Directive Shaping of Propagating Plane-Wave Fast Multipole Translation Operators

Two methods for the directive shaping of propagating plane-wave fast multipole method (FMM) translation operators (TLOPs) are discussed and compared. In the first approach, a real windowing function as often found in signal processing is considered within the truncated Legendre-polynomial series representation of the standard FMM TLOPs. The second approach utilizes Gaussian-beam (GB) TLOPs, which are obtained from complex-source theory. Both methods achieve considerable directive shaping and side-lobe suppression of the TLOPs and allow, thus, to drop a certain amount of plane-wave samples in the numerical integration of the corresponding planewave spectra in the FMM-based evaluation of time-harmonic electromagnetic integral operators. It is found that the GB TLOPs exhibit considerably better performance than TLOPs with standard real windowing functions. This is explained based on the observation that GB TLOPs can be interpreted as windowed TLOPs with a more appropriate complex windowing function, whose magnitude can even be larger than one. In order to achieve an accurate and efficient evaluation of the relevant integral operators, both approaches for directive shaping should work with an oversampled spectral representation as compared to the standard case together with a very careful choice of all relevant parameters.