Efficient field computation using Gaussian beams for both transmission and reception

An exact representation is presented for the field inside a sphere (the observation sphere) due to primary sources enclosed by a second sphere (the source sphere). The regions bounded by the two spheres have no common points. The field of the primary sources is expressed in terms of Gaussian beams whose branch-cut disks are all centered at the origin of the source sphere. The expansion coefficients for the standing spherical waves in the observation sphere are expressed in terms of the output of Gaussian-beam receivers, whose branch-cut disks are all centered at the origin of the observation sphere. In this configuration the patterns of the transmitting and receiving beams “multiply” to produce a higher directivity than is usually seen with Gaussian beams. The areas on the unit sphere, which must be covered by the transmitting and receiving disk normals to achieve a given accuracy, diminish as 1/(ka) for ka → ∞, where a is the disk radius and k is the wavenumber. This 1/(ka) behavior leads to a single-level method with O(N^3/2) complexity for computing matrix-vector multiplications in scattering calculations (N is the number of unknowns).