with high precision
and sensitivity.
