  The Annular Groove Phase Mask (AGPM) is a vectorial vortex phase mask. It
acts as a half-wave plate with a radial fast axis orientation operating in the
mid infrared domain. When placed at the focus of a telescope element provides a
continuous helical phase ramp for an on axis sources, which creates the orbital
angular momentum. Thanks to that phase, the intensity of the central source is
canceled by a down-stream pupil stop, while the off axis sources are not
affected. However due to experimental conditions the nulling is hardly perfect.
To improve the null, a Mach-Zehnder interferometer containing Dove prisms
differently oriented can be proposed to sort out light based on its orbital
angular momentum (OAM). Thanks to the differential rotation of the beam, a
{\pi} phase shift is achieved for the on axis light affected by a non zero OAM.
Therefore the contrast between the star and its faint companion is enhanced.
Nevertheless, due the Dove prisms birefringence, the performance of the
interferometer is relatively poor. To solve this problem, we propose to add a
birefringent wave-plate in each arm to compensate this birefringence. In this
paper, we will develop the mathematical model of the wave front using the Jones
formalism. The performance of the interferometer is at first computed for the
simple version without the birefringent plate. Then the effect of the
birefringent plate is be mathematically described and the performance is
re-computed.
