5.1 Theory
125
d = −
j
ωω z
k x k y ; δ =
j
ωμ
(μμ xy ω
2
+ k x k y ).
(5.8)
When J is a surface current confined to z = z , i.e., J = J s δ(z − z ), then
integration of (5.3) produces
e
(+)
− e
(−)
= U · J s .
(5.9)
The superscript (+) denotes the limit as z approaches z from above, and the
superscript (−) denotes the limit from below. Equation (5.9) is used to compute
the Green’s dyad for a layered workpiece.
Starting with these equations, Roberts [90] has developed a fairly complete
theory of normal modes of biaxial anisotropic media. This work is based on, and
extends, earlier work performed at Sabbagh Associates [92, 93]. From here on we
specialize the theory developed in [90] to the case to be considered here, in which
the media involved are transversely isotropic to the z-coordinate. The generalized
electric permittivity tensor, in its principal-axis coordinate system, then takes the
form
=
⎡
⎣
t 0 0
0 t 0
0 0 z
⎤
⎦ ,
(5.10)
where, for pure titanium t = 0 − j 2.205 × 10 6 /ω ≈ −j 2.205 × 10 6 /ω, , z =
0 − j 2.083 × 10 6 /ω ≈ −j 2.083 × 10 6 /ω.
The entries in S now become
a =
j
ωω z
k x k y ; α =
j
ωμ
(−k x k y )
b =
j
ωω z
(μμ z ω
2
− k
2
x ) ; β =
j
ωμ
(−μμ t ω
2
+ k
2
x )
c =
j
ωω z
(−μμ z ω
2
+ k
2
y ) ; γ =
j
ωμ
(μμ t ω
2
− k
2
y )
d = −
j
ωω z
k x k y ; δ =
j
ωμ
(k x k y ).
(5.11)
Let’s introduce some notation: k 2
x + k 2
y = k 2
t , ω 2 μμ t = Ω 2
t , ω 2 μμ z = Ω 2
z , , =
t // z . Then the eigenvalues of S are
λ 1 =
k 2
t − Ω 2
t λ 2 = −λ 1 λ 3 =
√
k 2
t − Ω 2
z λ 4 = −λ 3 .
(5.12)
The linearly-independent eigenvectors that correspond to these eigenvalues are:
125
d = −
j
ωω z
k x k y ; δ =
j
ωμ
(μμ xy ω
2
+ k x k y ).
(5.8)
When J is a surface current confined to z = z , i.e., J = J s δ(z − z ), then
integration of (5.3) produces
e
(+)
− e
(−)
= U · J s .
(5.9)
The superscript (+) denotes the limit as z approaches z from above, and the
superscript (−) denotes the limit from below. Equation (5.9) is used to compute
the Green’s dyad for a layered workpiece.
Starting with these equations, Roberts [90] has developed a fairly complete
theory of normal modes of biaxial anisotropic media. This work is based on, and
extends, earlier work performed at Sabbagh Associates [92, 93]. From here on we
specialize the theory developed in [90] to the case to be considered here, in which
the media involved are transversely isotropic to the z-coordinate. The generalized
electric permittivity tensor, in its principal-axis coordinate system, then takes the
form
=
⎡
⎣
t 0 0
0 t 0
0 0 z
⎤
⎦ ,
(5.10)
where, for pure titanium t = 0 − j 2.205 × 10 6 /ω ≈ −j 2.205 × 10 6 /ω, , z =
0 − j 2.083 × 10 6 /ω ≈ −j 2.083 × 10 6 /ω.
The entries in S now become
a =
j
ωω z
k x k y ; α =
j
ωμ
(−k x k y )
b =
j
ωω z
(μμ z ω
2
− k
2
x ) ; β =
j
ωμ
(−μμ t ω
2
+ k
2
x )
c =
j
ωω z
(−μμ z ω
2
+ k
2
y ) ; γ =
j
ωμ
(μμ t ω
2
− k
2
y )
d = −
j
ωω z
k x k y ; δ =
j
ωμ
(k x k y ).
(5.11)
Let’s introduce some notation: k 2
x + k 2
y = k 2
t , ω 2 μμ t = Ω 2
t , ω 2 μμ z = Ω 2
z , , =
t // z . Then the eigenvalues of S are
λ 1 =
k 2
t − Ω 2
t λ 2 = −λ 1 λ 3 =
√
k 2
t − Ω 2
z λ 4 = −λ 3 .
(5.12)
The linearly-independent eigenvectors that correspond to these eigenvalues are: