  We consider the proximity effect in a Topological Insulator sandwiched
between two conventional superconductors, by comparing s-wave spin singlet
superconducting pairing correlations and odd-parity triplet pairing
correlations with zero spin component orthogonal to the slab ("polar " phase).
A superconducting gap opens in the Dirac dispersion of the surface states
existing at the interfaces. An axial vortex is included, piercing the slab
along the normal to the interfaces with the superconductors. It is known that,
when proximity is s-wave, quasiparticles in the gap are Majorana Bound States,
localized at opposite interfaces. We report the full expression for the quantum
field associated to the midgap neutral fermions, as derived in the two-orbital
band model for the TI. When proximity involves odd-parity pairing, midgap modes
are charged Surface Andreev Bound States, and they originate from interfacial
circular states of definite chirality, centered at the vortex singularity and
decaying in the TI film with oscillations. When the chemical potential is moved
away from midgap, extended states along the vortex axis are also allowed. Their
orbital structure depends on the symmetry of the bulk band from where the
quasiparticle level splits off.
