Complex multipole pulsed beams and Hermite pulsed beams: a novel expansion scheme for transient radiation from well-collimated apertures

Abstract

We are concerned with an expansion scheme for radiation from well-collimated pulsed aperture distribution that uses a new set of pulsed-beam (PB) basis functions. Several beam-expansion schemes have been introduced recently. They are based on local matching, within a self-consistent phase space format, of beam propagators to the source distribution. For well-collimated source distributions, greater efficiency may be obtained by matching wide beams to the entire aperture distribution, so that the propagators exhibit the same radiation properties as the entire aperture. This is the underlying strategy of the well-known Hermite–Gaussian-beam expansion of time-harmonic fields. In the present study a type of PB propagator, complex multipole PB’s (CMPB’s), is introduced and utilized in a similar expansion scheme for time-dependent radiation. CMPB’s are exact, highly localized, space–time wave-packet solutions that can be modeled analytically in terms of radiation from time-dependent complex multipoles as a generalization of the recently introduced complex source PB. The CMPB’s have paraxial profiles that involve transverse oscillations expressed in terms of Hermite polynomials and temporal derivatives; hence they form a biorthogonal set of basis functions for the expansion of collimated time-dependent fields. It is shown that the representation is most efficient if both the wave-front curvature and the collimation of the CMPB’s are matched to those of the aperture so that they exhibit the same far-field properties (e.g., diffraction angle and decay rate). We therefore introduce criteria to determine the collimation (Fresnel) length of a given pulsed source distribution and thereby for the optimal expansion parameters. To establish the near- and the far-field convergence properties of the expansion under matched and unmatched conditions, we derive closed-form expressions for the expansion amplitudes for a general class of time-dependent distributions and examine extensive numerical results.