Translation operators based on Gaussian beams for the fast multipole method in two dimensions

An exact complex-space extension of the plane-wave Gegenbauer formula leads to a diagonal Gaussian translation operator for the fast multipole method (FMM) in two dimensions. The Gaussian beams are confined to the translation operator, and the fields are transmitted through plane waves as usual. The regions where the real source and receiver points can reside depend on the beam sharpness. As the beam gets sharper, the transverse dimensions of these regions get smaller. An arbitrarily high accuracy can be obtained with the Gaussian translation operator. The Gaussian translation operator makes it possible to disregard a large fraction of the plane-wave translations. The required sampling rate depends not only on the diameter of the source and receiver regions but also on the actual locations of the sources and receivers within those regions. For some common source–receiver geometries, the required sampling rate is below that of the standard translation operator. For other source–receiver geometries, the required sampling rate is greater than that of the standard translation operator. The theory is validated through numerical examples.

► Complex-space extension of the plane-wave Gegenbauer formula leads to a Gaussian translation operator for FMM.
► Gaussian beams are confined to the translation operator.
► The Gaussian translation operator makes it possible to disregard a large fraction of the plane-wave translations.