Translation operator based on Gaussian beams for the fast multipole method in three dimensions

• Complex-space extension of the plane-wave Gegenbauer formula leads to Gaussian translation operator for FMM in three dimensions.
• 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.

Analytic continuation of Gegenbauer’s addition theorems produces a diagonal Gaussian translation operator for the fast multipole method (FMM) in three dimensions. The Gaussian beams affect only the translation operator, and as usual the field computation is performed with plane waves. Sampling theorems determine the plane-wave sampling rate required by the Gaussian translation operator. The formulation is based on an exact identity, so arbitrarily high accuracy can be achieved. 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. The directionality of the Gaussian translation operator makes it possible to disregard a large fraction of the plane-wave translations. Numerical simulations reveal that for general source–receiver geometries the required number of plane-wave translations grows linearly with the diameter of the source–receiver groups.