Nonlinear planar forward and backward projection

Planar projection methods have been shown to rapidly relate fields between two planes. Such an approach is particularly useful for characterizing transducers, since only a single plane needs to be measured in order to characterize an entire field. The present work considers the same approach for the case of a nonlinear field in the presence of an arbitrary dispersion relation. Unlike traditional methods that use Fourier solutions of the time-domain wave equation, the approach forms a frequency-domain integral equation for waves in a dispersive medium. It is shown that a transfer function similar to that derived from time domain equations can be utilized. Both the forward- and backward-projection behaviors are examined and it is demonstrated that the approach is invariant to propagation direction.