Mathematical synthesis of the thickness profile of the waveguide Lüneburg lens using the adiabatic waveguide modes method

The paper describes the classical and generalized Luneburg lens in the 3D and planar waveguide implementation. We demonstrate the relation between the focusing inhomogeneity of the effective refractive index of waveguide Luneburg lens and the irregularity of the waveguide layer thickness generating this inhomogeneity. For the dispersion relation of the irregular thin-film waveguide in the model of adiabatic waveguide modes we solve the problem of mathematical synthesis and computer-aided design of the thickness profile of waveguide layer for the Luneburg thin-film generalized waveguide lens with a given focal length. The calculations are carried out in specially normalized coordinates convenient for computer calculations. The solution is compared with the one obtained using the cross-sections method.