JEOS RP ISSN01 | Page 311

306

J. Eur. Opt. Society-Rapid Publ. 21, 29( 2025)

Figure 1. 3D polarization states R that can be decomposed into a 2D state R 2D and a fully unpolarized 3D state R u�3D are called regular, and nonregular otherwise.

ð8Þ

with the intensity-normalized eigenvalues of the real part Re( R) satisfying ^a 1 ^a 2 ^a 3 0 and ^a 1 þ ^a 2 þ ^a 3 ¼ 1: The orthogonal rotation matrix Q O diagonalizes Re( R) [ 17, 21 ], viz.,

Q T O ReðRÞQ O ¼ I diagð^a 1; ^a 2; ^a 3 Þ: ð9Þ

Note that the intrinsic polarization matrix R O represents the same state as R, but with respect to the so-called intrinsic reference frame X O Y O Z O instead of XYZ for R. Agenuine property of R O is that its off-diagonal elements are purely imaginary and determine the spin vector n O ¼ I ð^n O1; ^n O2; ^n O3 Þ T of the state [ 17, 21, 40 ].

The quantities involved in R O are directly related to the six so-called intrinsic Stokes parameters of the state ðI; P l; P d; ^n O1; ^n O2; ^n O3 Þ [ 21, 27, 46 ]. The degree of linear polarization P l and the degree of directionality P d( a measure of the stability of the plane containing the polarization ellipse) are defined analogously to the IPP by replacing the eigenvalues ð^k 1; ^k 2; ^k 3 Þ of the polarization density matrix with the intensity-normalized principal intensities ð^a 1; ^a 2; ^a 3 Þ, i. e.,

P l ¼ ^a 2 � ^a 1; P d ¼ ^a 1 þ ^a 2 � 2^a 3; ð10Þ

which obey 0 P l P d 1. Moreover, the degree of circular polarization is given by P c ¼ j^n O j [ 21, 47 ], where ^n O ¼ n O = I ¼ð^n O1; ^n O2; ^n O3 Þ T. It is invariant under rotations of the Cartesian reference frame and consequently P c ¼ j^n j, with ^n being the intensity-normalized spin vector in the arbitrary reference frame XYZ. Note that, as occurs with the IPP, the descriptors P l, P d, and P c are intensity-independent.

For further analyses, we also mention the geometric representation of the polarization state in terms of the polarization object, constituted by the intensity ellipsoid( whose semiaxes are the principal intensities) together with the spin vector [ 27 ]. Other parameters that will be useful are the so-called degree of elliptical purity [ 44 ]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P e ¼ P 2 l þ P 2 c; ð11Þ

Note that in general P 1 P e and P 2 P d [ 38 ], while the above dual expression for P 3D implies the equality

P 2 e

� P2 1 ¼ 3 � P2 2 � P2 d:

3 Smart decomposition

The representation of the polarization density matrix in equation( 2) implies the so-called spectral decomposition

^R ¼ ^k 1 ^R p1 þ ^k 2 ^R p2 þ ^k 3 ^R p3; ð13Þ

where ^R pi ¼ ^u i ^u y i ði ¼ 1; 2; 3Þ: Here stands for the Kronecker product, the dagger indicates conjugate transpose, and the subscript p emphasizes that the state is pure. The spectral decomposition can be rearranged into the smart decomposition [ 38 ]

^R ¼ ^k 1 � ^k

3

^R p1 þ ^k 2 � ^k 3

^R p2 þ 3^k 3 ^R u�3D; � ð14Þ

^R u�3D ¼ 1 ^R

3 p1 þ ^R p2 þ ^R p3 ¼

1 I 3 3;

which in terms of the two IPP takes the interesting form [ 21 ]

^R ¼ P 1 þ P 2 2

^R p1 þ P 2 � P 1 2

Equivalently we may write

where the active component

^R p2 þ ð1 � P 2 Þ^R u�3D: ð15Þ

^R ¼ P 2 ^R a þ ð1 � P 2 Þ^R u�3D; ð16Þ

^R a ¼ P 1 þ P 2

2P 2

^Rp1 þ P 2 � P 1

2P 2

^Rp2 ð17Þ

encompasses all the information on the intensity and spin anisotropies of the polarization state up to the scale coefficient P 2.

Equation( 15) indicates that ^R is fully determined by the pair of IPP, which regulate the coefficients of the smart decomposition, together with the two first eigenvectors ^u 1 and ^u 2, associatedwiththetwomoresignificant eigenstates

specifying ^R p1 and ^R p2, respectively. The third eigenstate ^u 3 does not appear explicitly in the smart decomposition because it is fixed by the pair( ^u 1; ^u 2). In view of equations( 13) and( 14) the respective portions ^k 3 ^R p1 and ^k 3 ^R p2 have been subtracted from the two more significant pure components in order to group them with ^k 3 ^R p3, thus building the fully unpolarized component 3^k 3 ^R u�3D ¼ ^k 3 I 3, which in turn has a fully isotropic structure and therefore does not carry any information on polarimetric anisotropies.

Consequently, according to equation( 16), any polarization state( 3D in general) can be interpreted as a superposition of( a) a partially polarized state( the active