  Birefringent metasurfaces are two-dimensional structures capable of
independently controlling the amplitude, phase and polarization of orthogonally
polarized incident waves. In this work, we propose a in-depth discussion on the
mathematical synthesis of such metasurfaces. We compare two methods, one that
is rigorous and based on the exact electromagnetic fields involved in the
transformation and one that is based on approximate reflection and transmission
coefficients. We next validate the synthesis technique in metasurfaces
performing the operations of half- and quarter-wave plates, polarization beam
splitting and orbital angular momentum multiplexing.
