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where
2
arcsin
sin 2
E
m
1
E
m
sin
=
−
/
4
≤
≤
/
4
E
m
E
m
and
−
1
≤
E
m
≤
1
.
Thus, by the Swift-Eggers method we derive eight independent eigenstate
parameters:
|
ς
1
|
,ξ
1
=
ς
1
,
1
=
,
1
=
arg
E
1
E
1
(2
.
51)
|
ς
2
|
,ξ
2
=
ς
2
,
2
=
,
2
=
arg
E
2
E
2
which fill all eight degrees of freedom possessed by the matrix [
Z
].
There is a one-to-one correspondence between the impedance tensor and its prin-
cipal values, principal directions and eigenfield ellipticities. Given
ς
1
,
1
,
1
and
ς
2
,
2
,
2
, we can determine [
Z
]. The most simple are relationships between [
Z
]
and
ς
1
,ς
2
,
P
E
1
,
P
E
2
:
ς
2
−
ς
1
P
E
1
−
P
E
1
ς
2
−
P
E
2
ς
1
Z
xx
=
Z
xy
=
P
E
1
−
P
E
2
P
E
2
(2
.
52)
P
E
2
ς
2
−
P
E
1
ς
1
P
E
1
−
P
E
2
P
E
1
P
E
2
(
ς
2
−
ς
1
)
P
E
1
−
P
E
2
Z
yx
=
Z
yy
=
.
Here, with account for (2.9), (2.16) and (2.17),
E
m
e
i
m
P
E
m
=
tan
,
m
=
1
,
2
,
where
cos 2
E
m
=
cos (2 arctan
E
m
) cos 2
E
m
E
m
0
≤
≤
/
2
E
m
tan
=
tan (2 arctan
E
m
)csc2
E
m
E
m
0
≤
≤
if 0
≤
E
m
≤
1
E
m
<
0 f
−
<
−
1
≤
E
m
<
0
.
Now we examine indications of the Swift-Eggers method in the models of dif-
ferent dimensionality.
In the 1D-model, we have
Z
xx
=
Z
yy
=
0 and
Z
xy
=−
Z
yx
=
Z
where
Z
is
the Tikhonov-Cagniard impedance. Here
1
=
2
=
Z
and
P
E
1
,
2
=
0
/
0
,
E
1
,
2
=
0
/
0 . The principal values of the tensor [
Z
] coincide with Tikhonov-Cagniard's
