Information Technology Reference
In-Depth Information
where
m
is the complex principal value (eigenvalue) of the tensor [
M
(
r
|
r
B
)]. In
the
x
,
y
−
coordinates
H
xm
(
r
)
=
m
H
xm
(
r
B
)
H
ym
(
r
)
=
m
H
ym
(
r
B
)
m
=
1
,
2
.
(4
.
52)
Substituting (4.52) in (4.42), we get
(
M
xx
−
m
)
H
xm
(
r
B
)
+
=
M
xy
H
ym
(
r
B
)
0
(4
.
53)
M
yx
H
xm
(
r
B
)
+
(
M
yy
−
m
)
H
ym
(
r
B
)
=
0
,
m
=
1
,
2
.
Assuming the determinant of this uniform system of linear equations to be zero, we
obtain
m
−
(
M
xx
+
M
yy
)
m
+
(
M
xx
M
yy
−
M
xy
M
yx
)
=
0
,
(4
.
54)
whence
(
M
xx
+
(
M
xx
+
M
yy
)
+
M
yy
)
2
−
4(
M
xx
M
yy
−
M
xy
M
yx
)
1
=
,
2
−
(
M
xx
+
(4
.
55)
(
M
xx
+
M
yy
)
M
yy
)
2
−
4(
M
xx
M
yy
−
M
xy
M
yx
)
2
=
.
2
The principal directions of the tensor [
M
] are determined as directions of the
major axes of the polarization ellipses of the magnetic eigenfields
H
τ
1
(
r
) and
H
τ
2
(
r
). With (4.52) and (4.53), the polarization ratios for
H
τ
1
(
r
) and
H
τ
2
(
r
)are
H
ym
(
r
B
)
=
m
−
M
xx
H
ym
(
r
)
H
xm
(
r
)
=
H
ym
(
r
B
)
P
H
m
(
r
)
=
M
xy
(4
.
56)
m
−
M
yy
=
m
−
M
xx
+
M
yx
M
yx
=
m
+
M
xy
−
M
yy
,
m
=
1
,
2
.
Substituting (4.56) into (2.18), we evaluate angles
H
1
and
H
2
made by major axes
of the polarization ellipses with the
x
-axis:
H
m
cos
H
m
tan2
=
tan2
,
m
=
1
,
2
(4
.
57)
H
m
P
H
m
,
where tan
H
m
=
H
m
=
arg
P
H
m
.
The values of
are taken within quad-
H
m
H
m
rant I (0
≤
H
m
≤
/
2) if cos
≥
0 or within quadrant IV (0
>
H
m
≥−
/
2) if
H
m
<
0.
Finally we determine the ellipticity parameters
cos
H
1
and
H
2
. In accord with (2.19),
H
m
=
tan
H
m
,
m
=
1
,
2
,
(4
.
58)
where
