J. Eur. Opt. Society-Rapid Publ. 21, 29( 2025) 309
Reð ^R m Þ [ 39 ]. Now, by taking into account the relative weight( P 2 – P 1) of the discriminating component in the characteristic decomposition of equation( 18), the degree of nonregularity of the state R is given by
P N ðRÞ ¼ ðP 2 � P 1 Þ2 sin 2 v m3 ð24Þ
Thus, even though the state ^u m3 does not determine the remaining pair of eigenstates ^u m1 and ^u m2 of R m, the constituents of the active component of R m, itsellipticityangle governs the degree of nonregularity of the state R as seen from equation( 24). Conversely, since the orthonormal vectors ^u m1; ^u m2; ^u m3 constitute the columns of the unitary matrix that diagonalizes ^R m, the pair ^u m1; ^u m2 fixes completely ^u m3.
Furthermore, equation( 21) shows that the absolute value of the intensity-normalized spin vector ^n m of ^R m, which is invariant under arbitrary rotations of the reference frame, is given by
j^n m j ¼ cos v m3 sin v m3 ¼ sinð2v m3Þ 2 ð25Þ
which establishes the link between the nonregularity of R and j^n m jvia the angular parameter v m3. More precisely, as indicated above, the minimal value j^n m j ¼ 0 is a genuine property of regular states( v m3 = 0), while the maximum j^n m j ¼ 1 = 2 corresponds, uniquely, to so-called perfect nonregular states(| v m3 |= p / 4) [ 39 ]. It is remarkable that, contrary to the spin vector of a regular state whose direction is normal to the plane containing the two largest principal intensities, the vector ^n m exhibits transversal character in the sense that it lies along the axis X mO defined by the largest principal intensity of ^Rm. The analysis of the spin vector of R as the weighted sum of the spin vectors of the pure and discriminating components of R is studied in [ 48 ].
6 Conclusions
A simplified and meaningful interpretation of 3D polarization states can be attained from the( orthonormal) eigenstates ^u 1 and ^u 2 of R associated with the two largest eigenvalues, together with the indices of polarimetric purity. Such information determines the active component of R, which is introduced from the smart decomposition and provides a complete description of the spin and intensity anisotropies of the polarization state. Furthermore, the specific configuration of the pair( ^u 1; ^u 2) arbitrates the nonregularity and the transverse component of the spin vector by means of a single angular parameter. The concept of nonregularity was revisited in light of the indicated results and interpreted in terms of a distance of the state to a regular one.
Funding Research Council of Finland( 349396, 354918, PREIN 346518).
Conflicts of interest The authors have nothing to disclose.
Data availability statement No data were generated in this study.
Author contribution statement
J. J. G. developed the theory with contributions from T. S., A. N., and A. T. F. Original draft was prepared by J. J. G. and further edited by T. S., A. N., and A. T. F.
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