Towards a covariant formulation of electromagnetic wave polarization

2.2. GENERALIZED STOKES PARAMETERS 13
2.2.3 Three-dimensional polarization parameters
We want to obtain a more general form of Equation (2.53) applicable to an
arbitrary wave in three dimensions (to use Stokes parameters it is necessary to
know the direction of propagation). To achieve this we expand the full spectral
density matrix, S d , using the unit matrix, I 3 , and the generators of the SU(3)
symmetry group, λ i , (i = 1, . . . , 8), as given by Gell-Mann and Ne’eman [17]
(see Appendix A Equation (A.2)). The spectral density matrix is then formed
as a linear combination of these matrices in the following way
S d = 1 3 Λ 0I 3 + 1 8∑
Λ i λ i =
2
i=1
⎛
1
3
⎜
Λ 0 + 1 2 Λ 3 + 1
1
⎝ 2 Λ 1 + i 2 Λ 2
1
2 Λ 4 + i 2 Λ 5
2 √ 3 Λ 8
1
2 Λ 1 − i 2 Λ 2
1
3 Λ 0 − 1 2 Λ 3 + 1
1
2 Λ 6 + i 2 Λ 7
2 √ 3 Λ 8
1
2 Λ 4 − i 2 Λ ⎞
5
1
2 Λ 6 − i 2 Λ ⎟
7 ⎠ , (2.58)
1
3 Λ 0 − √ 1
3
Λ 8
where Λ i , i = 0, 1, . . . 8, are the generalized Stokes parameters. The trace of
this matrix, S d , is Λ 0 which is the energy density of the field. For the two
dimensional case this is the Stokes parameter I. The Λ i are found by comparing
Equation (2.48) with Equation (2.58), the explicit expressions for the Λ i are
found in Appendix A. In the case of a transverse wave propagating in the z-
direction, i.e. F z = 0, the spectral density matrix, Equation (2.58), reduces to
two-dimensional form
⎛
⎞
S d = 1 Λ 0 + Λ 3 Λ 1 − iΛ 2 0
⎝Λ 1 + iΛ 2 Λ 0 − Λ 3 0⎠ , (2.59)
2
0 0 0
where the Λ i have reduced to the Stokes parameters as in Equation (2.53). This
is just a rotation of the coordinate system into the one of the wave with the
z-axis being the direction of propagation.
There are different normalizations of the generalized polarization parameters.
Brosseau [5] uses 1 3 instead of 1 2
, which we have used, in front of the SU(3)
generators in Equation (2.58). If one uses 1 2
, Equation (2.59) (transverse wave
propagating in the z-direction) reduces to Stokes parameters, Equation (2.55),
which is not the case if one uses 1 3 .
2.2.4 Interpretation of the 3D polarization parameters
Every tensor can be decomposed into its symmetric and anti-symmetric parts,
Arfken & Weber [10]. Thus, Equation (2.58) can be written
S d = S S d + S A d = 1 2(
Sd + S T d
) 1( + Sd − S T )
d . (2.60)
2
The superscript T denotes a transposed tensor. From Equation (2.58) it is easily
seen that the symmetric part, Sd S, and anti-symmetric part, SA d
, are given by
⎛
1
3
Sd S ⎜
Λ 0 + 1 2 Λ 3 + 1
2 √ Λ 1
3
8
2 Λ 1
1
2 Λ ⎞
4
1
= ⎝
2 Λ 1
1
3 Λ 0 − 1 2 Λ 3 + 1
2 √ Λ 1
3
8 2 Λ ⎟
6 ⎠ (2.61)
1
2 Λ 1
4
2 Λ 1
6
3 Λ 0 − √ 1
3
Λ 8