Spectral analysis of complex-source pulsed beams: Reply to comment

By assigning complex values to the source coordinates and pulse-initiation time of the time-dependent Green function in free space, one may generate a field solution that behaves like a propagating pulsed beam. Although the conventional pulsed line source response is known in closed form, the complex extension cannot be performed directly thereon because of the nonanalytic behavior of the causal field. The analytic continuation is carried, out here by spectral analysis and synthesis, utilizing the recently formulated spectral theory of transients. This approach not only guarantees uniqueness but also elucidates the spectral content of the resulting waveform, which is composed of contributions from singularities in the complex spectral wave-number plane. By similar analytic extension of time-dependent Green functions for more complicated environments, one may construct directly the transient field produced in these environments by the incident pulsed beam.