Fig. 1. Schematic of a transformation relation. (a) The original space, which corresponds to the isotropic space w=x+iy (the red square grids). (b) The physical space, which corresponds to the anisotropic space w′=μyx′+iμxy′ (the red rectangular grids). (c) The transformation relations of Re(x′)−μy with μx=1 and μy≥0 will be changed to Im(x′)−μy when μy<0 with μx=1 and x=−2, −1, 1, and 2, respectively (black stipple lines). The contour curve of r′=1 relevantly changes from hyperbola to parallel straight lines to an ellipse with μy from −3 to 0 to 3.

Fig. 2. Hyperbolic Luneburg lens with a collimating effect. (a), (d), (b), (e), (c), and (f), respectively, are the geometrical light behaviors, electromagnetic wave pattern [Re(Ez)], and polaritonic wave pattern [Re(Ez)] from the point source at (0,1) or (1,2).

Fig. 3. (a) Schematic of the 2D model; (b) schematic of the 3D waveguide model; (c)  relation between the effective refractive index neff and the thickness d obtained from the dispersion relation Eq. (11).

Fig. 4. Thickness distribution d(x,y) of (a) hyperbolic collimating LL and (b) hyperbolic focusing MFL with an aerial view (left) and top view (right).

Fig. 5. Hyperbolic Maxwell’s fish-eye lens with the focusing effect. (a), (d), (b), (e), (c), and (f), respectively, are the geometrical light behaviors, the electromagnetic wave pattern [Re(Ez)], and the polaritonic wave pattern [Re(Ez)] from the point source at (0, 1) or (1,2).

Fig. 6. Polaritonic wave pattern [Re(Ez)] of the hyperbolic Luneburg lens with a collimating effect at a frequency 634  cm−1 where the maximum thicknesses dmax of the α–MoO3 layer are (a) 220 nm, (b) 130 nm, (c) 65 nm, respectively.

Fig. 7. Real-part permittivities of α–MoO3 where three different Reststrahlen bands of α–MoO3 are shaded in different colors.