A theorem on the outer product of input and output Stokes vectors for deterministic optical systems

Abstract: The Jones matrix transforms two dimensional complex Jones vectors into complex Jones vectors and accounts for the phase introduced by the deterministic optical system. On the other hand, the Mueller-Jones matrix of the deterministic optical system transforms four dimensional real Stokes vectors into real Stokes vectors which contains no information about the phase. Previously, a 4x4 complex matrix, Z, akin to the Mueller-Jones matrix (M=ZZ*) was introduced and it was shown that Z matrix transforms four dimensional real Stokes vectors representing totally polarized light into four dimensional complex vectors which contain the relevant phase besides the other information. In this note it is shown that, for deterministic optical systems, there exists a relation between the outer products of experimentally measured real input-output Stokes vectors and theoretically calculated four dimensional complex vectors that obtained as a result of the transformation of real Stokes vectors by the Z matrix.