Analytic design of refractive optical elements generating one-parameter directivity diagram

Design of refractive surface generating a one-parameter directivity diagram is considered for the case of collimated incident beam. Calculation of optical element generating a directivity diagram with an arbitrary shape is reduced to sequential solution of two explicit ordinary differential equations. The optical elements generating one-parameter directivity diagrams in the form of a line-segment, an arc, and a combination of two arcs are designed. The results of simulations demonstrate high performances of the designed optical elements.