Gradient-index lens design with quasi-conformal transformation optics for energy concentration

Transformation Optics (TO) has enabled new methodologies for the design and specification of gradient-index (GRIN) lenses for radio-frequency and optical applications by linking refractive index gradients to a mathematically equivalent change in geometry in another dimension. With the new mathematical design tools, there have been many interesting devices introduced in the literature, such as optical collimators and absorbers (optical "black holes"), GRIN couplers and bends for optical waveguides, and compressed or flattened collimating lenses for imaging and non-imaging applications. Many of the most interesting TO designs are not feasible for implementation, however, due to the complex anisotropic, inhomogeneous material parameters required by the full TO formulation. Instead, restricting the geometric transformations to be mathematically conformal or quasi-conformal (qTO) eliminates the anisotropic material requirements and allows implementation with an isotropic 2D or 3D GRIN profile, for which multiple fabrication methods exist in the RF and IR wavelength ranges and are under development for the complete optical spectrum. We present an overview of the usefulness of combining TO, qTO, and GRIN optics for energy concentration along with the associated design and analysis techniques. Moving away from traditional lenses to GRIN and TO optics for which, in general, no analytical geometric optics or full-wave solution exists, involves the development of new design strategies for individual lenses and systems of lenses. We demonstrate results obtained using advanced, multivariate optimizations that are tightly coupled to a fast, advanced inhomogeneous ray tracing engine for electrically-large lenses, and to an efficient body-of-revolution solver for electrically-small cylindrically-symmetric lenses.