Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams

It is well known that very high axial - or longitudinal - electric fields can be obtained in vacuo using very intense focused laser beams. If these fields are in the transverse magnetic M-01 mode, very high gains in energy (exceeding 1 GeV) can be obtained if sufficiently energetic ultra-relativistic electrons (say I GeV) are injected on-axis and near the optimum phase. This energy gain is obtained even though the phase velocity of the axial component of the TM01 laser beam is greater than the vacuum velocity of light. The result is apparently related to the Gouy phase shift at focus.