Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths

A rigorous method for finding the best-connected orthogonal communication channels, modes, or degrees of freedom for scalar waves between two volumes of arbitrary shape and position is derived explicitly without assuming planar surfaces or paraxial approximations. The communication channels are the solutions of two eigenvalue problems and are identical to the cavity modes of a double phase-conjugate resonator. A sum rule for the connection strengths is also derived, the sum being a simple volume integral. These results are used to analyze rectangular prism volumes, small volumes, thin volumes in different relative orientations, and arbitrary near-field volumes: all situations in which previous planar approaches have failed for one or more reasons. Previous planar results are reproduced explicitly, extending them to finite depth. Depth is shown not to increase the number of communications modes unless the volumes are close when compared with their depths. How to estimate the connection strengths in some cases without a full solution of the eigenvalue problem is discussed so that estimates of the number of usable communications modes can be made from the sum rule. In general, the approach gives a rigorous basis for handling problems related to volume sources and receivers. It may be especially applicable in near-field problems and in situations in which volume is an intrinsic part of the problem.