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This is to certify that the

dissertation entitled

TRANSIENT SCATTERING OF ELECTROMAGNETIC WAVES
IN AN OCEAN ENVIRONMENT

presented by

ADAM J. NORMAN

has been accepted towards fulfillment
of the requirements for

PhD degree in Electrical Engineering

 

 

Major professor
Date w I 6

MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

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MSU I. An Affirmative Action/Equal Opportunlty IMRWOH
Wanna-9.1

.________________.___—'———’—ri “

TRANSIENT SCATTERING OF ELECTROMAGNETIC WAVES IN AN
OCEAN ENVIRONMENT

By

Adam J. Norman

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1996

ABSTRACT

TRANSIENT SCATTERING OF ELECTROMAGNETIC WAVES IN AN
OCEAN ENVIRONMENT

By
Adam J. Norman

A short pulse of an electromagnetic wave is illuminated upon a simulated ocean
surface. The incident wave is then scattered resulting in complex, geometry dependent
electromagnetic fields. Frequency domain techniques are employed in the solution of the
scattered fields. The transient short pulse results are synthesized from the spectral
returns. Numerous ocean models are considered and compared. Experimental results
have also been obtained and validate the scattering theory developed. New findings
regarding the physics and nature of transient scattering from ocean surfaces are reported,

which could be of great interest to future researchers.

In memory of my Grandma Norman, who taught me the importance of school

iii

ACKNOWLEDGEMENTS

I would like to acknowledge my guidance committee members. The Big 3 Dr.
Chen, Dr. Nyquist, and Dr. Rothwell, whom not only taught and encouraged me, but
also were my friends. I would also like to give special thanks to Jack Ross and Ponniah
Ilavarasan whom went out of their way to welcome me and get me started. I would like
to acknowledge all the electromagnetics graduate students for all their help and ideas.

The most important factor in the completion of this dissertation was the unceasing
support of my family. My father and mother, John and Lynne, provided me with love,
support and guidance throughout my studies. And, my wife Jennifer, who keeps me

smiling all my days.

iv

Table of Contents

List of Tables ......................................... viii
List of Figures ......................................... ix
List of Abbreviations .................................... xvii
Chapter 1 ............................................ 1
Introduction ...................................... 1
Chapter 2 ............................................ 5
Experimental Measurements ............................. 5

2.1 Scattering Range Descriptions ......................... 5

2.2 Comparison and Analysis of Measurement Systems ............ 12

2.3 Conclusions. .................................. 22
Chapter 3 ........................................... 28
Transient Scattering from a Conducting Sinusoidal Surface ......... 28

3.1 Introduction ................................. 28

3 .2 TE Scattering from Conducting Sinusoidal Surface ......... 29

3.2.1 Infinite PEC Surface ....................... 30

3.2.2 Preliminary results of TE infinite case. ............ 40

3.2.3 TE Scattering from a finite conducting surface. ....... 48

3.2.4 Preliminary results of TE finite case. ............. 49

3.3 TM polarization .............................. 51

3.3.1 Infinite Surface .......................... 51

3.3.2 Preliminary results for the TM infmite case ......... 57

3.3.3 TM finite surface scattering ................... 63

3.3.4 Preliminary results for the TM fmite case .......... 63

3 .4 Cut-off frequency phenomena ...................... 65

3 .5 Results and Experimental Confirmation ................ 67

3.5.1 Infinite surface results ...................... 67

3.5.2 Finite surface results ....................... 84

3.6 Conclusions ................................. 88

Chapter 4 .

.......................................... 97

Transient Scattering from an Imperfectly Conducting Sinusoidal Surface . 97

4.1 Introduction ................................... 97
4.2 Floquet-Mode Matching Analysis ...................... 98
4.3 Integral Equation Formulation ........................ 106
4.3.1 Moment-Method Numerical Solution ............. 110

4.3.2 Scattered fields ........................... 112

4.4 Duality of TE and TM polarizations .................. 113
4.5 Numerical Examples ............................ 114
4.6 Experimental Results ........................... 137
4.7 Conclusions ................................. 141
Chapter 5 ........................................... 152
Transient Scattering from N on-Sinusoidal Surfaces .............. 152
5.1 Introduction ................................... 152
5.2 Extensions to theory to accommodate non-sinusoidal surface. . . . 153
5.2.1 Floquet mode-matching/Rayleigh Hypothesis ......... 153

5.2.2 Integral Equation Formulation .................. 155

5.3 Non-sinusoidal sea-surface models ................... 157
5.4 Preliminary numerical results ...................... 158
5.5 Scattering from PEC surfaces for TM excitation ........... 169
5.5.1 Experimental Results for TM Excitation ........... 169

5.5.1a TM Measurements performed in time domain . . . . 185

5.5.2 Theoretical Response for TM Excitation ........... 191

5.5.3 Comparison/Evaluation of TM results ............. 194

5.6 Scattering from PEC surfaces for TE excitation ........... 199
5.6.1 Experimental Results for TE Excitation. ........... 199

5.6. la Measurements performed in time domain ...... 208

5.6.2 Theoretical Results for TE Excitation ............. 212

5.6.3 Comparison/Evaluation of TE results. ............. 221

5.7. Conclusions ................................. 226
Chapter 6 ........................................... 227
Transient Scattering of a Beam from Infinite Periodic Surfaces ....... 227
6. 1 Introduction ................................. 227
6.2 Application of Scattering Theory .................... 228
6.3 Beam Synthesis Techniques ....................... 229
6.3.1 Examples of Gaussian Beam. .................. 233

6.4 Transient Scattering Results. ....................... 236
6.5 Conclusions ................................. 238
Chapter 7 ........................................... 250
Transient Scattering from a Three Layer Medium ............... 250
7. 1 Introduction ................................. 250
7.2 Floquet mode-matching/Rayleigh hypothesis. ............. 251
7.3 PGF IE Technique ............................. 255
7.5 Numerical Results ............................. 259
7.6 Conclusions ................................. 264

vi

Chapter 8 ........................................... 265

Conclusions and Future Study .......................... 265
8.1 Summary .................................. 265
8.2 Major contributions and findings. .................... 266
8.3 Future research ............................... 267
Appendix A .......................................... 269
Periodic Green’s Function Derivation ...................... 269
Appendix B .......................................... 272
Sea Surface Realizations .............................. 272
Appendix C .......................................... 275
Extinction theorem integral equation formulation ............... 275
Bibliography ......................................... 278

vii

List of Tables

Table 3.1 Low frequency cutoff and maximal frequency of backward F loquet
modes. L=O.1016m, 85° incidence angle. ............... 66

Table 3.2 Low frequency cutoff and maximal frequency of backward Floquet
modes. L=O.1016m, 67° incidence angle. ............... 66

viii

Figure 1.1
Figure 1.2

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5
Figure 2.6

Figure 2.7

Figure 2.8
Figure 2.9
Figure 2.10

Figure 2.11
Figure 2.12
Figure 2.13

Figure 3.1

Figure 3.2
Figure 3.3

List of Figures

Scattering terms in an ocean environment ................. 4

Qualitative plot of target detection capabilities in the ocean
environment. ................................. 4

Configuration of the ground-plane time-domain measurement

system at MSU. ............................... 9
Configuration of the anechoic chamber measurement system at

MSU. .................................... 10
Configuration of the free-field arch-range scattering system at

MSU. .................................... 1 1
Spectrum of an ideal baseband incident short pulse with 3 GHz

bandwidth. ................................. 15
Ideal baseband incident short pulses with 3 GHz bandwidth. . . . . 16
Spectrum and transient nature of ideal incident short pulse with 8

GHz bandwidth. .............................. 17
Spectrum and transient nature of ideal incident short pulses with

16 GHz bandwidth. ............................ 18
Block diagram model of measurement system. ............ 23
Graphical representation of the calibration procedure. ....... 24
Transient backseattered field from 6 inch sphere (background

subtracted). ................................. 25
Transient backseattered field from a 6 inch sphere. ......... 25
Spectrum of backseattered field from a 6 inch sphere. ....... 26
Transient backseattered field from a 1:72 scale B52 airplane. . . . 27
Infinite, conducting sinusoidal surface scattering geometry. . . . . 35
Graphical representation of F loquet-mode scattered waves ...... 35
Partitioning scheme for MoM numerical solution. .......... 39

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10
Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

Figure 3.18

Figure 3.19

Figure 3.20

Figure 3.21

Modal amplitudes for a mode-matching solution using differing
numbers of terms. .............................
Induced surface current on one period of infinite, conducting

sinusoidal surface for TB and TM excitation at 4 GHz. .......

Dependence of currents induced on a PEC sinusoid for TB
excitation for various incidence angles. ................
Dependence of currents induced on a PEC sinusoid for TB
excitation for various surface heights. .................
Dependence of currents induced on a PEC sinusoid for TE
excitation for various surface period lengths (L) ............

Dependence of currents induced on a PEC sinusoid for various TE
excitation frequencies. ...........................

Finite, conducting sinusoidal surface scattering geometry. .....
Comparison of induced surface currents on fmite and infmite
surfaces for TE excitation. ........................
Dependence of currents induced on a PEC sinusoid for TM
excitation for various incidence angles. ................
Dependence of currents induced on a PEC sinusoid for TM
excitation for various surface heights. .................
Dependence of currents induced on a PEC sinusoid for TM
excitation for various surface period lengths (L) ............

Dependence of currents induced on a PEC sinusoid for various TM
excitation frequencies. ...........................

Comparison of induced surface currents on fmite and infinite
surfaces for TM excitation. ........................

Magnitude of total scattered electric field from infinite, conducting
sinusoidal surface as a function of frequency for TB excitation. . .

Magnitude of backseattered electric field from infinite, conducting
sinusoidal surface for TB excitation as a function of frequency. . .

Synthesized pulses used for interrogation of conducting sinusoidal
surfaces. ...................................
Total scattered electric field created by a GMC pulse from infinite,

conducting sinusoidal surface for TB excitation. ...........

Comparison of transient total scattered electric fields created by a
GMC pulse for various incidence angles upon an infinite sinusoidal
surface for TB excitation ..........................

43

46

47

61

62

64

71

72

74

75

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25

Figure 3.26

Figure 3.27

Figure 3.28

Figure 3.29

Figure 3.30
Figure 3.31

Figure 3.32

Figure 3.33

Figure 3.34

Figure 3.35

Figure 3.36

Figure 3.37

Transient backscattered electric field created by a 1/ 8 cosine pulse
from infmite, conducting sinusoidal surface for TB excitation.

Comparison of transient backscattered electric fields created by a
short pulse of various incidence angles upon an infinite sinusoidal
surface for TB excitation ..........................

Comparison of transient backscaterred electric fields created by a
short pulse from various height infinite sinusoidal surfaces for TB
excitation. ..................................
Comparison of transient backscattered electric fields created by a
short TE pulse from infinite sinusoidal surfaces of differing
periods. ...................................
Total scattered magnetic field for TM excitation of a PEC
sinusoid. ...................................
Total scattered magnetic field created by a GMC pulse from
infinite, conducting sinusoidal surface for TM excitation. .....
Backscattered magnetic field for TM excitation of a PEC
sinusoid. ...................................

Transient backscattered magnetic field created by a 1/ 8 cosine
pulse from infinite, conducting sinusoidal surface for TM
excitation.

Construction of sinusoidal sea-surface model ..............

Experimental measurements of a PEC sinusoid with TE excitation.
The measured spectral response and the synthesized transient
returns are shown for numerous incidence angles. ..........

Spectral amplitudes of theoretical and experimental backscattered
electric fields from a finite surface for TB excitation. ........

Theoretical and experimental transient backscattered electric fields
created by a short pulse from a fmite surface for TB excitation.

Comparison of experimental backscatter (finite surface) results to
theoretical backscatter from an infmite surface for TB excitation.

Comparison of transient backscatter created by a short pulse from
an infinite surface (theoretical) and a finite surface (experimental)
for TE excitation. .............................
Comparison of transient backscattered electric field created by a
short pulse from an infinite and finite surface for TB excitation.

Comparison of spectral domain backscattered magnetic field from
an infinite surface (theory) and a finite surface (experiment) for
TM excitation.

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

xi

76

77

82

84

86

89

90

Figure 3.38

Figure 3.39

Figure 4.1
Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5
Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Comparison of transient backscattered magnetic field created by a
short pulse from an infinite surface (theory) and a finite surface
(experiment) for TM excitation. .....................

Theoretical transient backscattered magnetic field created by a
short pulse from an infinite surface and a finite surface for TM
excitation. ..................................

Geometry for two media scattering problem. .............

Surface field calculations using Rayleigh-mode-matching method
compared to MFIE PGF method for various h/L ratios. ......

Surface H-fields on an infmite sea-water sinusoid for a TM
incident plane wave. Excitation frequency dependence is shown. .

Surface H-fields on an infinite sea-water sinusoid due to TM
excitation at 2 GHz. The dependence on sinusoid interface
amplitude (h) is examined. ........................

Surface H-fields on smaller height sea-water sinusoids. .......

Surface H-fields on an infinite sinusoid for TM excitation at 2
GHz. The dependence on incidence angle is examined. .......

Surface H-fields on an infinite sea—water sinusoid due to TM
excitation at 2 GHz. The dependence on sea-surface period is
shown. ....................................

Surface H-fields on an infinite sinusoid due to TM excitation at 2
GHz. The effects of conductivity are shown. .............

Surface H-fields on a infinite sinusoid excited by a 6 GHz TM
wave. The effects of conductivity are shown. .............

Identification of a Brewster angle for TM incidence. Comparison
with TB excitation is shown. .......................

Identification of Brewster’s angle for lossless dielectric
(e=80,o=0) surface. ...........................

Identification of Brewster angle for lossy dielectric surface
(e=80,a=4) for various h/L. ......................

Identification of a Brewster angle for TM incidence. The effects
of excitation frequency are shown. ...................

Identification of a Brewster angle for TM incidence. The effects
of sea-water sinusoid amplitude are examined. ............

The backscattered H-field around the Brewster angle for TM
incidence. ..................................

xii

95

121
121

122

124

125

126

129

130

Figure 4.16

Figure 4.17

Figure 4.18

Figure 4.19

Figure 4.20

Figure 4.21

Figure 4.22

Figure 4.23

Figure 4.24

Figure 4.25

Figure 4.26

Figure 4.27

Figure 4.28

Figure 4.29
Figure 4.30

Figure 4.31
Figure 4.32
Figure 4.33

Backscattered H-field from an infinite sea-water sinusoid. The
effects of field point location are examined ...............

Backscattered H-field from an infinite sea-water sinusoid. The
effects of field point location are examined ...............

Forward scattered H-field from an infinite sea-water sinusoid. The
effects of field point location are examined ...............

Forward scattered H-field from an infinite sea-water sinusoid. The
effects of field point location are examined ...............

Comparison of Rayleigh and MFIE-PGF method for computing
total scattered field spectrum from imperfectly conducting surface.

Transient TM response of imperfectly conducting sinusoidal
surface. ...................................
Comparison of Rayleigh and MFIE PGF methods for computing
backscattered field spectrum for imperfectly conducting surface.

Backscattered transient response of imperfectly conducting
surface. ...................................
Configuration of
measurements.

sea-water sinusoid for experimental

Theoretical backscattered H-field from an infinite sinusoid.
Comparison of PEC and sea-water models for TM excitation at

Theoretical backscattered E-field from an infinite sinusoid.
Comparison of PEC and sea-water models for TE excitation at

Theoretical backscattered H-field from an infinite sinusoid.
Comparison of PEC and sea-water models for TM excitation at
85°. ......................................
Theoretical backscattered E-field from an infinite sinusoid.
Comparison of PEC and sea-water models for TB excitation at
85°.

Theoretical transient backscatter for TM excitation.

backscatter for TM

Experimentally measured transient
excitation.

Experimentally measured transient backscatter for TB excitation. .

Theoretical time-gated backscatter for TM excitation. ........

xiii

131

132

133

134

147
148

Figure 4.34
Figure 4.35
Figure 4.36
Figure 4.37

Figure 4.38

Figure 5.1

Figure 5.2
Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Experimentally measured backscatter for TM excitation. ......
Theoretical time-gated backscatter for TE excitation. ........
Experimentally measured backscatter for TB excitation. ......

Comparison of theoretical and measured TM backscatter from a
sea-water sinusoid. .............................

Experimentally measured forward scatter from a finite sea-water
sinusoid. Both transient and spectral results are shown of a number
of incidence angles. ............................
Partition scheme for sea-surface interface. ...............

Comparison of sea-surface models. ...................

Surface current on PEC double sinusoid with increasing ripple
height. ....................................

Surface current induced on the Stokes wave model for a TB
incident plane wave at 85° with frequency of 2 GHz. ........

Surface current induced on the double sinusoid wave model for a
TB incident plane wave at 85° with frequency of 2 GHz. ......

Comparison of induce surface currents on the Stokes and double
sinusoid wave models. ..........................

Spectrum of backscatter for TB incident wave at 85°, L= .3m at
field point x/L=0, z/L=20 .......................

Spectrum of backscatter for TB incident wave at 85°, L= .3m at
field point x/L=0, z/L=20 .......................

Transient backscatter from PEC surface with TB incident wave at
85°, L=0.3m, x/L=O, z/L=20 .....................

Transient backscatter from PEC surface with TB incident wave at
85°, L=0.3m, X/L=O, z/L=20 .....................

Angular dependence of TM scattered fields of the Stokes wave
model. ....................................

Angular dependence of TM scattered fields of the double sinusoid
wave model. ................................

Angular dependence of TM scattered fields of the Donelan Pierson
wave model. ................................

Angular dependence of TM scattered fields of the Donelan Pierson
wave model. ................................

xiv

148

150

151
154
160

161

162

163

167

168

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

Figure 5.21

Figure 5.22

Figure 5.23

Figure 5.24

Figure 5.25

Figure 5.26

Figure 5.27

Figure 5.28

Experimental TM scattering from various wave models for an
incidence angle of 70°.

Experimental TM transient scattering from various wave models
for an incidence angle of 70°. Synthesized with a 1/8 cosine

taper.

TM backscattered fields from the Stokes wave model.
Experimentally measured response at 60° and 80°. ..........

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

TM backscattered fields from the double sinusoid wave model.
Experimentally measured response at 60° and 80°. ..........

TM synthesized transient backscattered fields from the Stokes
wave model. Experimentally measured response at 60° and 80°.
Spectral returns were weighted with 1/8 cosine taper before
IFFT ......................................

TM synthesized transient backscattered fields from the double
sinusoid wave model. Experimentally measured response at 60°
and 80°. Spectral returns were weighted with 1/8 cosine taper
before IFFT. ................................
TM synthesized transient backscattered fields from the Donelan
Pierson wave model. Experimentally measured response at 60° and
80°. Spectral returns were weighted with 1/8 cosine taper before
IFFT ......................................

Transient TM backscatter response for the Stokes wave model. A
comparison of frequency-domain synthesis and time-domain
techniques. .................................
Transient TM backscatter response for the double sinusoid wave
model. A comparison of frequency—domain synthesis technique and
time-domain technique. ..........................

Transient TM backscatter response for the Donelan Pierson wave
model. A comparison of frequency-domain synthesis technique and
time—domain technique. ..........................

Spectrum of TM backscatter response for the Stokes wave model.
A comparison of frequency—domain synthesis technique and time-
domain technique. .............................

Spectra of TM backscatter response for the Stokes and double
sinusoid wave models. Measurements were performed using the
time-domain technique. ..........................

TM backscattered magnetic field for the Stokes wave model.
Comparison theoretical responses for 60° and 85° incidence. . . . .

Synthesized transient TM backscattered magnetic field for Stokes
wave.

0000000000000000000000000000000000000

XV

180

181

182

184

187

188

189

190

192

Figure 5.29

Figure 5.30

Figure 5.31

Figure 5.32

Figure 5.33

Figure 5.34

Figure 5.35

Figure 5.36

Figure 5.37

Figure 5.38

Figure 5.39

Figure 5.40

Figure 5.41

Figure 5.42

Figure 5.43

TM backscattered magnetic field for the double sinusoid wave
model. Comparison theoretical responses for 60° and 85°
incidence. ..................................
Synthesized transient TM backscattered magnetic field for double
sinusoid. ................................... 193
TM backscattered magnetic field for the Stokes wave model.
Comparison of theoretical and measured responses for 60°
incidence. ..................................
Synthesized transient TM backscattered magnetic field for the
Stokes wave model. Comparison of theoretical and measured
responses for 60° incidence. .......................
TM backscattered magnetic field for the double sinusoid wave
model. Comparison of theoretical and measured responses for 60°
incidence. ..................................
Synthesized transient TM backscattered magnetic field for the
double sinusoid wave model. Comparison of theoretical and
measured responses for 60° incidence. ................. 198
Angular dependence of TE scattered fields of the Stokes wave
model. ....................................
Angular dependence of TE scattered field spectrum of the double-
sinusoid wave model. ...........................
Angular dependence of TE scattered fields of the Donelan Pierson
wave model. ................................
Angular dependence of TE scattered fields of the Donelan Pierson
wave model. ................................
Comparison of synthesized transient scattering from the Stokes
wave and double sinusoid wave for near grazing incidence
angles. .................................... 205

Experimental TE scattering from various wave models for an
incidence angle of 70°. ..........................
Experimental synthesized transient TE scattering from various
wave models for an incidence angle of 70°. GMC weighting
applied to spectrum before IFFT. .................... 207

Comparison of frequency-domain synthesis measurement with
time- domain measurement. Both spectral and transient returns for
Stokes Wave interrogated at 60°. ....................
Spectra of TE backscatter response for the Stokes and double
sinusoid wave models. Measurements were performed using the

frequency-domain technique. ....................... 210

xvi

Figure 5.44

Figure 5.45

Figure 5.46

Figure 5.47

Figure 5.48

Figure 5.49

Figure 5.50

Figure 5.51

Figure 5.52

Figure 5.53

Figure 5.54

Figure 5.55

Figure 5.56

Figure 6.1
Figure 6.2

Figure 6.3

Figure 6.4

Transient TE backscatter response for the double-sinusoid wave
model. A comparison of frequency-domain synthesis the time-
domain techniques .............................. 211

Theoretical synthesized transient TE backscattered fields from the
Stokes wave model. The response at 85° incidence for two field

points is shown. .............................. 213
Theoretical synthesized transient TE backscattered fields from the
Stokes wave model. The response at 60° incidence for two field
points is shown. .............................. 214
Theoretical synthesized transient TE backscattered fields from the
double- sinusoid wave model. The response at 85° incidence for
two field points is shown. ........................ 215
Theoretical synthesized transient TE backscattered fields from the
double- sinusoid wave model. The response at 60° incidence for
two field points is shown. ........................ 216
Theoretical transient scattering from double-sinusoid wave at 85°,
with the effect of GMC window width examined. .......... 217
Theoretical transient scattering from double-sinusoid wave at 85°,
with the effect of GMC window center frequency examined. 218
Theoretical transient scattering from Stokes wave at 85°, with the
effect of GMC window width examined. ............... 219
Theoretical transient scattering from Stokes wave at 85°, with the
effect of GMC window center frequency examined. ......... 220
Comparison of synthesized transient scattering from Stokes wave
with theoretical transient scattering at 60°. Using 0.8-7.2 GHz
GMC windowed spectral content. .................... 222
Comparison of TE scattering from Stokes wave with theoretical
scattering at 60°. ............................. 223
Comparison of theoretical and measured transient scattering from
the double-sinusoid wave. Angle of incidence is 60°. ........ 224
Comparison of theoretical and measured scattering from the
Donelan Pierson wave. Angle of incidence is 60°. .......... 225
Geometry of Beam Synthesis Apertures. ................ 229
Beam synthesis and scattering geometry ................. 231
Spectral Amplitudes for a Gaussian Incident Beam Parameters are
L=5.0m and w =.5m. .......................... 234
Gaussian Incident beam created from 21 terms of the Fourier
Series, where L=5.0 m, w=.5 m, and 0 =70° ........... 235

xvii

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 6.11
Figure 6.12

Figure 6.13

Figure 6.14

Figure 6.15

Figure 6.16

Figure 6.17

........

Induced Surface currents on PEC surface, for 30° Gaussian beam
illumination. ................................ 240

Induced surface currents on PEC surface for 70° beam
illumination. ................................ 240

Induced surface currents on PEC surface for 30° beam
illumination. ................................ 241

Induced surface currents on PEC surface for 70° beam
illumination. ................................ 241

Frequency response of backscatter from a PEC surface for a single

plane wave. ................................ 242
Transient backscatter from a PEC surface for single plane wave

pulse illumination. ............................. 243
Transient backscatter for single plane wave incident. ........ 244

Spectrum of backscattered field for an incident Gaussian beam
(0=30°) ................................... 245

Frequency response of backscattered field for an incident Gaussian
beam (L=5.0m, w=0.5m, n=21 0=70°) ............... 245

Transient backscattered field for an incident Gaussian beam pulse
(0 = 30°). ................................... 246

Transient backscattered field for an incident Gaussian beam pulse
(0=70°). ................................... 246

Transient backscattered field for an incident Gaussian beam pulse
(L=5.0m, w=0.5m, 0=30°) ....................... 247

Synthesized transient backscatter response for Gaussian beam pulse
incident at 30° and 70°. (Beam parameters: n=21, w= .5m, L=5m
Surface parameters : P=0.1m, h=0.007m, PEC sinusoid )

.......................................... 248
Synthesized transient total scatter response for Gaussian beam
pulse incident at 30°. (Beam parameters : n=21, w=.5m, L=5m)
( Surface parameters : P=0.1m, h=0.007m, PEC sinusoid )
.......................................... 249
Configuration of three medium scattering problem. ......... 251
Comparison of induced surface fields for TM excitation of three
layered medium where the upper layer has a sinusoidal interface
(PGF IE). .................................. 260
Frequency response of backscattered fields for a TM plane wave
incident upon a three layer medium. .................. 261

xviii

Figure 7.4

Figure 7.5

Figure B.1

Figure C.1

Transient backscattered fields from a three layer medium where
the upper interface is sinusoidal. ....................

Transient total scattered field inside the middle layer of a three
layer medium where the upper layer has a sinusoidal interface. . .

The Donelan Pierson spectrum of wind driven waves for various

wind speeds. ................................

Generic configuration of a 2-D two-region problem. ........

xix

SP
CW
FMM
PGF
IE
MoM
EFIE
MFIE
BS
TS
FF’I‘
IFFT
GMC

List of Abbreviations

Ultra Wide-band

Short Pulse

Continuous Wave

Floquet Mode Matching
Periodic Green’s Function
Integral Equation

Method of Moments

Electric Field Integral Equation
Magnetic Field Integral Equation
Backscatter

Total Scatter

Fast Fourier Transform

Inverse Fast Fourier Transform

Gaussian Modulated Cosine

Chapter 1
Introduction

The radar detection and identification of targets in the ocean environment is a
venerable subject. The large and unpredictable (over time) clutter produced by the ocean
has limited the usefulness of conventional, narrow band (CW) radars [1-7], thereby
allowing targets (missiles, planes) to pass through radar coverage undetected. New hope
has arisen with the advent of feasible ultra-wideband/short-pulse (UWB/ SP) radar systems
[8,9]. UWB/ SP radar provides a much higher range resolution or target feature resolution
and may reduce the multipath effect. There is also hope that UWB/ SP radar may have
the ability to not only determine a target’s speed and location, but also identify the
target’s shape and size. The greater spectral content allows for entirely new identification
and detection schemes, including the use of time-frequency analysis.

A shipboard radar system, as shown in Figure 1.1, has to contend with a number
of scattering mechanisms [10,11]. The target response will be masked by clutter (or
backscatter) and could also be deteriorated by multipath scattering. The clutter term will
be most detrimental when the ocean is roughest. This is due to the large slopes occurring
on the ocean waves. The multipath effect is prominent for smooth ocean surfaces. A
qualitative plot of the problem facing radar target detection in the ocean environment is
shown in Figure 1.2. These are the problems associated with conventional CW type radar
used in detection. The ability of the UWB/SP radar to overcome these drawbacks will
be discussed throughout the following chapters.

Although the clutter description for CW type excitation in an ocean environment
has been studied by a great many researchers [1-7,12-36], the clutter descriptions for
UWB/SP have never been examined. This thesis will provide theoretical and

experimental descriptions of UWB/ SP scattering responses in an ocean environment.
These basic building blocks will allow future workers to fully exploit the WW SP radar
systems for target identification and detection.

The thesis will be broken up into chapters that logically develop the UWB/ SP
description of clutter in an ocean environment. Both the ocean models and interrogating
wave shape will be methodically evolved from simple (ideal) cases to complex (real-
world) models. The initial ocean sirnulation that will be considered in chapter 3, is the
perfectly electric conducting (PEC) sinusoidal surface illuminated by a plane wave. Both
a fmite and an infinite interface will be considered. A frequency-domain integral-equation
formulation is utilized, and is subsequently implemented numerically via the Method of
Moments (MoM). The transient scattered fields are obtained through the use of an
Inverse Fast Fourier Transform (IFFT). The infmite sinusoidal surface can also be
treated more classically by a mode-matching technique in conjunction with the Rayleigh
hypothesis. The important feature of the solutions is that there are no approximations
made, such as the Kirchoff approximation (physical optics). This rigor is needed to fully
include all the physical effects that could arise. It is this complete understanding of the
underlying physics of the problem that could open new avenues for target detection and
identification.

In chapter 4, the ocean surface model will be improved, and will be modelled
as a two-layer (non-PEC) system. The two layers will be air and sea (e,=80, 0 =4 S/m)
separated by a sinusoidal interface. This model relies on a similar frequency—domain
integral-equation formulation, which is also quite rigorous. The mode-matching technique
is also extended to the two-layer model. The effects of an imperfectly conducting layer
will be examined and compared to the PEC case. The possibility of a Brewster’s angle
phenomenon is also investigated and its implications are discussed.

The interface will then, in chapter 5, be allowed to have any (non-sinusoidal)

shape, but it must remain periodic for the infinite-surface case. The non-sinusoidal

surfaces will be constructed to more accurately depict real ocean waves. The analytical
techniques developed in the previous chapters are well equipped to handle the non-
sinusoidal surface. The enhanced range resolution of the UWB/ SP transient radar is fully
utilized with the finer structure of the non-sinusoidal waves.

The ideal incident plane wave restriction is removed in chapter 6, when a limited-
footprint beam wave is assumed incident. For the infinite surfaces this beam is
constructed from the superposition of a fmite number of plane waves; therefore the
scattering theory developed in the prior chapters will still hold. Beam excitation is an
important step in modelling a realistic radar system.

In chapter 7, the ocean model is once again improved with the addition of a third
layer, this layer could represent the ocean floor, a large submerged object, or a ducting
(guiding) region above the ocean surface. The additional layer is considered to be planar,
which minimizes the numerical complexity for a three layer structure.

In the chapters described above, experimental results will accompany many of the
theoretical findings. This will serve to validate both the theoretical and experimental
techniques that are developed. To facilitate these discussions on the experimental results,
a preliminary chapter is needed to introduce the measurement techniques and processing

methods used throughout the thesis. Chapter 2 will serve as this preliminary chapter.

Ship Radar Direct Path

 

 

 

 

 

   

Target
IT > <— /71
\l
Reflected Path
Stu-face Backscatter
W
Ocean Surface
Figure l. 1 Scattering terms in an ocean environment.
4.;
C
d)
E multipath clutter
é- , l/ I“I"uII‘IlIIIIIIIIIIIlium
II
C
.9

Detect

sea roughness

Figure 1.2 Qualitative plot of target detection capabilities in the ocean environment.

Chapter 2

Experimental Measurements

In this chapter the techniques employed in measuring the transient scattered fields
from ocean surface models will be examined. In-depth discussions of specific results will
be found throughout the thesis as the scattering theory is developed. This chapter will
expound on the technical aspects of performing such measurements.

There are three scattering measurement ranges located at MSU from which
transient results can be obtained. There are two free-field ranges (Arch Range and
Anechoic Chamber) which operate on the same basic principles, that is frequency—domain
synthesis [37-39]. There is also a large ground plane on which true base-band time-
domain measurements are performed. The arch range and the anechoic chamber may also
be used in performing true transient measurements, but the transmitting and receiving
horns are not optimal for such measurements. The operating principles of each range will
be explained and sample measurements will be performed on the systems for comparison

purposes and to illuminate the explanations.

2. 1 Scattering Range Descriptions

Historically transient measurements were performed solely in the time domain,
that is a pulse (or step) generator creates a train of pulses which is radiated by some
antenna, and is then received by another (or the same) antenna and measured by a
sampling oscilloscope. At Michigan State the experimental setup is as in Figure 2.1, of
note is the large ground plane which serves to reduce the physical size (imaging) of the
range and as shield for the feeding and receiving lines and connectors. A Picosecond

Pulse Labs (PPL 1000B) pulse generator creates the incident pulse and is radiated by a

8 foot Bi-Conical Antenna with a 16° apex angle. This type of complementary antenna
was chosen for its frequency independence, which is required for short pulse transmission
and ease of calibration. The receiving antenna is a short probe monopole, which acts a
differentiator [40], therefore a step incident wavefront would result in a pulse response
from the probe. A Tektronix 7854 Sampling oscilloscope is used to measure the return
signal, and is connected via HPIB to a PC. This data acquisition system allows for post
processing of the measured results.

The current system is currently limited to roughly a 1/2 nanosecond pulse, which
corresponds to a 2 Ghz wide baseband radar. This restriction is caused by the PPL
generator, which can only produce 2 GHz of bandwidth. However, the system does
produce high quality results over the operating bandwidth. The system is also an
excellent instructional tool due to the immediate results obtained and intuitive nature of
the time-domain reflected waves.

Due to the geometry of the ground plane and bi-conical antenna length, the
effective measurement time window of the system is roughly 12 nanoseconds. This
implies that the target must be sufficiently physically small and placed in close proximity
to the probe. The measurements that are performed later in the dissertation do not lend
themselves well to the ground plane system, therefore little emphasis will be placed on
it and the above discussion is only included for completeness.

The pulse generator and sampling oscilloscope can also be utilized in the anechoic
chamber to perform true short-pulse time-domain measurements. The transmitting and
receiving antennas are American Electronic Laboratories (AEL) model H-1734 TEM
wideband (0.5-6.0 GHz) horns. These antennas have non-ideal transfer functions that
reduce the fidelity of the measurement system. The anechoic chamber setup does,
however, lend itself well to the measurements to be performed later in the dissertation,
and therefore will be used as the primary comparison tool in this thesis. The non-ideal

characteristics of the horns must be compensated for through calibration and spectral

weighting. The short-pulse time—domain method in the anechoic chamber has roughly a
10 ns quiet zone (similar to ground-plane system), and this limits the physical size of the
scatterer.

In order to overcome the bandwidth limitations of the time-domain systems
mentioned above, a frequency-domain synthesis technique was developed [37,38]. This
development was aided by the advancement of highly-stable large-bandwidth CW signal
sources and similar amplifiers and detectors. The basic idea behind the frequency—domain
synthesis technique is to repeatedly measure the target (stationary) response at a large
number of frequencies. These frequencies should be chosen and weighted to simulate a
short pulse when synthesized to the time-domain using the inverse fast-Fourier transform
(IFFT). Therefore an IFFT performed on the measured return signal should accurately
describe the time-domain pulse response.

The principles and techniques behind performing these transient measurements
(synthesized) in an anechoic chamber are well established [37]. MSU’s anechoic chamber
(See Figure 2.2) is 12’ x 12’ x 24’ and is lined with pyramidal absorbers having a
pyramid depth of 6 inches. The frequency-domain measurements are performed using a
HP 8720B automatic network analyser (ANA) with a HP 8349B Amplifier, which has
a gain of 20 dB over a bandwidth of 2-20 GHz, connected between the ANA and the
transmitting antenna. Because of the wide bandwidth of needed measurements, a true
monostatic system is not feasible due to the large reflection at the transmit horn antenna.
Therefore a bi—static system consisting of two closely spaced AEL H-1734 pyramidal
horn antennas is used to simulate a monostatic system. The backscattered field is then
proportional to the S21 scattering-parameter measurement obtained with the network
analyser.

The general idea behind the measurement system is to measure the response of
an unknown target (ocean surface model) followed by a similar measurement for a known

target or calibrator (sphere). From the information obtained with the known target, any

effects due to the system response can be eliminated from the unknown target
measurement. The latter results are then taken to the time domain via an IFFT, yielding
the transient backscattered response due to a plane-wave incident pulse. A quantitative
analysis of this measurement technique can also be performed the details of which appear
in [37]. The procedure is summarized on p.19 where a sample measurement is performed
and analyzed.

The arch range (See Figure 2.3) is a large (20 ft diameter) horn-supporting
structure, which allows the horn—antennas to be arbitrarily located by rotation about its
periphery. The measurements are very similar to those of the anechoic chamber, in that
this is also a frequency domain system which utilizes the HP 8720B in the 82, mode.
Once again, due to the large bandwidth, a bi-static horn system is used to simulate a
mono-static radar. However, the arch range also offers the advantage of providing for
true bi-static measurements (large horn separation) with precise horn locations. However,
the lack of absorbers introduces many difficulties, associated with undesired reflected
waves, which aren’t present in the anechoic chamber. These difficulties include both the
unwanted reflections from the surrounding environment and the interaction of target with
that environment. These interactions limit the size and placement of the target. To
minimize the interaction of the horns with the supporting structure and floor, dielectric
lenses are used to collirnate the transmitted wave. Additionally, after much
experimentation, pyramidal absorbers where placed in the most problematic regions,

including the floor in front of the target and wall behind the target.

 

 

k 32' 'l

 
 

Bl—Conlcal Tx Antenna

\

O . 4’ Target
1

Rx Probe

    
 

 

     
  

20'

 

   

Ground Plane
(Side View)

  
    

 

 

 

 

 

 

 

 

 

PPL 1°00 ‘_ Tektronix
7854 .
[ pulse out Rx HPIB
trigger ! >—-_' trl er In
A

Figure 2. 1 Configuration of the ground-plane time—domain measurement system at
MSU.

antennas

target
/

<— podium

 

 

 

 

 

 

 

 

 

\l/ I \l/

CH1 CH 2 COMPUTER

 

 

 

 

 

 

 

 

I HP AMPLIFIER I L _‘ I S < I : (STORAGE a
____. 2‘ PROCESSING)
HP87208 ANA

 

 

 

 

 

Figure 2.2 Configuration of the anechoic chamber measurement system at MSU.

10

 

 

HP87ZOB If
ANA

— CH 2 .
4mm...“ Ej>

 

 

 

 

 

Figure 2.3 Configuration of the free-field arch-range scattering system at MSU.

11

2.2 Comparison and Analysis of Measurement Systems.

As a general statement regarding the performance of the three measurement
systems, the anechoic chamber (frequency domain) yields the highest quality, largest
bandwidth results. Therefore, most of the results in the following chapters are obtained
via the anechoic chamber. There are number of cases in which the arch range is the
system of choice, such as measuring forward scattered fields or any bi-static
measurement. The time-domain systems provide primarily a confirmation of the
frequency-domain synthesis measurement responses.

The frequency domain synthesis technique, although somewhat paradoxical in that
a number of sinusoidal steady-state measurements can yield a transient result, is
important to understand in a practical application context. There will be a great number
of synthesized results presented in the remainder of the thesis, and a quick refresher on
weighting, bandwidth and other related topics will be provided in the form of examples.
This will also serve to provide examples of measurements from the various systems.

The need for a weighting function or window, arises when there are
discontinuities at the endpoints of a waveform. The waveform must be weighted before
transformation into another domain (IFFT or FF’I‘). The discontinuities would produce
unwanted, non-causal oscillations. A weighting function that removes the discontinuities
would also remove the unwanted oscillations in the transformed waveform.

A weighting function usually tapers the waveform at the endpoints down to zero
magnitude, which would be continuous. The rate and shape of the roll-off greatly effects
the resultant transformed waveform, hence there are a great number of weighting
functions. There are two weighting functions that are extensively used throughout the
thesis, and they are the Gaussian modulated cosine (GMC) and the cosine taper.

In the frequency domain the GMC window is bell-shaped, where the center of the
bell and the width of the bell can be controlled by fc and r , respectively. The window

12

has a rather gentle roll-off which greatly dampens the oscillations due to the edge

discontinuities. The GMC window in the time domain is given by

w(t) = cos(7tfct) (“my (2-1)

and in the frequency domain (via a Fourier Transform) by

W( f) = r{ e'W-farl’ + (“WM } (2.2)

where fc is the center frequency of the window and t is the shape or width of the
window.

The cosine taper is rather rectangular looking, with a rather steep roll-off at the
edges. The rate of the roll-off is controlled by r , where a larger 1: results in a steeper
roll-off. The cosine taper will generally have more oscillations as compared to the GMC
window, but the cosine taper retains more of the spectral information. The cosine taper

or Tukey window in the frequency domain is given by

Sinzgflfxl) f> f—lFH or f< ——"

W(f) = 4 t 1: (2-3)
I 1 f< L117” and f> J
r

 

where F I, is the highest frequency of the band-limited spectrum and FL is the lowest
frequency, AF is F H - F L, and 1: is the shape or roll-off factor and must be even. The
standard notation for the cosine taper is : % cosine taper.

The effects of bandwidth and weighting functions can best be explained through
examining ideal incident short pulses. These are the pulses that an UWB/ SP radar would
be transmitting. A baseband short-pulse radar, which can be simulated with the ground-
plane time-domain measurement system, contains spectral information down to DC (0
Hz). In Figure 2.4 the Gaussian modulated cosine (GMC) weighted spectrum is
representative of a baseband radar and is very close to the output of the PPL pulse

generator. The actual shape of the resulting time-domain pulse is shown in Figure 2.5

13

(solid line), this pulse is compared to another ideal short pulse containing the same
spectral bandwidth but weighted differently. A cosine taper was applied to the spectrum
(See Figure 2.4), which eliminates any dc component (i.e. oscillations occur).

The GMC is a double-edged sword; it creates smooth transient results, because
of the gentle roll-off at the band edges, but there is also information lost at the band
edges. The cosine taper allows for more information (spectral content) to be preserved
but, due to the relatively steep roll-off at the band edges, there exist more oscillations
in the transient results. This point can be visualized by considering a rectangular pulse
weighted in the frequency domain which would transform to a sine function in the time
domain. The conclusion is that it is important to understand when and why to use the
various weighting functions. These points are further illustrated in the next figures.

In Figure 2.6 a larger bandwidth (8 Ghz) incident pulse is considered, and the
pulse is not baseband. Increasing the bandwidth results in a shorter pulse, which would
produce better feature resolution. The effects of the weighting functions discussed above
are also obvious. The GMC weighted pulse now contains some oscillations because it is
not a baseband pulse. The inset figure shows the respective spectra of the incident pulses,
and the differences in the roll-offs, which effect the oscillations, can be seen in this
figure.

To further examine the effects of bandwidth, in Figure 2.7 that bandwidth is
increased to 16 GHz. The resulting pulses are seen to be much shorter in duration. The
2-18 GHz bandwidth in this figure coincides with the limits imposed by equipment at
MSU radiation laboratory. Therefore, the pulses shown in Figure 2.7 represent the

shortest pulses that can be produced in our laboratory.

14

Spectrum of Incident Pulses

1'20 —— GMC 7'=.4, f,_.=0 GHz)
---- 1/8 osine Taper

1.00 , -------------------------------- i

0.80

Magnnude
O
8

 

0.40
0.20
II “
0.00 I I T I I I I I I I I I I I I I I I I I I I r I I I I I I I—I
0.00 1.00 2.00 3.00

Freq (GHz)

Figure 2.4 Spectrum of an ideal baseband incident short pulse with 3 GHz bandwidth.

15

S nthesized Incident Pulses
( —3 GHz BandWidth)

 

 

 

6E+009:
I f.‘
I ,‘I —- GMC (7'=.4, f¢=0 GHz)
: : '. ----- 1/8 Cosine Taper
4E+009: .' '.
l
a, 1 I' I.
‘s : I '
49 - ' i
._ _ I
< 2E+009: .' I:
<0 1 I' I
.2 . . :
*5 1 i I.
6 7 .' .
Er : A ' II "\
09000—4? ,", .' 5% 4: T\ A x. xx-
- I \ I ‘ ' I I \ I, \\ I \’/ \
-’ \ I I I I ' ‘ I \l
d \ I l I ' , ‘ ,
_ v | , I ' k
_ I I 1 I
- I I I '
_ \l l
_ \ 1’
—2E+009.TITIFIIITIIIj—IIIIIIIIIIIIIIIIIIIrIIIIII]
9.00 10.00 11.00 12.00 13.00
TIme (ns)
Figure 2.5 Ideal baseband incident short pulses with 3 GHz bandwidth.

16

Synthesized Incident Pulses
(1—9 GHz BandWidth)

mum-rerun-

 

    

 

 

 

 

 

 

: 1":
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—2E+0091 1”,.IH
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: I" I" ----- 1/8 Cosme Taper
'l l
—7E+oogqIIIIIIrIIrITjIIIIIIIIITIIIjTIlIIIIIIII—I—l
9.00 10.0 11.00 12.00 13.00
'anue (ns)

Figure 2.6 Spectrum and transient nature of ideal incident short pulse with 8 GHz
bandwidth.

17

S nthesized Incident Pulses
( —18 GHz BandWidth)

mocha-11m

3E+010

 

 

 

 

 

 

.1" ---- 1/8 osine Taper

j, 1":
+ :
: g '“' If 1.
: , .... I i,
‘ I: i
. w. .
25+010- p I i
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.t’ ‘ I:
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< 1E+010: I:
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> - I'
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B : I, :l
CK " I II" I' .10
OE+OOO_ swims - -
‘ fill'lll“
_ II'.II"
- "I'Il'u'
; i" “I — GMC 7’=.13, f,=10 GHz)

 

 

—1E+O1CIIIIIIfTTIIIIIIIIIIIIIIIIIIITT

9.50 I. 10.50 11.50 1250
Time (ns)

Figure 2.7 Spectrum and transient nature of ideal incident short pulses with 16 GHz
bandwidth.

18

An illustrative example of a measurement in the anechoic chamber (frequency
domain) will provide a glimpse at the quality of the chamber and is an excellent means
of explaining the calibration procedure. The end goal of a measurement is to find the
transfer function of an unknown target (Hs( f )). However, simply obtaining the
measurement response of the unknown target does not yield the transfer function. There
are numerous unknown transfer functions associated with the measurement system. These
transfer functions are given in a system block diagram in Figure 2.8. To determine the
unknown system transfer function another measurement is required, and this
measurement is called the calibration measurement.

Referring to Figure 2.8, a measurement (R( f )) consists of a number of transfer
functions acting upon an excitation (E( f )). There is also an additive noise factor, that
is negligible for a high quality anechoic chamber, such as the chamber at MSU.

A calibrated measurement in the anechoic chamber consists minimally of three
individual measurements; a background measurement response (R b( f) ), a calibrator
measurement response (R‘*”( f )), and finally a target measurement response (R‘ *b( f )).
Sometimes it is necessary to perform two background measurements if the background
has changed between calibrator and target measurements. These responses can be

mathematically described as

Rbtf) = SmlHam+Hcml 5m = HrmHmEm (2.4)
R“”(f) = $0) {Ham meg) +1150) + 113:0); (2.5)
“Mm = 50) {Ham +3.0) +HJU) +Hsim} (M)

where S( f) is the system transfer function and is composed of the antenna transfer
functions (H,( f ), H,( f )) and the excitation source transfer function (E( f) ). There is also
a clutter term (Hc( f )), a coupling term (Ha( f )) and an interaction term (Hsc( f )). It is

important to note that the system, coupling and clutter transfer functions remain the same

19

for all three measurements, therefore a background subtraction will eliminate the
coupling and clutter terms. The background measurements are then subtracted from the
respective calibrator and target returns (R c = R” - R b). There remains two resultant

sets of spectral waveform data, the subtracted calibrator (R ‘( f )) and the subtracted
target (R ‘(f))

Rem = sth:m+H.:m1 (2.7)

W) = sm1H.‘tn+H.‘.m} (2'8)

At this point the calibrator is used to remove the effects of the measuring system.
This is accomplished by using a calibrator that has a known response. A useful calibrator
is a sphere, since the scattered field from the sphere can be represented in closed form
by the Mic series, and the sphere is impervious to orientation/aspect/polarization.
Therefore a sphere is used for nearly all the subsequent measurements in this
dissertation.

The system can now be calibrated using the known response of the sphere. By
dividing (in the frequency domain) the measured sphere or calibrator response (R ‘( f)
or eqn. (2.7) ) by the theoretical model (H: (f) ), a calibration waveform is created. We
have assumed that the mutual interaction term (HS:( f )) is negligible, which is the case
for a high quality chamber. The calibration waveform (S( f )) represents the differences
between an ideal measuring system and the actual measuring system and is the transfer
function of the system. Those differences include transfer functions of the amplifier, the
horns, the connectors and the cables and the propagation paths.

sm = m (2.9)
H: U)

The calibration waveform or system transfer function is then divided out of the

subtracted target measurement response (R ‘( f) or eqn. (2.8) ), which results in the

theoretical target spectrum

20

' =R'U) (2.10)
11.0“) SO)

 

This set of steps is illustrated in Figure 2.9. A six (6) inch sphere is used as a
target in Figure 2.9, and a 14 inch sphere is used as the calibrator. The subtracted
spectrum of the 6 inch sphere is shown as the top waveform, after calibration with the
14 inch sphere the resulting calibrated return is shown in the middle waveform. It can
be seen that the frequency responses are much different. This is attributable to the system
transfer function which has been removed in the calibrated waveform. The lower
waveform is a "cleaned-up" version of the calibrated waveform. This clean-up process
was performed in the time domain, by time gating the causal noise in the chamber.

The transient response can be synthesized via an IFFT, and that waveform for the
subtracted 6 inch sphere data is shown in Figure 2.10 and corresponds the upper
spectrum in Figure 2.9. This transient result was obtained after applying a 1/8 cosine
taper to the spectral return, and is therefore the short-pulse response of an incident pulse
similar to Figure 2.6. This transient result can be compared to the calculated transient
result in Figure 2.11. In Figure 2.11 the measured 6 inch sphere response is compared
very favorably with the theoretical transient backscatter. There are noticeable differences
between the subtracted (Figure 2.10) and the calibrated (Figure 2.11) transient
backscatter from the 6 inch sphere. The shapes of these waveforms are different; this is
due to the non-ideal system transfer function still present in the subtracted waveform.
The differences in the spectral results (Figure 2.9) shows the actual effect of the system
transfer function. The time bases of the two measurements are also different; this is due
to the cables, which are eliminated in the calibrated response.

The transient result of the 6 inch sphere shows a large pulse which is a specular
reflection from the front of the sphere. Later in time there is another smaller pulse, this
pulse is due to the creeping wave and occurs at predicted time (21rr/c). Beyond that point

in time there should be negligible return, but the measured response contains noise (not

21

fully shown) after the first creeping-wave pulse, this noise corrupts the spectral result as
shown in Figure 2.8. This noise can be time gated with the a-priori knowledge of the
theoretical transient response, and then subsequently transformed (FFT) back to the
frequency domain. In Figure 2.12 the time-gated spectrum is compared very favorably
with the theoretical backscatter spectrum. The difference is attributable to imperfections
in the 6 inch sphere (constructed by hand) and measurement system errors.

The quality and resolution of the transient radar of .8-7.2 GHz bandwidth is
revealed in Figure 2.13. The transient response of a nose-on 1:72 scale B52 airplane is
shown, and the scattering centers are easily distinguishable. The calibration was
performed with the 14 inch sphere.

A quantitative discussion of the calibration procedure can be found in [37,38], but
the short summary and qualitative analysis above should suffice the needs for this
dissertation. A very similar procedure is used in calibrating the arch range measurement
system, but in that case time gating is essential to eliminate undesired scattering. When
performing a time-domain measurement in the arch range or the anechoic chamber it is
necessary to calibrate in the same fashion. This, however, can only be performed after
the initial transient measurements are transformed to the frequency domain. In the ground
plane transient measurements, a calibration is not required because there is no amplifier
transfer function, the antennas have known transfer functions and the cables are assumed

to be distortionless.

2.3 Conclusions.

The measurement systems and the calibration procedures, which will be used
extensively in the following chapters, have been described above. The important concepts
of synthesized transient response, bandwidth and weighting have been discussed and set

the stage for interpretation of the measured target responses in future chapters.

22

E(f)

Transmitted Field
H,( f) : >
am

Source:
CW Transmitting

or Antenna
~-

H(f)

n e 1111
Received Field

S N(f)

 

 

Figure 2.8 Block diagram model of measurement system.

23

Spectrum of BackScatter from 6 inch Sphere
A various points in the calibration procedure

 

 

 

Subtracted
Calibrated
Calibrated
and
Time Gated
l[TlllllllIIIIIIIIITIIIIIIIIIIII
0.80 2.80 4.80 6.80
Freq (GHz)

Figure 2.9 Graphical representation of the calibration procedure.

24

Transient Scattering from a 6 inch Sphere
(Synthesized from .8—7.2 GHz Return)
Background Subtracted Only (not calibrated)
2E+OO7

1E+007

5E+006

0E+OOO

 

“MW

 

-5E+006

Relative Amplitude

—1E+OO7

 

 

-2E+OO7

1111111111111“11111111111111linnunluuuiulnuuiuluutuuj

 

 

-2E+OO7 IrIllIIII]IUUUUli'l’ll‘ll'l'lI'IUIIIITI’j]

67.00 69.00 71.00 73.00 75.00
Time (ns)

Figure 2. 10 Transient backscattered field from 6 inch sphere (background subtracted).

Transient Scattering from a 6 inch Sphere
(Synthesized from .8—7.2 GHz Return)

   

BE+OO7

 

Measured
----- Theoretical

3E+OO7

Relative Amplitude

-2E+007

 

 

[11111111llllLllLJllllllllllllllllkl

 

 

-7E+007 IITTTIU'IIYIUITUUIIIITIliITTrTrTIIIFIYI]

10.00 11.00 12.00 13.00 14.00
Time (ns)

Figure 2. 11 Transient backscattered field from a 6 inch sphere.

25

Spectrum of BackScatter from 6 inch Sphere
Measured Data has been Time Gated.

 

 

0.02 '2
3 —— Measured
: I" """ Theory
1

.1 i ,
L 0.01 -1 1' ’
q) .1 ' l I
4.) _, 1 I \ I
+1 d ' 1 I 1 , \ ,a
O -4 I. I I ‘ I \ I \\ ‘ a-\
0 J ' I \ I \/ "’ \
(f) _ ’1 II 1‘ , z ‘\
x _ ' I I \
0 - , 1
U - 1 1
CD 0 01 - I ‘
a) .
U :
3
:1: I
C .1
CW -
C _
2 0.00 :

: I

-l

0.00 lfill'llIIIITIIIITIIIIIIIIIlllIl—I
0.80 2.80 4.80 6.80
Freq (GHz)

Figure 2.12 Spectrum of backscattered field from a 6 inch sphere.

26

Transient Scattering from a 852 (1:72 Scale)
(Synthesized from .8—7.2 GHz Return)
Nose On

3E+OO7

2E+OO7

nose l
1E+007 “

0E+000 MM

-1 E+OO7
1 l

—ZE+007 engine pods
w1ngs tail

 

 

 

 

 

 

 

 

 

 

 

 

Relative Amplitude

 

 

—3E+OO7

-4E+Oo7lrllIll]IIIIIIIIIIIIIIIIII'lllll'lll
9.00 1 1.00 13.00 15.00
Time (ns)

 

Figure 2.13 Transient backscattered field from a 1:72 scale B52 airplane.

27

Chapter 3

Transient Scattering from a Conducting Sinusoidal Surface

3.1 Introduction

In this chapter an in depth treatment of transient scattering from a perfectly
conducting (PEC) sinusoidal surface will be considered. The PEC sinusoidal surface is
a first approximation to an ocean surface, and is periodic. This is an excellent surface
to begin the thesis, it introduces many important concepts and phenomena that will also
be present in the more complex surface configurations. Floquet mode propagation and
cut-off and multiple scattering are two phenomena that will be introduced and discussed
in conjunction with the sinusoidal interface.

In addition to theoretical analyses, experimental results will be presented. These
experimental results will draw from the ideas of chapter 2, and will verify the developing
scattering theory.

The scattering of continuous wave (CW) radiation from a perfectly conducting
(PEC) sinusoidal surface has been analyzed by many authors [12,15,25-29], but few have
produced results on transient scattering [41,42,43]. This is due to the availability of CW
and doppler radar systems for which the clutter analysis therefore tended to become
statistical (averaging over time) in nature. The UWB/ SP radar allows for snap-shot
modelling of the clutter, due to the short pulse interrogation of the sea-surface. It is this
property that allows for static sea models to be used in analysis and in experimental
measurements.

Both finite and infinite PEC sinusoidal surfaces will be considered. The
approaches will include two different polarization states (TE and TM) and will use an

Integral Equation (IE) type formulation in conjunction with the Method of Moments.

28

The infinite surface can also be analyzed more classically using the Rayleigh hypothesis,
and results from this method will be included as a confirmation of the IE method. Both
methods yield frequency-domain solutions, therefore the transient results will be
synthesized using the inverse Fourier transform (IFFT).

Due to the periodicity of the infinite surface the analysis techniques for this
surface are centered around Floquet’s theorem. The Rayleigh hypothesis method directly
uses the theorem to expand the scattered fields, in the IE method the periodicity is
embedded in the periodic Green’s function. The infinite spatial extent of the surface can
therefore be reduced to just one period. This truncation solves one problem, the infinite
domain of integration is modified to a domain of one single period. The periodic Green’s
function, however, creates another problem in its convergence rate [44]. The
convergence problem of the PGF is accentuated for shallow (low height) sinusoids, this
is another reason for the inclusion of the Rayleigh-hypothesis method.

Research directed towards the finite surface has not been as active as the infinite
surface; this is due in part to the strictly numerical nature of the problem, and the lack
of apparent applications (sea surfaces, gratings, etc. can usually be considered infinite
for most purposes). There are few workers examining this and similar problems [45-47].

Insights are gained regarding the nature of scattering from a PEC periodic
surface, which can be extended to the other surfaces that will be considered in the
following chapters. For both TE and TM polarizations, the forward and backscattered
time and spectral domain fields are calculated. For the backscatter case, which is of
primary interest for many applications (sea clutter, etc) the results exhibit a periodic
retum of pulses dependent upon the period of the surface. This result was expected, and

illustrates the feature extraction capabilities of an UWB/ SP radar system.

3.2 TE Scattering from Conducting Sinusoidal Surface
Scattering of a plane wave with transversely polarized electric field [Ey(x,z)], as

indicated in Figure 3.1, is analyzed in the spectral domain. To simplify the problem

29

(numerically) the surface will be invariant along the y—direction. This approximation will
not be lifted throughout the thesis, and is an area of future interest. As stressed earlier
the transient response is to be obtained; this can be accomplished by utilizing the IFFT

on spectral results of appropriate bandwidth.

3.2.1 Infinite PEC Surface
The first scattering problem that will be analyzed is the case of a sinusoidal
surface of infinite extent. Two techniques are utilized in solving for the scattered fields

due to plane wave excitation.

Method 1) Floquet Mode Matching\Rayleigh-Hypothesis Analysis

The classical method known as the Rayleigh hypothesis can be applied to this
problem. Combining the Rayleigh hypothesis with Floquet’s theorem permits a solution
via a mode-matching technique. The frequency-domain Floquet analysis is found in
graduate level electromagnetics texts [12].

It is assumed that the ocean surface described by z = p(x) = -h cos(21tx/L) is
cosinusoidally periodic in the x coordinate with height h and period L (See Figure 3.1).
The y-invariance of the surface and orientation of the TB incident wave result in only one
non-vanishing component of the electric field [Ey(x,z)], this can be seen from Maxwell’s
equations. Therefore Ey(x,z) is the generating wave-function for the TB case, and the
non-vanishing magnetic field components can be found directly from Ey(x,z). The
generating wave function [Ey(x,z)] will be denoted by 1|:(x,z) for bookkeeping purposes.

The wave function (electric field) must satisfy Helmholtz’s equation in the air

region above the PEC sinusoid, where there are no sources

07248) 110.2) = o (3-1)

subject to 1|: =0 on the conducting surface S.

The incident plane wave can be written as

30

$1092) = Ace—1'13; = Ace-10mm) = A03 -j(px-qz) (3.2)

where B = kx = ksin6i, q = —k2 = kcosOi and k = didactic.
Given a periodic structure with period L along the x—direction, Floquet’s theorem

states that

w(x) = f: Ane-jfl't 51‘1“”? (3.3)

 

where 1|:(x) is a wavefunction and Ane '1' p "t is referred to as the nlh space or Hartree
harmonic. A proof of this theorem is available in [12].

The scattered wavefunctions are then written using Floquet’s theorem as

¢s(x,z) = i e-jp'xfna) (3.4)

n=—w

where B = ksin6; is necessary to match the phase progression of the scattered wave to

that of the incident wave. Substituting (3.4) into (3.1) results in

e113} 2 O (3.5)

 

 

.. (92];
2 I622 +(-Bi+k21f,.

Note also that

L
fem“? 'jfl’dx = L0” (3-6)
0

where 6“,“ is the Kroneker delta function. Therefore,

.32);

 

+ qun = 0 (3-7)

where q: = k2 - 8,2,. This differential equation has the usual solutions

fill) = Bne 'jqnz + Cnejquz (3.8)

31

_
I 1

 

Then, according to the Rayleigh Hypothesis, the scattered field consists of space
harmonics which travel only outward in the +2 direction, hence Cn=0. This

"approximation" is valid provided that

(it!) = 21’! < 0.448 (3.9)

There has been a considerable amount of controversy surrounding the legitimacy
of the Rayleigh hypothesis. Lord Rayleigh originally stated (without proof) the theorem
in conjunction with the scattering of sound waves from a corrugated surface. It was not
until fairly recently [34] that Rayleigh was questioned, with the argument that inside the
troughs there must exist a downgoing wave. The existence of a downgoing wave does
seem intuitively needed, and it was proven that the Rayleigh Hypothesis did indeed fail
when the surface slopes exceeded 0.448. However, the legitimacy of the Rayleigh
hypothesis still remained in question for surface slopes less than 0.448. Then, through
a number of analytical methods [31-33] the Rayleigh Hypothesis was finally proven to
be rigorous for slopes less than 0.448. It is, however, impossible to prove
experimentally, due to the fact that an ideal plane wave is required and a perfectly
periodic and smooth infinite surface is needed. But, the technique still remains to be quite
powerful, and numerically useful.

The scattered wavefunctions, using the Rayleigh hypothesis are therefore

1,0,2) = z; Bug—115,381” (3.10)

n=-~

which is periodic in x as expected.

Physically the scattered field is made up of an infinite summation of plane waves.
These plane waves are either propagating or evanescent in nature. The propagating
modes are characterized by a real valued q” or k > | [in | . The evanescent modes occur
when q,I is imaginary or k < | 13» | . Note that the branch out on q" is chosen so that the

scattered wave decays exponentially in the +2 direction. There are only a finite number

32

of propagating modes and that number depends on the sea-surface period (L), the
excitation frequency and incidence angle. The scattered modes can be thought of as plane
waves scattered at various angles (See Figure 3.2). The angles are determined by 13,. and
q" , which depend on the Floquet mode number (n). The n=0 mode is the specular
reflection (63 = -0‘. ). The backscattered field therefore consists of all the Floquet modes
that are scattered in the -x direction, which would be the n=-2 and n=-3 modes in
Figure 3.2. These topics and their implications will be discussed further in section 3.4.

In order to determine the unknown coefficients of all the scattered Floquet modes
(Bu) the boundary conditions must be used. Enforcing the Dirichlet boundary condition
(‘1'.- + in: = 0) at the surface (2 = p) results in

Aoeiqvev‘flx + 2 3”,, “I'm ’14»? = 0 (3.11)

One then divides out the e45" dependence and applies Galerkin’s method using a testing

operator of the form

f‘ 1...} eI'zW/L dx (3.12)
0
to yield
L . j21tmi °' L . j21t(m-n)i
A0 e’qpe de + 2 B" fe ’JQJ’e de = 0 (3.13)
o n=~~ o
This can be rewritten as
2 ENE" = A0,?" og|m|<oo (3.14)
where
_ L flnm-‘E
Am : —fejqpe de (3'15)
0

which can be shown to be equivalent to

33

Km = -Lj'"'I(-1)'"'|J|ml(qh) (3.16)
and

L
f e ""~ ’2’“ ”1.1111340

0

which can be seen to be equivalent to

Em = J- Im-nl Jlmlmnh) (3.13)

Both of the above integrations ((3.15) and (3.17)) where performed in closed form
because of the specialization of the surface (p) to a sinusoid. These integrals could be
numerically evaluated for any surface shape, provided the surface slope remains within
the Rayleigh hypothesis regime.

These simplifications result in

2 KmBn = ADAM OSIMI<°° (3.19)

with
A... = -i"""J|,,.,(qh) (3.20)
Km: = jlm—nl Jlm-n|(qnh) (3°21)

This infinitely large matrix equation can be truncated (-N to N) to obtain an approximate
solution for the spatial harmonic amplitudes. According to [12], if the height is small
compared with a wavelength and the period L is much greater than a wavelength, the
matrix can be truncated without too much error.

Subsequent to the determination of the spatial harmonic amplitudes the scattered

field in the air region can be found using (3. 10).

34

 

Incident Wave A

 

. . z
E ‘for TB case Fieidzg’omt .
HiforTMcase
/
111* I / 0i S or C
1 - P/p z=p=-hcos(211:x/L)
. m h

 

\ 1......- w V V
SorC |._L——-I K

Figure 3.1 Infinite, conducting sinusoidal surface scattering geometry.

Scattered Waves

Incident Bi
n=0 (-0i )

Figure 3.2 Graphical representation of Floquet-mode scattered waves.

 

35

Method 11) EFIE-MOM Analysis

A rigorous and more general treatment for the scattering of plane waves from a
PEC periodic surface has been developed. The technique sets no restrictions on the
surface slope and in fact the surface need not be sinusoidal. An integral-operator-based
analysis has been employed for this analysis, and will be referred to as EFIE-MOM
method. The currents induced on the PEC sea surface are calculated as solutions to an
EFIE with a periodic Green’s-function (PGF) kernel. With the surface currents known,
it is then possible to determine the scattered fields once again making use of the PGF.

The electric field Ey(x,z) associated with the two-dimensional (2-D) TE scattering

problem satisfies the 2-D Helmholtz equation

 

62E 62E
Y + Y +k2Ey ___ jpry (3.22)
6x2 622

where k = (in/[E = 211/A is the wavenumber in the space region above the ocean surface
and Jy(x,z) is the source current density. The incident wave is defined by (3.2).

As a consequence of the infinite periodicity of the ocean surface, the induced
current must be similarly periodic, except for a progressive phase shift of BL radians per
period to match that of the incident wave, such that Jy(x + mL,z) = exp( -jm BL) Jy(x,z) withm
integer valued. The induced currents maintain a scattered field having the similar
periodicity Ey(x +mL,z) = exp( -jm[iL) Ey(x,z).

The above nature of the induced current prompts the definition of a periodic
Green’s function to accommodate the periodicity of the scattered field. If a 2-D current
point source 6(x -x’) 6(z -z’) is located at (x’,z’) in the first period of the ocean surface,
then the periodic Green’s function is the field maintained by that point source as well as
its infinite translation exp( -ijL) 0[x -(x’ +mL)] 6(2 -z’) into the remaining ocean

periods for -oo<m<oo. The periodic Green’s function G(x,z |x’,z’) is defined by

36

520 .ifi +180 = — i e-J‘WLMx-(x’anMz—z’) (3-23)
6x2 522 m=-°°

 

and has periodicity property G(x +mL,z |x’,z’) = exp( —ijL) G(x,z Ix’,z’).
It is demonstrated in Appendix A that definition (3.23) leads to

e-ill.(x -x’) e -iq,.|z -2’l

G(lexazd = 31'2“: q (3.24)

 

where [in = B +2mt/L and q: = k2 - [3,2, so 4" = -j [3,2, -k2 where the branch out is chosen
such that Im{qn} <0. The periodic Green’s function innately accommodates the
periodicity of scattered field E;(x,z) , which is obtained by integrating that Green’s

function into the current which resides only on the first period of the surface to yield

E;(x.z) = -J'w11fs J,(x’.z’)G(x.z1x’.z’)dx’dz’ (3.25)
P

where S I, designates the cross section of the first ocean period.
The boundary condition at any point on the conducting ocean surface requires that
Ey(x,z) = E;(x,z) +E;(x,z) = 0 there. Using (3.2) for E; and the specialization of (3.25)

for E; maintained by the surface current Ky induced on the ocean surface leads to

f K(x’,z’)G(x,z|x’,z’)dl’ = .—A—e""”‘e""z V (x,z)eCP (3-26)
Cr y 1091‘

where C p designates the contour of the ocean surface in its first period. Exploiting

representation (3 .24) for the periodic Green’s function leads to

fc K,(x’,z’)5(x,zlx/,z’)dll = %e-ipxejm V (xz)€C, (3.27)
P

where n =1/p/e is the intrinsic impedance, and kernel G(x,z|x’,z’) is defined by

_ ~ e-ifl.(x-x’)e-Jq.lz-z’l
G(x.z|X’.z’) = Z

"‘3'” q"

(3.28)

 

Equation (3.27) is an EFIE for the unknown current Ky.

37

A MoM numerical solution is implemented by expanding the unknown currentKy

in basis set (qu as
N
K,(x,z) = {1 aqfthh) (3.29)

and point matching (3.27) at N points (ivy-p) on the first ocean-surface period CF. This
leads to the matrix equation

N
ZAmaq = Fp p=1,2,~-,N (3.30)
q=l

where the matrix elements are given by

Am = [Crfq(x’,z’)G-(§p,2p|x’,z’)dl’ (3.31)

and the forcing vector is

F : 2E e-jBEPejqu (3.32)

P

A pulse-function basis set was used to implement the numerical solution. The
first period 0 sx s L of the ocean surface is partitioned into N segments, each of length
Ax = L/N, between end points x" = (q - 1) Ax for q = l,2,---,N+1. Corresponding points
on the surface are located by zq = p (xq). The pulse functions are then defined by

l for xq<1c<xq+1

fq(x,z) = { (3.33)

O otherwise

For the purpose of evaluating the integrals in matrix elements (3.31), the ocean
surface is modelled as piecewise linear between partition end points xq with slope
m q = (xq+l-xq)/Ax. An element of displacement along the ocean surface is dl = S(x)dx
with 50:) = W such that within the q’th partition dl’ = sq dx’ with sq = 1 +mj.
The MoM partitioning scheme can be visualized in Figure 3.3. The matching points are

located by 5‘, = xp + Ax/2 , 2;, = zp +mpr/2. Matrix elements (3.31) are calculated from

38

x +Ax - _ _
qu = Sq L: G(xp,zp|x’,z’)dx’
,, -jp1;p (3.34)

e fxq+Ax ejpjft/e -jqnlzp-zq-mq(xl-xq)| dx/

"3'” q" xq

The integrals in (3.34) can be evaluated in closed form, leading to:
CASE 1: Epiifq, M = Bn+qnmq for Ep>z-q , an = [Sn-quit:q for Zp<§q
e -jBn(§p-§¢) e 'quGP '23) 2sin(an AXIZ)

4,. XM

(3.35)

 

Am = 5(1 X”

 

— " 1- 14,12,102 A 2
Mum-(4.582“ 24M e mm" x, )] (3.36)

+ jZBne “”1“” Sinai" Ax/2)}
Subsequent to solution of the MOM matrix equation (3.30) for a q , the scattered

field at points above the ocean surface can be obtained from expression (3.25) as

 

f: aqS 2”: e-J'B.(x-f¢)e whiz-5,) sin[(pn+qnmq) Ax/2] (3.37)

E,’(X.z) = -£"- q
L q=1 "3"” q" pn+qnmq

The backscattered field is obtained from (3.37) by restricting the sum over 11 to include

only those Floquet modes which are propagating (qn real) in the -x direction (13,50).

 

 

 

 

 

 

 

Z4\ (iavia) '
slopenm2
X
. z >
x1 x2 X3 x‘

Figure 3.3 Partitioning scheme for MoM numerical solution.

39

3.2.2 Preliminary results of TE infinite case.

The preliminary results are concerned with the currents which are induced on the
PEC sinusoid due to the TB plane wave excitation. The Floquet mode-matching technique
relies on matching the modal amplitudes of the scattered field modes (3.10) at the
sinusoidal boundary. There are an infinite number of these modes, but the numerical
solution requires truncation of the modes to a finite number of terms. From the
discussion on propagating and evanescent modes it might seem that the number of modes
could be truncated to contain only the propagating modes, which describe the far-field
scattering. This is incorrect because the evanescent modes do contribute to the surface
currents. If the surface currents are incorrectly obtained then the modal amplitudes of the
propagating modes must also be wrong. This idea can be seen in Figure 3.4, where the
number of terms (N) is varied. This figure shows that even if all the propagating modes
are considered (N=2 case), there are changes in the modal amplitudes as more modes
are included (N =6 and N =10) in the solution. For the surface characteristics that are
present in this thesis it was found that a N of 15 to 20 is sufficient.

As a preliminary check of the two methods developed above the surface currents
on a PEC sinusoid (L=O.1016 and h/L=0.0625) are compared for both methods. The
sinusoids were illuminated with a 6 GHz TE plane wave at an incidence angle of 85°.
The results in Figure 3 .5 show a very good agreement between the mode-matching and
integral equation techniques. This lends credibility to the methods and the numerical
implementations. There is also a noticeable shadowing effect in the trough area behind
the illuminated crest.

It will be assumed from this point that the FMM-RH method is used when the
sinusoidal surface is within the Rayleigh-hypothesis regime, unless specifically noted.
The IE method will be used for all other cases.

A further examination of the surface currents is provided. In Figure 3 .6 the

currents are compared for a number of incidence angles. The normally incident case

40

db

111

in.-

1h:

30

ex;

[01

CV;

displays a symmetric current distribution as expected. As the incidence angle is increased
the maximum current moves towards the illuminated region and the shadow zone
becomes apparent. The magnitudes of the current decrease as the incidence angle is
increased. This was an unexpected result. The expectation was that the magnitudes would
remain constant for all the incidences, but would shift maximal locations. The results,
however, indicate that the current tends to redistribute itself and since the higher
incidence angles intercept less surface area there is less total current (energy) induced.

The dependence of induced current on sinusoid height (h) is observed in
Figure 3.7. As the height of the sinusoid is increased the induced current is seen to grow
at the crest of the sinusoid. The shadow region also becomes more defined at the height
is increased. These observations were expected and increase the confidence in the
numerical solution.

In Figure 3.8 the period length (L) of the surface is varied with the resulting
induced currents being compared. The results are quite intuitive. As the period length is
increased the slope of the sinusoid is decreased, which results in a smaller maximum
magnitude. The shadow region becomes less distinct as L is increased, which is
expected.

The last figure in this section (Figure 3.9) is quite important. This figure shows
the dependence of the induced surface current on excitation frequency. Since the eventual
scattered fields will require this large bandwidth of returns it is interesting to observe the
effect on the surface currents. As the excitation frequency is increased the magnitude of
the current maximum increases and the shadow region becomes more defined. This is
expected because in the physical-optics limit (high frequency) the current would vanish
in the shadow region. The frequencies around 1.5 GHz and 2.95 GHz show a slight
roughness in the current. This phenomenon is due to a F loquet mode moving from

evanescent to propagating (or vice-versa) and will be discussed in section 3.4.

41

The preliminary results lend faith to the theoretical developments, through the

agreement with physics and intuition and the convergence of the two analytical methods.

Modal amplitudes for TE excitation
L=.1m, h=.007m, Freq=6 GHz, 0:=85°
FMM—RH Solution ~

 

 

0.301
1 1
A 1 1
, 1
q 11“
- ----- N=2 ‘1‘
--——N=6 1
‘e—e—e—e—eNz1O \‘
- 1
~ 1
03C ‘ ‘
1
0.20fi 1
a) - I ‘
‘0 "‘ I, ‘\
j -1 I ‘
it: - I 1
_ , 1
Q. d I \
E ~ 1 1
O - 1
- 1
6 1
‘ 1
'0 ._ 1
00°10 ‘
E " \
'1
0.001111111tlltllrlllltlrtlllllllIIttrrllltlllllltlltl

-5 -4 -3 -2 —1 0 1 2 3 4 5
Floquet Mode number (n)

Figure 3.4 Modal amplitudes for a mode-matching solution using differing numbers
of terms.

42

Induced Surface Current (K, for TE , Kc for TM)
Freq = 4 GHz, L = .1016 m

h/L = .0625 01 = 85°

Infinite Sinusoidal Surface

 

 

 

 

1'20 ? nun Current (K) for TM Excitation (MFIE—PGF)
: Current K, for TM Excitation Floquet Mode-Matching)
I; 1
0 0.80 1
.t‘ —
o _
E i
L ..
0 -
C '-1
v 10 ,1
+1 0.40 4 , . ‘10-’64
S - 00000 Current (K 3 for TE Ex01tat1on (EFIE—PGF
t : ---- Current 16, for TE Excitation Floquet ode-Matching)
3 —I
o - " _ \
_ , \
_ /
8 " / \ \
O - / \
‘t -0.00 - / \
a I / \
: ,// \\
j ‘ — - Surface Profile
—Oo40 -1rTrrlIllIIIIIIIITTIIIIIIIIIIIIIIIIIUFTIIUITITT111
0.00 0.20 0.40 0.60 0.80 1.00

location (x/L)

Figure 3.5 Induced surface current on one period of infinite, conducting sinusoidal
surface for TB and TM excitation at 4 GHz.

43

Induced Surface Current for TE excitation
L=O.1016m, h=0.005m, Freq.=2.0 GHz

 

   

 

65—0035
*5 i
8 4E-003"
L _.
3 ..
0 :
.0 -
a) -
O -
3 :
3 312—003:
a) I
“o -
.3 I
'E _‘
01 ..
o .1
2 5E-004':

E H-O-H Surface Profile (Amplitude/4)
-2E-003:-rn'rrrrrrrnTrrn-n1-rrrrrrrmfi'rrrrrrrn'n'n'm'm
0.00 0.20 0.40 0.60 0.80 1.00

x/L location

Figure 3.6 Dependence of currents induced on a PEC sinusoid for TE excitation for
various incidence angles.

 

 
 
  

 

 

 

 

Induced surface current for TE excitation at ZGHZ
L=.1016m, Incidence Angle=85°
1.0
8
31
1D
. 0
8’8‘
2 o'
q:
Figure 3.7

Dependence of currents induced on a PEC sinusoid for TB excitation for
various surface heights.

45

Induced Surface current for TE excitation at ZGHz
h=.005m, 0;=85°

 

 

6.0E-004
5.5E-004
5.0E-004
(D
'0 I
:3 ../I
+4 ’l
'E 4.5E-004 ("I
O’ I
U3 I
o 1, , 1‘
2 ;,x — L=.1016m 1.
,1 ----- L=.2032m 1 _, ."
4.0E-004 e—e—e—e—e L=,3048m \\ - .- . -. ‘ i ’1’
M L=.4064m ‘~---/
3.5E-004
300E—004 IIlIlrlIll]Il[Ir]IIIIIIIIIIIIIIITITIITIIIIIIllllll'
0.00 0.20 0.40 0.60 0.80 1.00

x/L location

Figure 3.8 Dependence of currents induced on a PEC sinusoid for TB excitation for
various surface period lengths (L).

46

 

Induced surface current for TE excitation
L=.1016m, h=.005m, Incidence Angle=85°

 

(.0
‘ Q 9.: ’5';.' ”’I "
50. . e. :.:‘:::3'5,:::';::2zz§;2!£!4’11::33.
g 0 ::’:::::::'£:‘:::’:::’e”’t’f" -:::::é:.=:: ’03.
“‘o:::0‘..: ..'.:-.:.:°" ‘ 7:. ::““‘:\::§§:::R:\‘$‘
. ‘ O. Q. o , ,- ‘ ““ \“ ‘ ‘\\ \\\\‘ \\
o. $‘.¢‘.¢’.’~;¢’¢fi‘ ““ ‘\\ \\\\ “ \\\\\\\\\\\\ \
O. O O O I I 1"“ “‘ “ ““ \\\\
, . .,.«:.‘o°:‘::o:‘::a2:2a2"Q3“‘8{\\\\“8§3\\“3§ 3“ ‘33:;
' 7“:o‘:'::’.,:;”2’~’" I ‘\ “‘ “\ \“ “\\\“ ‘\\\\“‘:“\\\‘.‘“
0. ' crave” \\\‘ \\\“ \ \“‘ \\\‘“ \\“‘\\
' - -9: \“ \\“ \\“ \\\\‘ \\\“‘ \\\\\\\
¢ 4.6,, “‘\\\ “\\\ ‘\\\ ‘\\\\ ‘ \\\\ “\\\
“‘ ““‘ “\“ “‘\‘ ‘\\ ‘ ‘\\‘ \\
\\“:\\\\\‘ ‘\\\ \\\\ “s“ “\\\.«‘;:\
“‘ ‘\\‘ “\\ “\‘ “\““\“ 3“
a “‘ ‘\\“ “\‘ ‘\\ “\‘ “\‘ “‘ “““‘.‘r‘.
... a~ “\\ “\\ “‘\\ “\\ “\\ “s\“““‘:
:= \\\“ \\\‘ ‘\\ ‘ \“ a“ 0“ “¢“ \
““ ‘\“ ““ “‘ “ “ ‘Q '1.
at.) Q, \\ “\\ “\\ “‘0 “v‘ ‘
‘11::«1::‘::8::8‘ ‘ x
O ‘1.
9 \:“::‘ ‘:““ Q, ”9.65“
\ ‘ Q
Q0 Q '

 

 

 

Figure 3.9 Dependence of currents induced on a PEC sinusoid for various
excitation frequencies.

47

3.2.3 TE Scattering from a finite conducting surface.

This section is devoted to briefly describe the techniques used to find the scattered
fields from a finite-sized, two-dimensional PEC surface. There are no restrictions to the
surface curvature (i.e. no slope limitations); the physical size of the scatterer is limited
only by numerical consequences (memory, and cpu time). This is due to the fact that the

entire surface must be partitioned not just one period as in the PGF case.

 

 

incident wave A
Ei for TB case Z
Hi for TM case
0 /
6i
V r1
9W X
/\/\//\ A >
V V V‘

 

 

 

Figure 3.10 Finite, conducting sinusoidal surface scattering geometry.

The general approach is very similar to the infinite—surface case, in that an IE for
the induced surface current must be solved prior to the typical determination of the
scattered field using the 2-D free space Green’s function.

The configuration and the nomenclature is the same as the infinite case except that
the surface is truncated (See Figure 3.10). Since the E-field is polarized transversely, the
induced surface current is also only transversely directed as I?(x,z) = y‘Ky(x,z). This

current then can be used to find the scattered E-field,

48

 

ampo -o —.
E,(x.z) = -y 4 fr Ky(x’,z’)H§2’(k|p-p’|)dz’ (3.38)

where p’ is the position vector in the x-z plane, I‘ describes the contour of the surface,
and H?) is the second kind Hankel function of order zero.

To obtain an integral equation for the currents, the boundary condition on the
electric field is applied at the conductor surface. Substituting the scattered field into
boundary condition yields the integral equation for the unknown surface current

distribution Ky(x,z) as

/ (2) ..__./ I: i -'p:: -' .
eryOc ,z’) H, (klp p |)dl knAoe , e m (3 39)

A MoM numerical solution is then implemented to solve the above integral
equation. The current is expanded in a set of pulse basis functions and point matching
is implemented. This results in a square matrix equation for the unknown current
coefficients. In typical fashion the scattered field (3.38) can be determined numerically
subsequent to evaluating the induced currents. For a far-zone scattered field, the
asymptotic form of the Hankel function can be used to simplify the numerical integration
[48]. Results comparing the infinite surface scattered fields with the finite surface
response are provided, and in addition direct comparisons with experimental results can

be made, because the finite surface is physically realizable.

3.2.4 Preliminary results of TE finite case.

The induced current for a finite PEC sinusoid is compared to the current induced
on an infinite sinusoid. In Figure 3.11 the currents induced on a 5 period sinusoid, a 3
period sinusoid an the current for one period of an infinite sinusoid are seen to agree
quite well. The effects of the edge of the finite surface are also evident (note that the
incident wave is coming from the left). This result helps to establish the idea that the
response from a periodic finite surface would match that of an infinite surface provided

there is a sufficient number of periods for the finite model. This idea is used extensively

49

for the scattered field responses, which are obtained experimentally from a finite surface

and are compared favorably to the responses from an infinite surface.

Induced Surface Current Ky
=.1016 m h/L = .125
Frequency = 4 GHz O.=—67°

 

 

  
 

0.02 -
- - - - - Finite surface 5 periods long;
- Finite surface .3 pertods long
_ 00m Infinite surface one period shown
-1 I
_ l
l
0.01 — i
7 1
>~ " I
x s 1
a: . I
.0 -4 I
3 1
.4: ‘ '
5. - '
o 'I i
E '1 t
0.01 -1 I
1 1'
q \I
0.00 IIIIIIllr]llllIlll[[UIIIIFITTITIIIIIIUIIIIIIIIIIIIIIIIIIF I

 

 

—0.:50 —0.20 —0.10 0.00 0.10 0.20 0.30
x—Iocation (m)

Figure 3.11 Comparison of induced surface currents on finite and infinite surfaces for
TB excitation.

50

3.3 TM polarization
T 0 complete the theoretical analysis, the other orthogonal polarization state is

required. The scattering of a TM incident plane from a PEC sinusoidal is considered.

3.3.1 Infinite Surface

The surface configuration for this polarization state is the same as previously (see
Figure 3.1) except the H-field is now directed transversely as 17(x,z) = yHy(x,z) with the
E-field parallel to the plane of incidence. Therefore by using Hy(x,z) as the generating
function in the same fashion as Ey(x,z) was used for the TB polarization, similar analysis
techniques may be employed. The electric field components can be found simply utilizing
Maxwell’s equations for this two-dimensional problem. The infinite PEC surface will be
considered first, and once again it is possible to utilize the Rayleigh hypothesis for an
adequately shallow sinusoidal surface, while a magnetic field integral equation (MFIE)
will be developed for the infinite case.

Method 1) Floquet Mode Matching\Rayleigh Hypothesis Analysis

Letting w(x,z) represent the generating wave function (Hy(x,z) ), an identical set
of steps can be followed as in the TB development ((3.1) to (3.10)). That is, the total
magnetic field is subjected to the 2-D Helmholtz equation and the Rayleigh hypothesis
is invoked.

The total field is decomposed into the superposition of incident and scattered

components as

1|:(x,z) = 11:,(x,z) + ¢s(x,z) = incident wave + scattered wave (3-40)

where the incident plane wave takes the form

¢,(x,z) = Ace-if? = «(firm = Ace-ioxeiqz (3.41)

and the scattered magnetic field (via Rayleigh hypothesis)

51

i: Bne _jwej” (3.42)

tII,(x.z) =

with identical definitions of [3, q, [3", q" and k as the TB case
The difference between the TB and TM polarization is manifested in the boundary
condition (Neumann). The tangential electric field must vanish at the conductor and this
condition is satisfied if the normal derivative of the magnetic field vamshes at the
conducting surface therefore, the boundary condition at the ocean surface requires
at the sea surface z=p . (3-43)

%—"’= n-+V(1:. 1:.)=o

The normal vector ti to the ocean surface can be constructed in the followmg manner

Cons1der the 2-D scalar field <I>(x,z) = z - p(x), then 0 = 0 defines the surface 2 p(x)

The required surface normal is consequently

 

z‘ - £98
. V4) 0
n = '76- = (3.44)
6p 2
1 + —
I I .)
and satisfaction of boundary condition (3.43) consequently requires
_ipa . = (3 45)
6x6 T11” 4"” 5+) ;(‘l’i 1453)
Combining (3.41) and (3.42) leads to
tlii + Ills = Aoe‘jlixej‘lz + Z Bne-jp’e (3-46)
na-co
so condition (3.45) requires
-fl. —jBAoe 'jpxejqz + 2 _jpane jpnxe .quZ)
61: n=-~ (3.47)
0

+ (quoe -jflxejqz +2 _janne-iwe 1'9"!) =

52

Since [in = [3 + 2mr/L , then (3 .47) carries a common factor of exp( -j[3x) , and implement-

ing the boundary condition at z = p = -h cos(21tx/L) provides

- -‘ 6 -
2 B j2mu/L 14.9 = _ A P + m (3.48)
M q "8) e 0(B ax q)e

The factor exp( -jqnp) in (3.48) destroys the orthogonality of the exp( -j2mtx/L).
Galerkin’s method is therefore applied with the testing operator (3. 12) leading to

2B no‘f (”nix _ _q) e j2n(m-n)x/L e 721.9 dx

ns-c

(3.49)
L 69 '21unx/L '
- -A — + e’ e’qux
ofo (‘3 6x q)

This result is an infinite matii'x equation for the unknown Bu , and can be written as

X 11,"an = A01) v Os|m|<oo (3.50)

n--~

where the various coefficients are defined by

_J' Leif [1% _qnieflflm-an-e €44.de (3.51)

eflmx/L 6149 (it (3.52)

 

L 8p
—+
Bax q

     

The integrals in the definitions of coefficients (3.51) and (3.52) can be evaluated in

closed form to obtain

«1113...- ,1. m 7:
HM" = _[]| 1|J|m_ml'(qnh) jl l"1|m-n- 1|(qnh)] (3.53)
-jqnj|m-n|J|m-n|(qnh)

-jqj"""J.,,,,(qh)

53

The infinite matrix equation (3.50) can be truncated to obtain an approximate solution
for the complex amplitudes of the spatial harmonics. Convergence of the harmonic series
must be tested numerically, and depends upon the relative height and period of the ocean

surface normalized to the wavelength of the incident radiation.

Method II) MFIE-MOM Analysis

The scattering of TM polarized plane waves by a perfectly conducting, periodic
surface can also be analyzed using an integral—operator—based method. The governing
integral equation is in terms of the unknown H—field at the PEC surface, and the kernel

consists of the normal derivative of the periodic Green’s function

 

Fix—’9 -PVf H(x’,z’)aG(x’z|x/’zl)dl’ = H,(x,z) ...v {x,z} e C (3.55)
2 Cr an’ p

where the PGF G(x,z|x’,z’) is given by (3.24), and Hi(x,z) is the incident plane wave.
The notation PV indicates that the integral must be evaluated in the principal-value
sense. The integration path CF is one period of the surface, because the PGF accounts
for the inherent periodicity of the solution. Refer to Appendix C for the details on
obtaining the above integral equation.

Subsequent to solution of MFIE (3.55) for surface field 1|: , the far-zone scattered
field can be calculated as

$0.2) = 1333 c. 1150 33-5—5 dz’ . (3-56)

A MoM numerical solution to MFIE (3.55) is implemented by expanding the

unknown surface magnetic field H in basis set {qu as

N
HOW) = Z aqfq(x,z) (357)

(M

and point matching the MFIE (3.55) at N points (xp,yp) on the first ocean-surface period

C p , leading to the matrix equation

54

2 Ameq = F p=l,2,---,N (353)

where the matrix elements are given by

 

f0? ,2) 6G(x ,z |x’,z’)
A = q P -PV x/ P P dl’ (3.59)
m 2 fcpfq( .z’) an’
and the forcing vector is
pp = Hoe'il’ivem» (3.60)

A pulse—function basis set was used to implement the numerical solution. The
first period 0 sx 5L of the ocean surface is partitioned into N segments, each of length
Ax = L/N, between end points x(1 = (q - l) Ax for q = l ,2,---,N+l . Corresponding points
on the surface are located by 2,1 = p (xq). The pulse functions are then defined by (3.33).
The partitioning scheme is the same as was used for the TB case and can be visualized
in Figure 3.3.

For the purpose of evaluating the integrals in matrix elements (3.59), the ocean
surface is modelled as piecewise linear between partition end points xq with slope
m q = (zq,,-zq)/Ax. An element of displacement along the ocean surface isdl = S(x)dx
with S(x) = W such that within the q’th partition dl’ = sq dx’ with sq = 1 +m:.
The matching points are located by 5p = xp + Ax/2, {p = 2,, +mpr/2. Matrix elements

(3.59) are consequently calculated from

= f;(xp ’zp) xq +Ax 6G(xp 92p Ix I‘ixazi) . (3.61)
2

an/

AP?

 

—s WI

The normal derivative of the periodic Green’s function can be evaluated as

 

 

.62 = -"p p’(x I) +sgn(z-z) e”jp'(x'xl)e‘jq"""' (3-62)
671’ 2LSl(xl) n=-o

and expression (3.61) leads to

55

A zgé’i—i °° e-jB'C-Pffox

[3 m
”9 2L "M,

”Li + sgn(z-p-z’)
q

emf/e -jq,,lz',,-z'1 dx , (3.63)

xv

 

 

where z’ =zq+mq(x’—xq) for xqsx’s(xq+Ax).

Finally, the integrals in (3.63) can be evaluated in closed form, leading to:

CASE 1: 2,424. am = [in+s,gn(Zp—Zq)qnmq

(3.64)

0 1 a. p m _ _ _-B (i _f) _- I‘ "-1 Sin(a AX/Z)
A =_fl—_ __fl q+s nz —z 8‘," P qelqnzp 7.4 "q
M 2 L n;..[ q" g (p 9)] a”

.-- -_/_-/_- --_
CASE2. zp zq, Izp zl—mqlx qu With mq—lmql

 

A _ 6 sgn(mq) °° e‘jPIGP-fq)

- —”-‘l + 113,032 +1)
m 2 L "3'" 13121 -(qn"—1q)2 [ q
+ [1: + q")fiqe _jq'fivAxnsin[—B"Ax\ (3-65)
4,. 2 I

 

 

-J'B,.(r7:§+1)e 'jq'fi‘mcos[ MAX)

The infinite series in case 2 consists of the sum of three terms, the first of which

has no exponential decay and is therefore very slowly convergent. It is consequently

necessary to accelerate the convergence of the series associated with that term. The
summand of that first term is

f = sgn(mq) e 734;, 2?.)
n L [32- I; 2
n (n a)

which for large |n| has the asymptotic form

10,0330) (3.66)

' - it 3 --i [L
fa = Jsgn(mq) e 12 n(, t) (3.67)
n 21: n

and decays only as n '1. The convergence of the first term in the series of (3.65) can be

accelerated by subtracting and adding (for n it 0) the asymptotic form (3.67) in its

56

summand. The subtracted term renders the resulting series rapidly convergent, while the
series involving the added term can be summed in closed form. Combining the terms

arising from m and -n leads to

1.. 6+ 6 _ sgn(mq) '° sin[21m(§p-§q)/L]

§ (f. f4.) n gl | n (3.68)
it - II

sgn(mq) 211: sgn(tli)

 

 

where 1|: = 211(1?p -iq)/L, and use has been made of the summed series

 

z M = “-x for OsxsZn . (3°69)
k=1 k 2

Subsequent to solution of MoM matrix equation (3.57) for the a q , the scattered
field at points above the ocean surface can be obtained from expression (3.56) as

N °° " 1'5 -' z-E . 3+ 11 A 2 3.70
1156.2) = l X a, X [—_p;mq +l]e 113.1 1),, 14.1 .) smlwfi 34m: xl ] ( )
" n n q

q=l ns-w

The backscattered field is obtained from (3 .70) by restricting the sum over 11 to include

only those Floquet modes which are propagating (qn real) in the -x direction (Bn<0).

3.3.2 Preliminary results for the TM infinite case

Preliminary results for the surface fields induced on a PEC infinite sinusoid for
a TM incident plane wave are given below. These results are analogous to the TB results
in 3.2.2. The comparison of the Floquet mode-matching and MFIE methods are shown
back in Figure 3.5. The magnitude of the induced surface currents can be seen to match
for both of the analytical methods. This serves as check for the numerical implementation
of the methods and as a confirmation of the theory. It should be noted that currents
induced by a TM incident plane wave will have both x- and z—components. This is
because I? = it x Silly , which is a vector tangential to the sinusoidal surface. A vector
6 is defined as ii x y and is the direction of the induced current (Kc). The following

plots are all of the magnitude of Kc or equivalently H.

57

In Figure 3.12 the induced currents dependence on incidence angle is shown.
Once again as the incidence angle is increased the side of the illuminated sinusoid has
a maximal current distribution. There is also an interesting effect due to the Floquet
mode resonances. Just as the propagating Floquet modes can be turned on and off as the
frequency is swept (See Figure 3.15) the same phenomena occurs for the incidence
angles. Therefore a Floquet mode must be located around 30°, which explains the
increased magnitude.

The effect of increasing the sinusoidal surface height is shown in Figure 3.13. In
the limit of zero height (flat surface) the magnitude of the H-field should be double the
incident field at PEC surface. This fact is observed in the figure for the smallest heights,
where the unit incident field is doubled to a strength of 2.0. As the heights increase the
familiar distribution is seen, with the shadow region becoming more defined.

In Figure 3.14 the sinusoidal surface period lengths (L) are varied. The results
are rather intuitive in that as L is increased the surface appears to become flatter, which
results in a current distribution tending towards a constant 2.0. For the shorter surface
period lengths an opposite effect occurs and shadowing becomes more noticeable.

As alluded to earlier the effect of varying the frequency of incident TM field are
shown in Figure 3.15. The Floquet mode resonances are more apparent and have a more
dramatic effect on the induced current. Those modes occur at frequencies of 1.5, 2.95
and 4.4 GHz (See Table 3.1 on page 66).

These preliminary results once again give us confidence in the theoretical

developments of the preceding sections.

58

Induced Surface Current for TM excitation
L=O.1016m, h=0.005m, Freq.=2.0 GHz

 

 

3.20 5
+4 1 -
0C) 1
t 2.20 :
3 c-
O -
'a I
a) a
0 —1
3 1
E 1.20 -:
0 Z
'0 _
3 :
'E q
a) -
a -
2 0.20 I

3 14+.» Surface Profile (Amplitude*64)
_0'80 :W'UW‘WWWW
0.00 0.20 0.40 0.60 0.80 1,00

x/L location

Figure 3.12 Dependence of currents induced on a PEC sinusoid for TM excitation for
various incidence angles.

59

Induced surface current for TM excitation at ZGHz
L=.1016m, Incidence Angle=85°

      
 
 

- \\
- ; - \\\\\
;- - .0 \‘\\\\\\\\

- .0- 0 ~‘ \\\\\\\\

o .9 9 .‘g‘ s“‘\\\\\\\\\\‘\
\\\\\\ \

    
   
   

  

     
      
 

 

   
  

0.
d \
00 \‘\‘{‘\\$\\
(' ‘33$\\\\\\\\\\\\:
6' to
O l
j 1’
Q9
Q
Q27 \
\\\\\\“ ‘ Q)‘
\\\\\\
\
dub, ‘\‘\\\\\\\‘ ‘2 f
\\\\\ ‘ ‘33
‘\\\\\\\\ ° w

Figure 3.13 Dependence of currents induced on a PEC sinusoid for TM excitation for
various surface heights.

60

Induced Surface current for TM excitation at 2GHz
h=.005m, 01:85°

2.20

2.00

1.80

 

Magnfiude
a

  
    

— L=.1016m
----- L=.2032m
1.40 mo-e L=.3048m

H...“ L=.4064m

 

1.20
1.00 [llTTlIIIIIIIlllIIIITTITIlIIIIIIIIIIIIIIIIIIIIIIII—l
0.00 0.20 0.40 0.60 0.80 1.00

x/L location

Figure 3.14 Dependence of currents induced on a PEC sinusoid for TM excitation for
various surface period lengths (L).

61

Induced surface current for TM excitation
=.1016m, h=.005m, Incidence Angle=85°

 

Mag

 

[I’ll]!!!

 

 

 

 

 

 

 

 

Figure 3.15 Dependence of currents induced on a PEC sinusoid for various TM

excitation frequencies.

62

3.3.3 TM finite surface scattering

The analysis for the TM plane wave scattering from a finite PEC sinusoidal
surface follows closely to techniques employed for the other integral-operator methods.
Considering the geometry in Figure 3.10 with an incident H-field in the transverse
direction, the scattered H-field can be determined, if the induced surface fields are

known,

_ j " l (2) /
H,(x.z) — I [Might ,z’)xVHo (kR) dc (3.71)

The contour needed for the MFIE is a closed contour and is denoted by C +I‘ , and 6 is
the unit vector tangential to the surface. The induced surface current 133(x,z) can be
rewritten as é(x,z)Kc(x,z) and the gradient of the second kind Hankel function can be

evaluated (note R is distance between source and observation points) yielding,

Hs(x,z) = 1.4’5 [Nico ’,z’) 11,011.11) [6 1x11] dc’ 13.72)

In order to determine the induced current a MFIE is implicated of the form

Kc(x’z) jk / 12) . / ~ /
__ + 7 Pvmecor ,z’) H1 (kR) [c(x ,z’)xR] dc = H,(x.z) (3.73)

...V (x,z) e C+I‘

Where the boundary condition on the tangential H-field has been applied.

A MoM numerical solution is then implemented to solve the above integral
equation. The current is expanded in a set of pulse basis functions and point matching
is implemented. This results in a square matrix equation for the unknown current
coefficients. The scattered fields (3 .71) can numerically be determined subsequent to

performing the MoM numerical solution for the induced surface currents.

3.3.4 Preliminary results for the TM finite case
A comparison of the induced surface currents on a finite PEC sinusoid and an

infinite PEC sinusoid are shown in Figure 3.16. The agreement is remarkable given the

63

construction of the finite surface. In order to create a MFIE for the finite surface a
closed contour was required. This contour is shown in the figure. The edge effect of the
finite surface is minimized by putting a curved endcap on the finite surface. The good
agreement will serve as an argument for using the infinite surface responses in

comparison to the experimentally obtained responses from a finite surface.

Surface Current for TM excitation
L=.1016 h/L = .125 0.=—60°
Frequency = 2 GHz

 

 

 

 

 

 

 

1.18 E
i """" Surface Contour. .
: Surface Field (Finite-MFIE
Q 98 ; °°'°°*3 Surface Field lnfinite—MFI )
3 0.78 E
x I
4.: ..
C :
“t’ a
3 0.58 —_
U -I
0 :
8 :
‘C i
3 0.38 E
m _
..,
:1
0.18 E
5
-—0.02 -‘-‘ t _________________________________________________________ '

 

'anmm
—0.40 —0.30 —0.20 -0.10 0.00 0.10 0.20 0.30
)1 location (meters)

Figure 3.16 Comparison of induced surface currents on finite and infinite surfaces for
‘ TM excitation.

3.4 Cut-off frequency phenomena

As alluded to in the previous sections, an interesting cut-off phenomenon is
associated with the scattered fields of the infinite periodic surface. The scattered Floquet
modes from the infinite surface become evanescent in nature if “3"] > k; there will be
no real power flow away from the surface for such modes. This results in a cut-off

frequency for each Floquet mode given as

(0" = L(llfs|icn6i) (3'74)
The n=0 mode (the specular reflection) has no low frequency cutoff, but all the other
Floquet modes will exhibit a low frequency cut-off. As more Floquet modes begin to
propagate and carry real power, an interference pattern is generated in the spectrum.
Also associated with the cut-off frequency is a maximal frequency for
backscattered waves. For a Floquet mode to propagate backwards (-x direction) [in < 0,
and also the mode must be above cut-off frequency. Therefore, there exists a range of
frequencies in which backward travelling Floquet modes can propagate. The maximal
frequency is given by
(fm), = LIT’iInCF (3.75)
Of note in the above equations is the lack of dependence on surface shape or
height, only the period of the surface and incidence angle are involved. For L=0. 1016
m, and an incidence angle of 85° a table of maximal and cut-off frequency for the
backward travelling Floquet modes has been provided, in Table 3.1. A similar table

(Table 3.2) is also provided for the case of a 67° incident angle.

65

Table 3.1

Low frequency cutoff and maximal frequency of backward Floquet modes.

L=O.1016m, 85° incidence angle.

 

 

 

 

 

 

 

n (1,)n GHz (f,,,,,)n GHz
-1 1.479 2.964
-2 2.958 5.928
-3 4.438 8.892
4 5.916 11.86
-5 7.395 14.82

 

 

 

Table 3.2

Low frequency cutoff and maximal frequency of backward Floquet modes.
L=O.1016m, 67° incidence angle.

 

 

 

 

 

 

 

n (1,)n GHz _ (rm), GHz
-1 1.537 3.117

-2 3.075 6.235

-3 4.613 9.526
.4 5.126 11.60

-5 9.226

 

 

19.25

 

66

 

 

3.5 Results and Experimental Confirmation

The primary interest is in the transient scattered field response of a sinusoidal
surface. This can be obtained via a Fourier synthesis of the frequency domain results
from the previous sections. Much physical insight is provided in the time domain that is
not available in the frequency domain.

The goal of this chapter is to not only understand the transient scattering from
infinite and finite sinusoidal surfaces, but to make a connection between the two. In the
preliminary results a basis for this connection was established with the favorable surface
current comparisons. This connection will serve as link between the experimental results
for the finite surface and the hypothetical experimental results from an infinite surface.
The theoretical fuiite surface scattering and the experimental finite surface results can be
directly compared, then the theoretical finite and infinite surfaces can be compared in

order to establish this link.

3.5. 1 Infinite surface results

Two methods of analysis were employed for the infinite surface, the Floquet
mode-matching and the integral equation (IE) method. As stated in the previous sections
these methods have overlapping regimes of validity. The Floquet mode regime is limited
by the surface slope being less than 0.448, meaning fairly smooth surfaces. This
restriction should be strictly enforced for the near-zone fields, however we have noted
that the far-zone fields are less affected by this restriction. Extending the Rayleigh
hypothesis regime is not the subject of this thesis so we will stay within the strict limit.
The IE method has no theoretical limits on surface slope or period length, but there are
computational limits, such as matrix size, and matrix ill-conditioning.

The induced surface currents have been examined in the preliminary result
sections. These currents give rise to the scattered fields, which are the primary interest
of this thesis. These scattered fields are what an actual UWB/SP radar would receive.

Both spectral and transient scattered field responses are examined and are interconnected.

67

This combination of time and frequency information is one of the strong points of
UWB/ SP radar.

The spectral domain scattered fields from a conducting sinusoidal surface
(L=O.1016m, h/L =0.03565) generated by a TB plane wave with an incidence angle of
85° are calculated at an off-surface field point (x/L =0, z/L= 20) as a function of
frequency over the bandwidth of 1-7 GHz. The spectral amplitudes of the total scattered
field are computed by summing all the Floquet modes and the result is shown in
Figure 3.17. The spectral amplitude of the backscattered field is computed by summing
all Floquet modes propagating in the negative x-direction, and the result is shown in
Figure 3.18. There are some interesting phenomena in these results. First, there are the
cut-off and band-pass phenomena for the backscattered field as discussed in section 3 .4.
Also of note are the apparent nulls in the frequency domain of the total scattered field,
demonstrating the frequency selectivity of the surface. The results shown in figures are
produced by the F loquet mode-matching and IE methods, and they give almost identical
results.

To find the time-domain, transient scattered field from the conducting sinusoidal
surface created by a short pulse, the spectral results of the scattered field are inversely
Fourier transformed. The short pulse used in this study is synthesized by inverting a
uniform spectral response over a bandwidth of 1-7 GHz with a 1/8 cosine taper or
Gaussian Modulated Cosine (GMC) windowings. The shapes of these two short pulses
are shown in Figure 3.19. Consequently, the time-domain, transient scattered fields
created by the short pulse can be obtained by inversely Fourier transforming the spectral
results for the scattered field shown in Figure 3.17 and Figure 3.18 with the same
weighting. The total scattered transient field created by a short pulse consists of large
specular reflection followed by a small non-specular reflection after a time delay, as
shown in Figure 3.20. For various incidence angles this time delay has been seen to

change (see Figure 3.21). The non-specular reflections are stationary for the various

68

incidence angles. This is because the reflections all originate from directly below the
field point and are in fact backscatter. The specular reflection is similar to that generated
by a flat surface, although not all the energy is reflected.

The backscattered, transient field created by a short pulse, which is of major
interest in this study, is shown in Figure 3.22. It is observed in this figure that the
backscatttered response of a short pulse exhibits an expected periodic nature with the
period correlating to the two-way transit time of the reflected wave between two crests
of the sinusoidal surface. This response is dominated by the reflections of the short pulse
from crests of the sinusoidal surface.

In Figure 3.23 is a close up of the non-specular reflection, referred to in the total
scatter discussion regarding Figure 3.21, and is seen to actually be backscatter. Very
interesting is the noticeable change in return pulse periods; this can be explained
geometrically by examining the path lengths to the crests nearest to the field point that
are contributing to the backscatter. By increasing the height of the sinusoid the
backscattered field is enhanced as expected and shown in Figure 3.24. In addition, for
the larger height sinusoids noticeable pulses are seen within the main crest pulses. This
can be explained by the Rayleigh hypothesis. As these sinusoids are outside of the regime
of validity, there must be multiple reflections occurring in the trough. Therefore
additional reflections can be seen. It is worth noting that the nature of the total scattered
field and the backscattered field are entirely different but they are consistent with
physical intuition.

The dependence on surface period length is examined in Figure 3.25, where the
transient backscattered fields from two surfaces of differing periods (L= .2032m and
L= .4064m) and equal crest heights are compared. The separation distance between
neighboring return pulses corresponds to the period length, but differing observation
heights (z/L) lead to the unexpected return pulse period. Although the two surfaces have

equal crest heights, it is interesting that the shorter period surface creates larger

69

amplitude return pulses. This phenomena can also be observed in the frequency domain
(not shown). The difference in surface slopes lead to a physical explanation. The
backscattered field is enhanced with the increased surface slopes, also there are
noticeable multiple reflections on the shorter period surface.

Similar results are obtained for the TM excitation, including the Floquet modes
cut-off frequencies, and the frequency selectivity. The total scattered magnetic field at
a point located at z/L=20 is shown in Figure 3.26. The corresponding synthesized
transient response for a GMC incident pulse is shown in Figure 3.27. The backscattered
field for the same case as shown in Figure 3.28 (spectrum) and Figure 3.29 (synthesized
transient response). The same phenomena as discussed for the TE excitation is observable
for TM excitation. There will be more significant difference between the TE and TM
cases when the sea-surface model is extended to be imperfectly conducting. This case is

considered in the next chapter.

70

Total Scattered Electric Field
L=O.1016m, h/L=0.03565, theta=85
1'04 x/L=0, z/L=20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.02 iMiiiiifl
A1.00
0)
.0
.3
iII
010.98 U
U
a II
9 WWW
20.96
h.
11
30.94
L” ———Fl0 uet Mode
------- EFI—PGF
0.92
0'90IIIIIIII-ll'lllillIIIII'IIIIIIIIIIIIIIIIII
0.00 2.00 4.00 6.00 8.00

Frequency (GHz)

Figure 3. l7 Magnitude of total scattered electric field from infinite, conducting
sinusoidal surface as a function of frequency for TB excitation.

71

0.03

 

 

 

 

 

 

T. Backscattered Electric Field
2 L=O.1016m, h/L=0.03565, theta=85
Z x/L=0, z/L=20
II
0.02 E
: Flo uet Mode
1? : ----- 151-'1 —PGF
'0 -
3 -
-H —1
'E 0.02 -
0‘ I
O -
2 _
V 1—
E Z
0 ..
1.: 0.01 2
o "'1
‘c I
+1 -
O ..
2 -
“J 1
0.01 :-
: M
0'00 IIIIIIIIIIIIIFIIII[[llllIlTllllITIIIIII]
0.00 2.00 4.00 6.00 8.00

Frequency (GHz)

Figure 3. l8 Magnitude of backscattered electric field from infinite, conducting
sinusoidal surface for TE excitation as a function of frequency.

72

Incident Pulses synthesized from
1—7 GHz Bandwidth

 

 

 

 

1
9.0E+009 E
_i
q
: — 1/8 Cosine Taper
'_‘ ------ GMC window (7'=. 4, f =4 GHz)
3’ 4.0E+009 :
2 -
3 3 i
:6 : I:
35 I ........ :iiiit .........
—1.0E+009 E ii ii
-6-OE+009 :WTWWWW
7.00 9.00 1 1 .00 13.00 15.00 17.00

Time (ns)

Figure 3.19 Synthesized pulses used for interrogation of conducting sinusoidal
surfaces.

73

Transient Electric Field Scattered

L=O.1016m, h/L=0.03565, theta=85

2.009009 x/L=0, z/L=20

1 DOE-+009

111111111l111111111|

 

 

 

 

 

 

 

 

 

A
G)
>
'43 0.00E-l-OOO _ ~44 i\v— Av-W
'76 .1
L —
v _
'g 2
.92 2
“- -1.001:+009 :
.9 “
L -
4, _
o —1
2 I
L” 1

-2.001:+009 —

3 — Flo uet Mode
3 ------ EFI -PGF
-3'OOE+009 TIIITTIIIITIIIIIII[[lllllllllrIIlTTllTlillITIIIFT]
0.00 2.00 4.00 6.00 8.00 10.00
Time (ns)

Figure 3.20 Total scattered electric field created by a GMC pulse from infinite,
conducting sinusoidal surface for TB excitation.

74

Transient Total Scattered E—Field
L=O.1016m, h/L=.125
x/L=0, z/L=20 EFIE-PGF

 

 

 

 

 

 

 

 

 

 

1.5E+009 E
0 5.0E+008 -:
'O -t
3 —
:t‘ 1
E- .
—i
< I
0)) —5.01-:+008 :
E 1
0 2
OZ _
—1.5E+009 : :1 1| 9:85.
; .I I ---- o=80°
- ii , — — e=60°
: '1'
_2-5E+OogqllllIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
0.00 2.00 4.00 6.00

Time (ns)

Figure 3.21 Comparison of transient total scattered electric fields created by 8 GMC
pulse for various incidence angles upon an infinite sinusoidal surface for
TB excitation.

75

Transient Backscattered E—Field
L=O.1016m, h/L=0.03565, 01:85°
x/L=0, z/L=20

 

5.0E+006

Flo uet Mode—Matching
3.0E+006 coco-0 EF| —PGF
1.01-:+006

 

—1 .OE+006

 

 

 

 

 

Electric Field (Relative Amplitude)

llllllllllllllJJllllll lllllllllllllLllllllllllllJ

 

 

-3.0E+006
"53544305 TlllllllilllllllllllTlllllllllllTllllllllllTlllll]
45.00 46.00 47.00 48.00 49.00 50.00

Time (ns)

Figure 3.22 Transient backscattered electric field created by a 1/8 cosine pulse from
infinite, conducting sinusoidal surface for TB excitation.

76

Transient Backscattered E-Field

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L=O.1016m, h/L=.125 £501,838»
x/LT, z/L=20 EFIE—PGF — e=45°
1.0E+009
1i .
:1 ,1 15"
'i i :i 1
| I I l ‘
0 7: ii i i r i ‘
,3. 2 UN} '11"; ii i ‘1‘."
:3- 0.0E+000 T. ii‘i, ' '1. 11‘H i’
e 41'" ' “I I I
< 111° :i‘; : 15 I.
"‘ I
.8 : IE I 1
E - I: i,‘
o - . q
01 .4 '
—1.0E+009 — ii
-2oOE+009-IIIIIIIIII'IIIIIIIUIIIII1IIIlfl'lll
7.50 8.50 9.50 10.50
Time (ns)

Figure 3.23 Comparison of transient backscattered electric fields created by a short

pulse of various incidence angles upon an infinite sinusoidal surface for
TB excitation.

77

Transient Backscatter E—Field

[FD-1016f”. 91=85° vomit/L=.0625
X/L=O. z/L=20 EFIE—PGF ~m-tii/L=.125
---- /1_=.25
I — h/L=.5

I

2.5E+007

 

 

 

 

 

 

111111111111111111

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q) '|
‘0 I
3 1
= 5.0E+006 a .
2' {nil ‘: I
<1: Ii ,4 J,
a.) 1“ ‘1 f
:5 II ll 1,
2 I I :
8 - I I :
-1.5E+007 : I : l
—305E+007-llll [1"] 111111 TIIIllTl'lIlllllIlitT—l
46.00 47.0 48.00 49.00 50.00

 

Time (ns)

Figure 3.24 Comparison of transient backscaterred electric fields created by a short
pulse from various height infinite sinusoidal surfaces for TB excitation.

78

Transient Backscatter E—Field
Oi=85° TE excitation
x/L=O, z/L=20 EFIE—PGF

— L=.2032m, h/L=.25

 
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

; ----- L=.4064m, h/L=.125
1
5.0E+007-
: a
- .‘g .
- l‘ I
(D 2 i" ‘t
'C i‘ l l
3 ‘4 | l
:L'i " l J“ |
a 0.0E+OOO- ' |\ , 4 I
E j ' l ‘11!
| v
< 1 l i
a) 4 'I
> - 'I
1.3 - 'I
2 - :;
5‘2 ' "
-5.0E+OO7- 'll
- l
- ‘l
|
J
‘1-OE+008 lllllllllllllTlllllllIIllllll]
40.00 42.00 44.00 46.00

 

Time (ns)

Figure 3.25 Comparison of transient backscattered electric fields created by a short TE
pulse from infinite sinusoidal surfaces of differing periods.

79

1 .20 .
Total scattered H—fleld

L=O.1016, h/L=.O3565, theta=85
x/L=O, z/L=20

1.00

 

0.80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(Magnitude)

WWW

Hy/H,
p
.p.
O

i

.0
m
o

—— Flo uet Mode (20 terms?
------- MFI —PGF (10000, 40, .e—5)

O

O)

O
llllllllllJJlljllllllllllllllllllllllllilllllljlllllllllllll

 

0.00 IIIIIIIIIIIlllllllllllllllllllllllllllj1

0.00 2.00 4.00 6.00 8.00
Frequency (GHz)

 

Figure 3.26 Total scattered magnetic field for TM excitation of a PEC sinusoid.

8O

 

 

 

 

 

2.00E+009 — Transient H—field Scattered
f L=O.1016m, h/L=0.03565, theta=85
; x/L=O, z/L=2o
1.009009 :
1
.4
{ 0.00E+OOO :1 LI
3:: I
.0 1
:13 I
‘T :
I _
—1.00E+009 :
3 —- Flo uet Mode
- ------- MFI —PGF
1
—2’OOE+m9 IIIIITIT11TIIIIIllllllTTlITTllrllTTlllT]
0.00 5.00 10.00 15.00 20.00
Time (ns)

Figure 3.27 Total scattered magnetic field created by a GMC pulse from infinite,
conducting sinusoidal surface for TM excitation.

81

1.20

 

  

 

 

 

: Backscattered H—field
* L=O.1015, h/L=.03555, thet0=85
1 x/L=O, z/L=20
1
L —— Flo uet Mode (20 terms
0'80 1 ------- MFI —PGF (10000, 40, ire—5)
’8 _
'0 —:
3 _
'E _
0" ..
U
E I
"' >.0.40 *
I
> i
s a 4
I _
0000 Illllillllllill‘leIIIIIIII
0.00 2.00 4.00 6.00 8.00

Frequency, GHz

Figure 3.28 Backscattered magnetic field for TM excitation of a PEC sinusoid.

82

Transient Backscattered H—Field
L=O.1016m, h/L=0.03565, Ol=85°
x/L=O, z/L=20

2.0E+008

 

”0% et Mode—Matching
09°06 MFI —PGF

1 .OE-l-008

11111l11111||14_l

b

 

 

 

 

Magnetic Field (Relative Amplitude)

 

 

ODE-+000:
_1 I
+
-1°OE+008‘l[FTIIIIIllllllllllIITIIIIIIIIIIIIIIIIIIIIIIIIIll]
45.00 46.00 47.00 48.00 49.00 50.00

Time (ns)

Figure 3.29 Transient backscattered magnetic field created by a 1/ 8 cosine pulse from
infinite, conducting sinusoidal surface for TM excitation.

83

3.5.2 Finite surface results

The theory developed for the scattering of a plane wave from a finite sea surface
is included primarily as a verification of experimental measurements. This is because the
sea surfaces used in the experimental measurements are finite. The PEC sinusoid sea-
surface model was constructed by adhering aluminum foil to a sinusoidally machined

piece of Styrofoam. This setup can be visualized in the figure below.

Aluminum Foil

//////

Foam Base

 

 

 

 

Figure 3.30 Construction of sinusoidal sea-surface model.

The period (L) of the surface is 0.1016m (4 inches) and the amplitude (h) is
1.27cm (0.5 inches). These dimensions where chosen to allow a reasonable number of
periods (6 and 11) to be placed on the surface. The surface was limited in size by the
measurement system characteristics (See Chapter 2).

The frequency domain synthesis technique in the anechoic chamber was used to
measure the PEC sinusoid model. A bandwidth of 1-7 GHz was used to create the
synthesized incident pulse. A qualitative look at the experimentally measured
backscattered fields is shown in Figure 3.31.

Both the spectral and synthesized transient responses are shown for TB incident
plane waves that vary from normally incident (0°) to grazing (90°). In the top plot the

spectral response is shown and the striking feature is the Floquet mode spikes. These
84

spikes occur at the Floquet mode cut-off frequencies (See section 3 .4). The dependence
of these spikes on the incidence angle is evident and is predicted by equations (3.74) and
(3.75). The cut-off frequencies were a by-product of the infinite surface theory and yet
the finite surface still exhibits these.

The synthesized transient responses are shown in bottom plot of Figure 3.31.
These returns were GMC windowed before transformation and then were individually
normalized. Normalization was required because the near grazing returns had far less
energy than the normally incident responses. This phenomenon was explained in the
preliminary results section and was worsened by the edge condition. The returns exhibit
a nice periodic spacing that corresponds to scattering from all the sinusoid crests.

These results indicate the high quality of the measurement system. A more
detailed examination of individual incidence angles will follow.

The spectral amplitudes of theoretical and experimental backscattered electric
fields generated by a TB plane wave at an incidence angle of 67° from finite sinusoidal
surfaces are shown in Figure 3.32. The finite sinusoidal surface used in the experiment
has 6.25 periods while that used in the theoretical calculations has only 5 periods.
Theoretical and experimental results agree quite well, especially at the locations of
frequency spikes. When the spectral results of Figure 3.32 are inversely Fourier
transformed with 1/8 cosine windowing, the time domain, transient responses of
theoretical and experimental backscattered electric fields created by a short pulse are
obtained as shown in Figure 3.33. The backscattered response of a short pulse from a
finite sinusoidal surface is a number of peaks representing the specular reflections of the
pulse from the crests of the surface. Theory and experiment agrees very well except in

the number of peaks; theory has 5 and experiment has 6.25.

85

TE backscatter from PEC sinusoid wave

 

1 2 3 5 6 7

4

Freq (GHz)
TE Transient backscatter from PEC sinusoid wave

80

60

Degrees
A
O

 

20

 

8 10 12 14 16 18
Time (ns)

Figure 3.31 Experimental measurements of a PEC sinusoid with TB excitation. The
measured spectral response and the synthesized transient returns are shown
for numerous incidence angles.

86

To seek a link between experimental results from a finite sinusoidal surface and
theoretical results based on an infinite sinusoidal surface, an enlarged experimental model
of a sinusoidal surface with 11 periods was constructed. There is also a problem in the
definition of backscatter from the finite and infinite surfaces. For the finite surface the
far-zone scattered fields can be calculated for any given angle in reference to the
sinusoidal surface. By choosing this angle to be the same as the incidence angle a
backscattered (monostatic) field can be specified. However, the infinite surface does not
allow for such an arrangement. The scattered fields are infinitely periodic in the x-
direction, therefore location of the field point is unimportant. What allows us to compare
the backscattered fields is their transient nature. By geometrically choosing the correct
angle from the field point to the surface the path length can be determined. This in turn
can be thought of as a time—delay. If the transient backscattered fields from the infinite
surface are time shifted by the correct amount a strong agreement should exist. Since this
method will only work for one point in space, there will be noticeable differences in the
periods of the return pulses.

The theoretical results on the backscattered electric field in spectral domain and
time domain derived on an infinite sinusoidal surface are compared with the
corresponding experimental results obtained from the enlarged experimental surface in
Figure 3.34 and Figure 3.35. A good qualitative agreement between theoretical results
from an infinite surface and experimental results from a finite surface is observed in
these figures. In Figure 3.34 the frequency spikes observed both in theory and
experiment occur at the cut-off frequencies of the Floquet modes signifying the excitation
of those modes. In Figure 3.35 both theory and experiment predict the backscattered
response of a short pulse from a sinusoidal surface to be a series of peaks representing
the specular reflections of the pulse from the crests of the sinusoidal surface. To further
make a connection between the results of an infinite surface and a finite surface, the

computed, transient backscattered electric fields from those two surfaces are compared

87

in Figure 3.36. The locations of the peaks are completely matched while a minor
discrepancy on the shape of the response is observed.

For completeness, the results for the TM excitation are also included. In
Figure 3.37 the frequency-domain backscattered magnetic fields, obtained from the
theory for an infinite surface and the experiment on a finite surface, are compared.
Figure 3.38 shows the comparison of the transient backscattered magnetic fields created
by a short pulse obtained from the theory on an infinite surface and the experiment on
an enlarged finite surface. Finally, theoretical, transient backscattered magnetic fields
created by a short pulse from an infinite surface and a finite surface are compared in
Figure 3.39. From the results there is generally good agreement between experimental
and theoretical results from an infinite surface and a finite surface. It is noted that the
MFIE employed in the analysis of a finite surface requires a closed surface contour. This

may degrade the accuracy of the results on a finite surface by TM excitation.

3.6 Conclusions

Much has been learned regarding the nature of transient scattering of a short EM
pulse from a conducting sinusoidal surface. In this chapter, theoretical analyses were
conducted for an infinite sinusoidal surface and a finite sinusoidal surface for both TE
and TM excitations. A series of experiments was also conducted on a finite surface
model to verify the theory. There are some interesting observations due to the cut-off and
band-pass phenomena of the Floquet modes excited by the periodic sinusoidal surface.
It is shown theoretically and experimentally that the backscattered response of a short
pulse from a conducting sinusoidal surface is a series of peaks representing the
reflections of the pulse from the crests of the surface. A link between the finite and
infinite surfaces was established. An argument for the validity of the Rayleigh hypothesis

was made by observing the multiple scattering by the higher height sinusoids.

88

1.20 Backscattered electric field

 

 

 

 

 

 

 

 

: L-0.1016m, ML -.25, ei- 67°
: — Measured response (6.25 periods)
_, ------- Calculated respsonse (5 periods)
a {i
m 1 H
s , i
3 d I
t 0.80 .. 1
C _
O) _
O I
2 ..
Q) " I
> _, I
a: 0.40 — 1'
2 Z .
(l) _
a: - g
‘ l t
0.00 iiiITliiiiiiiiiiiiiliiiiiiriiliiii‘iiii
0.00 2.00 4.00 6.00 8.00

Frequency (GHz)

Figure 3.32 Spectral amplitudes of theoretical and experimental backscattered electric
fields from a finite surface for TB excitation.

89

1, ()0 Backscattered electric field

 

 

 

 

 

 

 

 

 

 

 

 

? L=O.1016m, h/L=.25, 0i= 67°
(1) I l
"O _ 'I i“, 3 I
3 0.00 - . ‘ .' - -
.t.’ 4 '
E. I
E : ' '~
<1: _
(D ..
> .,
’4: --1.00 "
.9 I
(D -
D: : — Measured response ( 5-25 periods )
_ i ------- Calculated response ( 5 periods )
—ZoOO IIIIIIIIIIHIITIIUIIIIIIllllTllllTll'l‘j
0.00 2.00 4.00 5.00 8.00
Time (ns)

Figure 3.33 Theoretical and experimental transient backscattered electric fields created
by a short pulse from a finite surface for TB excitation.

Spectral Backscattered E-Field TE Pol.
L=.1016 m, h/L=.25, 0i=67°

1.20
— Finite Ex erimental
----- Infinite T eory
1.00 I
I
II
II
II
0.80

  

Relative Amplitude
o
a)
o

.0
.p.
O

 

 

 

 

 

 

0.20 . l
I ' |
ill”, ii' :i
i I

' lizfifliil'u‘imt‘fi'i i’n. ’i: ‘l.
0.00 I. I H ‘l'g‘VVV’V H
0.00 4.00 6.00 8.00

Freq (GHz)

Figure 3.34 Comparison of experimental backscatter (finite surface) results to
theoretical backscatter from an infinite surface for TB excitation.

91

1.00

0.50

0.00

—0.50

Relative Amplitude

-1.00

-1.50

Figure 3.35

Time Domain Backscattered Field TE Pol.
L=.1016 m, h/L=.25, 0.=67°

 

 

 

 

 

 

1

.,

i — Finite Ex erimental

: ----- Infinite T eory

: | - I. |‘ |

E {\l i ‘l ’l 'l‘ A 4‘" "I I“ 0|} '‘|"‘ 1? (I:

‘ u "lv“‘r|vllv||'lllvl" 'v' "u"

: ' , ' , '. i l

E l

E I

i

IlliiilillillililiilIliilllIlliiiiilillllilllliiil

.00 2.00 4.00 6.00 8.00 10.00
Time (ns)

Comparison of transient backscatter created by a short pulse from an
infinite surface (theoretical) and a finite surface (experimental) for TB
excitation.

92

finite surface (MOM)
------- infinite surface (EFIE—PGF)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—
.1
-i
c-

—l
u-
d
d
--|
0-1
--1
-i
-
an:

_
u-i
-
-
-
d
1
.1
-
an

and

fl
.1
d
d
c-
-
u-i
'i
—(
q
.1
an:
-
-I
1
I

0.0

O
0 2
0 4

mustache m>$a_mm

IIIIIIIIIIIUIIIIIIII]

llllIllIl

 

-1.20

6.0

4.0

2.0

Time (ns)

Figure 3.36 Comparison of transient backscattered electric field created by a short
pulse from an infinite and finite surface for TB excitation.

93

Backscattered H—Field TM Pol.
L=.1016 m, h/L=.25, 0.=67°

1.20

—- Finite Ex erimental
----- Infinite T eory

1.00

0.80

Relative Amplitude
o
a)
o

.0
.p.
O

 

 

 

 

 

b Iiiuiiiiilriiiiiiiiliiiiiiiiiliiiiiiiiiliiiiiiiiiliiiiiiiiil

0.20 l ,
I l' i J :|l
' r l l ' ' ‘
”#4 ' "1' m l " WW" i i . 'I
O OO ' H. ' | I” u H P: ‘VVVN “I ’.
0 0 2.00 4.00 6.00 8.00
Freq (GHz)

Figure 3.37 Comparison of spectral domain backscattered magnetic field from an
infinite surface (theory) and a finite surface (experiment) for TM
excitation.

94

Time Domain Backscattered Field TM Pol.
.1016 m, h/L=.25, 0i=67°

L

1.50

    

a'0

 

00‘0“

""""""""
IIII"

erimental

eory

ii

Finite Ex
------- Infinite T
i

'0"""'|"' 1‘! I

1 00
0 50
.0
—0 50
—1 00

onstage 020.2%

8.00 10.00

5.00
Time (ns)

4.00
95

2.00

llllllllll[llIllIIITIIIIIIIIIIIIIIIIIIIIIIIIIFITIII

pulse from an infinite surface (theory) and a finite surface (experiment)

for TM excitation.

 

0.00

— 1 .50
Figure 3.38 Comparison of transient backscattered magnetic field created by a short

Time Domain Backscattered Field TM Pol.
L=.1016 m, h/L=.125, 0i=60°

 
 
 

 

 

 

 

 

 

 

 

1.00?1
2 -—-—- Finite Theory
2 ----- Infinite Theory
I} l
0.50: i
_ I I
_ I I
1 i i
i I I,
a) 2 HI |i
g 0003 “ i.‘
:‘i . _ II I.
a : i.‘ :I
E I " I:
< .. :' :I
g : : I:
:—o.5o: . I
o - I I
E Z ' I
c: — I I
a I I i
I I
1 I
—1.00-_
1
_1.50 IIIIIFITITIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII]
5.00 7.00 9.00 11 .00 13.00 1500
Time (ns)

Figure 3.39 Theoretical transient backscattered magnetic field created by a short pulse
from an infinite surface and a finite surface for TM excitation.

96

Chapter 4
Transient Scattering from anS Imtperfectly Conducting Sinusoidal
lll‘ ace
4. 1 Introduction

In this chapter the ocean model is improved by replacing the perfectly conducting
sinusoidal surface by an imperfectly conducting one. The ocean model now consists of
two general layers (media) separated by a sinusoidal interface. The effects of the finite
conductivity will be examined and the possibility of a Brewster’s angle phenomenon will
be addressed. It is well known that a TM plane wave incident upon a flat dielectric
material will exhibit a Brewster angle, therefore only the TM case will be fully examined
for the sinusoidal surface. Both the Floquet-mode-matching and the Integral-Equation (IE-
PGF) methods are extended to this two-media case. The Floquet-mode-matching method
is once again limited to the Rayleigh hypothesis regime. However, it does provide a
verification for the rigorous IE—PGF method for the two-media case.

The numerical solutions for the two-media case use more computer resources,
both CPU time and memory, than the PEC case. This is due to the transmitted wave,
which is now present in the second medium. The modal amplitudes of the transmitted
wave and the scattered waves can be found through mode matching. However, the
transmitted wave introduces another unknown to the governing equations, which implies
that another set of equations is needed to specify the amplitudes. This additional equation
is established by another boundary condition. Therefore there is a set of coupled
equations created by satisfying the two boundary conditions at the interface of the media.

The goal of this chapter is to not only improve the ocean model, but to draw

comparisons between the PEC model and the two-media model. This comparison will be

97

made theoretically as well as through experimental measurements. The experimental
findings will also provide validation of the scattering theory. The experimental setup
physically requires a finite surface but the theory for the finite two-media case is not
presented and is not compared. The previous chapter established the link between the

finite and infinite cases and those ideas will be applied in this chapter.

4.2 Floquet-Mode Matching Analysis

The classical mode-matching formulation [12] for scattering from an imperfectly
conducting periodic ocean surface is considered.

The Rayleigh hypothesis allows simplification of the scattered and transmitted

fields, thereby reducing the computation time involved in finding a numerical solution.

A

. . z
E ,for TB case F1332]; amt T

H'forTMcase

Region 1

e./

V 1 Ci» z=p=-hcos (21tx/L)
I h

C |._ L—'I x
Region 2 ( e, u, a )

 

 

Figure 4.1 Geometry for two media scattering problem.

A TM polarized plane wave is incident at an angle 0,. upon a sinusoidally periodic
ocean surface. The region above the ocean surface is assumed to be free space, with
wavenumber Itl and the region below the air-ocean interface is ocean-water with the
nominal parameters (0 =4 S/m, e,=80) and with wavenumber k2. The geometry is shown

mfiwm4L

98

For TM excitation Hy(x,z) is the generating field component, which satisfies the

2-D Helmholtz equation

V3 H,,-(x.z) + k} Hyi(x,z) = o (4.1)
Ei(sz) = —j—)’AXV,Hy,-(X,Z) (4.2)
(OE.

where V, = ia/ax + z‘alaz is a 2-D transverse differential operator and i is the space
region (1 or 2). The boundary conditions require tangential magnetic and electric fields

to be continuous at the interface, thus (applying (4.2))

 

 

17,2 : 11,1 9 auq&2 =:'fé'auq;l (4"3)
’ ’ 6n t-:l 6n

Since the magnetic field has only a y-component, and the fields in region (1) are
composed only of incident and scattered waves, and the fields in region (2) are only
composed of transmitted waves, the subscripts y] and y2 are be dropped. The incident

magnetic field is assumed to be of a plane wave nature. Therefore,

H i(x,z) = Hoe 7131‘ej"z (4-4)

where B =kxl =klsin0,, and q = —kzI =klcos0r
By Floquet’s theorem'the scattered (i=1) and transmitted (i=2) magnetic fields
can be modeled as

H,<x.z)= f: (”1(2) (45)
H,(x.z)= 2 e"""f,.(z) (4-6)

n=-°°

where Bn = B +% , and [3 =kx, =klsin0, is necessary to match the phases of the incident

wave and the n=0 scattered and transmitted waves at the interface.

Both the scattered and transmitted magnetic fields obey the 2-D Helmholtz
equation (4.1). Therefore substituting (4.5) and (4.6) into (4.1), gives

 

 

 

E? 1:13 «mp/car, e"“*‘=o (“I
i [::+(-oi+k%if, e"°“=o (4-8)

 

where the partial derivatives with respect to x have been performed.

Due to orthogonality the e .1 9.x , the above equations are satisfied when

 

 

321;.qung (4.9)
(32
azfg.q;fm=o (4.10)

where q: 4:12-02, and midi—13,2, with branch cuts requiring Imlqnl<0 and Im{qm}<0.

The branch cuts are chosen to accommodate the physics of the problem by
requiring exponential decay of the non-propagating scattered and transmitted fields. These
branch cuts once again introduce the cut-off and tum-on frequencies associated with the
Floquet modes. This was discussed in chapter 3.

The general solutions to equations (4.9) and (4. 10) are

t<z>=B.e""~‘+C.e"'"‘ (“1)

fm(z) = Rule 7993 + Cmequz (4.12)

The Rayleigh hypothesis (see chapter 3) states that the scattered field can be
approximated by summing only the space harmonics that are travelling upward away
from the air-ocean interface, and the transmitted field can be approximated by summing

the space harmonics that are travelling downward away from the air-ocean interface.

100

Therefore, C,I =0 and B,” =0 which results in a simplified representation of the scattered
and transmitted magnetic fields

H,(x.z) = :3 3,,e ’I'We '1'” (4.13)

H,(x,z) = 2 gifted“ (4-14)

The first boundary condition at the air-ocean interface requires

Hi+Hs=Hr at z=p (4-15)

where, p = —hcos(2—%£) for a sinusoidal ocean surface. Substituting the values from (4.4),

(4.13), and (4.14) into (4.15) yields

~j2nnx .. -j21mx

Hoejq" = - 2 Rue-”"5 L +2: Cmejq"pe L (4°16)

 

where the common factor e ‘1'” has been eliminated.
The factors exp( -jqnp) and exp( -qup) in (4.16) destroy the orthogonality of the
exp( -j21rnx/L). Galerkin’s method is therefore applied, using the testing operator

L
feflm/Li...idx (4.17)
0
leading to
L .. L on 1-
H0 ejZImulLejqux = 2 _B [8 ’14.Pej2(m-n)1u/de+ z CmfejqupejZMnr-nwl. (4,18)
0 ""°° 0 ""‘° 0

To simplify the notation the integrals can be referenced as

L
Am = eflIrnuII-ejqux (4,19)
0
1.
Km = __ f e j21:(m-n)xll.e -1'q,.p dx (4.20)
0

101

L
K ___ _fej2n(m-n)x/Lequpdx (4.21)

mm
0
Therefore

”Mr 2 KM: 23 KMC... “'22)

For all three integrals (4.19), (4.20) and (4.21) let u=21tx/L ~ du=2T"dx then
21:

A = .1; f we -thcos(u)du (4.23)

m 21:0

Using Euler’s identity, and the fact that cos(x) is an even function and sin(x) is and odd
function, (4.23) reduces to

21:
L .
A». =— f cos Imu |e"'”‘”(“)du
1t
0

(4.24)

which has the closed form solution

Am = Lj '""J.,,,.(qh) (4.25)

because 1: j "Jn(z) = cos n(u)ejz‘“““)du.
0

In a similar manner (4.20) and (4.21) have the following closed form solution

Km = _Lj Im-nIJ|m_n|(qnh) (4.26)
Km“ = L j - lm-nlJ|m_nl(th) (4.27)
The second boundary condition at the air-ocean interface requires
e
w, (aims) = Jfi-VtH, at z=p (4.28)

62

where ti = (2“ 4323)] 1%?)2. Performing the dot product on the expanded gradient
x J it

results in

102

-@—(H +11 )+—(H +113): 51 (429)

6x6 62

__+__.

apaH 6H]
61: 6x 82

 

Substituting the values from (4.4), (4.13), and (4. 14) into (4.29) and performing the

partial differentiations, yields

- 0(B,:_:+q)ejqp = Z Bnn%__(Bejqupe—j2n1tx/L
"h" (4.30)
__ 612 C w(p a_p x+qm):l:mpe -j2nnx/L
52".-"

where the common factor e “1"” has been eliminated. The factors exp( -jqnp) and
exp( -qup) in (4.30) destroy the orthogonality of the exp( -j21mx/L) . Galerkin’s method

is therefore applied, using the testing operator (4.17) leading to

_H 0f(Baa: +q)ej2nnu:/Lejqux= 2 B nf(pn_aa_:_ ")8 )junej2(m-n)nx/de
"M” (4.31)
__2 Cmf(pnaxle—+qm)equpej21t(m-n).dex
62,..- _..

Using a form similar to the first boundary condition evaluation (4.22) gives

HOD,” 1;” LmBn-e—i'g; mem (4.32)
where
p ”10) (1‘13: (4.33)
me- 13.1(3’+4..1‘4’ (4.34)
D," = - 01,3,” -q1,§," (4.35)
and

103

L
[(1) = geequOn-nk/Le 'jqnpdx (4.36)

0 6x
L
If") :fej21:(m-n)XILe ‘jqnpdx (4.37)
o
L
Ifl=fflej2nm~nkflequpdx (4.38)
0 6x
L
IS; = fej2n(m—n)x/Leiqm9dx (4.39)
o
L
[(5) =j‘fl1ej2nmx/Lejqux (4.40)
"' 0 6x
L
1515) =fej21tmx/Lejqux (4.41)
0

These six integrals (4.36)-(4.41) can be evaluated to produce closed form solutions. Note
that (4.37),(4.39), and (4.41) are in the same form as (4.20),(4.21), and (4.19)

respectively, and have the following closed form solutions

[:2 = L J lm-nl J|m~n|(qnh) (4.42)
1,33,, = L j -lm—n | Jim-n|(th) (4.43)
1:," =Lj""".llml(qh) (4.44)

The remaining three integrals can be evaluated after the 6p/6x term is expanded. This

gives

15.1.3: ‘13"! jlm-n+1|J|m-n+ll(qnh)-j '""""'Jim-n-ii(qnh) (4'45)

104

1.533. = ~jnh

 

j- [in-n+1 lJ|m-n+l |(th) .1. — |m-n-l lJ|m-n-l |(qrnh)] (4.46)

1;,” = -j1rh

 

i""“'J.,..1.(qh)—jIm-IIJ,,,,_,I(qh)] (4.47)

The two boundary conditions (4.3) yield a system of equations, which can be
solved using matrix operations. This system can be written as

L?" """l [2113:] .4...

mu m m
where LM,LM,KM,KM are M xN matrices, and ENC"I are N element vectors, and
AND," are also N element vectors. The number N is related to the number of
propagating modes and is chosen to insure convergence of the series and is discussed in
chapter 3 and in [12].

Subsequent to determination of B" and Cm from the matrix in (4.48). The
scattered and transmitted fields can be evaluated with (4. 13) and (4.14) as

N, .
H,(x.z) = 2 Rue ””I’e ""4 (4-49)
n=-N_

N. n o
H,(x,z) = Z Cutie-”3’12”“z (4-50)

n=-N_
Where the summation range N _ ,N+ corresponds to the matrix in (4.48), such that
N=(N_+N++l).

The above analysis based on eqns. (4.15) to (4.48) was specialized to a sinusoidal
interface. A non-sinusoidal interface could have been specified, but would require
numerical integration to determine the matrix entries. The case of a non-sinusoidal

interface will be considered in the next chapter.

105

4.3 Integral Equation Formulation

The two-media problem can also be formulated using an integral-operator
technique. This method is rigorous an overcomes any limitations set by the Rayleigh-
hypothesis method. The transverse magnetic (TM) scattering problem is of greatest
interest, due to the absence of specular reflections at the Brewster’s incidence angle, so
only a formulation for that polarization state is presented here, although the TB problem
can be studied similarly.

The two-medium scattering problem consists of space and ocean-water regions
separated by a periodic ocean surface and illuminated by a plane wave incident from the
space region, as indicated in Figure 4.1. It is assumed that region (1) is located above
ocean-surface contour C, with permittivity e, = so. The ocean water is region (2), located
below ocean-surface contour C, with complex permittivity 62. Wavenumbers in the two
regions are k, = to p.06, for i= 1,2. Unit normal vectors to the interface contour C are
chosen as fil directed outward from region (1) and ti, directed outward from region (2)
such that 132: -fi,=r’i.

For the TM polarization, all EM fields can be expressed in terms of the
generating field Hy(x,z) , which satisfies the 2-D Helmholtz equation. Relevant field

relations are

V3H,,(x.z) + k,2H,,.(x,z) = -y-V,xj(x,z) for i=1,2 (4.51)

and

E,(x,z) = .56— y“ x V,H,,(x,z) (4.52)
where V, = ia/ax +£6/62 is a 2-D transverse differential operator.

The boundary conditions requiring continuity of tangential I7 and E (applying Eq.
(4.52)) lead to

106

H =H ’ 6H,, = :2- 6H,,
’2 ’1 an e, an

at points along the ocean-surface interface contour C. To simplify the notation, the

(4.53)

magnetic field is identified with a wave function such that Hy,(x,z) = 1|: ,(x,z).

The scattered field qr? in region (1) can be expressed in terms of the surface
values of qr, and 61p llé‘n at points along the interface contour C. It is consequently
desired to obtain a pair of simultaneous integral equations for the latter surface-field
quantities. Wave function ip, is the field in region (1) and at points approaching interface
contour C from the region (1) side. Similarly, wave function 1|:2 is the field in region
(2) and at points approaching the interface contour C from the region (2) side. The
required integral equations implicate two-dimensional (2-D) Green’s functions for
unbounded homogeneous media having wavenumbers kl and k2. They include both the

aperiodic 2-D Green’s function

6,05 iisi = 7’ Hé”(k.IrI —r>°’i> (454)

and the periodic Greens’s function (see Appendix A)

. .. -jb,.(x-x’) -J'q..|z-Z’I
6,-(5159 = —-L- e e (“'55)
2L na-w qin

 

for i= 1,2, where p‘ = ix+z‘z is the 2-D position vector, [3,, = B +2mt/L andqif, = k} - [3,2,
so q," = -j./T—k? with the branch cuts chosen such that Im{q,,,} <0.

For the purpose of calculating the field in region (1), region (2) is replaced by
equivalent polarization currents .72) = jun (62- 691752 immersed, along with primary current
.7, in a homogeneous medium having wavenumber k,. The polarization currents in
region (2) are then replaced by equivalent surface sources over contour C. Applying the
2-D Green’s theorem, the definitions for Green’s functions (4.54) and (4.55), and eqn.
(4.51) leads to (See Appendix C)

107

III(p)= w<5>+ f we”) Tacafi l)—G,(-°|5/) 5:41”, (4.56)

n

where C, is the ocean-surface contour in its first period for 0 sst, and w: is the

impressed field maintained by primary impressed current as
i105) = (Jay/Jimmie lb”) ds’ . (4.57)
For remote source currents, the incident field can be modelled as the plane wave

W103.) = Hoe -iflxe.iqz (4.58)

where B =klsin0, and q =k,cos0,.

Initially, the integral equation (4.56) domain was infinite and implicated the use
of the aperiodic Green’s function G,. The periodic Green’s function, however, allows
for the reduction of the integration domain to a single period of the sinusoidal surface.
This was discussed in chapter 3 and is the key step in creating a tractable problem.

When 5 is a point on C, accommodating the source-point singularity leads to

 

- , aG -
""2” = ¢1(5)+PVfC win—("J 6-G(riiri’)— 6“” dl’ “'59)
r an’ an

where PV indicates that the integral must be evaluated in a principal value sense by
excluding a small neighborhood centered at field point p" = 6.

Following a similar procedure, the field in region (2) is obtained by replacing
region (1) with equivalent polarization currents .72) =j0)(€1-€2)E1 immersed in a
homogeneous medium having wavenumber k2. The primary and polarization currents in
region (1) are then replaced by equivalent surface sources over interface contour C.
Applying the 2-D Green’s theorem, the homogeneous specialization of (4.51) and the
definitions for Green’s functions (4.54) and (4.55) leads, for field points 06C, to

108

 

 

‘I’2=‘I’1=‘I’ a _ = ___ = —— (4.61)

along contour C, where 1|: and air/6n are regarded as unknown. Implementing these

boundary conditions in (4.59) and (4.60) leads to

M-PVf
Cr

606' ”’) ‘° - _.
2 IlI(i5’) ‘ ‘1'" - 51“? Gl(p|p’)

aw = 111(6) (4.2)
an

 

 

6G2(fili>") __ g aims)
an’ 61 an’

dl/ ___ O (4.63)

 

 

“25’ +PVfc, was") 0mm

 

 

for all points p’eCP. These are a pair of coupled integral equations (IE’s) for the
unknown wave function and its normal derivative. Subsequent to obtaining a solution,

the scattered field in region (1) is obtained from Eq. (4.56) as

dl/ . (4.64)

 

6G " " —.
wins) = f, [w(b”) ‘(pm - “maimed

an’ an’

 

The formulation for TM wave scattering from a perfectly-conducting, periodic
ocean surface is obtained as a special case of IE’s (4.62) and (4.63). For a perfectly-
conducting ocean, 62 I61” -j<a<I such that IE (4.63) requires Gilt/an’ =0 and IE (4.62)

subsequently becomes independent in wave function W as

(a) _. 60(5Ii5’) _ i.. _
‘2—‘PVfCP‘I’(PI)—-la'n,—'dl/“I’1(P) - (465)

Integral equation (4.65) is identical to the result obtained in chapter 3.

109

The two-medium ocean scattering problem is studied through solutions to integral
equations (4.62) and (4.63). Based upon experience gained from the integral-equation
(with periodic Green’s function) formulation for the conducting ocean-surface model,
moment-method numerical solutions are obtained for the unknown surface field
quantities. The forward and backward scattered waves maintained by those surface fields
are subsequently calculated, leading to the various transfer functions required to calculate

the forward and backward scattered fields.

4.3. l Moment-Method Numerical Solution

In the previous section the field on the ocean-surface satisfied the pair of
simultaneous integral equations (4.62) and (4.63). A method-of-moments numerical
solution can be implemented by expanding the unknown field w(x,z) and its normal

derivative 2%?)- in basis sets (f,} and {g,,} , respectively, as

N
w(xx) = Zagqixxi (4456)
q-l
a¢(x.z) = N (4.67)
—an (IE; 15840.2)

A pulse-function basis set is used to implement the numerical solution. The

integral path for the first period of the ocean surface is divided into N parts where

21:):
L 4). The partitioning scheme is identical to that

 

Ax: 1,5,, x, =(q-1)Ax and z, = -hcos(
used in chapter 3 (e.g. Figure 3.3 ).
Point matching the integral equations at N points (Epip) on the first ocean surface

period C , leads to the matrix equation
" ’1 [:1
C D

where A,B,C,D are NxN matrices, and a,b,F are vectors with N elements. The matrix

F
0

(4.68)

 

 

 

elements are given by

110

- A;
xq+—

 

 

6G xp x’,z
qu=ésm - PVf 9:} I) squ’
11,-?
;.£
' 2
B PVf G,(J?p,z |x ,z’)S dx’
;.31
4 2
LE
2
66 x x
c ——e PVf 2(P’ZPI ’16s dx’
P? M an] q
1’32)"
35.3
t 2
e _ _
bps—é. PV f G,(x,,zp|x’,z’)s,dx’
- Ax
xq-T

- 1 p=q . -
where 0,, -{ 0 otherwise‘ The forcrng vector is

_ up? +qu
F p -H0e P P

(4.69)

(4.70)

(4.71)

(4.72)

(4.73)

Exchanging the implied summation of the PGF and then performing the principal

value integration results in the following

 

 

S °° 81
B =°_q l: P9,]
P9 JZLRZ'Z qln
1 1 "’ Bm
C =—0 +— [ + " " J
P4 2 P4 2L,.-.. f’Pq- q2,, 82"“-

_e_2j_S,, ,2": [82:]
D" 5—12—L

n=-m
There are three cases for jg“, and g,”

111

(4.74)

(4.75)

(4.76)

(4.77)

CASE I (2,32,)

2shill/113,. +q,,m,,)A—2— —]

J'(B,,+q,-,,mq)

(4.78)

e-J'B .05, -x4) jq...(z,- zq)

 

gi'pq,ll :fipqnze

CASE II (2:55,)

. . Ax
2smh — .m —
-jiI.(E,-f,)-jq,,(z‘,-E,) MB" 4”, q) 2 ] (4.79)

j(Bn-qinmq)

 

gipq,I = _fipq. :8

CASE III (2‘, =2,)

_ 2e -J'B,.(f,-x,)
8W..- 2 2 2

qunlmq |+e 1,, m" 2 [13 ”sin(finA ——) -jq,.,,|m,, |cos(p —)]} (4. 80)

 

n—qinmq
_ - -jB,.(§,-i,) _. m A!
fin”: ZJSSZEIP-ljgmz 113,58 1%.] 4| 2 [Bncos(pn%)+jq,n|mqlsm(pnA2x.)]} (4.81)
1 x>1
where i=1,2, and sgn(x) = 0 x= .
-l x<l

4.3.2 Scattered fields
Once 1|: and dip/6n have been obtained, the scattered fields in both regions may
be determined by

 

 

S—. _ _. aGl(pIp’)- a¢1(pl) ##Jdll 482

¢1(P)-fc'[‘1’1(p, I) an, an G,( |’)dl (. )
a -

Wok-f, ”Lian —2———aG(","’2— 6—2 ¢2(‘,)’)Gz(5|§’)]dl’ (4.83)
an 61 an

Exchanging the implied summation in the PGF and the principal value integration in
(4.82) and (4.83) gives

112

”5(a) = ¢i(x,z) = X 7.3-”“2822": (4'84)

 

 

Hy'3(x.z)= 1130.2) = Z C,,e'22"2‘e""2"z (4-35)
where
as. (x’.z’) 5410’s)
7,. = —— C[III (x’,z’)——’—'—— — g ,(x’,z’)—— dl’ (4.86)
1.2241,, Cr 1 an, 1 an,
1 682,.(x [22,) 52 / all! 1(x [’1 I) ,
C" =--—- Clll[ (x’.z’)—— - —g (x ,z’)——— dl (437)
1.214an P “’1 an’ 61 2» an
and g,,,(x’.z’) = eiw'equ” g2n(x’,z")=ejp"tle "192.1,.

The scattered field in region (1) is seen to be comprised of an infinite number of
plane waves. Only a finite number of these carry power to the far field, and the rest are
evanescent in nature. The modes that propagate must have a real 4,, , for i=l,2.
Therefore, in region (1) |13,,|<kl , and in region (2) |0n|<k2.

Subsequent to the solution of the MoM matrix equation (4.68) for a q,b ,, , the
scattered field in both regions off the ocean surface can be determined from (4.84) and
(4.85). The backscattered field is obtained from by restricting the sum over I: to include

only those modes which are propagating in the -x direction (0,50).

4.4 Duality of TE and TM polarizations

The above analysis was performed assuming a monochromatic TM polarized
incident plane wave. Through the use of the duality theorem [49] the TB polarization
state can be analyzed. The incident magnetic field [Hy] is replaced with an incident
electric field [Efl] and the constitutive parameters are also exchanged (u, e ). The
unknowns for TE case integral equations will be Ey and %. A table of the

relationships between the TB and TM cases is shown below.

113

 

“TM 6 u. H E E aHy/an ll
"TB is e E -H -H airy/an ll

 

 

 

 

 

 

 

 

The resulting integral equations for the TB case are

e

 

2'5) _ PVfc,

 

60 8 ‘2 '2 i ..
“1(9) 103/19,) _ 641(85G1wlpo51dl/ = 411(9) (4.88)
an an

 

 

~ _, 6G (5 in”) 1* ‘° .. _.
P I

2
where W, now represents the electric field [Ey] and the Green’s functions remain the
same. This set of coupled integral equations can also be solved by a similar MoM

numerical method.

4.5 Numerical Examples

This section presents theoretical numerical results for several cases of interest.
Where possible, the results from the PGF method are compared to the results of the
Rayleigh mode-matching analysis. As a starting point, the parameters are chosen to
match those of the experimental periodic surface discussed in Chapter 3. The spatial
wavelength is chosen as L=O.1016m. The parameters of the lower medium are chosen
as the nominal ocean parameters e=80, a=4.

The first set of results to be examined will be theoretical responses for TM
excitation. The TM polarization state is of primary interest because of the possibility of
a Brewster’s-angle phenomenon. The manner in which the results are to be shown will
mirror the chapter 3 treatment. The induced surface fields will be compared in order to
confirm the theoretical methods and to build upon the phenomenological base started in

chapter 3. Following the surface field results will be an examination of the Brewster’s-

114

angle phenomenon. The theoretical transient scattered fields, which are the ultimate goal,
are briefly examined before an in-depth treatment in section 4.6.

Figure 4.2 shows a comparison of the magnitude of the surface fields computed
using the PGF and Rayleigh methods. It is clear that for smaller heights the two methods
agree very well, while for larger heights the Rayleigh method does not match the
predictions of the PGF method. This result lends strength to the numerical methods. The
knowledge gained from the PEC case has also contributed to the confidence in the
methods. Note, although not shown, there was also excellent agreement between the
other unknown surface field quantity (the normal derivative of the magnetic field).

The effects of the various problem parameters on the surface fields are shown in
the following figures. Many of the same features noted in the PEC case are also evident
for the imperfectly conducting case. There are also some new findings that will be
highlighted. In Figure 4.3 the frequency dependence of the surface fields for a TM
incident wave are shown. The surface is an infinite sea-water sinusoid with L=O.1016m
and h/L=.125. There is more variation with frequency as compared to the PEC case
(Figure 3.15). This is due to the increased sinusoid height.

The dependence of sinusoid height is examined in Figure 4.4 and Figure 4.5. The
higher amplitude surface (Figure 4.4) require the MFIE formulation and exhibit some
interesting variation. The lower height sinusoids (Figure 4.5), which are all in the
Rayleigh-hypothesis regime, do not exhibit a similar dependency. This is reasonable since
there is multiple scattering in the troughs of the higher amplitude surfaces. These
multiple scatters manifest themselves in the redistribution of the surface fields.

The effect of incidence angle, as shown in Figure 4.6, is very similar to the PEC
case. The normal incidence has a symmetric distribution as expected and a shadowed
region appears for the increasing incidence angles.

In Figure 4.7 the period length (L) of the sea—water sinusoidal surface is increased

from 0.1016m to 0.4064m. The resultant induced surface fields reveal the same

115

tendencies as observed in the PEC case. The surface becomes relatively flatter since the
amplitude (h) is not changed which results in less shadowing and a field distribution
approaching a constant, which is expected for a flat surface.

The new feature with the two-media case is finite conductivity. The effects of
conductivity on the surface fields, which lead directly to scattered fields (by Huygen’s
principle), are examined in Figure 4.8 and Figure 4.9. In Figure 4.8 the excitation
frequency is 2 GHz and as conductivity is increased the magnitude of the surface fields
is also increased. It is interesting that sea-water (o = 4S/m) is closer to a perfect
dielectric than a PEC. In Figure 4.9 the excitation frequency is increased to 6 GHz. The
results indicate a more defined shadow region, and the conductivity effects are reduced.
These phenomena are expected, as the frequency increases the imaginary part of the
complex permittivity is reduced, thereby reducing the conductivity effects. Physically the
polarization currents become dominant over the conduction currents. The effects on the
TE case are similar but are not shown.

Information about the Brewster angle phenomenon can be extracted by varying
the incidence angle and observing the forward scattered amplitudes. It is well known that
a TM polarized plane wave incident upon an infinite planar dielectric will exhibit a
Brewster’s angle. From the classical theory the Brewster angle for a air—dielectric
interface is given by 0b = tan'1 (JG—r). If the dielectric had a relative permittivity of 80
then the Brewster’s angle would be 83.6°. However, the problem that has been analysed
deviates from the ideal planar/ dielectric case. The interface is now sinusoidal and the
lower medium can be lossy. The expectation is that there will no longer be a true
Brewster’s angle but possible a minimum in the same range of incidence angles. In
Figure 4.10 a comparison of TE and TM forward scattered fields are shown for
incidence angles around the expected Brewster’s angle. The TM case does indeed exhibit

a minimum and is also compared with the case of a lossy flat surface. This is a very

116

favorable result and gives hope that the TM incidence could be of great importance. A
detailed examination of the TM case follows.

Figure 4.11 shows the variation of the forward scatted H-field vs incidence angle
for various wave heights for a lossless dielectric (0 =0). It is clear for a flat surface that
the forward scattered field is zero at the Brewster’s angle (approximately 83.6° ). As the
wave height is increased, the forward scattered H-field no longer goes to zero, but
instead has a minimum near the Brewster angle. For the lossy case, there is no true
Brewster angle even for a perfectly flat surface. There is however, a well defined
minimum as shown in Figure 4.12.

The frequency dependence of the Brewster’s angle is shown in Figure 4.13. The
lower frequencies, where there is only one propagating Floquet mode, exhibit a smooth
variation and reveal the Brewster angle minimum. For the higher frequencies the location
of the Brewster’s angle minimum is less obvious. The additional propagating Floquet
modes produce an interference pattern; the fundamental mode (specular) has the largest
amplitude so there still remains the general minimum around the Brewster’s angle.

The effects on the Brewster’s angle minimum with respect to the sea-water surface
height is shown in Figure 4.14. This figure is a different representation of Figure 4.12
and shows the variation of the minimum angle as the height is increased. The Brewster’s
angle minimum is seen to move from roughly 84° to around 78° as the sea-water height
is increased. An explanation is found when the effective permittivity of the sea-water is
considered. The increasing sinusoid amplitude results in more backscatter (Figure 4.15)
and this would act like a change in effective permittivity. The increasing sinusoid height
would reduce the effective permittivity and this would change the Brewster’s angle
correspondingly.

The scattered field dependence on field point location is shown in the next set of
figures. The response for a TM incident plane wave at 85° upon an infinite sea-water

sinusoid with L=O.1016m and h/L=.125 is shown. In Figure 4.16 the backscattered H-

117

field is shown for various locations of the field point along the x-axis. The expected
periodic nature in the x—direction is noted as well the Floquet mode spikes. A variation
of the field point along the z-axis is shown in Figure 4.17. For small values of 2 (near
the surface) the effects of evanescent waves and the number of propagating Floquet
modes are observable.

The forward scattered H-fields for the same cases as above are shown in
Figure 4.18 and Figure 4.19. The Floquet mode locations are once again obvious and the
periodic nature of the fields along the x-direction is also seen. The frequency dependence
is quite interesting but there is little physical interpretation.

Next, the consistency of the two methods (MFIE and Floquet mode-matching) for
calculation of the total scattered field is evaluated. The parameters chosen are
h/L=0.0625, 0,.=85° , and the observation point is x/L=0, z/L=20. The field was
computed for frequencies between 0 and 12 GHz. The two methods are seen to agree
well in spectral magnitude as shown in Figure 4.20. The transient response was
synthesized using the F loquet mode spectral response and is shown in Figure 4.21.
Similar behavior was observed for the perfectly conducting results in chapter 3.

The backscattered field was also computed using both methods. The agreement
between the methods is very good as shown in Figure 4.22. The backscattered field for
the ocean surface behaves similarly to the perfectly conducting case shown in the
previous chapter. The synthesized transient response was also computed. A small portion
of the temporal response is shown in Figure 4.23. The two methods are seen to agree

reasonably well in overall amplitude and waveshape.

118

 

 

 

"LL”: h/L=.070 FMM/RA
...... h/L=.125 F'MM RA
0 o o t 4- h/L=.125 MFI /MOM
h/L=.07O MFIE/MOM
1.2 —
A? :
I>s 1.0 : v’, \ ‘24.."
\ q x” ‘2": '\ '.
>4 - “ I '
I j .
E 0.8 .. \ ,I (,I .2"
.22 - " ,
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E ; a.=80.o ,
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0 0.2 -—
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L ..
CC) _
0.0 illllllllllllllllIrIIIlllfiIiI]
0.0 0.2 0.4 , 0.6 0.8 1.0 1.2
x location (x10 cm)

Figure 4.2 Surface field calculations using Rayleigh-mode-matching method compared
to MFIE PGF method for various h/L ratios.

119

Surface H-Field for TM excitation
L=O.101 6m, h/L=.125,'0.,=85°

a=4 S/m.er=80

V.

 

 

Mag.

O0.0 1,2 2.4 3,-5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.3 Surface H-fields on an infinite sea-water sinusoid for a TM incident plane
wave. Excitation frequency dependence is shown.

120

Surface H—Field for TM excitation at 2 GHz
L=O.1016m.19,,=85°

    
      
    
   

 
 

     
  

 

 

    

  

    

 

    
   
   
      

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Figure 4.4 Surface H-fields on an infinite sea-water sinusoid due to TM excitation at
2

GHz. The dependence on sinusoid interface amplitude (h) is examined.

Surface Field due to TM excitation at 2 GHz
L

=O.1016m,19i=85°
a..." (£r=80, 0:4 S/m)

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Figure 4.5 Surface H—fields on smaller height sea-water sinusoids.

121

Induced Surface Fields for TM excitation at 2 GHz
L=O.1016m, h/L=.125, s,=80, o=4 S/m

3.00

2.50

2.00

Magnhude
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0.50

0.00 ITIIIIIIIIIIIIIIIIIIIIIlIlllllllllllIIIIIIITIIIIII]
0.00 0.20 0.40 0.60 0.80 1.00

x/L Locafion

Figure 4.6 Surface H-fields on an infinite sinusoid for TM excitation at 2 GHz. The
dependence on incidence angle is examined.

122

Induced Surface Fields for TM excitation at 2 GHz
h= .0127m, e,=80, o=4 S/m, 01=8S°

1.00

0.80

Magnfiude
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4;
O

 

— £18.13“
o—e—e—e—e =. m
0°20 mu L=.3048m
M... L=.4064m

 

0.00 IllllIIIIIITIITITIIIITTTIIIIIIIITIIIIIIIIT—Illll1lll

0.00 0.20 0.40 0.50 0.80 100
x/L Locafion

Figure 4.7 Surface H-fields on an infinite sea-water sinusoid due to TM excitation at
2 GHz. The dependence on sea-surface period is shown.

123

Induced Surface Fields for TM excitation at 2 GHz
L=O.1016m, h/L=.125, c,=80, 0i=85°
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Figure 4.8 Surface H-fields on an infinite sinusoid due to TM excitation at 2 GHz.
The effects of conductivity are shown.

124

Induced Surface Fields for TM excitation at 6 GHz
L=O.1016m, h/L=.125, s,=80, 0i=85°

 

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Figure 4.9 Surface H-fields on a infmite sinusoid excited by a 6 GHz TM wave. The
effects of conductivity are shown.

125

Forward Scatter Field Strength
L=O.1016,er=80,0‘=4 S/m

1.0 -
(l9 L————————"‘TTT——————————‘—’———‘—————#——————'fl—‘~
0.8 - —- TE Incident on h/L =.125 sinusoid I

- — - - TM Incident on h/L=.125 Sinusoid I’
0.7 —6— TM Incident upon Flat surface ,

3 0.5
3

 

 

 

50 53 55 59 72 75 78 81 84 87 90
Incidence Angle (13:) degrees

Figure 4.10 Identification of a Brewster angle for TM incidence. Comparison with TB
excitation is shown.

126

1.00 Rayle=i<ihG a prox

 

 

 

E Fre z, a=0
: z/Lq=20, x/L=0
0°80 3 flat

2 W h/L=.001

u ' - - h/L=.01

: ”H... h/L=.05

: H—fl-H h/L=O7 I
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degrees

Figure 4.11 Identification of Brewster’s angle for lossless dielectric (e=80,o=0)
surface.

127

 

 

 

 

1,00 — Forward scattered H-field
I Freq =2 GHz, 0:4
2 flat Z/L=?0,X/L=O
: o-oo-oe h/L=.O7 Rayleigh Approx.
3 ----- h/L=.001
0.80 1 m“ h/L=.O1
: hfiti-fi h L=.1
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65.00 70.00 75.00 80.00 85.00 90.00

degrees

Figure 4.12 Identification of Brewster angle for lossy dielectric surface (e=80,o=4)
for various h/L.

128

TM forward scattered H— Field
L=O.1016m, h/L= .0625,

(£r=80, o=4 S/m) 1"" lull"

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Figure 4.13 Identification of a Brewster angle for TM incidence. The effects f
excitation frequency are shown.

129

           

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Figure 4.14 Identification of a Brewster angle for TM incidence. The effects of sea-

water sinusoid amplitude are examined.

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TM backscattered H-Field at 2 GHz
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Figure 4.15 The backscattered H—field around the Brewster angle for TM incidence.

130

 

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(’0‘ 0.98 0 .\5

 

 

TM backscattered H-field at z=2m
L=O.1016m, h/L=.O625.19a=85°

(e.=80, 0:4 S/m)

 

 

 

Figure 4.16

Backscattered H-field from an infinite sea-water sinusoid. The effects of
field point location are examined.

131

TM backscattered H—field at x=0m
L=O.101 6m, h/L=.0625, 19i=85°

(er=80, a=4 S/m)

0‘3

0.2

l

l

Magnitude
0.1

 

gill

 

Figure 4.17 Backscattered H-field from an infinite sea-water sinusoid. The effects of
field point location are examined.

132

TM forward scattered H—field at z=2m
L=O.1 01 6m, h/L=.0625, 03:850

' (er— 80, 0:4 S/m)

III
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Figure 4.18 Forward scattered H-field from an infinite sea-water sinusoid The effects
of field point location are examined.

133

TM forward scattered H—field at x=Om
L—O.1016m,h/L=.0625, 19i=85°

   
   

   

(er=80, 0'=4 S/m)

 
    

 
 

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Figure 4 19 Forward scattered H-field from an infmite sea water smuso1d The effects
of field point location are exammed

134

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0'80 E Scattered H—f el=d H l'é/Hé)
- L=.1016m./ oi é.=85
: x/L =
‘O 3
—q3 0-60 T: MFIE—PGF
-— ‘ - - - Floquet Mode
“— :
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0.0 . 8.00
Freq (GHz)
Figure 4.20 Comparison of Rayleigh and MFIE-PGF method for computing total
scattered field spectrum from imperfectly conducting surface.
8.0E+008 : o
- e,=ao. o 19=85.O
: a= 4.0 s/m L=.1016 m
: h/L =0.0625
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l— -
.1-2E+OogdI'llIllIIIIIIIIIIIIIIIIIIIIIII IIIIIIIII
0.00 5.00 . 10.00 15. I00 20. I00
Time (ns) -
Figure 4.21 Transient TM response of imperfectly conducting sinusoidal surface.

135

Backscattered H—field (H b“ H,‘)
L=.1016m. h/L=.0625. a‘{= 5
x/L =0. z/L =20

----- Flo uet Mode
MFI —PGF

 

Normalized H—field

 

 

0.00 2.00 4.00 6 O 8.00 10.00 12.00

Freq . (GHz)

Figure 4.22 Comparison of Rayleigh and MFIE PGF methods for computing
backscattered field spectrum for imperfectly conducting surface.

 

 

 

1.0E+008
sr=80.0 13=85.0°
a=4.0 S/m L=.1016 m
h/L =0.0525
:‘3
OT 5.0E+OO7 ,
C 1
(D 1
L
4—1 l
(I) 'I I
_ ‘ ‘ ' “ I
O 0.0E+OOO l ‘ . ‘ ‘ ~ ‘ ' || ‘
C " l l ‘ \’ |
CW 2 q I
.(7) Z I
(1) I
45 3 ----- MFIEéMoM
(D I
L d
—1-OE+008_ 'Ilr" IIIIIIIIII]Ur‘UIIIIFII'II'Ir'ITTUUU

 

 

180.00' 181'.00 182.00 183.00 184.00 185.00
Tlme (ns)

Figure 4.23 Backscattered transient response of imperfectly conducting surface.

136

4.6 Experimental Results

The measured responses of a sinusoidal sea-water wave are more difficult to
obtain experimentally than the PEC sinusoid model. The construction of the sinusoidal
wave, which must once again be finite, is not as straightforward as constructing a PEC
sinusoid. The measurements must be performed using the arch range scattering system,
which was described in chapter 2, because of the nature of the sea-water wave. The wave
was constructed by creating a cavity in a block of expanded polystyrene ("Styrofoam").
The cavity was made in order to hold the water and had one side machined as a sinusoid
(ll crests) with L=O.1016m and h/L=.125. A diagram of this structure is shown in
Figure 4.24.

 

 

L

 

Styrofoam Walls /

  

 

    

H20 Cavity

 

 

 

Styrofoam Walls

 

Figure 4.24 Configuration of sea-water sinusoid for experimental measurements.

The Styrofoam cavity is placed vertically (cavity opening facing up) and filled
with sea-water. The measurements are then performed in the arch range, which can

produce backscatter and forward scatter responses.

137

There are a number of problems associated with the sea-water structure and the
arch-range system. The styrofoam cavity leaked and therefore required a thin plastic
lining. The thin lining had a negligible effect on the scattered fields. The finite depth of
the sea-water holding structure could also be disregarded due to the decay of the
transmitted fields inside the sea-water. There was an additional problem concerning the
antenna patterns of the arch-range system. The antennas (and lenses) create a TEM plane
wave, but it has a rather small footprint. The footprint is gaussian in nature and is
physically smaller than the sea-water sinusoid. The effects of the antenna pattern is
noticeable in the results.

The arch-range system operates in the same fashion as the anechoic chamber
system that was used in the chapter 3 measurements of the PEC model. The frequency-
domain synthesis technique is employed and calibration is performed with a sphere. The
details of this method can be found in chapter 2.

The first set of measurements consisted of backscattered field responses from the
sea-water model and also the PEC for comparison purposes. The wave models were
interrogated with both TE and TM incident waves at roughly 60°.

The theoretical backscattered fields from both a PEC and a sea-water sinusoid are
shown in Figure 4.25 for the TM polarization. In this figure an incidence angle of 60°
was assumed and the surface parameters match the experimental model parameters. The
results indicate that some of the energy is being transmitted into the sea-water. The
spectral content however is nearly identical (other than magnitude). The synthesized
transient response is shown in Figure 4.29 and will be discussed later.

The TE case is shown in Figure 4.26. The parameters and geometry are the same
as the TM case. These results also reveal that the effects of the sea-water seem to only
reduce the magnitude of the backscattered field. The synthesized transient response is

shown in Figure 4.31.

138

When the incidence angle is increased (near grazing) it is expected that the TM
case will exhibit some sort of Brewster’s angle phenomenon. In Figure 4.27 the incidence
angle is increased to 85° there is a noticeable change between the PEC and the sea-water
models as compared to Figure 4.25 (60°). This is due to the Brewster’s angle minimum.
As a confirmation of this the TB case is also considered. In Figure 4.28 the theoretical
backscattered responses for a PEC and sea-water surfaces are compared for TB incidence
at 85°. The results in this figure when compared with the 60° (Figure 4.26) returns
confirms the above assertion.

The measured synthesized transient responses of the PEC and sea-water surfaces
are shown in Figure 4.30 for TM excitation at 60°. The effects of the edges (finite
models) are seen and the limited footprint (gaussian beam) antenna pattern are also
obvious. These results are compared with the theoretical synthesized transient results in
Figure 4.29. Note that the theoretical results assume an infinite surface and an ideal plane
wave. The results match quite well. This agreement solidifies many of the gray areas.
Those gray areas include the effects of the lining, the effect of the finite surface (also
discussed in chapter 3) and the gaussian beam illumination. It should be noted that a 1/8
cosine taper was utilized for all (TM and TE) the synthesized transient results.

The TE case was also measured with a similar degree of success. The synthesized
transient responses for the PEC and sea-water sinusoids are compared in Figure 4.32
(measured) and Figure 4.31 (theoretical). There are increased difficulties with TB
interrogation. The TE incident wave polarization is such that the edges and any wrinkles
in the lining are excited. This injects unwanted pulses and noise, which can be seen in
the returns.

The spectral responses, which are what is actually measured, are also compared.
In Figure 4.33 and Figure 4.34 the theoretical and measured frequency-domain responses
are shown. In Figure 4.33 the theoretical results for both the PEC and sea-water wave

are shown for 60° incidence. The spectra were obtained by time gating the synthesized

139

transient responses and then transforming to the frequency domain. The portion of the
transient response that was time gated is shown in Figure 4.29 and corresponds to an 11
crest wave. The actual measured backscatter response for the PEC and sea-water
sinusoids is shown in Figure 4.34. The agreements between the measured and theoretical
responses is excellent. The measured response does, however, contain the effects of the
gaussian beam illumination. A direct comparison between the measured and theoretical
backscatter response from a sea-water sinusoid is shown in Figure 4.37. The strong
agreement between the measured and theoretical responses for both PEC and sea-water
sinusoids is an excellent indicator of the correctness of both the theory and the
measurement techniques.

The “spectra for the TB case are shown in Figure 4.35 (theoretical) and
Figure 4.36 (measured). As expected from the transient results there is good agreement
between measured and theoretical results. The theoretical results have once again been
time gated before transforming back to the frequency domain. The measured results
exhibit a slight inconsistency in the lower frequencies. This is probably due to the
problems associated with TB excitation explained above.

The arch-range system does allow for forward scatter measurements. The methods
for properly calibrating these measurements has not been conceived. Therefore a rough
method, which does not use a calibration will be implemented here.

The TM spectral and synthesized transient forward-scattered responses for the sea-
water sinusoid (L=O.1016m and h/L= .125) are shown in Figure 4.38. In the upper plot
the spectral returns between 4.8 and 6.2 GHz are shown for near-grazing angles. There
is a wide minimum located between 75° and 80°. The minimum is migrating towards 82°
as the frequency is increased. This minimum agrees with the theoretical prediction for
h/L=.125 as shown in Figure 4.11. The bottom plot is the magnitude of the synthesized

transient forward scatter for the same case as the top plot. This plot also shows the

140

Brewster’s angle effect in the 75° to 82° area. These plots and the methods of the forward

scattered measurement were quite rough and there is a need for further investigation.

4.7 Conclusions

In this chapter the sea model was improved to include the effects of an
imperfectly conducting sea-surface. The results presented in this chapter, therefore,
focused on the differences or likenesses of the transient scattering from PEC and lossy
sinusoidal surfaces. One of the major findings was that there are very few differences in
the scattering phenomenology between the PEC and lossy surfaces. The transient and
spectral responses were nearly identical except for a scale factor. The scale factor was
due to the portion of the incident wave that was transmitted by the lossy surface thereby
reducing the scattered energy. This was shown both theoretically and experimentally.

The Brewster’s-angle phenomenon, however, is a physical effect that was not
predicted with the PEC case. The theoretical and experimental search for a possible
Brewster’s angle for the TM polarization was another main concern of this chapter. The
results were fairly positive. It was shown rather conclusively with the theoretical results
that indeed a Brewster’s-type angle did exist for a lossy sinusoidal surface. The
conclusion was that the angle is reduced (from 83 .6° to 78°) when the surface is changed
from flat to sinusoidal (h/L=.125). The forward scatter at the Brewster’s angle is no
longer zero, as in the flat lossless case, but a sharp minimum occurs for the sinusoidal
(and/or lossy) case. The theoretical findings were substantiated with experimental
measurements. The measurements were performed in the frequency domain using the

arch-range system and provided a rough glimpse at the Brewster’s angle phenomena.

141

Theoretical backscattered magnetic field from an infinite sinusoid
L=.1016m, h/L=.125, 0.=60° TM polarization

2.50

 

Backscattered field from PEC sinusoid
---- Backscattered field from sea water sinusoid

2.00

1 .50

 

     
     
    

  

l l

 

Magnitude Magnetic Field

       
      

  

lllllLllJlllllLllllllllllljlllljlllllllIllllllllll

   

 

 

 

\ ' ”i i
\\ IiHH :1 .
0.50 ‘~- 4i“ HM”...
. . “I“ ' 1 '1
. “l ' t
‘1 I H ‘ I. |
0.00 If UI1IIIIT'I'TII1lrIllTTjIlIllllllll'l'll'tl'll
1.00 3.00 5.00 7.00 9.00

Freq. (GHz)

Figure 4.25 Theoretical backscattered H-field from an infinite sinusoid. Comparison
of PEC and sea-water models for TM excitation at 60°.

142

Theoretical backscattered electric field from an infinite sinusoid
L=.1016m, h/L=.125, 0.=60° TE polarization

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.80 -

_l

‘ Backscattered field from PEC sinusoid

: ---- Backscattered field from sea water sinusoid

j
.0 0.60 :1 I
.79 : 1
LL - I l

-. ' |1
0 - I:
'C _ . '1
*5 ‘ . l i "
20.40- i lil't'I
LIJ : i l f ‘1‘ H I I
a) I .|,' ' I :- l
.0 _ I I”|
D " I I
i": “ 1’ I Hi. |
C : ” | I! I . . l
30.20 — ,z’ I =
2 Z '

: ' ' III .' g‘ I

0.00—ll IIIIIllllIlllllllllllIIIIIIIIIIIIIITIIIIITII
1.00 3.00 5.00 7.00 9.00

 

Freq. (GHz)

Figure 4.26 Theoretical backscattered E—field from an infinite sinusoid. Comparison
of PEC and sea-water models for TB excitation at 60°.

143

Theoretical backscattered magnetic field from an infinite sinusoid
=.1016m, h/L=.125, 01=85° TM polarization

 

 

   

 
   

 
   
     

0.80 -
‘ Backscattered field from PEC sinusoid
: ---- Backscattered field from sea water sinusoid
20m:
.9 ‘
LI- 2
o I
1; _
Q) .1
S) ..
0040?
2' -
a) I
'0 -
:5 .4
:1: q
s. 2 ~ *
III
2 4 .‘!'1'5 0 i
: ”55.1 1 ' ‘fl
3 -‘ 'w‘“ “H”
: 1111 " medhly";" “I |l'a"‘
0.00 llllIITIIIllllllIll]!IIIIITIIIIIIIIIIITIIIIIIll

 

 

 

1.00 3.00 5.00 7.00 9.00
Freq. (GHz)

Figure 4.27 Theoretical backscattered H-field from an infinite sinusoid. Comparison
of PEC and sea-water models for TM excitation at 85°.

144

Theoretical backscattered electric field from an infinite sinusoid
=.1016m, h/L=.125, 0.=85° TE polarization

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.10 '3
: Backscattered field from PEC sinusoid
3 ---- Backscattered field from sea water sinus id
0.08 -:
E 3 I
.9 I 1
LL : l'
.9 0.06 E
L _
+4 _
O _
_‘2 I
LIJ - l
I I
(D _ I
.0 0.04 1
3 -
.4: - I
C I
01 ..
0 :
20%: u
3
0.00- llllllIerlTIIIIIIlIITTTlll11llllIIIIIIITIIIII
1.00 3.00 5.00 7.00 9.00

Freq. (GHz)

Figure 4.28 Theoretical backscattered E-field from an infinite sinusoid. Comparison
of PEC and sea—water models for TB excitation at 85°.

145

Transient backscattered field gue1o TM incident pulse
L=O.1016m. h/L=.125. 0.=60. 8 cosine taper
Theoretical response for infinite 1sinusoid

 

 

 

 

8E+008
. I. I .
' ‘ I :1 I: .
4E+008 H " I 'I, .1 '1‘
l
3 ~ , ~
~13: OE+OOO .
o— \ \ \ \I
E v \ \
I I t ' { 9
q’ 1 I 5' '.' 1
.2 —4E+008 l . .
H I 2
2
d)
a:
—8E+008

—- Sea Water (5,-80. 0-4 S/ mé
----- Perfect electric conductor /(P C)

 

—1E+009 j T I I I I I I I I I I I I I I r F I I f I I I f I
15.100 17.00 19.00 2700

Ti m e ( n s)
Figure 4.29 Theoretical transient backscatter for TM excitation.

Transient backscattered field ue o TM incident pulse
=0 16m. h/L=.125. 0.860.1 8 cosine taper

Experimentally measured

     

 

 

 

 
 

 

 

4‘

 

4E+008
3E+008
I I
D n I
II II ‘
u
2E+008 l1' 1 l .
:l : 1 1|
0 I . l I. ."I. I.
g 1E+008 . l . I . g . .
H ' ' in i i I I" "
s. l
I I ”I II, I , I l I 1 I1 '\ H ‘
E 05+000 l ' '|:" 'l I' l 111 I": I."..’
I | \ I
: :I'll‘. .:I':~.I\.I
0)) : ' N: I“: I u. \I|I " I" ‘v
2% —1E+008 1.: l g l I". t" a
5’ " i i
I

In:
'I-

— Sea Water (e,=80. 0-4 S/ mé
----- Perfect electric conductor /(P C)

 

—4E+008 I I T I I T I I I I I—T I I I I I I r I r I I I I I I I I I I I
8.80 10.80 12.80 14.80

Time (ns)

Figure 4.30 Experimentally measured transient backscatter for TM excitation.

146

Transient backscattered field due to TE incident pulse
L=O.1016m. h/L=.125. 0.=60°.1/8 cosine taper
Theoretical response for infinite sinusoid

SSE-r008

2E+008 i l
1E+008 .
OE+000

—1E+OOB

 

 

 

 

 

 

 

 

 

 

 

 

 

—2E+008 U y i

 

Relative Amplitude

 

 

l
-3E+008 '

-—4E-I-008 —- Sea Water (er—80, a—4 "1%
----- Perfect electric conductor /(P C)

 

300 1 17.00 10.00 ' ' '2'1.'00
Tkne (ns)

Figure 4.31 Theoretical transient backscatter for TB excitation.

Transient backscattered field due1 to TE incident pulse

 

L=O.1016m. h/L= .125. 0.=60°.1/8 cosine taper
Experimentally measured
2E+008
. i
1E+008 . : g g '
I I l I I I 4
I I I I I l
I. ‘ : | I. I . ~
Q) I
g ”‘5',“ ii]. i: "l ‘ II I r .I' ‘ {I , I.
a: OE+OOO l:‘l::‘ :l‘IL ‘ ' \.:'\ \lx'l
a 'I: I. :- mi 1.1; I ‘I ~ I. "
‘E i i. V: : i. g '
'| | l I
v ' I
Q) I I
.2 —1E+OOB
4.)
.2
a)
o:
—2E+008
-— Sea Water (er-80. 0—4 mE
----- Perfect electric conductor /(P C)
-3E+008 I I I I I I I I I I I I I Tfi I Ifi I I I I I I I I I I ‘Ij I I
8.80 10.80 12.80 14.80

Thne (ns)

Figure 4.32 Experimentally measured transient backscatter for TB excitation.

147

Theoretical backscattered magnetic field from an infinite sinusoid

L=.1016 h/L=.125. 0.860 TM polarization
Time gated to include only 11 crests
0.80 -

 

Backscattered field from PEC sinusoid
---- Backscattered field from sea water sinusoid

5.
Freq. (GHz)O

.0
m
o

 

.0
N
O

Magnitude Magnetic PICK]
0
.k
0

 

~
------
.
—-—
o‘-

0.00

 

blillllilllllllIIJJIlllllllllllllllllllll

L.

Figure 4.33 Theoretical time-gated backscatter for TM excitation.

Backscattered field due to TMo incident wave
L=O.1016m. h/L-——.125.G>.=-60°
0.40 Experimentally measured

—-—-—- Sea Water (er.=80 a=4 S/ m%
----- Perfect electric conductor /(P C)

'8‘

0.30

Mogntude
O
N
O

   

 

6.00
Freq (GHz)

Figure 4.34 Experimentally measured backscatter for TM excitation.

148

Theoretical backscattered electric field from an

infinite sinusoid
L—.1016m. h/L—.125. G).-60° TE polarization
Time gated to include only 11 crests
0.25

 

Backscattered field from PEC sinusoid
----— Backscattered field from sea water sinusoid

0.10

 

 

Magnitude Electric Field

 

0.00
1.I0 5.IO 5.IO 7.IO 9.IO
Freq. (GHz)

Figure 4.35 Theoretical time-gated backscatter for TB excitation.

Backscattered field due to TEO incident wave
L=O.1016m. h/L=.125. 9.=-60°

0.12 Experimentally measured
Sea Water (e..==80. a4= mé
'I ----- Perfect electric conductor /(F’ C)

o“’

I
I
I
I
I
I
I
I
I
I
I
I
I
I
I

Magnitude
O
O
O)

.0
O
4;

’.
a"‘
'a

 

  

0.02

 

'
o"-“

   

Freq (Gl—lz)

Figure 4.36 Experimentally measured backscatter for TB excitation.

149

Normalized backscattered magnetic field from a sinusoid
L=.1016m, h/L=.125, 0.=60° TM polarization
Theoretical response time gated to include only 11 crests

1 .20

 

Experimental response
---- Theoretical resposne (infinite)

1 .00

0.80

Magnitude Magnetic Field

 

 

 

 

0.20

 

 

 

0.00

b Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

 

A

5.00 7.00 9.00
Freq. (GHz)

Figure 4.37 Comparison of theoretical and measured TM backscatter from a sea-water
sinusoid.

150

TM forwardscatter from sea water sinusoid

 

5 5.2 5.4 5.6 5.8 6
Freq (GHz)

Transient forwardscatter from sea water sinusoid

i

    

 

91.6 91.9: 92 92.2 ”92.4 92.6
Time(ns)

Figure 4.38 Experimentally measured forward scatter from a finite sea-water sinusoid.
Both transient and spectral results are shown of a number of incidence
angles.

151

Chapter 5

Transient Scattering from Non-Sinusoidal Surfaces

5. 1 Introduction

This chapter will further expand the ocean models. Thus far only a sinusoidal
interface has been considered. Anyone who has seen an ocean (or lake) would agree that
the waves are far from sinusoidal. The sinusoidal surface was merely a first
approximation but it did impart a considerable amount of useful information, which will
be used in this chapter. The surface models that will be considered in this chapter will
make an important step towards a realistic sea-surface. The clutter (or backscatter) from
these non-sinusoidal surfaces greatly varies from the sinusoidal surface. It will provide
an excellent showcase for the attributes of a UWB/ SP radar system.

The theory developed in the previous chapters for the sinusoidal interface is not
limited to sinusoidal interfaces. The Floquet-mode-matching/Rayleigh-hypothesis method
is not limited to a sinusoidal surface, but the matrix entries that determine the modal
amplitudes do have closed form analytical solutions for a sinusoidal surface. To extend
the Floquet-mode-matching method requires numerical integration for the matrix entries.
The Rayleigh hypothesis still places demands on the surface slope of the interface which
limits the usefulness of this method. The integral-equation (IE) formulations are already
numerical in nature and are quite suited for any arbitrary interface. If the interface is
infinite then the use of the periodic Green’s function is still required and constrains the
non-sinusoidal surfaces to be periodic. The IE method does, however, suffer numerically
when the sea-surfaces are highly oscillatory (rough). There are a number of techniques
that have been developed to overcome the numeric difficulties of these rougher sea-

surface realizations [50,51].

152

The knowledge gained in chapters 3 and 4 will be put to use in this chapter. This
would include experimental methods that can now be applied to the non-sinusoidal
surfaces, phenemonoiogy associated with the short pulse scattering from periodic
surfaces, and the effects of sea-water surface as compared to conducting surface. It was
found in chapter 4 that the differences between PEC and imperfectly conducting surfaces
was minimal (only the Brewster’s angle phenomenon). Therefore, the results and

experiments in this chapter will focus on PEC non-sinusoidal surfaces.

5.2 Extensions to theory to accommodate non-sinusoidal surface.
As alluded to in the introduction it was necessary to make a number of numerical
adjustments to theory developed in chapters 3 and 4. These adjustments were needed to

accommodate the non-sinusoidal interface.

5.2. l Floquet mode-matching/Rayleigh Hypothesis

In the FMM/RH method the coefficients for the modal amplitudes are integrals
arising from the application of Galerkin’s method. As an example the TE case for the
PEC interface has (3.17) as the coefficients for the scattered modal amplitudes. Note that
the other integrals for the TM case and the two-media case are quite similar. That
integral is rewritten here

L x
K = f e -jq"pej2 m de (5'1)

0
where p (x) is the functional description of the interface.
For the sinusoidal case this integral had a closed form solution consisting of a
Bessel function. The specialization of p (x) to be non-sinusoidal requires a numerical
integration. This integration can be performed by standard numerical integration routines

(Romberg, trapezoidal rule, etc. .) or by treating the interface as piecewise linear. If the

interface is partitioned to be piecewise linear that does introduce discontinuities in the

153

first derivative of the surface. The Rayleigh hypothesis is therefore violated, but the error

can be minimized to an allowable tolerance by reducing partition lengths.

ZA

    

 

slope -‘m2

 

 

 

 

 

\/

Figure 5. 1 Partition scheme for sea-surface interface.

The partition scheme is identical to that used for the MoM solutions of the
integral equations of chapters 3 and 4, and is shown in Figure 5.1. The interface is
partitioned into N segments where the slope on the q’th segment is m q. With the

piecewise linear interface shape the integral (5.1) can be rewritten as

— N-l I _. ( g (_ ) 127“"3'"):
Kmn : 2] We 14" 7" qu x98 de (5'2)
=0 9

The integral over the q’th partition can be evaluated to yield

_ N'l . . Zuni-n) __ -2!(fl‘l)

K = Elem“ [ 42""? """"‘" — .J—‘L ‘4 (5.3)
mu

=0

IQ

where Ax = xq,1 -xq.

154

The matrix element integrals and the forcing vector integrals are the only changes
that are needed for the non-sinusoidal surface. The scattered fields are found subsequent

to determination of the modal amplitudes (matrix inversion).

5.2.2 Integral Equation Formulation

The integral equation methods are rigorous for non-sinusoidal surfaces and need
no adjustments for any such surfaces. The numerical implementation of the MoM
solution for the infinite (periodic) case is affected by non-sinusoidal surfaces. The rough
surfaces create a problem with the convergence of the periodic Green’s function.

In chapters 3 and 4 the MoM matrix entries were calculated by exchanging the
implied summation of the PGF with the integration over a partition. The integral in this
case can be computed in closed form. The resultant summation was then performed until
a convergence criterion was met. If the surface becomes rough this summation is very
slowly convergent for certain testing points. The convergence, however, can be
accelerated.

The method of acceleration used in this thesis can be found in [44]. In this
method the summation of the Green’s function is performed first and then the integration
over the partition is performed numerically. The alternate version of the PGF used in this

acceleration method [44] is

 

 

G(x,z lx’,z’) = G+ + G_ - £H§2)(k(/(x-x’)2+(z-z’)2) (54)
where
G.(x,z lx’,z’) = flfweflbflwzwdau “2 +21.16114 (5.5)
1i: 0 (1-e"“2efl‘)(/uz—+2j
where

a = k(z-z')

155

b = -b_ = -(x-x’)/L

I

s =kL
t. = -L(k+i3)
t- = -L(k-l3)

k = (Oi/F050

The integrals in (5.5) can be evaluated numerically and converge rapidly for most cases.
For the TB PEC case the MoM matrix entries are found by integrating G over a
partition (see Figure 5.1). The integral (3.34) that utilizes G is of the form
A — f’q’”G(i ‘ |x’ ’)dx’ (5 6)
pq _ p’zp ,z '

III

The MoM matrix entries (5.6) are numerically integrated. The numerical integration over
the q’th partition requires the successive evaluation of G (see eq. (5.4)). This may seem
to be a less effective method of determining the MoM matrix entries, but for the non-
sinusoidal surfaces this method is vastly superior to the original method in chapter 3.

The TM case and the two-media problem require the normal derivative of the
PGF (ea/an ’). If the interface is modelled as piecewise linear (as is the case) the normal
derivative can be performed analytically under the integrals (5.5). This eliminates the
need for numerical differentiation, which can be very noisy. The normal derivative of
the PGF over the q’th partition (See Figure 5.1) is given by

i-fm
93 = fi’-V’G = ‘1 -v’G (5,7)

2
4.
1m,

 

Performing the dot product yields

156

l

‘/l+m:

The partial derivatives are taken inside the implied integration of G. The derivative on

6G 6G
m—

3? “ax’

 

g, = (5.3)

 

the argument of a Hankel function of order zero results in a first order Hankel function.

The partial differentiations of G+ and G_ are

 

 

 

 

 

aG ej(b’s-f:) II -.t(btol)n2 . ( m
2 = e smau u +2} 5.9
I It I -u2 1‘: k‘udu ( )
62 o (l-e e )
aG -j(b:s-‘:) II -.r(b:.l)u2 2 .
t = $f;__f e 2008(aui/u +21>ki(1 + “2) du (5.10)
ax’ 7‘ o (l-e‘" e"*)\/u2+21

The integrals for the partial derivatives of G. are computed numerically. The MoM

matrix entries, given by (3.61) for the TM PEC case, are of the form

qu = [W 606.511.525de (5.11)
x, 6n

still must be numerically integrated. Once again the numerical integration over the

partition length requires successive evaluations of 6G] an ’. It has also been found that

this method is superior to the chapter 3 method for some troublesome cases.

The additional computational efficiency gained with the convergence accelerators,
which were discussed above, allow for the rigorous IE formulation of non-sinusoidal
rough (periodic) surfaces. The results for a number of non-sinusoidal surfaces will be
shown in the following sections. These results will be accompanied by experimental

findings that further confirm the theory developed.

5.3 Non-sinusoidal sea-surface models
The wave models have been chosen to test a number of scattering mechanisms.
There is a Stokes-type wave [52], that has a very large slope (at the crest), and double

sinusoid wave that simulates two-scale roughness, and a realistic aperiodic Donelan-

157

Pierson [53-56] wave. The Donelan-Pierson wave is constructed from actual ocean
statistics and is a function of wind-speed, a periodic swell is superimposed upon the wind
roughened waves. A comparison of the three wave models that will be used extensively
in this chapter is shown in Figure 5.2. In Appendix B is further discussion regarding the

development of these sea-surface models.

5.4 Preliminary numerical results

The effects of the change in surface shape will be considered with the examination
of the induced surface currents. This was the starting point for the chapter 3 and 4
numerical results for the sinusoidal surfaces. The results for these more complex surfaces
can be compared and contrasted with those of the sinusoidal surface. The IE method was
used for the analytic results, due to the nature of sea-surface models.

In Figure 5.3 the induced surface currents on a sinusoid are compared to the
currents on double-sinusoid models. The double sinusoids that are compared have
increasing ripple height. The noticeable feature is the shadowing that occurs for each
individual crest. As the secondary ripple height is increased deep nulls appear in the
troughs. It is important to note the excitation frequency to surface wavelength ratio. If
the frequency had been lower the effects of the ripple would be reduced. This effect will
further examined in the analysis of the scattered fields.

The induced surface current of TE excitation of the Stokes wave (L=O.1778m)
is shown in Figure 5.4 for an incidence angle of 85°. This figure displays the effects of
the large surface slope of the Stokes wave model. At 2 GHz the Stokes wave could be
fairly accurately described by a line source located at the crest of the wave. When the
frequency is increased there is a noticeable change in the current distribution. This
change could be attributable to the onset of multiple scattering or possibly a secondary
scattering center. In the following sections the transient backscattered field response will

further illuminate the possible scattering mechanisms.

158

In Figure 5.5 the currents induced on the double sinusoid model (L=O.1778) are
displayed. The multiple shadow regions are obvious once again for the secondary ripples.
A comparison of the currents induced on the double sinusoid model and the Stokes wave
model is shown in Figure 5.6. The current is seen to be larger in magnitude for the
Stokes wave model. The increased current is due to the surface slope of the stakes wave
model. This effect will lead to enhanced backscattered fields for the larger-sloped waves.

The enhanced backscatter for both the Stokes and double sinusoid wave models
can be observed in the frequency domain backscattered fields. In Figure 5.7 the
theoretical backscattered fields from a sinusoidal and double sinusoidal wave are shown
for an 85° TE incident wave. It is obvious that as the frequency is increased there is a
stronger response from the double sinusoid model. This effect is due to a number of
mechanisms. The surface slopes are greater for the double sinusoid model and also the
secondary ripples are in resonance (Floquet mode). There is a similar enhancement for
the Stokes wave model which is shown in Figure 5.8. For the Stokes wave the
enhancement is due entirely to the increased surface slope. It is interesting to note that
at the lower frequencies the sinusoid model is quite similar to the Stokes and double
sinusoid models. This effect was briefly explained above, and is due to the frequency to
surface wavelength ratio.

The synthesized transient field responses for the above spectral results are shown
in Figure 5.9 and Figure 5.10. The results reflect the backscatter enhancement and also
reveals the strength of the UWB/ SP radar. The increased feature resolution can be seen
with the double sinusoid response (Figure 5.9), where the secondary ripples are easily
observable. The response from the Stokes wave model seems to indicate multiple

scattering centers on the high peaked crest (Figure 5.10).

159

Sea—Surface Models

Double
Sinusoid

Stokes

 

Donelan
Pierson ,

 

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIllllTTle

0.00 0.30 0.60 0.90 1.20 1.50 1.80
x location (m)

Figure 5.2 Comparison of sea-surface models.

160

Induced Surface Current $510 Ghz)
9=85°, L=.1m, TE Pol. P C

 

    
   

 

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- l
‘ A R z='i M 27"74% 02 2010: L)
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t I
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d. 4.15-004 -
o -
2 1
9.05-005 J
-l ,
4.35-004 4
0.00 50.00 100.00 150.00 200.00

partition

Figure 5 .3 Surface current on PEC double sinusoid with increasing ripple height.

161

induced surface current for TE excitation of PEC Stokes wave

 

 

 

  

 

 

01=85°
1.15 -
. Induced current (Ky) Freq=2 GHz
- moo Freq = 8 GHz
0.95 -
1
;} 075-j
v .1
+5 2
a) 0.55 A
L. ..
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3
C9 1
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a) -
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6 -
E 0.15 -
L ‘1
0
Z I
-0.05 -
: ------------- Surface Profile ---------
_O.25 IIIITIITTIIIITTIIII|llllllIIIIITIIIIIIIIFTTIITIII]

 

0.00 0.20 ' 0.40 0.60 0.80 1.00
x/ L Location

Figure 5.4 Surface current induced on the Stokes wave model for a TB incident plane
wave at 85° with frequency of 2 GHz.

162

Induced surface current for TE excitation of PEC double sinusoid
0.=85°, Freq.= 2 GHz

1.15

0.95

 

Induced current (Ky)

----- Surface profile

.0
\l
01

 

.0
01
01

0.35

0.15

Normalized Current (Ky)

-0.05

IIIIJIIIIIIIIIIIIIIIII111114111111]

 

I
.0
N
01

I

 

o
o
o
.0
N
o

0.40 0.60 0.80 1.00
x / L Location

Figure 5.5 Surface current induced on the double sinusoid wave model for a TB
incident plane wave at 85° with frequency of 2 GHz.

163

Induced surface current for TE excitation of PEC wave models
01:85", Freq.= 2 GHz

 

 

 

SEE—003 E Double sinusoid wave
: ----- Stokes wave
2.5E—003 E
a ,.
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0.0E+OOO—TIIIIIIII]IIIIIIIIIIIIIIIIIIIIIITIIIIIIIIIIIIIIIII
0.00 0.20 0.40 0.60 0.80 1.00

x/L Location

Figure 5.6 Comparison of induce surface currents on the Stokes and double sinusoid
wave models.

164

0.16

0.12

0.04

0.00

Figure 5.7

Spectral Response of TE Backscatter from PEC surface
O= 85°, L=. 3m,
Field Point: x/L=O, z/L=20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'1
2 - .1 2 L +. 3 6 L
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1 fl
1
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_ L . :H 'i l
I

.40 2.40 4.40

Freq (GHz)
Spectrum of backscatter for TB incident wave at 85°, L= .3m at field

point x/L=0, z/L=20

165

Spectral Response of TE Backscatter from PEC surface
0=85°, L=.3m,
Field Point : x/L=0, z/L=2O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.20 :
: z=Stokes wave
: ---- z=.1cos(21rx/L)
0.15 —:
x Z
3 I
L” 0.10 '2
0‘ 2
C ..
2 :
0.05 f
i n
: \Inul
0.00IITIIIIIIjI'IIIIIIIIFIIIITTIIIj
0.40 2.40 4.40

Freq (GHz)

Figure 5.8 Spectrum of backscatter for TB incident wave at 85°, L= .3m at field
point x/L=0, zlL=20

166

Transient Response of TE Backscatter from PEC surface
0=85°, L=.3m,
Field Point : x/L=O, z/L=20

.- - - :zzl£2:2$39*A§>+-3C°s<6wt>>

l

1.3E+007

 

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lllllllllllllllllllllllllllljlllllllllJlIIIIIIIIIJ

 

 

 

 

 

Relative EbCk

 

-7.0E+006

 

 

 

 

 

 

 

 

 

 

 

"I-ZE'I'W I I I I I I I I I I T I I I I I I I I I I I I I I I I I I I
90.00 92.00 . 94.00 95.00
TIme (ns)

Figure 5.9 Transient backscatter from PEC surface with TB incident wave at 85°,
L=0.3m, x/L=0, zlL=20

167

Transient Response of TE Backscatter from PEC surface
0=85°, L=.3m,
Field Point : x/L=0, z/L=20

    

 

 

 

 

 

 

 

3.090071
1
1.09007:
x -
8 ‘ I
“J ' '.
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(D ' I '
> - I l
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01 -4 I I!
—l i I
I
‘ z=.1 cos 21rx L +.3co.s 61rx L
: ---- z=StS>kes( Wav/e) ( / ))
m3'0E4-m7mTIIIIIIIIlrIIIIIIIIIIIIIIIIIT‘]
90.00 92.00 94.00 96.00
Time (ns)

Figure 5 .10 Transient backscatter from PEC surface with TB incident wave at 85°,
L=O.3m, x/L=0, zlL=20

168

5.5 Scattering from PEC surfaces for TM excitation

In general there was very good agreement between the theoretical and measured
transient backscattered fields from all three sea-surface models. It will be shown later
that the theoretical backscattered fields from the infinite surface are in excellent
agreement with the measured results, and in fact results are better than theoretical ones
with the finite surface. This unexpected result can be attributed to the far-field
approximation made in calculating the backscattered field from the finite surface. This
approximation could have been lifted at greater numerical costs. The theory for the
infinite surface did not contain any such approximations and therefore could better model
the cylindrical scattered waves and the relative path length variation along the surface,
which manifested itself in the spreading of the Floquet mode spikes.

The experimental measurements consist of true time-domain interrogation with a
high voltage PPL pulse generator, and of synthesized frequency domain excitation (via
a HP 87208 Network Analyzer). The frequency domain synthesis method allows for a
much wider bandwidth and a larger dynamic range, resulting in higher quality
measurements. These experimental measurements will be examined in the time and
frequency domains and will be compared to theoretical models previously developed.

There are a number of scattering mechanisms present in the non-sinusoidal models
that will be closely examined. These mechanisms include the effect of large surface
slopes, rougher two-scale surfaces, the aperiodicity of the Donelan-Pierson wave and the
onset of multiple scattering. It is hoped that the combination of experimental and

theoretical results will encourage the use of UWB/SP radar.

5.5. 1 Experimental Results for TM Excitation

The measurements were performed in the anechoic chamber at Michigan State
University (MSU). The measurement techniques were described in chapter 2. Specific
applications of the frequency domain synthesis technique were provided in chapters 3 and

4. The frequency domain synthesis measurements of the non-sinusoidal surfaces were

169

performed in an identical manner as in chapters 3 and 4. The true time-domain short-
pulse measurements were also performed in the chamber as described in chapter 2.

The wave models for the experimental measurements were constructed by
adhering aluminum foil to precision machined polystyrene. The physical size of the three
wave models is 1.78m by 0.91m. The Stokes wave consisted of 10 periods with period
length (L) of 17.8cm and height of 5.5cm.

The Stokes wave is cycloidal in nature with a large surface slope at the crest of
the wave. It is the large slope that is of primary interest; this slope is expected to
enhance the backscattered field, and create multiple scattering within a trough.

The double sinusoid has 10 periods (L= 17.8cm) with a shorter wavelength
(L=3.6cm) superimposed. The height of the large swell is 5.5cm, while the smaller
wave is 1.1cm. The double sinusoid model simulates a two-scale roughness ocean model;
the finer structure should be resolved by the UWB/SP radar.

The Donelan-Pierson wave was constructed with 10 swells having an aperiodic
wind-driven wave superimposed, therefore the DP surface is not periodic. The Donelan-
Pierson model is the most realistic, and is constructed from actual ocean statistics, where
the wind was assumed to be 10 m/s. This model contains two-scale roughness and large
surface slopes, which will provide a stringent test of the UWB/SP radar.

The first results to be considered (somewhat qualitative) investigate the angular
dependence of the scattered fields, which were measured via the frequency-domain
synthesis technique. The normalized synthesized transient TM backscatter from the
Stokes wave model is shown in the bottom plot of Figure 5.11 for incidence angles
ranging from near grazing (6,. = 90") to normal incidence (6, = 0"). Normalization was
required to enhance the near grazing backscatter, which contained far less energy than
the normally incident backscatter. Observable in the bottom plot are the time—domain
backscatter from each individual peak of the wave, and the change in the two-way transit
time versus the incidence angle. The finer structure, such as multiple scattering, are hard

I.

170

to discern in this plot. The remaining transient results provide a better glimpse at
individual incidence angles.

Accompanying the transient results are the corresponding spectral returns (actual
measurements). In the top plot of Figure 5.11 the Floquet-mode spikes are evident, and
the interference patterns for higher frequencies can be seen. Of great interest is the
location and movement of the Floquet-mode spikes as the incidence angle is rotated from
near grazing to normal incidence. This phenomenon is expected and attributable to the
periodic nature of the surface, as discussed in previous chapters. The clarity of the modes
indicate the high quality of both the measurement system, and the sea surface
construction.

There are similar results for the other wave models. In Figure 5.12 the
experimentally measured response from double sinusoid wave is shown. The spectral
returns (upper plot) look remarkable similar to the Stokes wave, this is due to periodicity
of the wave. The differences between the two wave models can be seen the transient
response, which is of primary interest. The synthesized TM transient response, shown
in the bottom plot, reveals the non-shadowed ripples on the double sinusoid model.

The response from the Donelan Pierson surface is shown in Figure 5.13 and
Figure 5.14. The Donelan Pierson model is aperiodic, therefore it is not expected to
produce the Floquet mode spikes. Thus, the spectral returns (upper plots) are quite
amazing, and actually reveal a Floquet mode type pattern for the lower frequencies. This
is due to the periodic swell that what superimposed upon the wind-driven surface waves.
The transient results (lower plots) also show the effects of the aperiodicity. Due to the
variance in the wave crest heights there is a strong shadowing effect evident at the near
grazing angles. In Figure 5.2 the Donelan Pierson wave is shown and the highest peaks
in the wave are quite visible in the transient response. The smaller wave crests are only

visible for the angles nearing normal incidence.

171

The TM scattering from the three waves at an incidence angle (0,.) of 80°
are compared in Figure 5.15, where the measured frequency-domain results are
compared. The large spikes in the spectral returns are due to periodicity of the surfaces
and are called Floquet-mode spikes. These spikes are dependent upon the incidence angle
and the spatial period of the sea surface. Note that the Donelan Pierson surface lacks
these highly defined spikes due to the aperiodic nature of the surface. The overall
strength of the returns at this incidence angle are greatest for the Stokes wave, because
of the large slope near the crest of the wave.

The short-pulse transient response can be obtained by performing an IFFT on the
frequency-domain responses. The short-pulse interrogation of the three wave models for
the incidence angle of 70° is shown in Figure 5.16. These waveforms represent the
returns an actual UWB/ SP radar would produce if it was endowed with the bandwidth
of the corresponding spectral responses (Figure 5.15). The differences between the three
wave models is quite evident in the transient response. The double-sinusoid and Stokes
wave models produce a periodic set of return pulses, while the response from the
Donelan-Pierson model is erratic. The additional surface features of the double-sinusoid
and the Donelan-Pierson models produce the multiple-scattering-center return pulses.
These transient responses were weighted with a % cosine taper before transformation.

A further investigation continues in Figure 5.17 and Figure 5.18. The TM
backscattered response for two incidence angles is shown for the Stokes (Figure 5.17)
and double sinusoid (Figure 5.18) wave models. When the incidence angle is lowered to
60° there are visible changes in the spectral responses. The well—defmed single Floquet
mode spike is no longer present at the higher frequencies, where the spectra become
erratic. This indicates that small structures are nearing resonance, or that multiple
scattering is occurring.

The corresponding synthesized transient responses are shown in the following

figures. The differences in the responses between the three wave models is of primary

172

interest for describing the underlying physics. The TM transient response from the Stokes
wave model for 80° and 60° incidence is shown in Figure 5.19. The periodic return of
pulses is expected because of periodic nature of the wave. The effects of multiple
scattering, which was predicted from the spectral returns, is more pronounced for the 60°
case. This is due to the erratic high frequency content observable in the frequency
domain.

The transient response for the double sinusoid model is shown in Figure 5.20.
The major difference (with the Stokes wave) is the additional return pulses from the
smaller scale sinusoids. This was also predicted by the spectral returns. Another
interesting feature is the amplitude differences between the two incidence angles. There
is a much greater difference for the double sinusoid case. The cause for this effect is the
surface slope. The Stokes wave has a much larger surface slope, therefore the near
grazing angles still produce a large backscatter.

The response from the Donelan-Pierson model shows the shadowing effect
produced by the first crest of the surface as indicated in Figure 5.21. The first crest is
by far the largest (see Figure 5.2 and note that the wave was actually measured from the
right) and creates the easily observable shadowing of the remaining wave crests. Note
that the transient responses were normalized and the 60" incidence response was actually

much larger (as expected) than the 80" return.

173

TM backscatter from PEC Stokes wave

.-Q .7

    

Freq (GHz)
TM Transient backscatter from PEC Stokes wave
80 g; . ‘ . . .

   

TIme (ns)

Figure 5.11 Angular dependence of TM scattered fields of the Stokes wave model.

174

TM backscatter from PEC double sinusoid wave

 

129

4
Freq (GHz)
TM Transient backscatter from PEC double sinusoid wave

    

9 10 12.124 16 18
Tlme(ns)

Figure 5.12 Angular dependence of TM scattered fields of the double sinusoid wave
model.

175

TM backscatter from PEC DP wave

    

4 5 6 7;
Freq (GHz)

TM Transient backscatter from PEC DP wave

1 2; 3

   

8 10 1270114 16 18
Tlme(ns)

Figure 5.13 Angular dependence of TM scattered fields of the Donelan Pierson wave
model.

176

 

' ‘ : -' as
1 2 3 4 5 6 7
Freq (GHz)

TM Transient backscatter from PEC DP wave
0:; "If,

   

Tlme (ns)

Figure 5.14 Angular dependence of TM scattered fields of the Donelan Pierson wave
model.

177

TM backscattered field for 70° incidence
Experimentally measured.

Stokes

 

 

Double
Sinusoid

 

 

Donelan
Pierson

 

. i
[IIIIIIIII IIIIIIIIIIIIIIIIIII

1.00 6.00 1 1.00 16.00 '
Freq. (GHz)

Figure 5. 15 Experimental TM scattering from various wave models for an incidence
angle of 70°.

178

Synthesized TM transient backscatter for 70° incidence
Experimentally measured.

Stokes

Double
Sinusoid

 

Donelan
Pierson

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIllIIIIIIrr

7.00 9.00 11.00 13.00 15.00 17.00
Time (ns)

Figure 5.16 Experimental TM transient scattering from various wave models for an
incidence angle of 70°. Synthesized with a 1/8 cosine taper.

179

TM backscattered H—field from Stokes Wave
Experimentally measured.

 

 

91:800

 

 

 

 

 

 

e,=60° ' L

 

 

 

 

 

 

 

 

 

IIIIIIIIIIIIIIIIIIIr'IIIIIIIIIIIIIII

2.00 5.00 10.00 14.00
Freq. (GHz)

Figure 5. 17 TM backscattered fields from the Stokes wave model. Experimentally
measured response at 60° and 80°.

180

TM backscattered H—field from Double Sinusoid
Experimentally measured.

 

 

01=80° N

 

 

 

0i=60°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IIIIIIIIIIIIIIIIITTIIIIIIIIlllillIIEI

2.00 6.00 10.00 14.00
Freq. (GHz)

Figure 5.18 TM backscattered fields from the double sinusoid wave model.
Experimentally measured response at 60° and 80".

181

TM transient backscattered H—field from Stokes Wave
Experimentally measured. (synthesized, normalized)

 

01:80o

 

 

 

61:600

 

IIIIIIITI]IIIIIIIIIIIIIITIIIIITIIIIIIII]

7.00 9.00 1 1.00 13.00 15.00
Freq. (GHz)

 

Figure 5. 19 TM synthesized transient backscattered fields from the Stokes wave
model. Experimentally measured response at 60° and 80°. Spectral returns
were weighted with 1/8 cosine taper before IFFT.

182

TM transient backscattered H—field from Double sinusoid
Experimentally measured. (synthesized, normalized)

9|=80°

91:600 1 1 I

 

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIITIITI1111

7.00 9.00 1 1.00 13.00 15.00
Freq. (GHz)

Figure 5.20 TM synthesized transient backscattered fields from the double sinusoid
wave model. Experimentally measured response at 60° and 80°. Spectral
retums were weighted with 1/8 cosine taper before IFFT.

183

TM transient backscattered H-fieid from Donelan Pierson Model
Experimentally measured. (synthesized, normalized)

91:800

 

0,=60°

 

 

IIIIIIIIIIIIIIIIIIIIIITITIIII[IIIIIIII—I

6.00 8.00 10.00 12.00 14.00
Freq. (GHz)

Figure 5.21 TM synthesized transient backscattered fields from the Donelan Pierson
wave model. Experimentally measured response at 60° and 80°. Spectral
returns were weighted with 1/8 cosine taper before IFFT.

184

5.5. la TM Measurements performed in time domain

The experimental results up to this point have all been obtained through the
frequency domain synthesis technique. An actual UWB/ SP radar system would not
operate in this fashion. Instead the UWB/ SP radar would transmit and receive and an
actual short-pulse. This near instantaneous interrogation method is the primary reason the
ocean surface models can be treated as stationary. Therefore it is important to perform
some true short-pulse measurements of the sea-surface models. The importance lies not
only in establishing the feasibility of an UWB/SP radar system, but also as another
verification of both the theoretical and the frequency—domain synthesis results.

The first set of results are the TM transient backscatter responses of the three
wave models. The responses acquired via the frequency domain synthesis technique are
composed of roughly the same effective spectral bandwidth as the true time-domain
system. The true time-domain system produces a 1/2 ns pulse or a roughly 2 GHz
bandwidth (baseband). In Figure 5.22, Figure 5.23, and Figure 5.24 a comparison of the
two measurements techniques is shown for the Stokes, double sinusoid, and Donelan
Pierson models, respectively. The agreement between the two methods is quite
remarkable. The effect of the reduction in bandwidth is noticeable in the reduced
resolution. It is difficult to determine the differences between the Stokes and double
sinusoid transient responses. This is a direct consequence of the effective bandwidth of
the measurement systems. The lower frequencies do not excite the small-scale features
of the double sinusoid model and therefore only the fundamental frequency is seen. The
Donelan Pierson response does differ from the others, as expected, because of the
aperiodicity.

The spectral responses for the three models are considered next. The true time-
domain measurements had to be transformed to the frequency domain to be compared
with the actual measurements obtained by the frequency domain synthesis system. In

Figure 5 .25 the spectra generated by the two measurements techniques are compared.

185

There is an excellent agreement until roughly 2 GHz. The spectral limitations of the true
time-domain measurement system can be seen beyond 2 GHz.

A comparison of the TM spectral backscatter from the Stokes and the double
sinusoid in Figure 5.26 support the conclusion made above. That is, over this frequency
band the two wave models are quite similar and therefore produce similar transient
responses as shown above. This fact will also be shown in the theoretical results in the
following section.

The favorable results of this section strengthen confidence in the frequency
domain synthesis technique and have also illuminated some the phenomenology associated
with transient scattering. The performance of the true time—domain system, which

simulates an UWB/SP radar system, was also remarkable given the spectral limitations.

Transient TM Backscatter from the Stokes Wave Model
Experimental. O.=60°, 0.4-2.96 GHz

F1
+
O
0
a1
i

I

l

5E+OO7

OE+OOO

 

 

 

 

 

 

 

 

 

—5E+OO7

 

 

 

 

 

-1E+008

 

 

Relative Amplitude

 

Frequency Domain Synthesis
----- Transient Measurement

—2E+008

llllllllllJlllLlLLlJllllllllllllllJlLLlllllllllllllllllLlll
.
c.—
c
t
'
sg

 

 

—2E+008IlllIIIIllIllTlllll'I I
5.00 10.00 15 00 20.00

Time (ns) .

Figure 5 .22 Transient TM backscatter response for the Stokes wave model. A
comparison of frequency-domain synthesis and time-domain techniques.

186

Transient TM Backscatter from the Double Sinusoid

Experimental, 0.=60°, 0.4-2.96 GHz

 

 

 

 

 

 

 

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8
O
O
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i 10
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Time (ns)

187

A comparison of frequency-domain synthesis technique and time—domain

Figure 5.23 Transient TM backscatter response for the double sinusoid wave model.
technique.

2E+008

1E+008

5E+007

0E+000

IIIIIIIIlIIIIIIIIIiIIIIIIIIIlIIIIIIIIII

—5E+007E

Relative Amplitude

—1E+008

-2E+008

rrrrrrrrlIIIIIIIIIirrIIIIr

—2E+008‘

Transient TM Backscatter from the Donelan Pierson Model
Experimental, 0.=60°, 0.4—2.96 GHz

’

‘

\

 

--

 

 

 

-—
‘—:
-_ .
_-

 

Frequency Domain Synthesis
----- Transient Measurement

 

IIIIrIIII IIIIIITIIIIIIIIITII

 

5.00 10.00 15.00 20.00

Time (ns)

Figure 5.24 Transient TM backscatter response for the Donelan Pierson wave model.
A comparison of frequency-domain synthesis technique and time-domain
technique.

188

TM Backscatter from the Stokes Wave Model
Experimental, 01:60°, 0.4— 2.96 GHz

 

 

 

1.0 —_
Z — Frequency Domain Synthesis
: ----- Transient Measurement
0.8—j
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i , ’ I ' '
—l Ivi ‘
0.0 lll-llllllllIIlllIIIIIIIIIIIIIIIIIIIIIIIIIlllllllilll
0.40 0.90 1.40 1.90 2.40 2.90
Freq (GHz)

Figure 5.25 Spectrum of TM backscatter response for the Stokes wave model. A
comparison of frequency-domain synthesis technique and time-domain
technique.

189

TM Backscatter
Experimental, 0.=60°, 0.4-2.96 GHz

 

  

 

 

 

 

1.0:
i — Stokes Wave Model
I ----- Double Sinusmd
0.8-3
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000 4IIIIIIrIIII IIIIIIIIIIIIIIIIyIIIIIIIIIIII[IIIIIIIIIIII
0.40 0.90 1.40 1.90 2.40 2.90

Freq (GHz)

Figure 5 .26 Spectra of TM backscatter response for the Stokes and double sinusoid
wave models. Measurements were performed using the time—domain

technique.

190

5.5.2 Theoretical Response for TM Excitation

The theoretical TM backscattered fields from the wave models that have been
experimentally measured will be briefly examined. The comparison and validation of the
theoretical and measured responses are included in the following section. The theoretical
results were obtained using the IE formulation supplemented with the PGF acceleration
technique described in section 5.2.2. The results are for infinite (periodic) PEC surfaces
and therefore only the Stokes and double sinusoid waves are considered in this section.

In Figure 5.27 and Figure 5.28 are the spectral and synthesized transient
responses for TM excitation of the Stokes wave. The figures compare the effect of
incidence angle (0,.) upon the backscattered fields. The spectral returns (Figure 5.27)
illustrate the change in Floquet mode location with incidence angle. The synthesized
transient response (Figure 5.28) produces many of the same effects that were observable
in the experimental results. The periodic return of pulses from the single crest of the
Stokes wave appears along with an additional multiple scatter as seen in the trough
region. The effects of incidence angle on the transient response is difficult to discern.

The response from the double sinusoid model is shown in Figure 5.29 and
Figure 5.30. The strength of the return is greater for the 60° aspect as seen in the
spectral responses of Figure 5.29. This phenomenon has been explained in previous
sections and is intuitively expected. The synthesized transient responses reveal the
periodic nature of the wave and the resolution capabilities of the narrow interrogation
pulse. The transient results are shown in Figure 5.30. The shadowing effect is also
noticeable for the 85° incidence; this effect was seen the experimental results and will be
compared in the following section.

The spectral returns for both the Stokes and double-sinusoid wave models have
amazing similarities. This was also found with the experimental results and is therefore
substantiated by these theoretical findings. The periodicity of the wave models is what

causes the remarkable similarities in the spectra.

191

TM backscattered H—field for Stokes Wave
Infinite Theory. Field point ; x/L=0. z/L=1O

— 191=600
- - - - 191:850

 

 

‘ 31‘" M111
\1 \. ”lit": ‘1‘» 11'9", v J‘H‘“ 0" ‘1': fiiimflrl" ‘ ‘,il‘“‘i" v
j I

0.801.44 2.08 2.72 3:56 4 0.42.1564 5.28 5.92 6.56 7.20
Freq.

Figure 5 .27 TM backscattered magnetic field for the Stokes wave model. Comparison
theoretical responses for 60° and 85° incidence.

Transient TM backscattered H—fieid for Stokes Wave
Normalized infinite Theory, Field point : x/L=O, z/L=1O

 

 

 

 

 

 

 

 

 

191-60°
---- 1918850
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I [I‘ g g I I I
' i: ‘l ', fl ‘1 i' "I
‘I I' '1 I' ‘I I' I
l 1 IL L I I a; I l l l l I
18 19 2O 21 25 24 25 26

Time (ns)

Figure 5.28 Synthesized transient TM backscattered magnetic field for Stokes wave.

192

TM backscattered H— field for double sinusoid model
Infinite Theory, Field point; x/L= O. z/L=1O

 

1.91:600
- — - - 191=85°

 

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Figure 5.29 TM backscattered magnetic field for the double sinusoid wave model.
Comparison theoretical responses for 60° and 85° incidence.

Transient TM backscattered H— field for double sinusoid model
Normalizedlnfinite Theory, Field point: x/L= O. z/L= 10
,=

———— o1= 85°

 

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‘II
II” I"; ll! I‘li‘ll|
\l ”1’ ll ‘l

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18 19 2O 21 22 25 24 25 26
Time (ns)

Figure 5.30 Synthesized transient TM backscattered magnetic field for double sinusoid.

193

5.5.3 Comparison/Evaluation of TM results

The theoretical and experimental results for TM excitation shown in the previous
sections provide a qualitative comparison. The basic scattering mechanisms are apparent
in both the theoretical and experimental results and give hope for a positive comparison.
There are a number of difficulties in performing a comparison between the theoretical
and experimental responses. These difficulties have been addressed in the previous
chapters and will only be briefly listed again. The experimental models are of course
finite and comparison to theory for the infinite surfaces must recognize this fact. The
incident wave in the experimental system is not an ideal plane wave. The incidence angle
is also affected by the placement of the wave model in the anechoic chamber which
creates additional uncertainty. The wave model itself has small imperfections and cannot
be perfectly periodic.

Nonetheless, even with all the possible error contributions, the experimental and
theoretical results are seen to match quite well. In Figure 5.31 the backscattered field
spectra for measured and theoretical techniques are compared. The theoretical responses
were found using the theory for an infinite surface. The agreement is readily evident and
the spectra are seen to be dominated by the Floquet-mode spikes. Note that the
theoretical response was obtained by transforming the original spectrum to the time
domain and then time gating the transient response to the equivalent finite surface length.
The time-gated transient response was then transformed back to the frequency domain.

The synthesized transient response for the Stokes wave model is shown in
Figure 5.32. The agreement is once again quite amazing considering the possible
detractors. There is similar agreement for the scattered fields from the double-sinusoid
wave model. The spectral and synthesized transient responses of the double-sinusoid
model for TM excitation are shown in Figure 5.33 and Figure 5.34. The overall strong
agreement once again strengthens the confidence in both the theoretical and experimental

methods.

194

TM backscattered H-field at 60°
Stokes Wave Model

 

 

  

 

1.00 :
I —— Measured . .
: ----- Theoretical (IannIte)
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I."
0.00[IIIITIIIIITFTI[TWII'IIIITIIIITIII
0.80 2.80 4.80 6.80
Freq (GHz)
Figure 5 .31 TM backscattered magnetic field for the Stokes wave model. Comparison

of theoretical and measured responses for 60° incidence.

195

TM synthesized transient backscatter at 60°
Stokes Wave model (1/8 cosine taper)

2E+009

— Measured
----- Theoretical (Infinite)

1 E+009

5E+008

   
 

-----_‘
—
--—-—--

-,

OE+OOO

 

 

 

 

 

\
———-——-——
~“

Relative Amplitude

 

n
—2E+OOU|ITTFIIIIIIIIIIIIIII IIIIIIIIIIIITIIIII

7.40 9.40 1 1 .'40 13.40
Time (ns)

Figure 5 .32 Synthesized transient TM backscattered magnetic field for the Stokes wave
model. Comparison of theoretical and measured responses for 60°
incidence.

196

TM backscattered H—field at 60°
Double sinusoid model

 

 

 

  

 

 

 

 

1.00 1
2 1 Measured
Z ----- Theoretical (Infinite)
1
0.80 '3
E I
. II
_ 1|
I " I 11
m 0.60 : :I 1. .1 1
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g : I ‘ 1'1" I 5‘ I'l'l
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0.20 1 l .1 . 1 l '1: ','. l’" '.
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I I 5' I ll - 1
' 1.11'1’1 ,. '1" 5:: .. 1:. 1 1: I;
‘ I. '1' 1 11 1'1
0.00 IIIIIIIIIIIIIIIIIIII'IIIIIIjlllllvl
0.80 2 80 4 80 6 80

Figure 5.33 TM backscattered magnetic field for the double sinusoid wave model.
Comparison of theoretical and measured responses for 60° incidence.

197

2E+009

1 E+009

5E+008

OE+000

—5E+008

Relative Amplitude

—1 E+009

—2E+009
7.40

TM s nthesized transient backscatter at 60°
Doube sinusoid model (1/8 cosine taper)

 

Measured
1 ----- Theoretical (Infinite)

  
 

-
‘
-—--——--—
-

 

 

 

 

 

 

 

 

a
":

8.40 9.40 10.40 1 1.40 12.40
Time (ns)

Figure 5.34 Synthesized transient TM backscattered magnetic field for the double

sinusoid wave model. Comparison of theoretical and measured responses
for 60° incidence.

198

5.6 Scattering from PEC surfaces for TE excitation.

In section 5.5 the scattered fields for TM pulse interrogation of the Stokes,
double-sinusoid and Donelan-Pierson wave models were examined. This section contains
the analogous TE excitation results. Since the wave models are PEC the scattered fields
for both TE and TM excitations are similar. Therefore only brief explanations are needed
in the discussions of the scattering mechanisms. The differences between the polarization

states will be highlighted.

5.6. 1 Experimental Results for TE Excitation.

The frequency and synthesized transient responses for the three wave models are
shown in Figure 5.35 to Figure 5.38. The upper plot is the spectrum for various
incidence angles and the lower plot is the corresponding transient response. These plots
exhibit the same qualities and scattering mechanisms that were shown for the case of TM
excitation in section 5.5.1. These results indicate the quality of the measurement system
and reveal the resolution capabilities.

In Figure 5.39 is a comparison of the experimentally measured synthesized
transient responses from the Stokes and double-sinusoid wave models. The near grazing
angles are shown. The additional features (wave crests) are clearly visible in the double-
sinusoid case. The double sinusoid exhibits considerably more structure (dark vertical
bands) due to the two-scale roughness. However, the Stokes wave model has a more
pronounced multiple scatter, which is evident in the trough regions. The trough regions
of the Stokes wave are located between the sets of double dark vertical lines. There are
two dark vertical lines for every crest, which represent the specular backscattered pulse.
These two observations are intuitively satisfying and will be shown later to agree with
the numerical results.

The spectra of the measured backscattered fields for the three wave models at a
single incidence angle of 70° are shown in Figure 5.40. The Floquet-mode spikes are

once again obvious for the two periodic wave models, while the Donelan-Pierson model

199

does not exhibit the strong spikes. The backscattered field from the Stokes wave model
contains the most energy and this is most likely due to the high surface slope of that
wave model.

In Figure 5.41 the corresponding synthesized transient results are compared. The
time-domain signals are obtained by performing an Inverse Fast Fourier Transform
(IFFT) after a Gaussian Modulated Cosine (GMC) window has been applied in the
frequency domain. Only a portion of the transient results are shown in order to enhance
the resolution of the plot. The plots reveal the differences between the three models. The
Stokes wave has the largest return pulses (due to the large slope) and a multiple scatter
can be observed in the trough. The multiple scatter spectral content is in the higher
frequencies and could be enhanced in the time-domain with a different weighting function
that emphasizes the higher frequencies.

The synthesized transient backscatter from the double sinusoid wave exhibits the
scattering center reflections from all of the non-shadowed crests. There is also a multiple
scatter that could be enhanced with a different weighting function.

The Donelan-Pierson wave has the smallest return amplitude; this is due to the
orientation of the wave. The biggest crest was placed at the front edge, which effectively
shadows the remaining crests. The return from the first hump is not shown, but is much
larger than the other crests. The transient backscatter does indicate locations of scattering

centers, which do match the physical surface shape.

200

TE backscatter from Stokes wave

 

12 3‘ 5 6 7

4
Freq (GHz)
Transient backscatter from Stokes wave

 

'I'Ime (ns)

Figure 5.35 Angular dependence of TE scattered fields of the Stokes wave model.

201

TE backscatter from PEC double sinusoid wave

 

1 2 '3’ 4 H 5 6 7
Freq (GHz)
TE Transient backscatter from PEC double sinusoid wave

 

TIme (ns)

Figure 5.36 Angular dependence of TE scattered field spectrum of the double-sinusoid
wave model.

202

 

4 , ,. 5.. 6..
Freq (GHz)

TE Transient backscatter from PEC DP wave

   

8 10 12 "‘14 16 18
TIme(ns)

Figure 5.37 Angular dependence of TE scattered fields of the Donelan Pierson wave
model.

203

backscatter from PEC DP wave

    
  

1
-20
0'8
8 40 .x 0.6
2
I? -60 I 0.4
0.2
-80
1 2 3 4 5 6 7
Freq (GHz)

Transient backscatter from PEC DP wave

 

10 12 14 16 18
TIme(ns)

Figure 5.38 Angular dependence of TE scattered fields of the Donelan Pierson wave
model.

204

Transient backscatter from double sinusoid

 

 

9 :10 '11 H 12
TIme(ns)

Transient backscatter from Stokes wave

Degrees

‘5
<1

8 9 10 11 12
TIme(ns)

 

 

 

 

Figure 5.39 Comparison of synthesized transient scattering from the Stokes wave and
double sinusoid wave for near grazing incidence angles.

205

Experimental TE BackScatter at 70°
(Magnitude)

 

 

 

 

Stokes
Wave

 

 

Double
Sinusoid

 

 

 

Donelan
PIerson

 

Til lllrlllllll lllll

6.510 8.60 11.50 13.'4c')'
Freq (GHz)

Figure 5.40 Experimental TE scattering from various wave models for an incidence
angle of 70°.

206

Experimental Transient BackScatter at 70°
(Synthesize, Normalized)

 

Sto kes /

Wave

 

 

 

Double
Sinusoid

 

Donelan WWW
PIerson

 

 

IIIIIIIIIIUIrTIII

.10 10.60 11.80 13.00
Time (ns)

@-
N-
O

(D-

Figure 5 .41 Experimental synthesized transient TE scattering from various wave
models for an incidence angle of 70°. GMC weighting applied to spectrum
before IFFT.

207

5.6. la Measurements performed in time domain

The synthesized transient results discussed so far are all of very-large bandwidth
(i.e. very-short pulse). The method of frequency domain synthesis works extremely well
for stationary (non-time varying) targets, but is less effective for non-stationary targets.
A true short-pulse radar can be just as effective on non-stationary targets provided the
bandwidth is the same. As explained in section 5.5. la, the time—domain system at MSU
lacks the large bandwidth of the synthesis system. However, that bandwidth is large
enough the make some useful and validating comparisons.

Figure 5.42 compares the transient and spectral returns for both the time—domain
and frequency-domain systems. The Stokes wave model was used for this comparison.
The effective bandwidth of the PPL pulse generator is from DC to roughly 2 Ghz, and
the radar horns are effective down to around 0.5 Ghz. In Figure 5.42 the spectral returns
are compared for both methods, with the time—domain measurement transformed using
the FFT. The agreement between the two methods is excellent over the bandwidth of 0.5-
2.0 GHz. The large spike at 2.1 GHz is caused by calibration performed outside of the
pulse generator’s effective bandwidth. Similar agreement can be seen in the transient
results. In this case a GMC window was applied to both spectra and then transformed
to the time domain. The transient results are seen to contain 11 return pulses. The first
pulse is due to the leading edge of the wave model and the remaining 10 pulses are from
the wave crests and are separated by the two-way transit time between neighboring
crests.

The backscattered field spectra for double-sinusoid and Stokes wave models are
compared in Figure 5.43. The measurement was performed using the frequency-domain
system. The comparison reveals that the same low-frequency effect that was observable
in the theoretical results is present in the measurements. The effect is that the lower

frequencies can only "see" the fundamental harmonic of the double sinusoid model. The

208

fundamental harmonic of the double-sinusoid model is the same as the Stokes wave
period and therefore the spectra are very similar.

The results for the transient backscatter response from double-sinusoid wave
model are shown in Figure 5.44. The agreement is once again very positive. These
results help to confirm the validity of the synthesis method and provide support for the

future use of short pulse radar.

TE Backscatter from Stokes Wave
Incidence Angle=600, 0.4—2.96 GHz

 

 

 

 

1 O TransientSoatter
' 1 4.7 i: — s nthesized
0.9 r 307 . 1 :I “" Ime Domain
I 2.7 i g . '
0.8 ‘ 1'7 1 1 l 1| i 1:
L 34" ' 1 l ‘ 1 “1‘ II
0.7 - -506 ‘ ., I II
P 5407 f :I
g 0.6 - -2.7 . 1 1:
3 h -3o7 , I II '
'E 0.5 r 4.7 i ll '
81 _w .. ............... :1 :
20.4: 6.6 8.3 10.1 11.91131qu 17: 19.1 20.9 :: :
0'3 : I1 I‘ 1
1 ll 1
0.2 I \ ‘\ I ’1 I
01 h 1’ ”‘1 J 1' -
. f 111' ' ‘ K ‘- \. \~A'
0.0 I I I l l I I 1--- ~.. 1 I l I I I 1 I . I
0.4 0.7 0.9 1 .2 1 .4 1 .5 1 .9 2.1 2.3 2.6 2.8
Freq (GHz)

Figure 5 .42 Comparison of frequency-domain synthesis measurement with time-
domain measurement. Both spectral and transient returns for Stokes Wave
interrogated at 60°.

209

TE Backscatter
Experimental, O1=60°, 0.4-2.96 GHz

1.0

— Stokes Wave Model
----- Double Sinusoid

.0 .0
O) on

.0
.p

:h_1|111IJIIIIIILIIIIIlIIIIIIIIIIIIIIIILIIIIIIIIIIIII

Magnitude (BackScatter)

.0
N

 

 

 

.0
0°

 

IIIIIITTTIIIIIIIIFT

0 0.90 1.40 1.90 2.40 2.90
Freq (GHz)

Figure 5.43 Spectra of TE backscatter response for the Stokes and double sinusoid
wave models. Measurements were performed using the frequency-domain
technique.

210

Transient TE Backscatter from the Double Sinusoid
Experimental, 01=60°, 0.4—2.96 GHz

2E+008

1 E+008

5E+OO7

 

 

 

 

Relative Amplitude
0
F1
+
O
O
O

 

 

Frequency Domain Synthesis
----- Transient Measurement

 

_2E+008|erIIIIII IIIIIUTTTTjilliIIIIj

5.00 10.00 . 15.00 20.00
TIme (ns)

Figure 5.44 Transient TE backscatter response for the double-sinusoid wave model. A
comparison of frequency-domain synthesis the time-domain techniques.

211

5.6.2 Theoretical Results for TE Excitation.

The synthesized transient backscattered fields for the Stokes wave and the double-
sinusoid wave are shown in the following figures. The results were computed assuming
an infinite PEC surface and utilizing the theoretical methods developed earlier in the
chapter.

The transient backscatter from the Stokes wave model are shown in Figure 5.45
and Figure 5.46 for 85° and 60° incidence, respectively. In both figures the observation
point is located at two differing heights. The transient backscattered fields depict the
periodicity of the wave model as seen by the periodic return pulses. There is also a
multiple scatter present in the trough regions between the specular reflections. It is quite
interesting to note the difference that the observation point has on the backscattered
fields. When the observation point is in close proximity to the wave crest the evanescent
Floquet modes contribute to the backscattered fields. This was also noted in the
sinusoidal wave models.

There are similar results for the double-sinusoid model. In Figure 5.47 and
Figure 5.48 the transient backscattered fields from the double-sinusoid model are shown
for the respective incident angles of 85° and 60°. There is a noticeable difference between
responses for 85° and 60°. The increased effect of shadowing is evident for the 85° case.
This can be seen in the strength of the specular reflections from the wave crests. There
is also a prominent multiple scatter for the lower observation height.

The effects of spectral content are examined in Figure 5.49 and Figure 5.50. The
synthesized transient backscatter from the double sinusoid wave for an increasingly
narrower GMC window are shown in Figure 5.49. The incident pulse can be seen to
become very narrow as 'c becomes smaller, this results in enhanced resolution. In
Figure 5.50 the effects of moving a constant 1: GMC window (roughly 1.5 GHz wide)
through the spectral response is shown. Different parts of the wave are more active for

different frequency bands. For example, the lowest band produces scattering from the

212

main swell, but as the window is moved up in frequency (fc increased) the small-scale
ripples on the wave become more active and produce reflections.

The same time-frequency analysis is performed on the theoretical response from
the Stokes wave. In Figure 5.51 the effect of GMC window width is shown and in
Figure 5.52 the effect of GMC window center frequency is examined. The results for the
Stokes wave model exhibit the same qualities that were seen with the double-sinusoid
model. These results would simulate a less robust UWB/SP system. The loss in
bandwidth (or resolution) has an obvious detrimental effect upon wave clutter

identification.

TE Backscatter from Stokes Wave
Theoretical (Infinite) O1=BS°

Observation
Height

 

 

 

 

 

 

 

 

 

 

 

IIIIITIIIIIITIIIIIFTIIII[Ill

52.00 54.00 . 56.00 518.100

 

 

 

 

 

 

Figure 5.45 Theoretical synthesized transient TE backscattered fields from the Stokes
wave model. The response at 85° incidence for two field points is shown.

213

TE Backscatter from Stokes Wave
Theoretical (Infinite) O1=GO°

Observation
Height

“1W 01

 

 

 

 

 

 

 

 

Z=O.1778m m

 

 

TilIIIIlleIllerlllrIllllllI

68.00 70.00 . 72.00 74.00
TIme (ns)

Figure 5.46 Theoretical synthesized transient TE backscattered fields from the Stokes
wave model. The response at 60° incidence for two field points is shown.

214

TE Backscatter from Double Sinusoid
Theoretical (Infinite) 01=85°

Observation
Height

Z=1.778m

 

 

 

 

 

Z=O.1778m

 

 

 

 

 

 

 

 

IllllllIIIIIIIIIIIITIIIIIIIII1

70.00 72.00 . 74.00 76.00
TIme (ns)

Figure 5 .47 Theoretical synthesized transient TE backscattered fields from the double-
sinusoid wave model. The response at 85° incidence for two field points
is shown.

215

TE Backscatter from Double Sinusoid
Theoretical (Infinite) O1=60°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Observation
Height
Z=1.778m
Z=O.1778m W
50.061 FT I I l I512.l061. I I r r1 1514.661 T I I I I ggjoo
TIme (ns)

Figure 5 .48 Theoretical synthesized transient TE backscattered fields from the double-
sinusoid wave model. The response at 60° incidence for two field points
is shown.

216

Transient TE Scattering from the Double Sinusoid Model
F161d Point is at z=1m, 01:85°

Incident Wave Normalized Backscattered Field

|
I
l
GMC Window '
f,=7.4, ¢=.14 ‘ "

GMC Window A
fc=7.4, T=.23 AV

 

 

 

 

 

 

IMIII
MI

 

 

GMC Window
I,=7.4, T=.32 1

 

 

 

 

 

 

1 IV

I

I

I

GMC Window '
fc=7.4, 'r=.41“"’\ H
I

I

I

I

€197.11,“ affirms "“41 H

Illllllll [IIITIIIlllIIIIIIIIIIIIIIIIIII

10.00 12.00 . 14.00 16.00 18.00
TIme (ns)

: E E E E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5 .49 Theoretical transient scattering from double-sinusoid wave at 85°, with the
effect of GMC window width examined.

217

Transient TE Scattering from the Double Sinusoid Model
Feld Point is at z=1m, 0;=85°

 

 

 

 

 

 

 

 

 

 

Incident Wave I Normalized Backscattered Field
I
I
GMC Window
fc=11.8, T=.5
I
I
I
GMC Window ,
fc=9.6, T=.5
I
I
GMC Window '
f¢=7.4, 1:5
I I
I
GMC Window
f¢=5.2, r=.5 ‘ |
I
I
I
GMC Window '
f.=3 GHz;r=.5 M
I
IIIIIIIII [IIIIIIIIIIIIIIIIIIIIIIIIrIIIII
10.00 12.00 14.00 16.00 18.00
TIme (ns)

Figure 5.50 Theoretical transient scattering from double—sinusoid wave at 85°, with the
effect of GMC window center frequency examined.

218

Transient TE Scattering from the Stokes Wave Model
Field Point is at z=1m, Oi=85°

Incident Wave Normalized Backscattered Field

GMC Window
fc=7.4, T=.14 ‘

 

 

GMC Window
fc=7.4, T=.23

 

GMC Window
fc=7.4, T=.32

 

f¢=7.4, T=.41 ‘

 

 

 

 

GMC Window
f.=7.4 GHz,r=.5

 

__I
_IV
GMC Window _\l\
_.I

 

 

 

 

 

 

 

 

 

 

l
IIIIIIIIIII IIIIIIIIIIIIIIIIITIIIIIIIII]

10.00 12.00 14.00 16.00 18.00
TIme (ns)

Figure 5.51 Theoretical transient scattering from Stokes wave at 85°, with the effect
of GMC window width examined.

219

Transient TE Scattering from the Stokes Wave Model
l-"Ield Point is at 2: 1m, 8. =85°

Incident Wave Normalized Backscattered Field

GMC Window
fc=1 1.8, 'r=.5

GMC Window
f¢=9.6, 'r=.5

 

 

GMC Window
f¢=7.4, T=.5

 

 

 

 

 

 

 

 

 

fc=5.2, T=.5
l I

II I

|
I
l
IIIIIIFII [IIIIIIIIIIIIIIIIIITIIIIIIIIII]

10.00 12.00 . 14.00 16.00 18.00
TIme (ns)

lI—ll

GMC Window
fc=3 GHz,'r=.5

 

 

 

 

 

“ii

—¢\i

GMC Window 4
I

 

 

Figure 5.52 Theoretical transient scattering from Stokes wave at 85°, with the effect
of GMC window center frequency examined.

220

5.6.3 Comparison/Evaluation of TE results.

As stated earlier, there is excellent agreement between the theoretical and the
experimentally measured backscattered fields. In Figure 5.53 the synthesized transient
backscatter from the Stokes wave model is examined. Only a few periods of the wave
are shown, but there is very good agreement between the theoretical (infinite periodic
surface) backscatter and the frequency-domain synthesis measurements. These results
were obtained by weighting the spectral response with a GMC then transforming into the
time-domain. Although not shown, the theoretical transient backscatter from the finite
model also matched well with the measured data.

In Figure 5.54 the spectral returns are compared. Once again there is excellent
agreement. The peaks are due to the periodicity of the wave (Floquet modes) and are
slightly spread out due to the apparent period change as viewed from the field point. The
theoretical backscatter from the finite length surface did not exhibit this spreading, due
to the use of the far-field approximation.

The double sinusoid wave is considered next, and once again there is a strong
agreement between the theory and the measurements. In Figure 5.55 the synthesized
transient backscatter is compared. The additional wave structure is evident and accurately
depicted by the theoretical results. The spectral returns are not shown but are also in
good agreement.

The synthesized transient backscatter from the Donelan Pierson wave is compared
with the theoretical backscatter from a finite wave. The infinite (periodic) theory can not
accurately model the backscatter due to the large shadow produced by the first crest. The
finite theory does a fairly good job as seen in Figure 5.56, but due to the far-field
approximation the two-way transit times between crests are not accurately depicted. The
spectral comparison is also hampered by the far-field approximation, but does show

general agreement.

221

Stokes Wave

Incident Angle o? 60 deg.

TE Backscatter Usin

 

 

 

 

 

 

 

 

 

 

 

 

 

- - - - - Measurement

Theory (Infinite)

11.80

0

IIIIIIIIIIIII

i 0.8

9.80
Time (ns)

8.80

IIIIIIIIIIIIIIIIIIIIIIIIIII

 

o n_U

ov3=aE< oN__oE._oz

0
0.
_

 

 

7.80

—1.50

theoretical transient scattering at 60°. Using 0.8-7.2 GHz GMC windowed

spectral content.

Figure 5 .53 Comparison of synthesized transient scattering from Stokes wave with

222

1.20

1.00

.0
oo
0

Relative Magnitude
p o
3 8

.0
N
O

0.00

Figure 5.54

TE Backscatter Usin? Stokes Wave

 

 

 

 

 

 

 

Incident Angle o 60 deg.

E ----- Measurement
5 — Theory (Infinite)
z ,, I a .
: I' ,
E I :1 1" I: -
: I g l l I . I .,
Z I I I I l I I
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I II I I I I II ‘
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I I I I I I I I I I I I I I I I I I I [j I I I I I I I I l I I I I I I I
.00 2.00 4.00 6.00

Frequency (GHz)

Comparison of TE scattering from Stokes wave with theoretical scattering
at 60°.

223

Transient TE Backscatter from Double Sinusoid
O.=60°, GMC Window (1—14 GHz)

 

 

 

1.0 1 Theoretical élnfinite)
- I ; ---- Measured ( xperimental)
- I
I I
- I I
- I I
: ' I I
0.5 — '2' ,.
Q) .. P ,I . .
U " I: i I
3 - ‘ I| ‘ I
+1 - I I I
'2 - II'I , I
Q - 1‘ ”II'I I
E - :~ :th .
00: ' II'I' I" |::II.‘ ‘1
*0 ° - ‘I ~' illili‘i ':-u::vv I
0) ~ I III'I ' 'II,
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6 - ::I :E' '
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o : I: ‘
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_ ' I
- ' I
_ " I
-‘Io0-IlIIIlTIIlll[IllllllllllIllIIlllITIIIW
4.20 5.20 6.20 7.20

Time (ns)

Figure 5.55 Comparison of theoretical and measured transient scattering from the
double-sinusoid wave. Angle of incidence is 60°.

224

TE Scattering From Donelon Pierson Wove
Incidence Angle of 60°, 0.8-7.2 GHz

Spectrum

 

 

 

 

 

 

 

 

- —— The ticol
1.00 : 1.01 . ---- Meagfred
0.88 --u 1 5
.II 0.3- 11 g ' :'
0.76 -:: 1 515:. ' .' .~
0 '11 :5 0-5’ 1' fit“: {'5 1: :1 51”“:
g 0.64 ”H E. 91:". fififl i, i; «igsiifl'
.3 -:: g 04 “i i: i ., :5
g- O.52 :II | "I" 1: .1 ! i i I...
< 'i 01' '. ‘ ..
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_020 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 “—1
7.0 7.9 8.8 9.7 10.6 11.5 12.4 13.3 14.2 15.1 15.0
TIme(ns)

Figure 5.56 Comparison of theoretical and measured scattering from the Donelan
Pierson wave. Angle of incidence is 60".

225

5.7 . Conclusions

The experimental measurements presented in this chapter help to validate the
theoretical techniques, and they offer new insights into the scattering from these surface
models. The time-frequency nature of the scattering was only qualitatively discussed and
will be examined in detail in the future. The identification and analysis of scattering
mechanisms was possible through the use of UWB/SP radar on the specifically chosen
sea-surface models. These mechanisms include scattering centers (or specular reflection),
multiple scattering and sub-structure scattering. The increased bandwidth of the UWB/SP
radar scheme is the crucial factor in the improved resolution.

The transient backscatter obtained by the frequency-domain synthesis technique
was seen to agree with the true time-domain short-pulse measurements. This furthers the
confidence in the theoretical techniques and the measurement techniques. This experiment
reveals many of the future strengths of the short-pulse radar; the sea surface can be
modelled as a static wave, the extraction of wave features is enhanced, and the spectral-

time characteristics are of new importance.

226

Chapter 6
Transient Scattering of a Beam from Infinite Periodic Surfaces

6. 1 Introduction

The preceding chapters have emphasized the improvement of the ocean models.
The oceans models included PEC sinusoidal, lossy sinusoidal and non-sinusoidal
surfaces. Transient scattering from all of these surfaces was examined for an incident
plane wave. In this chapter, the improvement is made in the incident—wave model. The
incident wave thus far has been an ideal plane wave, which is of infinite spatial extent.
A realistic radar system produces a non-uniform near plane wave of finite extent. The
coverage area of the incident wave is called the footprint. Therefore this chapter will
examine the transient scattering from a limited-footprint or beam radar.

The incorporation of a non-planar incident field into the previously developed
scattering theory is quite simple for the fmite length ocean models. However, the
formulation for scattered fields from the infinite surface models with beam interrogation
is not so trivial. The difficulty arises with the use of the periodic Green’s function. The
PGF allows for truncation of the problem to just one period of the ocean surface.
Therefore the incident beam would also be repeated for every crest of the ocean surface
with a progressive phase shift. This chapter will utilize a plane wave expansion to create
an incident beam for use with the infinite length ocean models. The plane wave
expansion will allow for the continued usage of the scattering theories developed
previously for the infmite surfaces.

The preceding chapters introduced two frequency-domain methods to determine
the scattered fields from an infinite ocean surface for plane wave excitation. One

technique, known as the Rayleigh hypothesis, is only valid for sufficiently smooth

227

surfaces. A more encompassing integral-equation theory was also developed to provide
solutions for rougher Surfaces. These two methods can be modified, to a more general
non-uniform (beam) incident wave, by properly choosing a number of plane waves. The
technique in which a sum of plane waves is formed into a beam is known as the plane-
wave expansion. The idea is therefore to repeatedly solve for the scattered fields using
specific plane waves. The plane waves are chosen so that the resultant sum is a beam.
Thus, the summation of scattered fields from each constituent plane wave results in the
scattered field produced by that incident beam.

There are a number of ways in which the constituent plane waves can be
determined. One method is to Fourier synthesize the plane waves needed to create a
predetermined aperture field, this method is similar to antenna array beam synthesis.
Another method, although less general, is to simply create a square wave using Fourier
series. This square wave could then be space windowed to create the effect of a single
beam. Both methods would ideally involve an infinite number a plane waves to exactly
reconstruct the intended pattern. A feasible solution would involve a finite number of
plane waves with the smallest number of plane waves needed to accurately create the
beam as an optimal choice. By setting this limitation a slightly imperfect beam is
expected, in fact the aperture field method will create a periodic beam, which will also

need to be space windowed, and the two methods somewhat degrade into one.

6.2 Application of Scattering Theory

The two-dimensional surfaces of the previous chapters will be considered, and
also only TE incident fields are analysed. In addition only a sinusoidal surface will be
considered; this limitation eases the computational time involved by allowing for the use
of the Rayleigh-hypothesis method. The integral equation method could have just as
easily been employed, and for rough surfaces, but the computational time involved is

rather phenomenal.

228

The computations are not only repeated for various incidence angles, but must
also be iterated over a large bandwidth with a relatively small frequency step size. The
frequency domain results are then transformed to obtain the desired transient response.
The analytical technique uses two spectral transformations which results in a high
computational load. In order to retain any of the valuable techniques developed, the non-
uniform beam of interest must be composed of a finite sum of plane waves. This point
was brought up in the introduction, and the essential steps in creating a non-uniform

beam are presented below.

6.3 Beam Synthesis Techniques

In the introduction two methods for creating a non-uniform beam are mentioned,
the first case to be considered will be the more general case, which is the aperture field
method. The basic idea behind this method is to Fourier transform one of the spatial
variables (x,z) out of the field representation, then match the fields at the aperture to

obtain a set of spectral amplitudes that will recreate the same aperture field.

Z

 

 

‘ ‘ _o .‘
k 0 E = E06 ]F r
” b i: = ko(9 sine-’2‘ c030)
IIIIIIIII .0
opqque E

    

 

Figure 6.1 Geometry of Beam Synthesis Apertures.

229

The Fourier transform pair for either component of the field will be denoted

asE(x,z) ~e(x,kz) or E(x,z) ~e(kx,z). The Fourier transform is defined here as

E(x,z) = i f: e(kx,z) ejk‘x dkx
e(kx,z) = f“ E(x,z)e-jk‘x dx

(6.1)

The transformation that is used will be dependent on the aperture orientation.
Consider a plane wave incident upon an x-directed aperture (See Figure 6.1). Taking the
Fourier transform of the governing Helmholtz equation and solving the resulting

differential equation, the spectral domain fields are described by

e(kx,z) = A(kx)ejk‘z + B(kx)e'jk‘z (6.2)

where kz = «k: -k,,2 and k0 = to no ea .

For a beam propagating in the negative z—direction B(kx) = O , and the space

domain field can now be found by taking the inverse transform of e(kx,z).

E(x,z) = i f “A(kx)ej (11,141, "de (6.3)
21: -~

A prescribed aperture field is used to determine the unknown spectral amplitudes A (k).
Note if the aperture is placed in the 2:0 plane then e(kx,0) = A(kx) or

A(kx) = de(x,0)ejk" dx (6.4)

The aperture field can be selected by many means; one simple way would be to have the

 

aperture field be equal to the incident plane wave field Ea = E0 e jun“. For this case
the spectral amplitudes can be directly integrated to be
2 E e11(C+d)I2
A(k,) = ° . sin(v(d-c)/2) (“-5)
H

where y = kx-kosine. The spectral amplitudes will be maximal at y = 0.
In order to perfectly reconstruct the beam needed to produce such an aperture

field an infinite number of spectral components are needed, but by properly choosing a

230

finite number of spectral components a reasonable beam can be created, this beam
however will have a periodic nature. The idea of a periodic beam can be more simply

realized by just considering a Fourier series, which is the second method.

1? Z

t(x)
1!

X

Figure 6.2 Beam synthesis and scattering geometry.

In the Fourier series method the incident field is assumed periodic in the x-

direction. This periodicity permits the following representation of the incident field

at m 0 m

E i(x,z): 2 a,'e’.k”'”"ejk”rz k =n-E , k2 =k2-k2 (6-6)

"3*”

where 2L is period length of the incident field along the x-direction. This field could be
thought of as a summation of propagating and evanescent plane waves. The key is to pick
a finite number of these plane waves to recreate the periodic incident wave specified.
At this point a similar field matching procedure is performed to determine the
Fourier coefficients "1.- Matching at the z=0 plane results in the most compact field

representations. At z=0 the incident field is given by

231

E‘(x,0) = i anejknx (6-7)

The field must now be matched to a known distribution to determine the Fourier
coefficients. The field distribution that is chosen is non-unique and can be thought of as
a transmittance function t(x) (See Figure 6.2). Thus at the z=0 matching plane the two
field distributions can be equated, resulting in

Eot(x)e""ox‘ = X anejk’"x (6.8)

where E0 and km = k0 sine describes a plane-wave field incident upon the transmittance
screen at 2 =0.

The Fourier coefficients are then found in the usual manner utilizing the
orthogonality of the circular exponential function. This is accomplished by integrating
both sides of (6.8) by

ff; {f(x)l e‘W‘ dx (6-9)
which results in
E L -j(k +mr/L)x
a = —-‘-’- t(x)e “ dx (6-10)
" 2L 4-

There are a number of interesting transmittance functions, but the Gaussian
transmittance function has proven to be the most useful and easiest to implement. The

Gaussian transmittance function is given by

,(x) = e'(:) (6.11)

where w is a width factor and w<L.
It is interesting to note the similarities between the two methods, which is

completely analogous to the connection between the Fourier integral and Fourier series.

232

The discrete Fourier coefficients an of Eqn. (6.10) and continuous coefficients A(kz) of
Eqn. (6.4) are analogous. The aperture method could also be replaced with a

transmittance screen which would result in a spatial Fourier transform.

6.3.1 Examples of Gaussian Beam.

In Figure 6.3 the amplitude of the Fourier coefficients (an) are shown for a
Gaussian beam with a width of 1.0 meter (w = 0.5m). The incidence angle is 70° which
would correspond to km = -37 r/m for an excitation frequency of 2 GHz. A spatial
Gaussian beam has a Gaussian shaped Fourier transform where the width of the Gaussian
curves are inversely proportional in each domain. An analogous effect occurs in the
discrete case and is obvious in the figure. The 21 strongest modes are plotted. The small
amplitudes at the curve ends indicate that the spatial beam should be well constructed.

The reconstructed beam can be seen in Figure 6.4. The beam is well represented
by the 21 modes that are included. The width of the Gaussian beam and the repetition
period of the beam are seen to correspond to the predicted values of 2L = 10m and
2w = 1.0m. If a narrower beam was required a broader spectrum would be needed and
the number of important modes would increase. For this case 21 modes is sufficient to
create a beam with small side—lobes. The implication is that scattered fields will have to
computed 21 times, for each individual plane wave that makes up the beam.

An important detail regarding Figure 6.3 is the classification of evanescent and
propagating modes. These are with respect to the z-direction are a consequence of km.
When Ikxn | > Ice the plane waves become evanescent in the z-direction and therefore the
placement of the transmittance screen becomes important. The transmittance screen
should be placed as close to the ocean surface as possible, as shown in Figure 6.2. In
this fashion the effects of dispersion and decay are minimized and a good beam can

illuminate the surface.

233

Amplitude of spatial frequency coefficients
O1=7O°, L=5m, w=.5m, n=21, fo=2 GHz

 

 

 

 

ox=—37 r/m
0.10 j ko=-42 r/m
: propagating modes
0'08 _- evanescent
; modes
0 I
‘0 a
3 :
'E :
80.04 —_
2 ..
0.02 -'_~‘
3 k, k...
0.00 - IIIIIIIIIIFTTIII IIIIII TWIIIIIllllllTllIlllllTI]
-75.00 -70.00 -65.00 -60.00 -55.00 —50.00
n

Figure 6.3 Spectral Amplitudes for a Gaussian Incident Beam Parameters are
L=5.0m and w =.5m.

234

Amplitude of Incident Gaussian Beam (z=O)
G);=70°, L=5m, w=.5m, n=21, fo=2 GHz

1.20
1.00

0.80

.0
a)
O

Magnitude Ey

0.40

0.20

 

 

 

 

lllllJlIllllllLJlllllllllllllll

 

0.00
-11.00 —6.00 —1.00 4.00 9.00

x ( meters)

Figure 6.4 Gaussian Incident beam created from 21 terms of the Fourier Series,
where L=5.0 m, w=.5 m, and 0 =70°

235

6.4 Transient Scattering Results.

The induced surface currents for a TB incident Gaussian beam are examined in
Figure 6.5 and Figure 6.6. For both examples the Gaussian beam is 1.0m wide with a
repetition period of 10.0m. The beam is illuminating a PEC sinusoidal surface with a
period of 0.1m and a height of 0.007 m and the excitation frequency is 2 GHz. In
Figure 6.5 the beam is incident from 30° and the resulting induced surface currents are
shown. The plot reveals the width of the beam through the fact that the induced currents
are strongest where the beam is of greatest amplitude. In Figure 6.6 the response for an
incidence angle of 70° is shown. The beam is composed of 21 modes and was shown in
Figure 6.4. The induced currents reveal an increased shadowing effect which is expected
for the near grazing incidence angle.

In Figure 6.7 and Figure 6.8 are the induced currents for just the central
illuminated regions of the two previous figures. These plots highlight the differences
between the incidence angles of the Gaussian beams. The case of 70° incidence exhibits
a much greater shadowing for each individual wave crest and for the overall illuminated
wave packet. It is also interesting to note the difference in current amplitude. The 30°
response exhibits a greater amplitude; this effect was noted in previous chapters and was
expected for beam illumination as well.

The backscattered fields for a single plane wave (not a beam) are shown in both
the frequency and time domains in Figure 6.9 and Figure 6.10. These results will be
used as a comparison with the responses for beam illumination. In Figure 6.9 the
backscatter spectra from a PEC sinusoid for incidence angles of 70° and 85° are shown.
The synthesized transient responses of the respective spectra are shown in Figure 6.10.
Indicated in this figure by T is the two-way transit time between consecutive wave
crests. This two-way transit time does agree with the expected time for the ocean surface
period of 0.1m. In Figure 6.11 is the extended transient response for an incidence angle

of 85°. This plot is important because it reveals that a plane wave uniformly illuminates

236

the entire (infinite) sinusoidal surface. The transient response due to a beam, which will
not illuminate all of the wave crests, should reflect the non-uniform illumination.

The frequency response for Gaussian beam illumination is shown in Figure 6.12
and Figure 6.13. The beam and PEC sinusoidal surface parameters are the same as those
of the induced current plots (Figure 6.5 and Figure 6.6). In Figure 6.12 the response for
a 30° beam is shown for various field point locations. The field points are varied along
the x-direction and quite a large variation between the respective responses. The
spectrum produced by beam illumination is also quite different than that produced by a
single plane wave. It is interesting that beam illumination produces spectra that resemble
the response from a finite surface, with the well defined and spread out Floquet-mode
spikes. This is intuitively expected because the limited footprint beam effectively reduces
the infinite surface to a finite structure. The 70° case is shown in Figure 6.13 and
exhibits the same phenomena as the 30° case.

The corresponding synthesized transient responses are shown in Figure 6.14 and
Figure 6.15. These results show the transient backscatter due to a Gaussian beam
illumination. In Figure 6.14 the interrogating beam has an incidence angle of 30°. The
figure shows the transient backscatter at the three field points displayed in the frequency
response plot of Figure 6.12. The plot reveals that only the illuminated portion of the
sinusoidal wave is producing backscatter. The change in field point location exhibits the
expected behavior, which is a time shift in the backscatter wave packet. The decay in the
return wave packet amplitude is due to the increased distance the scattered wave had to
travel. Although it is not shown, the incident wave is periodic with a spatial period of
10m and backscatter would also be produced from the additional incident beams.

In Figure 6.15 the incidence angle is increased to 70° (near grazing). The
envelopes of the backscatter wave packets are somewhat distorted in shape. This is
explained by comparing the induced surface currents of Figure 6.5 and Figure 6.6. The

induced currents on the 70° incidence have a different distribution than the 30° case.

237

The difference between the single plane-wave excitation and the beam illumination
is quite evident in the transient responses. This is highlighted in Figure 6.16, where the
transient response for a 30° beam is shown. If this surface had been illuminated with a
single plane wave the three location points would have identical backscatter responses due
to the infinite spatial extent of both the incident wave and sinusoidal surface. But it is
clear from this figure that beam illumination induces non-uniform surface currents, which
create the footprint-limited wave-packet backscatter.

The synthesized transient backscatter for an incident Gaussian beam is shown in
Figure 6.17. The field point location is varied along the vertical axis and the log
magnitude of the transient backscatter response is plotted versus time. The upper plot is
for a 70° incident beam and is contrasted with a 30° incident beam in the lower plot. The
backscatter from the 30° beam is greater in magnitude and less spread out than the 70°
case. This was seen previously and is expected since the 700 beam has a larger footprint.

In Figure 6.18 the total transient scatter from a PEC sinusoid wave is shown for
a Gaussian beam incident pulse. The figure displays the backscatter as well as the
prominent specular forward scatter. The forward scatter are the narrow diagonal lines
with a positive slope. The forward scattered waves that occurs at the later times is due

to the periodicity of the incident beam.

6.5 Conclusions

In this chapter the scattering of a spatially limited incident beam by an infinite
periodic surface was examined. The construction of the beam was such that the previous
theoretical developments could be still used to determine the scattered fields. This was
accomplished with the idea of a plane-wave decomposition. The beam was decomposed
into a finite number of plane waves, which can be used with the existing scattering
theories. The restriction of the number of constituent plane waves resulted in a spatially

periodic beam. This unwanted characteristic could be mitigated by spatial windowing.

238

The results that were obtained were not verified with experimental measurements
but the confidence level is high regarding their correctness. The information gained from
previous experiments and theoretical results from the finite surfaces was closely related
to the beam excitation. The correlation between the two was noted in both the time and
frequency domains. The most notable, and expected difference between the single plane
wave and beam excitation is the spatial non-uniformity of the scattered fields. This was
shown by examining the spectral and transient responses at a number of field points in
the horizontal plane.

The usefulness of the beam excitation is its the ability to model a limited footprint
radar system in a infinite sea-surface environment. In the future this model will be used

for realistic radar simulations in which targets in the ocean environment are sought.

239

Induced surface current for incident Gaussian Beam
Beam Parameters : 01=30° , L=5m. w=.5m, n=21

Surface Parameters : h= .007m. P=.1m. PEC
0.06 f°=2 GHz

 

 

 

 

 

0.00 IAAAAAAAAAH‘ A‘A‘.““‘u‘
—10.00 —5.00 0.00 5.00 10.00
x (meters)

Figure 6.5 Induced Surface currents on PEC surface, for 30° Gaussian beam

illumination.
Induced surface current for incident Gaussian Beam
Beam Parameters : 0.=70°. L= 5m. w=.5m, n=21
Surface Parameters : h= .007m. P=.1m. PEC
0.04 ': f°=2 GHz
0.03 E
I
_J

 

Current
0
0
re

   

.00 'l‘luuluann..‘1.l I l“iAALaALLAll
—10.00 —5.00 0.00 5.00 10.00
x (meters)

Figure 6.6 Induced surface currents on PEC surface for 70° beam illumination.

240

 

 

 

 

 

 

 

 

 

 

 

 

x (meters)

Induced surface currents on PEC surface for 70° beam illumination.

241

 

m . m
G e n e
1 I 61 I
%2 t.” 1 .m 82 2:” I
__ o 1 __ o 1
r O r O
.l. nE I .I u t
8 mp t 6 I'll! To 1 8 mm t 8 ll---" '0
S n C ||||||| \ f n S ”C ||||||||||
ms 1 we at W5 1 mo rrrrrr 1
f rill r . f llllll .1
rr “‘\ rr IIII \
t 1 CS """ I _l t 1 CS lllllll U .l
n I: _ a |||||||| 1 m n 0= . s ||||||||||| I
me . 11111111 1 3 “ML . 111111111111 1
mam. " "HHHH- .o 1m.m__ m. " AHHHHH .o
.IL7 . iiiii 1 I2) .IL7 . 11.11”" 12
\\\i . S w n1 \ rrrrr .
row . 11111111 10 r a wow Na. 11111111 10
m0 . ttttttt \ 1 e m :10 . ttttttttttt \ .1
3 = fl lllll I t 7 = llllll I
.m __1.n ....... . . e K .m __1h With”. .
me . 11111111 1 H m P m0 u. 11111111111 1 fl
U rs- “ ||||||| r ( n U m H lllllllllll f
cse 1111111 1 0 cse U 11111111111
rt ||||| _l x w rt 1
cam 1111111 . 15 e cam 13.
Mma 1 111111 10 n Mma 10
r r I iiiiii _ ..u r r _
uma 111111 1 1 c uma 1
SOP Z 5”,- I w 8 CD! .I
P H 111111 1 1 a P 1
a s ...... . .n a w .
can ..u
cmo (I'll I S cmo .l
uocr-Z .11.... U C."
deu __ I am deU 1.
n o n
IBSf '0 u '88 U
.8 .m n
_quqd____du.u.__q_‘_-q____aq_1_d.._ O. I “11.441111344433313
o. a. o. o. 7. 0 o o. o o
o o o n_o w o o o o 0
€850 me Emtao

 

Figure 6.8

0.20

0.15

bck

>,0.10

E

Mag.

0.05

0.00

Figure 6.9

Backscattered Electric Field
h=.007m , L=.1m,
Field point : x=0m, z=1m

 

 

 

 

 

 

I

I
I
1” |
I I\
" I
I”
I
I

n , ,\ ,’\ I \
I \ \I \l
fiend ' \l I I

111L111]Illlllllllllllllllllllllllllllll

‘
a

fl”’
I,,— I .
nu\\

 

l
lllllllillllillll[ITIIIIIITTIIITTIIIIIIIIlIllIIIIIIUIIIIIITTTIIIIII[I]

 

 

.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

Freq (GHz)

Frequency response of backscatter from a PEC surface for a single plane
wave.

242

Backscattered Electric Field
h=.007m , L=.1m,

 

 

 

2-5E+003 : Field point : x=0m, z=lm
: —— 01:700
1.5E+008 : ----- 01:85°
§ 2
0T 5.054.007 : 1’ .
\
a) : I \ I \ I \\
g 2 I, \ "\ I, \\ I ‘\ I, ‘ -"\
E : I, \l ‘\ I, \\ I, \
g- : ’I \\ I] \ [I \‘
_. _ \J
< 5.013007: ,1
a, _
.2 I
+‘ -
_o -
o I
0‘ -1.5E+008 3
1
I
_205E+008.IIlllllIIIIIITIWIIIIIIIIIIIIII
4.50 5.00 5.50 6.00
Time (ns)

Figure 6.10 Transient backscatter from a PEC surface for single plane wave pulse
illumination.

243

Backscattered Electric Field

h=.007m , L=.1m,

Time (ns)

244

 

 

 

 

O

1o.

I O

I 7
r
1

1 m

m .o.

1 6
__ I
Z l
m 1
O ‘
= .1

X o

5 I

O C 8 I O

1m _._1 I0

.m e .1 O

p _ I 5

_

d _ l
I. _ I
m . -
Emmi-01%|. WI...IJI.I..|II."I ....... .l

lifluflwIIflIIflWIMINI ttttttttttttttt I 0

?~ . _ . q q _ . iijiffivglhfihkiii InU.

6 5 5 6 O

0 0 O O 4
0 0 0 O
+ + + +
E E E E
5 0 0. 5
1 5 5 1_I

1.». auntie/a 9,52%

Figure 6.11 Transient backscatter for single plane wave incident.

Backscatter Frequency Respone for a Gaussian Beam
h=. 007m. P= ..1m w=. 5m. L=5. 0m. =50°
Field point z= 1m plane

 

0.20 1
0.15 E

a, I
3 2
”0.10 —
g -1
.3 3
E. I
E _
< .1
0.05 E
0.00 1
1.5

 

Figure 6.12 Spectrum of backscattered field for an incident Gaussian beam (0=30°)

Backscatter Frequency Respone for a Gaussian Beam
h=. 007m. P= ..1m w=. 5m. L=5 0m. 0=70°
Field point z-1m plane

0.05

0.04

.0
0
0

Amplitude E,”
.0
0
N

0.01

 

3. 0 4. 0
Freq (GHz)

Figure 6.13 Frequency response of backscattered field for an incident Gaussian beam
(L=5.0m, w=0.5m, n=21 0=70°)

245

 

 

 

Transient Backscatter of a Gaussian Beam Pulse

 

 

 

 

 

 

 

 

 

 

 

 

 

=. 07m, P= ..1m w=. 5m, L=5.,0m 0=30°
Field point z=1m plane
.
5.0E+006 : — — - x=—-9m
- x=—1.8m
: ------ x=-2.7m
I I
: III:
. urn
1.0E+006 - 111,111.11
2 1.. Intro“;
: ., 1W
—.3.0E+006-:
1
—7.0E+006—
39.00 44.00 49.00. 54.00 59.00 64.00
TIme (ns)
Figure 6. l4 Transient backscattered field for an incident Gaussian beam pulse (0=30°).

1.0E+006

 

 

Transient Backscatter of a Gaussian Beam Pulse
h=. 007m. P= ..1m w=. 5m. L=5. 0m. 0=70°
Field point z=1m plane

 

 

 

 

 

 

 

 

j ----- x=—.9m

:1 x=-1.8m

j - — — x-'—2.7m
5.0E+005—: L!

.1 g ' I

3 ~ I ' I : I - I t I I 'I

: :Iglgi"‘|r.ln.1|1'1" 1

: ' ' .r‘liiiiiiiiul 11‘1..i1'ii'
0.0E+OOO -_ :11..qu 1:1“ 31;! III 1) I; r t I}; II‘IHJ'

Z '..'1 1 . '1' I """I

- :{IIJI II‘I! II'III

5 1.: :I.::1Wit"iii"“ii"'

: H: :1": i I i I I
—5.0I-:+005 -; g 1’- 11 '1' I -

: 1 1 .I i

: ' 1 '

- I
—1.OE+006‘IIIIIIIILIIIIIIIIIFII IIIIIIIIIIIIIIII

62.00 67.00 72. '00 77.00

Time (ns)

Figure 6.15 Transient backscattered field for an incident Gaussian beam pulse (0=70°).

246

Transient Backscatter of a Gaussian Beam Pulse
h=.007m, P=.1m, w=.5m, L=5.0m, 0=30°
Field point z=1m plane

 

 

 

 

 

 

 

 

 

 

 

5.0E+006 -
1.0E+006 —
—3.0I:+ooe - I. 5
2 ~‘ '1‘}. v
j
—700E+006 I I I I I I I T I Fl I l I I I I I I l I I I fiTj I I I ITT I I I I I I I
56.00 57.00 58.00 59.00 60.00

Time (ns)

Figure 6.16 Transient backscattered field for an incident Gaussian beam pulse
(L=5.0m, w=0.5m, 0=30°)

247

TE Transient backscatter for 70 degree beam
‘ '11 ‘ ‘1 *

   
   

I
1 9
—~ or

I
M

x location (m)
I
01

I
[\J
or

.r' 1;, I I
6 8 10 12 14 16 18
TIme(ns)

TE Transient backscatter for 30 degree beam

    

x location (m)
.r'o . .i~ . 3'3
(TI N 01 —h 01

 

' a 12 146 18
TIme(ns)

Figure 6.17 Synthesized transient backscatter response for Gaussian beam pulse
incident at 30° and 70°. ( Beam parameters : n=21, w=.5m, L=5m)
( Surface parameters : P=0.1m, h=0.007m, PEC sinusoid )

248

TE Transient total scatter for 30 degree beam

      

o

x location (m)

" 1o

 

 

5 1o 15 20 '2 30
TIme(ns)

Figure 6.18 Synthesized transient total scatter response for Gaussian beam pulse
incident at 30°. ( Beam parameters : n=21, w=.5m, L=5m)
( Surface parameters : P=0.1m, h=0.007m, PEC sinusoid )

249

Chapter 7

Transient Scattering from a Three Layer Medium

7. 1 Introduction

The electrical characteristics of the ocean have been considered in numerous
places throughout this thesis. The most elementary model was the perfectly conducting
(PEC) surface. The ocean model was later improved to be a lossy layer, in which the
conductivity and permittivity were parameters. In this chapter the composition of the
ocean is once again upgraded. A planar interface is added within the ocean layer. This
interface separates two electrically dissimilar layers and creates a three layer geometry.
The new geometry is shown in Figure 7.1.

The additional layer allows for the modelling of a number of ocean attributes. The
most obvious would be the ocean floor, but is rather uninteresting except in very shallow
water. A more interesting scenario is that of a ducting layer, which is a physical ocean
effect that has the possibility of trapping electromagnetic waves [57,58]. The third layer
could also model a salinity or temperature gradient, where the electrical characteristics
of the ocean may be varying.

The problem has been simplified by introducing just an infmite planar layer. The
general case of an additional periodic layer is possible but greatly intensifies the
computational difficulties. The computational complexity is relatively unchanged with the
addition of a planar layer. The three layer problem can be solved by extending both of
the frequency domain methods used for the PEC and two-layer cases. Namely, the
classical mode matching in conjunction with the Rayleigh hypothesis and the integml

equation methods.

250

Region 1 ( 60)

z = p = -h cos (21tx/L)
I h

 

|-—L—-l

 

1 =-d

\
Region 3 (53) W3.

 

 

Figure 7.1 Configuration of three medium scattering problem.

7.2 Floquet mode-matching/Rayleigh hypothesis.

The extension of the Floquet mode-matching method is straightforward. The
additional layer neither creates nor destroys any of the modes induced by the periodic
interface. Therefore a mode by mode reflection and transmission occurs at the planar
interface in which the typical reflection and transmission coefficients hold. There still
must be a mode matching procedure performed at the periodic interface which will now
contain an additional upward space harmonics (1|rr).

The geometry is shown in Figure 7.1. A TM plane wave is assumed incident upon

the sinusoidal interface at an angle of 91- The generating field quantity is Hy(x,z) and

251

will be denoted as 1|: (x,z). The corresponding electric field quantities can be found via
Maxwell’s equations. The sinusoidal interface, which separates air and sea, is represented
by p (x). The planar interface is located below the sinusoidal interface at z=-d meters.

The TM incident wave is given by

llr‘. = Ace'jp" ej‘" (7.1)

where B = kosinBi, q = kocosfli and k0 = w eouo.
The fields in each of the regions can be represented by the Floquet mode
expansion. This is due to the expected periodicity of the fields and was discussed in

detail in chapter 3. The fields in each region are therefore given by

¢,(x.z) = "if. e'jfl"’f,,,(z) (7.2)
11:2,(x,z) = :9 e"'"~"f2m(z) (7.3)
\II,(x.z) = "5:, e'jt-‘fmm (7.4)
W3,(I,Z) = 2: e"“-’jg,,(z) (7.5)

where B" = B + 2n1t/L.

The above wave functions must all satisfy the homogenous Helmholtz equation
in their respective regions. The application of the Helmholtz equation upon the wave
functions determines the functional z-dependence of the unknown field quantities. This
operation was performed in chapter 3 and chapter 4. The resulting field representations

are

1:. = Z B,e'”-’e‘”1~‘ (7-6)

252

1w 2 c, e11» .w (7.71

113. = i“ a. up” .113» W) (7.9)
n.-.
where
41,. - k3 - Bf.
42,, = ’92 - 13,2.
43,. = 1032 - Bi

k2=w€2|10 ’ k3=w€3|10

and the branch cuts are chosen such that the required exponential decay occurs.

There are four unknown modal coefficients, and this implies that four boundary
conditions are needed to completely specify the problem. There are two boundary
conditions associated with the sinusoidal surface as well as two with the planar interface.
The boundary conditions are the continuity of the tangential magnetic and electric fields

across the boundary. The boundary conditions in terms of the wave function are given

by
61p] 6 am,
_ = ._0 — ; : at = (7.10)
5 62 a 1:, 112 z p
6% 62 51113,

where 11:, = 1p, + 1115 , 1|:2 = 1|r2, + qr, and Ba; denotes normal derivative.

253

The remaining steps are identical to those performed in chapter 3 and chapter 4.
Namely, the field representations of equations (7.6) through (7 .9) are substituted into the
auxiliary equations (7.10) and (7.11). Galerkin’s method is then applied to isolate the
unknown modal coefficients. This was demonstrated in earlier chapters and is

accomplished by applying the following operator
[0‘ 1...} e’ ”1"" dx (7.12)

where L is the period length of the sinusoidal surface.
Note that for the z=-d boundary condition (7.11) the equations are orthogonal.
Therefore, it is possible to solve for D" and G" in terms of C". The relations are given

by

D = e" “I” Rn C" (7.13)

G = (”211" T. c" (7.14)

Where,

R = 63 q2n ‘62 q3n (7.15)

E31 ‘12» + 52 ‘13:.

 

r = 2 e3 ‘12" (7.16)

n
63 ‘12:: + 52 ‘13::

 

and coefficients Rn and T" are the typical reflection and transmission coefficients
associated with a planar interface, except that they could be different for any given mode.

The remaining unknowns B,I and Cu are found from the boundary condition at
the sinusoidal interface (7.10). These coefficients are coupled and the application of

Galerkin’s method results in the following set of equations

23 B.K....-2 C.<K....+R.e”""‘K,...)=-A.A.. (“7)

254

G e co -.
2 Bit Luv! ——0‘ Z Cn(me +Rnejqzndl’mnr) =Ao Fm (7.18)
E -

 

 

na-oo 2 n—-m
where,
_ -1’ q d
Lmnr — e 2" Lmn
qln = an
_ -1' q d
Km — e 2" Km
qu. = 92,.
and Lm , Lm, Km , Km, Am and F”' remain the same as the two layer case of chapter
4, section 2.

The numerical implementation of (7.17) and (7.18) is accomplished by truncating
the infinite number of equations and creating a square matrix. A square matrix is
produced by restricting the number of m and n entries to be the same. The convergence

of the solution is discussed in chapters 3 and 4.

7.3 PGF IE Technique

The extension of the integral equation technique to the three layer problem is
simplified by the restriction of the additional layer to be planar. The planar layer can be
handled analytically, whereas a periodic layer requires a numerical approach. The
approach is similar to that taken with the two layer problem of chapter 4.

The region below the sinusoidal surface is replaced with the upper region
medium, thus creating a homogenous medium. An equivalent volume polarization current
is placed in the lower region. This polarization current induces the same fields in the
upper region that the original configuration would. The key aspect is that a homogenous
Green’s function can be employed to find the scattered fields produced by the
polarization current (replaced by equivalent surface fields). The Green’s function for this
upper region case remains unchanged ( from the two layer problem ) and is known as

the primary Green’s function, which is given by

255

- °° -J'13,.(x-x’) -J'qa.|z-z’
G.” x x’,z/ = -—J— e e
.( ,zl ) 2 L ":2" ‘16.

(7 . l9)

 

where i designates the region (1 or 2) and q," is the same as in section 2.
In fact the integral equation (IE) obtained from closure in region 1 remains

unchanged (See Chapter 4 and Appendix C), and is given by
P _. ..
1- ..aG(1>|p’) “a“ ..
—¢(p) - PVf w(pO—‘————G,’(p|p’)—‘”—(fl dl’ = Mp) (7-20)
2 C, an/ an!

However, when the upper region medium is replaced with the region 2 medium
and polarization currents are placed in the upper region, there is an additional planar
layer (Region 3) that must be taken into account. The medium is no longer homogenous
and therefore the primary Green’s function does not adequately describe the induced
fields. The Green’s function for region 2 must be augmented to include the effects of the
additional planar layer below. This can be done analytically because of the ideal
geometry of the additional layer. The new Green’s function (Gs) is composed of the

primary Green’s function and a reflected Green’s function (G ’)

G2 = G; + Gr (7.21)

where

-111,.(x-x’) -}q (z+z’+2d)
e e "' (7.22)

 

G'x,zx” =-—L R
( |,z 2L2": ,. ‘12..

In this manner the problem can once again be thought of as a homogenous
medium, where the reflected Green’s function accounts for region 3. Therefore, the same
procedure that was used in chapter 4 for the two-layer problem can be used here to
obtain the other coupled integral equation.

The IE equation obtained by closure in the lower region remains unchanged in
form from the two-layer problem. The Green’s function kernels must be appropriately

changed in the IE, which is given by

256

1 _. + _. 302(fi'l5’)_ 62 _, _, away) ,_ 7.23
2",(0) PVfC, “KM—67— 6—0G2(P|P’)7 d1 -0 ( )

I

The derivation of the reflected Green’s function is straightforward. In region 2

the homogenous Helmholtz equation must hold, and is given by

(V 2 + k? ) G'(x.z|x'.z’) = o (7.24)

Because the scattered field will be periodic in the x-direction, G ’ can be

represented using Floquet’s Theorem as

G'(x,z|x’,z’) = 2: g,,'(z|x’,z’)e'jp" (7.25)

Substituting the reflected Green’s function (7.25) into the Helmholtz equation (7.24)
yields

2: {lg +qzznlgn'}e’”~’=o 026)

Using the orthogonality of e 115,}, ( i.e. {..} = O )

6ng
622

 

+ 4:. g; = o (W)

The known general solution of the above differential equation can be simplified
by the physics of the problem. The solution for the modal reflected Green’s functions are
given by

g.’(z Ix’.z’) = A,(x’,z’) (fie-‘2‘") (7.28)

Following a similar development for the transmitted Green’s function, which is

located in region 3,

g,.‘(z|x’,z’) = Bn(x’,z’) em“) (7.29)

257

The Green’s functions represent the magnetic field produced by a line source.
These fields must still obey the boundary conditions at the z=—d plane. The boundary

conditions are given by

P I“
602 + 66 = :2. 99.: ; gf+Gr=Gt atz = —d (7.30)
6‘7. 62 e3 dz

 

 

The unknown reflection and transmission coefficients ( A" and B" ) can be
determined by applying the boundary conditions at z = -d. Substituting the definition of

the Green’s functions into the boundary condition equations result in

 

.. . 16.x’ -' (’+d) .
)3 [1 e e ”a” +4 _ B 14’” = o (7.31)
"'2'” 2L q" n n

°° -J' 1'13..x’ -jqz,(z’+d) 62 ] ~jB.x _
— e e — A --—- B e - 0 (7-32)
2 [2L q2u n 63 q3n n

The orthogonality of e in" simplifies these equations to

16.x’ 'iqz.(z'+d)

 

:1: e ‘3 +A — B = 0 (7-33)
2L q" I! ll
‘1' ib.x’ -jqz,.(z’+d) 62
— e e — A -— B = 0 (7-34)
21. q2n n E q3n n

for all n.
The two equations for each set of unknown modal reflection and transmission

coefficients can easily be solved to yield

jplxe -jq25(zl+d)

 

A = R 1. e (7.35)
n n2L qzn
. x _. /+
B = T -j 8111,. e 192,.(2 d) (7.36)

 

n nfi 42,.

where Rn and T" are given by (7.15) and (7.16), respectively.

Therefore, the reflected Green’s function can be written as

258

e 713.0: -x’)e -iqz.(z+z’+d)

(7.37)

 

G' x,z x’,z’ = :1- R
( I ) 2L "32-“ n qzn

A coupled pair of integral equations (7.20) and (7 .23) are created by enforcing
the boundary conditions at the sinusoidal interface. A MoM numerical solution can be
implemented by expanding surface field quantities with a pulse function basis set and
point matching. The procedure for these steps are found in chapter 4, section 3.

Subsequent to the numerical solution for the surface fields 1|: and air; I an , the

scattered magnetic field in the middle layer can be determined by

H2(x.z)=-fL 114/211033311 6.16161 41' (7.38)

an/ eoan/

where 62 = G; +G’.
The methods described above are frequency domain techniques. The desired
transient response must be synthesized from a representative spectral response. This is

the identical approach taken in chapters 3 and 4.

7.5 Numerical Results

The case of a sinusoidal surface (wavelength L, amplitude h) located above a
perfectly conducting layer will be considered numerically. In Figure 7.2 the induced
surface fields on one period of the sinusoidal surface for a near grazing (85°) TM
incident plane wave is compared with the induced field when the third layer is absent.
The PEC third layer is located (d/L= .1) below the non-conducting sinusoidal layer,
whose parameters are L=O.1m and h/L=0.04. The additional third layer has a marked
effect on the induced fields. A change in the induced surface fields will result in different
scattered fields. This can be seen with frequency response of the backscattered fields
shown in Figure 7.3. In this figure the parameters of the sinusoidal surface and the third

layer are the same as in Figure 7.2.

259

In Figure 7.4 the corresponding synthesized transient backscattered fields are
shown. The comparison of a PEC third layer and no third layer is shown. The effect of
third layer is very evident in the additional peaks of the backscattered returns. The peaks
are due to multiple reflections (trapped waves) between the sinusoidal layer and the PEC
third layer. The initial return pulses are very similar for the two cases and then deviate
after the lOns point. This is expected because of the time delay of the incident pulse in
traversing the middle layer.

In the final figure (Figure 7.5) the transient total scattered field (backscatter +
forward scatter) in the middle layer, is depicted for the same case. Once again, the effect

of the PEC third layer is seen by the enhanced returns after the initial pulse passes.

Surface Fields for a three layer medium
d/L=.1. L=.1m, Freq= 2 GHz, O.=85°

innocent»

ilililililililililllililnliLililJJ

044—244—114

0.8

 

Amplitude Hy

 

No 3rd layer
----- PEC 3rd layer
o-o-o-o-o Surface shape

 

 

 

—Ou1 IIIIrTTII'IIIIIIIIOIAIOIIIIIIII‘IIIIIIIaIéIOIIIIIITII

o o 20 0 60100
x/L (Location)

.0
c

Figure 7.2 Comparison of induced surface fields for TM excitation of three layered
medium where the upper layer has a sinusoidal interface (PGF IE).

260

0.80

0.60

.0
4:.
O

Magnitude (Hbs1)

.0
N
o

0.00

Figure 7.3

Backscattered field from a three layer medium
d L=O.1, L=O.1m, 9.=85°, h/L=0.04
leld Point: x/L=O, z/L=20 (FMM—RH)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: -— PEC third layer

3 ----- No third layer

‘ - 1 1

1 h

d ‘s .- \r~1

-II ITIIII'IIIIIIIII:I-I-I-III:I-T|::I;::v:l::|’:IIIIIII|:‘]
.20 2.20 3.20 4.20 5.20 6.20

Freq. (Ghz)
Frequency response of backscattered fields for a TM plane wave incident

upon a three layer medium.

261

Transient backscattered field from three layer medium
d L=O.1, L=O.1m, 9.=85°, h/L=0.04
leld Point: x/L=O, z/L=20 (FMM—RH)

3E+008

2E+008 A A

 

1 E+008 h

 

.3

—~

 

 

‘_--
-

OE+OOO

 

 

§
___—

 

 

 

 

‘
L

 

 

 

 

g:

V
—
-
‘-

 

 

 

-1E+008

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Amplitude

 

-2E+008

 

 

-— PEC third layer
—3E+008 ----- No third layer

 

O
—4E+OOU|IIIIIIllI]IIIIIIIIIIIIIIIIIIIIIIIII

7.00 9.00 11.00 13.00
Time (ns)

Figure 7 .4 Transient backscattered fields from a three layer medium where the upper
interface is sinusoidal.

262

Transient total scattered field inside middle layer

d L=O.1. L=O.1m, 0.=85°. h/L=0.04

IeId Point: x/L=O. z/L=-0.08 (FMM—RH)
8E+009

6E+009

4E+009

2E+009

 

 

Relative Amplitude
O
rn
+
O
O
O

 

 

   

— PEC third layer
----- No third layer

 

—6E+009
—8E+009|IIIIIIIIIIIIIIIIIIIlIrIIIIIIfi
9.00 1 1.00 13.00 15.00
Time (ns)

Figure 7 .5 Transient total scattered field inside the middle layer of a three layer
medium where the upper layer has a sinusoidal interface.

263

7 .6 Conclusions

A number of improvements have been made concerning the theoretical scattering
models. A three layer ocean model has been analysed. An additional planar layer which
can be located above or below a rough periodic surface and is composed of a general
homogenous medium has been added to the existing theory. This additional layer could
possibly model a number of physical ocean characteristics, such as a temperature gradient
layer, or salinity gradient layer, or a ducting region or even a large submerged
conductor. Not only has the ocean model been improved, which will produce better
results, but the possibility of surface wave (or trapped wave) propagation can now be
examined. Which could shed new light on the target detection problem.

Two frequency domain methods have been employed to solve for the scattered
fields from a three layer periodic medium. Both methods are direct extensions to the
techniques developed for the two layer problem. The classical Floquet Mode-Matching
method has been extended as well as the rigorous Integral Equation (IE) method.

264

Chapter 8
Conclusions and Future Study

8. 1 Summary

In this thesis a basic study of the applicability of a shipboard UWB/SP radar in
the ocean environment was investigated. The study was both theoretical and
experimental. The experimental measurement techniques were summarized in chapter 2.
A detailed survey of the measurement, calibration and post-processing methods was
included along with a comparison of the time-domain and frequency-domain-synthesis
measurement techniques. The experimental measurements were crucial in verifying the
analytic methods.

A realistic UWB/ SP radar would emit an actual transient pulse. This would be
very similar to the time-domain measurement system at MSU. However, the time-domain
system at MSU lacks the bandwidth a UWB/ SP could attain. Therefore, a frequency-
domain-synthesis measurement facility was created, which can yield the necessary
bandwidth. The legitimacy of the frequency domain synthesis technique was, however,
in question. The favorable comparison of the time-domain and frequency—domain-
synthesis measurement techniques in this thesis was, therefore, quite satisfying.

The remaining chapters concentrated on the elucidation of electromagnetic
scattering from an ocean surface. The Floquet mode-matching technique with the aid of
the Rayleigh hypothesis was described in detail for use with infmite sinusoidal surfaces.
Integral equation methods were also developed, in chapters 3 and 4, for use with infinite
periodic and finite ocean surfaces. The numerical implementation of both methods was
discussed and a wide range of results shown. The rigorous IE method was the primary

choice for most applications. The FMM/RH technique provided numerous results and

265

served to validate the IE theory. These frequency-domain methods all required an IFFT
to yield the desired transient response. The experimental measurements were compared
with great success with the theoretical response from the ocean models utilized in this
study.

The ocean models were chosen to systematically test the analytic techniques and
to methodically increase the physical understanding. A PEC sinusoid was the starting
point. The ocean model was improved to a lossy sinusoidal interface. A planar third layer
was added to further improve the ocean modelling capabilities. The final and most
important addition was the case of non-sinusoidal and aperiodic surfaces. The transient
responses were examined in the hopes of improving the fundamental understanding and
to provide possible applications of UWB/SP radar.

In chapter 6 a limited-footprint UWB/SP radar system was simulated. The
incident beam was constructed from monochromatic plane waves. The construction of
the illuminating beam allowed for simple use of the previously developed scattering

theories and was applicable to any of the ocean surface realizations.

8.2 Major contributions and findings.

The most important and new contribution is the short-pulse transient response of
these ocean-like surfaces. The underlying physics has never been examined for such
excitation. Therefore, the rigorous IE method was needed to include all possible physical
effects. The bulk of the analytic techniques were for the first time rigorously
implemented. The earlier workers in this field had relied on numerous assumptions to
decrease the numerical complexity of the problems. Analytically, the application of beam
excitation to an infinite surface was a new contribution, along with the extension of the
FMM/ RH and IE method to two and three layer structures.

The transient responses from the ocean models revealed many new attributes
associated with the short-pulse excitation. The backscattered fields were seen to be

dominated by surface slope and wave crests. The resolution of the MI SP system

266

allowed for individual wave crest reflections to be seen. The larger sloped waves, such
as the Stokes wave, had enhanced backscatter and multiple specular reflections. The
resolution capabilities of the UWB/ SP radar would also mitigate multipath effects.

The wide bandwidth of spectral response, inherent in the transient returns,
provided new and interesting findings. The F loquet-mode spikes associated with periodic
surfaces were unmistakable in both the theoretical and experimental results. The
frequency selectivity of the total scattered field points to a possible application with target
detection. A target would be illuminated by this total scattered field and its response
would be affected by this selectivity.

The comparison of the various wave models was new and enlightening. The
frequency and transient response from the double-sinusoid model was compared with
sinusoidal, Stokes and Donelan Pierson wave models. The correlation between the
frequency band and the scattered fields from the wave models were quite amazing. The
lower frequencies could resolve the main swell, while the high frequencies were
responsible for multiple scattering or small features.

The success of the experimental measurements and the validation of the theoretical
models was a major contribution. The measurements also provided the impetus for
seeking a link between the infinite and finite surface responses. The theoretical and
experimental scattering from the finite surfaces were compared quite favorably with the
response from an infinite surface. The beam excitation of an infinite surface was also

seen to resemble the response from a finite surface.

8.3 Future research

There are numerous avenues in which these fundamental ideas can be expanded.
The utilization of approximations for the theoretical methods should be examined. The
results in this thesis have all been rigorous and it would be numerically prudent to

research possible approximations. These approximations would also be needed if the

267

wave length of the ocean was to be increased. The ocean models could also be improved
by including some randomness or aperiodicity.

The transient scattered fields from the ocean models are the first step in
developing new target detection and identification techniques. The incorporation of a
target in the scattered response is now necessary in order to proceed in developing target
detection schemes. The future workers in these areas will benefit from the ideas set forth

in this thesis.

268

Appendix A

Appendix A
Periodic Green’s Function Derivation

The periodic Green’s function (PGF) is defined as the solution to

 

626 +29 +k2G = — i: e”’°”'”"d[x-(x’+mL)]6(z-z') (A-l)
8x2 622 m.--

and consequently has the periodicity property G(x + mL,z |x ’,z ’) = exp( -jm BL) G(x,z |x ’,z ’) .
As a result of the latter periodicity, G(x,z|x’,z’) has the Floquet-mode series

representation

G(x,z|x’,z’) = f: gn(x’,z|z’)e'jp“’ (A.2)

where the g" are Floquet-mode amplitudes and B" = [3 +2mr/L. Since representation

(A.2) has the requisite periodicity, then it is sufficient that G satisfy

22$

6 2 622 +k2G = -6(x-x')6(z-z’) (A-3)
x

in the first period O<x< L of the ocean surface. Substituting representation (A.2) into

definition (A.3) leads to

i h,,(JC’,ZIZ/)e-jfl"t = ~6(x-x’)6(z-z’) (A.4)

where h" is defined by
62
[01—2- +Q:]8n(x/,Z|Z3 = hn(x’,zlz’) (A.5)
with wavenumber parameter 4" defined by q: = k2 433,.

269

Expression (A.4) leads to

i“, hnfo‘ejt-r‘e‘mdx = —6(z-z’) [0" 6(x-x’)e""-1'dx (A.6)

and exploiting the Floquet-mode orthogonality property
fo‘eft-re‘wax = Lam (41.7)

provides

1133’

 

h,.(x’.z|z’) = -e 5(z-z’) . (A3)

Using result (A.8) in definition (A.5) results in

I

62 2 / _ em"
(Q +q.)s,.(x,z|2’) - - L

 

6(z -z’) 0"”

which is a differential equation for the Floquet-mode amplitude coefficients gn(x’,z lz’).
A solution to (A.9) is obtained by the Fourier transform technique, in which the
transform §n(x’,C Iz) = .9; {gn(x’,z |z’)} , where C is the spatial-frequency transform

variable, is introduced through the transform pair

§n(x’,C lz’) = f: gn(x’,z |z’)e '1'“ dz

.. . (A.10)
8,.(x’,z|z’) = 51; f4 g",(x’.C|z’)e"’dC

Transforming (A.9) term by term leads, subsequent to application of the differentiation

theorem, to

jp‘t’e 'jCZ/

’(Cz ’43)8".(x’.C|2’) = - e L (41.11)

Solving for g'" and inverse transforming yields

270

163’ Mr")
, . £— . e d (41.12)
gn(x ’zlzl) 211-L "° (C -q,,)(c +qn) C

 

where the only singularities of the integrand are simple poles at C = iqn. Here
q" = —j [3,2, -k2 with branch cut chosen such that Imlqn} <0 to insure that +4» resides
in the lower half of the complex C plane. The integral in (A. 12) can be evaluated by
contour integration. Since exp[jC(z -z ’)] = exp[-Cl.(z -z’)]exp[jC,(z —z’)] , the real axis
integration contour should be closed along an infinite semicircle in the upper half plane
(enclosing the pole at C = “1») for z>z’ , leading to the residue contribution

. 1112’ e -Iq..(z-z’)

_ I e 214 (A.l3)

g..(x’.z|25 =

If z<z’ , then closure should be made along an infinite semicircle in the lower half plane

(enclosing the pole at C =qn), leading to the residue contribution

- I _ /_
2115,; e Jq..(z z)

 

 

xi = -1 (A.l4)
g,.( .zlz') 214"
The expressions in (A.l3) and (A.l4) can be combined in the form
- iw’ -iq.|z-Z’l
8,.(1’,Z|Z') = _,. 8 141-15)

214,,

and exploiting result (A. 15) in Floquet-series representation (A.2) provides the desired

periodic Green’s function

1» e -il3,.(x-x’)e -iq.|z-z’l

Gx x’ = -l-
(xl .2) 2L,,;.. q"

(A.l6)

 

An alternate derivation of the Periodic Green’s Function can be found in [15].

271

Appendix B

Appendix B

Sea Surface Realizations

There has been, and continues to be, an enormous amount of research directed
at accurately and dependably depicting ocean waves. This appendix will provide some
of the details of the sea surface models employed throughout this thesis. The most
primitive model is the sinusoid, this model is not only periodic, but also very smooth,
with relatively small surface slopes.

The multiple sinusoid model, is also smooth and periodic, but it contains some
larger surface slopes and mimics two-scale roughness. Two-scale roughness is an
important feature to test the scattering theory against. Many theoretical methods use
approximations that ’fall apart’ when confronted with two-scale roughness.

The Stokes Wave [52] has also been extensively examined, this wave is rather
smooth and periodic, but it contains extremely large surface slopes. The large surface
slopes are also an important wave feature. Once again some scattering theories break
down for larger surface slopes (e.g. Rayleigh Hypothesis). The Stokes wave is also a

form of a cycloidal wave, and can be expressed mathematically by,

kx = Bk sin(O) + 6
(13.1)

n = —13cos<e) + glam-3116‘ cos(zkx)
where n is the surface function (vertical displacement) and x is the horizontal
displacement. 6 is the parameter of this parametric curve. The B,k are wave shaping
constants.
The final sea surface realization that has been considered is the Donelan-Pierson

wind driven wave [53-56]. This model has been derived from actual ocean

272

measurements, and represents the most accurate model in consideration. The Donelan-
Pierson spectrum is the spectral representation of the spatial frequencies contained in the

wave, and this spectrum is dependent on wind speed.

 

 

M) = sum] ¢"’"K"'K'%"'"K’\J “alumna?

where C(K) is a complex Gaussian random function, with unit variance, and no
correlation between disjoint spatial wave numbers (K).

The Donelan-Pierson spectrum (D or is given by,

 

 

 

- - _i__
¢Dp(|K|) : 1.62x10 3 U . e m2 (1.217)‘, 1.7F((7,|K|) . h (3.3)
“('35 8.5
where
- 2
F(U,|K|) = exp -1.22 (1'2 U VIK' - 1) J
1/E
KP = ——g _
( 1.2 U )2
1.24 o < |K|/Kp < 0.31
h = 2.61 (|K|/Kp)"5 0.31 < |K|/Kp < 0.9

2.28 (Kp/lKl)“ 0.9 < liq/Kp < 10.0

The constants g and (7 are 9.81 m/s2 and average wind speed at 10m above ocean
surface, respectively.

A linear realization of the Donelan Pierson spectrum is to apply,

 

,famw) AK (3.4)

as a spatial filter to a collection of uncorrelated, unit-variance complex variables.
In Figure 8.1 the Donelan Pierson spectrum (or spatial filter) is shown for a

number of wind speeds. The spatial frequency K 9 marks the peak of the spectrum. The

273

wind wave will be composed of an infinite continuous spectrum of spatial frequencies,
but depending on wind speed the majority of spatial frequencies will come from a
particular spatial frequency band. As the wind speed increases the peak in the spectrum
is decreasing in spatial frequency, but is increasing in magnitude, this implies that larger
wavelength and larger amplitude waves are being driven. This is consistent with practical
experience; when the wind speed is low, only small ripples are seen, and as the wind

speed increases larger waves appear (whitecaps).

Donelan Pierson Spectrum for various windspeeds.

 

 

 

§ 1 K,=.14 1/m
E I
0.80 -_' '
_ l ----- U = 5 m/s 11.2 mph;
2 : —— U = 7 m/s 15.7 mfih
Z I WU=3m/s 67mp)
E 3 1 K,=.27 1/m
:1 _. l 1
413 0.60 1 I 1
o - 1 1
8 I l 1
(n i '
1 I I
Q_ - I 1
Q 0.40 g | 1
1 1 I
; I ,’I\\\
: I I, I \\
: I I I \
0-20 t i,’ : \ K,=.75 1/m
E y 1 “~‘ I
_ l ‘~~-1
I ’l | -~
: [II I 1 2 - I q‘ ----
0.00 """ I'lllrttllllltllllrlltlfi
0.00 0.40 0.80 1.20

K (l/m)

Figure B.1 The Donelan Pierson spectrum of wind driven waves for various wind

speeds.

274

Appendix C

Appendix C
Extinction theorem integral equation formulation

This appendix provides the details for utilizing the extinction theorem in the
formulation of the integral equations used throughout this thesis. This subject is covered
in many graduate level electromagnetic books [49] , but is included here due to the
extensive reliance upon the technique.

The configuration of the two-dimensional two-region problem is shown in
Figure C.1. There is a source region (Q(p')), two surface regions separated by a closed
contour (C) and a small surface containing the observation point surround by C1. Region
1 is characterized by wavenumber k1 and the scalar wave function 1111(6) , while region
2 is characterized by k2 and 1|12(p°).

The wave function in region 1 must obey the forced Helmholtz equation due to

the location of the source in region 1,
(V2 + k?) 1.16) = -Q(ii) (C.1)
The Green’s function for that region can be defined as
(V’ + him 66’) = -6(i>° - a) (c.2)

At this point the use of the 2—D Green’s theorem over the entire surface area of

region 1 is necessary. The 2-D Green’s theorem is given by

fs(u Vzv - szu)ds = [C(ug — vgflnwl (03

By letting u = w(B) and v = g1('p’,p”) and utilizing equations (Cl) and (C2),

Green’s theorem can be seen to yield

275

..,
11.16): lubed-i313 gl(5,51)§1_1:a(;g_)]d,, (Q4)

where 411(5) = Low gl(p°,p "’ ) Q(p’)ds’ and IS the surviving surface integral. The line
integral at infinity (Ca) has no contribution due to the radiation condition.

The location of the observation point (B) will effect the evaluation of the inteng
in (C4).

5 6 region 1 1|11(B)
p’ 6 region 2 0 = (C5)
6ec law)

681(p.6 ’)_

Mp) + PVf [1m (6’) gas 13")—

where PV indicates a principal-value integration and is only needed if B is on C , where
there is a source-point singularity.

When 6 is in region 2 there is a null field and this is known as the extinction
theorem. Since there is no field in region 2 for this equivalent problem the medium in
region 2 is arbitrary. Throughout this thesis region 2 is replaced with the same medium
as region 1, which allows for the use of a homogenous Green’s function. Equation (C5)
is also a statement of Huygen’s principle.

A similar development can be followed for the fields (wave function) in region
2. In this case there is no source and therefore no incident field. The resulting integral
equation for the wave function is given by

p’ 6 region 2 412(5) ‘°’
.. . (06)
p 6 region 1 0 = PVfC[1|12(f5/)%Z ‘ 32( 5,5,)a¢_2(f—) Idl,

:5 e C $1.15)

Equations (C6) and (C5) form the backbone of the integral equation formulation used

throughout this thesis. The typical application is to enforce the two equations at the sea-

276

surface interface (C) with the boundary conditions linking the field quantities. This
results in a pair of coupled integral equations for the unknown surface field quantities.
The special case of a PEC interface decouple the equations and only requires the

use of (C5).

 

”'1'". "I...“
W...»- SOUI'CGG “tn“...
'1'” Region 1 (101 k1) ‘~\
av ..
P c 7; 1.
E: ! observation:g
3‘ point 5:?
'a_ i
3“ Z 3 :i
“\ ,1":
,4
"‘1...“ I X 'I',1I"'””b 0°

Figure C.1 Generic configuration of a 2-D two-region problem.

277

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