A semi-analytical method is presented for the design of gradient index (GRIN) flat lenses. Closed-form expressions are derived to define the refractive index distribution of the lens, for several cases: collimating lenses with on-axis feed, collimating lenses with off-axis feed, lenses converting spherical wavefronts with different wavenumbers, lenses changing the focal number of a quasi-optical system, and Fresnel zone lenses. The design equations are validated by ray-tracing simulations in inhomogeneous media, implemented by numerical solution of the Eikonal equation.

This work aims to provide guidelines on the design of wideband flat lenses based on artificial dielectric layers (ADLs). Planar lenses based on metasurfaces are typically narrowband, due to the phase wrapping over the period of 2\pi that is strongly frequency-dependent. On the contrary, true-time-delay (TTD) planar lenses, which do not resort to phase discontinuities, can achieve large bandwidths. One convenient way to design wideband TTD lenses is by means of ADLs, which are stacks of subwavelength-period patch arrays embedded in a host medium to increase its effective permittivity. Tradeoffs including bandwidth, focal ratio, lens diameter, and thickness are discussed and related to the manufacturing constraints of artificial dielectrics, such as the smallest features realizable in printed circuit board (PCB) technology, which define the maximum achievable effective permittivity. An example of design is also presented, operating from 30 to 60 GHz and experimentally validated.