Fast source imaging using a multilevel non-uniform grid algorithm in oblate spheroidal coordinates

The Rayleigh-Sommerfeld (RS) back-propagation method applied to a single component of the measured electric field provides a good approximation for the field back-propagated from the measurement plane toward the source. The comparison between the field distribution reconstructed from the measurements and the desired one can be used for the localization of source anomalies. Since the measurement and, consequently, the integration involved in the RS back-propagation, are performed over a finite planar region, the practical implementation of this method is limited by truncation errors originating in a finite size of the measurement domain, which makes the method more suitable for the metrology of directional arrays or large reflector antennas. The usual practice of field back-propagation utilizes the plane wave spectrum (PWS) decomposition. Powered by the fast Fourier transform (FFT) algorithm, the PWS achieves a high computational efficiency, however this technique is limited to planar reconstruction surface and identical sampling steps in both input and output domains. On the other hand, the direct numerical calculation of the RS integral overcomes these limitations at the expense of a high computational complexity, i.e. proportional to the 4th power of the electrical size of the problem. In order to accelerate the RS integration, we propose to use a modified version of the multilevel nonuniform grid algorithm (MLNG), that achieves a high computational efficiency similar to that of the FFT.