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