Maxwell's fish-eye lenses under Schwartz-Christoffel mappings

In this paper, we designed Maxwell's fish-eye lens in a square shape from a conformal mapping and constructed it into a waveguide and a photonic crystal. We studied the unidirectional self-imaging effect from both geometric optics and wave optics. Ray tracing and full-wave simulations were performed to validate this effect. Specifically, the designed waveguide is of a finite width while having the same imaging properties as the ideal infinite waveguide-the Mikaelian lens, the refractive index profile of which is 1/cosh (x) [Mikaelian and Prokhorov, Prog. Opt. 17, 279 (1980)].