Interpretation of three-dimensional polarization states through the smart decomposition of the polarization matrix | Journal of the European Optical Society-Rapid Publications

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Figure 4

Example of a family of configurations of the polarization eigenstates ${\hat{u}}_{1}, {\hat{u}}_{2}, {\hat{u}}_{3}$ ${\hat{\mathbf{u}}}_1,{\hat{\mathbf{u}}}_2,{\hat{\mathbf{u}}}_3$ of the polarization density matrix, with the respective eigenvalues taken in decreasing order ${\hat{λ}}_{1} \geq {\hat{λ}}_{2} \geq {\hat{λ}}_{3}$ ${\hat{\lambda }}_1\ge {\hat{\lambda }}_2\ge {\hat{\lambda }}_3$ . The intrinsic axes X₁ Y₁ Z₁ of ${\hat{u}}_{1}$ ${\hat{\mathbf{u}}}_1$ are chosen as the reference frame for these representations. Although all the states correspond to the same point in space, for the sake of clarity they are represented separated. In the first configuration of the sequence, the polarization planes Π₁, Π₂ of the eigenstates ${\hat{u}}_{1}, {\hat{u}}_{2}$ ${\hat{\mathbf{u}}}_1,{\hat{\mathbf{u}}}_2$ involved in the smart decomposition coincide, while the respective normalized spin vectors are opposite, ${\hat{n}}_{2} = - {\hat{n}}_{1}$ ${\hat{\mathbf{n}}}_2=-{\hat{\mathbf{n}}}_1$ , and ${\hat{n}}_{3} = 0$ ${\hat{\mathbf{n}}}_3=\mathbf{0}$ . As the angle θ₂ subtended by the planes Π₁, Π₂ increases, the ellipticity angle of ${\hat{u}}_{2}$ ${\hat{\mathbf{u}}}_2$ takes different values until the last configuration, where ${\hat{u}}_{2}$ ${\hat{\mathbf{u}}}_2$ is linearly polarized along the Z₁ axis, orthogonal to Π₁.

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