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This is to certify that the
thesis entitled

Scattering from a Coated Curved Surface of Varying
Thickness with Application to Nondestructive Evaluation

presented by

Jonathan Wierzba

has been accepted towards fulfillment
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M. S. degree in Electrical Engineering

 

 

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6.101 c:/ClRC/DateDue.p65-p.15

 

Scattering from a Coated Curved Surface of Varying

Thickness with Application to Nondestructive Evaluation
By

Jonathan F. Wierzba

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirmnents
for the degree of
MASTER OF SCIENCE

Department of Electrical and Computer Engineering

2005

ABSTRACT

Scattering from a Coated Curved Surface of
Varying Thickness with Application to

Nondestructive Evaluation
By

Jonathan F. Wierzba

The extinction pulse (E—pulse) has been used to diagnose changes in the elect ro-
magnetic parameters of a coated surface. The E—pulse technique has been successfully
applied to a flat surface and constant curve surface with a constant thickness. An
important extension is to apply the E-pulse technique to a surface with a varying
thickness and curvature.

The coated surface investigated in this thesis is an elliptical metallic cylinder with
a circular dielectric coating. An incident plane wave with TEz and TMz polarization.
is reflected off the coated cylinder having either a vertical or a horizontal major axis.
The monostatic results generated in the frequency domain are transformed using an
FFT, providing the time domain data for convolution with the E—pulse.

The E~pulse for a planar coated surface is applied to the data obtain for the ellip—
tical metallic core with a circular dielectric coating. The placement of the major axis
and the type of polarization chosen has a greater impact on the E-pulse. Discrimina-

tion Number (EDN) as the radius decreases in size.

.. "I ur—n—fl:

 

‘IT

 

Copyright © by
Jonathan F. Wierzba

2005

To my parents, Gregory and Joan

 

iv

ACKNOWLEDGMENTS

I would like to thank my family for the support during the thesis.

Also, I would like to thank Dr. Ed Rothwell for his help on the thesis.

 

1-

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

1

2

6

7

Introduction
Background

Equations for Scattering

3.1 TMz Case .................................
3.2 TMz Case Comparison ..........................
3.3 TEz Case .................................
3.4 TEz Case Comparison ..........................

Computational Considerations

4.1 Computation Method ...........................
4.2 Flat Case .................................
4.3 Circular Case ...............................
4.4 Echo Width of Elliptical Metallic Core .................

Results for the Elliptical Case
5.1 Frequency Response ...........................
5.2 Time Domain Results ...........................

E—pulse Results

Conclusion

APPENDICES

A TMz Case Source Code

B TEz Case Source Code

vi

viii

ix

43
43
6.3

93

115

118

118

133

 

C Flat Case Source Code 148

BIBLIOGRAPHY

" """F

 

'1‘

vii

LIST OF TABLES

6.1 EDN values calculated for flat case and for cylinders of various radii in

meters ...................................

viii

 

3.1

3.2

4.1

5.7

\J‘
CI

CI!
CC

LIST OF FIGURES

Orientation of the elliptical metallic core with the x-axis being the
major axis. ................................
Orientation of the elliptical metallic core with the y-axis being the

major axis. ................................

Flat case normalized field magnitude vs. frequency with d = .1 111. Sr
2 9, [tr 2 1 ................................
Flat case phase vs. frequency with d = .1 m, Sr 2 9. ,ur : 1 .....

Flat case normalized amplitude vs. time with d = .1 111, 8r 2 9, [1,. = 1
Echo width vs wave number, x as major axis, a1 = 1.4 m, 212 = 1.3 m.

Echo width vs wave number, y as major axis, a1 = 1.4 In, 212 = 1.45 111. ~

TMz normalized field magnitude vs. frequency. a = .9 m. al 2 .8 In.
a2 = .7 m, with x as the major axis. ..................
TMz normalized field magnitude vs. frequency. a = 1.5 m. al 2 1.4
m, a2 = 1.3 m, with x as the major axis .................
TMz normalized field magnitude vs. frequency. a = 2 In. a1 = 1.9 m,
a2 = 1.8 m, with x as the major axis ...................
TMz normalized field magnitude vs. frequency, a = .9 m, a1 = .8 m.
a2 = .85 m, with y as the major axis ...................
TMz normalized field magnitude vs. frequency. a = 1.5 m. a 1 = 1.4
m, a2 = 1.45 m, with y as the major axis. ...............
TMz normalized field magnitude vs. frequency, a = 2 In. a1 = 1.9 In,
a2 = 1.95 m, with y as the major axis. .................
TEz normalized field magnitude vs. frequency, a = .9 m. a1 = .8 m.
a2 = .7 m, with x as the major axis. ..................
TEz normalized field magnitude vs. frequency. a = 1.5 m, al 2 1.4 In.
a2 = 1.3 m, with x as the major axis ...................
TEz normalized field magnitude vs. frequency. a = 2 m. a1 = 1.9 in.

a2 = 1.8 m, with x as the major axis ...................

ix

0

35
30
37

40
47
-18
49
so

51

5.10
5.11
5.12
5.13
5.14
5.15

5.16
5.17

5.19

5.20

5.21

5.22

5.23

5.24

5.25

5.26

5.27

TEz normalized field magnitude vs. frequency, a = .9 1n. a1 = .8 m,
a2 = .85 m, with y as the major axis ...................
TEz normalized field magnitude vs. frequency, a = 1.5 in, a1 = 1.4 m,
a2 = 1.45 m, with y as the major axis. .................
TEz normalized field magnitude vs. frequency, a = 2 m, a1 = 1.9 m,
a2 = 1.95 m, with y as the major axis. .................
TEz normalized field magnitude metallic elliptical cylinder ......
TEz normalized field magnitude difference ...............

TMz normalized field magnitude metallic elliptical cylinder ......

TMz normalized field magnitude difference ...............
TMz phase vs. frequency, a = 1.5 m, a1 = 1.4 m, a2 = 1.45 m, with
y as the major axis .............................
The electromagnetic pulse multiple reflections, x is the major axis of

the ellipse ..................................
The electromagnetic pulse multiple reflection, y is the major axis of
the ellipse ..................................
TMz normalized amplitude vs. time, a = 1.5 m, a1 = 1.4 m, a2 = 1.3
m, with x as the major axis. .......................
TMz normalized amplitude vs. time, a = 1.5 m, a1 = 1.4 m. a2 = 1.45
m, with y as the major axis. .......................
TEz normalized amplitude vs. time, a = 1.5 m, a1 = 1.4 m, a2 :2 1.3
m, with x as the major axis. .......................
TEz normalized amplitude vs. time, a = 1.5 m, a1 = 1.4 m, a2 = 1.45
m, and with y as the major axis. ....................
TMz normalized amplitude vs. time with a = .9 m, 1.5 m, 2 In and
with x as the major axis ..........................

TMz normalized amplitude vs. time with a = .9 m, 1.5 m, 2 m and
with y as the major axis ..........................
TEz normalized amplitude vs. time with a = .9 m, 1.5 m, 2 m and

with x as the major axis ..........................
TEz normalized amplitude vs. time with a = .9 m, 1.5 m. ‘2 m and
with y as the major axis ..........................
TMz normalized amplitude vs. time comparison of flat with a 2 .9 m
and with x as the major axis. ......................
TMz normalized amplitude vs. time comparison of flat. with a. = 1.5

m and with x as the major axis ......................

CH
\1

(11 CI!
)C

9

60

61

62

66

66

67

69

70

71.

5.31

5.32

5.36

5.37

5.39

5.40

5.42

5.43

5.45

6.1

TMz normalized amplitude vs. time comparison of flat with a = 2.0
m and with x as the major axis ......................
TMz normalized amplitude vs. time comparison of flat with a = .9 1n
and with y as the major axis. ......................
TMz normalized amplitude vs. time comparison of flat with a = 1..)
m and with y as the major axis ......................
TMz normalized amplitude vs. time comparison of flat with a. = 2.0
m and with y as the major axis ......................
TEz normalized amplitude vs. time comparison of flat with a z: .9 in
and with x as the major axis. ......................
TEz normalized amplitude vs. time comparison of flat with a = 1.5 in
and with x as the major axis. ......................
TEz normalized amplitude vs. time comparison of flat with a : 2.0 111
and with x as the major axis. ......................
TEz normalized amplitude vs. time comparison of flat with a = .9 m

and with y as the major axis. ......................

TEz normalized amplitude vs. time comparison of flat with a = 1.5 III
and with y as the major axis. ......................
TEz normalized amplitude vs. time comparison of flat with a = 2.0 m

and with y as the major axis. ......................
TMz normalized amplitude vs. time comparison of x vs y as the. major
axis, a = 1.5 m ...............................
TEz normalized amplitude vs. time comparison of x vs y as the major
axis, a = 1.5 m ...............................
TMz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a = 1.5 m and x
as the major axis ..............................
TMz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a = 1.5 m and y
as the major axis ..............................
TEz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a = 1.5 m and x
as the major axis ..............................
TEz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a 2 1.5 in and y
as the major axis ..............................

E—pulse for the planar coated conductor vs time .............

-I

*1

80

81

82

83

84

86

87

88

90

91

96

6.2
6.3

6.4

6.5

6.6

6.7

6.8

6.9

6.10

6.11

6.12

6.13

6.14

6.15

6.16

6.17

6.18

Convulated E-pulse vs. time, flat case. .................
TMz convolved E—pulse vs. time, a = .9 m, a1 = .8 m, a2 = .7 m. with
x as the major axis .............................
TMz convolved E—pulse vs. time, a = 1.5 m, a1 = 1.4 m, aQ = 1.3 m,
with x as the major axis ..........................
TMz convolved E—pulse vs. time, a = 2 m, a1 = 1.9 m, a2 = 1.8 m,
with x as the major axis ..........................
TMz convolved E—pulse vs. time, a = .9 m, al = .8 m, a2 = .7 m, with
y as the major axis .............................
TMz convolved E—pulse vs. time, a = 1.5 m, a1 = 1.4 m. a2 = 1.3 m.
with y as the major axis ..........................
TMz convolved E—pulse vs. time, a = 2 m, a1 = 1.9 m, a2 = 1.8 m.
with y as the major axis ..........................
TEz convolved E-pulse vs. time, a = .9 m, a1 = .8 m. a2 = .7 m, with
x as the major axis .............................
TEz convolved E—pulse vs. time, a = 1.5 m, a1 = 1.4 m, 212 = 1.3 m.
with x as the major axis ..........................
TEz convolved E-pulse vs. time, a = 2 m, a1 2: 1.9 m, 212 = 1.8 m,
with x as the major axis ..........................
TEz convolved E—pulse vs. time, a = .9 m, a1 = .8 m. a2 = .7 m. with
y as the major axis .............................
TEz convolved E—pulse vs. time, a = 1.5 m, a1 = 1.4 m, a2 = 1.3 in.
with y as the major axis ..........................
TEz convolved E—pulse vs. time, a = 2 m, a1 = 1.9 m. aQ = 1.8 m,
with y as the major axis ..........................
TMz subtracted convolved E—pulse vs. time, a = .9 m, al 2 .8 m, 212
= .7 m, with y as the major axis. ....................
TEz subtracted convolved E—pulse vs. time, a. = .9 m. a1 = .8 m. a2
= .7 m, with x as the major axis. ....................
EDN vs outer radii, subtracted vs non-subtracted, x as major axis.
XD indicates that it is the subtracted case with x as the major axis.

98

99

100

101

102

103

l 04

105

106

107

108

109

110

111

X indicates that it is the non—subtracted case with x as the major axis. 112

EDN vs outer radii, subtracted vs non-subtracted, y as major axis.
YD indicates that it is the subtracted case with y as the major axis.

Y indicates that it is the non-subtracted case with y as the major axis.

xii

113,

CHAPTER 1

Introduction

In this thesis the field scattered from an elliptic cylinder coated with a. circular
dielectric material is computed. This provides a test case for the E~pulse technique,
which can be used to diagnose changes to the coating parameters (thickness and
dielectric constant). The coated elliptical presents a case with a varying surface
curvature and thickness, which approaches the planar case studied in detail in [9]
when the radius becomes large. The goal is to determine whether the E-pulse
technique remains valid when the surface curvature and coating thickness is \~'arying.

The background of this thesis is covered in Chapter two. The E—pulse technique
origins show how the technique was first applied to coated surfaces. Also, the
origins of the equations that are used to derive the scattered electric field equation
and the calculation of the echo width of the elliptical cylinder are covered.

The equations for the scattered field are derived in Chapter three for both
transverse electric. and transverse magnetic incident plane-wave fields. The
derivations for the two polarizations start by describing the incident, scattered. and
dielectric region electromagnetic fields and then apply the boundary conditions at
the conductor and dielectric interfaces to determine the scattered electric field. Use.
of the point-matching method leads to a matrix equation for the expansion

coefficients used to compute the scattered field. These coefficients are compared

against the coefficients for the circular case derived in [10].

Chapter four contains the computation method for the computer code and the
results of the flat case, circular case. and the calculation of the echo width of the
uncoated elliptical cylinder. In the computation section, the logic and layout that is
used in writing the computer code is shown. For the flat. case. the results are shown
for the standard that will be used against the elliptical case. The circular case
shows a matching of the data when the ellipse becomes a circle. The echo width of
the elliptical cylinder is calculated as a check to see if the equations are derived
correctly.

In Chapter five, the scattered field data is plotted in the frequency and time
domains. The frequency domain plots show the effects of the various size of the
radii and the polarization on the scattered electric. field. The time. domain plots.
which are obtained using the FFT show the amplitude of the reflections of the
electromagnetic wave from the various cylinder configurations. Also, the time
domain plots are used to compare and contrast the effects of different geometries
and polarizations on the multiple reflections of the waves trapped within the
dielectric layer.

In Chapter six, the E—pulse for the planar coated surface is convolved with the
time domain scattered field waveforms, to determine the effectiveness of the E-pulse
technique on a coated surface with varying curvature and thickness. The various
shapes of the convolution are shown and compared to the flat case. The E-pulse
Discrimination Number (EDN) is calculated and compared for (,lifl'crcnt. sizcd radii
of the coated cylinder.

Finally, Chapter seven provides a summary and some concluding remarks.

CHAPTER 2

3‘
Background !
i
The E—pulse technique was first developed to identify radar targets based on g
their temporal scattered-field responses [6]. When a radar target. is illuminated by a is.

short electromagnetic pulse, the induced currents will oscillate in the natural
resonance of the target. Since the resonant frequencies are unique to the target. the
time-domain scattered field set up by the induced currents can be used for
identification. An E—pulse is a time domain waveform that is constructed based on a
specific set of natural frequencies. When the E-pulse is convolved with a natural
mode series containing these same natural frequencies a zero late-time waveform is
produced. When the same E—pulse is convolved with a natural mode series
consisting of different natural frequencies, a non-zero late—time response is produced.
allowing an identification decision to be made. The decision is quantified using the
E-pulse discrimination number (EDN) [3].

Recently, the E—pulse technique has been applied to layered media. When an
electromagnetic pulse is reflected by a layered medium, the time domain response
consists of a number of peaks produces by the multiple reflections within the layers.
This response may be written as a natural mode series, with resonant frequencies
dependent on the electromagnetic properties and thickness of the layers. Thus the

E—pulse technique may be used to determine when layer properties have changed [9].

For any given surface, there is a unique E—pulse that is created and if the surface
parameters change in anyway, the E-pulse will no longer match the reference and a
change is detected. The greater the difference in surface parameters, the larger EDN
value from the application E-pulse technique.

An important extension of the E-pulse technique is to apply it to a coated
curved surface with a constant thickness and curvature. This was investigated in
[11], where it was shown that under certain circumstances, the E-pulse technique
may be used successfully with curved layered surfaces.

In this thesis, the E-pulse technique is applied to a curved coated surface with a
varying curvature and thickness, by examining the time-domain response of an
elliptic cylinder coated with circular dielectric. The equations that describe the
scattered electric field with a circular outer region with arbitrary metallic core.
cross-section were derived by Yee using the point-matching method in the 1960‘s
[13]. To provide a varying thickness and curvature of the coating, an elliptical
cross-section was chosen. The varying thickness is chosen since it provides a more
realistic model of a curved coated surface where a constant coating thickness is not.
always possible.

The point-matching method has also been used to compute the field scattered
by an uncoated elliptical cylinder [5]. There, the echo width of an elliptical cylinder
was computed for various illumination conditions. In this thesis, the echo width of
the elliptical metallic cylinder is used as a check to see if the equations are derived
correctly, since high-frequency the limit of the echo width is merely 7r times the

radius of curvature at the contact contract point of the incident wave.

CHAPTER 3

Equations for Scattering

3.1 TMz Case

The electric field scattering equation is needed for finding the frequency
response. With the frequency response, applying the Fast Fourier Transform to the
data, a time domain plot can be obtained to see how the electromagnetic wave is
scattered off the cylinder. The plane electromagnetic excitation wave is chosen to be
incident on the cylinder along the -x axis for the derivations. With the elliptical
shape of the metallic core there are two choices for the major axis, the x—axis or the
y-axis. Figure 3.1 shows the cross section of the circular dielectric layer around the
elliptical metallic core with the x-axis being the major axis. In Figure 3.2, the
y-axis is the major axis. The placement of the major axis has no impact on the
derivation of the electric scattered field equation. However, the major axis

placement greatly affects the numerical calculations as shown later.

The incident plane electric field in region 2 is represented using the last
expression in the sequence from [2, eq. 11-81b]:

00
E; = Eoejkoa: = Eoejkorcosqb = E0 2: j"Jn..(f\"o7')(i"fm'§. (3.1)

n.=—oo

<2 32
W

89“ 80,110

 

Figure 3.1. Orientation of the elliptical metallic core with the x—axis being the major
axis.

1

W’
.,../

/ 80,u0

 

Figure 3.2. Orientation of the elliptical metallic core with the y-axis being the major
axis.

Since the electric field will be calculated in a numerical form, breaking up the
equation into summation of terms is the next step. The desired end result for this
equation is to combine the positive and negative indexed terms in order to reduce
the size of resulting the matrix by half. So the next equation shown is the breaking

of the summation into positive, zero, and negative terms:
0 —1 . m .
E: 2 E0 2 jan(z.-0r)ef"¢’ + Jofko’r) + Z j"Jn(A'(,r)eJ”f) . (3.2)
n = —OC n z 1

The following equation has the modification to the summation that contains the
negative terms by replacing ”n” with ”-n”. Also, the J —n is replaced by (—-1)an,

as found in [2, eq. IV—8]:

oo 00
E; = E0 2 j""(—1)"Jn(k0r)e‘1"¢ + Jo(ke7‘) + Z j"‘.1,,(/.~,,,')c.7"'0’
n = 1 n = 1
(3.3)
The j_n(—1)n product is equal to j". The next step is to combine the two positive

summations:

m . .
E: 2 El) J0(k07‘) + Z jan(k0T)(C‘]n(‘D + e—JNO) . (3.4)
n = 1

The final equation below is created by combining the exponentials into a cosine

function:

00

. 1

E: = EU 2 jnean(k0r)cos(n¢) 5n = - (3-5)
n = 0 2 n % O

The incident magnetic field is calculated by taking the curl of the electric field:

1 1 E’; . -
JW/ie Jwflo 7

Hi: VxE§=—

 

Substituting the derivatives of the electric field (3.5) gives:

1 E 00 . 00
12—0- : anan(k07‘)-nsin(n¢)) — (onko Z jlnentlhtkor)cos(n.(.‘))

'- . r
Jig/‘0 =0 n:0

 

(3.7)

 

The 7‘“ term is not needed when the boundary conditions are. applied, leaving only
the (5 term. Also, k0 and 1 / ( jay/10) combine to form 1/170:

H7275-=j7f§97720571jnl1nfk0fl(105("(5l- (3.8)

The scattered electric field in region 2 is given by:
°° (2)
E2 = E0 2 Can (1.:0r)cos(n¢) (3.9)

To find the scattered magnetic field in region 2, the curl of electric field needs to
calculated. As before, only the (2) component of the magnetic field needs to be

calculated:

 

 

H8: 07135: E0

(2 ) .. . .. ,
a 9'pr z- Mm n§0( 6M Can (k07)cos(n¢))). (3.10)

After taking the curl of the electric field, the scattered magnetic field is:

E .
Hg): -J— 0 Ion: CnH H792) (k()7‘)cos(nq)). (3.11)
=0
The. electric field in region l (dielectric region) is givcn by:
00
13:;! = E0 2: [Aan(kr) + Bur/nan) cos(n(.>). (3.12)
n = O

The magnetic field in this region is calculated by taking the curl of the electric field.

As above, only the ((3 component is needed:

d __ rid ‘
H¢— — j:.u'aT WE“ _ julfl Eonzoar( [Aan( ICT) +Bnl’n(k7‘)](OS(n (5)). (3.13)

The magnetic field (5 component is:

d_ J___E0

11¢- ’7 Z [AnJ’( A7)+Bnl77(l.7)]cos(no) (3.14)

n=0

Now that the electric and magnetic fields have been calculated in the various
regions, the next step is to apply the boundary conditions on the tangent electric

field which is continuous at r = a:

oo 00

E0 2 jnean(/€ga)cos(n¢) +E0 Z CnH7(12)(k0a)cos(na>) :
n, = 0 n = 0
00

E0 2 [An.]n(ka) + BnYn(/m)] ('()S(n(;)). (3.1.5)
72 = 0
By multiplying the equation by cos 7729‘) and integrating the result is eliminating the

summations to isolate An, Bn, and Cn terms through the orthogonality relation:
/cos(n¢) cos(m(/))d(,-‘) = WEmdmn. (3.16)

The Em, term is created when the integral is evaluated, with the result of two when
m equals zero and one when m is not equal to zero. The 6m" term is the Kronicker

delta given by:

2 m = O - 1 m = n. 7 _
077m = . (3.11)
1 m 75 O O m # 71

ml
3

Multiplying (3.15) by cos(m<f>) and applying the integral gives:

00
[(E0 2 jnean(k0a)cos(nq§)cos(mq’))+

7120

00 2
E0 2 CnH7(l )(koa)cos(n¢)cos(-m(p))d() =

n. = 0
00
/(E0 2: [Aan(ka.) + Bn,Yn(ka)]cos(nt/))cos(7no))dq). (3.18)
n = 0

After evaluating the integral, the summations drop out, leaving the ft.)ll(:)wing

equation:
.' 2 7 I
ananUCOG.) + CnH7(l )(kga) = AanUCtL) + 8,11,7(A‘U). (“319)

To create another equation to use to eliminate a variable, the boundary condition

for the tangent magnetic field being continuous at r = a is applied. giving:

9

Ebnzo 571.77 llil(fi00)(08( IIQ)‘ $1220 CanZ) ((100,) )(TOS(IIC)) :

7/0

E .
~J 0 Z [AnJ 77 (ku)+Bn1"' 77( (ka )](()s( 11(5)) (3.2.0)
77 n— — O
Repeating the same process above that eliminated the summation results in:

7' I 2 I 7. .
annJ77(koa) +0511), ) (Icon) :1" Hunt, Ira) + 8,7177(Au)] (3.21)

With the two equations above, (3.19) and (3.21), the next step is to solve for
An, and En in terms of Cn. The first step is to find to An in terms of Em and Cu
using (3.21):

7 ,, 2 ' .
AnJhUCG) = ‘L EnjnJh,(k0a) + Canl ) (130(1) — B711'7’77(k(l).
[)0

A
w
IQ
Iv

‘V

After solving for An, the next step is to solve the other equation, (3.19) for Bn in
terms of An and Cn:

1
B :
n Yn (Ira)

 

2 . .
[anan,(kQa) + CTLH7(1, )(kua) - Aan(k(1)] . (325)

After having An and En isolated, substituting Bn into (3.22) gives An in terms of

just Cn:

fin-"MW = % [Enf‘Jttkoo + Can) (koa)[ ..
mam) In A (2) .. 7 7 7
YnUCG.) [.7 €an(I\)Oa) + (3an (It-00.) — An.]n(A(l)] . (3.2-1)

 

The next step is to combine all the like terms and factor out An:

An [n,,(kr)J;,(ka) — Y,’,(ka)Jn(k-a)] = 57,7" [lvn(ka.)J;,(/.-.,a) — 17',(1a).),,(1.-.,uj)[

71 (2 ) _ ('3) . - ...-
+Cn n—OYn(ka)H77 (koa) 177(A(1)H77 (Ami) . (32.))

10

The following equation from [2, eq. 11-95], reduces the expression multiplying A" to

a constant:

2
Yneawttka) — seamen) = —— (3.26.)

aka. .

Thus An is now in terms of just Cn:

 

E.
aka , 7 ,
An = ——2— [ffnjn [U—IY71(ka)J;7(koa) — Yn(k(L)Jn(k()(l):l [
0
I r
+Cn [3Yn(ka)H7(72) (koa) — Y7'7(A:(1)H7(72)(/~'(7(1)H . (3.27,)

e .

After solving for An, the same equation can be solved for B77: .
[r

Bnlfn(k(l.) Z jnean(k0a) + CnH7g2)(k0(l)

J (ka) 7) . I 2
— infka) i— [Enjn'lnfk0“) + (771.1178,

710

I

)(kna)[ - Bnl’7’7(ka)[ . (3.28)

 

Combining like terms gives:

B" ll’ilfka'wfdk“) _ Jnfkalyvifkall 2 anIt [Jffkallrszoal — #Jnfffalelidffnflfl
e

+ Cn [J77(ka)H7(72)(k0a) — %Jn(ka)H7(72) We‘ll] . (3.29)

Replacing the Bessel function expression with the constant from [2. eq. 11—95] gives

Bn 2

B5,. = — [can [aneawteoe — Jh<ka):’"("‘°“)i +
0

0n, [%J7l(ka)HT(L2)I(A'()Q.) — J;7(fi'(l.)H,(72)(fi'()(l):l] . (3.30)

11

Since there are three unknowns and two equations so far, another equation is
needed to solve for C77,. The last equation comes from setting the tangent electric
field equal to zero where the dielectric material interfaces with the metallic core:

00
155? = E0 2 (47,17,037) + smear-)1 (5030.3) = 0. (3.31)
n = 0
Define the expressions fn, 777, gn, and fin to represent parts of the equations that

describe An and Bn:

(n = — [%Yn(ka>Ht2)'(koa) — s’.(ka)H5?)(A-ou>[ (3.322)

in = %Jn(ka)H7(22)I(koa) — Jakthzmw) (3.33)
971 = —€njn [%Y71(ka)‘]fl(k0a) — Kifkaflnfknafl (3-34)
:93. = Enjn [%J77(ka)J77(k0a) — .I;..<ka)Jn.<Aw)[ (3.3.3)

Then An and Ben can be simplified using the fn, f 77, gin, and ’g'77 expressions.

aka aka
An = ”72,011. + 9n) ‘2— Bn = [777,071. + .572) —2__ (33(3)
Substituting these into (3.31) gives:
(1 aka 00 —
E3 : EDT 2 [lCnfn + 9n] Jnfkr) + [Cnfn + fin] 1320”” 0050165) Z 0-
n = O

(3.37)

The boundary condition on the conducting cylinder is satisfied at a finite number of

points (rm, (6777) with m = O,l,2,...,M (point matching)
(1 aka 00
E2 : EDT 2: [lCnfn + 9n] Jnfkrm) + [Cnfn + fin] Yn(k'l‘m)] c0s(n(,’),717) : O
n = 0

(3.38)

The next step is to combine like terms, starting with fn, and 7.77.

12

1

 

__ .7 2 I
fanUC'rm) + fund/Wm) = —;]’—H7(7, ) (koal)Ynfka)Jn(k7‘m)+
0

(2). ,. _ (2),. .k 11,, (2),,

H” (lt-()0.)Yn(fta).]n(k7‘m) HT). (.oa)J77( (l)Yn,(ft-Inl)+ T) J”( .(l.)H77 ( ()(l)
0

(3.39)

Factoring the Hankel functions out of the equations gives:

__ 7 2 ’
fnerk’rm) + fnY7L(k7"m.) = #Hfz ) (kea) [Jn(ka)Yn(k7‘m) — Yr:(k(1)Jn(k7'm)l ‘
0

H7g2)(k()a) [J77(ka)Yn(krm) —— Y7'7(ka).ln,(krm)] . (3.40)

Define F mn and Emn to simplify the equations:

Fnzn = Jn(k(L)Yn(kT7n) — Yn(ka).]n(k7’7n) (3.41)
an = Jh(l€(l)Yn(/€Trn) — Y;l(ka).]n(fi77‘yn) (3.42)

J k — Y k —_77_H(2)’ k —H(2) .. ‘1 ..
fn nf Tm)+fn n( Tm) — 7)0 n (ea)an n (Ariafl'mn- (34-3)

After finishing the rearrangement of the fn and f n expressions, the next step is to

combine the gn and 577 expressions:

7
. 7 ,
QanfkTm) + gnYnfkT‘m) = Ean [—#OynfkaflnfkrmflnfA700) +
Y/dkaflnfkrmfln((900) + EEJnfkaankTmWM/Voa) — Jiszn)Yn(kI'm)sz(11‘0“) -

(3.44)

Next factor out the terms from the Bessel functions with koa as the argument:

gn.]n(ki7‘m) +§nYn(kTm) = Enjn n10[Jn(ka)Yn(kTm) — l%,(f£(l.)Jn(/ifl'n))[.];7(111700.)—

[Yfi(ka)Jn(kT7n) — J;7l(ka))zn,(ffl7‘7n[Jn,(f{()(l.) . (3.45)

13

Replace the term that multiply Jn(k0a) and J’n(k0a) with an and Fm":
.(J-an(kT'm) + gnYnUCTm) = Enjn [ganJMkoal — anJnU‘UG) - (346)
0

Substitute in the expressions for fn, f n: gn, and En with new expressions that. have

been simplified:

7rlca 0° 7; (2y — (2)
E21 : En—é—l Z [Cn [Ty—H” (I)?0(l)Frnn — F7nan (k()(1)] ('OS(II(,J)]”) 'l”
n = O 0

. 7 , — . ..
<5an [Egjrdkoalan — anJnUt‘nal] C05(719”m)] —_— 0° (34")

This can be written as:

00 7/ (2)’ — (2)
Z Cn lTHn (1160(1)an — anHn Meal] (705(7199m) =
n = O 0

00 . 7 . —. . , . .-
— 2571]” lJ‘Jhwoaflnm — 1“ nmJnUVuall FOSU'Om ). (345)
":0 L00
By truncating the sum at n=N, and choosing M=N matching points. this ("an be.
written as a matrix equation of the form 2 AmnCn = Bm which can be solved for
Cn. Then C", can be plugged into (3.9), to give the scattered field:
0° (2)
E; = E0 2 Can (ko'r)cos(vn.d)). (3.49)
n = 0

For k0r>>1 The Hankel function can be approximated using the following equation:

HT(12)(k07‘) : ___2_€_j(kor — ”/4 "' 'ILW/Q) r350)

Wkor

Substituting this into (3.49) gives:

 

oo
Eg = E0 2 63—30%“ — 7r/4) E C” ('os(lzg§)()_3"7r/2 (3.51)
WkOT' 0 ‘ ‘
n 2

14

3.2 TMz Case Comparison

The TMz case comparison is to check the derivation for the elliptical case with
the circular case that was derived in Heike Vollmer’s thesis [10]. The comparison
starts with the equation that describes the metallic core interfacing with the
dielectric layer and then having tangential electrical field going to zero at the

interface. To simplify the comparison, only the nth term in the summation is used.

‘4
Also, krm is now a constant value, with rm = b forming a circle. Beginning with
(3.48) and substituting for an and fmn gives:
!
Z Cn[—[Jn( ka )Yn(ka)— Yn(ka)Jn( (mm) )]Hn 2) (koa) !_

 

n— - 0
—[J;1(ka)Yn(krm)— Yfl(ka).ln(krm)]H,(22)(1:00)] cos('n(:>m) :

00
1.0: mint—l Jn( (ka)Yn(k-rm)— Yuk-m.(trusty.(to...)

- [Jh(ka)Yn(k7‘m) — Yfi<ka)Jn(kT1n)]]n(k0(l)) COS(IICJ))7)). (3.52)

The orthogonality of the cosine functions can now be applied since rm 2 b is a

constant. This gives:

C7),[%[.In(ka)Yn(ka)— Yn(ka)Jn( krm) )lHn (2) (Icon)
I I ‘2
_ l']n(k“)Yn(k7'm) _ Yn(ka')JTl(krnl)lH7(L )(kn‘m
= —€njn[1]10[Jn(k(l)Yn(k7m) — Yn(k(L)Jn(kl‘7n)]J;l(k0(l)

— [Jh(ka)Yn(kTm) — Y7,21(k(l).]n(k7"7n)]Jn(lx‘()(l)]. (3.5-3)

Define D and L to simplify the fractions:

JnUCa) YnfkalJn (k'rm)

D = ——
Jn(koa) YTL(kT77L)J72-(k()a)

 

 

(3.5.4)

15

 

 

L— fiUca) _ YT’L(ka)Jn(an1)

_ , , 37.3)

This expression matches with the expression for On from Heike Vollmer‘s thesis (in
[10, p. 21] with minor corrections, since radii are named by different methods.

Solving for Cn then gives:

. J k—
€an [D — «an ——(’-‘2()—°i’—
an H” “7”") . (3.56)

 

 

(2)’ a . a
DHTEZ) (k0 )Jntko ) _ ,/—L
Hn (koa)J;l(k0a)

16

3.3 TEz Case

Using the same figures in section 3.1, the incident magnetic field assumed to be
z-directed and is incident on the cylinder along the -x axis. The incident magnetic-
field in region 2 is represented as a summation from negative infinity to positive
infinity, so the magnetic field can be calculated numerically. Using the expression
from [2. eq. ll-8lb], the incident field can be written as:

00

H2, = €261,901? = f’flcjkor COSW) = .110 Z jTLJn/(ko.,.)ejne (3.57)
’20 770 770 _
n — —00
The summation is broken up into negative and positive terms to simplify the.
summation:
. E ‘1 . 00 .
H2 = —‘3 Z j‘"Jn(kovr>eJ"¢ + J0(k0r) + Z JianMOTVJm/D - (3.58)
7/0
n = —00 n = 1
Now take the negative indexed terms and replace “n” with “-11” to gives:
H: = 7—9 Jan—Mktfl'k—Jnd) + J()(k'0r) + Z J'an.“0")“me - (3'59)
,0 n = 1 n = 1
The J_n(k0r) term is equivalent to (_1)n JnUmr) [2, eq. IV-8], giving:
- E 0° . 00 .
H; = _9 Z: j_"'(——1)an(/€07‘)€_Jn(:) + J()(k()r) + Z jnJ”(A‘U'I'hi-lm’)
")0
n = 1 n. = 1
(3.00)
Combining the summations gives:
H3 = ——° 2 janUcor) [ca-9’“? HMO] + .I(,(Ar.,r) . (3.51)
710

n = 1
The exponentials can be combined into a cosine function and En is needed since.
there is a two factored out of the exponentials for n greater than zero. For n equal

to zero, there is only one exponential, so only one is factored out. This gives:

17

 

1' E0 .71 1 'n : 0 . _:
Hz 2 — an] Jn(k0r)cos(no) an 2 . (3.02)

710n=0 2 n # 0

The electric field is calculated by taking the curl of the magnetic field:

_)'. 1 . 1 . A . ,
E2 = , 5 [23 X th2] = —_—]E [7'1 ;0(D H; + (,DarHé (3.0”

 

Applying the derivatives gives:

—+' 1,:_E__0
El 2 ———;—— Z Enj "Jn(k07)nsin(n¢) +
JWE r770?
n: 0
E 00
(AS—”3 E:enjnkoJMkor)cos(ng§) (3.0.4)
0

n=0

The f“ part of the electric field is not needed in the calculation for the tangential
electrical field in the boundary conditions. Simplifying gives:

J— E0 ’90
UouJEn

15?: 20 snjnJ,’,(k0r)cos(n¢)= jEO Z enj’ 1’ n(km‘)cos(m;i) (3.0.5)

n=0

The magnetic scattered field in region two is given by the equation:

E
Hg— — ——0 7707120 CnH A07)cos(na)) (3.06)

The electric scattered field is calculated by taking the curl of the magnetic scattered

field. As before, only the <13 component is needed:
E 2 ._,
Egg: ——67~H z — ——w—-—6r7’ O on: CnH Hr(z )(k‘(,7)(os(m")) . (3.01,)
=0

Taking the derivative and simplifying gives:

'kE 00 (2) ’
Es _ _J_2 2 Can (1W) COSW, )—on Z: CnHI1,2(,f)(A0r)ros(no) (3.68)
0

(1) 7700.150
n = n— — O
The magnetic field in region one (dielectric region) is given by the following

equafion:

18

 

E 00
Hd = 7,9 Z [Aan(kr) + Bur/Mm] cos(nq5). (3.60)
n = 0

The electric field in the dielectric region is calculated by taking the curl of the
magnetic field equation in the dielectric material. As in the pervious cases, only the

as component is needed:

 

 

E
Ed = —_—0er— _ ———ar—9 n: [Aan( kr) + BnYn(A1)](os(nd)) (3.70)
99 Jane 3w: 3
=0 i
Applying the derivative gives:
. k E 00
Ed = 7 gm" 2 [AnJ,’,(kr) + BnJ;,(Ar)]cos(nqs) (3.71)
nvn=0
After the derivative is applied, the constants in front of the summation are y
simplified to give:
00
Egg = jEO Z [AnJ;,(Ar) + BnJ;l(lt:r)]cos(m;)). (3.72)
n = 0

After finding the electric and magnetic fields in the various regions, the next:
step is to apply the boundary conditions. The first boundary condition that is

applied is that the tangential electric field is continuous at r = a.

jEO Z Enjn Mime) cos(nq>) )+jE0 Z CanZ) (k0a)cos(n(,6) :
n- — 0 n— — 0

(X3
”30 Z [An'Jiz(’*'ti‘l')+BnYy’ztka‘Hcostw) (3.73)
n = 0

Applying orthogonality Of the 0050“?) functions gives:
I
ggenjnJMkoa) + gCnHTSQ) (koa) = [AnJfi(ka) + B‘N-Yr’ilktlfl . (3.74)
o

The next boundary conditions that is applied is the continuity of tangent. Illi‘lgllt‘lit‘

field being at r = a:

19

—" it): EnJ TMJTI(A()O)COS("¢) )+E — AER: CnH H(2 n)(A‘Ua)cos(mf2) =
0—72 — 0 -— ()00

E
0 ”Z {/1an(110) + Bnyn( (An) ](os( m2) (3.75)

"72:0

Usmg the same process described above, the summation is remm ed leax mg the An

B", and On terms:

1 ,. 'I 2 '
TIL€anJ7z.(/foa) + n—ICTLHAL )ikoa) = AanAk”) + BRA/"(kw
o 0

Since there are two equations, (3.74) and (3.76) two of the variables (an be

elmnnated by solving for them in terms of the third variable The first stt p is to

solve for An from equation (3.76):

 

 

1 n 7] (2) . . ..
A» = —€ nJ A a +—C H A‘ a cos no — 8-)", An. 3.”
n Jn(A'a.) 770 nJ n(0)+ 770 n n (0) (9) nn( ) (H)
Next, substitute in place An into (3.74) to give:
-n I . (2), _ 2 ,.
5n] Jn(Ana) + (7an (koa) — BnYn(Aa)
' J’ A'.
+ [I—IEOEnjnJMkOa) + ECnH£2)(Afoa) — BnYn(A‘a) n( ‘0). (3.78)
770 n0 Jn,(A(I)
Next factor the common terms:
, 7
£5an [J;l(k0(l).]n(ka) — é];z(ka).]n(k0(l)
I
Cu [71,?) (koa)Jn(ka)— 7’ —H.§,2)(A(,a,).1;,(Au,) =
’00
Bn [Y,Q(k(i).ln,(A:(1) — Y,,(A:(1,)J;,(Ara)] (3.7.0)

The term that multiplies Bn is equivalent to the following equation [2 tq 11 92]

Y7’1(ka)JTl(ka) —' Yn(k(l)J;l(ka) =

 

. (3.80)
’ITA’YI. ’

20

(3.70)

. ' l‘fijy

 

Finally, solving for Bn gives:

7rA‘a. , ,
Bn 2 T [571]" [J,,,(k0a)Jn(ka) — 3Jn(k0a).];l(ka)] +
0

I , ‘)
on (Hf?) (k(,a)Jn(Aca) - #115,“)(A~1,u).1,’,(A-a)]] . (3.31)
()

After solving for B”, the next step is to solve for An in terms of C". First solve
for B", from equation (3.76):
_ II -n . n (2). , . .
Bin — ———_—)’ FER] Jn(A0a) + 7(7an (A100) (0510“,?) — xln,Jn(Ax(l) . (3.52)
‘ o o

Equating this with the Eu in equation (3.74) gives:

I

. I 2 I
EanJnfk-oa) + CnHA ) (kW) = Aan(A'a)+

Yr [(3 . ‘ .
n( a) [7111572,jan(/c0a)+#CHHA2)(A‘M) *AanAft'”) - (3-83)
0 0

WWW)

Factoring out the common terms gives:

- I 7 /I
Enjn [Jn(k()(l)Yn(k(l) — #Jn(k0a)ln(Aa)] +
0

on (HA2)'(Aoa>Yn<ka>-7:1HA2)<koa>YA<ka> =
To

An [.];L(A"(L)Yn(k(l) ‘— )7,(k(l).]n(lf(l)] . (3.84)

Using equation (3.80) to simplify the expression multiplying A n gives:

—77ka , , 7 ,,
An = 2 [577]” [Jn(k()a)Yn(k(l) - n—IJ71I(k0(l)}n(k(l.):l
0

 

I

+ [H£?)I(koa)Yn,(ka.) — %H,(12)(A‘Al)l",’l(A'ti)]] . (3.85)

21

 

The derivatives of the magnetic field are needed for various equations later in

the derivation. The derivative of the magnetic field with respect to {/2 is:

1 1 E 0° . ..

72:0

01'

00
$63ng = —% 5% Z [Aan(kr) + BnYn(A1-r)] nsin(n<p) . (3.87)
n = 0

Also needed is the derivative with respect to r:

00
(9ng = 6T @770 Z [Aan(kr) + BnY-n(kr)] cos(n¢2) (3.88)
n = O
or
d ED 00
(9er = k [37— ZOMnJhUC'I‘) + BnY73(k7')]cos(n¢) . (3.89)
n :-

The last boundary condition to be applied is that of zero tangential electric
field at the metallic surface. Finding the tangent to the ellipse is necessary because
the tangential electric field is not the 2 component. The tangent vector is calculated
by taking the derivative of the position vector describing the points on the ellipse
divided by the magnitude of the same vector. Let 7”((,'2) describe the posititm

vector to points on the ellipse. Then the tangent. vector is:

 

 

—> 7"(75) ‘ ,
T = _, 3.90
(‘1’) Ir «(2)1 ‘ )
Where
?(¢) = ital cos(¢) + gag sin(<,b) (3,91)
and
7”(¢>) = (1:15),,» (3.92)

The derivative of the position vector and its magnitude are:

22

 

 

7"(11) = 41111 5111(1) + 171.2 cos(¢2)) (77(3)) = \flz,‘f(sin(3))2 + age-011(0))2

Thus

-a1 sin(¢) + l/ (12 cos(c2)

‘ \flz¥(sin(q2))2 + a%(cos(qb))2 \/a¥(sin(cx2))2 + (13(1“os(¢2))2

z 1,1+1,,1. 13.94)

 

 

 

 

By the. boundary condition the dot product of the. electric field vector and the
tangent vector is equal to zero on surface of the conductor. Also, the electric field is

equal to the. curl of the magnetic field, so
A _(1 A A A1 d A A A r d a F'
t' E '3 (tlfl‘ + tyy) ' r—BCsz +(t1‘1' + tyy) : (3%)er = 0. (3.90)
7‘ . ‘

Since the curl of the magnetic field is in cylindrical coordinates and the tangent
vector is in rectangular coordinates, the cylindrical coordinate unit vectors need to

be converted to rectangular coordinates [7, eqs. D44, D45]:

. . . . . 1
(112 +111) - (1111(1) + ysm<¢>>;a¢H§+
(115: + tyg) - (——1‘.~.sin(¢) + 3)cos(d2))07'H:i = . (3.90)
Multiplying out the unit vectors gives:
1
(tr cosh-D) + ty sin((p));0¢H§ + (—t;p sin(o) + 1y cos(o))0,-Hg = 0. (3.97)

The derivatives of the magnetic field are inserted to the equation from equations

(3.87) and (3.89) to give:

23

 

 

 

(a —a )sin(¢2 cos(¢
2 110“;
Tn [Aan( ffI') +BnYn( A'I) )II 5111 (71(1))
)+ 77

 

 

 

 

 

 

(/1,(111(1)2+1g(101(.i»2: =
(11(sin(d2))2 + (12(cos()E000
— :0 [Aan( (AT) + Bnin( (A7) )]A.(os( 77(12)— — O.
\/a%(sin(nd2))2 + a%(cos(9)5))2 "n =
(3.93.)
Define C(92) and 5(9)) to simplify the previous equation:
C(05) : (a2 — a1)sin(¢)cos(9t2) ' (3.99.)
(Memes)? + a3<cos<c>>>2
— (11(sin(92))2 + (12(cos((D))2
0(7)) = ‘ ‘ (3.100)
\/a%(sin(nq$))2 + c13(cos(92))2
Also define {7111 7,11, gnl, and gm as:
2 ’ 7 2 ., _ _
fnl : —H7(1 ) (k()a)Yn(ka) + #Hfl )(A:0(1)Y,,(A'u) (3.101)
0
— 2 ’ 7 2 , , ,
fnl = HTS ) (k00)Jn(ka) —' 5271172 )(AC()(I.)J7II(A7(I) (3.10.2)
977.1 : —enjn [Jh(A'0a)Yn(ka) + r;—’J;;(A'0(A)Y,',(A'u,)) (3.103)
0
' I I .
ynl : 571]" [.]h(Atoa.)Jn(/€a) + éJn(A'Oa)Jn(A'u)) . (3.104)
Using these, An and En can be written as:
aka rka ._
An = — 2lCnfnl + 9711i 371— — *iC’nfnl + 9721] (3'10”)

Equation (3.96) becomes

24

 

27'
n =
— 7rka 00 —
G(9’>)—§—k Z [(071nt + gn1)J,'1(k7) + (Cnfnl + §n1)Y,’7l(kr)] cos(n(;'>) =0.
n = 0

(3.106)

The common terms drop out and the point matching technique used in section 3.1 is

used to give: E

G(¢>m,) Cnfnl + gn1)Jn(kr) + (anf—nl + §n1)lr'}z(kr)] vn.sin(n9)m)+

‘SIH

 

€(¢m)k [(Cnfnl + gn1)Jh(kT) + (0717711 + §n.1)Y7,2.(k7')] (305010777) 2 0.

oo
2 [(
n=0
00
2
n=0
(3.107)

The terms of the fnlv 7711, gnl, and am are combined separate the Cu terms and

non Cn terms. The first expression in the first summation can be written as:

fnlJnUCTm) "l" THIYn(kTm) =
— H7(z2)'(koa)Yn(ka)Jn(kTm) + #211792)(k00)Y7’l(A‘a)Jn(A‘I‘7n)+

’10

Combining like terms gives:

fnlJn(k'rm) + 7n1Yn(k7'm) = H192) (koa) [Yn(k7m)o]n(k0) ~— yin(k(l).]n([17"];)1 '—

%H7(z2)(koa) [Yd/«mitten - Y7<ka>Jn<A~rm>1. (3.10!»

The last step is to simplify the equation by using an and Fm,” defined in (3.11)

25

and (3.42), since the terms appear in multiple expressions allowing a simplification

of the expression:
In,1Jn(krm) + fn1Yn(krm) — anHn (koa) - TanH" (7.0a) (3.110)
0

Simplifying the second expression in first summation gives:

977,1 Janffm) + EniYnWm) =

Enjn — Jh(k0(l)yn(ka)Jn(kTm) + g—Jn(koa)Y;L(k(l.).]n(Af’I‘-7n)‘i‘
0

J;1(k0a)Yn(kTm)Jn(ka) — E.]n(f€0(l)lfn(k7777)IfiUia) . (3.111)

770

Combining like terms gives:

gnljnlkrm) + §n1Yn(krm) :

572.7” szlkoa) [YnfkT7n)J77(ka) “’ Ynlk‘llhdlfrin)1 —

"EOJnf/COG) [Yn(kTm)Jfi(ka) — Y'zli.(k”)']n(k7‘=7n)l ]' (3112)

The last step is to simplify the expression by using F mn and Fm":
_ . 7 —: , . .
gnlJnfkrm) + 9n1Yn(k7'm) = 571]” szfkoa)an — #Jnmuflif 77m] - (3113)
0

The next expression to be simplified is the first expression of the second suininat ion,

giving

fnlJfi(k'rm) + 7711177027771) =
_ (2), I 77 (2) . ,, . I ,. .
Hn (koa)Yn(ka)Jn(kT7n) + ’0an (koa)Yn(Aa,)Jn(A7m)+
2 I f 2 I I 7
H5. ) (koa)Yn(k7"m)Jn(ka) — 71,3115. ’seamlsrmiln(kn). (5.114)

26

 

Combining the like terms gives

’ — ’ 2 I I f I
fnlJn(k7'rn) + fn1Yn(kTm) = Hg ) (koa) [Yn(krm).]n(ka) — T71,(AT(I)J"(A'I‘77))] -

1H,?)(koa) [Wl(k7'm).],’1(ka) — Y5(7a)1;,.(tr,,,,)] . (3.11.5)

"I0

The last step is to simplify the equation by defining Flmn and Flmnz

 

.1;
ntJfiAkT'm-I + Tnlyfiwrm) = Flman7(72),(koa) _ %F177777H7(f2)(k00) (3-115) ’
Flmn = K3(krnL)Jn,(ka) ‘ Yn,(A‘a).I;,A(A‘7‘m) (3-117)
Flmn = YMkrmNMka) — Yfi(fi’(lv)t]7"z(k7‘7n) (3.118) L
The last expression to be simplified is the second expression of the second

summation:

9711 Jfi (k-I'm) + .6771 Hz (krm) =

. 7 I I
Enjn —Jh(Atoa)Yn(ka)J;l(k7‘n-L) + n—IJn (Aroa)Yn(A'u.)J,L(A:7'm )+
0

Jh(k()(l)Y7’l(kTm)Jn/(ka) '— l.]n(k()(l)y'7ll(A'I'nl).l;l(kn) . (3.11.0)

7I0 ' ’

Combining like terms gives

911,1J71.(I‘7Tm)+§n_lyrli(krm) 2 571]” [JhUVoCU [lesz‘r7nlln(k0) — yhlkfll-Ih (AT/72)] *—

”lanUcoa.) [Yr’sz'IVnWfiUYU ‘- l/7I1(A3(1,)J;I(A‘I’7n)JJ . (3.120)

Simplifying the equation using Flmn and 7177177 gives

. _ , 7 —, ,
gnlJi-dm‘m) + 9n,1Yri(k7‘m) = €an [Jilkoawmn — #1710770” 177m) - (3.131)
0

27

Using each of these expressions, equation (3.105) becomes

)1 :00 l— (2)
C71, III) -— [I‘WmnH(’( (kofl )— 77 —FTIlan (1:00)] 7‘1811‘1(IIC))73)
0

(a

72:0
00 7— (‘2) . . .
(7195",)sz 171",an 2)(zcoa)— 37,771",an (7.00,) 004,799,”) =

00
. . 1 7 — . ,
— E0677)" GW'm); E [anJhlk-oa) '7 afianJnUfnllfl "811101997721
77 = 0

oo
+5(9bm)k Z [Flmnjfz(k0“)—%Flnin,J71/(k'0”)] cos(77957-n) . (3.122)
n = O
Truncating the sum at n=N and using N+1 matching points, (3.122) can be written
as a matrix equation of the form 2 AmnCn = Bm, which can be solved for C”.
After finding On, the last step is to insert the value of Can into elrwtric scattered
field (3.68):
I

00
E3 =jE0 Z CnHT?) (11:07:) cos(n<,a) (3.123)

For k0r>>1 The Hankel function can be approximated using the following equation:

(2), _ _ 2j n+1( 3—Jk0, 7
Hn (Is-or) — (Ark—0T7 (3.124)

Substituting this into (3.123) gives:

Eg = E0(/ #1607”: C nj 6 11‘0" cos(7m) (3.12:3)

=0

 

28

 

3.4 TEz Case Comparison

The TEz case comparison is to check the derivation for the elliptical case wit 11

the circular case that was derived in Heike Vollmer’s thesis [10]. The ellipse is now a

circle with constant radius rm = b. As done in the TMz case, only the nth term will
be used in the comparison for simplicity. Since the metallic core is now a circle. and

the coating has radius a1 = a2, the C(96) expression now gives:

(a1 — a1) sin(9b) cos(96)

 

 

 

 

C(99) = = 0. (3.126)
(Meme)? + a¥<cos<¢>>2
For C(95), letting a1 equal to a2, gives:
m) 2 “1(Sin(¢))2 + a1<cose>>2 = __1__ = 1. (3.12,,

Va? (sin(7791)))2 + a%(cos(9’9))2 0.112

Starting with the tangent electric field equal to zero at the dielectric material

interface with the metallic core and solving for Cn, gives:

710

’71, ((0); Z EanHn (1600,) — anHn (Aoa) Tl SlIl(II-(,f)7n)
n = 0

w I
+5(e)k Z [‘n—OFlmanz?) (koa) — FlmnH£2)(k0a)] cos(n(pm) =
n = O 7)
- 1 00 710 —1 -
— E0577]n C(72); : [ganJMkoW “ 1‘77272Jn(k0")] "7 5111("(757771
n = 0

00
+a(¢)k Z [ngFlanh(k0a) —‘ Flann(/€()a):l COS(II(,-1)'fn) . (3.128)
n = 0

Simplifying the equation using (3.126) and (3.127) gives:

29

 

Ti H . ' '.a

oo
' . 2 ’ — 2
Cn ’6 2 [2911771an ) (koa) — F1mnH7g )(koa)] cos(m,’)m):| =

 

n = O 77
00 "I _
_ Eoenjn k E [ToFianWW — Flann(k0a)] coswm) . (3.129)
n = 0 I
Orthogonality of the cosine function can be applied since rm 2 l) is a constant.
Also, substituting for Flmn and flmn from equations (3.41) and (3.42) with the E
original Bessel functions for the comparison gives: 7‘
|
. <2)’ ’
on :—:)[Y,’l(krm)Jn(/ca) — Yn(ka)J,’1(krm)]Hn (kw)

— [Y7’1(krm)J;z(ka) — misting/«mu H722)(koa)] =

. 7 I I I
— EDEan [% [Yn(]€7‘nl).]n(ka) — Yn(k(l)Jn(k7'7n)] Jn(/~?0a)

— l}?g(krm)J;l(ka) — Y};(ATCI)J;I(kT'IIL)] J-nUE'MOJ (3.131))

1

Define D and L to simplify the fraction:

 

 

JRUW) Yn(ka)J;L(k7'm) ‘
D = — .

Jn(koa) Yfi(k1'm)Jn(k0a) (3 131)
— JWW) YMkakarm) (3.132)

 

 

L _ — .
#10900) Yr'i(k7'm)Jh(koal

Solving for Cn gives:

 

 

 

—E0€njn [D -— 1 L] __J7;(koa)
C “5 Hfl )(koa) .
,n = (2), . (3.1.33)
H72, (koa).]n(koa) 1
D (2) - fl]
Hn (koalJMkoa) ”r

This expression matches with the expression for Cu in Heike Vollmer’s thesis on [10,

p. 18] with minor corrections, since the radii are named by different methods.

30

CHAPTER 4

Computational Considerations

4. 1 Computation Method

 

The computer code written to calculate the frequency domain data for the T122
and TMz scattered electric field has the following structure: placement of the ellipse
matching points, calculation of the values of the Bessel functions, normalization of
the matrix entries, and evaluation of the matrix. Appropriate placement of the
points on the ellipse is necessary so that the matrix is not singular. Placement. of
the points on the ellipse is done using the following formula:

afifiiH-fiVZ—l), -i=0,1,...,N (4.1)
After the ellipse is set up and the step size and number of the frequency points
is chosen, the wave numbers are calculated. The Bessel functions for the matrix are
calculated using a routine provided by Dr. Edward Rothwell based on the formulas
in Abramowtiz [1]. The results from the Bessel functions are scaled down by
applying a natural logarithm to the calculated values to prevent a numerical

overflow when the actual value of the Bessel functions are used. For the T My. case.

. 2 ’ . . .
equation (3.48) has H79, ) (koa) factored from the expansion coefficient as shown:

31

00 (2)
I —-.; H
2 CR (1an ' I‘mn—fi-YM 0050299,”) 2:
n. = 0 770 Hn (koa)

OO
. 7 —w ‘
‘- Z 5.an [#Jh(k0a)an — I‘mn,.]n,(k()a) (TOS(‘I'I(§))n) (4.2)
n - 0 0

where as

2

4, = 6‘an Vega). (4.3)

And equation (3.51) becomes:

 

oo , _
E; 2 E0 2 e—JUW ‘ 71/4) ——C:"—— c()s(n¢>)e“J"7‘/2 (4.4)
7rk07‘ n = 0H7?) (koa)

This normalization allows the usage of larger arguments and orders of the Hankel

function without overflowing the variable in FORTRAN. Two other methods for

I
normalization that were explored are dividing through by H52) (1:00.) and dividing

I
through by H’max(k0a). Dividing equation (3.48) by H572) (13.00.) gives:

00 (2)

r — H k
2 Cn —,an an n2 ,( 0 ) COS("¢m) =

n = O 7’0 H’EL ) ((900)
00
_ J' k a — J , k. a ,
— Z enjn 1%an—an—(gérfl— cost/Wm.)- (4-5)
71 = O 710 Hn (koa) H72, (koa)

I
The division by H792) (koa) was rejected because the results did not always match

the case without normalization. Dividing equation (3.48) by H’,,m_.r(k(,a) gives:

00 (2) .
2 Cu gan —anHTg),(flg)-

Tl = 0 O Hmaa:(koa)

C05(’"¢m) :

°° -n n thkoa) —. Jn(k0(l)
— E 677,] n—W—Fnln—Pmn—TQT—
'71 = 0 0 HmaacUCOG) Hmaflkoa)

cos(nom ). (415)

32

 

I'XK'KZL“- hfi'h‘n L1

. . . 2 ’ . .
The d1v1sion by Hgnzlflkoa) created zeros for all pomts 111 the data range.

After the matrix is set up, the next step is to solve matrix using an LC
decomposition routine. Lastly the On term is inserted into equation (3.51) or
equation (3.125) to get the scattered electric field value. For the TMz case. the
FORTRAN code is in Appendix A, and for the TEz case, the FORTRAN code is in

Appendix B.

33

 

4.2 Flat Case

The coated planar surface or the flat case is the standard reference for the
comparison in the time domain using the E—pulse technique. The equation that is

used to create the scattered electric field for the flat case is [9]:

71— 170 _ e—ijd

 

R = ’7 _ _ (4.!)
11+ 7m

Here d is the coating thickness and 17 = Mpg/(574m). To compare to the coated
ellipse, values of d = .1 m and 5r = 9 are used. The equation is coded in FORTRAN

and is shown in Appendix C. Magnitude and phase plots of R from (4.6) are shown
in figures 4.1 and 4.2, respectably. The magnitude of the flat case is always 1
because the incident electromagnetic wave is entirely reflected back from the flat
surface. The phase plot shows the typical pattern of variation between 180 to 480
degrees. Applying a 32768 point FF T to the frequency domain data gives the time
domain data shown in 4.3. The series of pulses represents the multiple reflection

within the planar coating. Note that the waveform is zero in between the pulses.

34

 

IITT‘ uL :.. .

.s
N
l
t

 

A
l

 

.0
co

Normalized Field Magnitude
p o
A O)

.0
N

 

l
o- J
l l l l f - l 7.

 

0.0 0.3 0.6 0.9 1.2 1.5 1.9 2.2 2.5
Frequency (GHz)

Figure 4.1. Flat. case normalized field magnitude vs. frequency with d : .1 m. S;- 2
9: [tr 2 1

35

'3 (deg)

200

100

-100

-200

 

-1

.4

 

 

 

 

 

 

0.0

1

0.3

I

0.6

T

0.9

Frequency (GHz)

1

1.2

l

1.5

T

1.9

2.2

T

2.5

Figure 4.2. Flat case phase vs. frequency with d = .1 m, 5r = 9, pr = l

36

 

 

1.00 gm -_ W3

 

 

 

 

 

 

 

 

0.50 4
O
'3 1
3 o 00 A
g [ [l
<
'5 -0.50 4 ‘
d)
.5
'5 [ .
E -1.00 ~ ;
o 1
z a
-1.50 i
E
'2.00 T I r r r l T T _. ‘
0.0 4.0 8.0 11.9 15.9 19.9 23.9 27.9 31.8 35.8 39.8
Time (ns)
Figure 4.3. Flat case normalized amplitude vs. time with d = .1 m, Er = 9. pr 2 1

37

4.3 Circular Case

One way to check to see if the equations for the scattered electric field are.
derived correctly is to compare the On terms of the elliptical case to the circular
case. The circular special case occurs when the major axis and minor axis are. equal
in length. Since in Heike Vollmer’s thesis [10], the circular case data was already
calculated, the elliptical case formulas can be setup to match those of the circular
case. The equations (3.56) and (3.133) for the TMz case and the T1112 case.
respectively are used for comparison with the On term from Heike Vollmer‘s thesis.
As shown in sections 3.2 and 3.4, the On are the same. Data is calculated using the
elliptical case with the major and minor axes set to the chosen radius of the circular

case are found to be identical to the data shown in Heike Vollmer’s thesis.

38

4.4 Echo Width of Elliptical Metallic Core

Another way to check that the scattered electric field equations are derived
correctly is to calculate the two dimensional radar cross section or echo width of the
uncoated elliptical cylinder [5]. At high frequencies, the echo width of a curved
surface approaches 7r times the radius of curvature at the plane-wave contact point.
Let x((;b) and y(qb) describe points on the surface parameterized by the variable 0‘.

Then the radius of curvature is given by the following [12]:

((202 + (y’)2)3/2

 

R = W, _ 31,513,, (4.8)
The x,y coordinates, the first derivative. and the second derivative of x.y
coordinates are:
1: = a1 cos6 y = a2 sin6 (4.9)
a:' = —a1sin6 y' = a2 cos6 (4.10)
:23” = —a1cos6 y” = —a2 sin 6. (4.11)

Substituting into equation (4.7) the formulas for x, y and their derivatives gives:

((—a1 Sin a)? + (a2 0086)?)3/2

 

 

R = . . . (4.12.)
(—a]L srn6)(-a2 Sin 6) — (—a1 cos 6)(a2 cos 6) '
Simplifying the equation for the radius of curvature at any angle yields:
((al sin 6)2 + (a2 cos 6)?)3/2
R = . (4.13)
“1‘12
Along the x—axis where 6 = 00, the radius of curvature is:
3 2
a a,
R = —2— = ——2-. (4.14)
alag a1

Along the y-axis where 6 = 900, the radius of curvature is:

39

R z .—= — (4.1.3)
0.102 (1.2

 

The echo width 0 of the metallic cylinder can be calculated using the FORTRAN
program in Appendices A and B to find the value of B: using 5r 2 1 and
substituting the value into the definition [2, 11-102]:

r|E§|2

2
E0

a=27r

 

(4.10)

 

 

Figure 4.4 shows the echo width found using a1 = 1.4 111 and a? = 1.3 m, with
x as the major axis. The radius of curvature with x as the major axis using
equation (4.13) is 1.21 m, giving a high-frequency limit to the echo width of 3.79 111.
The value of the echo width for k = 4.0 l/m from Figure 4.4 is 3.89 111. Figure 4.5
shows the echo width found using a1 = 1.4 m and a2 = 1.45 m, with y as the major
axis. The radius of curvature with y as the major axis found using equation (4.14)
is 1.51 In, giving a high-frequency limit to the echo width of 4.72 m. The value of
the echo width for k = 4.0 l/m from Figure 4.5 is 4.79 m.

40

 

 

 

 

 

[—_‘m Tit/IE]
t' I " TE?’
O _.' r 1 r r T i
0.0 0.7 1.3 2.0 2.6 3.3 4.0
k (1Im)

Figure 4.4. Echo width vs wave number, x as major axis, a1

41

= 1.4 111, a2 = 1.3 111.

 

 

 

 

 

(:— wry

Lt; TEA] :

o . . . . .

0.0 0.7 1.3 2.0 2.6 3.3 4.0
k(1Im)

Figure 4.5. Echo width vs wave number, y as major axis, a1 = 1.4 1'11, 21;; = 1.45 111.

42

CHAPTER 5

Results for the Elliptical Case

5.1 Frequency Response

The frequency domain plots of the TMz and the TEz cases are created using
either the x axis or the y axis as the major axis. The major axis that is chosen
greatly affects the results of the numerical calculations as shown in the plots. The
TMz and the TEz cases with the same major axis produce a similar result with the
largest difference occurring at the lower frequencies. After the results are calculated.
magnitudes are normalized by multiplying the data by \/r/_a/E0, for a normalized
field magnitude. The various sizes that are chosen shows the variance of the
different cases with a small, medium, and a large radius of the cylinders.

The frequency domain for the TMz cases uses equation (3.51) for the scattered
electric field. The On term is found by solving the matrix equation(3.48). The
FORTRAN software used to calculate the frequency domain data in appendix A.
For the TEz case, the scattered electric field is given by equation (3.125). The Cu
term for the T132 case is the matrix equation (3.122). The FORTRAN software used
calculating the frequency domain data is in appendix B. The dielectric thickness on
the x axis is 0.1 m and the dielectric constant, Er = 9. The number of points on the

ellipse is chosen to be 275, since the TMz cases at a = 2 m needs that. 111any points

43

to converge the data. The number of steps for the frequency domain data affects the
time needed in the time domain data. The x major axis uses 500 steps for the.
frequency domain since the wrap around in time domain does not occur. However,
with y as the major axis cases, the window in the time domain is not. big enough
with 500 steps, 2000 steps are needed to have the creeping wave decay out.

The magnitude of the TMz scattered electric field with x as the major axis, in
the frequency domain becomes less oscillatory as the radius increases as shown in
Figures 5.1, 5.2. and 5.3. Another characteristic shown in all the TMz scattering
electric field plots is when the frequency approaches DC, the magnitude goes toward
infinity. For the TMz scattered electric field plots with y as the major axis. Figures
5.4, 5.5, and 5.6 shows an increased rate of oscillations as the radius of the cylinder
increases. In Figures 5.7, 5.8, and 5.9, the T137. scattered electric field having x as
the major axis has a similar shape to the TMz with the same major axis, but the
lowest frequencies now approach zero instead of infinity. For the. T132 scattered
electric field with y as the major axis, the oscillations are occurring at a higher rate
than the TMz case as shown in Figures 5.10, 5.11, and 5.12. The phase plot shown
in Figure 5.17 is typical of all the phase plots with the major difference being that.
the slope of the phase is steeper for a larger cylinder radius.

The field scattered by a conducting elliptical cylinder without a dielectric
coating is needed to construct difference data. The difference data is created by
subtracting the field scattered by the uncoated cylinder from the field sct—rttcrctl by
the coated cylinder, and is used for comparison of the results of the E-pulse
technique. The two elliptical metallic core plots are shown as typical results for the
TMz and the TEz cases in Figures 5.13 and 5.15. In each of these plots, 5, was set.
to 1.01 to achieve zero coating. The TEZ scattered electric field inagnit udc from the
elliptical metallic core have the lowest frequencies approaching zero and at higher

frequencies approaching a constant. For the TMz scattered electric field from the

44

metallic core, the field at the lowest frequencies approaches infinity and the higher
frequencies approaching a constant. The magnitude difference shown in Figures 5.14
and 5.16 is typical of the difference data. The TMZ difference data no longer goes to
infinity as the frequency approaches zero. This result produces a time-domain
waveform that is closer to zero between reflections than is the original data. The
result of the difference data is converted to the time domain and the time domain
data has the E-pulse technique applied. The results of using difference data wit 11

the E—pulse technique are explored in the next chapter.

45

 

- .. -_‘-_—.-. fi-o... -mfimmo—Aw *4

Normalized Field Magnitude

 

 

 

o l l T 1 l T l
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

 

Figure 5.1. TMz normalized field magnitude vs. frequency, a = .9 1n, a1 = .8 m, 212
= .7 m, with x as the major axis.

46

 

 

 

e 5
13 .
3 f
.2 s
g, f
a 1 ‘
E
2 t
.2
I].
'U I
3 i
-_= 0.5 ~ 1
E 2
o
Z
0 l l l T l l 1 i
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.2. TMz normalized field magnitude vs. frequency. a = 1.5 111. a1 =2 1.4 m,
a2 = 1.3 m, with x as the major axis.

47

 

 

Normalized Field Magnitude

 

i

l

i
o 1 1 r 1 1 1 1 1

 

0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5
Frequency (GHz)

Figure 5.3. TMz normalized field magnitude vs. frequency, a = 2 m, a1 = 1.9 111. :12
= 1.8 m, with x as the major axis.

48

 

 

 

 

 

 

 

 

 

1
o 1
'g i
:1: 1.5 1
= 3
a t
I!
E
2
.2 1
u_ 1
1:
d)
.L‘
E 1
g 0.5 1 I 1
o 1 1
z 4
1
o f l T l l l l J
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.4. TMz normalized field magnitude vs. frequency, a = .9 m, 3.1 = .8 m, a2
= .85 m, with y as the major axis.

49

 

N
1
1

_|.
U"

 

.0
01

 

Normalized Field Magnitude

 

 

 

0 l T l l 1 F F.
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequnecy (GHz)

Figure 5.5. TMz normalized field magnitude vs. frequency, a = 1.5 m, a1 = 1.4 m.
a2 = 1.45 m, with y as the major axis.

50

 

_s
.
U1

1

 

 

Normalized Field Magnitude
O
'0. _.

 

 

 

0 l l T 1 T l 1 Ti
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.6. TMz normalized field magnitude vs. frequency. a. = 2 m. a] = 1.9 111, 212
= 1.95 m, with y as the major axis.

51

 

 

 

 

   

 

 

Normalized Field Magnitude

 

0 1 1 1 1 r 1 1 T 1‘
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

 

Figure 5.7. TEz normalized field magnitude vs. frequency, a = .9 m, a1 = .8 m, ag
= .7 m, with x as the major axis.

52

1 .5 “ "W“ "

 

 

 

Normalized Field Magnitude

 

 

0 l l l l l I l .
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

 

Figure 5.8. TEz normalized field magnitude vs. frequency, a z 1.5 m, a1 = 1.4 m,
a2 = 1.3 m, with x as the major axis.

53

 

 

 

 

o
'5
3 i
g ;'
C 3'
5’ 1 -‘
N .1
5 1
2
.2 1
“- ?‘
U 3
a 1 1
2: 0.5 a I
1'
E
0 i
z 1
0 l f l T T i T i

0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.9. TEz normalized field magnitude vs. frequency, a = 2 m, a1 =2 1.9 In. a2
= 1.8 m, with x as the major axis.

54

 

—l
U"
L

 

 

 

m........ -_.._,.

 

 

p
01
l

 

 

Normalized Field Magnitude

 

 

 

 

 

0 1 1 1 1 1 r 1 1 1
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.10. TEz normalized field magnitude vs. frequency. a : .9 m. al 2 .8 1n, a2
= .85 m, with y as the major axis.

55

 

.5
0'1
1

 
  
   

 

 

Normalized Field Magnitude
.0
01 .s

 

 

 

 

o T T T l l T T .
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.11. TEz normalized field magnitude vs. frequency, a = 1.5 m, a1 = 1.4 m,
a2 = 1.45 In, with y as the major axis.

56

 

 

.A
U'l

 

 

  

Normalized Field Magnitude
.0
o: .3

 

 

 

 

0 V l T T l T T .
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5
Frequency (GHz)

Figure 5.12. TEz normalized field magnitude vs. frequency, a = 2 111, a1 = 1.9 m, a2
= 1.95 m, with y as the major axis.

57

2 5 . .a.-....MM--..1.--.-.c---_-.-.---.-,
a I

 

 

 

 

 

“’ 3
g 2 . 1
:L' i
C )I
‘3’ 1
I! 1
2 1,5 1 1
u 1
"-
3 1 -
.L‘
g Nl/vww vvvvvvvvvvvvv '
o 0.5 -
z
0 T T T T l T T 7‘;
0.0 0.3 0.6 10.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.13. TEz normalized field magnitude metallic elliptical cylinder vs. frequency,
a = 1.5 m, a1 = 1.4 m, a2 = 1.45 In, with y as the major axis.

58

 

 

 

 

 

 

 

 

 

o 2 .

'u

3

'E

8’

2 1.5 4

E

.2

u.

u

.3 1‘

i

E

o

2 0.5 -1
O T T T T T 1' T T
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.14. TEz normalized field magnitude with metallic elliptical cylinder sub-
tractcd off vs. frequency. a = 1.5 m, a1 = 1.4 m, a2 = 1.45 m, with y as the major
axis.

59

 

.3
A
1

.3
N

A

.0
co
1

L

.0
0)

Normalized Field Magnitude
C
x

P
N

 

0 1 1 1 1 1 1 1 1
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

 

Figure 5.15. TMz normalized field magnitude metallic elliptical cylinder vs. fre-
quency, a = 1.5 m, a1 = 1.4 m, a2 = 1.3 m, with x as the major axis.

60

_L
.5
l
l
1

 

 

   

 

 

 

a 1.2 ~

1:

a

:2:

c 1-1

o:

N E

s

2 0.8 — é

o i

E 1 g

° 5

.5 f

E 0.4 -

o

2 0.2 ~
o I l l T T i i T?
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.16. T Mz normalized field magnitude with metallic elliptical cylinder sub-
tracted off vs. frequency, a = 1.5 m, 8.1 = 1.4 In, a2 = 1.3 m, with x as the major
axis.

61

 

200 mm?

100 +

 

 

 

 

 

 

Phase(deg)

 

 

 

 

 

 

 

 

 

-100 ~

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'200 l 1 1 1 1 1 1 1
0.0 0.3 0.6 0.9 1.2 1.6 1.9 2.2 2.5

Frequency (GHz)

Figure 5.17. TMz phase vs. frequency, a = 1.5 m, a1 = 1.4 m, 212 = 1.45 m, with y
as the major axis.

62

5.2 Time Domain Results

Once the frequency domain data has been created, the F F T is applied to get
time domain data. The program WaveCalc developed by John Ross
(www.johnross.com) has the FF T needed to calculate time domain data. The first
step is to apply a gaussian-modulated cosine of 'r : 0.5 and centered at zero
frequency to weight the frequency domain data properly. The FF T is applied using
zero padding to 32768 points and the time domain data is calculated. The next step
is to normalize the data with the amplitude of the initial reflection set. equal to one,
and to time-shift the waveform so that the initial reflections occur at. the same time.
With this setup, the relative values of the various amplitudes can be compared. and
the time origin is set to be at the point where the incident wave is contacting the
dielectric material. The succession of pulses seen in the time domain data. is
explained as a series of multiple reflection as shown in Figures 5.18 and 5.19.

In Figures 5.20, 5.21, 5.22, and 5.23, the data shows the full ranges of time
obtain from the F FT for four combinations of orientation and polarization. The.
TMz case of Figure 5.20 with x as the major axis has the smallest amplitude of the
creeping wave for the configurations considered. The TMz case with y as the major
axis has a creeping wave that is larger than with x as the major axis. The creeping
wave arrives earlier and has a strong repetition lasting longer than the case where x
is the major axis. This requires that the data be computed to 800 nanoseconds in
order to avoid wrap-around in the F FT. Changing the polarization results in a
larger creeping wave for TEz with the x axis as the major axis compared with TMz
case with the same major axis. The largest creeping wave occurs with TEz
polarization with y as the major axis, as shown in Figure 5.23.

The next set of plots shows the effects of cylinder size on the reflection of the.

pulse. The TMz case with x as the major axis has the most similar time waveforms

63

with the two larger outer radii cases almost matching, as shown in Figure 5.24.
Note that the creeping wave for the smallest arise returns earliest, and can be seen
in the plot. With y as the major axis for the TMz case, the curves are not. a. similar
as when x is major axis. For the TEz case with x as the major axis, the curves are
closer than in the TMz case with the same major axis; however, the creeping wave
starts to show up sooner in the smaller outer radii cases as shown in Figure 5.26.
The TEz case with y as the major axis shown in Figure 5.27, has a much greater
variance among various outer radii cases, comparably to the TMz case with the
same major axis. The creeping wave for the T132 cases arrives earlier in time than
for the TMz cases.

The planar surface case is the standard that is used in the E—pulse technique for
comparison with the various configurations of the cylinder. The field scattered
normally from a planar coated conductor with a dielectric thickness of 0.1 m and a.
dielectric constant of ET = 9 is computed using equation (4.1) and compared to the
field scattered from the coated cylinder as shown in Figures 5.28 to 5.39. The
amplitude of the flat case is normalized so that the amplitude of the initial reflection
is the same as in the cylinder case. With a = .9 111, all the plots have the creeping
wave appearing, since the time needed to travel around the cylinder is less than 40
nanoseconds. The TMz case with a = .9 m and x as the major axis has a smaller
relative amplitude compared with the flat case as shown in Figure 5.28. The TEz
case with x as the major axis and a = .9 m, as shown in Figure 5.34, is similar to
the TMz case, but the creeping wave is larger in amplitude. For the TMz and T137.-
cases with y as the major axis and a = .9 m relative amplitude of successive pulse is
larger than that of flat case as seen in Figures 5.31 and 5.37. For the TEz case, the
creeping wave is larger than that of TMz case. With a = 1.5 m, the creeping wave is
only appearing in the TEz case, since the traversal time needed for the TMz cz'tse is

longer than 40 nanoseconds. In Figures 5.29 and 5.35, the TEz and TMz cases with

64

x as the major axis and a = 1.5 m, the relative amplitude is still smaller than that
of the flat case. For the T132 and TMz with y as the major axis and a 2: 1.5 m,
relative amplitude is larger than that of the flat case but smaller than when a. = .9
m as seen in Figures 5.32 and 5.38. When a = 2 m, the creeping wave takes longer
than 40 nanoseconds, so no creeping wave appears in the plot. For Figures 5.30 and
5.36. the TEz and TMz cases have a relative amplitude that is smaller than the [lat
case, but getting closer to the flat case than with the smaller radii. The TMz and
T 1532 case with a = 2 m and y as the major axis is larger in amplitude than the flat.
case as shown in Figures 5.33 and 5.39. In all of these plots it is seen that as the
radius of curvature gets bigger, the field scattered from the cylinder gets closer to
the field reflected from the planar coated conductor. This suggests that it might be
possible to use the E—pulse for the flat case with cylinders of large radius.

A comparison of the difference between the placement of the major axis, as
shown in Figures 5.40 and 5.41, shows that the relative amplitude of the field with y
as the major axis is greater than that with x as the major axis. This occurs because
the y major axis has a greater radius of curvature than that of the x major axis. For
the T Ez case, the creeping wave occurs simultaneously, with the y major axis
having a larger amplitude than the x major axis as shown in Figure 5.41. The
subtraction of the metallic core data from the frequency data results in the time
domain having a flatter response at the cost of adding a false peak as shown in
Figures 5.42, 5.43, 5.44, and 5.45. With both TMz cases, the subtracted case is
closer to the flat case than that non-subtracted case. For the TEz cases, subtracting
off the metallic core does result in the time domain waveform being flatter than the
non—subtracted case. However, the subtracted case results in a larger EDN number

than the non-subtracted case, with the results being explored in the next chapter.

65

W ”11:;

 

Figure 5.18. The electromagnetic pulse multiple reflections, x is the major axis of the
ellipse.

32 a

e <9
/// ,f“

 

 

Hi

 

Figure 5.19. The electromagnetic pulse multiple reflection, y is the major axis of the
ellipse.

 

1.50 -- . ,

1.00 -

0.50 4

 

0.00 . rev:-
050 ~

-1.00 ~

Normalized Amplitude

 

4-:~‘—V........‘ .___.-,_ -_4

-1.50 «

 

 

-2.00 TI I I T I I I. 7
0 20 40 60 80 100 120 140 160 180 200

Time (ns)

I T

Figure 5.20. TMz normalized amplitude vs. time, a = 1.5 m, a1 = 1.4 m, 21.2 = 1.3
m, with x as the major axis.

67

 

1.50 m-“ . _ .

1.00 a

 

.0 .o
O 01
O O

L

-1.00

Normalized Amplitude

.150 ~ g

D
l
1
l
l
I

 

 

'2.00 I I I I I I I I I I
0 80 160 240 320 400 480 560 640 720 800

Time (ns)

Figure 5.21. TMz normalized amplitude vs. time, a = 1.5 m, a1 = 1.4 m, a2 2 1.4.3
In, with y as the major axis.

68

 

1.50 - M -_ w n.

1.00 4

0.50 4

 

-0.50 4

-1.00 ~

Normalized Amplitude

 

-1.50 -

I
l
5
l
l

'2.00 I I T I I I I . I 1‘
0 20 40 60 80 100 120 140 160 180 200

Time (ns)

 

 

Figure 5.22. TEz normalized amplitude vs. time, a = 1.5 m, a1 = 1.4 111, £12 2 1.3
m, with x as the major axis.

69

 

 

 

 

 

 

1.00
4’ i
1; 0.50~ :
.2.-
E-
0.00 W;
< s
'3 ?
_§ -0.50 ,
I
z I
-1.50 I
-2.00 I I I I I I I I I

0 80 160 240 320 400 480 560 640 720 800

Time (ns)

Figure 5.23. TEz normalized amplitude vs. time, a = 1.5 In, 81 = 1.4 m. a2 == 1.45
m, and with y as the major axis.

70

 

 

 

 

 

 

 

 

 

1 _ 00 H W‘ ‘ “ "2 " “_.;_ ' L14, ‘5
------ a=.9 m [
— — — a=1.5 m
_____?_=.__2_"1j
g 0.50
'5 i
3 i
.‘".'.' I
E 000 - i
I “‘10-
u i
a i
g -0.50 — i
g
é,
-1.00 -
'1.50 T I I I r I I I I 3

0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.24. TMz normalized amplitude vs. time with a = .9 m, 1.5 m, 2 m and with
x as the major axis.

71

 

1 50 W z . I

------ a=.9m

 

 

 

 

 

 

 

 

---a=1.5mi%

1.00 .. :'- ___---.a=2m_ H

.g
g 0.50 ~ I
TE: i
_ a - -W #53

< 0.00 I 3
3 I
,5 -0.50 - f
E i
z I
-1.50 - i

'* I

-2.00 I I I I I , , , T

0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.25. TMz normalized amplitude vs. time with a = .9 n1, 1.5 In, 2 1n and with
y as the major axis.

72

 

 

 

 

 

 

 

 

 

1 . 00 ”' ""F. 117::7‘ '1,’.I".".'.'..~‘. :2”
------ a=.9m I

- - -- a=1 5 m

____ ,_-a,,=2 m _}

a, 0.50 — 3
1: E
3 I
z . E
'5. .. ' M- I
E 0.00 I—-——‘ m f L‘~.‘.,‘.>g_.—"‘~?’I
< ‘
13
:3 :
E 050 ~ I
§ M
-1.00 - I

I

-1 .50 I I I I I I I I I
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.26. TEz normalized amplitude vs. time with a : .9 m, 1.5 m, 2 In and with
x as the major axis.

73

 

 

 

 

 

 

 

 

 

1 .50 ""‘M~:f"_"_:_"'" “""fl‘ ” E
’ ------ a=.9 m . .
:: —--a=1.5m1§
1.00 “ L H_. 3:231 3 i

3
< 0.00 V— ‘f s:’\ 4‘ ’4 ij‘,
'8 i
,5 -0.50 ~ :, ;
'g ‘s t
t
3 -100 ~ ?
z i
-1.50 .
t
i
”2.00 I v r T I I r 1 1‘

0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.27. TEz normalized amplitude vs. time with a = .9 m. 1.5 m, 2 In and with
y as the major axis.

74

 

1.00

 

 

 

 

 

 

 

 

0.50 ~
0
.5
3
E 0.00 ~
o.
E .
'o -0.50 -
o t
I: U :
g -1.00 ..
o
z

-1.50 ~

'2.00 I I I i I I I T Y

0 4 8 12 16 20 24 28 32 36 40
Time (ns)

Figure 5.28. T M7. normalized amplitude vs. time comparison of flat with a : .9 m
and with x as the major axis.

75

 

1.00 - —~———

0.50 4

0.00 -

 

-0.50 -

 

- - _. _.. ._ 2... ........~.._... . m»..- - . - . ....

 

 

 

400 ~ u

Normalized Amplitude

-1.50 -‘

 

—<..__.._..—.._‘ _——..“n. I. -4.— . r V 4

 

'2,00 I I I I T I I I I
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.29. TMz normalized amplitude vs. time comparison of flat with a : 1.5 m
and with x as the major axis.

76

 

 

 

 

 

 

 

Eiffégzm‘i I

_ ,-_._F'a‘_,_,i§

0.50 -- E

m
1:

a I

-.-.- 0.00 a . . - f. ...... *1 E

Q s

5 i

t -0.50 a

‘” +

.5

TI 9

g -1.00 a 1

2 I

-1.50 —‘ i

-2.00 I r I I , , f Y I

0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.30. TMz normalized amplitude vs. time comparison of flat with a = 2.0 m
and with x as the major axis.

77

 

1.50

 

 

 

 

 

 

 

.:~::—..;;g,.,~¢
i I
1.00 # l__-_'_:.'§[~__!
4‘;
3 0.50 ~
'2
< 0.00 ~ I
'3 :- ‘
,5 -0.50 — :: l
' I
'5 -1.00 — i
z i
-1.50 - .. i
-2.00 I I T T I I . . I
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.31. TMz normalized amplitude vs. time comparison of flat with a = .9 1n
and with y as the major axis.

78

 

150 .,-_-_-._--..-...-..--.

I ..... ' "5:115"! i

L_-____f_'?_‘___1

 

1.00 4

I
.'
l
l
l
l
l
i
I

0.50 ~

0.00 A

 

-O.50 -+

 

 

-1.oo .. I

Normalized Amplitude

-1.50 .

 

 

-2.00 I I I I I I I I
O 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.32. TMz normalized amplitude vs. time comparison of flat with a 2 1.5 m
and with y as the major axis.

79

 

1 .00 3'- "WW”””___'_"_"__': f, 2' E

0.50 —<

 

0.00 ‘
l

 

-o.5o -- , '~'

 

 

-1.00 —

 

Normalized Amplitude

-1.50 ~

J
T

 

T I l

.1

 

'2.00 I Y T T T
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.33. TMz normalized amplitude vs. time comparison of flat with a : 2.0 m
and with y as the major axis.

80

 

1 00 ---_---.._.....
I l
l

'_ -_ V_ l

 

0.50 ~

0.00 1

 

-0.50 «

 

 

 

 

4.00 - l

Normalized Amplitude

-1.5O --

 

-2.00 I I I I I I I I F 1
O 4 8 12 16 20 24 28 32 36 40

Time (ns)

 

Figure 5.34. T Ez normalized amplitude vs. time comparison of flat with a = .9 1n
and with x as the major axis.

81

 

 

1.00

 

 

 

 

 

 

 

0.50 ~
o
'o
3
= 0.00 1
o.
5
'o -o.50 ~
0 .f
.5 :i
'5 I
g -1.00 ~
o i
z a
-1.50 I
‘2.00 I I I I I I I I 1
O 4 8 12 16 20 24 28 32 36 40
Time (ns)

Figure 5.35. TEz normalized amplitude vs. time comparison of flat with a z 1.5 m
and with x as the major axis.

82

 

1.00 ~-—~-—~ -
0.50 -«

0.00 -

 

-O.50 ~

 

 

 

 

-1.oo ~

Normalized Amplitude

-1.50 ~

 

 

~45.—————.—-M. h. -_. . _. ._.. 24.. 4

”2.00 I I I I r I T I I
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.36. TEz normalized amplitude vs. time comparison of flat with a = 2.0 m
and with x as the major axis.

83

 

1 .50 ._...._-__----.-E

------ a=.9m &
Flat _

1.00 ~ 5; ~——————-

 

\

 

 

0.50 ..

0.00 -

~ u
D
.-.....-—~.~.—ab--—.-_ v..." . ~._ - .....

 

-o.50 . ii

 

 

 

Normalized Amplitude

-1.oo - l

-1.50 ‘4

 

 

-2_00 l l T l l l l l ' .
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.37. TEz normalized amplitude vs. time comparison of flat with a = .9 1n
and with y as the major axis.

84

 

 

1.50
pitta—I15";

L” flat .

 

1.00 ~

0.50 ~

 

0.00 ~

 

-0.50 —

 

 

-1.00 --

Normalized Amplitude

2-42.... -m.fi...¢......v. 2 .

-1.50 *4

 

 

'2.00 I I I I I I I I I ,
O 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.38. TEz normalized amplitude vs. time comparison of flat with a : 1.5 in
and with y as the major axis.

85

 

1 .50 --_........---._..--- .. .
1.00 .
0.50 ~

0.00 -

 

-O.5O -

 

 

-1.oo . ;

Normalized Amplitude

-1.50 -

 

b—~-_-.~.Hr,_... n“ -..-A —

 

-2.00 I
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

1 l T l l T T 1

_1

Figure 5.39. TEz normalized amplitude vs. time comparison of flat with a = 2.0 m
and with y as the major axis.

86

 

 

1 _ 50 .2-~.-_-_,_ _._..-..,-..-- ..._.-..._ .‘

{7? - x MajAor‘AxIsfiI;
Y M6170! Axis I 3

1.00 . --—--—

 

O.50 -

 

0.00 -

 

-O.50 -

 

 

-1.o'o I

Normalized Amplitude

-1.5o ~

 

 

'2.00 I I I I I I I I I I
O 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.40. TMz normalized amplitude vs. time comparison of x vs y as the major
axis, a = 1.5 m.

87

 

 

1.50 . 1 -__ ~; -.
l—---- XMajorAxis:
Y_M?l°'5x‘5_.i

 

 

 

 

 

 

 

 

1.00 ~ ~--—-- I

a:
E 0.50 ~
a
E 0.00 ~ Am
13
,5 -0.50 ~
g
O .100 ~
z
-1.50 - f
'2.00 I I I I I I I I T .

O 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.41. TEz normalized amplitude vs. time comparison of x vs y as the major
axis, a = 1.5 m.

88

 

 

 

 

 

 

 

 

 

 

 

 

 

2.00
3. ----- TMSubtracted l
1 50 a '2 -___ TM Nonsubtractedl
lg ..
3 1.00~
E
< 0.504 5‘
'3 z':
,5 0.00 ~ ;
E W _
—1.00- l i
l
-1.50 IIIII'
0 4 8 12 16 20 24 28 32 36 40
Time(ns)

Figure 5.42. TMz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a = 1.5 m and x as the major
axis.

89

 

 

----- TM Subtraétgdw ' I?

l
1 l

2.50 _-..-.-._-.,__.1
l

 

 

2,00 — ; TM Nonsubtracted

;
I
W'Nl
v

 

\
l

1.50 .
1.00 .
0.50 ~ .
0.00 - 33 . fawn- a
050 ~
-100 -
-1.50 -

-2.00 I 1 I I I I I I I 1
O 4 8 12 16 20 24 28 32 36 40

Time (ns)

- .._- ‘__.., -~.,I.....,... “v »—--~ - -

 

Normalized Amplitude

 

 

 

 

Figure 5.43. TMz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a = 1.5 Hi and y as the major
axis.

90

 

 

2.50 ~ -..
’ ----- TE Subtracted }
2 00 . Evgnwbtraezdj a

 

 

- _ “.3- # --.._..

...0‘~

1.50 -
1.00 ~
0.50 4

0.00

 

 

-0.50 ~

Normalized Amplitude

 

-1.00 —

 

 

‘1 .50 1 I I I I I I I I _1
0 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.44. TEz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a = 1.5 m and x as the major
axis.

91

2.50

 

 

 

 

 

 

 

 

Ti’TE $68336 "
2.00 ~ :; l “IE Nonsytzfiracwf
3 3' g
= 1.00 r :1 ?
g- :3 ?
< 0.50 E
'U
S 0.00 ~
'5 .
E -0.50 a I
h .
0 i
z -1.00 « g
-1.50 a
'2,00 T I I I I I I I I i
O 4 8 12 16 20 24 28 32 36 40

Time (ns)

Figure 5.45. TEz normalized amplitude vs. time comparison of standard cylinder
response with subtracted metallic core response with a = 1.5 In and y as the major
axis.

92

CHAPTER 6

E-pulse Results

In this chapter the E-pulse technique is used to measure the difference between
the field scattered by a coated elliptical conductor and a coated planar surface. The
E-pulse for the flat case calculated using the technique of Garrett Stenhohn‘s thesis
[8] is shown in Figure 6.1. The E—pulse of Figure 6.1 is convolved with data in
Figure 4.3 to calculate the waveform shown in Figure 6.2. As expected, after a time
TL, which is the sum of the E—pulse duration and the two-way transit time through
the coating, the convolution is almost zero. After the flat case waveform is
convolved with the E—pulse, the coated cylinder’s time domain data is convolved
with the E—pulse and the results are compared.

Figures 6.3 - 6.5 and 6.9 - 6.11 show the convolutions of the planar surface
E-pulse with the scattered field waveforms for coated ellipses with x as the major
axis. It is seen that the convolutions are in the late time, but not as flat as the.
convolution with the planar waveform. For the TMz and TEz cases with x as the.
major axis and a = .9 In, the response is small for the last 16 nanoseconds as seen in
Figures 6.3 and 6.9. As the outer radius increases to a = 1.5 in. the T137. and the
T.\=’Iz response become closer the flat case as seen in Figures 6.4 and 6.10. When a -:
2 in, the difference between the flat case and the TEz and Thlz cases with x as the.

major axis is even smaller as shown in Figures 6.5 and 6.11. Figures 6.6 - 6.8 and

93

 

6.12 - 6.14 shows similar results for the case where y is the major axis. When the y
axis is the major axis, the difference is sizable between the large and small cylinder
radii. For the TEz and TMz cases with y as the major axis and a = .9 In, the last
16 nanoseconds show large oscillations as seen in Figures 6.6 and 6.12. As the outer
radius increases to a = 1.5, the T132 and TMz cases become flatter, as seen in
Figures 6.7 and 6.13. When a = 2 m, the T1532 and the TMz cases in Figures (3.8 and
6.14 have only small oscillations occurring in the last 16 nanoseconds. As expected,
the convolution data becomes more flat as the cylinder radius increases, since larger
cylimler radii producing data that is closer to the flat case.

It was shown in Heike Vollmer’s thesis that subtracting the response of the
uncoated cylinder produces a smaller convolved waveform in the late time. The
convolution using subtracted cases shown in F igurcs 6.15 and 6.16 differ mostly in
the initial oscillation of the convolved plot. The standard convolution plot. has two
maximum and two minimums. while the difference convolution plot has four
maximums and three minimums. The last 15 nanoseconds have only a small
difference between the standard convolution plot and the difference convolution
plot. The reason that the difference is small compared to the difference in Heike
Vollmer’s thesis is that the E—pulse in Figure 6.1 is a DC E—pulse, while the li—pulse
used in Heike Vollmer’s thesis is a standard E—pulse. The DC E-pulse eliminates the
“tail” that is present in the fields scattered by the cylinders.

It is useful to quantify the value of the non-zero late-time convolved signal
using the signal energy. This done using the E—pulse discrimination number (EDN)
which is calculated from [11] using:
fi%+Ӵaau

EDN: ((3.1)
T W ‘
foTee 2(I (t)TLdtf L“ 1‘(t2()tdt

 

Here e(t) is the E-pulse waveform, r(t) is the time domain scattered field signal. and

94

CC is the convolution of e(t) and r(t). Also, T6 is the E—pulse duration, T L is the
start of the convolution late—time, and W is the window width of the energy
calculation. The EDN that is calculated using data with different major axes is
plotted in Figures 6.17 and 6.18. To the find the EDN value, T L was set to 8.7
nanoseconds. The ending time, T L + W, is chosen at the point where the
oscillations fade out, but before the convolution with the creeping wave arrives. For
the x as the major axis, the EDN value of the non-subtracted TEz case is closest to
the flat case. However, for TMz case, the subtracted case is closer that the
non—subtracted case to the flat case. The y major axis case is also like the x major
axis case having the T132 non-subtracted case closest. to the flat. case. For the TMz y
major axis case, the subtracted and non-subtracted cases are very close to the. same.
value for each size of the outer radius. In all cases, the EDN becomes lI'Iwer as the
radius of the cylinder increases, since the response becomes closer to that of the
planar case.

Table 6.] shows the EDN values for the different polarizations and plact-‘ment‘ of
the major axis. Since the E—pulse used for this data is different than what was used

in [11], the results are not comparable.

95

Amplitude

 

 

 

 

 

 

 

 

 

 

 

l [ 1 T

0.5 1 1.5 2
Time (ns)

Figure 6.1. E—pulse for the planar coated conductor vs time.

96

 

2.5

Amplitude

0.30

0.20 ~

0.10 ~

0.00

-0.10

-0.20

‘ -0.30

 

 

 

 

 

.1

 

 

 

l

 

l T l T

0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8
Time (ns)

Figure 6.2. Convulated E—pulse vs. time, flat case.

97

22.3 24.8

0.30

 

 

 

 

 

 

 

 

 

0.2o-«
0.104

¢

'0

3 F

a 0.00 v —f

E

< *
-0.10« l
-0.20~
“0.30 I I I I I I r I I I

0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.3. TMz convolved E—pulse vs. time, a = .9 m, a1 = .8 m, a2 = .7 m, with
x as the major axis.

98

 

 

 

 

 

 

 

 

0.30 --
0.20« I
0.10~ I

° :
.0 i
3 ll
'5. 0.00 + 2
E I
< i
-o.10« t
-o.2ot
'0.30 I I I I I I I I I IJ
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.4. TMz convolved E—pulse vs. time, a = 1.5 m, a1 = 1.4 m, a2 = 1.3 III,
with x as the major axis.

99

 

0.30

t
—.._.<--...~.-. “w_—-J

0.20 -*

0.10 . jl

0.00 T

 

 

 

 

Amplitude

-0.10 -

 

-0.20 I l

 

I I '

“0.30 I I I I I I I l
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.5. TMz convolved E—pulse vs. time, a = 2 m, a1 = 1.9 m, a2 = 1.8 m. with
x as the major axis.

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.30 .--------..e.. 3,
E
0.20 -
0.10 - .
3
'5. 0.00 &_
E 3
<
-o.10 . I
i
I
-0.20 « I
'0.30 T I 1 I 1 1 I I I l
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8
Time (ns)

Figure 6.6. TMz convolved E—pulse vs. time, a = .9 m, a1 = .8 m, a2 = .7 m, with
y as the major axis.

101

 

0.30 ffffffff 2 I...“

0.20 - I

 

0.10 ~

0.00 — Wv

-0.10

SE?! I“ '

 

 

Amplitude

 

J

 

 

L
t

-0.20

 

 

 

'0.30 I T I I I I I I I l
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.7. TMz convolved E—pulse vs. time, a = 1.5 m, a1 = 1.4 m, a? = 1.3 m.
with y as the major axis.

102

 

 

0.30

 

 

 

 

 

 

 

 

 

I
I .
I .-
t .
I
0.10 7 l 7- V
I» ~‘ g
13 E
3
'5 0.00— ,
E
<
-0.10~
-O.20~ l I
t
.0.30 I I I , , , , I I rI
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.8. TMz convolved E—pulse vs. time, a = 2 m, a1 = 1.9 m, a2 = 1.8 m, with
y as the major axis.

103

 

 

0.30

. -..__-.-.——..- -.—-'—

0.20 a

0.10 . f

0.00

 

 

 

Amplitude

-0.10 ~

 

-0.20 ~

 

 

 

'0.30 I I I I I I I I r I
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.9. TEz convolved E—pulse vs. time, a = .9 m, a1 = .8 m, a2 = .7 m, with x
as the major axis.

104

 

 

 

 

 

 

 

 

0.30 WI
0.20~ 5

I

0.10~

" z
1: I
3 F 2
E 3
< I
-o.10« I

i

-o.20~ l

I

.030 I . , , , , , , I gt
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.10. TEz convolved E—pulse vs. time, a = 1.5 m. a1 = 1.4 m. a2 = 1.3 m.
with x as the major axis.

105

 

 

0.30 ' “I

 

0.20 ~ 1
0.10 ~ jj
0.00

-0.10 A

 

 

Amplitude

(as. e

 

 

 

 

-0.20 —

 

I
-0.30 I I I I I I I I I Ii
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

 

Figure 6.11. TEz convolved E—pulse vs. time, a = 2 m, a1 = 1.9 m, a2 = 1.8 m, with
x as the major axis.

106

 

 

0.30

 

 

 

 

 

 

 

I

0.20~ I
0.10 « I

.3 I
= I
r: _ ~ ———~ J
a 0.00
E I
< ‘:
-o.10 ~ I
-o.20~ I

t

I

-0.30 I T I I I j I I I Ii
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.12. TEz convolved E—pulse vs. time, a = .9 m, a1 = .8 m, 212 = .7 m, with
y as the major axis.

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.30
0.20 ~ I I
I
0.10 - !
o E
1:
a J
'5. 0.00 w I
E
<
-0.10 ~
I
'0.30 I I I I I I | I I I
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.13. TEz convolved E—pulse vs. time, a = 1.5 m, 211 = 1.4 m. 212 z: 1.3 m.
with y as the major axis.

108

 

 

0.30 WW- ,
0.20 -« I

0.10 I

l
i
l
i
l
l
i
i
l
0.00 I L
(
I
I
i
E
I
I
i
l
l
i

 

Amplitude

 

-0.10 -

 

 

-o.20 - I

 

 

-0-30 1 l l l l l l T l 1%
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.14. TEz convolved E—pulse vs. time, a = 2 m, a1 = 1.9 m, a2 = 1.8 m, with
y as the major axis.

109

 

 

 

 

0.30

 

0.20 ~

.AAmw---‘—.-—.-—.-.... an

0.10 ~

 

 

0.00 ~ ~—

Amplitude

-0.10 I

 

 

 

 

-O.20 . I U

 

 

 

'0.30 l l T I if T l T I T
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.15. TMz subtracted convolved E—pulse vs. time, a = .9 n1, a1 = .8 Ill, 212 =
.7 m, with y as the major axis.

110

 

 

0.30

 

 

 

 

 

 

 

0.20 -« f
0.109 I

a: i
'0
3
'6. 0.00 ~ —v v— j.
E
< I
-o.10— :

I I ?

-o.20« ;
-o.3o . , . . . . T . .
0.0 2.5 5.0 7.4 9.9 12.4 14.9 17.3 19.8 22.3 24.8

Time (ns)

Figure 6.16. TEz subtracted convolved E—pulse vs. time, a = .9 In, a1 = .8 m, 212 =
.7 m, with x as the major axis.

111

 

1 . E-01 ‘ .._. .---.... ._ W _._......_  ..

j """ TEZ XD i
- - — - TMz XD I
— - — TEz X

 

 

 

 

 

 

1 . E‘04 T 1 —r T r i I

 

0.5 0.7 0.9 1.1 1.3 1.5 1.7 2

Figure 6.17. EDN vs outer radii, subtracted vs non-subtracted, x as major axis. XD
indicates that it is the subtracted case with x as the major axis. X indicates that it
is the non-subtracted case with x as the major axis.

112

 

. —
“qu ' . "
‘ _
'3

 

1.5-01 I ~,-;;,:__

3 ---7fE—zTIb I PI
1 —---TMz YD;

I —--TEzY II

 

 

1.E-02

111

 

|.'.‘n~_'-:I. _ ~
Etta-'1"! .

EDN

1.5-03 1

 

 

 

1.E‘04 I I I I l I I
0.5 0.7 0.9 1.1 1.3 1.5 1.7 2

a (m)

Figure 6.18. EDN vs outer radii, subtracted vs non-subtracted, y as major axis. YD
indicates that it is the subtracted case with y as the major axis. Y indicates that it,
is the non-subtracted case with y as the major axis.

113

Table 6.1. EDN values calculated for flat case and for cylinders of various radii in
meters

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a= Flat TMz X TMz Y TEz X TEz Y
9.71E-07
0.5 3.42E-03 9.64E-03 7.69E-04 1.03E-02
0.7 1.77E-03 5.83E-03 5.23E-04 5.33E-03
0.9 1.00E-03 3.44E-03 3.59E-04 3.05E-03
1.1 6.45E-04 1.99E-03 3.00E-04 1.62E-03
1.3 4.61E-04 1.14E-03 2.38E-04 8.96E-04
1.5 3.46E-04 6.40E-04 1.98E-04 4.86E-04
1.7 2.74E-04 3.65E-04 1.67E-04 2.77E-04
2 2.01E-04 1.65E-04 1.33E-04 1.31E-04
a= TMz XD TMz YD TEz XD TEz YD
0.5 4.56E-03 9.50E-03 1.47E-03 1.09E-02
0.7 1.65E-03 5.70E-03 1.02E-03 5.88E-03
0.9 9.04E-04 3.37E-03 5.76E-04 3.44E-03
1.1 5.87E-04 1.94E-03 4.48E-04 1.86E-03
1.3 4.26E-04 1.11E-03 3.34E-04 1.11E-03
1.5 3.23E-04 6.26E-04 2.71E-04 5.64E-04
1.7 2.60E-04 3.58E-04 2.20E-04 4.60E-04
2 1.93E-04 1.63E-04 1.70E-04 1.52E-04

 

 

114

 

Ilr'wr " -
is
1L

 

CHAPTER 7

Conclusion

This thesis has demonstrated that the E-pulse technique can be successfully
applied to a coated curve surface with a varying thickness and curvature using the
E-pulse of the coated planar surface. The E—pulse technique is applied to various
size radii of coated elliptic cylinders and the reflection from the surface is shown to
be a summation of damped sinusoids, which is the early—time natural response for
the coated curved surface. As the radius of the cylinder increases, the EDN value
from the cylinder become closer to the value of the coated planar surface. Amther
aspect of the increasing radius of the cylinder is the reduction of the effect of
changing the placement of the major axis or which polarization applied, TEz or
TMz; for the largest cylinder radius, the EDN values are similar for the four
different combinations of the polarization and placement of the major axis, since the
largest radius cylinders are closet to the planar case.

The variation of the scattered electric field for the various configurations of
polarization and the placement of the major axis has the largest effect. in the
frequency domain when the placement of the major axis is changed. In the time
domain data, changing y as the major axis produces larger multiple reflections than
in the planar case, while choosing x as the major axis produces smaller multiple

reflections. Thus, whether the thickness increases or decreases away from the

115

 

 

 

 

contact point has a large effect on the time domain response. EDN values for the.
various configurations of polarization and major axis placement are larger for
smaller cylinder radii and become smaller as the cylinder radius increases. This is
because larger radius cylinders have surface curvature closer to the flat case.

The methods that check whether the scattered electric field equations have
been derived correctly, are matching of the expansion coefficients of the elliptical

case to the circular case and computing the echo width of the elliptical metallic

 

core. The matching of the TEz and TMz expansion coefficients of the elliptical case
to the circular case shows that for the major and minor axes being equal, the

derived equations match a known equation. The echo width calculation shows that ”I

 

M
“Hm-”an“. -.

the point matching technique is valid. y
Future work in applying the E—pulse technique to various surfaces includes

changing the shapes of the metallic core and the dielectric coating. Shapes may be

chosen that will be more realistic than the coated planar surface, a constant coated

curved surface, or the varying coated curved surface examined in this thesis.

116

APPENDICES

117

 

1F“. _A=- _ "_ --

APPENDIX A

TMZ Case Source Code II

! TMZIQO
! FUNCTIONS: J

 

! TMZ
l****************************************************************************
! PROGRAM: TMZ

! PURPOSE: Calculating Scattering Electric Field
I****************************************************************************
program TMZ

parameter (pi=3.14159265358979,e0=8.85418781762e—12, u0=1.25663706144e-6, 8:
nsize=10000, nf = 2000, er = 1.01, ur = 1, a = 2., nmax 2: 275, ul 2 20,nbig=1000)
implicit real*8 (a—h,o—z)

integer ipvt(1:nbig),flag,isecho,nf1,i,m,n,l

real*8 e,n0,phi,phi0,u,n1,n2,en,Xn(0:nmax), Yn(0:nmax),ka1, ka‘2,k(1:nf)

real*8 k0(lznf),f(1:nf),Rm(O:nmax),Phim(0:nmax)

complex*16 arg,an,Fpmn,r,e_jk0p,a2, diag(1:nbig),j

(romplex*16 bjka(0:nsize),byka(0:nsize),bjpka(0:nsize), bypka(0:nsize)

complex*16 bhka(0:nsize),bhpka(0:nsize),bjk0a(0:nsize), byk0a(01nsize)

complex*16 bjpk0a(0:nsize),bypkOa(O:nsize),bhk0a(0:nsize), bhpk0a(():nsize)

118

 

complex*16 bjkrm(0:nsize),bykrm(0:nsize),bjpkrm(0:nsize), bypkrm(0:nsize)
complex*16 bhkrm(0:nsize),bhpkrm(0:nsize),A1(1:nbig,1:nbig),B1(1:nbig). Es('1:nf)
complex*16 Cn(1:nbig),W(1:nf)
j = (0.D0,1.D0)
phi = 0
u :2 ur * 110
e = er * e0
n0 2: sqrt(uO/eO)
111 = sqrt(u/e)
ka1=1.4 lX Axis
ka2=1.3 lY Axis
r = 1
e_jk0p == 1
n2 = n1 / n0
flag 2 1
isecho = 0
!Open the data file
open (ul, FILE=’temp10.txt’, STATUS=’OLD’)
!Creates N+1 points on an ellipse with axis lengths kal, ka2 in rectanglar and polar
lforms
dphi = pi/(nmax+1)
do i=0,nmax
phiO = i*dphi+dphi / 2.
Xn(i) = kal * cos(phi0)
Yn(i) = ka2 * sin(phi0)
R1n(i) = sqrt(Xn(i) * Xn(i) + Yn(i) * Yn(i))
Phim(i) = atan2(Yn(i), Xn(i))

119

“I". _- ‘

 

1m:

enddo

if(isecho 2: 1) then lNumber of data
nfl = 153

elseif(ise('-ho 2: 2) then
nfl = 191

else
nfl : nf

endif lcreating the range of frequencies
do i=1,nf1
if(nf 2: 500) then
f(i) = i * 5.e6
elseif(nf :2 1000) then
f(i) = i * 2.5e6
elscif(nf =2 2000) then
f(i) = i * 1.25e6
endif
enddo
lcreating the wave numbers
do i=1, nfl
k(i) = 2 * pi * f(i) * sqrt(u*e)
k0(i) = 2 * pi * f(i) * sqrt(u0*e0)
enddo
do i=1, nfl
Es(i) = 0
lCreating the arguement of the bessel functions and calling the
lfunction that creates the In of the bessel function

arg = k(i) * a

120

call logbjyh(arg,nmax,bjka,byka,bjpka,bypka,bhka,bhpka)
arg = k0(i) * a
call logbjyh(arg,nmax,bjk0a,byk0a,bjpkOa,bypk0a,bhk0a,bhpk0a)
lCreating the m x 11 matrix
do m=0, nmax
if(flag z: 4) then
arg = k0(i) * Rm(m) r.
else

arg = k(i) * Rm(m)

 

endif I
call logbjyh(arg,nmax,bjkrm,bykrm,bjpkrm,bypkrm,bhkrm,bhpkrm) I:
Bl(m+1) = 0

do n=0, nmax

if(n == 0) then

en = 1
else

en 2 2
endif

an = exp(bjka(n) + bykrm(n)) - exp(bjkrm(n) + byka(n))
Fpmn = exp(bjpka(n) + bykrm(n)) - exp(bypka(n) + bjkrm(n))
if(fiag == 0) then ldivide by H2p(n) normalization
A1(m+1,n+1) = (n2 * an - Fpmn * exp(bhk0a(n) - bhpk0a.(n)))
* cos(Phim(m) * n)
B1(m+1) = B1(m+1) - en * (j ** (n)) * (n2 * an * exp(b_j}')k0a(n) &
- bhpk0a(n)) - Fpmn * exp(bjk0a(n) - bhpk0a(n))) * cos(Phim(m)* n)
else if(flag 2: 1) then ! Factor H2p(n) Normalization

A1(m+1,n+1) 2 (n2 * an - Fpmn * exp(bhk0a(n) - bhpkOa(n)))

121

* cos(Phim(m) * n)
Bl(m+1) = Bl(m+1) - en * (j ** (n)) * (n2 * Frnn * 8;
exp(bjpk0a(n)) - Fpmn * exp(bjk0a(n))) * cos(Phim(m) * 11)
else if(flag == 2) then ldivide by H2p(nmax) normalization
A1(m+1,n+1) = (n2 * an * exp(bhpk0a(n) - bhpk0a(nmax)) - Fpmn * (Ki
exp(bhk0a(n) - bhpk0a(nmax)))*cos(Phim(m)*n)
Bl(m+1) = Bl(m+1)- en * (j ** (n)) * (n2 * an * exp(bjpk0a(n) 8;
— bhpk0a(nmax)) - Fpmn * exp(bjk0a(n) - bhpk0a(nmax))) * 85
cos(Phim(m) * 11)
else if(flag == 3) then 3110 normalization
A1(n1+l,n+1) = (n2 * an * exp(bhpk0a(n)) - Fpmn * exp(bhk0a(n)l)) kt
* cos(Phim(m) * n)
Bl(m+1) = Bl(m+l) - en * (j ** (n)) * (n2 * an * exp(bj};)k0e1(ll)) 8:.
- Fpmn * exp(bjk0a(n))) * cos(Phim(m) * 11)
else if(flag :2: 4) then lno normalization, no dielectric
A1(m+1,n+1) = exp(bhkrm(n)) * cos(Phim(m)*n)
B1(m+1) = Bl(m) - en * (j ** (n)) * exp(bjkrm(n))
* cos(Phim(m) * n)

endif

enddo

enddo

do 1:1, nmax+1

Cn(1) = 131(1)

end do

call DCOMPC (nbig,nma.x+1,A1,IPVT,DIAG)

call SOLVEC (nbig,nmax+1,A1,IPVT,Cn)
a2 = sqrt(2/(pi * k0(i) * r)) * e_jk0p * exp(pi *j / 4)

122

 

do 120, nmax

if(flag :2 1) then

Es(i) = Es(i) + Cn(l—+—1) * exp(0 - bhpk0a(l)) * cos (1 * phi) * &

exp(l *j * Pi / 2)
else
Es(i) = Es(i) + Cn(l+1) * cos (1 * phi) * exp(l *j * Pi / 2)
endif
enddo
Es(i) = Es(i) * a2
if(isecho :2 1) then
VV(i) = 2 * pi * Es(i) * conjg(Es(i))
write(ul,*) real(k0(i)),real(W(i)),a.imag(W(i))
write(*,*) i,real(W(i)),aimag(W(i))
elseif(isecho 2: 2) then
W(i) = 2 * pi * Es(i) * conjg(Es(i)) / (pi * Rm(1))
write(u1,*) real(k0(i) * Rm(1)),real(W(i)),aimag(VV(i))
write(*,*) i,real(W(i)),aimag(W(i))
else
lwrite(u1,*) f(i)*1.d-9,real(Es(i)),airnag(Es(i))
write(u1,*) real(Es(i)),aimag(Es(i))
write(*,*) i,real(Es(i)),aimag(Es(i))
endif
enddo
!prints out the points on the ellipse
ldo i=0,nmax
lwrite(u1,*) Xn(i),Yn(i),Rm(i), Phim(i)
lwrite(*,*) Xn(i),Yn(i)

123

fiJ

 

lend do

close(u1)

end program TMZ

lCalculating the In of the bessel functions subroutine
logbjyh(arg,n,blogj,blogy,blogjp,blogyp,blogh,bloghp)
implicit real*8 (a-h,o—z)

parameter (nsize-—-10000,tcst=1.d-10,gamma=0.57721566490153, S;
pi=3.14159265358979)

complex* 16 arg,resn,res,prod,zsum,zsum1,zj0,zy0
complex* 16 blogj (02nsize) ,blogy(0:nsize) ,ratio(0:nsize)
complex*16 blogjp(0:nsize),blogyp(0:nsize)

complex* 16 blogh(0:nsize) ,bloghp(0:nsize)
complex*16 ysum,ysum1

complex*16 zj,yj1,yj

lcomputes the log of J0-Jn for complex arguments
lfirst compute J0 using ratios Jn-l/Jn and

lequation 9.1.44 from Abramowitz

lalso compute Y0 using ratios Jn—l/Jn and

lequation 9.1.89 from Abramowitz

zj = dcmplx(0.d0,1.d0)

zsum = 1.d0

ysum = 2.d0*(zlog(arg)+dlog(0.5d0)+gamma) / pi
prod = 1.d0

iyflag = 1

izflag = 1 i z u

10 i = i + 2

zsuml = zsum

124

 

ysuml = ysum
call bratio(arg,i-1,rcs)
prod = prod*res
ratio(i—l) = res
call bratio(arg,i,res)
prod = prod*res
ratio(i) = res
i = i/2
if (izflag .cq. 1) then
zsum -—- zsum + 2.d0*(-1)**j/prod
if (abs(zsuml—zsum) /abs(zsum1) .lt. test) then
izflag=0
end if
end if
if (iyflag .eq. 1) then
ysum = ysum — 4.d0*(-1)**j/(prod*pi*j)
if (abs(ysuml-ysum)/abs(ysum1) .lt. test) then
iyflag=0
end if
end if
if ((izflag .eq. 0) .and. (izflag .eq. 0)) go to 20
zsum] = zsum
ysuml = ysum
go to 10
20 zj0 = cos(arg)/zsurn
zy0 = ysum*zj0

! finish computing ratios

125

 

 

if (i .lt. n) then
do j==i+l,n
call bratio(arg.j.res)
ratio(j) = res
end do
end if
! now compute log(Jn)
blogj(0) : zlog(zj0)
do i=1,n
blogj(i) = blogj(i-1)-zlog(ratio(i))
end do
1 now compute JnYn and then log(Yn)
! using Wronskian 9.1.16 from Abramowitz
yjl = zj0*zy0
blogy(0) = zlog(zy0)
do i=1,n
yj = yj1/(ratio(i)*ratio(i))-(2.d0/(pi*arg))/ratio(i)
blogyfi) = zlog(yj)-b10si(i)
yil = yi
end do
! next compute log(Jn’) from recursion formula
blogjp(0) = dcmplx(0.d0,pi)+blogj(l)
do i=1,n
blogjp(i) = blogj(i)+zlog(ratio(i)-i/arg)
end do
! next compute log(Yn’) from recursion formula

blogyp(0) = dcmp1x(0.d0,pi)+blogy( 1)

126

 

do i=1,n
blogyp(i) = -blogj(i)+zlog(exp(blogy(i-1)+ &
b10gj(i))-(i/ arg)*eXP(b10gj(i)+b10gY(i)))
end do
! next compute log(Hn(2)) from cross product
blogh(O) = zlog(zj0-zj*zy0)
do i=1,n
blogh(i) = zlog(exp(blogj(i)+blogh(i—1))- &
2.d0/(zj*pi*arg)) - blogj(i-1)
end do
! lastly compute log(Hn(2)’) from cross product
bloghp(O) = zlog(exp(b10sip(0))-2i*eXI)(b10gyp(0)))
do i=1,n
bloghp(i) = zlog(exp(blogjp(i)+blogh(i))- &
(2-d0*zj)/(pi*ars)) - blosifi)
end do
! reduce phase by increments of 2pi
do i=0,n
call reducepi(blogj(i))
call reducepi(blogy(i))
call reducepi(blogjp(i))
call reducepi(blogyp(i))
call reducepi(blogh(i))
call reducepi(bloghp(i))
end do
return

end

127

 

subroutine bratio(arg,n,res)
! computes the ratio of Jn-1(arg)/Jn(arg)
l to relative accuracy of test using
! continued fraction expansion of Lentz
! Reference: IEEE Trans on AP, vol. 48,
! no. 10, pp. 1528—1532, October 2000
! parameter (nmax=10000,test=1.d-14)
implicit real*8 (a-h,o-z)
complex* 16 arg,res,resl ,zvec(nmax)
nterms = 2
ii = -2
do i=1,nterms

ii 2 -ii

zvec(i) = ii*(n+i—1)/arg
end do
res = zvec(nterms)
do j=nterms-1,1,-1

res = 1.d0/res + zvec(j)
end do
resl = res
do k=nterms+1,nmax

ii 2 -ii

zvec(k) = ii*(n+k—1)/arg

res = zvec(k)

do j=k—1,1,—1

res = 1.d0/res + zvec(j)

end do

128

 

diff = abs(res-res1) /abs(res1)
if (diff .lt. test) go to 100
resl = res
end do
write (*,*) ’ERROR*** maximum number of terms exceeded’
write (*,*) ’ in subroutine bratio’
100 return
end
subroutine reducepi(z)
! reduces imaginary part of complex number
implicit real*8 (a—h,o—z)
parameter (tpi=6.283185307179586477)
complex*16 z
x = dreal(z)
y = dimag(z)
10 if (y .gt. tpi) then
yzy-tpi
go to 10
end if
20 if (y .lt. -tpi) then
y=y+qn
go to 20
end if
z=dcmplx(x,y)
return
end

subroutine DCOMPC (nbig,N,A,IPVT,DIAG)

129

! Decomposes matrix a to triangular form
! **** COMPLEX VERSION ****
! FROM: LINEAR ALGEBRA BY GILBERT STRANG
! parameter(nbig=101)
integer N,lPVT(nbig)
integer Nh~11,I,J,K,KP1,l\/l
complex*16 A(nbig,nbig),DIAG(nbig)
complex*16 P,T
IPVT(N) = 1
if (N :2 )GO TO 70
NMl = N-l
do 60 K=1,NMl
KPl = K+1
! find pivot p
M = K
do 10 I=KP1,N
10 if (abs(A(l,K)) g, abs(A(M,K))) M=I
IPVT(K) = M
if (M /= K) IPVT(N) = -IPVT(N)
P = A(M,K)
A(M,K) = A(K,K)

A(K,K) = P
DIAG(K) = P
if ( == 0.D0) go to 60

! compute multipliers
20 do 30 I=KP1,N 30 A(I,K) = -A(I,K)/P

l interchange rows and columns

130

— “773:3,

 

 

do 50 J =KP1,N

T = A(M,J)

A(M,Il) = A(K,.l)

A(K,J) = T

if (T :2 0.D0) go to 50

do 40 I=KP1,N

A(I,J) = A(I,J) + A(I,K)*T

40 continue

50 continue

60 continue

70 DIAG(N) = A(N,N)*IPVT(N)
return end

subroutine SOLVEC (nbig,N,A,IPVT,B)
! Solves matrix eqn AX=B with a decomposed by dcompc
! parameter(nbig=101)

integer N,lPVT(nbig)

integer NM1,K,KB,KP1,KM1,M,I
complex*16 A(nbig,nbig),B(nbig)
complex*16 S

! forward elimination

if (N =2 1) go to 30

NMl = N—l

do 10 K=1,Nl\/Il

KPl = K+1

M = IPVT(K)

S = B(M)

B(M) = B(K)

131

 

 

 

B(K) = S

do 10 I=KP1,N

10 B(I) = B(I) + A(I,K)*S
! back substitution

do 20 KB=1,NMl

KMl = N-KB

K = KM1+1

B(K) = B(K)/A(KIK)

S = -B(K)

do 20 I=1,KMl

20 B(I) = B(I) + A(I,K)*S
30 8(1) = B(1)/A(1,1)
return

end

132

 

-_..- n 11': 0%: -‘IT

 

APPENDIX B

TEZ Case Source Code

ELF-1- I“. Hi...

! TEZ.f90

j****************************************************************************

 

Im_.

! PROGRAM: TEZ

! PURPOSE: Calculating the scattered electric field of TEZ case
!****************************************************************************
program TEZ

parameter (pi=3.14159265358979,e0=8.85418781762e-12, 11021.256637061440-6, 8;
nsize=10000, nf = 2000, er = 9., ur = 1, a = 2., nmax = 275, 111 = 20.nbig21000)
implicit real*8 (a—h,o—z)

integer ipvt(lznbig),flag,isccho,nf1,i,m,n,1

real*8 e,n0,phi,phi0,u,n1,n2,en,Xn(0:nmax),Yn(0:nmax),kal,ka2,k(1:nf)

real*8 k0(1:nf),f(1:nf),Rm(0:nmax),Phim(0:nmax)

complex*16 arg,an,Fpmn,r,e_jk0p,a2,diag(1:nbig),j,W(1:nf)

complex*16 bjka(0:nsize),byka(0:nsize),bjpka(0:nsize),bypka(0:nsize)

complex* 16 bhka(0:nsize) ,bhpka(0:nsize) ,bjk0a(0:nsize), byk0a(0:nsize)

complex*16 bjpk0a(0:nsize),bypk0a(0:nsize),bhk0a(0:nsize), bhpk0a(0znsize)
complex*16 bjkrm(0:nsize),bykrm(0:nsize),bjpkrm(0:nsize), bypkrm(0:nsize)
complex*16 bhkrm(0:nsize),bhpkrm(0:nsize),A1(1:nbig.1:nbig), Bl(l:nbig),Es(1:nf)

133

complex*16 F1mn,F1pmn,G,Gp, one, two, three, four,Cn(1:nbig)
j : (O.D0.1.D0)

phi :'0

u = ur * 110

e = er * 90

110 = sqrt(uO/eO)

111 = sqrt(u/c) r-
kal=1.9 lX Axis
k212=1.95 lY Axis
r = 1

e_jk0p = 1

 

!r_- —-r-" .“l'

112 = 110 / n1
flag = 1
isecho = 0
! Open the data file
open (111, FILE=’ten1p10.txt’, STATUS=’OLD’)
lCreates N+1 points on an ellipse with axis lengths kal, ka2 in rcctanglar and polar
lforms
dphi 2 pi/ (nmax+1)
do i=0,nmax
phiO = i*dphi+dphi/2.
Xn(i) = kal * cos(phiO)
Yn(i) = ka2 * sin(phi0)
Rm(i) = sqrt(Xn(i) * Xn(i) + Yn(i) * Yn(i))
Phim(i) = atan2(Yn(i), Xn(i))
enddo

if(isecho :2 ) then lN umber of data

134

nfl = 153

elseif(isecho :2 ) then
nfl = 191

else
nfl = nf

endif !creating the range of frequencies
do i=1,nf1
if(nf == 500) then
f(i) = i * 5.e6
elseif(nf :2 1000) then
f(i) = i * 2.5e6
elseif(nf == 2000) then
f(i) = i * 1.25e6
endif
enddo ! creating the wave numbers
do i=1, nfl
k(i) = 2 * pi * f(i) * sqrt(u*e)

k0(i) = 2 * pi * f(i) * sqrt(u0*e0)

enddo
do i=1, nfl
Es(i) = O

! Creating the argument of the bessel functions and calling the
! function that creates the In of the bessel function
arg = k(i) * a
call logbjyh(arg,nmax,bjka,byka,bjpka,bypka,bhka,bhpka)
arg = k0(i) * a

call logbjyh(arg,nmax,bjk0a,byk0a,bjpkOa,bypk0a,bhk0a,bhpkl)a)

135

 

marten '

! Creating the m x n matrix

do m=0, nmax

if(flag == ' ) then
arg = k0(i) * Rm(m)

else

arg = k(i) * Rm(m)

endif

call logl)jyh(arg,nmax,bjkrm,bykrm,bjpkrnnbypkrm,bhkrm,hhpkrni)

B1(m+1) = 0

do 1120, nmax

if(n =2 ) then
en 2 1

else
en 2 2

endif

an = exp(bjka(n) + bykrm(n)) - exp(bjkrm(n) + byka(n))
F mm = exp(bjpka(n) + bykrm(n)) - exp(bypka(n) + bjkrm(n))
Flinn = exp(bjka(n) + bypkrm(n)) - exp(bjpkrm(n) + byka(n))
F 1mm = exp(bjpka(n) + bypkrm(n)) - exp(bjpkrm(n) + hyphen)
G = (kaZ-kal) * cos(Phim(m)) * sin(Phim(m)) / sqrt(kal**‘2 * &.
(sin(Phim(m)))**2 + ka2**2 * (cos(Phim(m)))**2)
Gp = (kal * sin(Phim(m))*sin(Phim(m)) + ka‘2 * cos(Phim(m))* &
cos(Phim(m))) /sqrt(ka1**2 * (sin(Phim(m)))**2 + ka2**2 *&
(008(Phim(m)))**2)
if(flag == 0) then ldividc by H2p(n) normalization

lNot done because the TMZ version did not work.

else if(flag 2: ) then ! Factor H2p(n) Normalization

136

 

"L3

 

F ?
‘V

 

one = (G/r)*(112*an - Fpmn*exp(bhk0a(n) - bhpk0a(n)))*n*
sin(n*Phim(m))
two = Gp*k(i)*(n2*F1mn - F1pmn*exp(bhk0a(n) - 1:)hpk0a(n)))*
cos(n*Phim(m))
three 2 (G/r)*(n2*an*exp(bjpkOa(n)) - Fpmn*exp(bjk0a(n)))*n*
sin(Phim(m)*n)
four 2 Gp*k(i)*(n2*F1mn*exp(bjpk0a(n)) - F1pmn*exp(bjk()a(n)))*
cos(Phim(m)*n)
A1(m+1,n+1) = one + two
Bl(m+1) = Bl(m+1) - en * (j ** (n)) * (four + three)

else if(flag 2: 2) then ldivide by H2p(nmax) normalization
!Not done because the TMz version did not work.

else if(flag == 3) then lno normalization
one = (G/r)*(n2*an*exp(bhpkOa(n)) - Fpmn*exp(bhk0a(n)))*n*
sin(n*Phim(m))
two = Gp*k(i)*(n2*F1mn*exp(bhpk0a(n)) - F1pmn*exp(blil<()a(n)))*
cos(n*Phim(m))
three = (G/r)*(n2*an*exp(bjpk0a(n)) - Fpmn*exp(bjk0a(n)))*n*
sin(Phim(m)*n)
four = Gp*k(i)*(n2*F1mn*exp(bjpkOa(n)) - Flpmn*exp(bjk()a(n)))*
cos(Phim(m)*n)
A1(m+1,n+l) = one + two
Bl(m+1) = Bl(m+1) - en * (j ** (n)) * (four + three)

else if(flag :2 4) then lno normalization, no dielectric

lNot done because comparison was not needed.
endif
enddo

137

 

 

enddo
do 1:1, nmax—H
Cn(l) = Bl(l)
end do
call DCOMPC (nbig,nmax+1,A1,IPVT,DIAG)
call SOLVEC (nbig,nmax+1,A1,IPVT,Cn)
a2 = sqrt(2/(pi * k0(i) * r)) * e_jk0p * exp(pi *j / 4)
do 120, nmax
if([lag :: 1) then
Es(i) = Es(i) + Cn(l+1) * exp(O - bhpkOa(l)) * cos (I * phi) * exp(l*j*pi/2)
else
Es(i) = Es(i) + Cn(l+1) * cos (1 * phi) * exp(l *j * pi / 2)
endif
end do
Es(i) = Es(i) * a2
if(isecho :2 ) then
W(i) :- 2 * pi * Es(i) * conjg(Es(i))
write(u1,*) real(k0(i)),real(W(i)),aimag(W(i))
write(*,*) i,real(W(i)),aimag(W(i))
elseif(isecho == ) then
W(i) = 2 * pi * Es(i) * conjg(Es(i)) / (pi * Rm(1))
write(u1,*) real(k0(i) * Rm(1)),real(W(i)),aimag(W(i))
write(*,*) i,real(W(i)),aimag(W(i))
else
lwrite(nl,*) f(i)*1.d—9,real(Es(i)),aimag(Es(i))
write(u1,*) real(Es(i)),aimag(Es(i))
write(*,*) i,real(Es(i)),aimag(Es(i))

138

PT

 

2r“. Lflr -_.. .

“‘LJIHJ": .

 

endif
enddo
!prints out the points on the ellipse
!(lo i=0,nmax
! write(ul,*) Xn(i),Yn(i),Rm(i), Phim(i)
! write(*,*) Xn(i),Yn(i)
lend do
close( u 1)
end program TEZ
! Calculating the In of the bessel functions
subroutine logbjyh(arg,n,blogj,blogy,blogjp,blogyp,blogh,bloghp)
implicit real*8 (a—h,o—z)
parameter (nsize=10000,test=1.d-10,gamma=0.57721566490153, &
pi=3.14159265358979)
complex*16 arg,resn,res,prod,zsum,zsum1,zj0,zy0
complex* 16 blogj (O:nsize) ,blogy(0:nsize) ,ratio(0:nsize)
complex* 16 blogj p(0:nsize) ,blogyp(0:nsize)
complex* 16 blogh(0:nsize) ,bloghp(0:nsize)
complex*16 ysum,ysum1
complex*16 zj,yj1,yj
lcomputes the log of JO-Jn for complex arguments
llirst compute J0 using ratios Jn-l/Jn and
lequation 9.1.44 from Abramowitz
!also compute YO using ratios Jn-l/ J n and
lequation 9.1.89 from Abramowitz
zj = dcmplx(0.d0,1.d0)

zsum == 1.d0

139

 

:Fhr - .
L
‘I.. __

 

 

ysum = 2.d0*(zlog(arg)+dlog(0.5d0)+gamma) / pi

prod = 1.dO

iyflag=1
izflagzl
i=0
10i=i+2

zsuml = zsum
ysuml = ysum
call bratio(arg,i—1,res)
prod = prod*res
ratio(i—l) = res
call bratio(arg,i,res)
prod = prod*res
ratio(i) = res
j = i/2
if (izflag .eq. 1) then
zsum = zsum + 2.d0*(-1)**j / prod
if (abs(zsuml-zsum)/abs(zsum1) .lt. test) then
izflag=0
end if
end if
if (iyflag .eq. 1) then
ysum = ysum - 4.d0*(—1)**j/(prod*pi*j)
if (abs(ysuml-ysum)/abs(ysum1) .lt. test) then
iyflagzO
end if
end if

140

 

Eu.
E

LD-A'. N. .'a' '.s.'_0..‘. .. t.‘

 

*Q

if ((izflag .eq. 0) .and. (izflag .eq. 0)) go to 20
zsuml = zsum
ysuml = ysum
go to 10
2O sz = cos(arg)/zsum
zyO = ysum’ksz
! finish computing ratios
if (i .lt. n) then
do j=i+1.n
call bratio(arg,j,res)
ratio(j) = res
end do
end if
! now compute log(Jn)
blogi(0) = zlog(zj0)
(lo i=1,n
blogj(i) = blogj(i-1)—zlog(ratio(i))
end (10
! now compute JnYn and then log(Yn)
! using Wronskian 9.1.16 from Abramowitz
yjl = zj0*zy0
blogy(0) = zlog(zy0)
do i=1,n
yj = yj1/(ratio(i)*ratio(i))-(2.d0/(pi*arg))/ratio(i)
blogyfi) = 7log(yjtbloai(i)
.le = yj

end do

141

 

 

! next compute log(Jn’) from recursion formula
blogjp(0) = dcmplx(0.d0,pi)+blogj(1)
do i=1,n
blogjp(i) = blogj(i)+zlog(ratio(i)-i/arg)
end do
! next compute log(Yn’) from recursion formula
blogyp(O) = dcmplx(0.d0,pi)+blogy(1)
do i=1.n
blogyp(i) = -blogj(i)+zlog(exp(blogy(i-1)+ &
bloai(i))-(i/arg)*exp(b10sj(i)+b10gy(i)))
end do
! next compute log(Hn(2)) from cross product
blogh(O) = zlog(zj0-zj*zy0)
do i=1,n
blogh(i) = zlog(exp(blogj(i)+blogh(i-1))- &
2.dO/(zj*pi*arg)) - blogj(i-1)
end do
! lastly compute log(Hn(2)’) from cross product
bloghp(O) = zlog(exp(b10gip(0))-2i*exp(b10gyp(0)))
do i=1,n
bloghp(i) = zlog(exp(blogjp(i)+blogh(i))- &
(2-d0*zj)/(pi*arg)) - b10gi(i)
end do
! reduce phase by increments of 2pi
do i=0,n
call reducepi(blogj(i))

call reducepi(blogy(i))

142

 

 

L}

 

 

call reducepi ( blogj p(i))

call red ucepi(blogyp( i) )

call reducepi(blogh(i))

call reducepi(bloghp(i))
end do
return
end
subroutine bratio(arg,n,res)
! computes the ratio of Jn-1(arg) / Jn(arg)
! to relative accuracy of test using
! continued fraction expansion of Lentz
! Reference: IEEE Trans on AP, vol. 48,
! no. 10, pp. 1528-1532, October 2000
parameter (nmax=10000,test=1.d-14)
implicit real*8 (a-h,o—z)
complex* 16 arg,res,res1,zvec(nmax)
nterms = 2
ii = —2
do i=1,nterms

ii =2 -ii

zvec(i) = ii*(n+i~1)/arg
end do
res : zvec(nterms)
do j=nterms-1,1,-1

res = 1.d0/res + zvec(j)
end do

resl :2 res

143

 

 

 

do k=nter1ns+1,nmax
ii 2 -ii
zvec(k) = ii*(n+k—1)/arg
res = zvec(k)
do j=k-l,1,—1
res = 1.d()/res + zvec(j)
end do
diff = abs(res—resl ) / abs(res 1)
if (dill .lt. test) go to 100
resl = res
end do
write (*,*) ’ERROR*** maximum number of terms exceeded’
write (*,*) ’ in subroutine bratio’
100 return
end
subroutine reducepi(z)
! reduces imaginary part of complex number
implicit real*8 (a-h,o—z)
parameter (tpi=6.28318530717958647?)
complex*16 z
x = dreal(z)
y = dimag(z)
10 if (y .gt. tpi) then y=y-tpi go to 10 end if 20 if (y .lt. -tpi) then
y=y+tpi
go to 20
end if

z=dcmplx(x,y)

144

 

 

 

return
end
subroutine DCOMPC (nbig,N,A,IPVT,DIAG)
lDecomposes matrix a to triangular form
!**** COMPLEX VERSION ****
!F ROM: LINEAR ALGEBRA BY GILBERT STRANG
!paramcter(nbig=101)
integer N ,IPVT(nbig)
integer NMl,I,J,K,KPl,M
complex*16 A(nbig,nbig),DIAG(nbig)
complex*16 P,T
IPVT(N) = 1
if (N =2 )GO TO 70
NMl = N-l
do 60 K=1,NMl
KPI = K+1
! find pivot p
M = K
do 10 I=KP1,N
10 if (abs(A(I,K)) > abs(A(M,K))) M=I
IPVT(K) = M
if (M /= K) IPVT(N) = -IPVT(N)
P = A(M,K)
A(M,K) = A(K,K)
A(K,K) = P
DIAG(K) = P
if (P 2: ODO) go to 60

145

 

I?

.- no.
.. I ..

 

L!

 

! compute multipliers
20 do 30 I=KP1,N
30 A(I,K) = -A(I,K)/P
! interchange rows and columns
do 50 J=KP1,N

T = A(M,J)

A(M,J) = A(K.J)

A(K,J) = T

if (T 2: ODO) go to 50

do 40 I=KP1,N

A(I,J) = A(I,J) + A(I,K)*T

40 continue
50 continue
60 continue
70 DIAG(N) = A(N,N)*IPVT(N)
return
end
subroutine SOLVEC (nbig,N,A,IPVT,B)
!Solves matrix eqn AX=B with a decomposed by dcompc
!parameter(nbig=101)
integer N,lPVT(nbig)
integer NM1,K,KB,KP1,Kl\/II,M,I
complex*16 A(nbig,nbig),B(nbig)
complex*16 S
!forward elimination
if (N == 1) go to 30
NMl = N-l

146

 

 

do 10 K=l,NMl

KPl = K+1

M = IPVT(K)

S = B(M)

B(M) = B(K)

B(K) = S

do 10 I=KP1,N
10 B(I) = B(I) + A(I,K)*S
!back substitution
do 20 KB=1,NM1

KMl = N-KB

K = KMl+1

B(K) = B(K)/A(KaK)

S = —B(K)
do 20 I=1,KMl
20 B(I) = B(I) + A(I,K)*S
30 B(1) 2 B(l)/A(1,1)
return

end

147

 

“-‘m‘ 11‘

 

 

 

APPENDIX C

Flat Case Source Code

! FLAT.f90
! FUNCTIONS:
! FLAT

|****************************************************************************

! PROGRAM: FLAT ! PURPOSE: Calculate the scattering field of the flat case
!****************************************************************************
program FLAT

parameter (pi=3.14159265358979, eO=8.85418781762e—12, u0=1.256637()6144e-6, 8;
nsize=10000, nf = 2000, er = 9., ur = 1, a = 1.1, nmax = 275, ul 2 20,nbig:1()t)0)
complex*16 P, R(1:nf), j, Gamma

real*8 e,u, d, f(lznf), k(l:nf),n0,n1

d = .1

j = (0.D0,1.DO)

u 2 ur * uO

e 2 er * e0

n0 2 sqrt(uO/eO) HI = sqrt(u/e)

lOpen the data file open (ul, FILE=’te1np10.txt’, STA'FUSz’OLD')

nfl = nf

148

 

w"

 

 

!creating the range of frequencies
do i=1,nf1
if(nf 2: 500) then
f(i) = i * 5.e6
elseif(nf == 2000) then
f(i) = i * 1.25e6
elseif(nf =2 2500) then
f(i) = i * 1.e6
endif
enddo
!creating the wave numbers do i=1, nfl
k(i) = 2 * pi * f(i) * sqrt(u*e)
enddo
do i=1, nfl
P = exp(-j*k(i)*d)
Gamma = (n1 - n0) / (n1 + n0)
R(i) = (Gamma - P ** 2)/(1 - Gamma * (P) ** 2)
write(u1,*) f(i)*l.d-9,real(R(i)),aimag(R(i))
!write(u1,*) real(R(i)),aimag(R(i))
write(*,*) i,real(R(i)),aimag(R(i))
enddo
close(u1)

end program FLAT

149

 

 

 

 

 

 

BIBLIOGRAPHY

150

 

BIBLIOGRAPHY

[1] Milton Abramowitz and Irene A. Stegun. Handbook of Ma[hemethical Ferric/ions.

Dover Publications, Inc, 1964.

[2] Constantine A. Balanis. Advanced Engineering Electromagnetics. John Wiley

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[3] P. Ilavarasan, J .E. Ross, Edward Rothwell, Dennis P. N yquist, and K. M. Chen.

[6]

[7]

[8]

[9]

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Jerrold E. Marsden and Anthony J. 'Iiomba. Vector Calculus. W. H. Freeman
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C. R. Mullin, R. Sandburg, and C. O. Velline. A Numerical Technique for the
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Edward Rothwell, Dennis P. Nyquist, K. M. Chen, and B. Drachman. Radar
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Edward J. Rothwell and Micheal J. Cloud. Electromagnetics. CRC Press LLC,
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Garrett J. Stenholm. Nondetructive Evaluation of Layer IVIaterials Using the
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Garrett J. Stenholm, Edward Rothwell, Dennis P. Nyquist, Lee C. Kempel.
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151

 

 

 

 

[10] Heike Vollmer. Natural Mode Analysis of the Early-Time Response of a Dielectric
Coated Cylinder. Junior Thesis, Michigan State University and University of
Kaiserslautern, 2003.

[11] Heike Vollmer and Edward J. Rothwell. Resonance Series Representation of
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[12] Eric W. Weisstein. Radius of Curvature. From MathWorld — A \\-'olfram Web
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[13] H. Y. Yee. Scattering of Electromagnetic Waves by Circular Dielectric-Coated
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tennas and Propagation, 131822—823, September 1965.

152

      

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