Propagating pulsed beam solutions by complex source parameter substitution

The complex source point method is extended to the time domain by considering the field due to an impulsive source point with complex space coordinates and a complex initiation time. The resulting solution, which is a particular analytic continuation of the free space time-dependent Green's function, describes a propagating pulsed beam that has a moving peak along the beam axis. Near the pulse maximum in the paraxial region, this new field type is the Fourier transform into the time domain of the time-harmonic paraxial Gaussian beam field but the method also accommodates, in closed form, observation points far from the beam axis and observation times long before and after the peak has passed. By corresponding analytic continuation of available space-time Green's functions for various propagation and diffraction environments, one may generate directly the response due to the pulsed beam incident on the environment.