Towards a covariant formulation of electromagnetic wave polarization

2.2. GENERALIZED STOKES PARAMETERS 15
expressions for the Λ i ’s, Equation (2.67), with Stokes’ parameter for circular
polarization, V, we have the relation (see Carozzi, Karlsson and Bergman [6])
|V| =
√
Λ 2 7 + Λ2 5 + Λ2 2 = |⃗ V |. (2.69)
We define left- and right-handed polarization as seen by the wave (opposite to
an observer looking into the wave). For a right-handed polarized wave, V ⃗ will
be parallel to the direction of propagation and anti-parallel for a left-handed
polarized wave. 15 Because measurements is generally made in one point, i.e.
we have the field components in one point, we really have two possibilities with
opposite signs (opposite direction of propagation). To get a correct sign on V ⃗
one needs further information about the direction of propagation of the wave.
2.2.5 The Storey and Lefeuvre six-quasivector
In order to obtain a straightforward description of the electromagnetic field with
all possible combinations of the electromagnetic field components, Storey and
Lefeuvre introduced a six dimensional quasivector, for the field components,
defined as
⃗E = (E 1 , E 2 , E 3 , cB 1 , cB 2 , cB 3 ). (2.70)
For a continuum of plane waves Storey and Lefeuvre introduces a generalized
electric field, e m i (⃗r, t), where the i-th component is the real part of the evolution
e m i (⃗r, t) = Re[e i exp i(ωt − ⃗ k · ⃗r)]. (2.71)
Here e i is the complex amplitude. The 6 × 6 matrix they define is constructed
as
a ij = e ie ∗ j
δ . (2.72)
These are constant since the complex amplitude is constant for the plane wave
in Equation (2.71). The six-vector, Equation (2.70), is not a vector, in the
geometrical meaning, more than in the sense that it has six entries. The first
three components, the electric part, together form the E ⃗ vector (polar vector)
and the last three form the B ⃗ vector which is a pseudo-vector (axial vector). So,
the six-vector defined here has three components transforming as a polar vector
and three components transforming as a pseudo-vector. 16 Anyhow, using this
six-vector we obtain a spectral density matrix
⎛
⎞
E1 2 E 1 E2 ∗ E 1 E3 ∗ E 1 B1 ∗ E 1 B2 ∗ E 1 B3
∗ E 2 E1 ∗ E2 2 E 2 E3 ∗ E 2 B1 ∗ E 2 B2 ∗ E 2 B ∗ 3
S d =
E 3 E1 ∗ E 3 E2 ∗ E3 2 E 3 B1 ∗ E 3 B2 ∗ E 3 B3
∗ ⎜B 1 E1 ∗ B 1 E2 ∗ B 1 E3 ∗ B1 2 B 1 B2 ∗ B 1 B3
∗ . (2.73)
⎟
⎝B 2 E1 ∗ B 2 E2 ∗ B 2 E3 ∗ B 2 B1 ∗ B2 2 B 2 B3
∗ ⎠
B 3 E1 ∗ B 3 E2 ∗ B 3 E3 ∗ B 3 B1 ∗ B 3 B2 ∗ B3
2
As seen from its structure, this matrix is symmetric in the real part and antisymmetric
in the imaginary part. It has 36 different entries of which 21 are
15 For more information on the ⃗ V and complex vector analysis see Lindell [14] [15].
16 More information on polar and axial vectors are found in Arfken & Weber [10]