The inscription of a tilted Bragg grating in the core of a standard telecommunication-grade fiber grants the latter a sensitivity to the refractive index (RI) of the fluid in which it is immersed. A gold-coated tilted fiber Bragg grating (Au-TFBG) is a transposition of the Kretschmann prism configuration onto a cylindrical structure. Taking advantage of the surface plasmon resonance (SPR), the RI sensitivity is further increased. In this work, we extract relative phase difference spectra from two orthogonal polarization states using the Jones formalism and use them to experimentally detect the insulin hormone at concentrations ranging from 0.1 ng/mL to 100 ng/mL.