Cross polarization in reflector-type beam waveguides and antennas

Using the paraxial ray approximation, simple formulas for the cross polarization introduced by curved reflectors are developed. In particular, when the reflectors are quadric surfaces of revolution with the center ray of the beam passing through the foci, the maximum cross-polarized field amplitude throughout a gaussian beam, relative to the on-axis copolarized field, is $C = {2\xi\kappa\bot\over \sqrt{e}}\sin \theta_{i},$ where e is the base of the natural logarithm, ξ is the 1/e power radius of the beam, ki is the curvature of the reflector perpendicular to the plane of incidence, and θi is the angle of incidence. For such reflectors, the beam fields are accurately represented by a superposition of just two gaussian modes for each plane of polarization: the fundamental mode, which corresponds to the co-polarized gaussian beam, and a higher-order mode, which accounts for the cross-polarized field and the amplitude “space” taper. Transformation of a beam through a general sequence of such reflectors is influenced by three factors: the curved reflectors, longitudinal propagation lengths, and rotations of the plane of incidence. The effect of each factor is described by a 4 × 4 matrix relating the input and output gaussian modes. Several typical beam-reflector systems are analyzed by this method. Theoretical cross-polarization patterns are shown to be in accurate agreement with measurements on a symmetrical dual-reflector system.