Spin Wave Propagation in Non-Uniform
Magnetic FieldsFirst Semester Report
Fall Semester 2006
by
James Derek Tucker
Michael Kabatek
Prepared to partially fulfill the requirements for EE401
Department of Electrical and Computer Engineering
Colorado State University
Fort Collins, CO 80523
Report Approved:
Project Advisor
Senior Design Coordinator
1
Abstract
There has been extensive work on spin wave dynamics in ferrite film structures for both
fundamental understanding and device applications. However, all previous work has mainly
involved spin wave propagation in spatially uniform magnetic fields. This work demon-
strates, for the first time, high resolution time and space resolved inductive probe imaging
of spin wave propagation in spatially non-uniform magnetic field configurations. Spin wave
propagation in spatially non-uniform fields allows for wave packet manipulations that involve
changes in both the carrier wave number and the wave packet group velocity. These spin
wave properties can be used to extend basic understanding of linear and nonlinear spin wave
dynamics, and to develop a new class of wavelength and velocity selective microwave devices.
This report describes the study of spin wave propagation in spatially non-uniform mag-
netic fields, as well as their possible applications for practical microwave devices. High
resolution time- and space-resolved imagining of spin wave propagation in magnetic thin
films under spatially non-uniform magnetic field configurations have been preformed. The
experiment was carried out with a yttrium iron garnet (YIG) film strip magnetized with a
spatially non-uniform magnetic field parallel to the length of the film strip in the magne-
tostatic backward volume wave (MSBVW) spin wave propagation configuration. Spin wave
pulses were excited with a microstrip transducer at one end of the strip. The propagation of
spin wave pulses was mapped with a movable, time- and space-resolved inductive magnetody-
namic probe. The wave number of the carrier for the spin wave pulses were found to increase
in a spatially increasing magnetic field and decrease in a spatially decreasing magnetic field.
The wave number change for a general spatially varying static field is reversible.
This report also demonstrates the use of spatially non-uniform magnetic fields in a mi-
crowave delay line structure to introduce a variable pulse propagation delay time. The
structure is used with a YIG film strip magnetized with a spatially non-uniform magnetic
field perpendicular to the length of the film strip in the magnetostatic surface wave (MSSW)
propagation configuration. Using two small ferrite magnets one can closely control the spa-
i
tial distribution of the magnetic field over the YIG film, this results in a variable pulse group
velocity by changing the spin wave dispersion relationship. Using a high temporal resolution
broad band oscilloscope, the delay time of the pulse can be viewed and altered by changing
the separation between two small ferrite magnets. These field dependent wave number prop-
erties present potential microwave signal processing applications with wave number selective
communication, wave number modulation, and specific delay line applications in microwave
devices.
ii
Contents
Abstract ii
Table of Contents iv
List of Figures v
I Introduction 1
II Spin Wave Propagation in Non-Uniform Magnetic Fields 3
II.a Experiment Hardware Design . . . . . . . . . . . . . . . . . . . . . . . . . . 3
II.b Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
II.c Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
IIINon-Uniform Spin Wave Delay Line 15
III.a Experiment Hardware Design . . . . . . . . . . . . . . . . . . . . . . . . . . 16
III.b Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
III.c Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
IV Proposed Applications 19
V Conclusions and Future Work 21
Acknowledgments 21
iii
References 22
iv
List of Figures
1 Non Uniform Spin Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Spin Wave Delay Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Non-uniform structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Structure Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 Finished Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6 Leakage Field Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7 Field Distribution Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
8 MSBVW non-uniform experimental setup . . . . . . . . . . . . . . . . . . . 10
9 Spatially non-uniform magnetic field configurations . . . . . . . . . . . . . . 11
10 Spatial Evolution of Spin Wave Pulses . . . . . . . . . . . . . . . . . . . . . 13
11 Non Uniform wave number as function of position . . . . . . . . . . . . . . . 14
12 Delay Line Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 17
13 Delay Line Field Configurations . . . . . . . . . . . . . . . . . . . . . . . . . 18
14 Delay Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
v
I Introduction
At the most basic level, a spin wave (SW) excitation consists of the spatial propagation of
the phase lead or lag for low amplitude spin precessions in a saturated ferromagnetic system
[1], [2]. Previous research on spin wave excitations has generally focused on excitation and
propagation in uniform magnetic fields. Most of the microwave devices developed over the
last 50 years have also involved spin wave excitations in uniform magnetic fields. Examples
of these devices include delay lines, phase shifters, and frequency selective limiters. The
matter of spin wave propagation in non-uniform magnetic fields largely represents an open
question. The basic propagation properties of spin wave excitations in spatially non-uniform
magnetic fields are of fundamental and practical importance. A basic understanding of the
properties of spin wave pulses in non-uniform fields can open up new radar and microwave
signal processing device possibilities, among others.
This paper reports on recent time- and space-resolved measurements of spin wave pulse
propagation properties in a YIG film strip magnetized to saturation with non-uniform mag-
netic fields. The experiment was performed with a long and narrow magnetic yttrium iron
garnet film strip. The non-uniform field was applied parallel to the long direction of the film
strip, in the MSBVW configuration. In this configuration, the frequency versus wave number
dispersion relationship for low wave numbers has a negative slope and the group velocity
for positive wave numbers is negative. The entire dispersion curve shifts up and down with
the local magnetic field because of the Zeeman interaction depicted in Figure 1. It is found
that, in a non-uniform magnetic field, the wave number of the spin wave carrier changes
while the carrier frequency remains constant. Specifically, the wave number increases in a
spatially increasing field and decreases in a spatially decreasing field. Moreover, if the field
changes back to its initial amplitude, the wave number also returns to its initial value. The
shift in the local wave packet carrier wave number with field is a reversible process. In spite
of these field dependent changes in the wave number, the group velocity of the spin wave
pulses remains relatively constant for small changes in field.
1
Figure 1: Non Uniform Spin Wave Theory
This paper also reports on how one can use a spatially non-uniform magnetic field to
manipulate pulse group velocity in a microwave delay line structure. The non-uniform field
was applied perpendicular to the long direction of the film strip, in the MSSW configuration.
In this configuration, the frequency versus wave number dispersion relationship for low wave
numbers has a positive slope and the group velocity for increasing wave numbers in this
configuration is positive. Again the entire dispersion curve shifts up and down with the local
magnetic field because of the Zeeman interaction shown in Figure 2. In this configuration
we can use a non-uniform magnetic field to shift the dispersion resulting in a change in the
pulse group velocity. By varying the magnetic field in a region of space one can significantly
manipulate the pulse group velocity resulting in a variable delay in the microwave delay line.
Figure 2: Spin Wave Delay Theory
The organization of this report is as follows; Chapter II explores new work on spin wave
propagation in spatially non-uniform magnetic fields, Chapter III explores the application
2
of non-uniform spin wave propagation to a microwave delay line structure, and Chapter IV
explores the potential and possible applications of non-uniform spin wave propagation. Con-
cluding comments and suggestions for future progress will be provided in Chapter V.
II Spin Wave Propagation in Non-Uniform Magnetic
Fields
In this work we study the propagation characteristics of spin waves in spatially non-uniform
magnetic fields. The experiments utilized a long and narrow yttrium iron garnet film strip
magnetized with a spatially non-uniform magnetic field parallel to the strip axis. A microstrip
transducer was used to excite spin wave pulses while inductive probe imaging techniques [3]
were used to map the spatial evolution of spin waves during propagation. In a non-uniform
magnetic field, the wavelength of the spin wave carrier changes while the frequency remains
constant. Specifically, the wavelength increases in a spatially decreasing field and decreases
in a spatially increasing field, see Figure 1. This response is associated with the spatial
changes of the spin wave dispersion response which results in a wave number and velocity
change of the propagating pulses.
II.a Experiment Hardware Design
For this project many different types of work were required including mechanical design,
machine shop metal work, computer modeling, and programming. The mechanical design
and machine work was required in order to fabricate a structure which we could use to
study spin waves propagation in spatially non-uniform magnetic fields. The design work
was done in Turbo Cad. The design was made such that probing devices could be placed
near the sample surface in order to probe spin wave excitations, as well as measure magnetic
field distributions. The structure designed consists of a stage, and two arms which house
3
(a) Structure 3d view
(b) Structure Top View
Figure 3: 3D and top views of the structure in Turbocad.
4
(a) Arm dimensions
(b) Stage dimensions
Figure 4: Structure dimensions.
5
Figure 5: Finished Structure
two microstrip line transducers/pads which connect to two SMA ports for input/output of
microwaves. The SMA ports are used to couple microwave radiation to the transducers which
in turn will excite spin waves in our YIG sample. A schematic created in turbo cad is shown
in Figure 3. The dimensions of the structure are shown in Figure 4. The Stage and arms
of the structure is composed of scrap aluminum (no-cost) and obtained from the physics
machine shop. The transducer pads were obtained in the lab (no-cost), and the SMA ports
were purchased from Coaxicom for P/N: 3215-5CC-1 $9.99 ea. The assembly of the structure
was done via holes tapped for 2-56x3/16 with and non-magnetic socket head screws from
Fastener Express P/N ASHC0203. The structure was fabricated using non-magnetic parts.
A picture of the finished structure is shown in Figure 5. This structure allows for transducer
input and output, and enables one to position probing mechanisms near the sample surface.
The system used for in this project consists of a three axis positioning system (OWIS),
and several types of hardware including an oscilloscope (Agilent DSO 81204A -12GHz),
Microwave source (HP 83623B), Pulse Generator (SRS DG535), Spectrum Analyzer (Agilent
6
Figure 6: Leakage Field Scanner
E4440). For this project positioning and data acquisition software was needed in order to
position probing mechanisms near the sample surface and record data. The two probing
mechanisms include a custom inductive loop probe for leakage field detection near the YIG
sample surface, as well a standard Magnetic field Hall probe for measuring the working field
distribution. Also numerical computation software was required to decode the acquired data.
Positioning and data acquisition software were written in Labview. Two different programs
were required for the two types of data acquisition. One scanning program was written to
scan the leakage fields near the sample surface. A second program was written to measure the
field distribution near the sample with a Hall probe. In Both cases the probes are mounted
on the three axis positioning system. Scans involved both 2-D spatial scans, and 1-D spatial
scans. In all cases data acquisition was archived through the use of GPIB connections to
desired instruments. Field Distributions were measured using the Labview program and a
FW Bell 9500 Gauss meter.
All measurements for the first experiment were preformed with the inductive loop probe.
7
Figure 7: Field Distribution Scanner
8
Data computation software was written in MATLAB. Once data files were obtained
from the instrumentation MATLAB was used to extract important features from the data.
Figure 6 and Figure 7 are screen shots of the inductive probe scanner program for detecting
leakage fields, and field scanner programs form measuring the field distributions.
II.b Experiment
Figure 8 shows a schematic of the experimental setup. The magnetic medium was a yttrium
iron garnet (YIG) film strip cut from a larger single crystal YIG film grown on a gadolinium
gallium garnet substrate by standard liquid phase epitaxy technique. The YIG strip was
magnetized to saturation by a static magnetic field H(z) parallel to the length of the film
strip. This film/field configuration supports the propagation of backward volume spin waves
[1]. The magnetic field was provided by an electromagnet. Different non-uniform field
configurations were realized through the adjustment of the positions of magnet N-S pole
pieces relative to the film/transducer structure. A microstrip line transducer was positioned
over and at one end of the film strip to excite the spin wave pulses. The spin wave pulses
propagating along the film strip were detected with a time- and space-resolved inductive
magnetodynamic probe [3]. The detected signals were measured with a broadband real time
microwave oscilloscope. The spatial scanning of the probe and the sequential spatio-temporal
data acquisition were automated with the use of a computer.
For the data presented below, the YIG strip was 7.2 µm thick, 2 mm wide, and 30 mm
long. The film had unpinned surface spins and a narrow ferromagnetic resonance linewidth.
The input transducer was 50 µm wide and 2 mm long. For the spatial scans, the resolution
was about 100 µm. The spin wave pulses were excited with input microwave pulses with
a temporal width of 35 ns at a carrier frequency of 5.515 GHz. The nominal input power
applied to the transducer was about 32 mW .
The experiment was preformed using three specific non-uniform H(z) configurations.
These consisted of fields that (a) increase with z, (b) decrease with z, and (c) have a decrease
9
Figure 8: Schematic of experiment set-up. The yttrium iron garnet (YIG) film strip is
magnetized to saturation by a z - dependent static field H(z) . The microstrip transducer
is used for the excitation of spin wave pulses. The time- and space-resolved inductive probe
is used for the detection of spin wave signals.
10
Figure 9: Magnetic field H as a function of position z for three field configurations, as
indicated.
followed by an increase, or a sag. A standard Hall effect magnetic field probe mounted on
the scan stage in place of the pickup loop was used to map the H(z) profile over the YIG
strip. Figure 9 gives quantitative maps of the three H(z) profiles. The z = 0 reference point
corresponds to a position about 1 mm away from the input transducer. For the decreasing
and increasing configurations, the overall field change over the length of the strip was about
20 Oe. For the sagging configuration, the amount of the sag was about 5 Oe. While these
field changes are relatively small, the data below will show that they lead to significant
changes in the carrier wave numbers for the spin wave pulses.
11
II.c Results
Figure 10 shows the spatial evolution of spin wave pulses for the three different field con-
figurations and at two different times, as indicated. The plots show normalized spin wave
signals as a function of position. The waveforms in the left column were recorded at time
t = 18.7 ns, relative to launch, while those in the right column were taken at t = 125 ns.
The input microwave pulses for all the configurations had the same temporal width, peak
power, and carrier frequency as given above.
The data in graph (a) show that, after propagation over about 4 mm or 106 ns in an
increasing magnetic field environment, the pulse carrier wave number increases significantly.
An analysis of the wave forms shows an increase from 175 rad/cm at about 1 − 2 mm to
275 rad/cm at about 5 mm. In contrast, the data in graph (b) show a significant decrease
in the carrier wave number. Here, the change is from 175 rad/cm at about 1 − 2 mm to
50 rad/cm at about 5 − 6 mm in a spatially decreasing field. The two waveforms in graph
(c), however, show similar carrier wave numbers at both time points. Note that, for the data
in graph (c), one has H ≈ 1270 Oe at the 1 − 2 mm and 5 − 6 mm positions. The data in
Figure 10 also show that, in all three cases, the spatial width of the pulses remains more or
less constant in spite of the non-uniform H(z).
The analysis of spin wave pulse data similar to those in Figure 10 for a full range of
times yielded quantitative profiles of wave number k versus position z for the three configu-
rations. Figure 11 shows representative results on k(z) for the three field cases. The profiles
demonstrate the qualitative observations noted from Figure 10. In graph (a), one sees that
k is a clear increasing function of z , while graph (b) shows a decrease and graph (c) shows
a sag. These k(z) wave number versus position profiles demonstrate a nice match with the
H(z) profiles in Figure 9. In contrast with these self consistent k(z) and H(z) results, it is
found that the carrier frequency of the spin wave pulses are largely unaffected by the field
changes. This was ascertained from temporal wave packet signal measurements for a range
of fixed positions along the strip and a Fourier transform analysis of these data. This result
12
Figure 10: Spatial waveforms for the spin wave pulses for two different times, t = 18.7 ns
and t = 125 ns, relative to launch, and the three field configurations, as indicated. The input
microwave pulses had common durations and carrier frequencies of 35 ns and 5.515 GHz,
respectively.
13
0 2 4 6 8 110
120
130
140
c) Sagging field
Position (mm)
50
100
150
200
b) decreasing field
Wav
e nu
mbe
r (r
ad/c
m)
200
250
300
350
a) increasing field
Figure 11: Spin wave wave number k as a function of position z for the three field configu-
rations, as indicated.
is significant. It means that the spin wave pulses generally will maintain the same carrier
frequency, even as it freely propagates in a changing field environment.
It is important to note that these results are for H(z) profiles in which the overall change
in field was relatively small and the fields were always co-linear with the strip axis. As a
consequence, the group velocity of the spin wave pulse is relatively constant over the more-or-
less linear propagation path. It is also important to emphasize that these results are for the
specific case of backward volume spin waves. The response would be completely opposite in
the case of surface spin waves, for example, that applies when the static field is in-plane and
transverse to the direction of propagation, or for forward volume modes for which the static
14
field is perpendicular to the plane of the film. In these two cases, the spin wave frequency ωk
is an increasing function of k . It will be important to examine the spin wave propagation
dynamics for the gamut of possible vector field H(z) profiles and the wide range of ωk(k)
responses that can be achieved by these variations.
Large field changes and propagation geometry variations may be useful for wave number
control, group velocity control, and dispersion control, not to mention options for the tuning
of the nonlinear magnetodynamic response. These effects have important possibilities for
new classes of microwave devices for radar and signal processing applications. Examples
include wave compression or wave expansion for microwave and millimeter wave pulses,
pulse chirping, and other microwave pulse manipulations.
III Non-Uniform Spin Wave Delay Line
Using the experimental results from Chapter II we have proposed to use the properties of spin
wave pulse propagation in non-uniform magnetic fields to manipulate the pulse group velocity
in order to achieve a variable delay in a microwave delay line structure. In this work we use
the propagation characteristics of spin waves in spatially non-uniform magnetic fields in our
microwave delay line structure. The experiments utilized a long and narrow yttrium iron
garnet film strip magnetized with a spatially non-uniform magnetic field perpendicular to the
strip axis (MSSW). A microstrip transducer was used to excite and detect spin wave pulses.
In the MSSW configuration, the frequency versus wave number dispersion relationship for
low wave numbers has a positive slope and the group velocity for increasing wave numbers
in this configuration is positive. These dispersion properties allow one to control the group
velocity by varying the magnitude of the magnetic field.
15
III.a Experiment Hardware Design
For this work the same system was used, except here data acquisition was preformed via
transducer to transducer (not the inductive loop probe). The microwave delay line structure
was modified to include two small ferrite magnets which introduce field non-uniformity in the
MSSW configuration. The ferrite magnets were purchased at Force Field (in Fort Collins)
for $2.12, and cut on a diamond saw in the magnetics laboratory. Data acquisition was done
using the Oscilloscope, and Gauss meter, and Labview program previously described. The
modification to the delay line structure included two aluminum housings to hold the small
secondary ferrite magnets, and a brass rod for secondary magnet position adjustment.
III.b Experiment
Figure 12 shows a schematic of the experimental setup for the non-uniform microwave delay
line. The magnetic medium was a yttrium iron garnet (YIG) film strip cut from a larger
single crystal YIG film grown on a gadolinium gallium garnet substrate by standard liquid
phase epitaxy technique. The YIG strip was magnetized to saturation by a static mag-
netic field Hz(x) perpendicular to the length of the film strip. This film/field configuration
supports the propagation of surface spin waves [1]. Two magnetic field configurations were
used to demonstrate how one can use a spatially non-uniform magnetic field to introduce a
variable delay to the microwave structure. The primary magnetic field was provided with an
electromagnet and is spatially uniform. A small secondary magnet was used to introduce a
spatial non-uniformity to the field along the x direction. A microstrip line transducer was
positioned over and at one end of the film strip to excite the spin wave pulses. The spin wave
pulses propagating along the film strip were detected with another microstrip transducer at
the other end of the film. The detected signals were measured with a broadband real time
microwave oscilloscope.
For the data presented below, the YIG strip was 15 µm thick, 2 mm wide, and 30 mm
16
Figure 12: Schematic of experiment set-up. The yttrium iron garnet (YIG) film strip is
magnetized to saturation by a z - dependent static field Hz(x). The microstrip transducer
is used for the excitation and detection of spin wave pulses.
long. The film had unpinned surface spins and a narrow ferromagnetic resonance linewidth.
The input and output transducers were 50 µm wide and 2 mm long and separated by 4.5 mm.
The spin wave pulses were excited with input microwave pulses with a temporal width of
31 ns at a carrier frequency of 5.536 GHz.
The experiment was preformed using two non-uniform Hz(x) configurations. One of
the configurations is uniform while the other is non-uniform. Again a standard Hall effect
magnetic field probe was mounted on the scan stage and used to map the field profile over the
YIG strip. Figure 13 gives quantitative maps of the two profiles. The spin wave propagation
takes place between 10.5 mm and 15 mm. For the uniform field configuration the magnitude
of the magnetic field is constant at 1124 Oe, while in the non-uniform configuration the
magnitude of the field change is approximately 140 Oe. This 140 Oe change in the magnetic
field results in a significant change to the pulse group velocity. Altering the magnitude of
this non-uniform field change allows for manipulation of the delay time in our microwave
delay line.
17
Figure 13: Magnetic field Hz(x) as a function of position x for two field configurations, as
indicated.
18
III.c Results
Figure 14 shows the temporal evolution of the spin wave pulses for the two different field
configurations as shown in Figure 13. The plots show normalized spin wave signals as a
function of time with the input microwave pulses for both spin waves having the identical
temporal width, and carrier frequencies. Both Figure 14 and Figure 13 show temporal traces
for a propagation distance of 4.5 mm.
The data in Figure 14(a) shows the original pulse in the spatially uniform magnetic field.
The delay of the pulse from input to output in the uniform magnetic field is τ = 115 ns.
The data in Figure 14(b) shows the pulse which is delayed as a result of the non-uniform
magnetic field. The delay of the pulse from input to output in the non-uniform magnetic
field is τ = 335 ns. Thus we have shown that by using a spatially non-uniform magnetic
field it is possible to increase the delay by 300%.
Using a non-uniform magnetic field it is also possible to decrease the delay in our mi-
crowave delay line. This can be achieved by reversing the polarity on the secondary set of
magnets.
IV Proposed Applications
The physical principles outlined in Chapter II and delay line described in Chapter III lead to
possible new innovative microwave signal processing applications using magnetic microwave
systems. One possible application lies in fabricating an integrated (on-chip) version of the
microwave delay line described in Chapter III. Using micro-piezoelectric positioning and
existing chip fabrication techniques it is possible to realize an integrated, variable microwave
delay line, which is voltage controlled. The on-chip device fabrication is possible using exist-
ing semiconductors processing techniques and existing material deposition techniques. This
device has potential applications in the area of high speed on-chip mixed signal processing.
19
(a) Original Pulse
(b) Delayed Pulse
Figure 14: Temporal Evolution of Spin Wave Pulses in Delay Line Structure.
20
Other potential applications include wave number modulation, and other wave number
manipulations for use in a new class of wave number selective devices using non-uniform
magnetic fields.
V Conclusions and Future Work
The high resolution time- and space-resolved imagining of spin wave pulse propagation under
spatially non-uniform magnetic fields has been achieved. It is found that, in a non-uniform
magnetic field, the wave number of the spin wave carrier changes while the frequency of the
spin wave carrier remains constant. Specifically, the wave number increases in a spatially
increasing field, decreases in a spatially decreasing field, and is reversible for a more general
re-entrant field change. These field dependent wave number properties present potential
microwave signal processing applications.
Also we have demonstrated spin wave pulse velocity manipulations using spatially non-
uniform magnetic fields. We have shown that it is possible to use a non-uniform magnetic
field to control the delay time in a YIG microwave delay line. By varying the magnitude of
the non-uniform region of the magnetic field we can decrease the group velocity of a pulse
propagating along the delay line. This property of spin waves allows one to vary the delay
in a YIG microwave delay line without changing the transducer separation. These spin wave
properties could prove to be useful in integrate microwave delay line structures for on-chip
mixed signal processing.
Acknowledgments
The authors of this report would like to thank Dr. Mingzhong Wu and Dr. Carl Patton of
the Department of Physics at Colorado State University for their expertise, help, support,
and use of the laboratory. Also special thanks to Pavol Krivosik for the help with theory
21
and graphics and Kevin Smith for general help in the lab. This work was also supported
in part by the Army Research Office (W911NF-04-1-0247) and the Office of Naval Research
(N00014-06-1-0889).
References
[1] D. Stancil, Theory of magnetostatic waves, 1st ed. Springer-Verlag, 1993.
[2] P. Kabos and V. Stalmachov, Magnetostatic Waves and Their Applications, 1st ed.
Chapman & Hall, 1994.
[3] M. Wu, M. A. Kraemer, M. M. Scott, C. E. Patton, and B. A. Kalinikos, “Spatial
evolution of multipeaked microwave magnetic envelope solitons in yttrium iron garnet
thin films,” Phys. Rev. B, vol. 70, no. 054402, pp. 1–9, 2004.
22