3.22. LONGITUDINALLY MAGNETIZED INFINITELY EXTENDING THIN SLABS 115
Concerning the frequency of this mode, we take the limit of (3.106) for ˛d ! 0 (note that the
second term requires the application of De L’Hospital’s rule), which yields !.n D 0 ! !
i
. at
is, for very thin slabs (e.g., films) the lowest-order mode is a symmetric one, primarily z-directed
or transverse to the film
N
k k
zi
Oz having ! ! !
i
. is is also confirmed in [17].
e other limiting case occurs when slab thickness becomes very large kd >> 1 and is
likely to approach the plane-wave propagation in an infinite medium. is is first verified by
taking the limit ˛d >> 1 in (3.106), which yields to !.n D 0/ !
p
!
i
.!
i
C !
m
/. is is iden-
tical to the plane wave case in (3.81) for D 0
ı
, which means that the n D 0 symmetric mode
propagates in the z-direction even for very thick slabs. However, a very thick slab supports an
infinite number of MSFVW, both symmetric and anti-symmetric.
3.22 LONGITUDINALLY MAGNETIZED INFINITELY
EXTENDING THIN SLABS
Damon and Eshbach [13], originally studied the magnetostatic modes of a thin ferrite slab,
magnetized to saturation along its plane. Once again, the coordinate system is modified in order
to preserve the notation of the previous paragraphs so that the analysis can be covered by the
Walker’s theory. For this purpose, the x-axis is oriented perpendicularly to the slab plane as
shown in Figure 3.15 and the DC biasing field, as well as saturation magnetization, is again
aligned along the Oz-axis.
H
0
X
Y
Z
+
d
/2
-d/2
Figure 3.15: Infinitely extending thin ferrite slab longitudinally biased to saturation.
Firstly, in order to examine the internal DC magnetic field H
i
recall that the demagneti-
zation factors parallel to the slab are zero N
y
D N
z
D 0 while normal to that is unity, N
x
D 1.
However, only N
z
is involved since Oz-bias is assumed. erefore,
N
H
i
D
N
H
0
N
z
N
M
s
D
N
H
0
or
!
i
D
0
H
i
D !
0
unless the anisotropy field is accounted for in a way like that of Eq. (3.66).
e separation of variables according to Eqs. (3.82) to (3.85) can be employed again, but
the orientation of the different coordinate needs to be taken into account. Following the same
reasoning as in the previous section, the boundary conditions for the coordinates parallel to the
slab surface require the same longitudinal eigenfunctions inside and outside the slab, that is,
116 3. FINITE FERRITE SAMPLES
g
i
.y/ D g
e
.y/ D g.y/ and h
i
.z/ D h
e
.z/ D h.z/. erefore, Eq. (3.82) reads as follows:
i
D f
i
.x/g.y/h.z/ (3.113a)
e
D f
e
.x/g.y/h.z/: (3.113b)
Along the x-direction, normal to the slab, the solution (now denoted f .x/) must be like
that of the normally biased case, that is, exponentially decaying outside the slab, which means
that it vanishes at infinity .x ! ˙1/ and has a standing wave form (co-sinusoidal or sinusoidal)
inside the slab. Hence, in a way similar to that of Eqs. (3.89) and (3.91) we may write
f
i
.x/ D A
x
sin
.
k
xi
x
/
C B
x
cos
.
k
xi
x
/
for d=2 x d=2 (3.114)
f
e
.x/ D
(
0e
e
jk
xe
.
x
d
2
/
D
0e
e
˛
.
x
d
2
/
for x d=2
0
0e
e
jk
xe
.
xC
d
2
/
D
0
0e
e
C˛
.
xC
d
2
/
for x d=2
)
: (3.115)
Note that k
2
xe
D ˛
2
< 0 or k
xe
D j˛.
e wavenumbers involved in the eigenfunctions g.y/ and h.z/ parallel to the slab, are
identical inside and outside the slab. Also, must be k
ye
D k
yi
D k
y
and k
ze
D k
zi
D k
z
since the
boundary conditions at x D ˙d=2 must hold at any arbitrary .y; z/ point. e common practice
is to consider solutions for g and h that represent waves propagating at arbitrary directions
along Oy; Oz, like e
j
N
k
t
N
, where
N
k
t
D k
y
Oy C k
z
Oz and N D y Oy C z Oz. is is because the slab is
infinitely extending in the above directions. A specific direction in the y–z plane is defined by the
ratio of k
y
=k
z
, which for an infinitely extending slab could be arbitrary, allowing propagation to
occur in all directions along the plane. However, this statement is not general enough. Obviously,
it holds for isotropic slabs, but it could be refuted in anisotropic materials. e question here
is whether the boundary conditions at x D ˙d=2 impose any restrictions on the k
y
and k
z
values and the corresponding eigenfunctions g.y/ and h.z/ through the anisotropy of the ferrite’s
susceptibility tensor ŒX or, equivalently, its permeability tensor []. To answer this question,
let us first examine the boundary conditions at x D ˙d=2.
(i) Continuity of the tangential magnetic field h
y
, h
z
at x D ˙d=2:
h
y
D
@
@y
$ f
i
.x/
dg.y/
dy
h.z/
ˇ
ˇ
xD˙d=2
D f
e
.x/
dg.y/
dy
h.z/
ˇ
ˇ
xD˙d=2
(3.116a)
h
z
D
@
@z
$ f
i
.x/g.z/
dh.z/
dz
ˇ
ˇ
ˇ
ˇ
xD˙d=2
D f
e
.x/g.z/
dh.z/
dz
ˇ
ˇ
ˇ
ˇ
xD˙d=2
: (3.116b)
e intuitive selection of identical g.y/ and h.z/ eigenfunctions inside and outside the
slab is required (or verified) by (3.116a) and (3.116b), which in view of this reduces to:
f
i
.x/
j
xD˙d=2
D f
e
.x/
j
xD˙d=2
: (3.117)
3.22. LONGITUDINALLY MAGNETIZED INFINITELY EXTENDING THIN SLABS 117
(ii) Continuity of the normal flux density component b
n
D b
x
, which according to (3.72)
and (3.73) is:
b
x
D
8
<
:
b
xi
.1 C X / h
xi
C j
0
k
1
h
yi
in the slab
j
x
j
d
2
/
b
xe
D
0
h
xe
outside the slab
j
x
j
d
2
:
(3.118)
At this point, it is interesting to take a closer look at (3.72) and (3.73) to gain some intuitive
understanding of the consequences of the different orientation of the . Ox; Oy; Oz/-axes with
respect to the slab. First, when the Oz-axis is normally oriented, the Ox and Oy-axes retain a
specific symmetry, both geometrically and in terms of the permeability anisotropy. us, in
this case, a wave is expected to propagate in both Ox and Oy directions. In contrast, when Ox (or
Oy)-axis is oriented normal to the slab, the Oy (or Ox) and Oz-axes are geometrically symmetric,
but there is an anisotropy along Oy (or Ox), while the material is isotropic along the Oz-axis.
is can be justified in this case through Eq. (
3.118), which involves h
yi
D @
i
=@y, but
not h
zi
D @
i
=@z. Moreover, remember that this anisotropy caused the “self-guidance”
effect observed in the case of magnetostatic modes propagating in an infinite (unbounded)
medium; in other words, each mode propagates along a specific direction.
To return to the topic under discussion, Eq. (3.118) can be recast in terms of the potential
functions as:
.1 C X /
@
i
@x
C j k
1
@
i
@y
xD˙d=2
D
@
e
@x
ˇ
ˇ
ˇ
ˇ
xD˙d=2
: (3.119a)
In view of (3.113), the above equation reads as follows:
.1 C X /
df
i
.x/
dx
g.y/ C jk
1
f
i
.x/
dg.y/
dy
xD˙d=2
D
df
e
.x/
dx
g.y/
ˇ
ˇ
ˇ
ˇ
xD˙d=2
: (3.119b)
Note that the eigenfunction
h.z/
is absent from (
3.119b) since it is common in all terms.
us, it could be arbitrarily selected. For example, the choice of Damon and Eshbach [13] was
h.z/ D cos.k
z
z/, but a propagating wave e
jk
z
z
could have been chosen as well. However, a
closer examination of conditions (3.115) at x D Cd=2 and x D d=2 reveal that g.y/ can only
be chosen as a propagating wave, e.g., g.y/ D e
jk
y
y
. is is also stated in [13], and a similar
condition is observed by Walker [9], who found that the only possible azimuthal dependence in
the spheroid case is e
j m'
. Moreover, in order to be compatible with [13], a positive sign in the
exponential is assumed (e.g., propagation in the negative y-direction). us,
g.y/ D e
jk
y
y
(3.120)
which yields
dg
dy
D Cj k
y
e
jk
y
y
D j k
y
g.y/ or
d
dy
! j k
y
: (3.121)
118 3. FINITE FERRITE SAMPLES
In view of the latter, g.y/ may also be eliminated from (3.119b), which now reads as
follows:
.1 C X /
df
i
.x/
dx
k
1
k
y
f
i
.x/
xD˙d=2
D
df
e
.x/
dx
: (3.122)
In turn, Eqs. (3.122) and (3.117) must be combined with the wavenumber condi-
tions (3.85) resulting for the separation of variables. For the present case, the latter can be
rewritten as:
.1 C X /k
2
xi
C .1 C X /k
2
y
C k
2
z
D 0 (3.123a)
˛
2
C k
2
y
C k
2
z
D 0: (3.123b)
Substituting (3.123b) into (3.123a) yields
.1 C X /k
2
xi
C X k
2
y
C ˛
2
D 0: (3.123c)
Let us now enforce the boundary conditions. Starting from the tangential field compo-
nents, (3.117) through (3.114) and (3.115) read as follows:
x D Cd=2 W A
x
sin
.
k
xi
d=2
/
C B
x
cos
.
k
xi
d=2
/
D ‰
0e
(3.124a)
x D d=2 W A
x
sin
.
k
xi
d=2
/
C B
x
cos
.
k
xi
d=2
/
D
0
0e
: (3.124b)
ese can be solved for A
x
and B
x
as:
A
x
D
0e
0
0e
2
1
sin
.
k
xi
d=2
/
(3.125a)
B
x
D
0e
C
0
0e
2
1
cos
.
k
xi
d=2
/
: (3.125b)
e procedure can be simplified by substituting A
x
and B
x
back into (3.114) to yield
f
i
.x/ D
0e
0
0e
2
sin
.
k
xi
x
/
sin
.
k
xi
d=2
/
C
0e
C
0
0e
2
cos
.
k
xi
x
/
cos
.
k
xi
d=2
/
: (3.126)
It is interesting to observe f
i
.x/ at x D ˙d=2, which is
f
i
.x D Cd=2/ D
0e
and f
i
.x D d=2/ D
0
0e
: (3.127)
3.22. LONGITUDINALLY MAGNETIZED INFINITELY EXTENDING THIN SLABS 119
e boundary condition (3.122) for the normal component can be written explicitly
through (3.115) and (3.126) as:
x D Cd=2 W
.1 C X /k
xi
‰
0e
‰
0
0e
2
cot
k
xi
d
2
‰
0e
C ‰
0
0e
2
tan
k
xi
d
2
k
1
k
y
‰
0e
D ˛ ‰
0e
(3.128a)
x D d=2 W
.1 C X /k
xi
‰
0e
‰
0
0e
2
cot
k
xi
d
2
C
‰
0e
C ‰
0
0e
2
tan
k
xi
d
2
k
1
k
y
‰
0
0e
D ˛ ‰
0
0e
:
(3.128b)
Some algebraic manipulation is required to eliminate ‰
0e
and ‰
0
0e
from (3.128a) and
(3.128b). For example, in each equation, all terms containing ‰
0e
can be moved to the left side,
and all terms with ‰
0
0e
to the right side. en, by dividing the two equations, we obtain the
modes characteristic equation:
.1 C X /
2
k
2
xi
C .1 C X / k
xi
˛
cot
k
xi
d
2
tan
k
xi
d
2
C ˛
2
k
2
1
k
2
y
D 0: (3.129)
For further simplification, recall the trigonometric identity:
tan.2z/ D 2=Œcot.z/ tan.z/ D 1= cot.2z/: (3.130)
en (3.129) becomes
2.1 CX / k
xi
˛ cot
.
k
xi
d
/
.1 C X /
2
k
2
xi
C ˛
2
k
2
i
k
2
y
D 0: (3.131)
Equation (3.131) is identical to Damon and Eshbach’s [13], even though different symbols
are used herein. e characteristic equation (3.131) along with conditions (3.123) determines
the mode spectrum. As in the previous section, the quantity .1 CX / may be either positive or
negative. Recall (3.100) as:
1 C X < 0 for !
i
< ! <
Œ
!
i
.
!
i
C !
m
/
1=2
: (3.132a)
1 C X > 0 for ! < !
i
or ! >
Œ
!
i
.
!
i
C !
m
/
1=2
: (3.132b)
In addition, from the definition of X in (3.90) it is obvious that
X > 0 for ! < !
i
; X ! ˙1 at ! D !
i
; and X < 0 for ! > !
i
:
Note that the infinity of X at ! D !
i
is a direct consequence of ignoring ferrite losses,
while the inclusion of losses results in a finite value. e behavior of both .1 C X / and X vs.
frequency is presented in Figure 3.16.
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