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

We extend the study of alternative phase-space formulations of time-harmonic radiation from extended but truncated aperture source distributions to the time domain. Included are nonwindowed continuous forms spanning the space–time (configuration) domains, wave-number-frequency (spectrum) domains, and windowed (local beam-type) continuous forms. Synthesized in the frequency domain by nonwindowed or windowed Fourier transforms, field synthesis in the time domain involves nonwindowed or windowed radon transforms combined with the theory of analytic signals. Because the properties of suitable wave objects used in the analysis and synthesis of the field are strongly tied to relevant configurational and spectral parameters, the incorporation of these aspects into the various formats is referred to as phase-space parameterization. In the continuous parameterization the resulting time-dependent field radiated from the aperture is expressed as a superposition of pulsed beams whose phase-space parameters are their initiation time, initiation location, and initial direction. The properties of these formulations are discussed in detail, within a rigorous format and also with more physically transparent asymptotic approximations. As in the time-harmonic case, major stress is placed on localization in the phase space, which is achievable with various alternatives, and on the corresponding implications. Specific examples include analytic δ windows that yield as propagators complex-source pulsed beams, and numerical implementation of field synthesis for nonfocused and focused pulsed aperture distributions.