Doppler effect of Laguerre-Gaussian beams propagating in left-handed materials

We investigate the Doppler effect associated with light beams carrying orbital angular momentum in left-handed materials (LHMs). The Doppler shift divides into four tapes of contribution: an axial contribution along the propagation direction, a contribution due to the Gouy phase, a contribution of the beam curvature, and finally an azimuthal contribution due to the screwing wavefront. We demonstrate that the axial Doppler effect in LHMs is reversed as expected. The additional contribution caused by Gouy phase shift and beam curvature should also be reversed. However the azimuthal Doppler shift, proportional to the orbital angular momentum of photons, is unreversed. The physics underlying this intriguing effect is the joint contributions of negative phase velocity and inverse screw of wave-front. We predict that the orbital angular moment of the photon in LHMs should remain its direction unreversed.