# Two theorems on the outer product of input and output Stokes vectors for deterministic optical systems

## TL;DR

This paper presents two theorems linking the outer product of input and output Stokes vectors to complex polarization matrices in deterministic optical systems, revealing phase information beyond traditional Mueller-Jones representations.

## Contribution

It introduces two new theorems that connect measured Stokes vectors with complex polarization matrices, enhancing understanding of phase information in deterministic optical systems.

## Key findings

- Established relations between outer products of Stokes vectors and complex matrices.
- Revealed phase information encoded in the outer product relations.
- Extended the theoretical framework of polarization state transformations.

## Abstract

$2\times2$ complex Jones matrix transforms two dimensional complex Jones vectors into complex Jones vectors and accounts for phase introduced by deterministic optical systems. On the other hand, Mueller-Jones matrix transforms four parameter real Stokes vectors into four parameter real Stokes vectors that contain no information about phase. Previously, a $4\times4$ complex matrix ($\mathbf{Z}$ matrix) was introduced. $\mathbf{Z}$ matrix is analogous to the Jones matrix and it is also akin to the Mueller-Jones matrix by the relation $\mathbf{M}=\mathbf{Z}\mathbf{Z^*}$. It was shown that $\mathbf{Z}$ matrix transforms Stokes vectors (Stokes matrices) into complex vectors (complex matrices) that contain relevant phases besides the other information. In this note it is shown that, for deterministic optical systems, there exist two relations between outer product of experimentally measured real input-output Stokes vectors and complex vectors (matrices) that represent the polarization state and phase of totally polarized output light.