Complex ray analysis of radiation from large apertures with tapered illumination

Near and far zone calculation of the radiation characteristics of large apertures with tapered illumination generally requires time-consuming numerical integraton over the aperture plane because geometrical theory of diffraction (GTD) asymptotic techniques are difficult to apply to focused beam systems. Asymptotic methods do, however, become feasible for complex ray systems since ray fields originating in a complex coordinate space yield beam type fields in physical space. In particular, rays emanating from a complex source point are known to generate a real beam type field without sidelobes, very similar to that produced by a Gaussian amplitude taper. These considerations are generalized here to one-dimensional taper profiles of the form exp [
-kI
], where
k
is the wavenumber and
I
has a polynomial dependence on the aperture coordinate. Profile shapes ranging from Gaussian to rectangular are accommodated thereby. The analysis is performed by complex ray tracing from the analytically extended complex aperture plane to the observer. Criteria are developed for inclusion of all relevant rays, and only these, and for their domains of validity. These criteria are based on saddle point analysis of the exact field integral, whose direct numerical evaluation furnishes the reference solution. Saddle points and steepest descent paths corresponding to various envelope polynomials are traced in detail for near and far zone fields. Numerical comparisons between saddle point asymptotics (completely equivalent to complex ray tracing) and the reference solution establish that simple complex ray theory is useful and accurate for Gaussian-like profiles with moderate envelope gradients but that the strong gradients in almost square profiles require an excessively large number of contributing complex rays. For the latter profiles, where edge-diffraction-like phenomena predominate, modification of conventional GTD would provide a better alternative. Thus, this study establishes the complementary utility of complex ray tracing for smoothly tapered, and of GTD for almost rectangular, aperture distributions.