Gradient-index Geodesic Lenses for Integrated Optics: A Uniform Theory

A new generation of planar waveguide lenses for integrated-optical circuits is proposed. They have potentially perfect focusing capabilities due to the assumed simultaneous action of the geodesic curvature of the thin film and of the gradient of its apparent refractive index. The ‘joint forces’ philosophy permits the range of the index variation, as well as the depth of the geodesic profile, to be considerably reduced or, at least, designed with a formerly inaccessible amount of flexibility, as compared with Luneburg and geodesic lenses. In the two-parts of this paper as a whole, a suitable theory of radially symmetric gradient-index geodesic lenses is formulated, the final equations of which provide a relatively simple and concise description of a family of possible full-aperture and non-full-aperture solutions with arbitrary focal length. Part I provides a detailed analysis of the full-aperture lens problem, together with two illustrative numerical examples. These two solutions are characterized by an analytically described moderate refractive-index variation, the first having an aspherical but smooth-transition complementary geodesic profile and the second involving spherical lens depression.