J. Eur. Opt. Society-Rapid Publ. 21, 29( 2025) 307
Figure 2. The active component of a polarization state admits an interpretation either as an incoherent composition of two pure states, the first two eigenstates, or equivalently as an incoherent combination of the pure and discriminating components of the state.
component), given by the incoherent composition of the two pure components ^R p1 and ^R p2 with respective weights( P 1 + P 2)/ 2 and( P 2 � P 1)/ 2, which in turn are determined by the two eigenstates ^u 1 and ^u 2 of more significant eigenvalues; and( b) a fully unpolarized 3D state weighted by 1 – P 2. Notice that the weight affecting R p1 is never smaller than that of R p2.
To further understand the smart decomposition, it is worth comparing it with the so-called characteristic decomposition [ 19, 44 ]
^R ¼ P 1 ^R p1 þ ðP 2 � P 1 Þ^R m þ ð1 � P 2 Þ^R u�3D ð18Þ
where the middle term, given by ^R m ¼ð^R p1 þ ^R p2 Þ = 2 and called the discriminating component [ 48, 49 ], consists of an equiprobable incoherent mixture of the two first eigenstates. Naturally, the active parts of the smart and characteristic decompositions coincide but are expressed through respective decompositions. The smart decomposition is illustrated in Figure 2 for a generic configuration.
From the very definition of the active component, its characteristic decomposition has the form
^R a ¼ P a1 ^R p1 þ ð1 � P a1 Þ^R m; P a1 ¼ P 1; P a2 ¼ P 2
¼ 1: P 2 P 2 ð19Þ
Therefore, the IPP of ^R a, denoted by P a1 and P a2, which obviously coincide with those of R a, are proportional to those of R via the coefficient 1 / P 2 1, i. e., P a1 = P 1 / P 2 P 1 and P a2 = 1 P 2. Consequently, the polarization object of the active component ^R a, whose principal intensities and intrinsic spin vector are denoted as ^a a1; ^a a2; ^a a3 and ^n aO, coincides with that of ^R except for the scale coefficient 1 / P 2 as illustrated in Figure 3.
Figure
3. The shapes and orientations of the polarization objects of the polarization matrix and its active component coincide but differ by a scale coefficient 1 / P 2 1as^a a1 ¼ ^a 1 = P 2, ^a a2 ¼ ^a 2 = P 2, ^a a3 ¼ ^a 3 = P 2, and ^n aO ¼ ^n O = P 2. Note that the case where P 2 = 0 is excluded because it corresponds to a fully unpolarized state, which lacks an active component.
4 Configurations of the active component
An algebraic and geometric representation of all possible configuration sets of 3D orthonormal states was described in [ 43 ], which can be applied to the three orthonormal
eigenstates ð^u 1; ^u 2; ^u 3 Þ of ^R. As an example, a representative family of configurations of the triad of eigenstates is shown in Figure 4, where the intrinsic axes X 1, Y 1, Z 1 of the first eigenstate are taken as the reference frame for the three eigenstates. All families share the property that, when the polarization planes of two eigenstates coincide, the third one corresponds to a linearly polarized state along the direction orthogonal to the said planes [ 43 ].
Since a fully unpolarized state R u�3D lacks spin, the smart decomposition shows that the normalized spin vector ^n can be expressed as a weighted sum of the normalized spin vectors ^n 1 and ^n 2 related to the two first eigenvectors ^u 1 and ^u 2,
^n ¼ P 1 þ P 2 2
^n 1 þ P 2 � P 1 2
^n 2; ð20Þ
so that ^n ¼ P 2^n a. We note that regular states are characterized by the fact that the eigenstates ^u 1 and ^u 2 have opposite normalized spin vectors, ^n 2 ¼�^n 1. Consequently, the normalized spin vector of a regular state is given by ^n ¼ P 1^n 1, in accordance with the fact that the entire spin originates from the pure component of the characteristic decomposition.
5 Interpretation of nonregularity
For a regular state the discriminating component ^R m is necessarily a 2D unpolarized state, while otherwise the state
is nonregular. Since ^R m, and hence nonregularity, depends on the relative configurations of the two more significant eigenstates ^u 1 and ^u 2, nonregularity has a direct relation to the smart decomposition. In particular, from the analysis performed in Sections 2 and 3, it follows that a polarization state R is regular if and only if any of the following condi-