Relativistic aberration and null Doppler shift within the framework of superluminal and subluminal nondiffracting waves

Abstract

We reveal relations between superluminal and subluminal spatiotemporally localized optical pulses and the phenomena of the relativistic aberration of a wavevector and the Doppler shift. It turns out that such waves are Lorentz-transformed versions of simple optical waves generated in another reference frame. The requirement of a null Doppler shift is shown to give rise to a speed associated with the relativistic velocity addition law of a double (two-step) Lorentz transformation. The effects of such a transformation are examined both in terms of four-coordinates and in the spectral domain. It is established that a subluminal pulse reverses its direction. In addition to a change in direction, the propagation term of a superluminal pulse becomes negative. The aberration due to a double Lorentz transformation is examined in detail for propagation invariant superluminal waves (X wave, Bessel X wave), as well as intensity-invariant superluminal and subluminal waves. Detailed symmetry considerations are provided for the superluminal focus X wave and the subluminal MacKinnon wavepacket.