Interpretation of the coherency matrix for three-dimensional polarization states

From an appropriate parameterization of the three-dimensional (3D) coherency matrix R that characterizes the second-order, classical states of polarization, the coherency matrices are classified and interpreted in terms of incoherent decompositions. The relevant physical quantities derived from R, such as the intensity, degree of polarimetric purity, the indices of polarimetric purity, angular momentum, degree of directionality, and degree of linear polarization, are identified and interpreted in light of the case study performed. The information provided by R about the direction of propagation is clarified and it is found that coherency matrices with $\mathrm{rank}\mathbf{R}=2$ do not always represent states with a well-defined direction of propagation. Moreover, the existence of 3D mixed states that cannot be decomposed into a superposition of a pure state, a two-dimensional (2D) unpolarized state, and a 3D unpolarized state is demonstrated. Appropriate representation and interpretation for all the different types of 3D coherency matrices are provided through physically consistent criteria. Under the approach proposed, the conventional 2D model arises naturally.

Figure 1

Intensity ellipsoid, with semiaxes $a_1\ge a_2\ge a_3$, representing a mixed state constituted by the incoherent superposition of three pure linearly polarized states ${\mathbf{R}}_{p1},{\mathbf{R}}_{p2},$ and ${\mathbf{R}}_{p3}$. The direction k along the reference axis $Z_O$ can always be chosen as the common direction of propagation of the pure components ${\mathbf{R}}_{p1}$ and ${\mathbf{R}}_{p2}$ (with respective intensities $a_1,a_2)$, whereas any direction orthogonal to k can be taken as the direction of propagation of the third pure component ${\mathbf{R}}_{p3}$ (with intensity $a_3)$.

Figure 2

Coherency matrix ${\mathbf{R}}_{p{Z}_O}$ of a generic pure state propagating along the axis $Z_O$, obtained from its intrinsic coherency matrix ${\mathbf{R}}_O$ through an arbitrary rotation θ around $Z_O$.

Figure 3

Representation of a linearly polarized pure state as the incoherent superposition of an arbitrary number of linearly polarized states with the same polarization axis along $X_O$ but with different directions of propagation perpendicular to $X_O$.

Figure 4

Arbitrary representation of a 2D mixed state (i.e., $r=2,t=2)$ as the incoherent superposition of two pure states with the same propagation direction $Z_O$.

Figure 5

Characteristic representation of a 2D mixed state (i.e., $r=2,t=2)$ as the incoherent superposition of a pure state and a 2D unpolarized state with the same propagation direction $Z_O$. $P_1$ is the first index of polarimetric purity (or degree of polarization).

Figure 6

Arbitrary representation of a mixed state R with $\mathrm{rank}\mathbf{R}=2$ and $\mathrm{rank}\left[\mathrm{Re}\left(\mathbf{R}\right)\right]=3$ as the incoherent superposition of two pure states with different directions of propagation. In spite of $\mathrm{rank}\mathbf{R}=2$, R corresponds to a 3D mixed state.

Figure 7

Characteristic representation of a mixed state R with $\mathrm{rank}\mathbf{R}=2,\mathrm{rank}\left[\mathrm{Re}\left(\mathbf{R}\right)\right]=3$ and $\mathrm{rank}\left[\mathrm{Re}\left({\mathbf{R}}_m\right)\right]=2$ as the incoherent superposition of a pure state and a 2D unpolarized state, with different directions of propagation. In spite of $\mathrm{rank}\mathbf{R}=2$, R corresponds to a 3D mixed state.

Figure 8

Characteristic representation of a mixed state R with $\mathrm{rank}\mathbf{R}=2,\mathrm{rank}\left[\mathrm{Re}\left(\mathbf{R}\right)\right]=3$ and $\mathrm{rank}\left[\mathrm{Re}\left({\mathbf{R}}_m\right)\right]=3$ as the incoherent superposition of a pure state and a 3D mixed state. In spite of $\mathrm{rank}\mathbf{R}=2$, R corresponds to a 3D mixed state.

Figure 9

Arbitrary representation of a mixed state R with $\mathrm{rank}\mathbf{R}=3$ as the incoherent superposition of three pure states with different directions of propagation.

Figure 10

Characteristic representation of a mixed state R with $\mathrm{rank}\mathbf{R}=3$, and $\mathrm{rank}\left[\mathrm{Re}\left({\mathbf{R}}_m\right)\right]=2$ as the incoherent superposition of a pure state, a 2D unpolarized state (with different direction of propagation), and a 3D unpolarized state.

Figure 11

Characteristic representation of a mixed state R with $\mathrm{rank}\mathbf{R}=3$, and $\mathrm{rank}\left[\mathrm{Re}\left({\mathbf{R}}_m\right)\right]=3$ as the incoherent superposition of two pure states (with different directions of propagation) and a 3D unpolarized state.