3.22. LONGITUDINALLY MAGNETIZED INFINITELY EXTENDING THIN SLABS 121
is concentrated along the top or the bottom of the ferrite-air interface and propagates along it
(in the y- and z-directions). erefore, these types of modes are termed “surface modes” and
symbolized as MSSW (magneto-static surface waves).
e question now is which type of mode is excited in each of the three regions identified
in (3.133) and what their specific characteristics are. For range I as well as range III defined
in (3.133), it is obvious that the only requirement that needs to be met for (3.123a) or (3.123c)
is for k
2
xi
< 0 or k
xi
to be imaginary, so these modes should be MSSW. In contrast, in range II,
the negative sign of .1 C X / enables the choice k
2
xi
> 0, so MSBVW are supported. In addition,
remember that range II is the extrapolated spin-wave band since it was previously found that
this is the only range supporting spin waves provided that the already mentioned wavelength
restrictions are met. erefore, outside this range k
xi
must be imaginary [13].
3.22.1 MAGNETOSTATIC VOLUME MODES .1 C X / < 0
As already noted, in range II, it is k
2
xi
> 0. is is a consequence of .1 C X / < 0. To examine
the characteristics, we may express the wavenumber in spherical coordinates as in [13]:
Ferrite region:
N
k
i
D k
xi
Ox C k
y
Oy C k
z
Oz (3.134a)
Air region:
N
k
˛
D k
xe
Ox C k
y
Oy C k
z
Oz where k
xe
D j˛ (3.134b)
and k
xi
D k
ti
cos '; k
y
D k
ti
sin '; k
ti
D k
i
sin ; k
z
D k
i
cos ; (3.134c)
where and ' are the angles of
N
k
i
with respect to the Oz and Ox axes, respectively.
In view of the above notation, (3.123b) reads as follows:
k
2
˛
D ˛
2
C k
2
y
C k
2
z
D 0: (3.135a)
is simply means that the total wavenumber in the air region is zero.
In turn, (3.123a) may be written in two similar forms:
.1 C X /k
2
ti
C k
2
z
D 0 ,
k
2
ti
k
2
z
D tan
2
D
1
.1 C X /
(3.135b)
or X k
2
ti
C k
2
i
D 0 ,
k
2
ti
k
2
D sin
2
D
1
X
: (3.135c)
Using the definition of X from (3.84), the latter yields
sin
2
D
2
2
H
H
,
2
D
2
H
C
H
sin
2
: (3.136)
Equation (3.136) reveals once again the frequency range of volume modes defined by
0 sin
2
1. eir lower frequency corresponds to D 0
ı
, where D
H
,
N
k
i
D k
z
Oz and
122 3. FINITE FERRITE SAMPLES
k
ti
D 0, which means that the wave propagates along the direction of DC magnetization (z-
axis). In contrast, D 90
ı
corresponds to their maximum frequency D Œ
2
H
C
H
1=2
and
k
z
D 0; k
ti
D k. Since the mode pattern in the x-direction is that of a standing wave, the wave
propagates in either positive or negative y-directions. Recall that both signs of k
y
are permitted
and a dependence of the form e
jk
y
y
is assumed. Since these two modes are degenerate in fre-
quency, they may be combined to form two new modes [13]. However, these combined modes
propagate in opposite directions on the two sides of the slab. It is also interesting to observe
the special case of modes appearing at D 0
ı
. From (3.123a), it can be seen that ˛ D 0. us,
these modes do not have an RF field outside the slab. On the other hand, from (3.123a) we can
see that k
xi
is arbitrary since .1 C X / D 0 at this frequency; in other words, these types of mode
have no variation along the longitudinal dimensions .y; z/ while their field pattern is arbitrary
in the transverse x-direction. Moreover, it will be proved next that for D 90
ı
these modes
become modes of the surface wave type (MSSW).
Density of modes: Equations (3.123) clearly show that the number of possible volume modes
is proportional to X and .1 CX ). ese quantities tend to infinity, and so does the number of
modes when !
H
. However, this phenomenon appears only at the ideal ferrite slab case
when ferrite losses do not exist. In actual practice, when ferrite losses are included, X becomes
finite at !
H
and the number of possible modes is drastically reduced.
Phase and group velocities: e phase (V
p
) and group velocities (V
g
) are expressed in a way
similar to that of Eq. (3.111), but here propagation occurs in both the
O
y
and
O
z
directions as
e
jk
y
y
e
jk
z
z
D e
j
N
k
N
, where
N
k
D k
y
Oy C k
z
Oz and N D y Oy C z Oz. erefore, k
t
in (3.111)
should be replaced by k
. is, in turn, is k
D ˛ from Eq. (3.123b). us, V
p
D !=˛ and
˛ D> 0 could be obtained from the solution of the dispersion equation (3.131). However, group
velocity V
g
D .d!=d k
) requires the differentiation of (3.131), which results in very compli-
cated expressions. In contrast, relatively simple expressions clarifying the properties of the modes
can be obtained for the particular cases when:
.i/ D 0
ı
and k
y
D 0 and the wave propagates along the z-direction and
.i i/ D 90
ı
and k
z
D 0 and the wave propagates in the y-direction.
Propagation in the z-direction: (k
y
D 0 $ MSBVW)
e phase and group velocities for this particular case read as follows:
V
p
D
!
k
z
and V
g
D
@!
@k
z
: (3.137)
Equations (3.123) for k
y
D 0 are reduced to:
˛ D k
xe
D k
z
and k
xi
D k
z
=
p
.1 C X /: (3.138)
3.22. LONGITUDINALLY MAGNETIZED INFINITELY EXTENDING THIN SLABS 123
Substituting back into (3.131), the dispersion equation is simplified to:
2.1 CX /k
2
z
p
.1 C X / cot
k
z
d=
p
.1 C X /
C .2 C X / k
2
z
D 0: (3.139a)
us, one possible solution is k
z
D 0, which has already been discussed as the case (k
z
D
0; k
y
D 0). By carefully handling .1 CX / < 0 terms, the interesting solution reads as follows:
cot
k
z
d=
p
.1 C X /
D C
1
2
.2 C X /=
p
.1 C X / D
1
p
.1 C X /
p
.1 C X /:
(3.139b)
Note that .1 C X / in the range of volume waves is negative. Also, the question of the
cotangent and its argument appears. First, a function of the form
1
y
y
with y D .1 CX / >
0 is obviously positive when y < 1, zero at y D 1 and negative for y > 1. erefore, this is also
the behavior of the cotangent in (3.139b). In turn, in range II volume waves are split into two
sub-ranges at .1 CX / D 1. Using the definition (3.86) for X , we have that:
.1 C X / D 1 $ D
A
D
2
H
C
1
2
H
1=2
: (3.140a)
ese sub-ranges can be set as follows:
Range II. a:
H
< D
r
2
H
C
1
2
H
<
A
$ .1 CX / < 1 negative cotangent (3.140b)
Range II. 0:
A
D $ .1 C X / D 1 zero cotangent (3.140c)
Range II. b:
A
< D
q
2
H
C
H
<
B
$ 1 < .1 CX / < 0 positive cotangent: (3.140d)
e characteristic equation (3.139b) is very simple at D
A
and can be readily solved
as follows:
D
A
$ cot
k
z
d=
p
.1 C X /
D 0
$ k
z
d=
p
.1 C X / D n
2
where n D 1; 3; 5:
(3.141)
Likewise, the argument of the cotangent falls in the second or fourth quadrant for range
II.a and in the first and third ones for range II.b.
124 3. FINITE FERRITE SAMPLES
In turn, group velocity can be obtained by differentiating (3.139) with respect to k
z
, but re-
member that X is a function of (!) given in Eq. (3.86). However, instead of dealing with (3.139),
it is preferable to transform it into a more familiar form. For this purpose, we may adopt the half
argument identity like that of (3.130) and solve (3.139b) by assuming that tan.z/ D 1= cot.z/
is the unknown. e resulting second-order polynomial equation has two distinct solutions:
tan
k
z
d
2A
D ˙
1
A
: (3.142a)
e above can also be written as:
cot
k
z
d
2A
D ˙A; (3.142b)
where A D
p
.1 C X /.
Now, in view of the identity (3.107a), it can be clearly seen that (3.142b) represents the
odd .n D 1; 3; 5; : : :/ and the even modes .n D 0; 2; 4; : : :/ of a dispersion equation of the form
tan
k
z
d
2
p
.1 C X /
n
2
!
D
p
.1 C X / n D 0; 1; 2; 3; : : : (3.143)
is dispersion equation is identical to the one given in [12], with a minor difference in the
definition of .n/. is can be solved graphically if it is recast in a way similar to that of (3.109)
and (3.110). An approximate solution of (3.143) for the lowest-order mode .n D 0/ is again
given by Kallinikos [38]:
!
2
D !
i
(
!
i
C !
m
1 e
k
z
d
k
z
d
!)
: (3.144)
e dispersion curves of D !=!
m
vs. k
z
d are shown in Figure 3.17 [12]. It is obvious
that the slope of these curves is negative and group velocity V
g
is also negative; hence, they are
characterized as backward waves (MSBVW).
e mathematical expression for group velocity can be obtained from the differentiation
of (3.144) with respect to k
z
[12], as follows:
V
g
D
d!
d k
z
D
2k
1
!
m
d
C
k
1
k
z
!
m
X
1 C X
: (3.145a)
e lowest-order .n D 0/ mode has the steepest slope and, thus, the most negative group
velocity. For this mode and for electrically thin slabs or films (k
z
d << 1), the above expression
can be approximated to
1
V
g
ˇ
ˇ
ˇ
ˇ
k
z
d <<1
4
!
m
d
p
!
i
.
!
i
C !
m
/
!
i
: (3.145b)
3.22. LONGITUDINALLY MAGNETIZED INFINITELY EXTENDING THIN SLABS 125
0 1 2
n = 0
n
= 1
n = 2
n
=
3
3 4 5
0.9
0.8
0.7
0.6
0.5
0.4
ω/ω
M
k
z
d
Figure 3.17: Dispersion curves for backward magnetostatic volume waves (MSBVW) [12].
It is obvious from (3.145b) that group velocity is negative; hence, the modes are correctly
termed “backward waves.” e inverse of group velocity .1=V
g
/ vs. D !=!
m
for the first four
modes is presented in Figure 3.18 [12].
0.5 0.6 0.7
n = 0
n = 1
n = 2
n = 3
0.8 0.9
1,000
800
600
400
200
0
1/Vg (ns/cm)
ω/ω
M
Figure 3.18: Inverse group velocity .1=Vg/ for MSBVW [12].
Before moving on to the analysis of magnetostatic surface waves it is worth examining
whether volume modes are possible when pure propagation only in the y-direction is assumed.
at is, consider k
z
D 0 and
N
k D k
xi
Ox C k
y
Oy. In this case, Eq. (3.123a) reads as follows:
.1 C X /k
2
xi
C .1 C X /k
2
y
D 0 $ k
2
xi
D k
2
y
$ k
xi
D ˙j k
y
: (3.145c)
It is then obvious that k
2
xi
should be negative or k
xi
should be imaginary when k
z
D 0 since
pure propagation in the y-direction requires k
y
to be real. However, this is the case of magne-
tostatic surface waves, (to be discussed next), as the imaginary k
xi
causes the field to decay very
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