Fig. 3: Double-layer lenses with one focus at r = 1.
From an initial point source at rs = 1, ϕ = π (blue dot) we trace the ray motion through the profiles defined through the implicit Eq. (30) for different values of M. To aid visualization we plot the rays launched in the +y direction as dashed lines, and those launched in the −y direction as solid lines. We give four examples, where the turning angle Δϕ is π plus a π/4; b 3π/4; c π; and d 2π. When M is integer, the double-layer lens has the functionality of a ordinary (for even numbers) and double-layer (for odd numbers) Luneburg lens, otherwise, it acts as a beam splitter fed by a point source.
