Phase-space beam summation for time-dependent radiation from large apertures: Discretized parameterization

In this last portion of a three-part sequence on phase-space formulations of radiation from distributed aperture sources, we deal with the discretized parameterizations of the phase space for the transient case, extending into the time domain the discretized formats and numerical examples of the time-harmonic treatment. We utilize Gabor discretization for both the spatial and temporal dependence of the initial time-dependent aperture field. The resulting basis functions are localized in both the time-frequency and the space-wave-number domains, and they generate localized pulsed beams (PB’s) that propagate away from the aperture plane along well-defined space-time trajectories. The time-dependent field is therefore described as a fourfold discrete superposition of shifted, rotated, delayed, and modulated PB’s. Because the Gabor lattice of initial points and radiation directions is frequency dependent, alternative choices for the lattice may yield either frequency-independent initial points or frequency-independent radiation directions but not both. The general formulation is then specialized to spacetime Gaussian windows, for which we derive explicit asymptotic expressions that can be interpreted in terms of space-time Gaussian PB’s. Numerical examples illustrate the properties of the discretized expansion.