Relativistic-flying laser focus by a laser-produced parabolic plasma mirror

The question of electromagnetic field intensification towards the values typical for strong field Quantum Electrodynamics is of fundamental importance. One of the most promising intensification schemes is based on the relativistic-flying mirror concept, which shows that the electromagnetic radiation reflected by the mirror will be frequency up-shifted by a factor of 4 gamma^2 (gamma: the Lorentz factor of the mirror). In laser-plasma interactions, such a mirror travels with relativistic velocities and typically has a parabolic form, which is advantageous for light intensification. Thus, a relativistic-flying parabolic mirror reflects the counter-propagating radiation in a form of focused and flying electromagnetic wave with a high frequency. The relativistic-flying motion of the laser focus makes the electric and magnetic field distributions of the focus complicated, and the mathematical expressions describing the field distributions of the focus is important. We present analytical expressions describing the field distribution formed by an ideal flying mirror having a perfect reflectance over the entire surface and wavelength range. The peak field strength of an incident laser pulse with a center wavelength of lambda_0 and an effective beam radius of w_e is enhanced by a factor proportional to gamma^3 (w_e/lambda_0) in the relativistic limit. Electron-positron pair production is investigated in the context of invariant fields based on the enhanced electromagnetic field. The pair production rate under the relativistic-flying laser focus is modified by the Lorentz gamma-factor and the beam radius-wavelength ratio (w_e/lambda_0). We show that the electron-positron pairs can be created by colliding two counter-propagating relativistic-flying laser focuses in vacuum, each of which is formed when a 180 TW laser pulse is reflected by a relativistic-flying parabolic mirror with a gamma = 12.2.