Towards a covariant formulation of electromagnetic wave polarization

2.2. GENERALIZED STOKES PARAMETERS 11
2.2 Generalized Stokes parameters
In this Section we give the definitions of the two-dimensional Stokes parameters
and the generalized polarization parameters introduced by T. Carozzi, R. Karlsson
and J. Bergman [6]. These polarization parameters are a generalization of
the Stokes parameters into three-dimensions. A similar set of parameters were
introduced by Brosseau [5]. The Poynting vector and the Maxwell stress tensor
are introduced and defined. Storey and Lefeuvre’s [23] six dimensional electromagnetic
quasi-vector and their representation of the quadratic quantities of an
electromagnetic field are also discussed.
2.2.1 Spectral density matrix
Consider an arbitrary vector field, ⃗ f(⃗r, t). The field is a superposition of all
waves and stationary fields at the point ⃗r. The spectral components of the field
⃗f(⃗r, t), one obtains by taking the Fourier transform, ⃗ F (⃗r, ω), of the field, defined
as
⃗F (⃗r, ω) =
∫ ∞
−∞
⃗f(⃗r, t)e iωt dt. (2.46)
The vector, ⃗ F , represents a complex valued vector field which can be separated
into three complex amplitudes (the x, y, z components), or into three real
amplitudes (F 1 , F 2 , F 3 ) and their real phases (δ 1 , δ 2 , δ 3 ), in the following way
⎛
⃗F (⃗r, ω) = ⎝ F ⎞ ⎛
⎞
x(⃗r, ω) F 1 (⃗r, ω)e δ1(⃗r,ω)
F y (⃗r, ω) ⎠ = ⎝F 2 (⃗r, ω)e δ2(⃗r,ω) ⎠ . (2.47)
F z (⃗r, ω) F 3 (⃗r, ω)e δ3(⃗r,ω)
The spectral density matrix, S d (⃗r, ω), is constructed by taking the tensor product
between F ⃗ = F ⃗ (⃗r, ω), and its Hermitian conjugate, F ⃗ † = F ⃗ † (⃗r, ω). This
gives
⎛
⎞
S d (⃗r, ω) = F ⃗ ⊗ F ⃗ F x F † x ∗ F x Fy ∗ F x Fz
∗
= ⎝F y Fx ∗ F y Fy ∗ F y Fz
∗ ⎠ , (2.48)
F z Fx ∗ F z Fy ∗ F z Fz
∗
where F ∗ denotes complex conjugate. We will assume (⃗r, ω)-dependence and
the notation S d = S d (⃗r, ω) will be used in the following.
2.2.2 Instantaneous Stokes’ parameters
Stokes’ parameters describe the polarization state of a transverse electromagnetic
wave in the polarization plane. One usually assumes that the wave propagates
in the z-direction and the polarization ellipse lies in the xy-plane (see
Brosseau [4]). Stokes parameters can be written (using the notation 11 of R.
11 The notation for Stokes’ parameters is not universal. The notation of Stokes [22] was
(A,B,C,D), another notation is (S 0 , S 1 , S 2 , S 3 ) used by e.g. Jackson [12] and Brosseau [4].