Duplex Mikaelian lenses and duplex Maxwells fish eye lenses

In this paper, we report two new kinds of absolute optical instruments that can make stigmatically 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 exponential conformal transformer of duplex Mikaelian lens with the ratio of different periods of its two half Mikaelian lenses a rational number, which we call duplex Maxwells fish eye lens. Duplex Mikaelian lenses have continuous translation symmetry with arbitrary real number, while duplex Maxwells fish eye lenses have continuous rotational symmetry from 0 to 2*Pi. Hence each duplex Maxwells fish eye lens corresponds to a duplex Mikaelian lens. We further demonstrate the caustic effect of geometric optics in duplex Mikaelian lenses and duplex Maxwells 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.