microwave magnetics 10 - ee.sharif.edu
TRANSCRIPT
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Microwave Magnetics
Graduate Course
Electrical Engineering (Communications)
2nd Semester, 1389-1390
Sharif University of Technology
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Magnetostatic waves and oscillations 2
General information
� Contents of lecture 10:
• Magnetostatic waves and oscillations
� Introduction
� Magnetostatic waves
� Magnetic potential
� Magnetostatic waves in metallized plates
� Normally magnetized plates
� Transversely magnetized plates
� Magnetostatic waves in tangentially magnetized free plates
� Surface waves
� Excitation of magnetostatic waves
� Magnetostatic wave devices
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Magnetostatic waves and oscillations 3
(i) Introduction
� Let us start our discussion with an example: an
unbounded magnetic film placed on top of a ground plane
� The film is magnetized parallel to its plane in the z-
direction
0 0,M Hx
y
z
d
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Magnetostatic waves and oscillations 4
(i) Introduction
� We look for solutions which represent waves
propagating in the y-direction, we assume them to be
uniform along the magnetization (z-direction)
x
y
z
( ) exp( ) ( ) exp( )x j y x j y� �� � � ���e e h h
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Magnetostatic waves and oscillations 5
(i) Introduction
� To analyze the structure we resort to equations we had
used before (inside the magnetic material)
x
y
z
� �2
2 20 02 2
0akx z z
� ��� � �� �
� �� �� � �
� � � � �� �� � �� �
2z z z
z
h h eh�
�
� � ��
� �2
2 20 02 2
0akx z z
� ��� � ��
��
� �� � �� � � � �� �� � �� �
��� �
�2
z z zz
e e he
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Magnetostatic waves and oscillations 6
(i) Introduction
� Two decoupled sets result.
� 1st set (inside magnetic material):
� �2
2 202
0d
kdx
�� ��� � ��
�zz
ee
2 20
1
( ) /
a
a a
j j
j j d dx
� � �
�� � � � �
� � � � �� �� � � � �� �
�� � � �� �� � � �� �
� �
� �
zx
zy
eh
eh
x
y
z
0� � ��� �x y ze e h
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Magnetostatic waves and oscillations 7
(i) Introduction
� 2nd set (inside magnetic material): x
y
z
0� � �� � �x y zh h e
� �2
2 202
0d
kdx
�� �� � ��
��z
z
hh
0
1
/
j
j d dx
�
�� �
� �� �� �� � � �� �� �� � � �
x z
y z
e h
e h
��
��
� 2nd set not affected by magnetic properties since �|| = 1 (ac
magnetic field parallel to static magnetization).
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Magnetostatic waves and oscillations 8
(i) Introduction
� 1st set: Region I: 0 < x < d
2xk
� �2
2 202
0d
kdx
�� ��� � ��
�zz
ee
� �( ) sin xx A k x��ze � � � �
0
cos sinx x x
Ak k x k x
j�
�� ��
� �� �� ��
yh
a�� ��
�
d
x
y
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Magnetostatic waves and oscillations 9
(i) Introduction
� 1st set: Region II (air): x >d
2,0xk
� �2
2 202
0d
kdx
�� � ��
�zz
ee
� �,0( ) exp xx B jk x� ��ze � �,0
,00
expxx
k Bjk x
��� � ��
yh
d
x
y
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Magnetostatic waves and oscillations 10
(i) Introduction
� Resulting propagation equation:
� � ,0cotx x xk k d jk� ��� �
� �2 2 2 2 2 20 0 0cot ak d k j k
��� � �� � � � �
�� � �� � � � �
� The left hand side is always real. Since
the right hand side should be real �0k� �
2 2 2 20 0k j k� �� � � � Why the minus sign?
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Magnetostatic waves and oscillations 11
(i) Introduction
� There are two classes of solutions. The 1st class satisfies
2 2 20 0k k� ���� � Condition: �� > 0
� These solutions are surface waves (why?) Like ordinary
(TE) surface waves on grounded dielectrics, they have a
frequency cutoff which depends on the film thickness,
dielectric constant, etc.
� The magnetic material, however, makes � dependent on
the direction of propagation.
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Magnetostatic waves and oscillations 12
(i) Introduction
� Numerical example:
2 GHz
6 GHz
4
1 mm
H
M
f
f
d
�
��
��
-1 (mm )�
(GHz)f
� In any case 0k� ����
0� �
� Note also that the phase velocities are comparable to
that of light (e.g. at 50 GHz: vp = 2�f / � ~ 2.85 x 108 m/s)
0� �
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Magnetostatic waves and oscillations 13
(i) Introduction
� However, there is a second branch of solutions in which
2 20k� ����
� These are also surface waves (why?). They only exist for
particular frequencies, but the range mainly depends on
magnetic properties
� �2 2 2 2 2 20 0 0coth ak d k k
�� �� � �� � � �
�� � �� � � � �
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Magnetostatic waves and oscillations 14
(i) Introduction
� Numerical example:
-1 (mm )�
2 GHz
6 GHz
4
1 mm
H
M
f
f
d
�
��
��
H Mf f�
2M
H
ff �
� These �’s are large compared to to k0 !
� Wavelengths (2�/�) are short compared to
electromagnetic wavelength 2�/k0.
f�0� �
0� �
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Magnetostatic waves and oscillations 15
(i) Introduction
� These waves are called slow waves
� In thinner films (here the thickness was 1mm) the velocity
and wavelength can be even 2-3 orders of magnitude
smaller than electromagnetic velocities!
� Note also that the phase velocities can be much smaller
than that of light: for example
• left moving wave at 7 GHz: vp = 2�f / � ~ 0.665 x 108 m/s
• right moving wave at 4.7 GHz: vp = 2�f / � ~ 0.45 x 108 m/s
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Magnetostatic waves and oscillations 16
(i) Introduction
� These slow waves can satisfy
� Dispersion equation may have been approximated by
2 20k� � ���
� �2 2 2 2 2 20 0 0coth ak d k k
�� �� � �� � � �
�� � �� � � � �
� �coth ad�
� � � � �� �� �
2 20k� �
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Magnetostatic waves and oscillations 17
(i) Introduction
� Comparison with the exact result (lines: exact, points:
approximate) show the accuracy of this approximation,
in particular for short wavelength’s (large �’s)
-1 (mm )�
H Mf f�
2M
H
ff �
f�0� �
0� �
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Magnetostatic waves and oscillations 18
(ii) Magnetostatic waves
� So, with magnetic materials, it is possible to have slow
waves with short wavelength’s
0pv c� 00
pp
v c
f f� �� ��
� What is more: to obtain these solutions we may
assume that the velocity of light is infinite
0 0pk v c� �� �
� This approximation means that we neglect all the
propagation effects!
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Magnetostatic waves and oscillations 19
(ii) Magnetostatic waves
� This means that, when dealing with slow wave
solutions, in Maxwell equations we are allowed to
neglect the displacement current term everywhere:
� �
00
0 0
d
d
j
j
���
� �
�� �
�� � � � � � �
�� � � �� � �
e h e
h e j h
�
�
� This leads to the magnetostatic approximation:
� � 0�� � �� � �h j h�
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Magnetostatic waves and oscillations 20
(ii) Magnetostatic waves
� In electromagnetic problems we are accustomed to the
idea that wave propagation requires the displacement
current.
� Clearly here we have waves which do not need the
electric field for propagation!
� These are called magnetostatic waves. They are, in fact,
waves of the magnetization propagating inside the
medium.
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Magnetostatic waves and oscillations 21
(ii) Magnetostatic waves
� To see the physical origin of these waves recall the
linearized Landau-Lifshitz equation
0 0
( , )( , ) ( , )
d tt t
dt� �� � � � �
m rm r H M h r
( , )tM r0M
( , )tm r
x
y
� � � �0x
s y y
dmM h H m
dt� �� �
� � � �0y
s x x
dmM h H m
dt� �� � �
M� H�
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Magnetostatic waves and oscillations 22
(ii) Magnetostatic waves
( , )tM r0M
( , )tm r
x
y
� Differentiation with respect to time:
22
2
yxH x M H M x
dhd mm h
dt dt� � � �� � �
22
2
y xH y M H M y
d m dhm h
dt dt� � � �� � � �
� The field h is the ac magnetic field which contains the
externally applied field, and the “demagnetization”
field generated by the magnetization itself.
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Magnetostatic waves and oscillations 23
(ii) Magnetostatic waves
� Let us assume that no external ac field is applied.
� Besides, since we are adopting the magnetostatic
approximation anyway, we use the approximation
0( , ) ( , ) ( ) ( , )M
V
t t G t dV� � � �� � �� � � ��h r h r r r m r
� If we neglect the effect of the boundary of the volume V
(in reality this is wrong but this is just a qualitative
argument)
� �0( , ) ( ) ( , )V
t G t dV� � � � �� � � � ��h r r r m r
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Magnetostatic waves and oscillations 24
(ii) Magnetostatic waves
� The demagnetization field is related to the change of the
magnetization in space (its second derivative)
� It can lead to wavelike behavior. For instance, imagine
for some reason we can neglect my and hy
22
2
yxH x M H M x H M x
dhd mm h h
dt dt� � � � � �� � � �
2
0 2
( , )( ) x
x
V
m th G dV
x
�� ��� �� � � ���� ��
rr r
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Magnetostatic waves and oscillations 25
(ii) Magnetostatic waves
� This is a wavelike equation (not a wave equation
because of the integration involved)
� Hence: change of magnetization in space induces
demagnetization fields which interact with the motion
(rotation) of the magnetization
� This leads to wavelike phenomena
2 22
02 2
( , )( ) 0x x
H M H x
V
d m m tG dV m
dt x� � �
�� ��� �� � � �� ���� ��
rr r
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Magnetostatic waves and oscillations 26
(ii) Magnetostatic waves
� Remarks:
• Whatever the mechanism, everything ‘is’ already covered by
the full Maxwell equations. The magnetostatic approach
does not bring about new phenomena such as slow waves:
these are already in the Maxwell equations.
• Magnetostatic approach only leads to a simplified formalism
which allows us to study a certain class of solutions.
• Yet, magnetostatic approach cannot cover all possible
solutions. Had we neglected the displacement current in our
example, we could not have found the conventional surface
waves with 2 2 20 0k k� ���� �
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Magnetostatic waves and oscillations 27
(ii) Magnetostatic waves
• Short wavelength magnetostatic waves can be used to
design compact devices at microwave frequencies (MSW
devices)
• Magnetostatic approach allows us to perform an
approximate, but simple analysis of these components.
• Magnetostatic waves are sometimes called spin waves, but
that is not completely accurate. They are, actually, the long
wavelength limit of the spin waves. A more accurate terms
is: non-exchange spin waves.
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Magnetostatic waves and oscillations 28
(iii) Magnetic potential
� The magnetostatic equations are often solved in
materials where the electric (conduction) currents are
negligible. Therefore
� �0 0�� � �� � �h h�
� Let us introduce
�� �h Magnetic potential
� Then the first equation is automatically satisfied
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Magnetostatic waves and oscillations 29
(iii) Magnetic potential
� 2nd equation:
� � 0�� � �h�
2 2 2
2 2 20
x y z
� � �� �� �� � �
� � �� �� � �� ��
� �a does not appear in this equation! But it affects the
problem through boundary conditions.
0
0
0 0
a
a
j
j
� �� �
�
� �� �� �� �� �� ��
�
Walker equation
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Magnetostatic waves and oscillations 30
(iv) Magnetostatic waves in metallized plates
� Consider an unbounded magnetic plate metallized on
both sides
� We consider two cases:
• Normally magnetized plate
• Tangentially magnetized plate
0 0,M H 0 0,M H
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Magnetostatic waves and oscillations 31
(iv) Normally magnetized metallized plates
� 1st case: normally magnetized plate
� We had seen similar systems before (transversely
magnetized waveguides, microstrips on normally
magnetized magnetic substrates)
� But now we will not restrict ourselves to solutions
uniform along z-direction
xy
z
0 0,M H
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Magnetostatic waves and oscillations 32
(iv) Normally magnetized metallized plates
� It suffices to consider waves along one direction only
(because of the rotational symmetry of the system)
� Solution written as
xy
z
yk
( , ) ( ) exp( )yy z f z jk y� � �
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Magnetostatic waves and oscillations 33
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Magnetostatic waves and oscillations 34
(iv) Normally magnetized metallized plates
� Boundary conditions on perfect metallized surfaces
xy
z
yk
( 0) ( ) 0z z d� � � �z zb b
d
�� � � ��b h� � z
���
�zb
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Magnetostatic waves and oscillations 35
(iv) Normally magnetized metallized plates
� Solution:
( ) cosn z
f z Ad
�� �� � �� �
� Such solutions only exist when �< 0:
2 2
2 2H
� ��
� �� ���
yk
1n � 2n �H� � ��� �
ny
nk
d
��
� ��
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Magnetostatic waves and oscillations 36
(iv) Normally magnetized metallized plates
� Dispersion curves
� �� �
2 22 2
221 /
n My H H
ny
nk
d n k d
��� � �
� �
� �� �� � � � �� ��� �
� Numerical example:
-1 (mm )yk
2 GHz
6 GHz
0.1 mm
H
M
f
f
d
���
Hf
f�
(GHz)f
1n �
2n �
3n �
1(3.5 GHz) 0.157mm!n� � �
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Magnetostatic waves and oscillations 37
(iv) Normally magnetized metallized plates
� Overall solution:
� �( , ) cos exp nn y
n zy z A jk y
d
�� � �� �� �
� �
� �( , ) cos expn ny n y
n zy z jk A jk y
d
�� �� � �� �� �
yh
� �( , ) sin exp nn y
n n zy z A jk y
d d
� �� �� � �� �� �
zh
� These waves have sinusoidal behavior inside the
sample. They are called volume waves.
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Magnetostatic waves and oscillations 38
(iv) Normally magnetized metallized plates
� Remarks about volume waves:
• Dispersion curves are independent of the propagation direction
because of the rotational symmetry of the problem. The same
result is found for waves propagating in the x-direction or along
any other direction.
• All modes can propagate between �H and �� . There is no
size-dependent cutoff frequency as in a classical
electromagnetic waveguide.
• The mode n=0 is not a solution. It results in a zero magnetic
field (constant �).
• The waves have a group velocity parallel to the phase velocity:
they are forward waves.
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Magnetostatic waves and oscillations 39
(iv) Normally magnetized metallized plates
� The electric field does not enter the magnetostatic
equations in the first place, but can be perturbatively
found after the magnetic field h is solved
00
d
j�
��� �
�� � � � � � �e h e�
� For simplicity assume there are no charges
(conduction or external)
0 0j���� � � � � � �e h e�
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Magnetostatic waves and oscillations 40
(iv) Normally magnetized metallized plates
� We expect
0ny a
djk
dz�� �� � �y
z y
ee h
xy
z
yk
d
0
dj
dz�� �� �x
y
eh
0nyk
��� �x ze h
0ny
djk
dz� � �z
y
ee
� �( ) exp nyz jk y� ��e e
![Page 41: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/41.jpg)
Magnetostatic waves and oscillations 41
(iv) Normally magnetized metallized plates
� �( , ) cos expn ny n y
n zy z jk A jk y
d
�� �� � �� �� �
yh
� �( , ) sin exp nn y
n n zy z A jk y
d d
� �� �� � �� �� �
zh
� �0( , ) sin exp nn yn
y
n n zy z A jk y
k d d
� ��� � �� �� �
� �xe
� We shall not compute the other components here!
� Note: the electric field satisfies the boundary conditions.
![Page 42: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/42.jpg)
Magnetostatic waves and oscillations 42
(v) Tangentially magnetized metallized plates
� Let us now turn our attention to a tangentially
magnetized, metallized plate
� Consider waves propagating with a wave vector
z
x
y0 0,M H
� � � �0, , 0, sin , cosy z k kk k k � �� �k
k�k
![Page 43: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/43.jpg)
Magnetostatic waves and oscillations 43
(v) Tangentially magnetized metallized plates
� We look for wave solutions of the type:
0 0,M H
k�k
� �( , , ) ( ) exp y zx y z f x jk y jk z� � � �
� Walker’s equation leads to
� �2
2 22
( )( ) 0y z
d f xk k f x
dx� �� � �
z
x
y
![Page 44: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/44.jpg)
Magnetostatic waves and oscillations 44
(v) Tangentially magnetized metallized plates
� Boundary conditions lead to:
0 0,M H
k�k
0 0 and aj x x dx y
� �� �� �
� � � �� �
z
x
y
d
0 0 and a y
dfk f x x d
dx� �� � � �
![Page 45: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/45.jpg)
Magnetostatic waves and oscillations 45
(v) Tangentially magnetized metallized plates
� General solution:
� � � �( ) sin cosx xf x A k x B k x� �
22 z
x y
kk k
�� � �
� Boundary conditions �
0x y ak A k B� �� �
� � � �sin 0y a x xk A k B k d� �� � �
![Page 46: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/46.jpg)
Magnetostatic waves and oscillations 46
(v) Tangentially magnetized metallized plates
� Setting determinant to zero
� 1st solution:
� � � �2 2 2 2 sin 0x y a xk k k d� �� �
� � 2 2 2 2sin 0 or 0x x y ak d k k� �� � �
222 z
x y
kn nk k
d d
� ��
� �� � � � � � �� �
![Page 47: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/47.jpg)
Magnetostatic waves and oscillations 47
(v) Tangentially magnetized metallized plates
222 2 cos
sin kk
nk
d
� ��
�� � � �� � �� � � �
� �� �
� Result:
� Regardless of the value of n, in order to have
propagation: 2
2 cossin 0k
k
��
�� �
� We should have µ < 0 like in the previous case;
otherwise this condition cannot be satisfied
![Page 48: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/48.jpg)
Magnetostatic waves and oscillations 48
(v) Tangentially magnetized metallized plates
� Even then, propagation only occurs for certain angles
2 1tan k� �
� �
Propagation region
� This limitation does not depend on thickness
![Page 49: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/49.jpg)
Magnetostatic waves and oscillations 49
(v) Tangentially magnetized metallized plates
� The width of the propagation cone at each frequency:
2 22
2 2
1tan H
k
� ��
� � ��
�� � �
�
H� �� � ���
� Propagation cone becomes very narrow near �H , but
covers the whole plane near ��
H� � ��� �
![Page 50: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/50.jpg)
Magnetostatic waves and oscillations 50
(v) Tangentially magnetized metallized plates
� Within the propagation cone the dispersion relation is:
222 2 cos
sin kn k
nk
d
� ��
�� � � �� � �� � � �
� �� �
� �� �
222
2
sin /
1 /k n
H H M
n
n k d
n k d
� �� � � �
�
� ��� �� �
�� �� �
� The dispersion curve depends on the propagation angle
![Page 51: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/51.jpg)
Magnetostatic waves and oscillations 51
(v) Tangentially magnetized metallized plates
� For every angle, the limit k � 0 corresponds to the same
frequency ��
-1 (mm )k
f�
f
� But, with increasing k,
frequency decreases
finally reaching
� �2sin
k
H H M k
�
� � � �
�� �
�
1n �2n �
3n �
� The higher n, the later this limit is reached
![Page 52: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/52.jpg)
Magnetostatic waves and oscillations 52
(v) Tangentially magnetized metallized plates
� Propagation along static magnetization (z)
-1 (mm )k
f�
1n �2n �
3n �
� �0k� �
Hf
0 0,M Hk
d
![Page 53: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/53.jpg)
Magnetostatic waves and oscillations 53
(v) Tangentially magnetized metallized plates
-1 (mm )k
f�
Hf
� Propagation nearly perpendicular to z � �/ 2k� ��
0 0,M Hk
d
� The dispersion curves become nearly flat
![Page 54: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/54.jpg)
Magnetostatic waves and oscillations 54
(v) Tangentially magnetized metallized plates
� Magnetic potential:
( ) cos sin
cos sin sin
y an
n ak
k dn x n xf x B
d n d
k dn x n xB
d n d
�� �� �
�� ��
� �
� �� � � �� �� � � �� �� � � �� �
� �� � � �� �� � � �� �� � � �� �
� �( , , ) ( ) expn y zx y z f x jk y jk z� � � �
( , , )x y z�� �h
![Page 55: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/55.jpg)
Magnetostatic waves and oscillations 55
(v) Tangentially magnetized metallized plates
� Note that for all angles the dispersion curves have a
negative slope: the group velocity is opposite to phase
velocity
� These waves are called backward volume waves
� The are called volume waves since fields have a
sinusoidal behavior inside the magnetic material
� How does the wave front of these waves look like if we
have an isotropic source?
![Page 56: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/56.jpg)
Magnetostatic waves and oscillations 56
(v) Tangentially magnetized metallized plates
� But, there also exist a second class of solutions:
2 2 2 2 2 20x y a z yk k k k� � ��� � � � �
� ky and kz are real numbers (related to propagation).
Therefore this solution exists only if
� �2 2
2 20 0M H
M H
� � �� � � � �
� �� ��
� �� � � � � � �
�
� Propagation
angle: � �
2 22
2 2
1tan k
M H
� ��
� � � ��
�
�� � �
� �
![Page 57: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/57.jpg)
Magnetostatic waves and oscillations 57
(v) Tangentially magnetized metallized plates
� Propagation angle is a function of frequency:
k
f� H Mf f�
f
k�
2
�
k�
![Page 58: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/58.jpg)
Magnetostatic waves and oscillations 58
(v) Tangentially magnetized metallized plates
� Note that this is a single mode (there is no mode
number n like in the previous case)
� Besides, apart from the propagation angle, nothing can
be said about the relation between the frequency and
the magnitude of the wave number k.
� This means the dispersion relation is flat: at any allowed
frequency, every value of k is allowed� flat dispersion
� Of course, this is an artifact of the magnetostatic
approximation. But it can be said that in the true case
the dispersion is “almost” flat.
![Page 59: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/59.jpg)
Magnetostatic waves and oscillations 59
(v) Tangentially magnetized metallized plates
� Also, note that2
2 2 2 2 2 20 ax y a x yk k k k
�� �
�� �
� � � � � � �� �
� Since ky is real, kx should be imaginary
� The corresponding magnetic potential and field should
have an exponential behavior inside the magnetic
material
![Page 60: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/60.jpg)
Magnetostatic waves and oscillations 60
(v) Tangentially magnetized metallized plates
� � � �( ) cos siny ax x
x
kf x B k x k x
k
��
� �� �� �
� �
a yx x x
kk jq q
��
� � � �( ) exp xf x B q x� �
� This type of solution is called a magnetostatic surface
wave because the field drops exponentially inside the
magnetic material
� It is concentrated near the top or bottom surface
depending on the sign of qx
![Page 61: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/61.jpg)
Magnetostatic waves and oscillations 61
(v) Tangentially magnetized metallized plates
� Summarizing the results:
• Between �H and �� we have volume waves. There are different
modes. At each frequency, the modes can propagate within a
certain range of angles with respect to the static magnetization.
All modes represent backward waves: their group velocity is
opposite to their phase velocity.
• Between �� and �H + �M we have a surface wave. To each
frequency their corresponds a specific propagation angle. But
the dispersion curve is flat. The field is concentrated at the top or
bottom interface (with the metal) depending on the propagation
direction.
![Page 62: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/62.jpg)
Magnetostatic waves and oscillations 62
(v) Tangentially magnetized metallized plates
� Remember, for normally
magnetized metallized
plates:
� Forward volume waves
between �H and ��
� Independent of the
propagation angle
0 0,M H
-1 (mm )yk
Hf
f�
(GHz)f
1n �
2n �
3n �
![Page 63: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/63.jpg)
Magnetostatic waves and oscillations 63
(v) Tangentially magnetized metallized plates
� Tangentially magnetized
metallized plates:0 0,M H
-1 (mm )k
f�
1n �2n �
3n �
Hf
H Mf f�� Angle-dependent, back-
ward volume waves
between �H and �� .
� Flat-dispersion surface
wave between �� and �H
+ �M . Frequency
depends on angle.
![Page 64: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/64.jpg)
Magnetostatic waves and oscillations 64
(vi) Tangentially magnetized free plate
� We now turn to the case of a tangentially magnetized
free plate (no metallization)
� This case is more difficult to solve, but it is important
due to its rich physics, and application in MSW devices
0 0,M Hk�
kz
x
y
![Page 65: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/65.jpg)
Magnetostatic waves and oscillations 65
(vi) Tangentially magnetized free plate
� Walker’s equation
0 0,M H
k�k
z
x
y
2 2 2
2 2 20
x y z
� � ��� �� � �
� � �� �� � �� �
� �( , , ) ( ) exp y zx y z f x jk y jk z� � � �
![Page 66: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/66.jpg)
Magnetostatic waves and oscillations 66
(vi) Tangentially magnetized free plate
� Inside the magnetic material (0<x<d)
d
x
y
� �2
2 22
( )( ) 0y z
d f xk k f x
dx� �� � �
� Outside the magnetic material (x<0, x>d) we have �=1:
� �2
2 22
( )( ) 0y z
d f xk k f x
dx� � �
![Page 67: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/67.jpg)
Magnetostatic waves and oscillations 67
(vi) Tangentially magnetized free plate
� Inside the magnetic material (0<x<d)
d
x
y
� � � �( ) sin cosx xf x A k x B k x� �
22 z
x y
kk k
�� � �
![Page 68: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/68.jpg)
Magnetostatic waves and oscillations 68
(vi) Tangentially magnetized free plate
� Outside the magnetic material (x<0, x>d):
d
x
y
� �exp ( ) ( )
exp( ) 0
C k x d x df x
D kx x
� � � ��� ����
� �2
2 22
( )( ) 0y z
d f xk k f x
dx� � �
2 2y zk k k� �
![Page 69: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/69.jpg)
Magnetostatic waves and oscillations 69
(vi) Tangentially magnetized free plate
� Boundary conditions at magnetic material/air interface:
: continuousy z xh ,h ,b
0 0,M H
k�k
z
x
y
: continuousy
z
jk
jk
��
�
�� � � �� � � �� �� � � ��� � � �
y
z
h
h
![Page 70: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/70.jpg)
Magnetostatic waves and oscillations 70
(vi) Tangentially magnetized free plate
0 ajx y
� �� � �
� �� �� �� �� �� �
xb
� Inside magnetic plate:
� Outside magnetic plate:
0 x
��
��
�xb
![Page 71: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/71.jpg)
Magnetostatic waves and oscillations 71
(vi) Tangentially magnetized free plate
� System
to solve:
0 x
��
��
�xb
x
y
� � � �sin cos exp( )x x y zA k x B k x jk y jk z� � � � �� �� �
0 ajx y
� �� � �
� �� �� �� �� �� �
xb
� �exp ( ) exp( )y zC k x d jk y jk z� � � � � �
0 x
��
��
�xb
� �exp exp( )y zD kx jk y jk z� � � �
![Page 72: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/72.jpg)
Magnetostatic waves and oscillations 72
(vi) Tangentially magnetized free plate
� Matching:
D B�
x y akD k A k B� �� �
� � � �sin cosx xC A k d B k d� �
� � � �� � � �
cos
sin
x y a x
x y a x
kC k A k B k d
k B k A k d
� �
� �
� � �
� �
![Page 73: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/73.jpg)
Magnetostatic waves and oscillations 73
(vi) Tangentially magnetized free plate
� Dispersion equation:
� �2 2 2 2 2
cot2
x y ax x
k k kk k d
k
� �
�
� ��
� � 2 21 1cot
2x x z yk k d k kk
���
� �� � � �� �
� �
k�k
z
x
y
![Page 74: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/74.jpg)
Magnetostatic waves and oscillations 74
(vi) Tangentially magnetized free plate
� Note that: 2
2 cossin k
x kk k�
��
� � �
� Hence, we distinguish two situations:
Volume waves: 2
: real
10, tan
x
k
k
� ��
� � �
Surface waves: 2
: imaginary
1tan
x
k
k
��
� �
22 cos
sin 0kk
��
�� � �
22 cos
sin 0kk
��
�� � �
![Page 75: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/75.jpg)
Magnetostatic waves and oscillations 75
(vi) Tangentially magnetized free plate
� Again, volume waves are
allowed within certain angles
1tan k� �
��
� � � �2 21 1
cot cos sin2k k k
k
kdgg
� � � �� ��
� �� � � �� � � �� �
� �
� For any angle, the dispersion equation of volume waves is
� � 2 2sin cos /k k kg � � � �� � �
![Page 76: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/76.jpg)
Magnetostatic waves and oscillations 76
(vi) Tangentially magnetized free plate
� Example: volume waves for
0k� �k
-1 (mm )k
f�
Hf
1n �
2n �
3n �� Again different modes
appear which are all
backward waves
0k� �
1 1cot
2
kd�
� �
� � � �� � � �� � � �� �� �� �� � � �
![Page 77: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/77.jpg)
Magnetostatic waves and oscillations 77
(vi) Tangentially magnetized free plate
� For surface waves we have
� �2
2 cossin 0k
k kg�
� ��
� � �
� � � �2 21 1
coth cos sin2k k k
k
kdgg
� � � �� ��
� �� � � �� � � �� �
� �
� Left hand side positive� there are solutions only if
2 21 1cos sin 0
2 k k� � ���
� �� � � �� �
� �
We can also take the minus sign, it does not matter
![Page 78: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/78.jpg)
Magnetostatic waves and oscillations 78
(vi) Tangentially magnetized free plate
� We should have2 2 2
22 2
cossin 0k
k
� ��
� � � �
� �� � �
�
2 22 2
2 2
1 1cos sin 0
2 k k
�� � �
� � ���
� � � �� � � � �� � �� �
� �� �2 2 21 1sin
2 2H M H M k H� � � � � �� � � � �
� �2 2sinH H M k� � � �� � �
![Page 79: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/79.jpg)
Magnetostatic waves and oscillations 79
(vi) Tangentially magnetized free plate
� If � � ��
� � � �� �2 21 1sin
2 2H H M H M H M k H� � � � � � � � �� � � � �
�� � � � Impossible (Why?)
� Only possibility: � � �� � �� � � �
� But then: �� � �
2sin Hk
H M
��
� ��
�Necessary condition
![Page 80: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/80.jpg)
Magnetostatic waves and oscillations 80
(vi) Tangentially magnetized free plate
� But these conditions are not enough, we should also have
� �2 21 1cos sin
2 k k kg� � � ���
� �� � � �� �
� �2 2 2 2
2 2 2 2
� �� � � �� �
� � ���
� �
� � � �� �22 2 2 2 2 2� � � ��� � � �� �
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Magnetostatic waves and oscillations 81
(vi) Tangentially magnetized free plate
� These relations lead to an upper frequency limit dependent
of propagation angle
� �� �2 2sin
2 sinH M H M k H
M k
� � � � � ��
� �
� � ��
-1 (mm )k
f�
1
2H Mf f�
2k
�� �
3k
�� �
4k
�� �
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Magnetostatic waves and oscillations 82
(vi) Tangentially magnetized free plate
� The opposite is also true: like volume waves, surface
waves on a free plate are only allowed within a certain
range of angles
Propagation region
2 2
sin kM H
� � ��
� ��� �
��
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Magnetostatic waves and oscillations 83
(vi) Tangentially magnetized free plate
� Summarizing:
• Free magnetized plate allows
backward volume modes in
the frequency range between
�H and �� . Between �� and
�H + �M/2 a surface waves
mode is allowed.
• But the actual dispersion
relation and frequency range
of propagation depends on the
propagation angle in both
cases. -1 (mm )k
f�
Hf
1n �2n �
3n �
1
2H Mf f�
Volume modes
Surface mode
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Magnetostatic waves and oscillations 84
(vii) Surface waves
� A particularly important case is that of surface waves
propagating perpendicular to the direction of static
magnetization
� We analyze this case in more detail
0 0,M H
2k
�� �
k
z
x
y
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Magnetostatic waves and oscillations 85
(vii) Surface waves
� The dispersion equation becomes
� � 1 1coth
2kd �
��
� �� � �� �
� �
2y yk k k� �
� �� �
22
22
12 1exp(2 )
2 1 1a
a
kd� �� � �
� � � � ��
�
� �� �� � �
� � � �
� �� �
22
22
11ln
2 1a
a
kd
� �
� �
� �� �� � �
� �� �� �
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Magnetostatic waves and oscillations 86
(vii) Surface waves
� These waves propagate in the range
k
x
y
2M
H
�� � �� � � �
� Magnetic potential inside the magnetic layer:
� � � �sin cos exp( )x x yA k x B k x jk y� � � �� �� �
2x y yk k jk� � �
2y yk k k� �
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Magnetostatic waves and oscillations 87
(vii) Surface waves
� From the matching equation it follows that
y y ay a a
x y
k kk k sAj
B k jk
�� �� � �
�� �� � � �
1 0
1 0
yy
y y
kks
k k
���� � �� ���
� � � �sinh cosh exp( )ay y y
sB k x k x jk y
��
�� ��
� � �� �� �
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Magnetostatic waves and oscillations 88
(vii) Surface waves
� We rewrite this as
� � � � � �1 sinh cosh exp( )a y
Bs kx kx jk y� � �
�� � � �� �� �
� � � �exp( ) exp ( ) expyjk y U k x d U kx� � �� �� � � � �� �
� �21aU s� �� � � � � �2
1aU s� �� � � �
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Magnetostatic waves and oscillations 89
(vii) Surface waves
� Plotting the magnetic potential, it is found that waves
moving in the +y direction (s=1) concentrate near the
bottom surface of the magnetic plate
� Those moving in the –y direction (s=-1) concentrate
near the top surface of the plate
yk
0M
yk
S =1 S = -1
x
y
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Magnetostatic waves and oscillations 90
(vii) Surface waves
� What about a half infinite magnetic plate?
� What is its dispersion equation?
� What about the field profile?
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Magnetostatic waves and oscillations 91
(vii) Surface waves
� Free plate is not the only structure whose surface waves
have been studied (in view of possible application)
� Various structures have been considered containing
metallic ground planes
� Solid lines: waves
moving in –y direction
(near top surface)
� Dashed lines: waves
moving in +y direction
(near bottom surface)
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Magnetostatic waves and oscillations 92
(viii) Excitation of MSW’s in magnetic films
� Open or half-open (grounded) magnetic plates form the
basis of MSW devices. But how are these waves excited?
� By conventional current carrying lines (transducers) on the
surface or close to the surface of the magnetic film
Microstriptransducer
Meandertransducer
Lattice transducer
Magnetic film
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Magnetostatic waves and oscillations 93
(viii) Excitation of MSW’s in magnetic films
� Roughly speaking, the amplitude of the excited MSW with
a (tangential) wave vector k is proportional to Fourier
transform of current density at that wave vector:
Microstriptransducer
� �( ) expV
j dV�� J r k r
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Magnetostatic waves and oscillations 94
(ix) MSW devices
� MSW devices: based on
excitation and reception
of MSW’s in a finite or
infinite magnetic film
� Mostly based on
excitation of surface
waves, but volume
waves used as well
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Magnetostatic waves and oscillations 95
(ix) MSW devices
� These devices benefit from the following properties of the
MSW’s:
• Broad frequency range
( )H H H M� � � � � ��� � � � Volume waves
H M� � � �� � � � Surface waves (grounded plates)
By applying very high dc magnetic fields (up to 2 Tesla by using
permanent magnets) or using materials with a high saturation
magnetization the range of 1-50GHz may be covered
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Magnetostatic waves and oscillations 96
(ix) MSW devices
• Properties of MSW devices can be tuned by changing the applied
magnetic field
• Wavelength’s are short, for instance for surface MSW’s propagating
perpendicular to the magnetization in a tangentially magnetized
free plate wavelength is proportional to the film thickness. Using
thin films leads to very short wavelengths
� �� �
22
22
11ln
2 1a
a
kd
� �
� �
� �� �� � �
� �� �� �
� �� �
22
22
124 / ln
1a
a
dk
� ��� �
� �
� �� �� � � �
� �� �� �
![Page 97: Microwave Magnetics 10 - ee.sharif.edu](https://reader031.vdocument.in/reader031/viewer/2022020701/61f851a4af32263b6d0d3852/html5/thumbnails/97.jpg)
Magnetostatic waves and oscillations 97
(ix) MSW devices
• It is possible to change the dispersion properties of MSW’s by the
choice of the wave type and by changing the layer thickness,
adding ground planes, etc.
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Magnetostatic waves and oscillations 98
(ix) MSW devices
• The losses are comparatively law (if single-crystalline high quality
films are used)
• Transducers (for exciting MSW’s) are easy to design
� For these reasons MSW devices were investigated in the
1970’s and early 1980’s:
• Delay lines (phase shifters)
• Filters
• Resonators
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Magnetostatic waves and oscillations 99
(ix) MSW devices
� Delay lines: consist of a transmitting transducer and a
receiving transducer
� The resulting time delay (phase shift) can be large because
MSW’s are slow (propagation constants are large)
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Magnetostatic waves and oscillations 100
(ix) MSW devices
� These devices can be tuned by changing the dc magnetic
bias. They can also be reciprocal or non-reciprocal
� The dispersion characteristics can be engineered to
realize true wideband delay lines (small dispersion over a
wide frequency band) or to have other properties
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Magnetostatic waves and oscillations 101
(ix) MSW devices
� MSW filters:
• Wide band filters can be built by using the natural propagation
ranges of MSW’s between transducers
• Narrow-band filters built by engineering the transducers. For
instance note that the amplitude of the wave is proportional to
� �( ) expV
j dV�� J r k r
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Magnetostatic waves and oscillations 102
(ix) MSW devices
� For a lattice transducer with N elements each carrying a
current density
J
xs
01
( ) ( , , )N
nn
J r J x x y z�
� � ���
0J
� � � � � �1
00
( ) exp ( ) exp expN
xnV V
j dV j dV jk ns�
�
� �� � �� �
� ��� �J r k r J r k r
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Magnetostatic waves and oscillations 103
(ix) MSW devices
� � � �� �
� � � �� �
1
0
1 expexp
1 exp
sin / 2 exp 1 / 2
sin / 2
Nx
xn x
xx
x
jk sNjk ns
jk s
k sNjk s N
k s
�
�
�� �
�
�� �� �
�
� Therefore, one can select
just certain values of wave
number (thus certain
frequencies) for excitation
xk
� �� �
sin / 2
sin / 2x
x
k sN
k s
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Magnetostatic waves and oscillations 104
(ix) MSW devices
� MSW resonators: utilize the formation of standing
MSW’s in a ‘finite’ magnetic sample excited by a
transducer
� The standing wave is formed by the reflection of the
MSW off the edges of the finite film
� Since wavelength is short, resonators are small
Standing wave
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Magnetostatic waves and oscillations 105
(ix) MSW devices
� For a more detailed overview see
W.S. Ishak, “Magnetostatic wave technology: a review”, Proceedings of
the IEEE, Vol. 78, Issue 2, 1988.
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