Analytical Computation of Conjugate Points' Location in 3D Parabolic Reflector Systems

The location of conjugate points plays a crucial role in understanding and predicting certain radiation characteristics of reflector antenna systems. In this manuscript, we present an analytical approach for determining the location of conjugate points for both broadside radiation and beam scanning. In the 2D case, our methodology is based on the well-known mirror formula. Moving to the 3D case, we demonstrate that for broadside radiation, reflected rays converge to a single point. However, during beam scanning, incident rays from different planes converge to different points after reflection. These points can be analytically determined by locally approximating the reflector as either an ellipsoid or a hyperboloid. Our findings highlight the limitations of the conjugate point model for larger scan angles.