Profile Inversion and Closed Form Formulation of Compact GRIN Lenses

New formulas for the design of cylindrical Graded-Index (GRIN) lens-Antennas with integrated feeder are presented. The possibility of integrating the feeder within the lens makes the system more compact, avoids complex mechanical design and alignment errors. The lens is characterized in analytical form by Geometrical Optics (GO). The expression of the radially varying refractive index is derived by applying the conservation of the momentum inside the lens and imposing parallel rays at the lens output interface. The mathematical procedure to retrieve the refractive index is based on the inversion of a truncated Abel transform. The Poynting vector at the lens aperture is derived by applying the conservation of energy in each elementary ray-Tube. By approximating the Poynting vector to a Gaussian function the total efficiency of the lens-Antenna is derived in a closed form, allowing for a quick lens design. The proposed formulation has been successfully validated by using ray-Tracing and a full-wave simulations. Finally, we present examples of practical design of GRIN lenses by using holes of different shapes in a dielectric ABS/Teflon host media.