Rotational Doppler effect in left-handed materials

We explain the rotational Doppler effect associated with light beams carrying with orbital angular momentum in left- handed materials (LHMs). We demonstrate that the rotational Doppler effect in LHMs is unreversed, which is significantly different from the axial Doppler effect. The physics underlying this intriguing effect is the combined contributions of negative phase velocity and the inverse screw of the wave front. We find that the additional Doppler effect caused by Gouy phase and wave-front curvature should be reversed, because of the negative index. In the normal dispersion region, the rotational Doppler effect induces an upstream energy flow but a downstream momentum flow. In the anomalous dispersion region, however, the rotational Doppler effect produces a downstream energy flow but an upstream momentum flow. We theoretically predict that the rotational Doppler effect can induce a transfer of angular momentum of the LHM to orbital angular momentum of the beam.