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)].

Figure 1

(a) Maxwell's fish-eye lens in $w$ space with a refractive index profile ranging from 1 to 2. (b) A square fish-eye lens in $z$ space with a refractive index profile ranging from zero to two. (b) A waveguide constructed by an array of square fish-eye lenses. In simulations, we will set the arbitrary unit of length as the meter.

Figure 2

The ray tracing simulation for a waveguide constructed by the square fish-eye lenses for a source at (a) (0, 0), (b) (0.5, 0), (c) (0.5, 0.5), and (d) (1, 0.7). In simulations, we set the arbitrary unit of length as the meter.

Figure 3

The ray tracing simulation for three sources at (0, 0) [the red (light gray) dotted curves], (0, 0.5) (the black solid curves), and (0.5, 0.5) (the blue dashed curves). In simulations, we set the arbitrary unit of length as the meter.

Figure 4

The electric-field patterns for a waveguide constructed by the square fish-eye lenses for a line current source at (a) (0, 0), (b) (0.5, 0), (c) (0.5, 0.5), and (d) (1, 0.7), for a wavelength of 0.25. In simulations, we set the arbitrary unit of length as the meter.

Figure 5

The amplitudes of the electric fields of a waveguide constructed by the square fish-eye lenses for a line current source at (a) (0.5, 0) for a wavelength of 0.25, (b) (0.5, 0.5) for a wavelength of 0.25, (c) (0.5, 0) for a wavelength of 0.15, and (d) (0.5, 0.5) for a wavelength of 0.15. In simulations, we set the arbitrary unit of length as the meter.

Figure 6

The electric-field amplitudes along the line $y=0$ for a source at (a) (0, 0) and (b) (0.5, 0), and two sources at (c) (0, 0), (0.5, 0) and (d) (0, 0,), (0.75, 0). The wavelength is 0.15. The corresponding FWHMs for each peak have also been plotted together. In simulations, we set the arbitrary unit of length (wavelength, coordinates, and FWHM) as the meter.

Figure 7

The ray tracing simulation for a photonic crystal for a source at (a) $(–1.6,–0.4)$ and (b) (1, 0.5), and the electric-field patterns for a photonic crystal for a line current source at (a) $(–1.6,–0.4)$ and (b) (1, 0.5), for a wavelength of 0.25. In simulations, we set the arbitrary unit of length as the meter.