J. Eur. Opt. Society-Rapid Publ. 21, 29( 2025) 305
component and their relations to those of the whole state are introduced here for the first time, allowing a deeper insight into the sources of the polarization properties of the state. The presented approach also allows us to interpret any given 3D polarization state in terms of its two more significant eigenstates( those with larger associated eigenvalues). In addition, the concept of nonregularity is revisited and interpreted as a distance of the state to a regular state as well as in terms of the spin of the discriminating component.
2 3D and 2D polarization states
In the most general 3D representation, the second-order polarization properties of random( stationary) light at a given point in space are fully characterized by the corresponding 3 3 polarization matrix
0 B
R ¼ @ he x ðÞe t x ðÞi t he y ðÞe t x ðÞi t he z ðÞe t x ðÞi t he xðÞe t y ðÞi t he yðÞe t y ðÞi t he zðÞe t y ðÞi t he xðÞe t z ðÞi t 1 he yðÞe t z ðÞi t C A; he zðÞe t z ðÞi t
where e e x( t), e y( t), e z( t) are the analytic signals [ 41 ] of the electric field components [ 14, 42 ] with respect to the given Cartesian reference frame XYZ [ 21 ]. In addition, the asterisk indicates complex conjugation and h... i represents time averaging. The polarization matrix can be expressed as R ¼ I ^R in terms of the intensity I = trR( tr representing the trace) and the so-called polarization density matrix ^R.
The matrix ^R is fully characterized mathematically by its Hermiticity and positive semidefiniteness together with the normalization condition tr ^R ¼ 1. It can be expressed as
^R ¼ Udiag ^k 1; ^k 2; ^k
3 U y; ð2Þ
where diag indicates a diagonal matrix, ^k 1 ^k 2 ^k 3 with ^k 1 þ ^k 2 þ ^k 3 ¼ 1 are the eigenvalues of ^R, and U is a
unitary matrix whose columns ð^u 1; ^u 2; ^u 3 Þ are the threecomponent Jones vectors of the corresponding orthonormal eigenstates [ 43 ]. The eigenvalues specify the structure of polarimetric purity-randomness, which is characterized through the pair of indices of polarimetric purity( IPP) [ 16, 19, 44 ]
P 1 ¼ ^k 1 � ^k 2; P 2 ¼ ^k 1 þ ^k 2 � 2^k 3: ð3Þ
The IPP are independent of intensity and satisfy the nested inequalities 0 P 1 P 2 1. The eigenvalues of R are given by k i ¼ I ^k i( i = 1,2,3).
In the particular case of 2D states, the reference axis Z can be taken orthogonal to the well-defined polarization plane XY, so that the polarization matrix can straightforwardly be reduced to its conventional 2 2 form
0 1
R 2D ¼ 1 s 0 þ s 1 s 2 � is 3 0 B
C @ s 2 þ is 3 s 0 � s 1 0 A! 1
s 0 þ s 1 s 2 � is 3
; 2
2 s 2 þ is 3 s 0 � s 1
0 0 0 ð4Þ
ð1Þ where s 0, s 1, s 2, s 3 are the traditional Stokes parameters of a2Dfield. Above, despite that common experiments or arrangements on 2D polarization states do not necessitate to consider explicitly the third axis Z, its existence is nevertheless implicitly assumed. In fact, the representation of a 2D state can be referenced with respect to arbitrary 3D reference frames, in which case nonzero components associated with the Z axis appear [ 45 ]. The transformation of the original reference axes XYZ to the new Cartesian ones X 0 Y 0 Z 0 is performed through a rotation represented by the corresponding orthogonal matrix Q. The polarization matrix R 0 2D expressing the state R 2D in the new reference frame is then given by
2 0 13
R 0 2D ¼ Q 1 s 0 þ s 1 s 2 � is 3 0 6 B
C7 4 @ s 2 þ is 3 s 0 � s 1 0 A5Q T; ð5Þ 2
0 0 0
where the superscript T indicates transpose. Explicit expressions for the above 3D representation of the polarization matrix of a 2D state can be found in [ 45 ]. States that cannot be represented as in equation( 5) are called genuine 3D states because they necessarily require a 3D representation, i. e., the intensities of the three components of the fluctuating electric field are nonzero regardless of the reference frame taken to represent the corresponding polarization matrix.
Let us now recall that the term regular has been coined to refer to those 3D states that can be represented as an incoherent composition of a 2D polarization state( pure or not) and a 3D unpolarized state [ 39 ]( see Fig. 1). It has been shown that such a property is not general but particular; 3D polarization states are generally nonregular [ 39 ]. Thus, regular states are those that can be expressed as
2 0 13 s 0 þ s 1 s 2 � is 3 0
6 R ¼ Q 1 B
C7
4 s
2
@ 2 þ is 3 s 0 � s 1 0 A5Q T þ R u�3D; ð6Þ 0 0 0 ½ R u�3D ¼ I u I 3 = 3Š;
where I 3 is the 3 3 identity matrix and R u�3D is the polarization matrix of a 3D unpolarized state with intensity I u, which is invariant under arbitrary rotations of the Cartesian reference frame, i. e., QR u�3D Q T = R u�3D. Accordingly, through an appropriate choice of the reference frame, regular states can be represented as
0 1
R ¼ 1 s 0 þ s 1 s 2 � is 3 0 B
C @ s 2 þ is 3 s 0 � s 1 0 A þ R u�3D: ð7Þ 2
0 0 0
On the other hand, 3D states that cannot be represented as in equation( 7) are nonregular.
A simplified and meaningful view of the polarization matrix of a generic polarization state in the 3D space( regardless of whether it is pure, 2D, regular, or nonregular) is given by its intrinsic representation [ 21 ]