Combined complex Doppler and Cherenkov effect in left-handed metamaterials

We derive the formula of the complex Doppler shift in a two-dimensional, dispersive metamaterial and we show that a moving, monochromatic radiation source generates multiple frequency modes. The role of the group velocity is stressed and the Doppler shifted radiation field exhibits features of the Cherenkov effect. The presented theory is also applicable to the case of a moving, nonoscillating charge and explains many peculiar characteristics of the Cherenkov radiation in lefthanded metamaterials such as the backward direction of power emission, the constant radiation angle and the lack of velocity threshold.