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 centered in 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 centered in 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. This leads to a fast method for computing matrix-vector multiplications in scattering calculations, as will be illustrated for a Dirichlet square plate.