Phase-space beam summation: A local spectrum analysis of time-dependent radiation

The phase-space beam summation is a general analytical frame- work for local analysis and modeling of radiation from extended source distributions. In this formulation the field is expressed as a superposition of beam propagators that emanate from all points in the source domain and in all directions. The theory is presented here for both time-harmonic and time- dependent fields: in the later case, the propagators are pulsed-beams (PB). The phase-space spectrum of beam propagators is matched locally to the source distribution via local spectral transforms: a local Fourier transform for time-harmonic fields and a "local Radon transform" for time-dependent fields. These transforms extract the local radiation properties of the source distributions and thus provide a priori localized field representations. Some of these basic concepts have been introduced previously for two-dimensional configurations. The present paper extends the theory to three dimensions, derives the operative expressions for the transforms and discusses additional phenomena due to the three dimensionality. Special emphasis is placed on numerical implementation and on choosing a numerically converging space- time window. It is found that the twice differentiated Gaussian-δ window is both properly converging and provides a convenient propagator that can readily be tracked in complicated inhomogeneous medium.