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Imaging the spatial variation of dielectric constant in materials using microwave near field microscopy

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IMAGING THE SPATIAL VARIATION OF DIELECTRIC CONSTANT
IN MATERIALS USING MICROWAVE NEAR FIELD MICROSCOPY
by
Jennifer Lynn Schlegel
A dissertation submitted to The Johns Hopkins University
in conformity with the requirements for the
degree o f Doctor o f Philosophy
Baltimore, Maryland
2003
й Jennifer Lynn Schlegel 2003
All rights reserved.
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UMI Number: 3030759
Copyright 2003 by
Schlegel, Jennifer Lynn
All rights reserved.
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ABSTRACT
This work presents an investigation o f high spatial resolution subwavelength
imaging o f conductivity variations in materials using microwaves. The purpose o f this
investigation is to gain a better understanding o f the imaging capabilities and contrast
mechanisms that affect microwave near field microscope measurements. Previous work
has attempted to separate changes in reflected microwave power resulting from
topography versus material property changes. The efforts in this study investigate how
topography influences reflected microwave power measured in the near field, how
material property changes (specifically changes in a material's dielectric constant)
influence reflected microwave power in the near field, and to determine if these effects
can be separated for high resolution imaging. M axwell's equations were solved for
electromagnetic fields at an observation point generated from a dipole above a half space
without limiting the solution to the far field. By avoiding the restrictions on the
observation distance, electromagnetic field solutions were derived for the near field. The
magnitude o f the total electric field was calculated at an observation point for dipoles
above a half space with varying conductivities to study how material property changes
affect the magnitude o f the total electric field and the contributing fields. The magnitude
o f the electric field was calculated for varying dipole distances above a metal half space
and a semiconduting half space to study how topography affects the magnitude o f the
electric field. A microwave near field microscope was constructed to measure approach
curves and create images o f various samples for study. It was determined that all the
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contributing fields, direct, image, and surface wave terms, are affected by topography
changes but only the surface wave terms are affected by changes in the material property
o f the half space.
Advisor: Dr. James B. Spicer
Reader: Dr. Robert E. Green, Jr.
iii
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ACKNOWLEDGEMENTS
I owe many thanks to many people for continually encouraging me to complete
this dissertation. First I?d like to thank my advisor. Dr. James Spicer, for his patience,
encouragement, and willingness to believe that I would indeed finish. I'd also like to
thank Dr. Green for his help with reading this work and initially supporting me in the
beginning o f my graduate studies. I owe a great deal o f gratitude to Marge Weaver and
Linda Eckhardt for helping me navigate through the sometimes treacherous
administrative processes.
Many thanks are also due to several colleagues from the Materials Science and
Engineering department. I'd like to thank Dr. Grover Whetsel for sharing his enthusiasm
about the world o f near field imaging. It made working in the lab fun. I'd like to thank
several people who helped me throughout my studies whether h was sharing equipment
or sharing knowledge. Thanks go to Doug Oursler, Michael Erhlich, Kevin Baldwin,
John Champion and Chris Richardson. I am also grateful for the machining expertise o f
Mike Franckowiak and Walt Krug, they could always make anything fit into your
experimental set up.
I would also like to thank my friends at AT&T for continually reminding me to
complete this work.
And last but not least, I want to thank my family for their support in every way
possible through it all.
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TABLE OF CONTENTS
ABSTRACT............................................................................................................................ii
ACKNOWLEDGEMENTS
.......................................................................................iv
TABLE OF CONTENTS..................................................................................................... v
LIST OF FIGURES............................................................................................................. vi
CHAPTER 1.............................
1
I n t r o d u c t i o n ...................................................................................................................................................... 1
L i t e r a t u r e R e v i e w ......................................................................................................................................... 4
O v e r v i e w ............................................................................................................................................................ 21
CHAPTER 2 .........................................................................................................................23
B a c k g r o u n d a n d T h e o r y ....................................................................................................................... 23
V e r t i c a l e l e c t r i c d ip o l e a b o v e a h a l f s p a c e ........................................................................... 2 6
H o r i z o n t a l e l e c t r ic d ip o l e a b o v e a h a l f s p a c e .....................................................................5 9
CHAPTER 3 .........................................................................................................................77
E x p e r im e n t a l M e t h o d .............................................................................................................................. 7 7
S a m p l e s ............................................................................................................................................................... 8 0
A p p a r a t u s ..........................................................................................................................................................8 0
CHAPTER 4 ........................................................................................................................ 84
R e s u l t s a n d D i s c u s s i o n ........................................................................................................................... 8 4
V e r t i c a l e l e c t r i c d ip o l e c o n d u c t iv it y c h a n g e s ................................................................... 8 6
C a l c u l a t e d a p p r o a c h c u r v e s f o r v e r t ic a l e l e c t r i c d i p o l e ......................................... 8 9
H o r i z o n t a l e l e c t r i c d ip o l e c o n d u c t iv it y c h a n g e s ..............................................................9 6
C a l c u l a t e d a p p r o a c h c u r v e s f o r h o r iz o n t a l e l e c t r ic d i p o l e ................................... 9 9
M e a s u r e d r e s u l t s ......................................................................................................................................105
CHAPTERS___________________________________________________________ 118
C o n c l u s i o n s ................................................................................................................................................... 118
F u t u r e d i r e c t i o n s ......................................................................................................................................120
REFERENCES_______________________________________
121
VITA___________
125
v
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LIST OF FIGURES
Figure 1-1
Measured and fitted resonant frequency as a function or tip-sample
distance for MgO single crystal as reported b C. Gao et al in Review o f
Scientific Instruments November 1998 issue. Approach curve from
Lawrence Berkeley image charge model........................................................14
Figure 1-2
Measured approach curves from M. Golosovsky et al
in Ultramicroscopy............................................................................................ 19
Figure 2-1
Field lines for TE mode waveguide reproduced from Miner......................24
Figure 2-2
Field lines for TM mode waveguide reproduced from Miner.....................25
Figure 2-3
Problem setup for the vertical electric dipole above half space..................26
Figure 2-4
Problem setup for positive image dipole for vertical electric dipole above a
half space........................................................................................................... 48
Figure 2-5
Agreement for the surface wave term, T, for all thetas calculated with both
the large argument approximation and the numerically calculated Fresnel
number................................................................................................................ 53
Figure 2-6
Magnitude o f the surface wave term calculated using both the large
argument approximation and the numerically calculated Fresnel number as
a function o f observation distance...................................................................55
Figure 2-7
Calculated directivity plots showing the agreement for a vertical electric
dipole operating at 10 MHz on the boundary between air and lake water.
The observation distance is 500 km ............................................................... 57
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Figure 2-8
Directivity plots as presented in King p. 101 and p. 103.............................58
Figure 2-9
Problem setup for horizontal electric dipole.................................................. 60
Figure 2-10
Problem setup for positive image dipole for horizontal electric dipole
above a half space.............................................................................................. 73
Figure 3-1
Microwave near field microscope configuration with probe acting as both
the transmitter and receiver.............................................................................. 77
Figure 3-2
Photograph o f microwave near field microscope with separate transmitter
and receiver locations........................................................................................ 78
Figure 3-3
Microwave diode detector voltage versus microwave power......................79
Figure 4-1
Magnitude o f the electric field for a vertical electric dipole on the
boundary o f a half space with various conductivities. The dipole is
operating at 10 GHz. The observation point is 5 cm with an observation
angle, theta, o f 85 degrees................................................................................ 87
Figure 4-2
Magnitude o f the electric field for a vertical electric dipole on the
boundary o f a half space with conductivities in the range for metals. The
dipole is operating at 10 GHz. The observation point is 5 cm with an
observation angle, theta, o f 85 degrees........................................................... 88
Figure 4-3
Calculated approach curve for vertical electric dipole above a perfectly
conducting half space with conductivity o f 1x 109S/m and a relative
dielectric constant o f 1. The dipole is operating at 10 GHz. The
observation angle, theta, is 85 degrees............................................................91
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Figure 4-4
Calculated approach curve for vertical electric dipole above a perfectly
conducting half space with conductivity o f 1x lO4S/m and a relative
dielectric constant o f 1. The dipole is operating at 10 GHz. The
observation angle, theta, is 85 degrees. These curves are expanded to
show dipole distances from 10 microns to below a nanometer...................92
Figure 4-5
Calculated approach curve for vertical electric dipole above a
semiconducting half space with conductivity o f 9 x 10?"*S/m and a relative
dielectric constant o f 11.9. The vertical electric dipole is operating at 10
GHz. The observation distance is 5 cm. The observation angle, theta, is
85 degrees........................................................................................................... 94
Figure 4-6
Calculated approach curve for vertical electric dipole above a
semiconducting half space with conductivity o f 9x lO '1S/m and a relative
dielectric constant o f 11.9. The dipole is operating at 10 GHz. The
observation distance is 5 cm. The observation angle, theta. is 85 degrees.
These curves are expanded to show the dipole distances from 10 microns
to below a nanometer........................................................................................ 95
Figure 4-7
Magnitude o f the electric field for a horizontal electric dipole on the
boundary o f a half space with various conductivities. The dipole is
operating at 10 GHz. The observation point is 5 cm with an observation
angle, theta. o f 85 degrees and observation angeL, phi, o f 0 degrees
Figure 4-8
97
Magnitude o f the electric field for a horizontal electric dipole on the
boundary o f a half space with conductivities in the range for metals. The
dipole is operating at 10 GHz. The observation point is 5 cm with an
observation angle, theta, o f 85 degrees and observation angel, phi, o f 0
degrees.................................................................................................................98
vui
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Figure 4-9
Calculated approach curve for horizontal electric dipole above a perfectly
conducting half space with conductivity o f 1x 109S/m and a relative
dielectric constant o f 1. The dipole is operating at 10 GHz. The
observation distance is 5 cm. The observation angle, theta. is 85 degrees
and the observation angle, phi, is 0 degrees.................................................100
Figure 4-10
Calculated approach curve for horizontal electric dipole above a perfectly
conducting half space with conductivity o f 1x 109S/m and a relative
dielectric constant o f 1. The dipole is operating at 10 GHz. The
observation distance is 5 cm. The observation angle, theta, is 85 degrees
and the observation angle, phi. is 0 degrees. These curves are expanded to
show dipole distances from 10 microns to below a nanometer................. 101
Figure 4-11
Calculated approach curve for horizontal electric dipole above a
semiconducting half space with conductivity o f 9x \Q~* S/m and a relative
dielectric constant o f 11.9. The dipole is operating at 10 GHz. The
observation distance is 5 cm. The observation angle, theta. is 85 degrees
and the observation angle, phi, is 0 degrees............................................... 103
Figure 4-12
Calculated approach curve for horizontal electric dipole above a
semiconducting half space with conductivity o f 9 x 10"4 S/m and a relative
dielectric constant o f 11.9. The dipole is operating at 10 GHz. The
observation distance is 5 cm. The observation angle, theta. is 85 degrees
and the observation angle, phi. is 0 degrees. These curves are expanded to
show dipole distances fromlO microns to below a nanometer.................. 104
Figure 4-13
Measured approach curve for copper sample withthree different probe tip
radii....................................................................................................................106
Figure 4-14
Measure approach curve for brass sample.................................................... 108
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Figure 4-15
Measured line scan for aluminum brass interface.......................................110
Figure 4-16
Measured line scan o f brass solder interface...............................................112
Figure 4-17
Contour plot o f percent reflected microwave power for brass solder
interface with the area believed to be the solder surrounded by a dashed
box................................................................................................................... 113
Figure 4-18
Contour plot o f the diode voltage for brass solder interface with the area
believed to be the solder surrounded by a dashed box..............................114
Figure 5-1
Dipole model extended to represent the effects o f probe tip radius
x
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120
Chapter I
Introduction
Rapid developments in the electronics industry regarding thin films has furthered
interests in determining material property variations with high spatial resolution.
Although much work has been done with optical techniques to image materials on the
nanometer scale, there is still a need to characterize materials at the micron and
submicron scales. Applying similar near field imaging techniques and principles with
microwaves provides a method to characterize materials on these length scales.
An image is constructed by replicating the distribution o f electromagnetic (EM)
energy in one plane to another. Optical imaging systems replicate the distribution of
light. This "replication" process is never perfect. Imperfections in the image are created
by the diffraction o f electromagnetic waves and the components used to transmit and
receive the electromagnetic energy. Diffraction o f electromagnetic waves occurs when
propagating waves encounter an obstacle. This phenomenon is demonstrated by Young's
double slit experiment. In this experiment, two slits or apertures are illuminated by plane
EM waves. The ability to image two distinct slits is dependent on the location o f the
maxima and minima o f the diffraction pattern. The distance between the slits is limited
by the location o f the maxima and minima o f the diffraction pattern. If the maxima o f the
diffraction pattern from one o f the slits occur at the minima o f the diffraction pattern for
the other slit then the two slits can be clearly resolved. If the distance between the slits
decreases and the maxima o f the diffraction patterns overlap than the two slits cannot be
clearly resolved. This dependence is known as the Rayliegh criterion for image
resolution. The maximum resolution for a diffraction limited system in air approaches
I
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A/2 . This is also known as the Abbe diffraction limit. Diffraction will govern the size o f
the features that can be imaged and is dependent on the wavelength.
When electromagnetic radiation encounters an aperture the radiation pattern far
away from the aperture plane is diverging and the spatial distribution is determined by the
Fourier transform o f the aperture. But radiation near the aperture has the spatial
dimensions o f the aperture itself. Unlike conventional (far field) imaging systems,
wavelength no longer governs the imaging resolution. Researchers have developed
methods to beat this diffraction limit to achieve better imaging resolution.
In the 1930?s Synge proposed ideas for exceeding the diffraction limits o f imaging
systems to achieve high resolution. One method he proposed was to use an aperture
smaller than the radiating energy. This sub-wavelength aperture would be scanned close
to the surface. In 1972, Ash and Nicholls demonstrated this concept using microwaves.
Using 10 GHz microwaves with a wavelength o f 3 cm radiating from a 1.5 mm diameter
aperture placed near an object Ash and Nicholls imaged gratings with line widths o f 1
mm. 0.75 mm, and 0.5 mm achieving an imaging resolution o f A /60. These concepts
have led to a large field o f study in optical near field microscopes.
To illustrate the concepts o f far and near fields, consider an aperture being
illuminated by plane waves. Placing a screen very close to the aperture shows an image
which is recognizable as the aperture. If the screen is moved further away from the
aperture the image becomes more structured with fringes. If the screen is moved an even
greater distance from the aperture the imaged pattern o f fringes creates an image unlike
the actual aperture. Moving the screen further away only changes the size o f the pattern
and not the shape. This example illustrates the difference between the near and far field
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diffraction patterns created by an aperture. The region where the shape o f the pattern no
longer changes is considered the far field. The region close to the aperture with no
visible fringes is considered the near field.
This work describes an investigation o f near field imaging o f conductivity
variations in materials with high spatial resolution using microwaves. Microwave near
field microscopy to image spatial variations in electrical properties has been reported by
researchers from the Center for Superconductivity Research at the University o f
Maryland, Lawrence Berkeley National Laboratory, the Racah Institute o f Physics at The
Hebrew University o f Jerusalem, and the Max Planck Institute for Biochemistry in
Germany. These researchers have attempted to separate changes in reflected microwave
power owing to topography versus material property changes. The efforts in the present
study are to investigate how topography influences reflected microwave power in the
near field, how material property changes, specifically changes in a material's dielectric
constant, influence reflected microwave power in the near field, and to determine if these
effects can be separated for high resolution imaging o f variations in dielectric constant
for materials.
For this investigation a microwave near field microscope was constructed to
investigate contrast mechanisms for microwave near field imaging, spatial resolution, and
the effects o f topography. A model was developed for our microscope measurements by
solving Maxwell's equations for an electric dipole above a half space. This solution was
tailored so that it would be valid in the near field as well as the far field by employing
numerical solutions rather than imposing assumptions on the variables that would limit
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the solution to the far field. Field equations were then used to model both conductivity
changes in a material as well as the change in probe to sample spacing.
Literature Review
Microwave near field microscopy literature has been generated by primarily four
groups o f researchers. In this review, we will discuss their microscope configurations,
contrast mechanisms used for imaging, samples selected for study, and their findings.
University o f Maryland Effort
In Applied Physics Letters published November 18, 1996, University o f Maryland
researchers reported a near field scanning microwave microscope with 100 micron
resolution operating in the X band (7.5 to 12.4 GHz). This microscope was constructed
from coaxial waveguides. A probe delivered the microwaves to a sample and detected
the reflected microwaves. The probe was constructed from a SMA (small miniature
adapter) connector and 10 mm o f 50 ohm coaxial cabling. The inner conductor o f the
coaxial cabling varied in diameter. The probe was silver plated copper wire surrounded
by a Teflon? dielectric and an outer covering o f stainless steel essentially a coaxial
cable. The reflected microwaves were measured using a directional coupler and a diode
detector that provided voltage measurements proportional to the reflected power. In their
configuration the sample was mounted on motion control devices and moved underneath
the probe. Using the reflected microwave power to create an image, the researchers
scanned an optical comparator with lines ranging in width from 20 to 160 microns.
From these images they concluded that the spatial resolution was determined by the
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diameter o f the inner conductor. In continuing efforts, they mixed a local oscillator
frequency and used lock-in detection techniques that measured the phase o f the reflected
microwave power.
The authors modeled their probe as a piece o f transmission line. They swept the
frequencies generated by the microwave source to obtain a standing wave pattern. This
standing wave pattern was then used to determine the resonant frequency o f the short
section o f coaxial cable. They would then tune their microwave source to supply
microwaves at the probe's resonant frequency. They also found that placing a sample
under the probe operating at resonant frequency caused the resonant frequency to shift
since the probe and sample were now acting as the transmission line rather than just the
probe.
The authors cite three contrast mechanisms in an IEEE Transactions On Applied
Superconductivity June 1997 that could be used for imaging with their microwave near
field microscope
?
resonant frequency o f the standing wave pattern created from the probe and sample
?
quality factor o f the resonant probe
?
amplitude o f the voltage from the diode detector indicating changes in reflected
power.
In the September 22, 1997 issue o f Applied Physics Letters the authors decided to use
the resonant frequency shift o f the probe and sample as a contrast mechanism to image
surface resisitivity changes. The researchers modulated the probe's resonant frequency
and used lock-in detection techniques to measure voltage changes which corresponded to
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shifts in the probe's resonant frequency owing to the presence o f the sample. To image
the shifts in resonant frequency caused by changes in a sample's surface resistivity, the
authors used a variable thickness thin film oxidized aluminum sample and a NIST 3.2
lines per mm resolution imaging target o f patterned chromium lines on a glass substrate.
In the February 16. 1998 issue o f Applied Physics Letters, the University o f Maryland
researchers chose the quality factor for quantitative measurements o f sheet resistivity.
Here they introduced a method to measure the quality factor, Q, o f the resonant system
created by the probe and sample. Once again they employed lock-in techniques, this time
at twice the modulation frequency o f the source. Voltage changes at twice the
modulation frequency provided an output that is related to the curvature o f the reflected
power versus frequency curve at resonance which is proportional to the quality factor.
The unloaded quality factor o f the probe was experimentally measured before the sample
was introduced. This required calibration o f the probe to determine the functional
relationship between the quality factor and the voltage changes. This also allowed the
researchers to simultaneously measure the shifts in resonant frequency and shifts in the
quality factor. The authors imaged a YBCO ( YBa^Cu^O^g) thin film deposited on a 5
cm diameter 300 micron thick sapphire disk. The thickness o f the thin film varied from
100 nm at the edge o f the disk to 200 ran in the center o f the disk. The authors also noted
that the sample's substrate was warped and attributed a variation o f a few microns in the
probe to sample separation during the scans. Based on measurements o f this sample, the
researchers concluded that the quality factor data was primarily sensitive to changes in
sheet resistance while changes in the resonant frequency were primarily sensitive to
changes in the probe to sample separation.
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The University o f Maryland researchers based their system models on transmission
line lumped circuit element theory to characterize the behavior o f their microscope. The
assumptions o f transmission line theory do not fully account for all the material
interactions for thin films radiated at these frequencies nor do they account for the near
field interactions occurring between the probe and the sample. The researchers base their
conclusions on experimental observations but they failed to control all the variables in
their experiments; for example, when measuring material property variation such as the
resistance they failed to control the probe to sample spacing. Their model is based on the
sample extending the length o f the probe as a transmission line segment but the distance
between probe and sample extends the transmission line as well. So their transmission
line models do not adequately predict how topography changes would cause the system
to behave versus how a material property change would cause the system to behave.
Max Planck Efforts
Researchers at Max Planck Institute began their work in microwave near field
imaging by measuring changes in the microwave transmittance rather than reflectivity
(Optics Communications, August 1, 1996). In their configuration a sharpened coaxial tip
was used to concentrate the electric field lines while a blunt coaxial tip was used to detect
microwaves transmitted through a sample.
In this case the imaging resolution was
determined by focusing the electric field lines with the sharpened coaxial tip. The sample
then perturbed the concentrated quasi static field distribution between the generating and
detecting conductors. The authors demonstrated this phenomenon by imaging the
microwave transmittance o f a commercially available polarizer foil that is 40 microns
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thick o f Mylar with less than one micron gold stripes. A 6 micron sheet o f Mylar was
placed on top o f the polarizer foil and slightly compressed between the probes.
Microwaves at 740 MHz clearly showed decreased microwave transmittance when
scanned over the metal wires in the foil. The contrast mechanism used to create their
images was perturbation o f the quasi static field distribution between the coaxial tips
which was measured by the amount o f microwaves transmitted through the sample.
In later work the researchers attempted to increase their imaging resolution by
sharpening the tips used for their measurements and they integrated their microwave
measurements with a scanning tunneling microscope. They changed their configuration
slightly so that microwaves were delivered to the back side o f a sample through the blunt
tip and the STM tip was used to detect the microwave transmittance through the sample
and the tunneling current. The STM tip was coupled to a waveguide system configured
much like a heterodyne interferometer to measure the microwave transmission amplitude
and phase with lock-in amplifiers. With this configuration (December 1996, Micron)
researchers simultaneously measured the topography with STM and imaged the
microwave transmittance. The sample used for this study was a 5 nanometer thick
platinum thin film deposited on a silicon substrate created with a mask that generated 2
nanometer deep depressions o f various diameters from 10 nanometers to one micron.
They found the contrast in the microwave images to be very weak. They also found the
measurements were extremely sensitive to any modulation or fluctuation such as
movement within the laboratory where the measurements were made. To improve the
measurement the researchers chose to use tip dithering to modulate the tip to sample
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distance thereby modulating the microwave transmission. They indicated this provided
images with better contrast.
The researchers described their results as follows "The experiment confirms
initial expectations that miniaturization o f the coaxial tip to a sub-micron dimension still
permits the taking o f meaningful microwave transmission data. This result is due. first of
all to the field concentration near the sharpened tip (lightning rod effect) and secondly
because the dither modulation o f the tip position was utilized. The achievable spatial
resolution, however, is not yet determined. Indeed (the researchers earlier work) using a
flat sample had demonstrated that a 10 micron wide tip yielded about a 10 micron
resolution. The present experiment with a tip more than one order o f magnitude smaller
did; however, use a sample with 2 nm high surface structures, the effect o f which on the
contrast formation needs theoretical modeling. Specifically it is not clear to which degree
the observed contrast giving an edge width o f 15 nm may be artificial, i.e. induced by the
topography."
In an Applied Physics Letters published in the May 19, 1997 issue, the researchers
reviewed the above experiment again and determined that the "correspondence to
topography is so good that we suspect that the observed electromagnetic contrast is
artificially induced by the fact that the tip follows the topography. This is more certain
after constant height scans, which we perform by interrupting the STM feedback in single
scan traces and lifting the tip up to a known height above the sample surface."
"The resolution o f the inherent contrast images in our experiments seems to be solely
defined by the width o f the probing tip." The basic concept o f a quasi-static field
distribution with field lines mainly longitudinal is generally applicable in the
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subwavelength domain. Therefore, it should account for contrast with microwaves as
well as infrared and visible. They state that their measurements demonstrate the very
high subwavelength resolution o f t-SNOMs and that the inherent contrast mechanism rest
on the longitudinal electric field component.
Lawrence Berkeley Effort
In the Applied Physics Letters published June 10, 1996, Lawrence Berkeley
researchers described their scanning tip microwave near field microscope. In their
configuration they used a sharpened metal tip (an STM tip) coupled inside a quarter
wavelength coaxial resonator. "As the tip radius decreases, the spatial resolution
increases due to localization o f the interaction between the tip and sample." In this
structure the field distribution is concentrated around the center conductor and the electric
field at the tip is at a maximum value. The researchers noted that due to the favorable
field distribution o f their probe, a small change in field distribution near the tip induced a
large change in the resonant frequency. These researchers also used the perturbation o f
the electric field distribution as the contrast mechanism. They measured the
perturbations o f the electric field distribution by measuring changes in the resonant
frequency o f a quarter wavelength coaxial resonator rather than the microwave
transmittance that the Max Planck researchers used. They estimated their probe tip radius
to be on the order o f 10 microns. "As expected, a direct correlation between tip spatial
resolution o f the scanned images and the radius o f the tip was observed." With this
microscope configuration the authors claimed a 5 micron spatial resolution as limited by
their motion control devices.
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In a June 27. 1997 issue o f Science, these researchers used this microscope to
nondestructive ly image the surface dielectric constant. Images o f the surface dielectric
constant allowed researches to see the periodic ferroelectric domain structure
(superlattice) in a yttrium doped LiNbCb single crystal. The surface o f the crystal was
examined with a profilometer to confirm optical quality smoothness. The probe tip was
an electrochemically etched tungsten tip with a probe tip radius o f approximately 0.1
micron. The probe tip contacted the surface using a soft spring. The images o f the
resonant frequency reflected variations in the dielectric constant associated with changes
in dopant levels, whereas the image o f Q (quality factor) corresponded to losses in
microwave energy, which are large at the ferroelectric domain boundaries (primarily as
the result o f movement o f the domain walls under the influence o f the microwave field).
The total loss tangent variation in the image o f the ferroelectric domain boundaries was
measured to be 1x 10 : . The loss tangent is proportional to changes in conductivity
(Hayt, p350). They also imaged an edge dislocation in this material. The changes in
local compressive and tensile strain around the edge dislocation caused changes in
dielectric constant which were detected with this microscope. These types o f lattice
distortions are not observable with optical microscopy using polarized light because o f
the lithium niobate's large birefringence.
In a later Applied Physics Letters published in September 29, 1997. the
researchers modified their microscope so that a soft tungsten tape cantilever was used to
hold the sample to provide soft contact between the sample and the tip. The estimated
force on the tip for typical operating condition was less than 20 microNewtons. The
authors claimed that such a soft contact force did not significantly deform the tip even if
ll
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the tip radius was less than one micron. Since the spring maintained an almost constant
soft contact while the tip scanned the sample surface, the microscope was not sensitive to
topographic features. The authors modeled shifts in the resonant frequency based on
perturbation theory. The theory predicted that the intrinsic spatial resolution o f the
microscope was proportional to the tip radius. The theory involved a constant that
depended on the geometry o f the cavity and tip assembly which was calibrated with
materials o f known dielectric constant.
With this tip cavity assembly and the soft
contact force, the researchers demonstrated a spatial resolution o f 100 nanometers (on a
dielectric material) with tip radius less than one micron. The authors dubbed this system
as a scanning evanescent microwave microscope (SEMM).
In an article in Review o f Scientific Instruments (November 1998), the researchers
reported a theoretical model to quantify the measurements o f dielectric properties with
their scanning evanescent microwave microscope. Their configuration which was
described as a tip structured probe used a high quality resonator and an efficient shielding
structure in the probe to shield off the far field components. The researchers derived an
analytic expression for the field distribution around the tip enabling quantitative
microscopy o f complex electrical impedance. The resonant system was analyzed using
an equivalent lumped series resonant circuit with effective capacitance, inductance, and
ds
resistance to establish a sensitivity to dielectric constant changes, ? . Using the circuit
e
theory, the authors developed an expression for the energy stored in the cavity. They
assumed the microwave diode was operating as a square law detector. They calculated
an output power caused by a change in frequency. They estimated the Johnson noise
(thermal noise) in the detector as the factor that limits the sensitivity. From their
12
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expressions they concluded that the sensitivity increased linearly with tip radius. "As the
sample is placed in the near field range o f the tip, and the tip radius and the effective field
distribution range are much smaller than the wavelength, the electromagnetic wave can
be treated as quasi-static, i.e. the wave nature can be ignored." This is the authors' basic
assumption for employing circuit theory and charge distributions to model their probe
and sample interaction. These researchers only evaluated homogeneous, isotropic,
dielectric materials. The sample was a dielectric material with a thickness much larger
than the tip radius. They represented the tip as a charged conducting sphere. They
described the tip and sample interaction as a redistribution o f the charged particles in the
probe tip caused by the sample. The dielectric material underneath the tip would become
polarized by the probe?s electric field and this polarization would then cause a
redistribution o f the charged particles in the conducting sphere to maintain the equal
potential surface o f the conducting sphere. They used image charges and boundary
conditions to satisfy Coulomb's law. In contrast to the dipole model o f tapered
waveguide probes for NSOM, where electric dipoles lie above and parallel to the sample
surface, this is a monopole model. The electric field here is concentrated in a very small
volume underneath the tip and almost perpendicular to the sample surface. The authors
then used perturbation theory to calculate the frequency and quality factor shifts caused
by the probe tip and dielectric sample interaction since the majority o f the energy was
concentrated in the cavity and not disturbed significantly by the sample interaction. The
authors measured an approach curve for a MgO single crystal and then used their image
charge model to fit the data. Their results are shown in Figure 1-1. This curve showed
the measured resonant frequency as a function o f the gap width between the sample and
13
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i
z
ft*
1*
ft*
?ft*
Figure 1-1 Measured and fitted resonant frequency as a function of tip-sample
distance for a MgO single crystal as reported by C. Gao et al. in
Review o f Scientific Instruments November 1998 issue.
the tip. The best curve was fitted using a parameter o f 1.71 x 10'3which was determined
by the geometry o f the probe-tip resonator assembly, a relative dielectric constant o f 9.5,
and a probe tip radius o f 12.7 microns. In their article they reported the relative dielectric
constant for MgO from a Journal o f Superconductivity to be 9.8. They also stated that
the agreement between measurement and theory indicated that the quasi-static and sphere
tip approximations were accurate enough for this particular application. The authors
acknowledged that the image charge approach is not valid for thin film samples due to
the divergence o f the image charges. The effective probing charge on the tip was
attracted downwards to the sample by the polarized dielectric sample. The higher the
dielectric constants, the shorter the effective charge-sample distance. As a result the field
distribution inside the sample was concentrated in a very small region just below the tip
apex with the polarization perpendicular to the sample surface. This is analogous to the
longitudinal electric field. Therefore, the resonant frequency shifts and the quality factor
shifts are dominated by the contribution from this small region. These experiments have
14
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shown 100 nm resolution can be achieved with a tip radius o f several microns but only on
dielectric materials with moderate dielectric constants.
In a May 3. 1999 issue o f Applied Physics Letters the authors developed a tip
sample distance feedback control for their SEMM. This SEMM was a scanned probe
microscope consisting o f a quarter wavelength coaxial resonator operating at a resonant
frequency o f 1 GHz with a typical quality factor o f a few thousand. The probe assembly
was coupled to a sharp tip that protruded from a small hole in a thin metallic shielding
layer coated on a sapphire plate. When the tip was brought near a sample, the resonant
frequency and quality factor shifted. By monitoring shifts in the resonant frequency and
the quality factor, the authors measured the electrical properties o f the sample. Since the
extent o f the field distribution was limited by the tip radius, this microscope was capable
o f sub-micron resolution. For a dielectric sample, the interaction was dependent on the
dielectric constant and tangent loss o f the nearby sample. For metallic samples, the
interaction was dependent on the surface resistance o f the sample. Previously to
minimize the effects o f topographic variations the authors maintained a constant tip to
sample distance by keeping the tip in soft contact with the sample. This soft contact did
introduce tip distortion which decreased spatial resolution and could damage the sample
and/or tip. For conductive samples, the shift in resonant frequency diverged as the tip to
sample separation decreased to zero. The authors enabled noncontact imaging by
developing a tip to sample distance control by regulating the resonant frequency o f the
cavity to maintain a constant separation.
15
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Hebrew University Efforts
In a March 11, 1996 issue o f Applied Physics Letters researchers at the Hebrew
University o f Jerusalem in Israel reported on their configuration o f a millimeter wave
near field resistivity microscope. They based their contrast mechanism on the fact that
the reflection o f electromagnetic waves from conducting surfaces is determined by the
sample?s resistivity. Thereby, scans o f the surface with a microwave antenna and
measurements o f the reflected power yields a resisitvity map. The authors stated that
near field microscopy employs a probe o f subwavelength size and an object that is
mounted in the near field o f the probe so that spatial resolution is determined by the
probe size rather than by the wavelength. The authors proposed using a narrow resonant
slit as an aperture rather than a probe tip. They claimed this type o f aperture did not
suffer from the cut off effects like a circular aperture. Their sensing probe was a narrow
slit cut in the end plate o f a TE waveguide. The length o f the slit was approximately
equal to half the wavelength in free space while the width was exceedingly smaller. The
authors cited the main advantage o f long and narrow resonant slit over the small circular
aperture was a high transmission coefficient in certain frequency ranges so that coupling
to the object was more effective. The resolution o f a resonant slit probe in the direction
perpendicular to the slit was determined by the width o f the slit and by the skin depth on
the material that the slit was fabricated. The author's configuration used a microwave
bridge with the probe assembly attached. The bridge was tuned so that it was balanced
without a sample present. A sample was then placed on an xy stage at a constant
separation o f 50 to 100 microns below the probe. The authors demonstrated the
operation o f their microscope on a NIST test target which was constructed from
16
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chromium lines deposited on a glass substrate. With a slit width o f 100 microns the
author's scans resolved five line pairs per millimeter and eight line pairs per millimeter.
Therefore, they claimed the spatial resolution o f the microscope was determined by the
slit width and not the wavelength.
The authors published the same microscope configuration in a July 1996 IEEE
Transaction o f Microwave Theory and Techniques. The probe was reconstructed so that
a dielectric insert (TeflonrM or high density polyethylene) with a convex or wedge like
shape tip protruded from the waveguide. The insert was then covered with a two micron
thick vacuum deposited coating o f silver. A slit was cut perpendicular to the curvature
axis o f the insert by using a 20 micron diameter gold wire as a mask. The width o f the
slit was approximately 50 microns and produced through lithography techniques. The
authors cited the advantage o f this slit aperture over coaxial resonators in that pulsed
microwaves could be used and claimed an ease o f fabrication over miniature coaxial
probe tips.
In a March 1998 article in Ultramicroscopy, the same researchers presented work
on the design o f their probe assembly and control o f the probe to sample separation.
They presented findings on the dependence o f the millimeter wave reflectivity o f
different materials on the probe to sample distance. The purpose o f their efforts was to
develop a microwave imaging system for high spatial resolution resistivity mapping. In
comparing their probe assembly to a coaxial probe they referenced the work o f the Max
Planck researchers stating that the coaxial probe was more or less a capacitive probe that
produced mostly longitudinal electric field and it probed the charge distribution rather
than the current distribution. They stated that the Max Planck probe distinguished well
17
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between metal and insulator but that it was not well suited for mapping o f resistivity in
conductors because it did not probe changes in electrical current distribution. The
University o f Hebrew's probe was based on an asymmetrical aperture rather than a tip.
The asymmetry allowed them to circumvent the transmission problems encountered by
the cut off frequency o f symmetrical apertures. The asymmetry permitted radiation to
propagate beyond the slit. They constructed their probe with an asymmetrical slit using
lithography techniques. Their current technique constructed a slit with a width o f about
20 microns in a silver coated PMMA extension o f a TE waveguide. The authors claimed
that the spatial resolution was determined by the width o f such a probe. The authors
constructed their microscope in the W band (75 to 110 GHz). They set up a microwave
bridge and balanced the bridge and then measured any reflected. The authors reported
that since the spatial resolution o f their techniques was several microns that surface
roughness or topography changes o f most samples were essentially flat on that scale.
They stated the main purpose o f their distance control was to avoid mechanical contact
between the probe and sample. Since the probe to sample spacing in their measurements
was several microns, the authors proposed two methods to measure the gap between the
probe and sample: an optical technique and a capacitive technique. The optical technique
could be used on all samples. The capacitance technique could only be used with
conducting samples. The authors then reported that changes in microwave reflectivity
were affected by changes in probe height. They took known material samples o f copper,
chromium, silicon, and glass and measured microwave reflectivity as a function o f probe
height creating approach curves. The researchers noted (see Figure 1-2) that at distances
above 50 microns the dependence o f the reflectivity on resistivity became complicated
18
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lu
X 3 8 mm
Cu
3
A
u
4*
Cr
*
z
10'
V v
o
4
2
6
8
Z - d u ta n c e (m m )
Cm
Cr
>
lr
c
ir
1<T
1?'
10*
Z*4tstancc (\im)
10*
10*
Figure 1-2 Measured approach curves from M. Golosovsky et aL U ltram icroscopy.
19
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and at 100 microns the near field reflectivities for such different materials as chromium,
silicon, and glass, almost coincided. At distance below 50 microns the authors claimed
reflectivity varied monotonically with resistivity. The authors reported that by measuring
the reflectivity in this extreme near field region, the contrast comes from variations o f the
resistivity o f the upper surface or from topography.
In continuing work reported in an Applied Physics Letters issue November 9,
1998 the University o f Jerusalem researchers showed measurements o f silver (Ag) films
o f varying thickness from 920 Angstroms to 78 Angstroms. They employed their
capacitive distance control mechanism which measured the change in resonance
frequency between the probe and the sample at 5 MHz. In this work, they demonstrated
changes in the magnitude o f microwave reflectivity as a function o f sheet resistivity for
silver films with varying thickness but there was little to no change in the phase o f the
reflected microwaves. The measured phase o f the reflected microwaves in the near field
only showed a linear dependence for the thinnest film when the probe to sample distance
was decreased. The authors reported that a considerable part o f the incident radiation
passed through the film and was reflected from the reverse side o f the substrate. These
measurements were carried out at 82 GHz. For the thicker films, the phase variation
with distance in the near field was very weak. The authors stated that this was an
extremely important feature tliat had not been appreciated in the context o f the near field
microwave imaging and that this feature originated from the fact that the Poynting vector
in the near field is imaginary, thus the electromagnetic field barely propagates. The
authors then go on to use a transmission line model to estimate reflectivity based on the
effective impedance o f the probe and the effective impedance o f the sample. They
20
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estimated the effective impedance o f the sample from the plane wave solution which used
the surface impedance o f an infinitely thick sample, the skin depth, and effective complex
impedance o f the substrate. They estimated the effective impedance o f the sample for
three regions, the thick film limit where the thickness was much larger than the skin
depth, the extreme thin film limit where the thickness was much smaller than the skin
depth, and the intermediate thin film region. In the thick film limit, the effective
impedance o f the sample was simply the surface impedance which was directly
proportional to the square root o f the surface resistivity. In the extreme thin film region,
both the magnitude and phase o f the reflectivity linearly depended upon the thickness. In
the intermediate region, the magnitude o f reflectivity was reciprocally related to the
thickness. But the phase o f the reflectivity had two terms where one increased with
thickness while the other term decreased with thickness. The authors then claimed that in
this region the phase hardly depends on thickness but that there could be two different
mechanisms at work. The plane wave solution used in transmission line models to
determine the reflection from multilayers may not be a suitable model/theory describing
this interaction.
Overview
To better understand the various observations regarding the imaging capabilities
and contrast mechanisms o f microwave near field microscopes modeling and experiments
have been carried out. In the next chapter the background and theory is presented for
solving Maxwell's equations for the electric field distribution in the near field for a
vertical and horizontal electric dipole above a half space to gain insight to the
21
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electromagnetic interaction between the microwave near field probe and a sample. The
observation distance is varied for each case to determine how the electric fields would
vary for horizontal and vertical electric dipoles. This is to help understand how
topography changes affect the measurements. The material properties o f the half space
are also varied to determine how the material property changes would affect the electric
field distribution. We then describe our microwave near field microscope configuration
and the samples used for our study. The experimental results obtained with our
microscope are presented. The last chapter presents a discussion on our experimental
results and observations from our models with a comparison to the work o f the other four
research groups.
22
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Chapter 2
Background and Theory
In this chapter we will provide background about microwave waveguides and
electromagnetic fields. We will describe a model based on this background for our
experimental setup. We chose to model our microwave near field microscope as an
electric dipole above a half space. The fields radiating from an electric dipole closely
resemble microwaves emanating from our sharpened probe tips. We calculated the
electric field distribution at an observation point away from the dipole. To understand
how changes in probe to sample spacing affected the electric field at the observation
distance, we varied the dipole distance above the half space. To investigate how changes
in the dielectric constant o f the half space affected the electric held at the observation
distance, we varied the conductivity o f the half space.
Microwaves emanating (however small o f a distance) from a near field probe tip
can be treated as radiation from a tiny antenna which can be modeled as a dipole.
Electric fields from a unit electric dipole radiating in free space are well known. Since
we are interested in the interaction between the probe and the sample material, we
evaluated a dipole above a half space. A dipole can be configured in a variety o f ways there are electric or magnetic dipoles which can be oriented horizontal or vertical to a
boundary. To determine which configuration best suits our model we looked at the field
distribution in the waveguides used for our study.
In general there are two types o f waveguides - one conductor and two conductor
systems. A pipe with a rectangular cross section that guides electromagnetic waves in
considered a one conductor system. Solutions to Maxwell?s equations require boundary
23
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conditions that need scalar potential on two distinct independent surfaces. Therefore, one
conductor systems can only support transverse electric (TE) or transverse magnetic (TM)
waves. In the interior o f a waveguide Maxwell?s equations can be divided into two basic
sets o f solutions or modes. For one mode, an axial component o f the electric field exists
but no longitudinal or axial magnetic field components exist. This mode is referred to as
the electric type, E mode, or transverse magnetic (TM). The other set o f solutions has an
axial magnetic field but no axial component for the electric field. This mode is referred
to as the magnetic type, H mode, or transverse electric (TE). These two modes are
illustrated in figures below.
1-7,
IA ? r -
l a -
m
-
?
?
T
-I
?r
?*
x.v . v -! ^ ;
...
------
i a
?
------
1 4
m
?
* 0*
*
I
(
r-t
W
/
W
<1
2
?
░i?r ----------- 1? i?i------------- 1?i
c :c :::
3 ?
II
I I I
II
0r ~ r ............ m - t
? ---i-i
t
4
---------- -i--i *
Figure 2-1 Field lines for TE mode waveguide.
24
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r.,
J>
?i i i ii
y
h
л? ii ?i ?╗ *i ii
I 'O '?
-v
1v s -
?t
ii
ii
ft i t r
I
4.
0
TM,,
Figure 2-1 Electric fields lines in TM waveguide.
It is important to note the field distribution within a waveguide to properly design a feed
system to establish propagating waves. It is also important to note the field distribution
to illustrate the environment the probe will experience. By examining the field
distribution in the waveguide we can examine the fields exciting the probe.
Dipole Type and Waveguide Field Distribution
To investigate how a dipole interacts with a material in the near field. Maxwell's
equations were solved for the field distribution o f an electric dipole above a half space.
A vertical electric dipole was used to model a TM waveguide. A horizontal electric
dipole was used to model a TE waveguide. In both cases, the electric dipole was placed
above a half space. Investigations included changes in field distribution caused by
material property changes and also the height above the half space. This was done to help
understand how topography versus material property changes affect the electromagnetic
fields. The electromagnetic field distribution for electric dipoles above a conducting half
25
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space is described in King's Lateral Electromagnetic Waves Theory and Applications to
Communications. Geophysical Exploration, and Remote Sensing. In his solution,
conditions are used to solve for the field distribution in the far field. In our solution, the
problem is reworked without the restricting assumption that limits the solution for
observation distances in the far field.
Vertical Electric Dipole
The following problem development is for a vertical electric dipole above a half
space. Electromagnetic fields observed at a distance from the dipole source are
calculated using Maxwell's equations. The total field at an observation distance , rn ,
consists o f three contributing terms: the direct field, the image field, and a field from the
surface wave.
Region 2
Negative
Image
Dipole
V?
r. = >Jp'
Vertical
Electric
Dipole
Observation
Point
R e g io n 1
Figure 2-3 Problem setup for vertical electric dipole
26
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Figure 2-3 describes the problem setup for a vertical electric dipole situated a
distance d below a homogenous isotropic half space with a negative image dipole. The
observation point ra is located in region 1 the same region as the dipole itself. The
distance between the dipole and the observation point is rt . The distance between the
image dipole and the observation point is r , .
The electromagnetic fields are calculated from M axwell's equation in a standard
.r. y , z coordinate system. The rectangular field equations are then transformed into
cylindrical coordinates p .9 ,z to take advantage o f the rotational symmetry present and
to integrate the field equations. The integrated cylindrical field equations are then
transformed again into spherical coordinate system r,9,<f> to describe the field
distribution o f the vertical electrical dipole.
The electromagnetic fields for a vertical dipole with unit electric moment
/(A /) = 1 Amp-m is located on the downward directed z axis at a distance d from the
origin residing at the interface between the two materials. The electromagnetic fields can
be determined from Maxwell's equations. Assuming the time dependence em> and that
both regions are nonmagnetic and can be represented by the complex wave number k l ,
the equations take the form:
V x E = ia>B
(2 . 1)
ik 2 V x 5 = ?^ E + p aJ
a)
27
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(2.2)
J =5 (x ) 5 ( y ) 8 { z - d ) z
(2.3)
k = COyjJdJ = CO\fi?
toн
co j
(2.4)
Representing Maxwell's equations in the various coordinate directions yields:
r 8E. 8Et . D
i : ? -------- = icoB.
Bv
Bz
, BEX BE. . _
j : ? ------ = icoB.
dz
Bx
k : - - ? ▒ = icoB.
Bx
Bv
r BB. 8BV
ik 2 n
: ? --------= ---------г ,
Bv
Bz
co
(2.5)
( 2 .6 )
(2.7)
i
, BBX
Bz
BB.
ik2
=
?
Bx
co E '
dBv BBX
ik 2 c
k : ----------- = ------- E. + u J .
Bx
By
co
(2.8 )
(2.9)
( 2 . 10)
These are solved using the two dimensional Fourier transforms
x
or
E ( x .y .z ) = j-p - f d g fd r je ?i~I*n' ,E (g ,rj.z)
5 (x . v .r) = ?
f d г jd r je 'Ul*?,y)B (г .rj.z)
( 2 . 11)
y ( x .* z ) - - L fa t
28
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Substituting the transformed equations in 2.11 into Maxwell's equations 2.5-10 and
accounting for the rotational symmetry such that B. = 0 , yields the following
dEx
irjE .
dz
= icoBz
(2.12)
dE
-?S--i$E . =icoB ,
(2.13)
E ' = ?г
= r
*9
(2.14)
-ik ' ?
BB,
(2.15)
SB,
-ik :
(2.16)
i4B x - irjBx =
-
ik 2
to
E. + ^ , 6 ( z - d )
(2.17)
Solving for the individual field components in both regions 1 and 2 yields:
г _ ia>4 dBx _ -ico <?гv
1 ? k 2rj dz ~ k 2 d=
k 2 dz
4
(2.18)
(2.19)
E: = -jp-[i$By ~ iT?B* - M ? d ( z - d ) ]
( 2 . 20 )
(2.21)
29
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In the following, the substitution
r = \lk 2 - f ' - n 2
(2.23)
will be convenient. Examining the x component o f the magnetic field at the boundary
z = 0 . propose the following solution to equation 2.21 for region 1, Bu (z ) when z > 0
and B2i ( z ) when z < 0.
5,
(z) = C2e~'r': + D2e'7-:
(2.24)
If both media are lossy, that is Im (y,) > Oand lm (y ,) > 0,as z ??▒oo then these fields
get exponentially large; therefore. D, = D, = Oand these terms are omitted. If one
medium or both are lossless, then the terms should still be omitted because o f the limit o f
the loss approaching zero or from the Sommerfeld radiation condition. Imposing these
physical limits on equation 2.24 yields
nMn' r, :-<!
-Y i
B2x{ z ) = C 2e ,7':
(2.25)
Using the boundary conditions such that
Bu (z = 0) = B ^ (z = 0)
(z = 0) = Elx ( z = 0)
30
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(2.26)
Gives the following solution for the constants
C, = C , + ^ e ?r'J
? 2r,
-H ?nk\e"J
f7
#7
y 2k { + y lk2
2
(2.27)
where
= J(a>2fi0s0гr - 4 2 -rf2) + i(ayi?<rj)
(2.28)
Substituting the constants into equation 2.25 and then into equations 2.18 - 2.22 gives the
field distributions for region 1 in the half space where z> 0 :
(2.29)
F (( Sл '7
n ---)) - ~ C░M"n
,2
)!
X
e'r' S~J
(2.30)
;-d
k\
1Y i k f + r M
2y ,
2Y\
T - V2yt - + V2y xy 2ks ; + y xk;
f J - V ri<*w)
=
j rA :+J)
(2.31)
(2.32)
jrr--a
r r ----- i -----+ ?
Yzk\ + Y\kl
2 Y\
2 Y\
(2.33)
(2.34)
For region 2 above the half space where z < 0 ,
31
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Yi \
Y\ 2
iy.J
лr ( p ?
-H,a>nY2*
-
-
X
-ly
e
Y2k ' + Y ^ 2
(2.36)
Y2k i +Y\^2
(2.37)
I 2
╗ /g
(2.35)
-iv,;
g '
y
k
2+
y
k
/:* i ^ /i* :2
-
+Y\^2
(2.38)
(2.39)
B2.(4,rj,z) = 0
(2.40)
Using the inverse Fourier transform in two dimensions for Cartesian coordinates we can
transform equations 2.29 to 2.40 from the frequency domain back to the direct space to
determine the electromagnetic fields in Cartesian coordinates.
E(x.y,z) =?
( 2;r)
| j ?e'u ' +,-v)f ( г 17,*)г/гл/;7
fl(x,y,z) = t~~tt J Je'(,?^ ' 5 (^,^z)c/^c/ 7 *
(2/t )
To simplify the integration the transformed field distributions are put in cylindrical
coordinates using the following substitutions
32
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(2.41)
X = p co s^
y = p s in ^
(2.42)
T > ?>
P~ =x~ + y
г = Xcos0'
7 = A sin^'
(2.43)
A2 = f + n :
d^drj = Xd<f>dk
gx + rjy = A /? c o s (^ -^ ')
(2.44)
г p = г t cos^ + г vsin^
B4 = - B l sii\<j>+Bvcos<f>.
(2.45)
In cylindrical coordinates, the following field components are zero for a vertical electric
dipole.
S .( ^ ,z ) = 0
B ,(p .$ .z ) = 0
(2.46)
г ,(a < M = 0
The following relations for the integral representations o f the Bessel functions are also
used:
?
J
2^r 0
For region 1 where z > 0 below the half space,
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2-47)
C.
?i
i
^\
л-
*r
~(░M?
i n kr
X
=? i
Jl ( X p ) A 2dA
7
?
2
*,V:+*:Vi
~>y
V 2r,1
2/,
k; y2 +k; yt
~>7rk 2 ?
╗<
*
(2.48)
J0( Xp ) X2dX
(2.49)
/
lYx
2r,
2/,
k 2y 2 + k2y ]
J x( A p ) X 2dA
(2.50)
For region 2 above the half space where z < 0 ,
(2.51)
W
in k*
( A *? - ) ?=
2*
/ ' W
"
(2.52)
i,r;*
J l3 ? T 7 - J, ( - W V .I
ii k { y 2+k ; y x
(2.53)
The above six equations are the general integrals for the electromagnetic fields in the
regions above and below the half space owing to a vertical electric dipole located at
: = d in region 1 when the observation point is also in region 1 below the half space.
Integrating the above equations provides the electromagnetic fields in cylindrical
coordinates.
For the region containing the dipole (in this case, region 1) the total
electromagnetic field can be divided into three parts the direct field, the reflected field
from an ideal image, and the surface wave fields. The surface wave field has two
contributions one from the surface wave itself and the second is a correction term for the
reflected field when it is not an ideal image.
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E(p,<!>,z) = EJ ( p , 0 , z ) + E ? ( p , # . z ) + EL ( p , 0 , z ) + Ec (p,</>,z)
B i p . f r z ) ^ # ( p ^ z ) + B? (p,<f>,z) + Bl (p,<f>,z) + B? ( p. f i . z)
(2.54)
The direct field is the entire field from the vertical electric dipole observed at the
observation point as if there was no boundary and medium 1 filled all o f space. The
direct field component for the p directed electric field in region 1 is
Z> d
E l (p.j.z) =
**7tKy
( A p ) A !J A
q
0<z<d
(2.55)
Using the integral o f the form below.
/ ? ( p , t . z ) = j [ y 0 (Ap) + J 2( A p ) y ^ kl^ : k ' d k
/ r (/7 .^ .r) = - p , ( A p ) e - '^ J^ A 2</A
pi
(2.56)
/ 22( / , , * r ) = - 2t e - r | 4r + 4r - 4r j
gives
( ik;2
r)=
e?V'
f 4xk;
\ r,
.
Ar
3*,
3i
rr
'i j \
?/
г,J, ( p .
\
T\ )
' :-d'
v
ri
j
(2.57)
The direct field component for the z-directed electric field is
(p-*- -
-*71Ky) Q =J \
( xp ) x ?d x
The following two integrals will be used in representing the solution:
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.58)
L ( p . z ) = ) - 7 J = r [ j 0 ( p k ) + J 2( p A ) y ^ ' ^ X' d k
o \ k~ ?k~
/.(p ..-) = - i )
o
1 , [ j 0 (pX) + J, (pA)]e-'
\ lk ' - k '
(2.59)
A
/, ( p . . - ) 1
P o\k
/- ( p ..- ) = - 2k
x'dx
?A
n
7 +^
n ?
|e
x 'j x
/, ( p - - ) =
I , ( p - - ) - - i \ - n = ^ [ J ? ( p X ) - J 1 ( p i ) > - " ? :-?: A>dX
o v k~ - k~
Ia( p . : ) = - 2 k
ik
r
2
r2
2/
kr'
z-(ik
rz r
3
r2
(2.60)
3i
kr3
Re-writing the direct field component o f the z-directed electric field in region 1 with the
above integrals provides:
%
( * * ?
=
) ( p- : - < o + ; . ( p - - j >]
(2 .6 1 )
or
k,
/
(
i k?
3/fc,
3 i 'j
f i2
f i3 J
u
4 rtk2
i k2
1
H
(1>P? ^i*,r,
I
fi
J
I
Integrating the direct field portion o f the magnetic field for region 1 yields:
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.62)
* Ir, Z~d
Wo re
j г ( p , * . : ) = - г - { ------- J x{Xp)ArdX
Y\
o
B f.(p.t.z)= 'J▒гl,(p.z-d)
(2.63)
(
\
p1
47t
f \%
I n
i N
?>
rr )
The equation below summarizes the direct field contributions for the electromagnetic
fields in region 1. The direct field contributions depend on the wave vector for region
1. ^ , and the observation distance from observation point to the VED. r ,. There is no
contribution from the wave vector in region 2.
' ik;
3*,
s. r i
rf
\
p_ ( z ~ d )
r.3 , Kr\ J I r l J
3 /'
/
iky
( z - d \ f ik{
vi __21____/
2
>
r\ n
rx
'I y v ri
4/TAC,
f
P
\
/г,
vr. yv r.
3kx
3/^
ri
r, y
(2.64)
1
r( y
The ideal reflected field is the field observed at ( p , z ) from an image antenna at
(O.c/) when there is no boundary and as if the properties o f medium 1 characterized all o f
space. It is given as follows:
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(p.*.z) =
(Xp)X'dX
1 0
(2.65)
f \
~io)Mo vV: 'f*L _ ^ L?> 3,' l г r z + d '
Ank,2
< r2 r; ~ r ? ) ^ 2 , { rz J
( 2 .66 )
?>
- (░Mu r ?V .
K ( p ^ = ) = -A n kf
ik~
r2
i
2
r2
z +d
2
r2
J
{ r2
ik^
3Ar,
3 /'
r2
2
r2
2
r2 ;
I
B ^ ( p . + .l ) - = j l ▒ j - ------- J, ( Xp ) X' d X
An
~?Mu P
An 2
(2.67)
/
4;T
г
S i
\
X
'ik y
,
r2
1
r2
'
X
The equations below summarize the image field contributions for region 1. The ideal
reflected fields involve only the wave number. k} .and the observation distance. r2, from
the observation point to the image dipole:
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
l0>Vo Jk,r. ' i k;
< r2
f< t ~ j.
r2 , S 2 ,
k,
1
r2 ,
( z + di \ ?>' ik;
i
V'
\
P
, r2 /
r2
r?c
c
~(op? ,V; ikj2
4 xk ;
fг M
3k, _ 3 r
I
r:
J
I r2
3k,
2
r2
3i '
1
r2 )
(2.68)
/
-r: - Vr; )J
The other fields represented in equation 2.54 are the surface wave fields. The
surface wave is also known as a lateral wave if the electromagnetic wave is generated by
a dipole on or near a boundary between two media. If one o f the media is air. the lateral
wave is generally referred to as a surface wave. Evaluating the lateral and correction
fields for region 1 yields:
- r,e ?'
F ? ( p . t . z + d ) = k l ( - г -----F - J , ( A p ) * !dA
0J *, n + k 2r\
*г
=
Ink;
(2.69)
(p .+ .z + d )
Fu ( p, *. z + d) ?k; U
- r - J n ( A p ) A }dA
>k;y2+k;r,
(2.70)
fl,', ( p . * . z ) + Bt, { p . * . : ) = '?&-F?(p.t.z + d )
L7Z
F ? { p . t , : + d) = k ; ) - f ? ^ - J , ( A p ) X :dA. ?
0 k , y 2+k2y,
(2.71)
Equations 2.69-2.71 provide general expressions for the surface wave fields at an
observation distance r? generated from a vertical electric dipole above a half space. In
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
order to solve for the integral expressions contained in the equations above certain
assumptions are invoked. In order to rewrite the denominator o f the integral expressions
in equations 2.69-2.71 limitations are placed on the value o f the wave vectors o f the
media. The above integrals can be evaluated by considering the inequality
(2.72)
which can be approximated by
|* ,|> 3 |* ; |
(2.73)
This assumptions invokes the dielectric limit. This requires the dielectric constant o f
medium two to be at least three times greater than the dielectric constant o f medium one
which contains the dipole. This limitation holds true for boundaries such as air and
water, air and earth, and water and earth as well as air and polymers, air and metals, and
air and semiconductors. Using this condition the denominator o f the integrals in
equations 2.69-71 can be re-written with the following expression
1
1
k \ Y i + k lYx
?+
*,V: I k \ Y i +
k iY\
k Wz
(2.74)
Substituting equation 2.74 provides
Fu, ( p .* . : + d ) = г
(Ap) A'dA + G,? ( p . +. z )
m o Yz
(2.75)
F,_. (p .* . z + d ) = |i -
(Xp) X' dX + G1; ( p . t . z )
*i o Yz
(2.76)
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where the G functions can be approximated by
Gx?
~ Gim
m = <j>,p,z
(2.78)
Since contributions to the integrals in the G functions occur when the variable of
integration. A ,is o f the order o f magnitude o f k2, the wave number in region 2 (much
smaller than k } by 2.73). it is then a good approximation to set
Y\ = ^ k 2 - A 2
(2.79)
Using the above approximations
G ^ p ^ = ) ~ e k^ G Xp{ p)
ylJ ]( A p ) A 2dA
it
(2.80)
i l k\Yi + klYx k\Yi
Gx*{p,<P*z)~ e?k'(:*J)GX4{ p )
( p ) = *1 I f TJ? '- n ----- - p - V (?*p ) * ' d A
Gx:{ p ) = k ; \
J0(Ap) A' dA-
(2.81)
(2.82)
k>2r 2 +kl r x kxr 2
Rewriting equation 2.75 such that
Fl? (/,.z + J ) = г f / A( p . ; + d ) +e ? - 'G ╗ ( p )
(2.83)
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where I A and Glp are as follows
h (/> -') =
( A . p ) y ??A'dA
o Yi
I , ( p . z) = - p - J , { \ p ) e ?r':X2dX
PoY2
(2.84)
1
I a { P ' z ) ~ ~-K
3/
+
P3
-
2*,p4 y
e*'r
(2.85)
and
R=
2k;
( 2 . 86 )
Substituting equations 2.84-86 into equation 2.83 yields.
- 2 k.
?
'
2
P
1
3/
P*
2*,p4
-
3
P
(2.87)
Oh. / n c,k-pc?p
?F ( p )
k; yjk2p
The lateral wave and correction parts comprising the p component o f the surface wave
electric field in region 1 is
(2 .88)
p 2 + 2 k, p '
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where
/ ( p : k ,.k ,) =& - - L - г [ ^ - ? ' F ( p )
p
p
K V *2p
p =
kip
2V
(2.89)
where C\ and S 2 are the Fresnel Integrals defined in The Mathematical Handbook by
Abramowitz and Stegmen. The following equations evaluate the lateral and correction
fields for the z component o f the electric field.
k; 1
AC,
(2.90)
where
l , , ( p . : ) = j - [ J Q( X p ) - J , ( X p t y - X ' J X
0 Y2
5i
e*'r - k ,
8kxp 3,
r iky
ik,
V'
/
p
2p 2
2/
P
p2
e?k:k?elkr
(2.91)
*;P 3
I? ( p . : ) = | - [ y 0 (X p ) * J , ( X p ) Y ?-,i'dX
o Yi
i, (p .r
) =
-
J ?
j ,
(2.92)
/
t
.
\
*l + _L
2 ~3
P
VP?
+
2/
.
k,
-
?
\P . vP
3/
\
+
ik.r
e '
2P : /
and
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.93)
Substituting equations 2.91-93 into equation 2.90 with that result substituted into
equation 2.70 provides
K (p.--) =
e ;=( p - z )
=
Ink
-i(op?k; f z + d ' f ik:
?
InK
P
k.
7/
h- +
r
8p
(2.94)
where
n ( P:kr k; ) = ^ - ^ T - - ^ - ^ l - f - e " ' F ( p )
p
p- k2p
\ k2p
(2.95)
Evaluating the lateral and correction fields for the magnetic field component
F:, ( p . : + d ) = г г i ? ( p . : + <t)+ G ? ( p y ?'i' j '
i -
(2.96)
using equations 2.92 and equations 2.81 yields
(2.97)
e -f
-ie- tk. r.
ik L__ 1_ *2
. P P1 K \
( z + d ) ^k]__
3 1
V P )V
.P V J
such that
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.98)
Summarizing the contributions o f the lateral fields:
ik-,
1
kl I 7Z
P
P2
* l \ k 2P
ik-,
1
P
P~
ik2
1
P
P1
/
k 2p }
rF (p )
kl I jt
e ' PF { p )
kx \ k 2p
(2.100)
kl ' K
e PF ( p )
kx \ k2p
where
kip
F ( p ) ~ \ j 2 ^ dt ~ 9 0 + /) Q >(p)
*S2 ( p )
(2 . 101)
Summarizing the contributions o f the correction factors needed for the non-ideal image
field:
E
M
_ ?6>P?kl
? )=
,
E ;,(p.t.z)=
3/
1
P2
2 *,p3J
icop?kl
i Z + d ) f ik?
2 itk? I P
{ P
*>o*2
B \ A p *+*z ) = 2 xk?
/
.\
Z + */
I p J
(
+ 7/ \
2p 2 8 p s J
K
3 ]
**?╗*%
2 p 2J
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( 2 . 102 )
Combining contributions o f the direct, image, lateral, and correction fields
provides the complete integrated general expressions for the electromagnetic fields in
region 1 when the dipole is also in region 1. This combines the expressions in equations
2.64, 2.68, 2.100. and 2.102.
-(o/u,
2 ;rkr
-ie? ik.'r -,
M. e ik'A z + J ) e ik ^-po ik,
k,
P
P2
kl
K
e 'rF(p)
kx V k2P
3/
2 k}p '
pr ik]
3/t,
v r.
n
\
p r :-d'
ri / \ r\ j \ r \ /
e ' ? f ik]
3
f \
3 /N P ( z + d1 \
1
l
ik.r*
2
1
.
3/
\f
r2 ,
I
r2
J
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.103)
Equations 2.103-2.105 are the electromagnetic fields at an observation distance r? from a
vertical electric dipole above a half space as depicted in Figure 2-3. This configuration
provides a negative image dipole. Our experimental setup is more accurately represented
by a vertical electric dipole located in air above a half space as shown in Figure 2-4. This
provides a positive image dipole.
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
P ositive
im ag e
r
?Jp:
R e g io n I
D ip o le
Figure 2-4 Problem setup for positive image dipole for vertical electric dipole above a
half space.
When the dipole is located below the interface surface in the earth then the surface wave
is also known as a lateral wave. When the dipole is located above the boundary into the
air then the surface wave is generally called the Norton surface wave. Equations 2.1032.105 can be modified by changing the subscript 1 to 2 and replacing
= ?z and
<f> = -<f>. Then the expressions for the field values above the half space in region 2 are
(2.106)
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E 2; ( P ,t.Z ') =
etk"-r- ik 2
r,
cop,,
2 xk,
1
ik,
Je
n
/
J
k,r.3
rr2
1
i k ▒ _ 2 ___ 3i_'
kzri
k
/*,
r,
r,
V
.
3
r,*
/
I
3/
k,r,'
V ri
r.:
V ."
,
\
(2.107)
г 3 I jr ( r t )
?e ** K2 I ?
vr:
/
p
\
'/Jfcj
<rl> I r.
' ik,
1^
r2
r2 J
1'
?>
rf J
+ -
v r: /
(2.108)
where
rt ~ \j P 2 +{- ~ d )2
(2.109)
r2 = J p 2 + ( - +d ) 2
( 2 . 110)
and
P
-
k 2r: '
k 2r, +k^(z' + d)
k,p
2 " 2 k;
( 2 . 111 )
e?
F (* ) =
= j 0I + 0 - C 2 (?P;)"?о: ( л >
( 2 . 112 )
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The expressions in equations 2.106 to 2.112 are for the situation depicted in
Figure 2-4 where the VED at z = d creates a positive image dipole at z = - d with the
observation point above the half space.
Note from these equations when the half space is a perfect conductor, <r, ?>░o, as
does
?> oo. so the surface wave terms with
in the denominator go to zero and the
entire field is given by the contribution o f the direct and image fields.
Using the following substitutions, the fields in cylindrical coordinates can be
converted to spherical.
E2r (r;,,<t>.#) = E2p ( p . 0 \ z ' ) s i n 0 +
( p ,^ ',- ') c o s #
(2.113)
Eio
= E2p ( p . 0 '.z')cos 0 -
( p . f i '. z ') sin#
(2.114)
B ? ( r ? O . 0 ) = B u .(p.4r.z-)
(2.115)
sin # = ?
r"
(2.116)
Tt
COS# = ?
r" .
(2.117)
Having the electromagnetic fields in spherical coordinates allows calculation o f
directivity plots which show the magnitude o f the fields atparticularobservation points
for all thetas.This type o f presentation shows the direction with the highest field strength
which implies the direction the power is ?radiating?.
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
King calculated directivity plots for a vertical electric dipole on the surface o f a
half space when d = 0 . By substituting 2.106-108 into the equations 2.113-2.115 and
setting d = 0 provides
2
2/
. -kl
' + ---t ╗;? sin 0 ?
*?
= ^ j ~ e?k'r j cos0 ?
r: k, r ?
k, V k,r
Eir
(2.118)
sin
0 ^ - sin#
k,
E 2<>{r.,-0 ) =
ik,
r?
2 nk.
/ ik,
-cos # ? sin #
k,
v r,
kl
1_
r?
71
-
1-
k,r,
kx \ k 2r?
(2.119)
e"F(r?)
k\ \ k 2f
sin#
ik.
k, I 7Z _,/> , _ .
~ ~ T \ \ T ~ r e F ( p >)
r. j k\ V k 2r╗
1
v r,
k l r , ( k,n.+k,z,s
v: fri 1
P.. =
k2p
" 2*f
( 2 . 120 )
(2 . 121)
Define the surface wave term as
1 ╗ 2"
.
(2. 122)
Using the Fresnel number defined in 2.112 and the asymptotic expansion o f the error
function, the surface wave term can be approximated as
1
1
?+ 1
kj Vk2r╗ yl2 JtPn 2P
j'f _ a 2
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.123)
when the absolute value o f Pu > 4 (as defined in 2.121). Substituting the expression for
P0 and converting p and z to spherical coordinates provides the following expression
T'{r?.0) = -
ik, sin 0
k; sin-? 0
1+
1+
k,
cos 6
1C O S 0
(2.124)
k,
The fields in 2.106 to 2.108 and 2.118 to 2.120 are valid for all observation distances.
The far field limitations are only invoked when equation 2.124 is used to represent the
surface wave term. To avoid this limitation on observation distances, the Fresnel number
was numerically calculated for equation 2.122.
In order to use the field expressions in equations 2.106-2.108 for the field values
when the observation distance is very close to the dipole i.e. the near field values
expressions from Abrambomitz and Stegman's The Mathematical Handbook were used to
numerically calculate the Fresnel number. Using the expression for the Fresnel number
f W
? o + o -Q w -tfin i)
(2.125)
C, and 5, are the respective Fresnel integrals which can be approximated by fractional
order spherical Bessel functions.
C 2 (P?) = J l {Pn) + Ji {Pn) ^ J ^ (P0) + J l3 (P?) + ...
(2.126)
s 2 ( f , ) = j 3 (pn) +J 7 (P0 ) + J u { p. ) + j ? (PD)+...
(2.127)
To demonstrate the validity o f the above expressions, calculations for the surface
wave term approximated in equation 2.124 were compared to the surface wave term
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
calculated by substituting equations 2.125 -127 into equation 2.122. The sample
calculation used in King for a vertical electric dipole above lake water with a relative
dielectric constant o f 80 and a conductivity o f 0.004 S/m at 10 MHz with an observation
distance o f 500 km was used to validate the surface wave term for all thetas. This is
shown in Figure 2-5. Note the number of Bessel function terms needed for agreement for
all thetas is 5000.
These directivity plots show that in the extreme o f the large argument
approximation the numerically calculated Fresnel number behaves as predicted by the
large argument approximations. Therefore, the numerically calculated Fresnel number
can also be used to investigate field values for small arguments or small observation
distances.
.70ti-009
.7* ?
09
09
IS
IS
. .0
4 --
Figure 2-5 Agreement for the surface wave term, T. for all thetas calculated with both the
large argument approximation and numerically calculated Fresnel number.
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2-6 shows the magnitude o f the surface wave term as the observation
distance goes from 5 meters to 0.5 microns for a vertical electric dipole at 10 GHz that is
on the boundary o f an air and lake water interface where the relative dielectric constant o f
the lake water is 80 and the conductivity is 0.004 S/m using both the large argument
approximation for the Fresnel number and the numerically calculated Fresnel number. If
the large argument approximated Fresnel number was used to investigate the field values
at small observation distances the surface wave term would be greatly overestimated. At
an observation distance o f one micron the surface wave term calculated with the large
argument approximation is on the order o f 10u while the surface wave term numerically
calculated Fresnel number is on the order o f1 0 6. Clearly the Fresnel number using the
large argument approximation must not be used to investigate near field expressions. The
surface wave term using numerically calculated values o f the Fresnel number must be
used to assess the near field behavior.
The large argument approximation limits how close the observation point can be
to the dipole source. For a dipole located on the surface d=0.
(2.128)
where
(2.129)
By using the numerically calculated Fresnel number, the location o f observation
distance relative to the source location is not restricted and the observation distance can
then be close to the dipole source.
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
? ? ? N um erically calculated Fresnel num ber
Magnitude of the Surface Wave Term, T, in V/m
??? Large argum ent approxim ation
9
,-8
?7
,-6
?5
.-4
3
2
1
,0
observation d istan ce in m eters
Figure 2-6 Magnitude o f the surface wave term calculated using both the
large argument approximation and the numerically calculated Fresnel
number as a function o f observation distance.
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
To verify the total field calculations for the derived expressions, we compared our
total field expression using the numerically calculated surface wave term to field
calculations described in King. In Chapter 4, King shows directivity plots for a vertical
electric dipole on the surface o f an air and lake water boundary. The dipole is excited at
10 MHz with an observation distance o f 500 km. The electrical properties used to
represent the lake water with a relative dielectric constant o f 80 and a conductivity o f 4.0
S/m. In Figure 2-7. the magnitude o f the radial and angular electrical field as a function
o f the observation angle, theta. varies from 0" to 90". The left two graphs are the
electric fields using the large argument approximation for the surface wave term while
the right two graphs use the numerically calculated Fresnel number to determine the field
values. This figure shows the agreement between the numerically calculated and
approximated surface wave term used to calculate the total electric fields. Figure 2-8
shows the directivity plots as presented in King's Chapter 4 page 101 and 103.
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
s r*tri╗u
1
9
л
2 .1 ? 9 J ╗-0 ? 9
9
120
19
I JO,
#
2 .!?< (? -0 09
09
1Jл
181,
1
Z rtxFitU
'
90
1
801*1-0
9
01
181,
I
1 . 8 0 1 i i - 0 03
18
270
27 0
Figure 2-7 Calculated directivity plots showing the agreement for vertical electric dipole
operating at 10 MHz on the boundary between air and lake water. The left two plots
show the surface wave term calculated using King's far field assumptions. The right two
plot show the surface wave term calculated using the numerical solution for the Fresnel
number. The observation distance is 500 km.
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ji. r>? -r
i ll S m
U4
JO l
o .u w
1
|
* 1 ; TCP- S ? 2 .3 6
12
TT 0 j 2 .3 1
* . 6 6 ' U l> 7
, ╗ i.v r, ! i * ,
1-
r0*3?- ?
1
* 10 s V . m j 1.73 ╗ l ( r л \
? 10 1
: 1 73 . 10 T
< III- 1
11 74 * 111 '
. io 3
; i.9 3 * i r r *
t
I lr*ITr 1 2 3 <"..mv>lrtr f a l l ui / ? - rn H ╗ |nr m t k a l tU|auk* ii) %ir on boumiarv
>"*^*<v>n air aiwi IA *m-a w.-urr h . M i rnrth. Ic I (try rarth. atrl d Irkr waUT.
I'i-╗iuл*-y /
10 M H i radta- dw tanrr t } - 300 Lui liar >Uaiw*t n m . ╗ f,4
й - 7T72
й= TT/2
yX
---? й- ir/?
)?
(a)
(╗>)
(O
(л*)
"i
<tr
4 .0 S /m
0.4
0.04
0.004
80
12
ft
80
ir/2
г ^ < r a ,ir /2 )
2.0 4
6.4 4
2.0 4
2 .1 5
*
x
x
x
1 0 * ' V /m
10-*
10"*
1 0 -*
F i f u n 4 . 2 1 Ctaaiaietc fid d o f |г г ,( r a , 6 ) | far vertical dipotr in air o n bntm lw jr
brtw rra air and (a) л?╗ w ater, (b ) wot t a n k , (c ) dry cartk. and (d ) lake water.
Fteqecncy / = 1U M ila; radial d a ta o c e r? a 500 tun.
Figure 2-8 Directivity plots as presented in King p. 101 and p. 103.
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Horizontal Electric Dipole
The picture in Figure 2-9 shows the problem setup for calculating the
electromagnetic fields from a dipole oriented parallel to the half space boundary. When
the dipole is in this configuration it is referred to as a horizontal electric dipole. This
field distribution would model electric fields radiating from a TE waveguide. The
solution closely follows that for the vertical electric dipole. Again the electromagnetic
fields are expressed as integral transforms and substituted into Maxwell's equations.
Using the boundary conditions all six o f the electromagnetic field components are
expressed as general integrals. The horizontal problem is more involved than that o f the
vertical electric dipole due to the absence o f symmetry about the vertical axis; therefore,
all six field components are needed. The general integrals for the six electromagnetic
field components are again converted to cylindrical coordinates and then integrated for
general solutions. As with the vertical electric dipole, the horizontal dipole is located a
distance d from the interface boundary and the fields are calculated at an observation
point r? . In this configuration, the horizontal dipole is directed along the x-axis.
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Region 2
(metal, dielectric, etc)
Horizontal
Electric
Dipole
Region 1
(Air)
Observation
Point
Z-d
Z
Figure 2-9 Problem setup for an x- axis directed horizontal electric
dipole above a half space.
Maxwell's equations in the half space region are
V x г , = ieoB2
V x B2 =
(2.130)
(-io )г2 E 2 + J tx)
(2.131)
60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where the volume density o f current in the dipole is normalized to a unit electric moment
o f /(A /) = 1Amp-meter.
J = d ( x ) d ( .v ) г ( r - < /) x
(2.132)
Solving Maxwell's equations
/:
dE.
dy
cE,
dEt
dz
dE.
dx
= icoB.
CZ
dEv dEx
k :?
= icoB.
dx
dv
- dB.
1
(2.133)
(2.134)
(2.135)
dB,
.
? ? = -uoH.e, Ew+ J z
(2.136)
dBx
J-?
uz
dB.
rj =
cx
dB':----------=
6 8 ? -i(ou?s^E.
?
~r
ki : ?
dx
d\?
(2.137)
(2.138)
using the following Fourier transforms
30
x
E (x.y.z) =- ^ j
(2.139)
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and substituting equation 2.139 into Maxwell?s equations 2.133-2.138 yields
dEx
irjE.
dz
= icoB,
(2.140)
dE
??
x i4E. = icoB
dz
(2.141)
i ^ E v -ii] Ez = icoB.
(2.142)
-
dBv
- i k 2:
dz
co
Et + n ? S ( z - d )
(2.143)
dz
co
(2.144)
igBv - iqB, = - ^ - E .
co
(2.145)
Rearranging the above equation for the y and z components o f the electric and magnetic
fields in terms o f the x component gives
1
E. =
B =
B. =
? dBz
- n s E ' + i c o ? !dz
D
- S dEz
n<oBx + i4 ?r~
dz
k;-$l
-n$Bx-
k;-$'
f
1
\
U 2- ^ J r
dBx
dz
ik; dEt
co dz
k-n
< o E' \
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.146)
(2.147)
(2.148)
(2.149)
Substituting equations 2.146-2.149 into 2.141 and 2.143 provides the ordinary differential
equations
f dr
B= 0
(2.150)
where
^
k/ ; - k ; - 4 - -rj-
(2.151)
(2.152)
Solving the above differential equations with the following boundary conditions provides
expressions for the x component o f the electric and magnetic field which can then be
substituted into equations 2.146-2.149 for integral expressions o f all six components of
the electromagnetic fields:
г u (x ,y .0 ) = г 2t(jr,j\0 )
(2.153)
Ely( x .y . 0 ) = E2} (x ,y ,0 )
(2.154)
kxEu (x.y.O) = k2E2: (x.y.O)
(2.155)
B ,(x ,y ,0 ) = B2(x ,y ,0 )
(2.156)
The general solution for equation 2.150 in region 1 is
Bu = c y r': + C / r':
(2.157)
63
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With - > 0 , the exponential term with - lm ( / ,z ) ->ooas r -*
qo-;
thereforeC' = 0 .
Equation 2.157 becomes
Bu =
(2.158)
Using the same reasoning for the region where z < 0 and the boundary condition for the x
component o f the magnetic field, then the solution for the region above the interface is
b 2x = c y v
(2.159)
Equation 2.151, the ordinary differential equation for the electric field, is homogenous for
the region z < 0,
w
(2.160)
For the region z > 0.
г? = CV v +
,2.161)
Again for the physical limitations the constant C must vanish since the exponential term
would go to infinity as z
o . The boundary condition for the x component o f the
electric field, which requires the electric field in region 1 to equal the electric field in
region 2 at the interface where z = 0. provides the following expression for the constant C
C = C\+
( A:.2 - г 2)
K1,
2y'k]
? e'T'J
Substituting into 2.161 gives
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.162)
" лsiny,;
? ------ e╗?'
л
гIt= C ,e "
1y A
(2.163)
for 0 < : < d and
cov?(k{-f-) .
C , ----------- --------siny,rf e '
'M f
(2.164)
for d < z .
There are four remaining boundary conditions (continuity in the y and z
component at the interface where z=0) as noted in equations 2.154. 2.155. and the y and z
components o f equation 2.156. Using the conditions in equation 2.154 and the y
component o f 2.156 at r = Ogives the following equations:
A
k: - s
1
k ^2 - 'T
rjC2 + o y ,C ,) =
(2.165)
? Y t'i ~
k : - г 2 co
7ic z
co
oyyxC,)
1
i + M., ( k{ ~
)e
'
<r\J
. (2.166)
Rearranging to solve for the constants in matrix form
Yi
Y\
l
<? + l 2 _ -2
V *1
k 2 - 4:2
v * r-c :
(
\
Y
\
k
Y,
2 ? ^~2
k : ? c?
2
I
1
k;-g2
- +
/
1
k ; - c 2r '
-
4nc\-
?9
f 1v 1k*12
Yik-,
Cl
+ - . ? . ?
k 2 - g 2 k ; - g 2 co
1
r
l
гijC\
i : _ e2
co
*2
*5
i
{ k ; - g
2
^
k ; - g 2\
-
0
(2.167)
,yJ
= f.i? e ?1
(2.168)
гrj ' c ;
c
Y\k\ ^ Y2k 2
{k?-?
kg-g
=
0
=
tr.J
M?e '
. CO .
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.169)
Solving for the determinant o f the 2 X 2 matrix gives
?\{N
d etl ] = 7 n
(2.170)
where
M =Y\+Yi
(2.171)
N = k~y 2 + k:y,
(2.172)
and the constants are
<-\ =
C\ =
MX
-w '
(2.173)
M?e ?1
MX
(2.174)
These constants can be substituted into the above equations to provide the general
integrals for the components o f the electromagnetic fields in the two regions with a
horizontal electric dipole located above the interface. The following equations are the six
components o f the electric and magnetic fields.
In region 1 with 0 < r < d
r
г ? = -4 K
p r
r
---------------------------------e ' +?---- T -sin v .r
MX
i/X'
e '
siny.:
ii
X
X
Yl
N
(2.175)
ely,J
y
cos,yik:
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.176)
(2.177)
X
x
4*-ny)
f lr2 - I
v
X
X
B? = ^ j f j f f d n e ?' * " ?" cos y x: +
Ax
?
X
-X
X
Ax
/
X
?X
2
\
mn
(2.178)
* :V ~ ki ( k ; - 4 2 + / , / : )
MV
' * n\ I
B,
t
~x
.
e
A/
s in /,r
iy|
,r _.
e
(2.179)
\
e
(2.180)
In region 1 where d < :
X
X
~CQM? j d g f d r ? 1'-"'?" M t i - F b M Z - F )
Bu =
Ax -x -r
mv
e
x
x
x
k,: s : .
+ - ^ r s,n ^
e r'
X
x
sin y xd
e'
iyxk{
'
(2.181)
IY~
(2.182)
*v) r h . ?y<J I sin /i< ^
e 1
V
A| /
IV *
x
-i-2
?7' i
s ? = ^
(2.183)
MX
(2.184)
/
* ? - ;г = K
W
1"
x
5 ,,=
7>) /sin/,c/
'r,*/
Atfv
x
sin ^ c/
Z* * n y )
4;r:
JW \ d n n e ?{l
-x
?x
V
/ / ,I
v
e?
a/
(2.185)
/ y,r
tr'
(2.186)
In region 2 where - < 0
M*2 ~г2 )+M*r-г2)
.V/A'
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.187)
о
г ='
T-
4 x~ _il rfг W
*
1* " ?-
jv
( 2 . 188 )
t
(2.189)
л
X
* r-^ ;
(2.190)
X
X
r;
?
X
s : (^ : ~ * : )
A/ +
?
X
f. vc f
.j
(2.191)
'I4 * * n y) г
8 =-? = 7 f r J < ^ / ^ W * ?
?X
A/-V
-x
A/
(2.192)
Converting to cylindrical coordinates
A2
E^ = - p4/rC
T T cos<(,
Y\
(2.193)
o' ~
-T'l
|(*,V? ( i p ) - ^ - [ / , (^j) + V, (Ap )]l-< P ' :-JXdA
o
"if * ? [ > . . ( ^ ) + -A
O' ^
) Yi
^
*-/l
(-lp )--A (-Ip )]
e r'{:*J)AdA
4;r&
(2.194)
(2.195)
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.196)
B,a = - ? cos <j>
'*
4n
(2.197)
B.. = ^ s i n < * f ( e ^ - Pe'r'{:"J'\ ? J , U p ) X 2dX
An
o
>Yx
(2.198)
where
p = Yi ~Yx = ^2 ~Y\
M
^ ,+ x ,
_ k ; y 2 - k ; y x _ k{y 2 - k ; y x .
N
(2.199)
+ k^Y\
For equations with two signs, the upper sign is for z > d and the lower is for 0 < z < d .
For region 2 the expressions are
-p
= ? ? cosdi
4K
( 2 .200 )
г ,. = ?? sinй е ^ \J? (Xp) - _/; ( ,lp ) ] + г & [ ./? (Ap) +
4n
(A p )] y ^ 'X d X
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( 2 . 201 )
=-^ c o s i*
iV
J,
(2.202)
(2.203)
l(r'J-r'-:)AdA.
B, = ^ c o s
*' 4/r
(2.204)
N
B2:
= ^ s i n * J - i : J , ( k p ) e ?(^
U
2d k
(2.205)
Following the integration in King's chapter 5 sections 5 and 6, the integrated
expressions for the electromagnetic fields in region 1 from a horizontal electric dipole in
region 1 are as follows.
J)
( k
?>
v ri
(
+
/ ^ tk.r,
; e " +
ri y
/
,\
z +d (
3_ '
~ 2p'
I
K P
\
( r - J i V f .. ?>
iky
3ky
3/
v ri )
rr
' ?I y
z + dJ
\
I r: J
-
/
v ri
/A,2 3г,
r*
V r,
.
uk.r.
3/ N
e1*r,: /
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.206)
: .J )
E,t ( p ^ , z ) = - ^ Ts in ^ { k 2h ( p \k ,.k 2)e?k'->
?
r:
/
+/
J \
Z+d
I
J
r2
3/
rz
r2
\
e
ik .r ,
v ri
j
( ik ;
3k,
I P
Ip ?
*f
^3
+
11
I
.
3k,
(2.207)
r,;
,'V :
r/ ,
E\.-(P'&': ) =
(P -k ^ k 2) e?k:tle?k'{ J) ~
\
p
, rl , I n
/
\
+
' ik~
JV rx
1
>I
/
z-d
y
5/ )
ikl__ 3 ^ l_ 3/^
r2 ) V r,
'T
00
1 ik{
+?
">
/г,:
3k,
r~
A:,
p-
-k ,p s
3 /'
r,} J
/
(2.208)
ik.r*
r,
_p-
+
~vS
+
?i
j|-^
1
i
v r:
I r2 AV A
2p
?>
I
rx
) <
ri
rr J
e ik.r.
'? +
r;*
(2.209)
^3
+
11
?'a
i
11
' i k;
r,:
I
r2
J
*<?
,
r:
1e
tk,r
1?
r 2* J
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
NOTE TO USERS
Page(s) not included in the original manuscript
are unavailable from the author or university. The
manuscript was microfilmed as received.
72
This reproduction is the best copy available.
UMI
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the half space, the subscripts I and 2 must be interchanged and the following
substitutions must be made: z = - r in region 2 with E,. =
and B. = - B . . In
addition, the associated angle must be changed such that <j>'= -<f> with E4- = ~Ef and
fl, = - B 0. The configuration depicted in Figure 2-9 involves a negative image dipole
rather than a positive and causes the coefficients Q +1 and P -1 instead o f Q - \ and
P +1 where P and Q are defined in 2.199. The positive image dipole configuration is
shown in Figure 2-10.
Horizontal
Electric
Dipole
Region 2
Observation
Point
jr.
Image
Dipole
Region 1
Figure 2-10 Problem setup for a positive image dipole for a
horizontal electric dipole above a half space in region 2.
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The electromagnetic field equations are altered for the observation point in region 2
above the half space when the horizontal electric dipole is also in region 2 above the half
space as follows:
?>
E2 A P '0 ''= l = - ^ f cos<t>'
> 2
/. -
1
2
2/
1
rr
2k,
ik*r*
2k;
2
r: J
l r:
/
г,r;;
( P-&'?*') =
-e
r /*,
f \
A:,' r.
A:, KP)
sin
z' + d ' V i k ,
r,
r:
2/
- ;
r,- L
k,r,
r2
3/ N
J
I
n
]
2k {=' + d ' \ f ik,
3/
i *
k,r, )
kx
'i
V
I r. J
1^
(2.216)
7T -i/'.
V
ik,
r,
\e
1
r~
e ik,r,
??
k,rx
ik,
i
1
2
I
r
1
r2
, V /'A:-,
r: y r,
: ?+ d
v
3
.
J I
3
VT
2k; /A:,
+r.
k?
r ? + c/O
f /A2
k2rx;
L
1
- e - <?2 +i- '
r, k,r.
2/
ikyr.
e -
3
r,:
(2.217)
3/
A:,r,'
2ikt ( r;' \
k\P < p \
k,r,
/
v'i yv
+e
ik.r.
2k,
k,
?
= -e
{ p \ ( z ' + d , \ / ik,
r,
\ r2 7 v
r:
y
r
^ ik ,
3
2
r2
1
r.
.
ik,
3
\ r\
ri
3/
k^r: y
3/
i I
*2r2
(2.218)
I 7T
? ?
\ r: y\ r:
rf y
k
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f z ' - d ' \ '/it,
B,
4/r
I
rt
1'
' /*,
ik.r.
?> - e - - f Z>+ ^ 1
rr )
I r: J I r:
J I r\
1'
^ J
(2.219)
?
+?
2/
<A,r,
??
r2
k2r2
/A:^ ( r; ^
kxp P~ )
/A,r
e ? 'F (P 2) +
( z ' + d '^
v
'/it,
f
1'
-< ?
*>
I
ri
J I r\
ri
<*,r.
* *
)
r\
-
J
/ .
ik*,
3
v r:
r:
( + t/,l ' lit,
I
JI
3/
1 '
z ?
r:
r :
?)
^ )
( 2 .220 )
/
+2e
;A,r,
P
\
\b
U
j_
/V
J/
fc,/v
\ e ?k'-
^7Z
( : ' +* )
I
ri
3_
k;r*
'lib.
J v r:
3
r2
e ? 'F (P 2)
1^
/
** P
?г
\
(/A ,
1)
i
?^1
B2:\p .< p \z') =
\
P ( ik,
|
r
tk*r.
k*r*
.r* j
k U r*
k*rk. \y /
ik*
r, r;
tk * ry
1
-^1
2k*
3/ \
k2r2 ,
( 2 .2 2 1 )
r =' + J '^ r ik2 _6_
?>
r,
r,?
V
r:
y
.
\
^ 2 r/
J
15/
where
P =S,
S 2 + Z '+ P ' '
R
C? _? ^:r:
╗3i
? 2*r
( 2 . 222 )
(2.223)
lr 2 - ?
Z' = ^ ~
2k.
R = kM
2k;
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.224)
(2.225)
2k,
P, =
k lr 21 klr, + ksk;z' + k,kid' ^
kip
2kf
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.226)
(2.227)
Chapter 3
Experimental Procedure
This chapter describes the investigation o f contrast mechanisms for subн
wavelength microwave imaging o f dielectric constant variations in materials with high
spatial resolution. A microwave near field microscope modeled after near field scanning
optical microscopes (NSOMs) was constructed. Samples with known conductivities
were used to characterize the microscope performance. Experiments were conducted to
investigate the effect o f the tip to sample spacing indicative o f topography by studying
approach curves. Experiments were also conducted to investigate the changes in material
conductivity.
The microwave near field microscope designed for this investigation used a subн
wavelength probe tip to scan a sample?s surface.
The probe tip was positioned a few
microns above the sample's surface. Samples are placed on motion control devices. The
interaction between the probe tip and the sample surface was monitored by changes in the
|
I
HP 43SA
PniKrt
?л -
?
o 47" OD Semi-rigid
Coax Cable
Coax
Adapter
[MPA IT
Diode Microwave
Source
SMA Connector
Probe
Sample
to dB Directional Couplers
Motion
Controller
Computer
Figure 3-1 Microwave near field microscope with probe acting as
both transmitter and receiver.
77
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percent o f reflected microwave radiation as sensed by microwave power detectors and
diode detectors. At low power these sensors operate much like square law detectors. In
the configuration depicted in Figure 3-1 the probe acted as both the transmitter and
receiver o f microwave energy. Microwave power sensors were used to monitor the
percentage o f microwave power that was reflected from the sample.
a:
Figure 3-2 Photograph o f Microwave Near Field Microscope with separate
transmitter and receiver locations.
The reflected microwaves were also measured using a horn antenna.
In this
configuration the probe tip acted as the transmitter but the horn was the receiver. This
configuration is depicted in Figure 3-2. A microwave diode detector measured a voltage
proportional to the amount o f microwave energy reflected from the sample.
The
microwave detector performance is shown in Figure 3-3 as a plot o f diode detector
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
voltage versus transmitted microwave power as measured by the power sensors.
Experiments were conducted using the probe as both transmitter and receiver as well as
using the probe as the transmitter and the horn as the receiver.
4 .0 -
╗'
3.0 -
>
2.0
-
0.0
-
>
0
20
40
60
80
P e r c e n t o f R e f l e c te d M ic r o w a v e P o w e r
Figure 3-3 Microwave diode detector voltage versus microwave power
79
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100
Samples
To investigate the material's conductivy as a contrast mechanism for resisitivity
imaging, measurements were made on samples o f known conductivities. Several
conductive samples were purchased from Goodfellow with thicknesses much greater than
the electromagnetic skin depth at our operating frequency o f 9.35 GHz. These samples
included aluminum, copper, brass, niobium, and titanium with the following resistivities.
Material
Resistivity in /jfi - cm
Copper
1.69
Aluminum
2.67
Brass
6.2
Niobium
16
Titanium
54
Table 1 Resistivity o f samples
All samples were two millimeters thick and mounted in epoxy for ease o f handling and
polishing. All samples were diamond polished using the Buehler system to achieve a flat
surface. Two samples simulating conductivity variations were also investigated. One
sample was an aluminum brass interface that had been mechanically joined and used in
near field optical microscopy work. The other sample was created by cutting the brass
sample in half and soldering the two pieces together. This sample was also polished flat.
Apparatus and Instrumentation
The microwave near field microscope was modeled after optical scanning near
field microscopes. In the optical case, light emanates from a fiber and then reflected light
is collected. In our design, a sharp probe tip is used to transmit microwaves. In the first
80
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configuration the sharpened tip is also used to receive the microwaves reflected from the
sample's surface. In a later configuration, a microwave horn antenna is used to collect
microwave energy reflected from the sample's surface.
Probe tips were constructed from electrochemically etched tungsten wire. Wire
pieces several millimeters long were etched using a potassium hydroxide solution. Tips
were then placed in an optical microscope to investigate the sharpening process.
Photographs o f the tips were taken so that measurements o f the probe tip radius could be
made. Selected probe tip radii ranged from 300 to less than 10 microns. Sharpened tips
were then glued into 22 gauge hypodermic tubing using epoxy. This size tubing fit
snugly into an SMA (small miniature adapter) jack to jack connector. This arrangement
allowed changing probe tips without re-using the SMA connector. The probe assembly
was then viewed under the microscope again to ensure the sharpened tip was not
damaged and that the wire was placed straight in the connector. The SMA connector was
then attached to an adapter assembly that connected a microwave SMA to a type N
connector and then to a TE waveguide. The adapters and the sharpened tip comprised the
probe assembly. Adapters were joined to a waveguide and fixed in place with waveguide
holders to prevent movement. The probe assembly remained stationary in our
configuration while the sample moves.
The microwaves were delivered to the probe assembly utilizing transverse electric
(TE) waveguides. Types o f waveguides included passive straight and curved sections
and two 10 dB directional couplers to sample transmitted and reflected power from the
probe. Power sensors were used to measure the power. For this work, we employed
Hewlett Packard 8618 sensors. A Hewlett Packard 938A microwave power meter
81
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measured the transmitted and reflected power yielding a percentage o f reflected power
via GPIB interface to the data acquisition system. In the configuration with separate
transmitter and receiver, a microwave horn collected microwaves transmitted from the
probe tip. The horn antenna was placed so that the center line o f the horn opening was
parallel to the sample surface. The horn was connected to a section o f waveguide that
contained a diode detector. A NP23 diode was placed in the detector yielding an output
voltage proportional to the intensity o f microwaves. This diode detector behaved much
like an optical square law detector. Measurements were made by adjusting the output of
the microwave source to determine a linear region o f the diode detector. A digital
voltmeter recorded the diode?s output. A GPIB interface was used to record the voltage
via the data acquisition system.
The microwave source used to deliver microwaves to the probe assembly was a
9.35 GHz IMP ATT diode. An isolator was placed between the source and the waveguide
system to prevent reflections from interfering with the source.
The microwave components used to deliver the microwaves to the probe tip as
well as the probe assembly were secured to a vibration free surface (floating optics
bench). The waveguide and waveguide holders were securely fastened to the table to
prevent shifting during measurements. A stage was constructed to hold the epoxy
mounted samples. The stage was then attached to a Newport translation stage with a
micrometer for coarse sample movement on the order o f tens and hundreds o f microns.
This stage permitted changing samples underneath the probe without disturbing the probe
assembly. A three axis translation stage was also mounted to the coarse stage. The axes
o f the three dimensional translation stage had stepping motors in place o f manual
82
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micrometers to allow for automated micron and submicron movement. During our
experiments we noted that an even finer motion control was needed in order to detect
some near field transitions. The motion control o f our microwave near field microscope
was later modified to include a piezoelectric scanning tube. The scanning tube was
constructed with Macor? mounting plates on the top and bottom o f the tube. The
scanning tube assembly was calibrated using optical techniques to move at a 13.7 nm per
volt increment. A DC voltage source was automatically controlled to supply increasing
amounts o f voltage to produce nanometer step increments.
83
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Chapter 4
Results and Discussion
Field Equations fo r VED and HED above a H alf Space
In Chapter 2 Maxwell's equations were solved for the electromagnetic field at an
observation point generated from a VED and HED above a half space. These solutions
provided field equations that were not limited to the far field. These field equations can
be used to investigate the behavior o f the electric field when the observation point is very
close to the dipole i.e. in the near field.
The field equations at an observation point for a VED above a half space are
given in equations 2.106-108. The field equation for the z-component of the electric field
for the VED above a half space is repeated here for reference.
~ '-d ) '( ik :
2
r,
rr
3
k,r*
where
2k? {
k,p
84
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3/ '
The first line o f the equation represents the direct field, the second line represents the
image field, and third line represents the surface wave field. The direct and image field
contributions depend on the observation distances, rt and r , , from the observation point
to the direct and image dipoles. The direct and image fields also depend on the wave
vector in air, the region above the half space. k2(i.e. wavelength in air). The direct and
image field contributions do not depend on the material property o f the half space. The
field contribution from the surface wave terms also depend on the observation distances.
But the field contribution o f the surface wave term depends on the material property o f
the half space, fc,. Therefore, all the field values, direct, image, and surface wave,
depend on topography, but only the surface wave terms depend on the material property
o f the half space. These dependencies also hold true for the field equations for a HED
above a half space which are presented in equations 2.216 to 2.221.
These equations for the electric fields can be used to investigate effects o f
topography and material property changes to understand the probe and sample
interactions in microwave near field imaging. Researchers at Max Planck Institute and
Lawrence Berkeley identified the perturbation o f the electric field as the contrast
mechanism for microwave near field imaging. Max Planck researchers selected this
contrast mechanism based on their experimental microwave transmission measurements.
They concluded that a sample perturbed the electric field lines transmitted from a probe
to a detector. Researchers from Lawrence Berkeley also attributed their scanning tip
microwave near field microscope interactions to perturbation o f the concentrated electric
field lines present at their probe tip. They chose to model their probe tip as a conducting
sphere, essentially a monopole. They represented the probe to sample interactions as
85
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follows: the tip acting as a conducting sphere had a charge distribution , the dielectric
sample when placed in close proximity to the conducting sphere becomes polarized and
causes a redistribution o f the charged particles in the conducting sphere. Their model
utilizes image charges and Coulomb's law. The Lawrence Berkeley researchers
acknowledged that their model is not valid for thin film samples due to the divergence o f
the image charges nor is it valid to model the probe to sample interaction for metal
samples. The Lawrence Berkeley researchers described the electric field as being
concentrated in a very small volume under the tip and almost perpendicular to the sample
surface. The electric field solutions presented in Chapter 2 do allow investigation for
metallic samples as well as dielectrics.
VED Conductivity Changes
To model the change in material properties o f the sample, the conductivity o f the
half space is varied for ranges representing conductors, semiconductors, and insulators
(dielectrics). Typically good metal have conductivities from 106 -1 0 8S/m while
semiconductors exhibit conductivities from 104 -1 O'* S/m and insulators/dielectrics have
conductivities below 1(T4 S/m. In Figure 4-1. the magnitude o f the z-component o f the
electric field is calculated for a VED on the boundary o f the half space. The
conductivity o f the half space is varied from 10"6 - 1 0 9S/m. The VED is operating at 10
GHz. The observation point was 5 cm from the dipole with an observation angle, theta.
o f 85 degrees as depicted in Figure 2-4. The change in the z-component o f the electric
field is most sensitive to conductivity changes from 10~? -1 0 3S/m. This
86
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o\
?
01
2 5x10 ~
0*1-
o?ooamxsmxoKX^?
2.0x10 -
-
? Total Field
~A? Direct Field
Image Field
-o? Surface Wave Field
1 5x10 -
MUM
S
jj
ij
?s
1 0x10 ~
5 0x10 ?
u
-3
3
00 _
10
10
10
10
10
10
10*
Conductivity in S/m
10 '
10?
10'
10
Figure 4-1 Magnitude o f the electric field for a VED on the boundary o f a half space with
various conductivities. The VED is operating at 10 GHz. The observation point is 5 cm
with an observation angle, theta, o f 85 degrees.
87
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248300
2000
1900
248290
1800
248280
1700
248270
1600
1500
248260
Mugiutude of / component of the lilcctnc Held in V/m
1400
248250
1300
248240
1200
1100
248230
? o? Total Field
?o ? Surface Wave Field
248220
248210
1000
900
800
700
248200
600
248190
500
248180
400
300
248170
200
248160
100
?oooa>
248150
10
ft
10 '
108
Conductivity in S-m
10
1010
Figure 4-2 Magnitude o f the electric field for a VED on the boundary o f a half space
with conductivities in the range for metals. The VED is operating at 10 GHz. The
observation point is 5 cm with an observation angle, theta, o f 85 degrees.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
conductivity range corresponds to values typical for semiconductors. To investigate
conductivity changes relative to metallic samples, the plot is expanded in Figure 4-2 to
show the total field magnitude for the z-component o f the electric field and the
contribution o f the magnitude o f the surface wave term. The magnitude o f the total field
is displayed on the left axis while the right axis displays the scale for the magnitude o f
the surface wave contribution. In these figures, the magnitude o f the total z-component
o f the electric field decreases as the conductivity decreases while the magnitude o f the
surface wave contribution increases as the conductivity decreases. At a conductivity o f
109 S/m the half space is essentially a perfect conductor. The surface wave term is
essentially zero at 109 S/m. This agrees with King's statement that when the half space
approaches a perfect conductor the wave vector in the half space approaches infinity such
that the surface terms with the wave vector in the denominator vanish and the entire field
is determined by the direct and image fields. Calculated wave vectors are shown in the
table below.
er
a [ S/m ]
*[/.]
Metal
1
l xl O9
6.28 x l 0 6 + 6.28x106i
Semiconductor
11.9
9x10^
7.23xl02 +4.91x 10"2/
Air
1
0
209.58
Calculated Approach Curve For VED above a Metal H alf Space
An approach curve shows how the field strength at an observation point changes
as the dipole approaches the surface. These curves help to understand the interaction
89
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between the probe to sample spacing and the field value. This qualitatively models the
effect o f topography. Figure 4-3 shows a calculated approach curve for a VED above a
metallic half space with a relative dielectric constant o f 1 and a conductivity o f
1x 10? S/m. This models the half space as a perfect conductor. The VED is operating at
10 GHz. The observation point is 5 cm with observation angle, theta, o f 85 degrees. The
dipole distance was varied from 10 meters to below a nanometer. In Figure 4-3 the
magnitude o f the z-component o f the electric field is plotted for distances above the
surface. The total field value is plotted along with the field contributions from the direct,
image, and surface wave terms. In this case the direct and image fields are the major
contributors to the total electric field. There is large change in the magnitude o f the
electric field from a meter to a millimeter. In our experiment length scales o f microns
and sub-microns are o f interest. Figure 4-4 expands the previous figure to illustrate the
field behavior from 10 microns to below a nanometer above the surface o f the half space.
In this figure the surface wave term shows no change below a micron. As previously
stated for perfect conductors the surface wave terms with k2 in the denominator vanish
and the total z-component o f the electric field is determined by the direct and image
fields. The magnitude o f the total z-component o f the electric field is carried out to the
fourth decimal place in order to see any change below 10 microns.
90
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250000 -
200000
? a ? Total Field
? a ? Direct Field
-
Image Field
?o? Surface Wave Field
>
150000 -
100000 S
50000 -
0-
1 0 13 1 0 12 10'" 10'1C 10* 10* 10'" lO* 10'3 10* 10'3 10'2 10''
10░
io ?
102 103
distance to surface in meters
Figure 4-3 Calculated approach curve for VED above a perfectly conducting half space
with conductivity o f 1x 109S/m and relative dielectric constant o f 1. The VED is
operating at 10 GHz with an observation distance o f 5 cm and
observation angle, theta, 85 degrees.
91
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124146.0 - i
20 434
248289 345-
124145 9 248289 340124145 8 -
248289 335-> 20 433
г
>
124145 7 248289 325-
|
124145 6 U
T?
C
J
U
\
124145 5 H
t
124145 4 -1
|
^
? . -m
.
10i: 10'" 10'?░ 10" 10* 10''
i m л^
lo"
10''
20 432
No o o a
<*&
a? Total Field
3? Direct Field
Image Field
3? Surface Wave Field
124145 3 -
2
124145.2 -
A
20.431
tk
124145 1 -
124145 0 -f -vi rm ap i *i i n i l
1 0 11
10 ':
10?
i r iin^
i m i^
1 0 10
i i i um)
1C4
I1O'*
(
H1HJ I It 111^ I HUM 20 430
10'"
10*
10'*
distance to surface in meters
Figure 4-4 Calculated approach curve for VED above a perfectly conducting half space
with conductivity o f 1x 109S/m and relative dielectric constant o f 1. The VED is
operating at 10 GHz with an observation distance o f 5 cm and observation angle, theta, of
85 degrees. These curves are expanded to show dipole distances from 10 microns to
below a nanometer.
92
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Calculated Approach Curve For VED above a Semiconducting H alf Space
In Figure 4-5. the same calculations are made for a semiconducting half space
with a relative dielectric constant o f 11.9 and a conductivity o f 9 x 1O?*S/m, values
typical for silicon. Again the dipole is operating at 10 GHz and the observation distance
is 5 cm with an observation angle, theta, o f 85 degrees. The dipole distance above the
surface is varied from 100 meters to below a nanometer and the magnitude o f the zcomponent o f the electric field is calculated. Figure 4-5 shows the magnitude o f the zcomponent o f the electric field when the dipole approaches a semiconducting half space.
This plot shows the total field value along with the contributions from the direct, image,
and surface wave terms. There is again a large change in magnitude o f the z-component
o f the electric field when the distance changes from meters to millimeters. Figure 4-6 is
expanded to investigate the behavior o f the magnitude o f the electric field values below
10 microns. The total z-component o f the electric field is shown in the inset. The left
axis is for the direct and image field values and the right axis shows the surface wave
field values. The change in magnitude o f the total field follows the changes in the surface
wave fields. These field values are carried out to only one decimal place in order to see
the field value change.
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
160000 -
140000 -
120000
-
u.
u
1V
100000 -
HI
2
o
tг
4)
O
? a? Total Field
? A? Direct Field
Image Field
60000 -
? л? Surface W ave Field
V
N
4>
fi
O
?0o9
.a
cra
10
s
20000 -
0-
-20000
?5
10': 10'' 10░ io' 10: 105
distance to the surface in meters
1 0 13 1 0 i: 10'? 1 0 1010* lO* 10* lO* lo ' 10?* 10
Figure 4-5 Calculated approach curve for VED above a semiconducting conducting half
space with conductivity o f 9 x 1(T4S/m and relative dielectric constant o f 11.9. The VED
is operating at 10 GHz with an observation distance o f 5 cm and
observation angle, theta, 85 degrees.
94
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124146 0 ?.
153004.5
124145 8
153004.0
Total Field
Direct Fidd
Image Field
Surface Wave Field
124145 6
124145 4
153003 5
153003.0
24145 2
124145 0 -
153002.5
124577 4-
153002.0
124577 224144 8
1530015
124577 0124576 8-
?3
124144 6
3
&
s
2
153001.0
124576.6153000 5
124576.4124144 4 124576 224144 2
153000.0
124576 0
152999 5
10
124144 0
152999 0
ii
.-10
10'
distance to the surface in meters
Figure 4-6 Calculated approach curve for VED above a semiconducting conducting half
space with conductivity o f 9 x 1(T* S/m and relative dielectric constant o f 11.9. The VED
is operating at 10 GHz with an observation distance o f 5 cm and observation angle, theta,
o f 85 degrees. These curves are expanded to show dipole distances from 10 microns to
below a nanometer.
95
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HED C onductivity Changes
The same calculations presented for the VED were executed for the HED above a
half space. The electric field at an observation point was calculated for a HED above a
half space while the conductivity o f the half space was varied to represent metals,
semiconductors, and dielectrics. The following figure shows the magnitude o f the zcomponent o f the electric field for a HED on the boundary o f the half space for various
conductivities. The HED is operating at 10 GHz. The observation distance is 5 cm with
an observation angle, theta, o f 85 degrees and an observation angle, phi, o f 0 degrees as
depicted in Figure 2-10. Much like the VED, the greatest change in the magnitude o f the
z-component o f the electric fkld occurs within the range typical for semiconducting
materials. There is little change in the magnitude o f the electric field for conductivities
typical o f metals. This is also attributed to the large wavevectors associated with metals
that cause the terms with the wavevector o f the half space, kx, in the denominator to go to
zero. For the HED the behavior o f the total z-component o f the electric field is largely
determined by the surface wave field.
For the z-directed electric field component o f the HED there is essentially no
measurable electric field when the observation angle, phi, is 90 degrees because the HED
is directed along the x-axis and phi at 90 degrees is along the y-axis. The waves
emanating from an x-directed dipole orientation are more easily seen with an observation
angle o f 0 degrees due to the polarization o f the transmitted field.
The dependency on
the observation distance with respect to phi is not seen with the VED because o f the
cylindrical symmetry associated with a VED.
%
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70000
? a ? Total Field
? A? Direct Field
60000
Image Field
? o? Surface Wave Field
50000
s
40000
e
30000
a
20000
m m mm
10000
0
10 5
io:
t
4
10
13
Conductivity in S/m
Figure 4-7 Magnitude o f the electric field for a HED on the boundary o f a half space
with various conductivities. The HED is operating at 10 GHz. The observation point is 5
cm with an observation angle, theta, o f 85 degrees and observation angle, phi, o f 0
degrees.
97
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
700
650
6 0 0 -A
550
.E
>
500
? a ? Total Field
? o? Surface Wave Field
450
400
350
I
I
300
E
250
-f
200
1
150
100
00(
X
.6
.9
Conductivity in S/m
Figure 4-8 Magnitude o f the electric field for a HED on the boundary o f a half space
with conductivities in the range for metals. The HED is operating at 10 GHz. The
observation point is 5 cm with an observation angle, theta, o f 85 degrees and observation
angle, phi, o f 0 degrees.
98
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Calculated Approach Curve fo r HED above a Metal H alf Space
A calculated approach curve for the HED above a metal half space is shown in
Figure 4-9. The HED is operating at 10 GHz, the observation distance is 5 cm, the
observation angle, theta, is 85 degrees and the observation angle, phi, is 0 degrees. The
relative dielectric constant o f the half space is one with a conductivity o f 1x 109S/m
representing a perfect conductor. The magnitude o f the z-component o f the electric field
varies greatly for dipole distances from a meter to 100 microns. This plot shows the total
electric field at the observation point as well as the contributions from the direct, image,
and surface wave terms. This plot is expanded to investigate the dipole distances from 10
microns to below a nanometer in Figure 4-10. The magnitude o f the total z-component o f
the electric field is shown in the inset o f Figure 4-10. The magnitudes o f the surface
wave terms are small because o f the wavevector in a perfect conductor; however, the
surface wave terms are increasing as the distances to the surface decrease. This is
opposite o f the behavior o f the magnitude o f the total field shown in the inset. The
magnitude o f the z-component o f the total electric field is decreasing as surface is
approached which is similar to the behavior o f the image field. The majority o f the total
z-component o f the electric field is determined by the direct and image fields for a HED
above a metal half space.
99
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90000 -
80000 -
>
7(XXX) -
g
-o
?o ? Surtace Wave Field
? a ? Direct Field
Image Field
? a ? Total Field
B
50000 -
c
gj
4(XMX) -
i.
v
N
5i
?a
20000 -
10000
-
0-1 (XXX)
10
13
10
10
10 ?3
distance to surface in meters
10
.7
10
10 I
10
,3
10
Figure 4-9 Calculated approach curve for HED above a perfectly conducting half space
with conductivity o f I x 109 S/m and relative dielectric constant o f 1. The HED is
operating at 10 GHz with an observation distance o f 5 cm and
observation angle, theta, 85 degrees and observation angle, phi, o f 0 degrees.
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IKK) -i
5.9321
1090 -
>
-3
1080 -
? Surface Wave Field
? Direct Field
Image Field
?? Total Field
5.9321
1070 5.9320
1060 60
5.9320
1050 -
|
i
30
1040 5.9319
10
1.
10
It
10
Ifi
10
10-<
10
10 *
10
1030
10
10 ?
10
10
10
10
10
10
10 '
distance to surface in meters
Figure 4-10 Calculated approach curve for HED above a perfectly conducting half space
with conductivity o f 1x 109S/m and relative dielectric constant o f 1. The HED is
operating at 10 GHz with an observation distance o f 5 cm,
observation angle, theta, 85 degrees, and observation angle, phi, o f 0 degrees. These
curves are expanded to show dipole distances from 10 microns to below a nanometer.
101
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Calculated Approach Curve fo r HED above a Semiconducting H alf Space
The calculated approach curve for a HED above a semiconducting half space is
shown in Figure 4-11. The magnitude o f the z-component o f the electric field is shown
for varying dipole distances above a semiconducting half space. The half space has a
relative dielectric constant o f 11.9 with a conductivity o f 9x 10~* S/m. The HED is
operating at 10 GHz. The observation distance is 5 cm with an observation angle, theta.
o f 85 degrees and an observation angle, phi, o f 0 degrees. The magnitude o f the total
electric field o f the observation point is shown in addition to the contributions o f the
direct, image, and surface wave fields. Again the magnitude o f the electric field varies
greatly between a meter and 100 microns. Figure 4-12 shows an expanded plot o f the
electric field when the dipole distance varies from 10 microns to below a nanometer
above the surface. For the semiconducting half space, the total z-component o f the
electric field increases as the dipole distance decreases while the magnitude o f the zcomponent o f the surface wave field decreases as the dipole distance decreases.
102
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90000 -
80000 -
>
70000 -
60000 -
? o? Surface Wave Field
?A? Direct Field
Image F id d
? ?? Total F id d
50000 -
40000 ST
V
1
1
i
30000 -
20000
-
10000
-
0-10000
10
13
10
10
10
10
10?3
10''
10
1
10
distance to surface m meters
Figure 4-11 Calculated approach curve for HED above a semiconducting half space with
conductivity o f 9x I O'4S/m and relative dielectric constant o f 11.9. The HED is
operating at 10 GHz with an observation distance o f 5 cm, observation angle, theta. 85
degrees, and observation angle, phi, o f 0.
103
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1100
3 6 8 4 7 .0
1090
?o? Surface Wave Field
?^ ? Direct Field
Image Field
?a? Total Field
1080
3 6 846 8
36 846 6
1070 -
>
1060
36 846 4
36850-
o
1050 -
8
36846 2
N.
1040
O
1
1030 -
368460
11020
368458
1010
-
11000
36845 6
10 ii
!0,J
10
10 '
10
10.*
10 '
10
distance to surface in meters
Figure 4-12 Calculated approach curve for HED above a semiconducting half space with
conductivity o f 9 x KT1S/m and relative dielectric constant o f 11.9. The HED is
operating at 10 GHz with an observation distance o f 5 cm. observation angle, theta. 85
degrees, and observation angle, phi, o f 0. These curves are expanded to show dipole
distances from 10 microns to below a nanometer.
104
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Measured Results
As reported in the literature the probe tip radius will affect the spatial resolution
for microwave near field imaging. Sharper probe tips are needed for better imaging
resolution so that smaller features can be imaged. Measurements showed that sensitivity
to topography also increased with sharper probe tips. Approach curves were measured
for a copper sample with three different probe tips. Measurements were made with the
microwave near field microscope configuration using the probe as both the transmitter
and the receiver. The percent o f microwave reflected power was measured as a function
o f the probe distance above the copper sample for three different probe tips. The copper
sample was two millimeters thick which was sufficiently greater than 0.66 microns, the
skin depth at the operating frequency o f 9.35 GHz for the microscope. The probe tip
radii were measured using optical micrographs and determined to be 300 microns, 150
microns, and less than 10 microns. Figure 4-13 shows the measured approach curves for
all three probe tips. For the sharpest probe tip, a step size o f 0.287 microns produced a
change in the percent o f microwave reflected power o f 16.5. For the 150 micron probe
tip. a corresponding change in percent o f reflected microwave power from 50 percent
reflected power to 35 percent reflected power required 12 microns o f travel. While the
300 micron probe tip needed 25 microns o f travel to change the percent o f reflected
power by 12. So although smaller probe tips provide better spatial resolution they are
also more sensitive to topography changes and require good probe to sample distance
control.
105
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60 -
52
**
w.
58
50
Percentage of reflected microwave power
48
46
44
42
48
40
?c ? Probe radius = 300 microns
46
38
Probe radius < 10 microns
x V
? * ? Probe radius = 150 microns
44
36
42
34
0
5
10
15
25
distance in microns
Figure 4-13 Measured approach curves for a copper sample with
three different probe tips.
106
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The directivity plots in Chapter 2 that were used to verify the field values solved
from Maxwell's equations indicated that the majority o f the reflected microwave power
was not reflected perpendicular to the surface. The microwave near field microscope was
then augmented to measure reflected microwave power from directions other than normal
to the surface by using a side hom antenna. This provided a microwave near field
microscope that could simultaneously measure microwave power reflected normal to the
surface and microwave power reflected in other directions.
Approach curves were measured with the augmented microwave near field
microscope for a brass sample. The brass sample was also two millimeters thick with a
resistivity o f 6.2 f г l - c m . The thickness is still much greater than the skin depth at 9.35
GHz. The probe tip used in these approaches had a radius o f curvature o f 150 microns.
Figure 4-14 shows percentage o f reflected microwave power measured by using the
probe as both the transmitter and the receiver and a diode voltage measured by the side
hom using an alternate receiver location as the probe approaches the surface o f the
sample. In this measurement, the percentage o f reflected microwave power decreases as
the probe distance to the surface decreases. But the diode detector voltage increases as
the probe distance to the surface decreases.
107
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t>() -|
I) 3 0
58 -
0 28
0 20
0 22
?=? Percent reflected microwave power (left axis)
?o ? [>odc voltage (right axis)
0 20
50 -
0 16
48 -
0 14
0
5
10
15
20
distance in microns
25
.
sample
surface
Figure 4-14 Measured approach curves for brass sample.
108
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I )iode Voltag<
Reflected Microwave Power
0.24
54 -
A line scan o f the aluminum brass interface sample was also measured with the
microwave near field microscope configured to use the probe as both the transmitter and
the receiver. Probe tip radius used for this line scan was approximately 40 microns. The
results o f the line scan are in Figure 4-15. There are two different values o f percentage
o f reflected microwave power for the aluminum surface and the brass surface that vary by
4 percent. There is also an interface area with varying percentage o f reflected microwave
power. To further investigate this sample profilometer measurements were made. A
profilometer was used to measure the flatness o f the aluminum brass sample. The Alpha
Step 200 profilometer manufactured by Tencor Instruments at the Applied Physics
Laboratory was used to measure the topography o f the aluminum brass interface.
Profilometer scans revealed the mechanically fastened interface to extend about 150
microns wide. They also determined that the area scanned with the MNFM had a
topographic variation o f about 1.5 microns between the aluminum and brass surfaces. It
is uncertain if the change in percent o f reflected microwave power was caused by the
change in conductivity between aluminum and brass or the 1.5 micron topographic
variation. The 150 micron wide gap is clearly detected in the microwave near field scan.
109
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Percentage of reflected microwave pow er
60
r-
46
44
40
0
50
100
200
150
N um ber o f 5 m icron steps
Brass
Aluminum
Figure 4-15 Measured line scan o f aluminum and brass interface.
110
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A brass solder sample was also investigated. These measurements were made
with the augmented microwave near field microscope. Measurements o f reflected
microwave power using the probe as a transmitter and receiver were obtained
simultaneously with measurements o f the reflected microwave power using the side horn
as a receiver. The reflected microwave power received by the side horn was measured by
a diode. These measurements were made using a probe tip o f approximately 40 microns.
A line scan is shown in Figure 4-16 with one micron step sizes.. This figure shows that
the percentage o f microwave reflected power recorded from the probe tip was noisy and
had a small change in reflected power across the interface. The measurement o f the
diode voltage did resolve the solder interface. By making several line scans, a contour
plot o f the brass solder interface was constructed to demonstrate the i m a g i n g capability o f
the microscope. The image in Figure 4-17 shows the contour plot o f the reflected
microwave power as detected by the probe . This image has low contrast so that it is
difficult to ascertain the solder presence. However, the contour plot o f the diode voltage
as measured by the separate receiving hom antenna shown in Figure 4-18 clearly resolve
the brass solder interface.
Ill
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distance in mm
Figure 4-16 Measured line scan of brass solder interface.
112
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Brass Solder Interface Probe 30
microns
Figure 4-17 Contour plot o f percent reflected microwave power for
the brass solder interface with area believed to be the
solder surrounded by a dashed box.
113
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Brass Solder Interface Probe 30
0.09
0.085
0.08
microns
0.075
0.07
0.065
0.06
0.055
microns
Figure 4-18 Contour plot o f diode voltage for the brass solder interface with the area
believed to be the solder surrounded by a dashed box.
114
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In summary the electric field equations showed that all the contributing fields,
direct, image, and surface wave fields, depend on topography but only the surface wave
field contribution depends on the material property o f the half space Le. the sample. For
a VED on the boundary o f the half space the magnitude o f the z-component o f the total
field decreases as the conductivity decreases but the magnitude o f the surface wave field
increases as the conductivity decreases. For a HED on the boundary o f a half space both
the magnitude o f the z-component o f the total electric field and the contribution o f the
surface wave field increases as the conductivity decreases.
The calculated approach curves showed that the VED above a perfectly
conducting half space had no measurable change in the magnitude o f the z-component of
the total electric field nor did it have a measurable change in the magnitude o f the surface
wave field contribution as the distance to the surface decreased below 10 microns. The
VED above a semiconducting half space showed the magnitude o f the z-component of
the total field and the surface wave fields increased as the distance to the surface
decreased below 10 microns. The HED above a perfectly conducting half space showed
the magnitude o f the z-component o f the total electric field decreased as the distance to
the surface decreased below 10 microns. This corresponded to the behavior o f the image
field. The surface wave field showed no measurable change until magnitudes to the
fourth decimal placed were examined which showed a slight increase in magnitude for
decreasing distance to the surface. The HED above a semiconducting half space showed
the magnitude o f z-component o f the total electric field to increase as the distance to the
surface decreased. This behavior corresponded to the direct field. The magnitude o f the
115
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surface wave term decreased as the distance to the surface decreased below 10 microns.
In the measured approach curves, the diode voltage increased as the distance to the
surface decreased while the percentage o f reflected microwave power decreased as the
distance to the surface decreased. This type o f behavior is qualitatively represented in the
HED over a metal half space but the shape o f the curves is distinctly different. This
suggests that the dipole model over a half space does not account for all the mechanisms
in the probe to sample interaction. In the line scans o f the aluminum brass interface and
brass solder interface, differences in the scans are recognizable but it is difficult to
attribute the changes to topography or conductivity without a measurement method to
ensure constant probe to sample spacing. The diode voltage measured from brass solder
sample can be used to create images o f the sample but it is unclear if these are
topography induced changes or conductivity induced or a combination o f both.
116
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NOTE TO USERS
Page(s) not included in the original manuscript
are unavailable from the author or university. The
manuscript was microfilmed as received.
117
This reproduction is the best copy available.
UMI
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 5
Conclusions
Solving the electromagnetic field equations at an observation point for a dipole
above a half space without restricting the solution to observation distances within the far
field allowed investigation o f changes in the magnitude o f the electric field at observation
points in the near field o f the dipole. These field equations were then used to investigate
the behavior of the magnitude o f the z-component o f the total electric field and the
contributing fields, direct, image, and surface wave, when the conductivity o f the half
space was varied to simulate conductivity changes in a sample. The behavior o f the
magnitude o f the z-component o f the electric field was also investigated when the dipole
distance above the half space was varied to simulate topography changes. A microwave
near field microscope was constructed to compare measurements to the calculated field
changes.
The field equations showed that all the contributing fields, direct, i m a g e , and
surface wave, depend on topography but only the surface wave field depends on the
material property o f the half space, i.e. the sample. This dependence is associated with
the wavevector o f the half space in the field equations. The surface wave terms are the
only terms containing the wavevector o f the half space. The wavevector is in the
denominator o f this field term as well as the argument used in the numerical calculation
o f the Fresnel number. The wavevector associated with conductivities o f metals is large;
therefore; the surface wave terms for metals are small. So conductivity changes in metals
produce small changes in the magnitude o f the total electric field. Wavevectors
118
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associated with conductivities o f semiconductors are smaller; therefore, the surface wave
terms are larger. The conductivity changes in semiconductors produce larger changes in
the magnitude o f the total electric field. For the VED on the boundary o f a half space the
magnitude o f the z-component o f the total electric field decreased as the conductivity
decreased but the magnitude o f the surface wave field increased as the conductivity
decreased (since the wavevector decreased). For the HED on the boundary both the
magnitude o f the z-component o f the total electric field and the surface wave field
increased as the conductivity decreased.
These field equations were also used to calculate approach curves for the different
dipole configurations above conducting and semiconducting half spaces. For the VED
above a perfectly conducting half space, the magnitude o f the z-component o f the total
electric field and the contributing surface wave field showed no change for dipole
distances below 10 microns. The VED above a semiconducting half space showed the
magnitude o f both the total z-component o f the electric field and the surface wave field
increased as the dipole distance decreased below 10 microns. For the HED above a
perfectly conducting half space, the magnitude o f the total z-component field decreased
as the dipole distance decreased below 10 microns. This behavior followed the image
field. The surface wave field showed a slight increase as the dipole distance decreased
below 10 microns once the magnitudes were expanded to the fourth decimal place. The
HED above a semiconducting half space showed the magnitude o f the total z-component
o f the electric field increased as the dipole distance decreased below 10 microns. This
behavior followed the magnitudes o f the direct field. The magnitude o f the surface wave
field contributing to the z-component field decreased as the dipole distance decreased
119
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below 10 microns. In the measured approach curves, the diode voitage increased as the
probe height decreased and the percentage o f reflected microwave power decreased as the
probe height decreased. This qualitatively follows the same behavior as the HED above a
perfectly conducting half space but the shapes o f the curves are distinctly different.
Changes in the measured diode voltage can be used to create images but it is unclear if
these images are induced by topography, conductivity, or a combination o f both.
Future Directions
The dipole above a half space model does not account for the radius o f curvature
o f the probe tip. There is a definitive effect in the experimental measurements of
approach curves with different probe tip radii on the same copper sample that is not
represented in our dipole over a half space model. The literature also reports a definitive
association between probe tip radius o f curvature and imaging resolution. The dipole
above a half space model represents the tip as a unit dipole. An extension o f our model
may be used to investigate the effects o f probe tip radius by summing the electric field
contributions o f dipoles with many heights above the surface to simulate the probe's
radius o f curvature. This concept is illustrated in the following figure.
d* cfc
d?
Figure 5-1 Dipole model extended to represent the effect o f probe tip radius.
120
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By representing the samples as half spaces, the field equations are not valid for
investigating thin films. This model assumes that the thickness o f the sample is greater
than the penetration depth. This would not be the case for measurements on thin films
particularly metallic thin films. The penetration depth and surface resistivity would have
to be considered. Additionally, surface waves created at the interface o f the bottom o f a
thin film and the substrate would also have to be considered. Effects from the waves
reflected from the interface o f the thin film and substrate would also need to be included
in the field equations. The model could be extended by considering a multilayer sample
underneath the dipole.
121
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REFERENCES
University o f Maryland Researchers:
Vlahacos, C.P., R.C. Black, S.M. Anlage, A.Amar, and F.C. Wellstood, Near Field
Scanning microwave microscope with 100 micron resolution. Applied Physics Letters,
69(21), 18 November 1996.
Anlage, S.M., C.P. Vlahacos, S. Dutta, and F.C. Wellstood, Scanning Microwave
Microscopy o f Active Superconducting Microwave Devices, IEEE Transactions on
Applied Superconductivity,7(2), pp. 3686-3689, June 1997.
Steinhauer, D.E., C.P. Vlahacos, S.K. Dutta, F.C. Wellstood, and S.M. Anlage, Surface
resistance imaging with a scanning near field microwave microscope. Applied Physics
Letters, 71(12), 22 September 1997.
Steinhauer, D.E., C.P. Vlahacos, S.K. Dutta, B.J. Feenstra, F.C. Wellstood, S.M. Anlage.
Quantitative Imaging o f sheet resistance with a scanning near field microwave
microscope, Applied Phy !rs Letters, 72(7), 16 February 1998.
Vlahacos, C.P., D.E. Steinhauer, S.K. Dutta, B.J. Feenstra, S.M. Anlage, and F.C.
Wellstood, Quantitative topographic imaging using a near field scanning microwave
microscope. Applied Physics Letters, 72(14), 6 April 1998.
Max Planck Researchers:
Keilmann, F., D.W. van der Weide, T. Eickelkamp, R. Merz, D. Stockle, Extreme
Subwavelength resolution with a scanning radio frequency transmission microscope.
Optics Communications, 129(1996), p. 15-18, 1 August 1996.
Knoll, B.,F. Kielmann, A. Kramer, R. Guckenberger, Contrast o f Microwave Near Field
Microscopy, Applied Physics Letters, 70(20), 19 May 1997.
Lawrence Berkeley Researchers:
Wei, T., X. -D. Xiang, W. G. Wallace-Freedman, and P.G. Schultz, Scanning tip
microwave near field microscope, Applied Physics Letters, 68(24), 10 June 1996.
Lu, Y., T. Wei, F. Duewer, Y. Lu, N. -B. Ming, P.G. Schultz, X. -D. Xiang,
Nondestructive Imaging o f dielectric constant profiles and ferroelectric domain with a
scanning tip microwave near field microscope, Science, Volume 276(5321), pp. 20042006. 27 June 1997.
Gao, C.. T. Wei, F. Duewer. Y. Lu, X. -D. Xiang, High spatial resolutions quantitative
microwave impedance microscopy by a scanning tip microwave near field microscope,
Applied Physics Letters, 71(13), 29 September 1997.
122
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Takeuchi I., T. Wei, F. Duewer, Y.FC Yoo, X-D. Xiang, V.Tabyansky, S.P. P al G.J.
Chen, and T. Venkatesan, Low temperature scanning tip microwave near field
microscopy o f YBaCuO films. Applied Physics Letters, 71(14), 6 October 1997.
Gao, C. and X-D Xiang, Quantitative microwave near filed microscopy o f dielectric
properties. Review o f Scientific Instruments, 69(11), pp. 3846-3851, November 1998.
Konaka. T., M. Sato, H. Asana. and S. Kubo, Journal o f Superconductivity, 4(283), 1991.
Duewer, F., C. Gao, I. Takeuchi. X-D Xiang, Tip sample distance feedback control in a
scanning evanescent microwave microscope, Applied Physics Letters, 74( 18), 1 May
1999.
The Hebrew University Researchers:
Golosovsky, M. and D. Davidov. Novel millimeter wave near field resistivity
microscope. Applied Physics Letters, 68(11), 11 March 1996.
Golosovsky, M., A. Galkin, D. Davidov, High spatial resolution resistivity mapping o f
large area YBCO films by a near field millimeter wave microscope, IEEE Transactions
on Microwave Theory and Techniques, 44(7). pp. 1390-1392, July 1996
Lann A. F.. M. Golosovsky, D. Davidov, A. Frenkel, Combined millimeter wave near
field microscope and capacitance distance control for the quantitative mapping o f sheet
resistance o f conducting layers. Micron, 73(19), p. 2832, 9 November 1998.
Golosovsky, M., A. Lann, D. Davidov, A millimeter wave near filed scanning probe with
an optical distance control Ultramicroscopy, 71, pp. 133-141, 1998.
Additional References:
Abramowitz. M. and I. Stegun, Handbook o f Mathematical Functions, Dover
Publications. New York, 1972.
Ash. E.A. and G. Nicholls. Super resolution aperture scanning microscope. Nature (237).
pp.510-512, 1972.
Han. H.C . and E.S. Mansueto, Thin Film Inspection with millimeter wave re Hectometer,
Research in Nondestructive Evaluation, 7, pp. 89-100, 1995.
Hayt, Jr., W.H., Engineering Electromagnetics, 5th Edition, McGraw-Hill New York.
1989.
Hecht. E.. Optics, 2nd Edition, Addison -Wesley Publishing Company, Reading, MA,
1987.
123
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
King, R.W.P., M. Owens, T.T. Wu, Lateral Electromagnetic Waves Theory and
Applications to Communications, Geophysical Exploration, and Remote Sensing,
Springer-Verlag. New York, 1992.
Miner, G.F., Lines and Electromagnetic Fields fo r Engineers, Oxford University Press,
New York. 1996.
Paesler, M.A. and P.J. Moyer, Near Field Optics: Theory, Instrumentation and
Applications, John Wiley and Sons. 1996.
Shackelford, J.F., Introduction to Materials Science fo r Engineers, 2nd Edition,
Macmillan Publishing Company, New York, 1988.
124
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VITA
Jennifer Schlegel was bom in Baltimore, Maryland on November 1, 1970. She
graduated from Sevema Park High School in 1988. She satisfied the degree requirements
for a Bachelors o f Science in Engineering Science and Mechanics from the Virginia
Polytechnic Institute and State University in December o f 1992. She then worked at
Baltimore Gas and Electric in the Nondestructive Evaluation Unit until she matriculated
at The Whiting School o f Engineering's Department o f Materials Science and
Engineering at the Johns Hopkins University in January o f 1994. She received a Masters
o f Science in Engineering from the department in May 1996. She is currently employed
as a Principal Technical Staff Member at AT&T where she began working in June o f
1999.
125
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
he x component o f the
electric field, which requires the electric field in region 1 to equal the electric field in
region 2 at the interface where z = 0. provides the following expression for the constant C
C = C\+
( A:.2 - г 2)
K1,
2y'k]
? e'T'J
Substituting into 2.161 gives
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.162)
" лsiny,;
? ------ e╗?'
л
гIt= C ,e "
1y A
(2.163)
for 0 < : < d and
cov?(k{-f-) .
C , ----------- --------siny,rf e '
'M f
(2.164)
for d < z .
There are four remaining boundary conditions (continuity in the y and z
component at the interface where z=0) as noted in equations 2.154. 2.155. and the y and z
components o f equation 2.156. Using the conditions in equation 2.154 and the y
component o f 2.156 at r = Ogives the following equations:
A
k: - s
1
k ^2 - 'T
rjC2 + o y ,C ,) =
(2.165)
? Y t'i ~
k : - г 2 co
7ic z
co
oyyxC,)
1
i + M., ( k{ ~
)e
'
<r\J
. (2.166)
Rearranging to solve for the constants in matrix form
Yi
Y\
l
<? + l 2 _ -2
V *1
k 2 - 4:2
v * r-c :
(
\
Y
\
k
Y,
2 ? ^~2
k : ? c?
2
I
1
k;-g2
- +
/
1
k ; - c 2r '
-
4nc\-
?9
f 1v 1k*12
Yik-,
Cl
+ - . ? . ?
k 2 - g 2 k ; - g 2 co
1
r
l
гijC\
i : _ e2
co
*2
*5
i
{ k ; - g
2
^
k ; - g 2\
-
0
(2.167)
,yJ
= f.i? e ?1
(2.168)
гrj ' c ;
c
Y\k\ ^ Y2k 2
{k?-?
kg-g
=
0
=
tr.J
M?e '
. CO .
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.169)
Solving for the determinant o f the 2 X 2 matrix gives
?\{N
d etl ] = 7 n
(2.170)
where
M =Y\+Yi
(2.171)
N = k~y 2 + k:y,
(2.172)
and the constants are
<-\ =
C\ =
MX
-w '
(2.173)
M?e ?1
MX
(2.174)
These constants can be substituted into the above equations to provide the general
integrals for the components o f the electromagnetic fields in the two regions with a
horizontal electric dipole located above the interface. The following equations are the six
components o f the electric and magnetic fields.
In region 1 with 0 < r < d
r
г ? = -4 K
p r
r
---------------------------------e ' +?---- T -sin v .r
MX
i/X'
e '
siny.:
ii
X
X
Yl
N
(2.175)
ely,J
y
cos,yik:
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.176)
(2.177)
X
x
4*-ny)
f lr2 - I
v
X
X
B? = ^ j f j f f d n e ?' * " ?" cos y x: +
Ax
?
X
-X
X
Ax
/
X
?X
2
\
mn
(2.178)
* :V ~ ki ( k ; - 4 2 + / , / : )
MV
' * n\ I
B,
t
~x
.
e
A/
s in /,r
iy|
,r _.
e
(2.179)
\
e
(2.180)
In region 1 where d < :
X
X
~CQM? j d g f d r ? 1'-"'?" M t i - F b M Z - F )
Bu =
Ax -x -r
mv
e
x
x
x
k,: s : .
+ - ^ r s,n ^
e r'
X
x
sin y xd
e'
iyxk{
'
(2.181)
IY~
(2.182)
*v) r h . ?y<J I sin /i< ^
e 1
V
A| /
IV *
x
-i-2
?7' i
s ? = ^
(2.183)
MX
(2.184)
/
* ? - ;г = K
W
1"
x
5 ,,=
7>) /sin/,c/
'r,*/
Atfv
x
sin ^ c/
Z* * n y )
4;r:
JW \ d n n e ?{l
-x
?x
V
/ / ,I
v
e?
a/
(2.185)
/ y,r
tr'
(2.186)
In region 2 where - < 0
M*2 ~г2 )+M*r-г2)
.V/A'
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.187)
о
г ='
T-
4 x~ _il rfг W
*
1* " ?-
jv
( 2 . 188 )
t
(2.189)
л
X
* r-^ ;
(2.190)
X
X
r;
?
X
s : (^ : ~ * : )
A/ +
?
X
f. vc f
.j
(2.191)
'I4 * * n y) г
8 =-? = 7 f r J < ^ / ^ W * ?
?X
A/-V
-x
A/
(2.192)
Converting to cylindrical coordinates
A2
E^ = - p4/rC
T T cos<(,
Y\
(2.193)
o' ~
-T'l
|(*,V? ( i p ) - ^ - [ / , (^j) + V, (Ap )]l-< P ' :-JXdA
o
"if * ? [ > . . ( ^ ) + -A
O' ^
) Yi
^
*-/l
(-lp )--A (-Ip )]
e r'{:*J)AdA
4;r&
(2.194)
(2.195)
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.196)
B,a = - ? cos <j>
'*
4n
(2.197)
B.. = ^ s i n < * f ( e ^ - Pe'r'{:"J'\ ? J , U p ) X 2dX
An
o
>Yx
(2.198)
where
p = Yi ~Yx = ^2 ~Y\
M
^ ,+ x ,
_ k ; y 2 - k ; y x _ k{y 2 - k ; y x .
N
(2.199)
+ k^Y\
For equations with two signs, the upper sign is for z > d and the lower is for 0 < z < d .
For region 2 the expressions are
-p
= ? ? cosdi
4K
( 2 .200 )
г ,. = ?? sinй е ^ \J? (Xp) - _/; ( ,lp ) ] + г & [ ./? (Ap) +
4n
(A p )] y ^ 'X d X
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( 2 . 201 )
=-^ c o s i*
iV
J,
(2.202)
(2.203)
l(r'J-r'-:)AdA.
B, = ^ c o s
*' 4/r
(2.204)
N
B2:
= ^ s i n * J - i : J , ( k p ) e ?(^
U
2d k
(2.205)
Following the integration in King's chapter 5 sections 5 and 6, the integrated
expressions for the electromagnetic fields in region 1 from a horizontal electric dipole in
region 1 are as follows.
J)
( k
?>
v ri
(
+
/ ^ tk.r,
; e " +
ri y
/
,\
z +d (
3_ '
~ 2p'
I
K P
\
( r - J i V f .. ?>
iky
3ky
3/
v ri )
rr
' ?I y
z + dJ
\
I r: J
-
/
v ri
/A,2 3г,
r*
V r,
.
uk.r.
3/ N
e1*r,: /
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.206)
: .J )
E,t ( p ^ , z ) = - ^ Ts in ^ { k 2h ( p \k ,.k 2)e?k'->
?
r:
/
+/
J \
Z+d
I
J
r2
3/
rz
r2
\
e
ik .r ,
v ri
j
( ik ;
3k,
I P
Ip ?
*f
^3
+
11
I
.
3k,
(2.207)
r,;
,'V :
r/ ,
E\.-(P'&': ) =
(P -k ^ k 2) e?k:tle?k'{ J) ~
\
p
, rl , I n
/
\
+
' ik~
JV rx
1
>I
/
z-d
y
5/ )
ikl__ 3 ^ l_ 3/^
r2 ) V r,
'T
00
1 ik{
+?
">
/г,:
3k,
r~
A:,
p-
-k ,p s
3 /'
r,} J
/
(2.208)
ik.r*
r,
_p-
+
~vS
+
?i
j|-^
1
i
v r:
I r2 AV A
2p
?>
I
rx
) <
ri
rr J
e ik.r.
'? +
r;*
(2.209)
^3
+
11
?'a
i
11
' i k;
r,:
I
r2
J
*<?
,
r:
1e
tk,r
1?
r 2* J
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
NOTE TO USERS
Page(s) not included in the original manuscript
are unavailable from the author or university. The
manuscript was microfilmed as received.
72
This reproduction is the best copy available.
UMI
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the half space, the subscripts I and 2 must be interchanged and the following
substitutions must be made: z = - r in region 2 with E,. =
and B. = - B . . In
addition, the associated angle must be changed such that <j>'= -<f> with E4- = ~Ef and
fl, = - B 0. The configuration depicted in Figure 2-9 involves a negative image dipole
rather than a positive and causes the coefficients Q +1 and P -1 instead o f Q - \ and
P +1 where P and Q are defined in 2.199. The positive image dipole configuration is
shown in Figure 2-10.
Horizontal
Electric
Dipole
Region 2
Observation
Point
jr.
Image
Dipole
Region 1
Figure 2-10 Problem setup for a positive image dipole for a
horizontal electric dipole above a half space in region 2.
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The electromagnetic field equations are altered for the observation point in region 2
above the half space when the horizontal electric dipole is also in region 2 above the half
space as follows:
?>
E2 A P '0 ''= l = - ^ f cos<t>'
> 2
/. -
1
2
2/
1
rr
2k,
ik*r*
2k;
2
r: J
l r:
/
г,r;;
( P-&'?*') =
-e
r /*,
f \
A:,' r.
A:, KP)
sin
z' + d ' V i k ,
r,
r:
2/
- ;
r,- L
k,r,
r2
3/ N
J
I
n
]
2k {=' + d ' \ f ik,
3/
i *
k,r, )
kx
'i
V
I r. J
1^
(2.216)
7T -i/'.
V
ik,
r,
\e
1
r~
e ik,r,
??
k,rx
ik,
i
1
2
I
r
1
r2
, V /'A:-,
r: y r,
: ?+ d
v
3
.
J I
3
VT
2k; /A:,
+r.
k?
r ? + c/O
f /A2
k2rx;
L
1
- e - <?2 +i- '
r, k,r.
2/
ikyr.
e -
3
r,:
(2.217)
3/
A:,r,'
2ikt ( r;' \
k\P < p \
k,r,
/
v'i yv
+e
ik.r.
2k,
k,
?
= -e
{ p \ ( z ' + d , \ / ik,
r,
\ r2 7 v
r:
y
r
^ ik ,
3
2
r2
1
r.
.
ik,
3
\ r\
ri
3/
k^r: y
3/
i I
*2r2
(2.218)
I 7T
? ?
\ r: y\ r:
rf y
k
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f z ' - d ' \ '/it,
B,
4/r
I
rt
1'
' /*,
ik.r.
?> - e - - f Z>+ ^ 1
rr )
I r: J I r:
J I r\
1'
^ J
(2.219)
?
+?
2/
<A,r,
??
r2
k2r2
/A:^ ( r; ^
kxp P~ )
/A,r
e ? 'F (P 2) +
( z ' + d '^
v
'/it,
f
1'
-< ?
*>
I
ri
J I r\
ri
<*,r.
* *
)
r\
-
J
/ .
ik*,
3
v r:
r:
( + t/,l ' lit,
I
JI
3/
1 '
z ?
r:
r :
?)
^ )
( 2 .220 )
/
+2e
;A,r,
P
\
\b
U
j_
/V
J/
fc,/v
\ e ?k'-
^7Z
( : ' +* )
I
ri
3_
k;r*
'lib.
J v r:
3
r2
e ? 'F (P 2)
1^
/
** P
?г
\
(/A ,
1)
i
?^1
B2:\p .< p \z') =
\
P ( ik,
|
r
tk*r.
k*r*
.r* j
k U r*
k*rk. \y /
ik*
r, r;
tk * ry
1
-^1
2k*
3/ \
k2r2 ,
( 2 .2 2 1 )
r =' + J '^ r ik2 _6_
?>
r,
r,?
V
r:
y
.
\
^ 2 r/
J
15/
where
P =S,
S 2 + Z '+ P ' '
R
C? _? ^:r:
╗3i
? 2*r
( 2 . 222 )
(2.223)
lr 2 - ?
Z' = ^ ~
2k.
R = kM
2k;
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.224)
(2.225)
2k,
P, =
k lr 21 klr, + ksk;z' + k,kid' ^
kip
2kf
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.226)
(2.227)
Chapter 3
Experimental Procedure
This chapter describes the investigation o f contrast mechanisms for subн
wavelength microwave imaging o f dielectric constant variations in materials with high
spatial resolution. A microwave near field microscope modeled after near field scanning
optical microscopes (NSOMs) was constructed. Samples with known conductivities
were used to characterize the microscope performance. Experiments were conducted to
investigate the effect o f the tip to sample spacing indicative o f topography by studying
approach curves. Experiments were also conducted to investigate the changes in material
conductivity.
The microwave near field microscope designed for this investigation used a subн
wavelength probe tip to scan a sample?s surface.
The probe tip was positioned a few
microns above the sample's surface. Samples are placed on motion control devices. The
interaction between the probe tip and the sample surface was monitored by changes in the
|
I
HP 43SA
PniKrt
?л -
?
o 47" OD Semi-rigid
Coax Cable
Coax
Adapter
[MPA IT
Diode Microwave
Source
SMA Connector
Probe
Sample
to dB Directional Couplers
Motion
Controller
Computer
Figure 3-1 Microwave near field microscope with probe acting as
both transmitter and receiver.
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
percent o f reflected microwave radiation as sensed by microwave power detectors and
diode detectors. At low power these sensors operate much like square law detectors. In
the configuration depicted in Figure 3-1 the probe acted as both the transmitter and
receiver o f microwave energy. Microwave power sensors were used to monitor the
percentage o f microwave power that was reflected from the sample.
a:
Figure 3-2 Photograph o f Microwave Near Field Microscope with separate
transmitter and receiver locations.
The reflected microwaves were also measured using a horn antenna.
In this
configuration the probe tip acted as the transmitter but the horn was the receiver. This
configuration is depicted in Figure 3-2. A microwave diode detector measured a voltage
proportional to the amount o f microwave energy reflected from the sample.
The
microwave detector performance is shown in Figure 3-3 as a plot o f diode detector
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
voltage versus transmitted microwave power as measured by the power sensors.
Experiments were conducted using the probe as both transmitter and receiver as well as
using the probe as the transmitter and the horn as the receiver.
4 .0 -
╗'
3.0 -
>
2.0
-
0.0
-
>
0
20
40
60
80
P e r c e n t o f R e f l e c te d M ic r o w a v e P o w e r
Figure 3-3 Microwave diode detector voltage versus microwave power
79
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100
Samples
To investigate the material's conductivy as a contrast mechanism for resisitivity
imaging, measurements were made on samples o f known conductivities. Several
conductive samples were purchased from Goodfellow with thicknesses much greater than
the electromagnetic skin depth at our operating frequency o f 9.35 GHz. These samples
included aluminum, copper, brass, niobium, and titanium with the following resistivities.
Material
Resistivity in /jfi - cm
Copper
1.69
Aluminum
2.67
Brass
6.2
Niobium
16
Titanium
54
Table 1 Resistivity o f samples
All samples were two millimeters thick and mounted in epoxy for ease o f handling and
polishing. All samples were diamond polished using the Buehler system to achieve a flat
surface. Two samples simulating conductivity variations were also investigated. One
sample was an aluminum brass interface that had been mechanically joined and used in
near field optical microscopy work. The other sample was created by cutting the brass
sample in half and soldering the two pieces together. This sample was also polished flat.
Apparatus and Instrumentation
The microwave near field microscope was modeled after optical scanning near
field microscopes. In the optical case, light emanates from a fiber and then reflected light
is collected. In our design, a sharp probe tip is used to transmit microwaves. In the first
80
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configuration the sharpened tip is also used to receive the microwaves reflected from the
sample's surface. In a later configuration, a microwave horn antenna is used to collect
microwave energy reflected from the sample's surface.
Probe tips were constructed from electrochemically etched tungsten wire. Wire
pieces several millimeters long were etched using a potassium hydroxide solution. Tips
were then placed in an optical microscope to investigate the sharpening process.
Photographs o f the tips were taken so that measurements o f the probe tip radius could be
made. Selected probe tip radii ranged from 300 to less than 10 microns. Sharpened tips
were then glued into 22 gauge hypodermic tubing using epoxy. This size tubing fit
snugly into an SMA (small miniature adapter) jack to jack connector. This arrangement
allowed changing probe tips without re-using the SMA connector. The probe assembly
was then viewed under the microscope again to ensure the sharpened tip was not
damaged and that the wire was placed straight in the connector. The SMA connector was
then attached to an adapter assembly that connected a microwave SMA to a type N
connector and then to a TE waveguide. The adapters and the sharpened tip comprised the
probe assembly. Adapters were joined to a waveguide and fixed in place with waveguide
holders to prevent movement. The probe assembly remained stationary in our
configuration while the sample moves.
The microwaves were delivered to the probe assembly utilizing transverse electric
(TE) waveguides. Types o f waveguides included passive straight and curved sections
and two 10 dB directional couplers to sample transmitted and reflected power from the
probe. Power sensors were used to measure the power. For this work, we employed
Hewlett Packard 8618 sensors. A Hewlett Packard 938A microwave power meter
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
measured the transmitted and reflected power yielding a percentage o f reflected power
via GPIB interface to the data acquisition system. In the configuration with separate
transmitter and receiver, a microwave horn collected microwaves transmitted from the
probe tip. The horn antenna was placed so that the center line o f the horn opening was
parallel to the sample surface. The horn was connected to a section o f waveguide that
contained a diode detector. A NP23 diode was placed in the detector yielding an output
voltage proportional to the intensity o f microwaves. This diode detector behaved much
like an optical square law detector. Measurements were made by adjusting the output of
the microwave source to determine a linear region o f the diode detector. A digital
voltmeter recorded the diode?s output. A GPIB interface was used to record the voltage
via the data acquisition system.
The microwave source used to deliver microwaves to the probe assembly was a
9.35 GHz IMP ATT diode. An isolator was placed between the source and the waveguide
system to prevent reflections from interfering with the source.
The microwave components used to deliver the microwaves to the probe tip as
well as the probe assembly were secured to a vibration free surface (floating optics
bench). The waveguide and waveguide holders were securely fastened to the table to
prevent shifting during measurements. A stage was constructed to hold the epoxy
mounted samples. The stage was then attached to a Newport translation stage with a
micrometer for coarse sample movement on the order o f tens and hundreds o f microns.
This stage permitted changing samples underneath the probe without disturbing the probe
assembly. A three axis translation stage was also mounted to the coarse stage. The axes
o f the three dimensional translation stage had stepping motors in place o f manual
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
micrometers to allow for automated micron and submicron movement. During our
experiments we noted that an even finer motion control was needed in order to detect
some near field transitions. The motion control o f our microwave near field microscope
was later modified to include a piezoelectric scanning tube. The scanning tube was
constructed with Macor? mounting plates on the top and bottom o f the tube. The
scanning tube assembly was calibrated using optical techniques to move at a 13.7 nm per
volt increment. A DC voltage source was automatically controlled to supply increasing
amounts o f voltage to produce nanometer step increments.
83
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Chapter 4
Results and Discussion
Field Equations fo r VED and HED above a H alf Space
In Chapter 2 Maxwell's equations were solved for the electromagnetic field at an
observation point generated from a VED and HED above a half space. These solutions
provided field equations that were not limited to the far field. These field equations can
be used to investigate the behavior o f the electric field when the observation point is very
close to the dipole i.e. in the near field.
The field equations at an observation point for a VED above a half space are
given in equations 2.106-108. The field equation for the z-component of the electric field
for the VED above a half space is repeated here for reference.
~ '-d ) '( ik :
2
r,
rr
3
k,r*
where
2k? {
k,p
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3/ '
The first line o f the equation represents the direct field, the second line represents the
image field, and third line represents the surface wave field. The direct and image field
contributions depend on the observation distances, rt and r , , from the observation point
to the direct and image dipoles. The direct and image fields also depend on the wave
vector in air, the region above the half space. k2(i.e. wavelength in air). The direct and
image field contributions do not depend on the material property o f the half space. The
field contribution from the surface wave terms also depend on the observation distances.
But the field contribution o f the surface wave term depends on the material property o f
the half space, fc,. Therefore, all the field values, direct, image, and surface wave,
depend on topography, but only the surface wave terms depend on the material property
o f the half space. These dependencies also hold true for the field equations for a HED
above a half space which are presented in equations 2.216 to 2.221.
These equations for the electric fields can be used to investigate effects o f
topography and material property changes to understand the probe and sample
interactions in microwave near field imaging. Researchers at Max Planck Institute and
Lawrence Berkeley identified the perturbation o f the electric field as the contrast
mechanism for microwave near field imaging. Max Planck researchers selected this
contrast mechanism based on their experimental microwave transmission measurements.
They concluded that a sample perturbed the electric field lines transmitted from a probe
to a detector. Researchers from Lawrence Berkeley also attributed their scanning tip
microwave near field microscope interactions to perturbation o f the concentrated electric
field lines present at their probe tip. They chose to model their probe tip as a conducting
sphere, essentially a monopole. They represented the probe to sample interactions as
85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
follows: the tip acting as a conducting sphere had a charge distribution , the dielectric
sample when placed in close proximity to the conducting sphere becomes polarized and
causes a redistribution o f the charged particles in the conducting sphere. Their model
utilizes image charges and Coulomb's law. The Lawrence Berkeley researchers
acknowledged that their model is not valid for thin film samples due to the divergence o f
the image charges nor is it valid to model the probe to sample interaction for metal
samples. The Lawrence Berkeley researchers described the electric field as being
concentrated in a very small volume under the tip and almost perpendicular to the sample
surface. The electric field solutions presented in Chapter 2 do allow investigation for
metallic samples as well as dielectrics.
VED Conductivity Changes
To model the change in material properties o f the sample, the conductivity o f the
half space is varied for ranges representing conductors, semiconductors, and insulators
(dielectrics). Typically good metal have conductivities from 106 -1 0 8S/m while
semiconductors exhibit conductivities from 104 -1 O'* S/m and insulators/dielectrics have
conductivities below 1(T4 S/m. In Figure 4-1. the magnitude o f the z-component o f the
electric field is calculated for a VED on the boundary o f the half space. The
conductivity o f the half space is varied from 10"6 - 1 0 9S/m. The VED is operating at 10
GHz. The observation point was 5 cm from the dipole with an observation angle, theta.
o f 85 degrees as depicted in Figure 2-4. The change in the z-component o f the electric
field is most sensitive to conductivity changes from 10~? -1 0 3S/m. This
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o\
?
01
2 5x10 ~
0*1-
o?ooamxsmxoKX^?
2.0x10 -
-
? Total Field
~A? Direct Field
Image Field
-o? Surface Wave Field
1 5x10 -
MUM
S
jj
ij
?s
1 0x10 ~
5 0x10 ?
u
-3
3
00 _
10
10
10
10
10
10
10*
Conductivity in S/m
10 '
10?
10'
10
Figure 4-1 Magnitude o f the electric field for a VED on the boundary o f a half space with
various conductivities. The VED is operating at 10 GHz. The observation point is 5 cm
with an observation angle, theta, o f 85 degrees.
87
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
248300
2000
1900
248290
1800
248280
1700
248270
1600
1500
248260
Mugiutude of / component of the lilcctnc Held in V/m
1400
248250
1300
248240
1200
1100
248230
? o? Total Field
?o ? Surface Wave Field
248220
248210
1000
900
800
700
248200
600
248190
500
248180
400
300
248170
200
248160
100
?oooa>
248150
10
ft
10 '
108
Conductivity in S-m
10
1010
Figure 4-2 Magnitude o f the electric field for a VED on the boundary o f a half space
with conductivities in the range for metals. The VED is operating at 10 GHz. The
observation point is 5 cm with an observation angle, theta, o f 85 degrees.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
conductivity range corresponds to values typical for semiconductors. To investigate
conductivity changes relative to metallic samples, the plot is expanded in Figure 4-2 to
show the total field magnitude for the z-component o f the electric field and the
contribution o f the magnitude o f the surface wave term. The magnitude o f the total field
is displayed on the left axis while the right axis displays the scale for the magnitude o f
the surface wave contribution. In these figures, the magnitude o f the total z-component
o f the electric field decreases as the conductivity decreases while the magnitude o f the
surface wave contribution increases as the conductivity decreases. At a conductivity o f
109 S/m the half space is essentially a perfect conductor. The surface wave term is
essentially zero at 109 S/m. This agrees with King's statement that when the half space
approaches a perfect conductor the wave vector in the half space approaches infinity such
that the surface terms with the wave vector in the denominator vanish and the entire field
is determined by the direct and image fields. Calculated wave vectors are shown in the
table below.
er
a [ S/m ]
*[/.]
Metal
1
l xl O9
6.28 x l 0 6 + 6.28x106i
Semiconductor
11.9
9x10^
7.23xl02 +4.91x 10"2/
Air
1
0
209.58
Calculated Approach Curve For VED above a Metal H alf Space
An approach curve shows how the field strength at an observation point changes
as the dipole approaches the surface. These curves help to understand the interaction
89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
between the probe to sample spacing and the field value. This qualitatively models the
effect o f topography. Figure 4-3 shows a calculated approach curve for a VED above a
metallic half space with a relative dielectric constant o f 1 and a conductivity o f
1x 10? S/m. This models the half space as a perfect conductor. The VED is operating at
10 GHz. The observation point is 5 cm with observation angle, theta, o f 85 degrees. The
dipole distance was varied from 10 meters to below a nanometer. In Figure 4-3 the
magnitude o f the z-component o f the electric field is plotted for distances above the
surface. The total field value is plotted along with the field contributions from the direct,
image, and surface wave terms. In this case the direct and image fields are the major
contributors to the total electric field. There is large change in the magnitude o f the
electric field from a meter to a millimeter. In our experiment length scales o f microns
and sub-microns are o f interest. Figure 4-4 expands the previous figure to illustrate the
field behavior from 10 microns to below a nanometer above the surface o f the half space.
In this figure the surface wave term shows no change below a micron. As previously
stated for perfect conductors the surface wave terms with k2 in the denominator vanish
and the total z-component o f the electric field is determined by the direct and image
fields. The magnitude o f the total z-component o f the electric field is carried out to the
fourth decimal place in order to see any change below 10 microns.
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
250000 -
200000
? a ? Total Field
? a ? Direct Field
-
Image Field
?o? Surface Wave Field
>
150000 -
100000 S
50000 -
0-
1 0 13 1 0 12 10'" 10'1C 10* 10* 10'" lO* 10'3 10* 10'3 10'2 10''
10░
io ?
102 103
distance to surface in meters
Figure 4-3 Calculated approach curve for VED above a perfectly conducting half space
with conductivity o f 1x 109S/m and relative dielectric constant o f 1. The VED is
operating at 10 GHz with an observation distance o f 5 cm and
observation angle, theta, 85 degrees.
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124146.0 - i
20 434
248289 345-
124145 9 248289 340124145 8 -
248289 335-> 20 433
г
>
124145 7 248289 325-
|
124145 6 U
T?
C
J
U
\
124145 5 H
t
124145 4 -1
|
^
? . -m
.
10i: 10'" 10'?░ 10" 10* 10''
i m л^
lo"
10''
20 432
No o o a
<*&
a? Total Field
3? Direct Field
Image Field
3? Surface Wave Field
124145 3 -
2
124145.2 -
A
20.431
tk
124145 1 -
124145 0 -f -vi rm ap i *i i n i l
1 0 11
10 ':
10?
i r iin^
i m i^
1 0 10
i i i um)
1C4
I1O'*
(
H1HJ I It 111^ I HUM 20 430
10'"
10*
10'*
distance to surface in meters
Figure 4-4 Calculated approach curve for VED above a perfectly conducting half space
with conductivity o f 1x 109S/m and relative dielectric constant o f 1. The VED is
operating at 10 GHz with an observation distance o f 5 cm and observation angle, theta, of
85 degrees. These curves are expanded to show dipole distances from 10 microns to
below a nanometer.
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Calculated Approach Curve For VED above a Semiconducting H alf Space
In Figure 4-5. the same calculations are made for a semiconducting half space
with a relative dielectric constant o f 11.9 and a conductivity o f 9 x 1O?*S/m, values
typical for silicon. Again the dipole is operating at 10 GHz and the observation distance
is 5 cm with an observation angle, theta, o f 85 degrees. The dipole distance above the
surface is varied from 100 meters to below a nanometer and the magnitude o f the zcomponent o f the electric field is calculated. Figure 4-5 shows the magnitude o f the zcomponent o f the electric field when the dipole approaches a semiconducting half space.
This plot shows the total field value along with the contributions from the direct, image,
and surface wave terms. There is again a large change in magnitude o f the z-component
o f the electric field when the distance changes from meters to millimeters. Figure 4-6 is
expanded to investigate the behavior o f the magnitude o f the electric field values below
10 microns. The total z-component o f the electric field is shown in the inset. The left
axis is for the direct and image field values and the right axis shows the surface wave
field values. The change in magnitude o f the total field follows the changes in the surface
wave fields. These field values are carried out to only one decimal place in order to see
the field value change.
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
160000 -
140000 -
120000
-
u.
u
1V
100000 -
HI
2
o
tг
4)
O
? a? Total Field
? A? Direct Field
Image Field
60000 -
? л? Surface W ave Field
V
N
4>
fi
O
?0o9
.a
cra
10
s
20000 -
0-
-20000
?5
10': 10'' 10░ io' 10: 105
distance to the surface in meters
1 0 13 1 0 i: 10'? 1 0 1010* lO* 10* lO* lo ' 10?* 10
Figure 4-5 Calculated approach curve for VED above a semiconducting conducting half
space with conductivity o f 9 x 1(T4S/m and relative dielectric constant o f 11.9. The VED
is operating at 10 GHz with an observation distance o f 5 cm and
observation angle, theta, 85 degrees.
94
Reproduced with permission of the copyright owner. Further reproduction prohibit
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