Duplex Mikaelian and Duplex Maxwell's Fish-Eye Lenses

Here, we report two kinds of absolute optical instruments that can make stigmatic images for geometric optics in two-dimensional space. One is called the duplex Mikaelian lens, which is made by splicing two half-Mikaelian lenses with different periods. The other is an exponential conformal transformer of a duplex Mikaelian lens with a rational number of the ratio of different periods of its two half-Mikaelian lenses, which we call a duplex Maxwell's fish-eye lens. Duplex Mikaelian lenses have continuous translation symmetry with an arbitrary real number, while duplex Maxwell's fish-eye lenses have continuous rotation symmetry from zero to 2p. Hence, each duplex Maxwell's fish-eye lens corresponds to a duplex Mikaelian lens. We further demonstrate the caustic effect of geometric optics in duplex Mikaelian lenses and duplex Maxwell's fish-eye lenses. In addition, we investigate the Talbot effect of wave optics in the duplex Mikaelian lens based on numeric calculations. Our findings, based on splicing and exponential conformal mapping, enlarge the family of absolute optical instruments.