3.24. MAGNETOSTATIC WAVES ON MULTILAYER AND GROUNDED STRUCTURES 151
After some algebra, this concludes to:
V
g
D 2d !
m
.S/
h
2
.
H
C 1
/
2
i
Œ
S C
.
H
C 0:5
/
Œ
2
C 2 S
.
H
C 1
/
C
H
.
H
C 1
/
: (3.229b)
Note, that for this section to be compatible with the previous one and, in turn, with the
work of Damon and Eshbach [13], the Oy propagation was assumed e
Cjk
y
y
D e
j jk
y
jSy
. us,
S D 1 represents propagation toward positive Oy for which (3.229b) yields positive V
g
. at
is, MSSW are ordinary forward modes with group velocity pointing toward the propagation
direction. Also, it is worth noting that when the metallic ground plane is located on the up-
per face x D d of the slab instead of the lower face x D 0, then the characteristics of MSSW
propagating in the positive Oy and negative Oy directions are interchanged. An important result
shown in (3.229) is that the group velocity V
g
is proportional to the substrate thickness d . Since
“delay lines” represent a very attractive application the desired group delay t
d
D `=V
g
(` is the
structure length) is inversely proportional to the slab thickness t
d
/ 1=d . In other words, the
slab thickness may be exploited as a degree of freedom in controlling the group delay.
Further, it is worth comparing the MSSW wavenumber and group velocity with those of
the ungrounded ferrite slab, which from Eqs. (3.157b) and (3.158) [48], reads:
ˇ
ˇ
k
y
ˇ
ˇ
D
1
2d
ln
h
4
.
H
C 0:5
/
2
2
i
: (3.230)
V
g
D
!
m
d
h
.
H
C 0:5
/
2
2
i
: (3.231)
Comparing (3.231) with (3.229), one may see that in both cases the group delay .t
d
/
1=V
g
/ is inversely proportional to the ferrite slab thickness. However, the sign .S/ is absent
from the ungrounded ferrite slab relations (3.230)–(3.231), meaning that MSSW propagating
in both positive and negative y-directions have the same characteristics.
3.24.5 GROUNDED DIELECTRIC–FERRITE LAYERS
Continuing with the printed ferrite structures, the next to be studied is the grounded dielectric-
ferrite layers shown in Figure 3.29 [47–50]. Bongianni [47], who in 1972 followed the notation
of Seshadri [48], originally studied this. However, herein the notation and coordinate system of
the previous sections will be used. e almost ideal dielectric substrate for YIG film growth is
the gadolinium galium garnet (GGG), as pointed out by Ishak [50]. e practical ferrite YIG
film thickness ranges from 5–150 m, with linewidth as low as H 0:5 Oe.
152 3. FINITE FERRITE SAMPLES
ŷ
x̂
ẑ
x = d + t
x = t
x = 0
H̄
DC
⇒
↕
↕
d
t
Ferrite (YIG)
Dielectric
Figure 3.29: Infinite longitudinally magnetized two dielectric-ferrite layers.
Following the reasoning of the grounded ferrite section, the general solution for the mag-
netic potential can be written as:
Dielectric region:
d
D f
d
.x/e
jk
y
y
e
jk
z
z
(3.232a)
Ferrite region:
i
D f
i
.x/e
jk
y
y
e
jk
z
z
(3.232b)
Air region:
e
D
0e
e
˛.xd t/
e
jk
y
y
e
jk
z
z
: (3.232c)
e appropriate eigenfunction in the x-direction is:
f
d
.x/ D A
d
e
k
xd
x
C B
d
e
k
xd
x
(3.233a)
MS volume modes: f
i
.x/ D f
ix
.x/ D A
x
sin
.
k
xi
x
/
C B
x
cos
.
k
xi
x
/
(3.233b)
MS surface modes: f
i
.x/ D f
is
.x/ D A
0
x
e
˛
i
x
C B
0
x
e
˛
i
x
: (3.233c)
For the wavenumber relations, Equations (3.215) are still valid for the air and ferrite layers
as:
Air: ˛
2
C k
2
y
C k
2
z
D 0 (3.234a)
Ferrite: .1 C X /k
2
xi
C .1 C X /k
2
y
C k
2
z
D 0; (3.234b)
while for the dielectric layer it is:
Dielectric: k
2
xd
C k
2
y
C k
2
z
D 0 $ k
xd
D ˛: (3.234c)
3.24. MAGNETOSTATIC WAVES ON MULTILAYER AND GROUNDED STRUCTURES 153
A point of interest here is the expected field behavior in the dielectric region, which is
directly assumed to be similar to that of surface waves, that is, an exponential decay away from
the surface. is is dictated by the wavenumber relation (3.234) where k
y
and k
z
should be real,
(k
2
y
; k
2
z
> 0), representing propagation along the infinite Oy and Oz directions. is asks k
2
x
to be
negative for (3.234c) to be fulfilled as k
2
x
< 0 or k
2
x
D k
2
xd
and k
x
D j k
xd
. Comparing (3.234a)
and (3.234c), it is obvious that k
xd
D ˛. at is, the wave behavior in the air and dielectric are
quite similar. is is intuitively expected since the dielectric is a non-magnetic material (
d
D
0
). It is also expected that when the dielectric thickness tends to infinity .t ! 1/, then the
wave characteristics will tend to those of the ungrounded (free) ferrite slab. Likewise, when its
thickness tends to zero .t ! 0/, the structure will behave exactly as the grounded ferrite slab.
Let us now seek the magnetostatic volume or surface-wave modes characteristic equations
through the enforcement of the boundary conditions. Starting from the ferrite-air interface
x D d C t, the continuity conditions remain the same as in the previous section. Since the same
notation is retained, we may directly adapt (3.219c) to this case as:
.1 C X /f
0
i
.x D d C t/ D
k
1
k
y
˛
f
i
.x D d C t/: (3.235)
e normal magnetic flux density should again vanish at the metallic ground plane at
x D 0:
b
xd
D
0
h
xd
D
0
@
d
@x
ˇ
ˇ
ˇ
ˇ
xD0
D 0 $ f
0
d
.x D 0/ D 0 $ B
d
D A
d
: (3.236)
Substituting (3.236) to (3.233a), we have:
f
d
.x/ D 2A
d
cosh
.
k
xd
x
/
and f
0
d
.x/ D 2A
d
k
xd
sinh
.
k
xd
x
/
: (3.237)
e normal flux density in the ferrite region is given in (3.218b), so the continuity condi-
tion for h
y
, h
z
, and b
x
yields correspondingly:
f
d
.x D t/ D f
i
.x D t/ (3.238a)
.
1 C X
/
f
0
i
.x D t/ D k
1
k
y
f
i
.x D t/ D f
0
d
.x D t/: (3.238b)
Combine the above together through (3.237) to get:
.1 C X /f
0
i
.x D t/ D
k
1
k
y
k
xd
tanh
.
k
xd
t
/
f
i
.x D t/: (3.239)
Equations (3.235) and (3.239) can be readily combined in conjunction with the eigenfunc-
tions of (3.233) to yield the characteristic equations of either volume or surface magnetostatic
modes.
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