A mathematical framework for polarization algebra of deterministic optical systems

Abstract: In general, Jones formalism of polarization optics refers to deterministic optical systems represented by  $2\times2$ complex Jones matrices acting on totally polarized light states represented by two dimensional complex Jones vectors. On the other hand, Stokes-Mueller formalism deals only with real states, hence it cannot account for phases introduced by optical systems. But, Stokes-Mueller formalism has an advantage over the Jones formalism that it works with directly measurable parameters, therefore it is desirable to merge these two approaches into a single formalism that links the real-measurable states with complex states. In previous works it was shown that there exists a $4\times4$ complex matrix that can keep track of the phase introduced by the deterministic optical system. This complex matrix transforms real Stokes vectors into four dimensional complex vectors related with phase. Recently a theorem was introduced. The theorem states that, for deterministic optical systems, there exists a relation between the outer products of  real-measurable input-output Stokes vectors and four dimensional complex state vectors of totally polarized light. In this note these results are combined in a compact mathematical framework.