Towards a covariant formulation of electromagnetic wave polarization

2.3. IRREDUCIBLE REPRESENTATION 19
associate a vector to the anti-symmetric part of the matrix. This means that
we can write the spectral density matrix in the following way
⃗F ⊗ ⃗ F † = 1 3 Iδ ij + iɛ ijk v k + Q ij , (2.87)
where I is a real scalar, ⃗v is the real dual pseudo-vector (associated to the antisymmetric
imaginary part, see Eq. (2.63) ) and Q is a real symmetric tensor (the
anti-symmetric part is contained in ɛ ijk v k ). This way of representing a tenor is
termed an irreducible tensor representation (see e.g. Sakurai [20] and Arfken
& Weber [10]). Since I and Q are real valued we know that ⃗v must contain the
imaginary part. Hence,
⎛
⎞
0 Im[F 1 F2 ∗ ] Im[F 1 F3 ∗ ]
ɛ ijk v k = ⎝Im[F 2 F1 ∗ ] 0 Im[F 2 F3 ∗ ] ⎠ , (2.88)
Im[F 3 F1 ∗ ] Im[F 3 F2 ∗ ] 0
which gives
⃗v = ( Im[F 2 F ∗ 3 ], Im[F 3 F ∗ 1 ], Im[F 1 F ∗ 2 ] ) . (2.89)
To get ⃗v in terms of ⃗ F , we look at one component of ⃗ V , say V 3 (see Eq. (2.66))
V 3 = i(F 1 F ∗ 2 − F 2 F ∗ 1 ) = i2i(b 1 a 2 − a 1 b 2 ) = −2Im(F 1 F ∗ 2 ) = 2Im(F 2 F ∗ 1 ). (2.90)
And similarly for the other two components. We want ⃗v to be the imaginary
part and thus we get
⃗V = 2 ( Im(F 3 F ∗ 2 ), Im(F 1 F ∗ 3 ), Im(F 2 F ∗ 1 ) ) = −2⃗v. (2.91)
Rewriting ⃗v in terms of ⃗ F we have the following expression for ⃗v
⃗v = i [
(F3 F2 ∗ − F 2 F3 ∗ ), (F 1 F3 ∗ − F 3 F1 ∗ ), (F 2 F1 ∗ − F 1 F2 ∗ ) ]
2
= ( − Im(F 3 F ∗ 2 ), −Im(F 1 F ∗ 3 ), −Im(F 2 F ∗ 1 ) ) = − i 2 ⃗ F × ⃗ F † . (2.92)
Comparing ⃗v with V ⃗ , we have that ⃗v defines left-handed and right-handed circular
polarization as seen by an observer receiving the wave (standing in the
direction of propagation), the size of this vector, ⃗v is 1 π
times the area of the polarization
ellipse (see Lindell [14]). The relation between ⃗v and Stokes parameter
V is (compare with Equation (2.69))
√
|V| = 2 v1 2 + v2 2 + v2 3 = |2⃗v|. (2.93)
By subtraction we get Q from Equation (2.87), and we have
⎛
F1 2 − 1 3 I 1
2 (F 1F2 ∗ + F 2 F ∗ 1
1 )
2 (F ⎞
3F1 ∗ + F 1 F3 ∗ )
Q = ⎝ 1
2 (F 1F2 ∗ + F 2 F1 ∗ ) F2 2 − 1 3 I 1
2 (F 3F2 ∗ + F 2 F3 ∗ ) ⎠ . (2.94)
1
2 (F 3F1 ∗ + F 1 F3 ∗ 1
)
2 (F 3F2 ∗ + F 2 F3 ∗ ) F3 2 − 1 3 I
We see that ⃗v contains the imaginary part of the matrix ⃗ F ⊗ ⃗ F † and that Q is
real valued and symmetric.