Theory and group velocity of ultrashort, tightly focused laser pulses

The wave equation describing an ultrashort, tightly focused laser pulse in vacuum is solved analytically. Plasma dispersive effects are also included. Based on exact short-pulse solutions, analytical expressions are obtained for the pulse-length evolution, the pulse centroid motion, and the group velocity. Approximate shortpulse solutions are obtained to arbitrary order in the parameter ly2pL , 1, where l is the pulse wavelength and L is the length of the pulse envelope. Comparisons are made to the solutions of the paraxial wave equation and to numerical solutions of the full wave equation. The exact analytical expression for the pulse group velocity vg, which is correctly determined from the motion of the pulse centroid, is in excellent agreement with the numerical solution. In vacuum, 1 2 vgyc > sly2pr0d, where r0 is the laser spot size at focus. Estimates for the quantity 1 2 vgyc, based on the paraxial wave equation, are found to be in error by a factor of 2.