Fig. 1: Schematic of the 2D double-layer lens problem.
a Graded index double-layer lens defined under polar coordinate system (r, ϕ). Incident rays from a point source at r = rs in upper waveguide are first refracted by the profile n1(r) before encountering the mirror on the far side of the device. The mirror reflects the ray into a lower waveguide where it is refracted by the profile n2(r), focusing all rays to a point image at r = ri. The thickness of the waveguides is neglected in first approximation for the purposes of this development, considering it is small compared to the wavelength. b Equivalent geodesic double-layer lens, where the refractive index is placed with an out of plane deformation z(ρ) of the upper/lower waveguide described by the arc length s1,2(ρ) of the surface measured along the meridian from the axis of symmetry under a cylindrical coordinate system (ρ, ϕ, z). In both cases, the background medium outside the lens has refractive indices na and nb for the upper and lower waveguides, respectively.
