Ultrashort focused electromagnetic pulses

In this paper we present a closed analytical description for few-cycle, focused electromagnetic pulses of arbitrary duration and carrier-envelope phase. Because of the vectorial character of light, not all thinkable one-dimensional shapes for the transverse electric field or vector potential can be realized as finite energy three-dimensional (3D) structures. We cope with this problem by using a second potential, which is defined as a primitive to the vector potential. This allows one to construct fully consistent 3D wave-packet solutions for the Maxwell equations, given a solution of the scalar wave equation. The wave equation is solved for ultrashort, Gaussian, and related pulses in paraxial approximation. The solution is given in a closed and numerically convenient form, based on the complex error function. All results undergo thorough numerical testing, validating their correctness and accuracy. A reliable and accurate representation of few-cycle pulses is, e.g., crucial for analytical and numerical theory of vacuum particle acceleration.