Propagating Plane-Wave Fast Multipole Translation Operators Revisited—Standard, Windowed, Gaussian Beam

Propagating plane-wave fast multipole translation operators (TLOPs) convert propagating plane-wave spectra radiated from a source region of finite extent into incident propagating plane-wave spectra at a distant observation region of finite extent. This translation is diagonal, and the received fields in the observation region are obtained by integration of the received plane-wave spectra. In its standard form, the TLOPs exhibit some directivity but with strong sidelobes and the integration needs to consider the entire spectra. Windowed and Gaussian-beam (GB) TLOPs with improved directivity and reduced sidelobes have been considered in order to reduce the integration effort. Both these techniques are investigated in this article, and it will be demonstrated that GB TLOPs are considerably more powerful. It is shown that GB TLOPs can be interpreted as windowed TLOPs, where a complex windowing function is multiplied with the Legendre polynomial series coefficients of the standard TLOPs. The sampling requirements of the GB TLOPs are investigated, and a numerical procedure is presented for an error-controlled determination of the imaginary shift involved in the computation of the GB TLOPs and their orders. Numerical results obtained with an inverse source solver for several antennas with aperture sizes of up to 1000 wavelengths are presented.