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Assessment of the radiation characteristics of certain types of radome covered antennas poses the problem of transmission of amplitude tapered large-aperture fields through a curved dielectric layer. Modeling the amplitude taper by a Gaussian, the incident illumination can be generated by a source at a complex coordinate location. The radome problem is then addressed by tracing complex ray fields from the complex source point through the complex extension of the radome configuration to the real location of the observer. No integrations over equivalent apertures are required here. The complex ray tracing has been performed for conventional complex geometric trajectories as well as more accurate trajectories with lateral shifts at the radome interfaces, and takes into account multiple reflections inside the radome. As previously for real rays, the multiple internally reflected complex rays can be combined into a "collective ray" that is weighted with a curvature and (or) taper corrected slab transmission coefficient. Numerical calculations for various two-dimensional circular cylindrical and wedge-tapered prototypes and various beam illuminations demonstrate the feasibility of the complex ray method, the efficiency of the collective ray concept, and the adequacy of the ray model without lateral shifts for the radome application.