  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 Maxwell's
fish eye lens. Duplex Mikaelian lenses have continuous translation symmetry
with arbitrary real number, while duplex Maxwell's fish eye lenses have
continuous rotational symmetry from 0 to 2*Pi. 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.
