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tions holds:( a) Imð ^R m Þ¼0( i. e., ^R m is a real-valued matrix);( b) the angle h 2 subtended by the polarization planes of the eigenstates ^u 1 and ^u 2 is zero( see Fig. 4);( c) ^R m has zero spin;( d) ^n 2 ¼�^n 1;( e) ^a 3 ¼ 0;( f) the third eigenstate ^u 3 of R is linearly polarized;( g) P 1 = P e, and( h) P 2 = P d. In addition, we stress that states satisfying ^n ¼ 0
( thus susceptible to being represented as incoherent superpositions of linearly polarized states) are always regular. Conditions( a) to( f) follow directly from the analyses performed above. Property( g) comes from the fact that P d is defined in equation( 10) from the ordered eigenvalues ð^a 1; ^a 2; ^a 3 Þ of Re( R) while P 2 is defined in
equation( 3) from the ordered eigenvalues ð^k 1; ^k 2; ^k 3 Þ of ^R, so that necessarily ^a 1 ^k 3 and thus P 2 P d, with
the equality P 2 = P d corresponding, uniquely, to regular
states ð^a 1 ¼ ^k 1; ^a 2 ¼ ^k 2; ^a 3 ¼ ^k 3 Þ. These considerations together with equation( 12) imply P 1 P e and property( h)[ 38 ].
As shown in [ 38, 39 ], in its own intrinsic reference frame X mO Y mO Z mO, the discriminating component adopts thesimpleform
0 1
R mO ¼ 1
1 0 0
2 I B
0 cos 2 C
@ v m3 �i cos v m3 sin v m3 A; 0 i cos v m3 sin v m3 sin 2 v m3 ð21Þ
Figure
4. Example of a family of configurations of the polarization eigenstates ^u 1; ^u 2; ^u 3 of the polarization density matrix, with the respective eigenvalues taken in decreasing order ^k 1 ^k 2 ^k 3. The intrinsic axes X 1 Y 1 Z 1 of ^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 P 1, P 2 of the eigenstates ^u 1; ^u 2 involved in the smart decomposition coincide, while the respective normalized spin vectors are opposite, ^n 2 ¼�^n 1, and ^n 3 ¼ 0. As the angle h 2 subtended by the planes P 1, P 2 increases, the ellipticity angle of ^u 2 takes different values until the last configuration, where ^u 2 is linearly polarized along the Z 1 axis, orthogonal to P 1.
where �p / 4 v m3 p / 4 is the ellipticity angle of the third eigenstate ^u m3 of ^R m. It is remarkable that regular states are characterized by v m3 = 0, showing that R is regular if and only if ^u m3 represents a linearly polarized state. This property is similar to, but different from, condition( f) described above( which refers to ^u 3 instead of ^u m3).
A proper way to quantify the nonregularity of ^R m is to consider the distance from ^R m to ^R u�2D. Note that, since R m has been defined under the convention k 1 k 2 k 3, it necessarily takes the form R u�2D =( I / 2) diag( 1, 1, 0) for regular states( that is, the regular form of R m corresponds to an unpolarized 2D state whose electric field fluctuates in the common polarization planes of the eigenstates ^u 1 and ^u 2). On employing the Frobenius norm, which is invariant under orthogonal similarity transformations( i. e., rotations of the reference frame), we obtain
jj ^R m � ^R u�2D jj 2
2 ¼ 1 2 sin2 v m3; ð22Þ where | v m3 | p / 4 implies ^R m � ^R u�2D
2
1 = 4. By normalizing this squared distance in order to take values from
2
0( regularity) to 1( maximal nonregularity of ^R m) we define the degree of nonregularity of ^R m as
� P N ^R m ¼ 2 sin 2 v m3; ð23Þ
which coincides with the definition introduced previously in terms of the smallest eigenvalue ^m 3 ¼ð1 = 2Þ sin 2 v m3 of