A phase-space beam summation formulation for ultrawide-band radiation. II: a multiband scheme

The ultrawide-band (UWB) beam summation representation introduced in Part I of this two-part sequence, is extended here to make it more efficient for excitations with bandwidths that are larger than one octave. The main features of the basic formulation in Part I were: 1) utilization of overcomplete windowed Fourier transform (WFT) frames to construct the beam lattice (or skeleton) that is independent of the frequency; 2) the use of isodiffracting Gaussian beams (ID-GBs) provides the snuggest frame representation, and thus stable and localized expansion coefficients, for all frequencies; 3) the ID-GBs can readily be tracked in the ambient medium and, due to the ID property, their propagation parameters are calculated only once and then can be used for all frequencies. Although the basic formulation can accommodate large bandwidths, it becomes increasingly less efficient at the low end of the frequency spectrum where the overcompleteness increases. The self-consistent multiband beam-summation scheme presented here extends and generalizes the basic formulation of Part I by dividing the excitation band into one-octave sub-bands and applying the UWB beam expansion of Part I in each band. This is done, though, via a novel self-consistent scheme wherein the beam sets at the lower bands are decimated subsets of those at the highest band, so that only the set of beam-propagators at the highest band needs to be traced, while for the lower bands one uses properly decimated subsets. This approach requires less beam elements at the lower frequency bands and thus keeps the overcompleteness (oversampling) in all the bands below a given level. As in Part I, we provide the guidelines for choosing the expansion parameters and then demonstrate the effectiveness of the new scheme via a numerical example of UWB focused aperture whose frequency band spans several octaves.