124
5 An Electromagnetic Model for Anisotropic Media: Green’s Dyad for Plane-. . .
where J m and J e are anomalous magnetic and electric currents that account for the
presence of flaws, or anomalies, in the otherwise-uniform host material. From here
on we drop the subscript h on the generalized host permittivity and permeability.
Because of the material anisotropy, it is convenient to work with a matrix
formulation of these equations that has been useful in crystal optics, plasmas and
microwave devices [7–9, 17, 59–61, 115, 125]. If the body is homogeneous with
respect to (x, y), then Maxwell’s equations can be Fourier transformed with respect
to (x, y), and written as the following four-vector matrix differential equation in the
spectral domain:
d e
dz
= S · e + U · J
(5.3)
E z =
k y
z ω
H x −
k x
z ω
H y +
j
z ω
J ez
(5.4)
H z =
−k y
μω
E x +
k x
μω
E y −
j
μω
J mz ,
(5.5)
where the tilde denotes a function defined in the transform domain (k x , k y ), and
e =
⎡
⎢
⎢
⎣
E x
E y
H x
H y
⎤
⎥
⎥
⎦ ; J =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
J ex
J ey
J ez
J mx
J my
J mz
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(5.6)
The subscript e denotes an electric current, and m denotes a magnetic current. The
matrices in (5.3) are given by
S = −
⎡
⎢
⎢
⎣
0 0 a b
0 0 c d
α β 0 0
γ δ 0 0
⎤
⎥
⎥
⎦ ; U =
⎡
⎢
⎢
⎣
0 0 k x /ωω z 0 1
0
0 0 k y /ωω z −1 0
0
0 1
0
0 0 −k x /ωμ
−1 0
0
0 0 −k y /ωμ
⎤
⎥
⎥
⎦ .
(5.7)
The entries of S are given in terms of the entries of (5.1) by
a =
j
ωω z
k x k y ; α =
j
ωμ
(−μμ yx ω
2
− k x k y )
b =
j
ωω z
(μμ z ω
2
− k
2
x ) ; β =
j
ωμ
(−μμ y ω
2
+ k
2
x )
c =
j
ωω z
(−μμ z ω
2
+ k
2
y ) ; γ =
j
ωμ
(μμ x ω
2
− k
2
y )