2-D Niblett-Bostick magnetotelluric inversion - MTNet

Geologica Acta, 8(1), 15-30 (2010)
DOI: 10.1344/105.000001513
J. RODRÍGUEZ et al. 2-D Niblett-Bostick
29
Zyx modes, we have that ∆θ = 0, and θ is simply the strike
direction. ZS and ZP, the series and parallel impedances,
respectively, can be written as
(C.7)
and
(C.8)
Series impedance derivative
We can express the series apparent resistivity r a
S in
terms of the TE and TM apparent resistivities as
(C.9)
in this equation the three apparent resistivities are complex
quantities. The partial derivate of r a
S is
(C.10)
r j is the resistivity of the j-th block in the model. We also
know that
(C.11)
Using Eqs. (C10) and (C11), we have
(C.12)
Using a relation similar to Eq. (C11) for r a
TM and ra
TE ,
(C.13)
in terms of apparent conductivities this is:
(C.14)
The general expression for the elements of matrix A is
(C.15)
All quantities are obtained directly from the data.
Parallel impedance derivative
For the parallel data, it is better to deal with apparent
conductivities
(C.16)
The partial derivative with respect to σj is
(C.17)
Writing parallel apparent conductivity as
(C.18)
then
(C.19)
Using Eq. (C17)
(C.20)
Using a relation similar to Eq. (C11) but for σa
TM and
σa
TE , we have
(C.21)
The general expression for the elements of matrix A is
(C.22)
Again, all quantities are obtained directly from the
data.
4 Generalized Model in a Hopfield Artificial Neural
Network
It is necessary to define models with more realistic
conductivity values (not only 0’s and 1’s). Therefore, a
general model whose states are formed by an ordered set
of 0’s and 1’s is defined to represent a finite precision real
number. We will refer to this model as the general sequence
model. The general sequence model {s i
g }1
N has a typical
element:
6
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
| j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
xy
x
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
|
|
1
j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
x
y
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
M xy
Z
=
E yx
Z
=
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
S
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
j
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
TM
TE
a
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
|
|
1
j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
x
H
y
H
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
TM xy
Z Z
=
TE yx
Z Z
=
S
Z
P
Z
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
a
ρ
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
S
a
ρ
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
j
ρ
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
TM
a
ρ
TE
a
ρ
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
|
|
1
j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
},
,
,
,
{
}
,
,
,
{ θ
θ ∆
⇔ P
S
yy
yx
xy
xx
Z
Z
Z
Z
Z
Z (C.2)
,
)
2
(
2
/
1
2
2
2
2
yx
yy
xy
xx
S
Z
Z
Z
Z
Z
+
+
+
= (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
| j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
1
1
1
|
|
1
TE
a
TM
a
P
a
σ
σ
σ ∂
∂
∂
+
∂
∂
−
∂
∂
= )
|
|
|
(|
|
|
Re{
2
1
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
1
1
1
|
|
1
TE
a
TM
a
P
a
σ
σ
σ ∂
∂
∂
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}.
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.21)
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑
)
(
]
)
(
1
1
1
)
(
2
1
1
]
)
)
2
(
)
2
(
)
2
(
2
1
[
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
B
A
A
s
A
B
A
(a−term−independent−of−the−model). (D.2)
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}.
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.21)
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑
)
(
]
)
(
1
1
1
)
(
2
1
1
]
)
)
2
(
)
2
(
)
2
(
2
1
[
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
B
A
A
s
A
B
A
(a−term−independent−of−the−model). (D.2)
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}.
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.21)
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑
)
(
]
)
(
1
1
1
)
(
2
1
1
]
)
)
2
(
)
2
(
)
2
(
2
1
[
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
B
A
A
s
A
B
A
(a−term−independent−of−the−model). (D.2)
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}.
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.21)
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑
)
(
]
)
(
1
1
1
)
(
2
1
1
]
)
)
2
(
)
2
(
)
2
(
2
1
[
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
B
A
A
s
A
B
A
(a−term−independent−of−the−model). (D.2)
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}.
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑
)
(
]
)
(
1
1
1
)
(
2
1
1
]
)
)
2
(
)
2
(
)
2
(
2
1
[
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
B
A
A
s
A
B
A
(a−term−independent−of−the−model). (D.
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑ ]
)
(
1
1
1
)
(
2
1
1
)
)
2
(
)
2
(
)
2
(
2
1
[ kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
A
A
s
A
B
A
(a−term−independent−of−the−model).
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}.
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.21)
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑
)
(
]
)
(
1
1
1
)
(
2
1
1
]
)
)
2
(
)
2
(
)
2
(
2
1
[
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
B
A
A
s
A
B
A
(a−term−independent−of−the−model). (D.2)
7
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
P
a
j
TM
a
P
a
j
P
a
P
a
σ
σ
σ
σ
σ
σ
σ
σ
σ ∂
∂
+
∂
∂
=
∂
∂
(C.20)
TM
a
σ
TE
a
σ
+
∂
∂
−
∂
∂
=
∂
∂ −
)
|
|
|
|
1
(
|
Re{|
2
1
|
| )
(
j
TM
j
TM
a
TM
a
i
TM
a
j
P
a
i
e
TM
P
σ
φ
σ
σ
σ
σ
σ
σ φ
φ
)}.
|
|
|
|
1
(
|
|
)
(
j
TE
j
TE
a
TE
a
i
TE
a
i
e
TE
P
σ
φ
σ
σ
σ
σ
φ
φ
∂
∂
−
∂
∂
−
(C.21)
}.
Re{
2
1 )
(
)
(
TE
i
TM
i
I
e
I
e
TE
P
TM
P
φ
φ
φ
φ −
−
+ (C.22)
N
g
i 1
}
{σ
,
,...,
2
,
1
,
2
2
)
(
N
i
B
B
U
t
ij
j
U
D
j
g
i
=
−
= ∑ −
=
σ (D.1)
0
=
ij
B
ij
B
N
g
i 1
}
{σ
−
−
−
=
+
=
−
=
=
−
=
=
∑
∑
∑
∑
∑
)
(
)
(
)
(
1
1
1
]
2
[
2
1 t
jn
t
im
kj
ki
n
m
M
k
U
D
n
N
j
U
D
m
N
i
B
B
A
A
C
+
+
+
−
+
=
=
=
=
−
=
=
∑
∑
∑
∑
∑
∑
)
(
]
)
(
1
1
1
)
(
2
1
1
]
)
)
2
(
)
2
(
)
2
(
2
1
[
t
im
kj
ki
U
m
N
j
M
k
a
k
ki
m
M
k
t
im
ki
m
M
k
U
D
m
N
i
B
B
A
A
s
A
B
A
(a−term−independent−of−the−model). (D.2)
6
,
)
2
(
2
/
1
S
Z = (C.3)
,
)
(
2 2
/
1
2
2
2
2
yx
yy
xy
xx
yy
xx
xy
yx
P
Z
Z
Z
Z
Z
Z
Z
Z
Z
+
+
+
−
= (C.4)
),
arctan(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
=
∆θ (C.5)
).
arctan(
2
1
yx
xy
xx
yy
Z
Z
Z
Z
+
−
=
θ (C.6)
TM xy
Z Z
=
TE yx
Z Z
=
S
Z
P
Z
),
(
2
1 2
2
2
TE
TM
S
Z
Z
Z +
= (C.7)
).
1
1
(
2
1
1
2
2
2
TE
TM
P
Z
Z
Z
+
= (C.8)
S
a
ρ
),
(
2
1 TE
a
TM
a
S
a
ρ
ρ
ρ +
= (C.9)
S
a
ρ
),
(
2
1
j
TE
a
j
TM
a
j
S
a
ρ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
=
∂
∂
(C.10)
j
ρ
.
|
|
|
|
1
1
j
S
j
S
a
S
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.11)
).
1
1
Re(
2
1
|
|
|
|
1
j
TE
a
S
a
j
TM
a
S
a
j
S
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
=
∂
∂
(C.12)
TM
a
ρ
TE
a
ρ
+
∂
∂
+
∂
∂
=
∂
∂
)
|
|
|
|
1
(
Re{
2
1
|
|
|
|
1
j
TM
j
TM
a
TM
a
S
a
TM
a
j
S
a
S
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
},
,
,
,
{ θ
θ ∆
⇔ P
S Z
Z (C.2)
,
)
2
2
/
1
2
2
yx
yy
y
Z
Z +
+
(C.3)
,
) 2
/
1
2
2
2
yx
yy
xy
yy
xx
xy
Z
Z
Z
Z
Z
+
+
−
(C.4)
),
(
yx
xy
yy
xx
Z
Z
Z
Z
−
+
(C.5)
).
n(
yx
xy
xx
yy
Z
Z
Z
Z
+
−
(C.6)
),
2
2
TE
TM
Z
+ (C.7)
).
1
1
2
2
TE
TM
Z
+ (C.8)
),
TE
a
TM
a
ρ
+ (C.9)
),
j
TE
a
j
TM
a
ρ
ρ
ρ
ρ
∂
∂
+ (C.10)
.
|
|
| j
S
j
S
a i ρ
φ
ρ
ρ
∂
∂
+
∂
∂
(C.11)
).
1
1
j
TE
a
S
a
j
TM
a
S
a r ρ
ρ
ρ
ρ
ρ
ρ ∂
∂
+
∂
∂
(C.12)
+
∂
∂
+
∂
∂
)
|
|
|
1
j
TM
j
TM
a
TM
a
i ρ
φ
ρ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
6
},
,
,
,
{ θ
θ ∆
P
S Z
Z (C.2)
,
) 2
/
1
2
2
yx
yy
Z
Z +
(C.3)
,
) 2
/
1
2
2
yx
yy
yy
xx
y
Z
Z
Z
Z
+
+
−
(C.4)
),
yx
xy
yy
xx
Z
Z
−
+
(C.5)
).
yx
xy
xx
yy
Z
Z
+
−
(C.6)
),
2
TE
Z
+ (C.7)
).
1 2
TE
Z
+ (C.8)
),
TE
a
ρ
+ (C.9)
),
j
TE
a
ρ
ρ
∂
∂
+ (C.10)
.
|
|
j
S
j
S
a i ρ
φ
ρ
ρ
∂
∂
+
∂
(C.11)
).
1
j
TE
a
S
a
j
TM
a
r ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.12)
+
∂
∂
+
∂
∂
)
|
|
| j
TM
j
TM
a
M i ρ
φ
ρ
ρ
)},
|
|
|
|
1
(
j
TE
j
TE
a
TE
a
S
a
TE
a
i ρ
φ
ρ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
(C.13)
+
∂
∂
−
∂
∂
=
∂
∂
)
|
|
|
(|
|
|
Re{
2
1
|
|
j
TM
j
TM
a
TM
a
S
a
S
a
TM
a
j
S
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
σ
σ
)}.
|
|
|
(|
|
| j
TE
j
TE
a
TE
a
S
a
S
a
TE
a
i σ
φ
σ
σ
ρ
ρ
ρ
ρ
∂
∂
−
∂
∂
(C.14)
}.
|
|
|
|
|
|
|
|
Re{
2
1
TE
S
a
S
a
TE
a
TE
a
TM
S
a
S
a
TM
a
TM
a
I
I
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
+ (C.15)
).
(
2
1 TE
a
TM
a
P
a
σ
σ
σ +
= (C.16)
).
(
2
1
j
TE
a
j
TM
a
j
P
a
σ
σ
σ
σ
σ
σ
∂
∂
+
∂
∂
=
∂
∂
(C.17)
,
|
|
P
i
P
a
P
a
e φ
σ
σ
−
= (C.18)
.
|
|
|
|
1
1
j
P
j
P
a
P
a
j
P
a
P
a
i σ
φ
σ
σ
σ
σ
σ
σ ∂
∂
−
∂
∂
=
∂
∂
(C.19)