Spatiotemporal evolution of focused single-cycle electromagnetic pulses

We analyze exact solutions of Maxwell's equations that are capable of describing focused single-cycle electromagnetic pulses. These finite energy solutions are a subset of Ziollcowski's ''modified power spectrum'' pulse solutions [Phys. Rev A 39, 2005 (1989)]. They display substantial temporal reshaping, time reversal, and polarity reversals as they pass through the focus. The temporal profiles at the focus and in the far field are related by a Hilbert transform in time. These results are explained in terms of the Gouy phase shift of focused beams. We also show that these pulse solutions are natural spatiotemporal modes of an open resonator and propose methods for their practical realization.