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Numerical calculations of microwave scattering from dielectric structures used in vegetation models

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Num erical Calculations of Microwave Scattering from
D ielectric Structures used in V egetation M odels
by
P a o lo de M a tth a e is
L a u re a D egree in E lecrical E n g in e e rin g
J u n e 1991, U n iv e rsity o f R o m a “T o r V e rg a ta ” , R o m a , Ita ly
A d is s e rta tio n s u b m itte d to
th e F a c u lty of
th e S chool o f E n g in e e rin g a n d A p p lie d Science
of T h e G eo rg e W a sh in g to n U n iv e rsity
in p a r tia l fu llfillm en t of th e re q u ire m e n ts
for th e d eg ree of D o c to r of Science
J a n u a r y 31, 2006
D is s e rta tio n d ire c te d by
R o g e r H e n ry L a n g
P ro fe sso r o f E n g in e e rin g a n d A p p lie d Science
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UMI Number: 3203663
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A bstract
N um erical models for electrom agnetic scattering from individual cpm onents of a vegetation canopy
are studied. T he m odels are based on a surface integral representation of the electrom agnetic fields
on th e m aterial bodies representing th e vegetation constituents.
Surface integral equations are
obtained by using th e equivalence principles, and th eir unknowns are equivalent surface currents.
From the equivalent surface currents, the b istatic scattering coefficient is evaluated.
T he m ethod of m om ents is applied to solve these integral equations in the case of two-dim ensional
objects and of axisym m etric scatterers. T h e p a rticu la r geom etry of these bodies allows a sim plifica­
tio n of the num erical solution. T he results are validated for a num ber of shapes, for b o th perfectly
conducting and hom ogeneous dielectric bodies. T he num erical approach is then com pared to a n a ­
lytical approxim ate m odels for scattering from vegetation com ponents.
In order to take into account th e presence of th e ground under a vegetation canopy, a sem i­
infinite half-space is introduced in th e modeling^ a n d th e surface integral equations are m odified to
consider its effects. T h e resulting integral equations contain a dyadic G reen’s function for th e half­
space. In particular, axisym m etric objects are considered, an d the dyadic G reen’s function is w ritte n
in term s of cylindrical waves. T he solution by th e m eth o d of m om ents requires th e calculation of
Som merfeld-type integrals, an d a brief discussion of th e issues related to their evaluation is included.
T he half-space num erical approach is finally applied to th e problem of m odeling tree tru n k s
over a fiat ground. Its results are com pared w ith those of an analytical approach th a t neglects the
near field interaction of th e tru n k a n d th e surface of th e ground.
ii
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Acknowledgm ents
My sincere appreciation goes first to my advisor, Prof. Roger H enry Lang. W ithout his guidance,
patience a n d su p p o rt th ro u g h my doctoral studies, it would have been impossible for me to com plete
this work.
I would also like to th a n k Prof. W asyl W asylkiwskyj and Prof. W alter K ahn, for all w hat I
have learned in th eir E lectrom agnetics classes, an d extend my appreciation to all the m em bers of
the D issertation Com m itee for th eir insightful com m ents and advice.
M any thanks also to Dr. Allen Glisson for providing a useful num erical code for com parison of
results, and to fellow g rad u ate stu d e n t C uneyt U tk u for useful advice and discussions.
Last, b u t not least, I would like to express my g ratitu d e to my fam ily and friends who have
supported and encouraged me th roughout the m any years of stu d y leading to the com pletion of
this degree, and to th e E C E D epartm ent secretaries D eborah Sw anson and Sariane Leigh for their
adm inistrative help.
m
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D edication
This d issertatio n is dedicated to my dad A ugusto, and in m em ory of my m om Adele.
IV
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C ontents
A b stra ct
ii
A ck now ledgm ents
iii
D ed ica tio n
iv
C on ten ts
v
List o f F igures
ix
List o f Tables
xxii
1 In trod u ction
2
1
1.1
M otivation
........................................................................................................................................
1
1.2
B a c k g ro u n d ........................................................................................................................................
2
1.3
O v e r v ie w ...........................................................................................................................................
6
T he surface in tegral eq u a tio n form u lation
9
2.1
T he equivalence p r i n c i p l e .............................................................................................................
10
2.2
Surface integral equations in unbounded space
....................................................................
16
2.3
O ther forms of coupled surface integral e q u a tio n s .................................................................
19
2.4
A lternative form for th e dyadic G reen’s f u n c t i o n .................................................................
21
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2.5
S cattered field and scattering c o e ff ic ie n t..................................................................................
3 T w ord im en sion al b o d ies
21
27
3.1
T he coupled surface integral equations in two-dim ensions
.................................................
28
3.2
Far-field and scattering w i d t h .....................................................................................................
30
3.3
Perfect electric conducting two-dim ensional b o d y ................................................................
32
3.3.1
T M incident wave
.............................................................................................................
32
3.3.2
T E incident w a v e .................................................................................................................
41
Dielectric two-dim ensional b o d y .................................................................................................
50
3.4.1
.............................................................................................................
50
C o n c lu s io n s .......................................................................................................................................
61
3.4
3.5
TM incident wave
4 B od ies o f R ev o lu tio n
63
4.1
G eom etry of th e p r o b l e m ...........................................................................................................
63
4.2
T he electric field integral equations for a body of r e v o lu tio n ..............................................
65
4.2.1
F o rm u la tio n ............................................................................................................................
65
4.2.2
Separation of m o d e s ...........................................................................................................
67
4.2.3
M ethod of m om ents s o l u t i o n ..........................................................................................
70
4.2.4
Incident f i e l d ........................................................................................................................
73
4.2.5
S cattered field in the rad iatio n z o n e ............................................................................
75
V a lid a tio n ..........................................................................................................................................
77
4.3
4.4
4.3.1
Perfectly electric conducting bodies of r e v o l u t io n ....................................................
78
4.3.2
Homogeneous dielectric body of re v o lu tio n .................................................................
89
A pplication to scattering from vegetation
............................................................................
96
4.4.1
F in ite-length dielectric cylinder a p p r o x i m a t io n .......................................................
97
4.4.2
T apered cylinder approxim ation
.....................................................................................100
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4.4.3
R esults an d c o m p a riso n s.....................................................................................................103
4.4.4
C o n c lu s io n s ............................................................................................................................ 116
5 H alf-Space P ro b lem
117
5.1
T he dyadic G reen’s function for an infinite, homogeneous m e d i u m .................................... 118
5.2
T h e dyadic G reen’s function for sem i-infinite m e d ia .................................................................122
5.2.1
5.3
T he dyadic G reen’s function in th e far
S cattering from a B ody in a Sem i-Infinite M e d i u m ................................................................128
5.3.1
Expression of Surface Integral E quations for Sem i-Infinite M e d iu m .....................128
5.3.2
Solution of th e Surface Integral E quations for a B O R in H alf-S p ace.................... 132
5.3.3
Expression of “incident” field on th e surface S
5.3.4 C o m p utation of the scattering coefficient
5.4
f i e l d ....................................................125
..........................................................136
.....................................................................140
E valuation of sp ectral i n t e g r a l s .................................................................................................... 141
5.4.1 In tegration by T rapezoidal Rule
......................................................................................143
5.4.2 Tank T ra n sfo rm a tio n .............................................................................................................. 144
5.4.3 W eighted-Averages M e t h o d ................................................................................................145
5.4.4 V alidation of Integration M ethods
5.5
6
.................................................................................. 147
V a lid a tio n ............................................................................................................................................. 150
5.5.1
Perfectly conducting body of r e v o l u t i o n ...............................
5.5.2
Dielectric body of revolution
151
............................................................................................ 153
5.6
A pplication
to scatterin g from vegetation
.......................................................................154
5.7
C o n c lu s io n s ..........................................................................................................................................163
C onclusions
167
B ibliography
171
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A p p en d ix
A E lectro m a g n etic fields in u n b ou n d ed m edia
A .l
181
E lectrom agnetic fields generated by electrom agnetic s o u r c e s .............................................. 181
A.2 G reen’s theorem for a m ultiply connected r e g i o n ...................................................................184
A.3 D erivation of expression (2.17)
186
A.4 E valuation of surface integral in (2.22) at singularity p o i n t ..................................................188
A. 5 Equivalence of alternative dyadic G reen’s f u n c t i o n ............................................................... 190
B
D erivation o f tw o-d im en sion al G reen ’s fu n ction
193
C
B od ies o f R ev o lu tio n
195
D
C .l
Explicit form of operators a p q ....................................................................................................... 195
C.2
Explicit form of operators (3p q ....................................................................................................... 196
C.3
D erivation of operators a pq
.......................................................................................................... 199
C.4
D erivation of operators /3pq
.......................................................................................................... 201
C.5
D erivation of scattering a m p litu d e s ............................................................................................ 202
H alf-Space P ro b lem
205
D .l Expression of dyadic G reen’s function c o m p o n e n t s ............................................................... 205
D.1.1
Electric dyadic G reen’s f u n c t i o n s ....................................................................................205
D .l.2
M agnetic dyadic G reen’s f u n c t i o n s ................................................................................ 207
D.2 D erivation of scattering a m p litu d e s ....................
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208
List o f Figures
2.1
D efinition of surfaces 5 i and 52....................................................................................................
11
2.2
Equivalent sources configuration..................................................................................................
13
2.3
E xterior problem ...............................................................................................................................
15
2.4
Interior problem .................................................................................................................................
17
2.5
Definition of geom etry for scattered field calculation............................................................
21
2.6
Incident an d scattered plane waves on an o b ject....................................................................
22
3.1
Geom etry of a tw o-dim ensional b o d y ........................................................................................
26
3.2
Definition of scatterin g problem for a tw o-dim ensional b o d y ............................................
28
3.3
(a) D iscretization of curve C. (b) Basis (expansion) functions..........................................
31
3.4
TM and T E incident waves on a circular cylinder of radius a ...........................................
33
3.5
Surface equivalent current obtained w ith m eth o d of m om ents and exact solution for
k{]<a = 0.1 a n d num ber of points N — 12. N um ber of term s in analytical expression
sum m ation is M = 10.......................................................................................................................
3.6
34
Surface equivalent current obtained w ith m eth o d of m om ents and exact solution for
koa = 1.0 an d n um ber of points N = 20. N um ber of term s in analytical expression
sum m ation is M = 15.......................................................................................................................
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35
3.7
Surface equivalent current obtained w ith m ethod of m om ents an d exact solution for
koa = 2ir and num ber of points N = 100. N um ber of term s in analytical expression
sum m ation is M = 20.......................................................................................................................
3.8
35
Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
k^a = 87r and num ber of points N = 500. N um ber of term s in analytical expression
sum m ation is M = 40.......................................................................................................................
3.9
36
Scattering w idth obtained w ith m ethod of m om ents an d exact solution vs. koa, for
scattering angles 0 = 0° (backscattering) and 0 = 90° (forw ard scatterin g )....................... 36
3.10 Scattering w idth obtained w ith m ethod of m om ents a n d exact solution vs. <j>3, for
koa — 0.1, num ber of M OM points N = 12, and analytical expression tru n cated to
M = 10 term s......................................................................................................................................
37
3.11 Scattering w idth obtained w ith m ethod of m om ents an d exact solution vs. <f>s, for
koa = 1.0, num ber of M OM points N — 20, and analytical expression tru n cated to
M — 15 term s......................................................................................................................................
38
3.12 Scattering w idth obtained w ith m ethod of m om ents a n d exact solution vs. <f>s, for
koa = 2n, num ber of M OM points N = 130, and an alytical expression tru n cated to
M = 20 term s......................................................................................................................................
38
3.13 Scattering w idth obtained w ith m ethod of m om ents a n d exact solution vs. 0 S, for
koa = 87r, num ber of M OM points N = 500, and an alytical expression tru n cated to
M = 40 te rm s ...........................................................................................................................................39
3.14 Surface equivalent current obtained w ith m ethod of m om ents a n d exact solution for
koa = 0.1 and num ber of points N = 12. N um ber of term s in analytical expression
sum m ation is M = 10........................................................................................................................
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42
3.15 Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa = 1.0 and num ber of points N = 20. N um ber of term s in analytical expression
sum m ation is M = 15.......................................................................................................................
43
3.16 Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa — 2n an d num ber of points N = 130. N um ber of term s in analytical expression
sum m ation is M = 20.......................................................................................................................
43
3.17 Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa = 87r and num ber of points N = 500. N um ber of term s in analytical expression
sum m ation is M = 40.......................................................................................................................
44
3.18 C om parison betw een scattering w idth obtained w ith m eth o d of moments and exact
solution as functions of koa for scattering angles <f>— 0° (backscattering) and <f>= 90°
(forward sc atterin g )...........................................................................................................................
45
3.19 C om parison betw een scattering w idth obtained w ith m eth o d of m om ents and exact
solution for koa = 0.1 and num ber of points N = 12.. N um ber of term s in analytical
expression sum m ation is M —... 10................................................................................................
46
3.20 C om parison betw een scattering w idth obtained w ith m eth o d of m om ents and exact
solution for koa = 1.0 and num ber of points N = 20. N um ber of term s in analytical
expression sum m ation is M = ... 15................................................................................................
46
3.21 C om parison betw een scattering w idth obtain ed w ith m eth o d of m om ents and exact
solution for koa = 2 tt and num ber of points N = 100.. N um ber of term s in analytical
expression sum m ation is M =
20................................................................................................
47
3.22 C om parison betw een scatterin g w idth o b tained w ith m eth o d of m om ents and exact
solution for koa = 87t and num ber of points N = 500.. N um ber of term s in analytical
expression sum m ation is M =
40................................................................................................
xi
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47
3.23 Equivalent electric current, m ethod of m om ents solution and theory, vs. <f>, for a
lossless hom ogeneous cylinder w ith er = 2.56, for koa = 1.0 and koa = 5.0. N um ber
of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively.
. .
51
3.24 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <f>, for a
lossless hom ogeneous cylinder w ith eT = 2.56, for koa = 1.0 and koa = 5.0. N um ber
of points used in th e m ethod of m om ents are N — 20 a n d N = 100, respectively.
. .
51
3.25 Equivalent electric current, m ethod of m om ents solution a n d theory, vs. (f>, for a lossy
homogeneous cylinder w ith er = 2.56 —jO. 102, for koa = 1.0 and koa — 5.0. N um ber
of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively.
. .
52
3.26 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. (f>, for a
lossy hom ogeneous cylinder w ith er — 2.56, koa = 1.0 and koa — 5.0. N um ber of
points used in th e m ethod of m om ents are N = 20 and N = 100, respectively. . . . .
52
3.27 Equivalent electric current, m ethod of m om ents solution and theory, vs. <j>, for a
lossy hom ogeneous cylinder w ith er — 5.0 —j 1.0, koa = 1.0 and koa = 5.0. N um ber
of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively.
. .
53
3.28 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <fi, for a
lossy homogeneous cylinder w ith er = 5.0 —jl.O , koa — 1.0 and koa = 5.0. N um ber
of points used in th e m ethod of m om ents are N = 20 an d N = 100, respectively.
3.29 Equivalent electric current, m ethod of m om ents solution and
lossy homogeneous cylinder w ith er = 60 —j 59.9,
. .
53
theory, vs. (j>, for a
koa = 1.0 and koa = 5.0. N um ber
of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively.
. . 54
3.30 Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <f>, for a
lossy homogeneous cylinder w ith er = 60 —j’59.9,
koa = 1.0 and koa = 5.0. N um ber
of points used in th e m eth o d of m om ents are N = 20 and N = 100, respectively.
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. .
54
3.31 S cattering w idth, m ethod of m om ents solution and theory, size param eter koa, for
a lossless hom ogeneous cylinder w ith er = 2.56, for koa = 1.0 a n d koa = 5.0. Solid
line is backscattering (cf>s = 0°), and dashed line is forw ard scatterin g (<fis = 180°).
. 55
a lossy hom ogeneous cylinder w ith er = 2.56 —j'0.102, for koa = 1.0 and koa = 5.0.
Solid line is backscattering (<ps = 0°), and dashed line is forw ard scattering {<f>3 =
180°).......................................................................................................................................................
55
3.33 S cattering w idth, m ethod of m om ents solution an d theory, size param eter koa, for
a lossy hom ogeneous cylinder w ith er = 5.0 —j l.0 , koa = 1.0 a n d koa = 5.0. Solid
line is backscattering (<j>3 = 0°), and dashed line is forw ard scatterin g (<ps = 180°).
. 56
3.34 S cattering w idth, m ethod of m om ents solution and theory, size param eter koa, for
a lossy hom ogeneous cylinder w ith er = 60 —j 59.9, koa = 1.0 a n d koa — 5.0. Solid
line is b ackscattering (<j>s = 0°), and dashed line is forw ard scatterin g (<f>3 = 180°).
. 56
3.35 Scattering w idth, m ethod of m om ents solution a n d theory, vs. scattering angle <ps,
for a lossless hom ogeneous cylinder w ith er = 2.56, for koa = 1.0 and koa = 5.0.
N um ber of points used in the m ethod of m om ents are N — 20 and N — 100,
respectively...........................................................................................................................................
57
3.36 Scattering w idth, m ethod of m om ents solution a n d theory, vs. scattering angle <ps, for
a lossy hom ogeneous cylinder w ith er — 2.56 —j 0.102, for koa = 1.0 and koa = 5.0.
N um ber of points used in the m ethod of m om ents are N = 20 and N = 100,
respectively...........................................................................................................................................
57
3.37 Scattering w idth, m ethod of m om ents solution an d theory, vs. scattering angle <ps, for
a lossy hom ogeneous cylinder w ith er = 5.0 —j l .0 , kcja = 1.0 and koa = 5.0. N um ber
of points used in th e m ethod of m om ents are N = 20 and N = 100, respectively.
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. . 58
3.38 Scattering w idth, m ethod of m om ents solution and theory, vs. scattering angle <f>s ,
for a lossy homogeneous cylinder w ith eT = 60 — j59.9, koa = 1.0 and koa = 5.0.
N um ber of points used in the m ethod of m om ents are N = 20 and N = 100,
re s p e c tiv e ly ........................................................................................................................................
58
4.1
Body of r e v o l u t i o n ..........................................................................................................................
61
4.2
(a) Choice of discretization points {t*,} on the curve C. (b) Discretized curve C.
. .
66
4.3
Definition of expansion functions: (a)
(b) P m \ t ) , and (c) Tm(t)........................
68
4.4
G eom etry of various bodies of revolution...................................................................................
74
4.5
Surface currents vs. t /A on a P E C disk w ith radius a = 1.5A, for 0i = 0°. C om parison
between results of th e present m ethod (BO R ), W ilton-G lisson (WG), and M autzH arrington (MH) solutions.............................................................................................................
4.6
76
Surface currents vs. t /A on a P E C washer w ith radii oi = 0.4A and a\ = 1.2A, for
9i = 0°. C om parison betw een results of th e present m ethod (BOR), W ilton-G lisson
(W G), and M autz-H arrington (MH) solutions.........................................................................
4.7
76
Surface currents vs. t f A on a P E C open-ended cylinder of radius a = 0.4A and length
L = 1.2A, for 0i = 0°. Com parison betw een results of th e present m ethod (BOR),
W ilton-G lisson (W G ), and M autz-H arrington (MH) solutions...........................................
77
4.8 Surface currents vs. t j A on a P E C cone-sphere of rad iu s a — 0.2A and length L =
1.2A, for 0i = 0°. C om parison betw een results of th e present m ethod (BOR), W iltonGlisson (W G), a n d M autz-H arrington (MH) solutions..........................................................
4.9
77
Surface currents vs. t / A on a P E C sphere w ith size p a ra m ete r k^a = 1.0. Com parison
between results of th e present m ethod (BOR) an d Mie series (MIE) solution................... 78
4.10 Surface currents vs. t /A on a P E C sphere w ith size p a ra m ete r koa = 10.0. C om par­
ison betw een results of the present m ethod (BO R) a n d M ie series (MIE) solution. . .
xiv
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79
4.11 Surface currents vs. t / A on a P E C finite cylinder of radius a = 0.5A and length
L = A, for 6i = 0° (left) and 9{ = 45° (right). Com parison betw een results of the
present m ethod (BOR) and W ilton-G lisson (WG) solution................................................
80
4.12 Surface currents vs. t/X on a P E C finite cylinder of radius a = 0.5A and length
L = 2.5A, for 9i = 0° (left) an d 0i = 45° (right). Com parison betw een results of the
present m ethod (BOR) and W ilton-G lisson (WG) solution.................................................... 80
4.13 Surface currents vs. t/X on a P E C finite cylinder of radius a = 0.05A and length
L = 2.5A, for Qi — 0° (left) and 9t = 45° (right). Com parison betw een results of the
present m ethod (BOR) and W ilton-G lisson (W G) solution.................................................... 81
4.14 N orm alized b istatic scattering coefficient vs. 9S for a P E C disk of size param eter
koa = 10, for 9i = 0°. C om parison betw een results of th e present m ethod (BOR),
and Hodge solutions..........................................................................................................................
82
4.15 Norm alized b istatic scattering coefficient vs. 9S for a P E C open-ended cylinder of
radius a = 0.4A and length L = 1.2A, for 9i = 0°. C om parison betw een results of
th e present m ethod (BOR) and W ilton-G lisson (WG) solu tio n ..........................................
82
4.16 Norm alized b istatic scattering coefficient vs. 9S for a P E C cone-sphere of radius
a = 0.2A and length L = 1.2A, for plane wave incident at 9{ — 0°. Com parison
betw een results of th e present m ethod (BO R) and W ilton-G lisson (WG) solution. . .
83
4.17 Norm alized b istatic scattering coefficient vs. 9S for a P E C sphere w ith size param eter
koa = 1.0, for
— 0°. C om parison betw een results of th e present m ethod (BOR)
and Mie series (MIE) solution........................................................................................................
83
4.18 Norm alized b istatic scattering coefficient vs. 9S for a P E C sphere w ith size param eter
koa — 10.0, for 9i = 0°. C om parison betw een results of th e present m ethod (BOR)
and Mie series (MIE) solution........................................................................................................
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84
4.19 N orm alized b istatic scattering coefficient vs. 9S for a P E C finite cylinder of radius
a = 0.5A and length L = A, for &i = 0° (left) and Qi = 45° (right). Com parison
betw een results of th e present m ethod (BOR) and W ilton-G lisson (WG) solution. .
. 85
4.20 N orm alized b ista tic scattering coefficient vs. 9S for a P E C finite cylinder of radius
a = 0.5A and length L = 2.5A, for 9i = 0° (left) and 9{ = 45° (right). Com parison
betw een results of the present m ethod (BOR) a n d W ilton-G lisson (WG) solution. .
. 85
4.21 N orm alized b istatic scattering coefficient vs. 9S for a P E C finite cylinder of radius
a = 0.05A and length L = 2.5A, for 9i = 0° (left) and 9{ = 45° (right). Com parison
between results of th e present m ethod (BOR) and W ilton-G lisson (WG) solution. .
. 86
4.22 Equivalent surface currents vs. t j A on a dielectric sphere w ith koa = 1.0 and ec = 4.
C om parison betw een results of the present m ethod (BO R) and Mie series (MIE)
solution..................................................................................................................................................
87
4.23 Equivalent surface currents vs. t j A on a dielectric sphere w ith koa = 1.0 and ec —
4 —j .
C om parison betw een results of the present m eth o d (BOR) and Mie series
(MIE) solution.....................................................................................................................................
87
4.24 Equivalent surface currents vs. t /X on a dielectric sphere w ith koa = 1.0 and ec =
18 —j6. C om parison betw een results of th e present m eth o d (BOR) and Mie series
(MIE) solution.....................................................................................................................................
88
4.25 Equivalent surface currents vs. t / X on a dielectric sphere w ith koa = 10.0 and ec = 4.
Com parison betw een results of the present m ethod (BO R) and Mie series (MIE)
solution..................................................................................................................................................
88
4.26 Equivalent surface currents vs. t j A on a dielectric sphere w ith koa = 10.0 and ec =
4 —j .
C om parison betw een results of th e present m eth o d (BOR) and Mie series
(MIE) solution.....................................................................................................................................
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89
4.27 E quivalent surface currents vs. t/X on a dielectric sphere w ith koa = 10.0 and ec =
18 —j 6 . C om parison between results of th e present m ethod (BOR) and Mie series
(MIE) so lu tio n ....................................................................................................................................
89
4.28 N orm alized b ista tic scattering coefficient vs. 83 for a dielectric sphere w ith koa = 1.0
and ec = 4. C om parison betw een results of the present m ethod (BOR) an d Mie series
(MIE) so lu tio n ....................................................................................................................................
90
4.29 N orm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 1.0
and e c = 4
—
j . C om parison betw een results of the present m ethod (BOR) and Mie
series (M IE) solution.........................................................................................................................
90
4.30 N orm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 1.0
and s c = 18 — j 6. C om parison betw een results of th e present m ethod (BOR) and
Mie series (MIE) solution................................................................................................................
91
4.31 N orm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 10.0
and £c = 4. C om parison betw een results of th e present m ethod (BOR) and Mie series
(MIE) solu tio n .....................................................................................................................................
91
4.32 Norm alized b ista tic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 10.0
and £c = 4
—
j . Com parison betw een results of th e present m ethod (BOR) and Mie
series (M IE) solution ............................................................................................................................. 92
4.33 Norm alized b istatic scattering coefficient vs. 8S for a dielectric sphere w ith koa = 10.0
and ec = 18 — j 6. C om parison betw een results of th e present m ethod (BOR) and
Mie series (M IE) solution......................................................................................................
92
4.34 Geom etry of th e finite-length dielectric cylinder p r o b le m ..................................................
95
4.35
Tapered cylinder m o d e l ...............................................................................................................
97
4.36
Scattering p a tte rn of a c y li n d e r ..................................................................................................
99
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4.37 N orm alized b istatic scattering coefficient vs. scattering angle for finite dielectric
cylinder w ith L = 10.0A, a = 0.04A, and incident angle 9i — 20°..........................................102
4.38 Norm alized b ista tic scattering coefficient vs. scattering angle for finite dielectric
cylinder w ith L = 10.0A, a = 0.4A, eT = 18 — j 6, an d incident angle
= 20° (top)
and 0i — 80° (b o tto m ).........................................................................................................................103
4.39 A bsolute error vs. incident angle for finite dielectric cylinder w ith a = 0.04A, er =
18 — j 6, L = 5.0A, 3.0A, an d 1.0A, for hh- and w -p o la riz a tio n ..............................................104
4.40 A bsolute error vs. incident angle for finite dielectric cylinder w ith a =
0.4A, er =
18 —j 6, L = 10.0A, 5.0A, and 3.0A, for hh- and iw -polarization........................................... 105
4.41 Norm alized b ista tic scattering coefficient vs. scattering angle for tap ered dielectric
cylinder w ith L = 10.0 A, a = 0.7 A, b = 0.1 A, and incident angle 6t = 40°. hh- and
vv- polarization (left and right, respectively)..................
108
4.42 Norm alized b istatic scattering coefficient vs. scatterin g angle for tap ered dielectric
cylinder w ith L = 10.0 A, a = 0.5 A, b = 0.3 A, and incident angle 0, = 40°. hh- and
vv- polarization (left and right, re sp e c tiv e ly )............................................................................. 108
4.43 E rror vs. incident angle for tap ered dielectric cylinder w ith L — 10.0 A, (left) a =
0.7 A, d2 = 0.1 A, a n d (right) a = 0.5 A, b = 0.3A. Solid lines are hh-pol, dashed lines
vu-pol........................................................................................................................................................ 109
5.1
Definition of (k, h , v ) for waves traveling forw ard an d backw ards in the ^-direction.
5.2
Far-field in half-space m e d iu m ...................................................................................................... 122
5.3
M aterial body in half-space m edium ............................................................................................ 125
5.4
Exterior problem for half-space m edium ..................................................................................... 127
5.5
Plane waves incident on a body in a half-space m ed iu m ...................................................... 133
5.6
Reflection a n d transm ission of plane wave across a plane b o u n d a ry ............................... 134
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116
5.7
In teg ratio n p a th of I in the com plex A^-plane............................................................................ 139
5.8
A bsolute error [%] for th e Tank T ransform ation m ethod vs. num ber of integration
points N .............................
5.9
148
S cattering from a vertical P E C open-ended cylinder on a P E C half-space (left), and
equivalent problem (right) in free space........................................................................................ 149
5.10 C om parison betw een P E C open-ended cylinder of length L — 0.5A and radius a =
0.1 A on a P E C half-space, and equivalent image problem , for incident angle 9i =
60°. Left: surface electric currents vs. z/X . R ight: norm alized bistatic scattering
coefficient vs. 9S.................................................................................................................................... 150
5.11 Surface electric currents vs. t/X on a perfectly conducting cylinder of length L = X
and koa = 1.0 located a t a distance d = 0.2A above a hom ogeneous, lossy half-space
w ith relative dielectric constant 16 —j 16. Incident angle is 9i= 0°....................................151
5.12 Scattering from a vertical dielectric cylinder over a P E C half-space (left), and equiv­
alent problem (right) in free space.................................................................................................. 152
5.13 C om parison betw een dielectric cylinder of length L = 0.6A, radius a = 0.1A and a
relative dielectric constant 4 —j on a P E C half-space, and equivalent image problem ,
for incident angle 9i = 45°. E quivalent surface electric currents vs. z/X .
Right:
norm alized b istatic scattering coefficient vs. 9s .......................................................................... 153
5.14 C om parison betw een dielectric cylinder of length L = 0.6A, rad iu s a — 0.1A and a
relative dielectric constant 4 - j on a P E C half-space, an d equivalent image problem,
for incident angle 9i = 45°. N orm alized b istatic scatterin g coefficient vs. 9s....................153
5.15 N orm alized scatterin g coefficient cr/i/! vs. 9S. C om parison betw een analytical approx­
im ation an d num erical solution for a dielectric cylinder of len g th L = 2.5A and radius
a = 0.025A. R elative dielectric constant is 4 —j for the cylinder, and 10 —jo for the
ground. Incident angle is 9i = 30°................................................................................................... 157
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5.16 N orm alized scattering coefficient avv vs. 8S. C om parison betw een analytical approx­
im ation an d num erical solution for a dielectric cylinder of length L = 2.5A an d radius
a — 0.025A. R elative dielectric constant is 4 —j for th e cylinder, a n d 10 —j o for the
ground. Incident angle is 0* = 30°.............................................................................................
5.17 N orm alized scattering coefficient (T/jft, us. 9S. C om parison betw een analytical approx­
im ation an d num erical solution for a dielectric cylinder of length L = 2.5A and radius
a = 0.025A. R elative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the
ground. Incident angle is 6i = 60°.............................................................................................
5.18 N orm alized scattering coefficient a vv vs. 0S. C om parison betw een analytical approx­
im ation an d num erical solution for a dielectric cylinder of length L = 2.5A and radius
a = 0.025A. R elative dielectric constant is 4 — j for the cylinder, and 10 —j 5 for the
ground. Incident angle is 0* = 60°.............................................................................................
5.19 N orm alized scattering coefficient 07^ vs. 8S. C om parison betw een analytical approx­
im ation a n d num erical solution for a dielectric cylinder of length L = 4.0A and radius
a = 0.025A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —jb for the
ground. Incident angle is 0; = 30°.............................................................................................
5.20 N orm alized scatterin g coefficient a vv vs. 9S. C om parison betw een analytical approx­
im ation an d num erical solution for a dielectric cylinder of length L = 4.0A and radius
a = 0.025A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —jo for the
ground. Incident angle is 6i = 30°.............................................................................................
5.21 Norm alized scatterin g coefficient
us. 9S. C om parison betw een analytical approx­
im ation an d num erical solution for a dielectric cylinder of length L = 4.0A a n d radius
a = 0.025A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —j'o for the
ground. Incident angle is 8\ = 60°.............................................................................................
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5.22 Norm alized scattering coefficient avv vs. 9S. C om parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius
a = 0.025A. R elative dielectric constant is 4 — j for the cylinder, and 10 —j 5 for th e
ground. Incident angle is Qi = 60°.................................................................................................. 161
5.23 Norm alized scattering coefficient c r^ us. Qs. Com parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius
a = 0.25A. R elative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the
ground. Incident angle is 6i = 30°.................................................................................................. 162
5.24 N orm alized scattering coefficient a vv vs. 9S. Com parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius
a = 0.25A. Relative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the
ground. Incident angle is Qi = 30°.................................................................................................. 162
5.25 Norm alized scatterin g coefficient a^h vs- Qs- C om parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of len g th L = 2.5A and radius
a = 0.25A. R elative dielectric constant is 4 —j for th e cylinder, and 10 —j 5 for the
ground. Incident angle is Qi = 60°...................................................................................................163
5.26 Norm alized scatterin g coefficient a vv vs. Qs. C om parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of len g th L = 2.5A and radius
a = 0.25A. R elative dielectric constant is 4 — j for th e cylinder, and 10 —j 5 for the
ground. Incident angle is Qi = 60°..................
163
A .l
Definition of volum e V and surfaces S \ and S 2 for th e G reen’s theorem ..........................182
A .2
Definition of in teg ratio n dom ain.....................................................................................................186
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List o f Tables
3.1 A ccuracy of M OM com putation of TM scattering w idth for P E C circular cylinder
of various values of koa, w ith emax in % .....................................................................................
40
3.2 Accuracy of M OM com putation of T M scattering w idth for P E C circular cylinder
of various values of koa, w ith emax h1 %.....................................................................................
44
4.1
G round d a ta .........................................................................................................................................
93
4.2
G eom etric param eters used in tap ered cylinder approxim ations........................................... 107
4.3
C om parison of com putation tim e betw een analytical m odel and num erical algorithm
for four tap e red cylinders of length L a n d a = 0.6A, b = 0.2A............................................... 112
5.1
Results of the evaluation of I s using th e W eighted-Averages (WA) m ethod, Tanh
T ransform ation (T T ), and Com posite T rapezoidal (CT) integration, for various val­
ues of Z , R , an d k = 18 —6j . I b is th e exact value, e is th e absolute accuracy in %,
and T is th e com putational tim e in sec.........................................................................................146
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Chapter 1
Introduction
1.1
M otivation
In th e field of microwave rem ote sensing, the stu d y of scattering from vegetation is of great in ter­
est. P articularly w ith th e deploym ent of air- and satellite-born Synthetic A perture R adar (SAR)
instrum ents, it is especially im p o rta n t to have reliable models to characterize th e electrom agnetic
behavior of vegetation.
A vegetation canopy can be considered as a m ultilayered m edium above a ground surface. For
example, in a forest, its crown region m ay be chosen as one layer, and its tru n k region as another
layer. Furtherm ore, each layer generally contains more th a n one ty p e of scatterer. For instance, a
branch could be a scatterer, and a leaf another scatterer. T herefore, each layer is m odelled as an
ensemble of individual dielectric objects of different type, size, and orientation. Among th e m ost
common com ponents in a vegetated m edium are cylindrical stru ctu res, such as stem s, branches,
tru n k s or needles, an d disk-like stru ctu res such as leaves. A linear variation of the radius along the
axis of a cylinder, can also be introduced to m odel tru n k s an d branches m ore realistically. A long,
curved broadleaf m ay be m odeled by several sm all disks each positioned w ith a different slope to
sim ulate the entire leaf.
T he electrom agnetic scatterin g from a vegetation canopy can be studied by replacing each layer
w ith a random m edium whose sta tistic a l properties are related to th e physical and geom etrical
1
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characteristics o f th e vegetation layer. This random m odeling can be done in either a discrete or
a continuous fashion. In the discrete case, the layer is viewed as a collection of dielectric particles
whose size, position and orientation statistics are given. In the continuous case, the vegetation
is replaced by a continuous random m edium w ith a perm ittiv ity th a t is a random process whose
m om ents, such as th e m ean and correlation function, are known. In b o th cases, th e scattering from
the layer is obtain ed by perform ing an average - either discrete or continuous - of th e values of the
scattering coefficient of the individual constituents. T he availability of electrom agnetic m odels for
the single scatterers is therefore essential in order to investigate th e scattering from a vegetation
canopy.
T he approach followed in the stu d y of th e electrom agnetic scattering from an object can be
analytical or num erical. T he purpose of this dissertation is to develop num erical m ethods to eval­
uate the electrom agnetic scattering from individual vegetation com ponents. W hile the num erical
m ethods can provide exact solutions, their com putational cost often makes their use unpractical
in problem s involving large objects. Therefore, a m otivation for developing num erical m ethods is
th a t, particularly in th e case of large scatterers, th ey represent an essential tool to test the validity
of com putationally faster approxim ate analytical models.
1.2
Background
M odels of vegetation as a canopy composed of individual dielectric objects can be found in Fung
[1], Ulaby et al. [2], E ngheta and Elachi [3], and Lang an d Sidhu [4], U laby et al. [5] and Tavakoli
[6] have used vertical cylinders, representing th e stalks, and random ly oriented disks, representing
the leaves, to m odel a canopy. K aram et al. [7] have represented a forest canopy as a two-layered
m edium above a rough interface, while Lang et al. [8] have introduced tap ered cylinders to m odel
a forest. C hauan et al. have m odeled a boreal forest [9] an d cornfields [10]. Koh et al._ [11] have
considered the scattering and atte n u atio n th ro u g h a forest w ith dense foliage.
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T he problem of electrom agnetic scattering from the individual com ponents of the vegetation
canopy has been studied by several authors in the past w ith e ither analytical or num erical m ethods.
T he analytical m ethods were th e first to be developed due to lim ited com putational tools
available in th e past.
Historically, the first class of objects considered for analytical solutions
was th a t of infinite cylinders. E xact analytical solutions are available for scattering by perfectly
conducting and homogeneous dielectric cylinders, b u t not for other cross sections. W ait [12, 13]
was among the first ones to investigate and present an exact solution for scattering from an infinite
circular cylinder at oblique incidence. Ruck et al. [14] and B ow m an [15] also gave a sum m ary of
exact solutions for b o th conducting and homogeneous dielectric cylinders of infinite length.
U nfortunately, exact analytical solutions for the scattering from finite dielectric bodies exist only
in the case of the sphere [14, 15, 16]. However, for the scatterin g from objects such as finite-length
cylinders, finite-thickness disks, and spheroids, a num ber of approxim ations have been studied.
T h e Rayleigh-Gans approxim ation [17] is applicable to tenuous scatterers for which the phase
shift across the m axim um dim ension is small. A cquista [18] a n d Cohen et al. [19] extended the
use of the Rayleigh-Gans approach to particles w ith low polarizability and slightly larger phase
shift across them . Schiffer a n d T hielheim [20] introduced an approxim ation for cylinders w ith one
dim ension electrically sm all and shorter th a n th e other, i.e. eith er very th in or flat.
Shepherd and H olt [21] applied th e Fredholm integral equation m ethod to the scatterin g of
electrom agnetic waves by finite cylinders of circular cross section, b u t th eir m ethod also h ad lim i­
tatio n s at the increase of th e cylinder radius. K aram et al. [22, 23] used Schiffer and T hielheim ’s
approxim ation to m odel dielectric discs and cylinders. Stiles a n d Sarabandi [24] provided a solution
for th in dielectric cylinders w ith a broader range of validity, b u t still lim ited to small cross sections,
and showed th a t Schiffer and T hielheim ’s approxim ation is a specific case of their solution. Seker
and Schneider [25] have delveloped a physical optics ap proxim ation for dielectric cylinders of any
size, provided th a t th e length is m uch larger th a n th e radius.
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An analytical m odel for plane wave scatterin g from th in dielectric circular disks has been de­
veloped by Le Vine et al. [26, 27, 28] a n d W illis et al. [29]. T h eir m odel is based on a physical
optics approxim ation of the internal electric field of the disk. Very recently, Koh et al. [11] have
introduced a new approxim ate solution for th in dielectric disks of a rb itra ry size and shape th a t
perform b e tte r th a n the Le V ine’s m odel for edge-on incident wave.
T he num erical treatm ents of problem s of electrom agnetic scattering from perfectly conducting
a n d homogeneous dielectric bodies have usually been based on integral equation form ulations,
particularly surface integral equations, th e preferred approach to whose solution has been the
M ethod of M om ents (MOM) [30]. O ther approaches based on differential equations, such as the
F in ite Elem ent M ethod (FEM ) and the F in ite Difference T im e D om ain (FD T D ), are quite powerful
com putational tools, b u t require discretization of the entire co m p u tatio n al dom ain, com pared to
th e surface integral equation approach, which requires discretization of th e object surface only. For
th is reason they have been used m ainly for problem s involving sm all objects or inhomogeneous
bodies.
Richm ond [31] developed a num erical m eth o d to com pute th e scatterin g by a dielectric cylinder
of arb itra ry cross section, however his approach was based on th e solution of a volume integral
equation. T he problem can also be tre a te d using a surface integral equation, as shown in [32]. Raz
and Lewinsohn [33] investigated the volume a n d surface integral equation form ulations relevant to
the scattering and absorption of electrom agnetic waves by th in , finite, a n d lossy dielectric cylinders.
Papayiannakis et al. [34] tre a te d th e problem of scattering from a finite dielectric cylinder w ith
dim ensions com parable to th e w avelength of th e incident field, by solving an integral equation
containing the free-space G reen’s function over the cylinder volume by th e m ethod of m oments.
L ater, the same a u th o r [35] employed a tran sfo rm atio n to reduce th e three-dim ensional integrals
in th e equation into tw o-dim ensional ones. Dielectric disks have also been tre a te d using num erical
models. C hu and Weil [36, 37] have derived a volume integral equation, and solved it w ith the
4
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m ethod of m om ents.
Form ulations of the equivalence principle and the derivation of surface integral equations have
been presented by several authors, including Poggio an d M iller [38], Glisson [39], L eviatan [40],
and Peterson [32], These integral equations can be tre a te d by using th e M ethod of M oments.
Theoretically, the M ethod of M oments technique can solve any scattering problem , however its
m em ory and com putational requirem ents makes its use prohibitive for large problems, even when
only the surface integral equations are considered. For this reason only objects w ith p articu lar geo­
m etrical sym m etries are norm ally considered. T heir geom etrical properties lead to a sim plification
of the problem , w ith consequent reduction of com putational tim e and m em ory requirem ents.
One simple class of scatterers includes infinite cylindrical bodies for which all the electrom agnetic
quantities have no or only periodical variation along th eir axis. T his bodies can be tre a te d as twodim ensional bodies, as done by Richm ond [31].
A nother specific class of objects includes bodies th a t exhibit a ro tatio n al symmetry. For these
axisym m etric scatterers - or bodies of revolutions - th e problem can be reduced to the solution of
an integral equation along a curve by the m ethod of m om ents. Solutions of the surface integral
equations for bodies of revolutions have been presented by A ndreasen [41], M autz and H arrington
[42, 43, 44, 45], and W ilton an d Glisson [46, 47], am ong others.
T he problem of scatterin g in an half-space m edium was first investigated by B utler et al. [48, 49,
50] and X u et al. [51, 52] for th e two-dim ensional case of infinite cylinders parallel to the bou n d ary
betw een the two half-spaces. In [48], th e current on an infinite conducting strip located on the
interface between the half-spaces is com puted.
[49, 51] and [52, 50] study the surface currents
and the scattering from perfect electric conducting a n d hom ogeneous dielectric infinite cylinders,
respectively, parallel to th e half-space interface and located above or below it, or p artially buried in
it. In their approach, tw o-dim ensional surface integral equations are solved for the surface currents
by the MOM; the co m putations include th e evaluation of Som m erfield’s-type integrals.
5
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M ichalski and Zheng considered surface integral equations for bodies of arb itra ry shapes, pro­
viding the theory for layered m edia [53] and some results for the half-space case [54]. Cui et al. provided
the theoretical expressions of the volume integral equations for tw o-dim ensional objects buried u n ­
der m ultilayered m edia [55, 56], and later for th e three-dim ensional objects [57] w ith some num erical
results [58] using the MOM. V itebskiy et al. [59, 60, 61] considered axisym m etric perfect electric
conductors buried in a half-space. Geng et al. [62, 63, 64], and He et al. [65, 66, 67] have extended
their algorithm for a dielectric targ et in a m ulti-layered environm ent.
1.3
O verview
T he present C hapter 1 provides a background and m otivation for th e work in this doctoral disser­
tation. It also gives a brief overview of th e inform ation contained in each chapter.
In C h ap ter 2, a surface integral equation form ulation for electrom agnetic scattering is derived
sta rtin g from basic electrom agnetic theory and using th e equivalence principle. A lternative forms
of surface integral equations are also given.
In C h ap ter 3, the surface integral equations are solved by th e M ethod of M oments for the case
of two-dim ensional bodies. R esults are given for infinite cylinders of circular cross section, b o th
perfect electric conductors and dielectrics, and these are com pared to th e exact analytical solutions.
T his chapter is intended to be b o th an exam ple of solution of th e coupled surface integral equations
developed in C hapter 2, and a validation of the correctness of such an approach.
C hapter 4 illustrates th e application of the coupled surface in teg ral equations to axisym m etric
objects. T he M ethod of M om ents solution is used, which exploits th e ro tationl sym m etry of the
body to reduce the com plexity of th e problem . T he results are validated using objects of different
shapes, b o th perfect electric conducting and dielectric.
T his num erical procedure is th en used
to m odel common vegetation com ponents, and th e results are com pared to those obtained by
approxim ate analytical models.
6
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C hapter 5 extends th e application of the m ethod of m om ents to objects located in an half-space
m edium. T he surface integral equation approach is extended to include bodies of revolution located
in the upper half space, an d its solution is validated by considering equivalent cases. Finally, this
half-space num erical approach is applied to the problem of m odeling vegetation over a flat ground.
In C hapter 6 conclusions are draw n and recom m endations are m ade for future work.
7
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Chapter 2
T he surface integral equation
form ulation
T he purpose of this chapter is to introduce the integral equations th a t will be used to solve scattering
problem s in the rest of th is dissertation. T he use of such form ulation provides th e advantage of
reducing a three-dim ensional problem to the solution of a set of surface integral equations.
In
p a rticu la r situations, which will be fu rth er considered in the following chapters, such an approach
leads to th e reduction of th e num ber of independent variables from three to two, thus allowing a
less complex and m ore efficient num erical com putation techniques.
In section 2.1, th e electrom agnetic field generated by an a rb itra ry d istribution of sources in
an unbounded space is w ritten as a sum of th e contribution from th e sources inside and from the
sources outside a volume. T he second contribution is in th e form of a surface integral over the
b o undary surface of th e volume, which allows the definition of equivalent sources on this boundary.
In section 2.2, a m aterial body is considered, and th e expression derived in section 2.1 is applied to
th e exterior and interior of its b o u n d ary surface. T he resulting two integral equations are m atched
on the boundary surface to obtain a system of coupled surface integral equations.
Section 2.3
presents alternative form s of these surface integral equations. Finally, in section 2.5, expressions
for the scattered fields are given an d im p o rta n t quantities for th e stu d y of electrom agnetic scattering
are defined.
8
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2.1
The equivalence principle
In th is section, sta rtin g from M axwell’s equations, and applying equivalence principles, an integral
equation relating th e electrom agnetic fields them selves to th eir values on a closed surface shall be
derived.
From hereafter, a tim e harm onic variation of the electrom agnetic field of the form
u = 2 7i-/,
(2.1)
is assum ed, / being th e frequency of the electrom agnetic field. A n infinite TZ3 space is considered,
filled w ith a hom ogeneous and isotropic dielectric m edium w ith p e rm ittiv ity e and perm eability p,
given by
£ — £q £r
P — Po Pr
j
(^*^)
where £o and sT are respectively th e free-space p e rm ittiv ity and the relative perm ittivity, b e tte r
known as dielectric constant, of the m edium , po and p r are respectively th e free-space perm eability
and the relative perm eability of the m edium.
In such a m edium , th e electric and m agnetic fields, E an d H respectively, are related to the
prescribed electric a n d m agnetic sources J and M by M axwell’s equations:
V x E
= —j u j p H — M
(2.3)
V x H
= jtue E + J
(2.4)
= -
(2.5)
V -E
£
V -H
= 0
(2.6)
where p is the electric charge density, related to J by th e law of conservation of charge
V • J = —juip
(2.7)
As shown in A ppendix A .l, w hen b o th electric and m agnetic sources exists in a volume V, the
9
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solution to the M axwell’s equations (2.3)-(2.6) for r £ 7£3 is given by:
E(r)
= - j u n [ G (r, r') ■J(r') d V'
H(r)
=
Jv
J
K{r,r')-J{r')dV' - j u s
- [ K(r, r') ■M(r) dV'
(2.8)
J
(2.9)
Jv
G (r, r') • M(r') dF'
w here th e integration is carried out over the region V where th e sources J and M are located,
G(r,r')
=
g (r,r')I + V g^ r/)-V'
(2.10)
K(r,r')
=
V5( r , r ' ) x l
(2.11)
are th e electric and m agnetic dyadic G reen’s functions, respectively, and
““ ’^ / c j r
j
ff(r,r') = e 47r|r _ r/[ ' ’
k= . ^ i
(2.12)
is th e scalar G reen’s function defined in A ppendix A .I. Note th a t, technically speaking, th e dyadic
G reen’s function defined by (2.10) is actually a functional o p erato r and not a function.
C alculation of the electrom agnetic fields th rough (2.8) and (2.9) requires complete knowledge
of th e sources J and M a t any point, which is often not possible. However, if the electrom agnetic
fields
are already known over a closed boundary, it is th e n possible to determ ine them in the
entire space by considering equivalent sources on such b o u n d a ry surface. The equations relating
the electrom agnetic fields inside th e m edium w ith their value on th e boundary are called surface
integral equations a n d will be derived in the rem inder of this section.
T he purpose of th e following derivation is to find a solution for th e electrom agnetic field any­
where in space.
Let Si
and S2 be two closed surfaces, w ith S2 surrounding S i,asshown in Figure
2.1, and the sources J and M be known only in the volume V betw een S i and S2, w ith no sources on
Si or S2. Let n ' be the norm al u n it vector on the surfaces S i an d S 2, pointing inwards. E quations
(2.3), (2.4) can be com bined in th e following two wave equations for E and H:
V x ( V x E ) - f e 2E =
—ju/j, J —V x M
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(2.13)
V (known
unknown
unknown
J ,M * 0
Figure 2.1: D efinition of surfaces Si and 52V x (V x H ) — &2 H
=
-ju e M + V x J ,
(2.14)
Let E be an electric field th a t satisfies M axwell’s equations (2.3) and (2.4)anywhere in space. T he
G reen’s theorem , given in A ppendix A.2, can be applied to the volume V and surfaces Si and S 2
previously defined, w ith
P = E,
Q = ap
(2.15)
where a is an arbitrary, constant vector. T he result is th e following:
/ {E • [V' x (V ' x a <7)] - p a - [V' x (V ' x E)]} dV' =
Jv
p a x ( V ' x E ) - E x (V 'x a p )]- h 'd S '
(2.16)
S 12
where S 12 = S\ U S 2 , th e prim e is used for convenience to indicate th e integration variables, an d the
negative sign on th e right side of equation (2.16) is due to th e definition of the norm al u n it vector
pointing inwards instead of outw ards as in th e derivation of th e G reen’s theorem from A ppendix
A.2.
11
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Using M axwell’s equation (2.3) and the wave equation (2.13), and perform ing the calculations
described in A ppendix A .3, th e equality (2.16) can be w ritten as:
E(r) = - a •
f
Jv
( j u g J + V ' x M ) g(r, r ') + X - ^ V ' g
jue
dV' +
[jug, (H x n ') g + ( n x E ) x V'g + V'g (E ■ n ')] dS'
+ a •
JS ,2
(2.17)
T his equation is valid for any a rb itra ry value of a, and, as a result,
E(r) = - [ ( j u g J + V ' x M ) g( r, r') + —
V'g d V' +
Jv L
JUE
[ j u g (H x n ') g + ( n ' x E ) x V'g + V'g (E • n ')] dS'
+ (f
(2.18)
JSin
Since M = 0 on the surface S, th e application of Stokes’ theorem yields:
lv
[ V ' x M(r') ] g( r, r') d V '
=
J
V ' x [g( r, r') M(r') ] d V' -
=
/
g ( r , r ' ) M ( r ' ) ■h ' d S ' - /
=
- / [ V ' g x M ( r ' ) ] dV'
Jv
=
~ [
J
[ V' g x M(r') ] dV'
[ V ' g x M (r') ] dV'
[V'g(r,r') x I ] ■ M ( v ' ) d V '
(2.19)
JV
Use of the relationship
( 2 . 20 )
and the definition of electric an d m agnetic G reen’s functions (2.10) and (2.11) lead to the equation:
E (r) = —j u g [ G ( r , r ' ) - J ( r ') dV' — [ K (r, r ') ■M ( r ') dV' +
Jv
Jv
+ I
J Sl2
[ j u g (H x n ,) g + ( n / x E ) x V'g — V g ( E ■ n ;)] dS'
(2.21)
T he right side of (2.21) contains two types of integrals, w hich correspond to the contributions
to the electric field of th e known sources in the volum e V an d th e unknow n sources outside V,
respectively. T he two volum e integrals represent the field generated by the known sources J an d
M inside the volume V. T h e surface integral is the co n trib u tio n to th e field from the unknow n
12
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V (known J,M £ 0)
J ,M = 0
e ,|i
e.M-
J ,M = 0
Figure 2.2: E quivalent sources configuration.
sources located outside th e volume V. Its integrand suggests th a t such unknow n
replaced by equivalent sources H x n , n x E and E ■ n on th e surface S u ,
sources can be
which would give the
same contribution to th e electric field E(r) inside th e volume V as th e unknown sources.
T he intro d u ctio n of these equivalent sources im plies th a t an equivalent problem is indeed con­
sidered, where th e unknow n sources J a n d M inside S i and outside S 2 are set equal to zero and
replaced by equivalent sources J s and M s on th e surface S 12. T he sources J s and M s are surface
currents and are defined so th a t they generate th e sam e electrom agnetic fields w ithin the volume
V as the unknow n sources. This equivalent configuration is illu strated in Figure 2.2. This result is
often referred to as th e equivalence principle.
In expression (2.21), r is a generic point in space. In order to derive a surface integral equation
from it, the point r m ust be located on the surface S. However, in such a situation the integrand
of the surface integral is singular a t r = r'. It is shown in A ppendix A.4 th a t its integration by
means of a lim iting procedure leads to th e following resu lt (A .65):
(f
[jtdfj, (H x n ') g{ r, r') + ( n ' x E) x V '^ r, r') + V g { r , r') (E • n
')] dS' =
J S\ 2
13
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+ f
=
(H x
JS12
2
') 9(r, r') + ( n ' x E) x V'g - V g (E • n ')] dS'
(2.22)
where the asterisk indicates the principal value of th e integral. S u b stitu tin g (2.22)into (2.21) yields
^
=
2
+ f
- ju t* f G (r, r') • J(r') d V' - f K (r, r') •M(r') dV' +
Jv
Jv
\ju{* (H x n ') g(r, r') + ( n ' x E) x V'g - V g (E ■n ')] dS'
(2.23)
JS 12
Note th a t the expression above is valid only for points r on th e surface S\ 2 A n equivalent tan g en tial electric surface current J s and an equivalent surface charge density p s
can be defined on S 12 = S i U 52 as, respectively,
J s (r')
=
n'xH (r')
(2.24)
p s ( r')
= e E(r') • n '
(2.25)
Since in the equivalent configuration J = M = 0 on S 12 and outside V , then, the law of conservation
of charge yields
E - n ' = — = - V * ' Js
e
jlj£
(2.26)
Note th a t, since Js depends only on th e surface coordinates, V(. ■J s = V ' ■J s.
Using (2.24)-(2.26) as well as the definition of electric an d m agnetic dyadic G reen’s functions
(2.10) and (2.11), after some m anipulations equation (2.23) can be w ritten as follows:
^
2
=
-jut* [
Jv
-jwp, £
G ( r , r ' ) - J ( r ') d V ' - [ K(r,r') ■M(r') dV' +
Jv
G (r, r') ■J s(r') dS' — <j>
J s 12
K (r, r') ■[E(r') x n '] d S ' ,
r E S 12
(2.27)
JS12
In the following section, expression (2.27) will be used to derive a set surface integral equations
for a m aterial body in an unbounded space.
14
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e,fx
Figure 2.3: E xterior problem .
2.2
Surface integral equations in unbounded space
An homogeneous, isotropic dielectric body of p e rm ittiv ity e and perm eability /j,. surrounded by free
space, is now considered. Sources J , M / 0 axe assum ed to exist in th e free space, but not inside
the m aterial body. V is th e volume occupied by th e body, and S is the b oundary surface betw een
the dielectric and free space.
W ith reference to Figure 2.3, th e surfaces 5 1 an d S 2 previously introduced can be defined such
th a t S is inside Si , in w hat is called th e exterior problem .
If S 2 expands to infinity while at
the same tim e the o th er surface S 1 approaches S, th en the integral equation (2.27) assumes the
following form:
E (r)
!
Jv
f
Js+
G ° (r,r')'J(r ')d V '-
G ° ( r , r ') ■J s (r') dS' -
f
Js+
[ K ° ( r , r') • M (r ') d V ' +
Jv
K ° ( r , r') ■ [E (r') x n '] d S ' ,
r 6 S+
(2.28)
where the superscript “0” indicates th e free space G reen’s functions, which have the form (2.10)(2.12), w ith e — £0, and /j, = (j,q (eo being the perm ittivity, and /j,q th e perm eability, of free space),
15
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and S + is th e exterior side of the surface S. An “incident” electric field can be defined as
E « (r) = - j u n o f
Jv
G ° ( r , r /) • J ( r ') d V ' -
r GS
[ K ° ( r , r ' ) • M (r') d V ' ,
Jv
(2.29)
which corresponds to th e electric field induced by th e sources J and M on the surface S, a n d can
be determ ined from them . Using expression (2.29), equation (2.28) can be rew ritten as
= E « ( r ) - j w / i o r G ° ( r , r ' ) - J ^ r ' ) ^ ' - / * K ° ( r , r ' ) • [E (r')
x n ']
dS ' , r 6 5 + (2.30)
J s + JS+
2
which is the surface integral equation for the exterior problem .
T he interior problem is illu strated in Figure 2.4. In this case, b o th surfaces S\ a n d £2 are
assum ed to be enclosed by S, w ith S i —>0 and S 2 approaching S. Since there are no sources inside
S 2 , equation (2.24) becomes:
= juifi
G ( r , r ' ) - J s (r' ) d S ' + j> K ( r , r ') ■ [E (r') x n '] d S ' , r G S -
(2-31)
where S ~ is the interior side of S, and n " = — n ' is th e norm al unit vector to S ~ .
T he electrom agnetic field inside S is uniquely determ ined by the values of the tangential com po­
nent of the electric field over th e boundary S. Hence, th e relationship betw een the electrom agnetic
fields inside and outside S can be found by m atching the tan gential com ponents of E given by
(2.30) and (2.31), as it will be shown below.
T he tangential com ponent vt of a vector v can be determ ined as follows:
Vi = v — n ( n • v) = ( I — n n ) ■v
(2.32)
A pplying (2.32) b o th sides of equations (2.30) and (2.31), yields
= E^(r)
-
( I - n n ) - <f
G ° ( r , r ') • J s (r') dS' +
JS+
2
— ( I —h n ) - ( f
Js+
K ° ( r , r ' ) ■[E (r') x h ' ] d S ' ,
r GS+
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(2.33)
S,|d
Figure 2.4: Interior problem .
=
junil-nh)-^
G ( r , r ' ) ■J s( r ' ) dS' +
+ { l - h h ) - £ _ K ( r , r ' ) • [E (r') x n ' j d S 1,
r e S~
(2.34)
Because of th e continuity of th e electric field, E t m ust th e sam e on b o th exterior and interior
sides of S. In addition, since th e tan g en tial m agnetic fields are also continuous, then J s on S +
m ust be equal to J s on S~, and J s in (2.33) and (2.34) are th e same. Hence, expressions (2.33)
and (2.34) constitute a system of coupled surface integral equations th a t m ust be solved together
to find the unknow n functions J s, and E on th e surface 5 , given th e tangential incident electric
field e J ° on S.
Since (2.33) and (2.34) have been derived by m atching th e tan g en tial com ponents of the electric
field over the boundary surface S, th ey are called surface tan g en tial electric field integral equations
(E FIE ). They are also called ’’b o u n d ary integral equations” in [68]. T hey can be w ritten in term s
of equivalent currents an d tan g en tial incident field only, by introducing equivalent m agnetic sources
17
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on S.
Sim ilarly to th e definition of the equivalent surface electric current (2.25), an equivalent surface
m agnetic current on S can be introduced as
M s (r') = E(r') x n '
(2.35)
so th a t the tan g en tial electric field on 5 is given by:
Et(r') = M s(r') x n '
(2.36)
Hence, using (2.35) and (2.36), the two equations (2.33) and (2.34) m ay be w ritten in th e form:
--- = E ^ ( r ) - jufiQ (I - n n)-jf + G°(r,r') • J s(r') dS' +
- ( I - n n ) i ’ K°(r,r') • M s {r' )dS' ,
Js+
=
r G S+
(2.37)
r G S~
(2.38)
j u } f j , { l - h n ) - ^ _ G { r , r ' ) - J s{r ' ) dS' +
+ ( I - n n ) - j f ’ K(r,r') • M s {r' )dS' ,
E quations (2.37) and (2.38) involve only J s and M s as unknowns, and therefore from now on will
be referred to as th e coupled equivalent surface currents integral equations. However, even though
the electric field does not ap p ear explicitly in th em anym ore, they are still considered astangential
electric field integral equations. Unless noted otherw ise, these are th e integral equations th a t will
be solved in the rem ainder of this d issertation work.
2.3
Other forms o f coupled surface integral equations
Coupled tangential M agnetic Field Integral E quations (M FIE) m ay be derived by m atching the
tangential com ponents of th e m agnetic field over th e surface S th ro u g h a procedure sim ilar to the
one followed to find th e coupled tan gential E lectric Field Integral E quations (2.33) and (2.34).
18
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However, an easier way is to apply th e duality principle. In particular, substituting:
E
J
£
-*• H
-> M
(JL
H
m
fi
-* - E
-* - J
-> £
(2.39)
in (2.33) and (2.34) yields:
^
= H « ( r ) - ( I - n n ) . | ‘ K ° ( r , r ' ). [Ht(r') x n '] dS' +
—jus o ( I - n n ) ' /
Js+
=
( - ~~ “
G°(r,r') ■ M s(r') dS',
— ^r,r^ '
- jujs{l-hh)-f
Js+
r e S+
(2.40)
x fi/] dS1 +
G(r,r') ■ M s{r')dS',
r
G
S~
(2.41)
Note th a t by duality from (2.37) and (2.38), it m ay also be w ritten:
n x ^ r) = H Sl)(r) + ( I - n n ) - £ + K°(r,r') ■ J s(r')dS" +
- jusQ (I - n n ) - <£ G°(r,r') - M s(r') dS',
Js+
nxJj(r) _
r € S+
(2.42)
K(r,r') • J s{r')dS' +
+ juje (I —n n)'j>
G(r, r ' ) - M s(r')dS',
r € S~
(2.43)
This alternative form of th e coupled equivalent surface currents integral equations is useful when
the expression of the tan g en tial incident m agnetic field
is sim pler th a n the expression of the
tangential incident electric field E ) '.
In addition to th e E F IE form ulation (2.37),(2.38) and th e M FIE form ulation (2.43),(2.44), other
form ulations based on any linear com binations of those two system s of integral equations are pos­
sible and are generally called C om bined Field Integral E quations (C FIE ). T he accuracy of each
form ulation for a given scatterin g problem depends on the geom etry and dielectric properties of
the scatterer. In particu lar, it has been found th a t resonances can arise in problem s involving con­
ducting bodies when using th e E F IE or M FIE form ulation, b u t do not occur if a C FIE form ulation
is used [42, 69]. However, even for th e C F IE form ulation, some resonances can still occur [70].
19
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2.4
A lternative form for the dyadic G reen’s function
T he integral equations derived in this chapter are based on th e form (2.11) of the dyadic G reen’s
function. A different form of the dyadic G reen’s function, i.e.,
'G (r,r') =
(l-^P jg (r,r')
=
+
gfar')
(2.44)
is often encountered in th e literature and will be used later in the present work, specifically in
C hapter 5 for th e half-space problem.
T he m ain advantage in using the operator form (2.11) ra th e r th a n the above form (2.44) is th a t
the latte r perform s a double differentiation of the scalar G reen’s function g(r,r' ), This can result
into problem s w hen perform ing integration th a t require to evaluation of g{r, r') a t the singular
point r ' = r.
All the surface integral equations previously derived still apply if G ( r , r ' ) defined by (2.44) is
used instead of G ( r , r ') in (2.11). In order to prove it, it is sufficient to show th a t
[ G (r,r')-Fir^dV’ =
f
G (r, r ') ■F ( r ') dV'
(2.45)
G ( r , r ' ) ■J s (r') dS' = £
G (r, r') • J s (r ') dS'
(2.46)
Jv
Jv
for F = J , M , and th a t
f
Condition (2.45) ensures th a t the electrom agnetic fields generated by electrom agnetic sources are
still given by (2.9) and (2.10), while condition (2.46) assures th e equivalence between th e integral
equations w ith G ( r , r') and G ( r , r ') . P ro o f of th e equivalences (2.45) a n d (2.46) can be found in
A ppendix A.5.
2.5
Scattered field and scattering coefficient
Once the equivalent tan g en tial surface currents J s a n d M s on 5 are com puted by solving the
coupled integral equations, the scattered electric field generated by th em outside th e surface S can
20
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J,M = 0
Figure 2.5: Definition of geom etry for scattered field calculation.
be calculated. W ith reference to the geom etry in Figure 2.5, since r is not on the surface, th e n the
electric field E (r) can be found by applying expression (2.21). T here rare no sources w ithin the
volume V, hence th e only contribution to th e field outside 5 comes from th e equivalent sources.
Using definitions (2.24),(2.25), and (2.35) into expression (2.21) yields
EW ( r)
=
—jw/J-o j> G ° ( r , r') • J s (r') dS' —
5
K ° ( r , r ') ■M s (r') dS'
(2.47)
S
T he scattered m agnetic field can be found by duality, i.e.,
H (s)(r)
K ° ( r , r ' ) ■J s (r') dS' — jcoeo j> G ° ( r , r ' ) • M 5(r') dS'
=
s
(2.48)
s
For rem ote sensing problem s, th e knowledge of th e scattered electrom agnetic fields a t large
distances from the object is essential. T he general configuration of the problem is illu strated in
Figure 2.6. The object - also referred to as a scatterer - is located a t th e origin of
a coordinate
system . T he scatterer is assum ed to be illum inated by a plane wave representing
th e incident
electric field, having th e form
E w (r)
=
e ~ J' k o ^ ' r
(2.49)
T he u n it propagation vector k 8 th a t gives the direction of th e incident plane wave is :
kj = —sin
cos & x — sin
sin fa y — cosfy z
21
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(2.50)
Figure 2.6: Incident a n d scattered plane waves on an object.
where 6i, fa are th e elevation and azim uth angles of the incident wave, respectively, in a spherical
coordinate system centered a t the origin.
T he incident field (2.49) generates equivalent currents J s a n d M s on th e surface S th a t can be
evaluated using th e surface integral equations derived in the previous sections. T hen, the scattered
electrom agnetic fields associated w ith these equivalent currents can be determ ined using (2.47),
(2.48).
T he far-field, or rad iatio n zone, for a source w ith m axim um size D is defined as the region
of space where |r — r '| > 2 D 2/ \ , w ith A being the wavelength.
In such region the following
approxim ations can be used:
k0 |r - r '
ko r —ko r • r '
(2.51)
1
r
(2.52)
1
where r is the u n it vector into the direction of th e observation point.
T he scalar G reen’s function in th e far-field is therefore
(2.53)
22
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U sing (2.53) into expressions (2.11) and (2.12), th e dyadic G reen’s functions in th e far-field are
found to be approxim ately given by
G°(r,r')
~
+
eA , f - r '
(2.54)
47r r
K °(r,r')
~
"”**3 ko
- j k 0 ( 4> 9 - 9 4>) — A
\
/
47r
47Tr
i
e ^ ° f ' r'
(2.55)
S u b stitu tin g th e approxim ations (2.54) and (2.55) into expression (2.47), the scattered electric field
E (s)(r) can be w ritte n as a spherical wave
E^(r)
~
-jko
e, - j f a r
An r
-
f
Cof s ( e e
+ 4>4>)- j s (r')
f ' r ' dS' +
(4> 0 - 9 4 > y M,(r') e i k° f - r ' dS'
(2.56)
where the sym bol ~ indicates th a t the approxim ation is valid in th e far-field, and
Co
=
M
V £o
(2.57)
is th e im pedance of free space. A sim ilar expression for th e scattered m agnetic field in the rad iatio n
zone can be found using (2.54) and (2.55) into (2.48). T he observation point r is located a t elevation
and azim uth angles 9S, (fis, respectively, and defines th e direction of scattering. T he propagation
vector of the scattered wave is therefore
k s = sinf?s cos cf>s x + sin£?s sin cj)s y + cos9s z
(2.58)
Note th a t the unit vector r is equal to the u nit propagation vector k s.
Let the incident plane wave be polarized in the q direction, i.e.,
4 °
= Ef
(2-59)
and E ^ be the corresponding scattered field. T he bistatic scatterin g coefficient, also called b istatic
scattering cross-section, in th e polarization p for this (^-polarized incident field is defined as
23
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A n equivalent way of characterizing th e scattering from an object makes use of the so-called
scattering am plitudes. T he dyadic scattering am plitude f is im plicitly defined by the relationship
-jkor
E W(r) -
(2.61)
f&.kO-Ej
while the scalar scattering am plitudes f pq in the p- and (/-polarization are
/p ?(k s,k j) = p s • f (ks,kj) • q .
(2.62)
From (2.56), (2.59), (2.61) and (2.62), it follows th a t
/pg(ks,kj)
—
1 j k oCo *
Ps ■ £ ( § 6 + 4> 4 > y J ^ (r ') e J fc°
E (i) 4?r
( 0 0 - 6 4 > y M ^ (r ') e J k° ^s ' r' dS'
r' dS' +
(2.63)
where J ^ , M i9-* are the equivalent surface currents induced by the (/-polarized incident field (2.49),
(2.59).
T he scattering am plitudes f pq are related to th e b istatic scattering coefficients apq by the ex­
pression:
<rpq(9$,<f>s;8i,<i>i) = 4 7 r | / p9(ks,ki) |2
24
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(2.64)
Chapter 3
T w o-dim ensional bodies
T he integral equation form ulation derived in the previous chapter can be easily applied to twodim ensional problem s. If th e surface S of an infinitely long body in a three dim ensional space
is shift-invariant along a p articu lar axis, and all th e field quantities are also invariant, or exhibit
the same periodic variation, along th a t axis, th en th e problem reduces to two dimensions on an
a rb itrary cross section plane perpendicular to the sym m etry axis.
In particular, an infinitely long cylinder S in th e z-direction is considered, a section of w ith in the
xy-plane is depicted in Figure 3.1. T he cylinder is assum ed to have perm ittiv ity e and perm eability
/i,
and to be located in free space (perm ittiv ity
£o
an d perm eability po). Sources J and M are
present somewhere in th e free space, and generate an incident field E j ’ and the equivalent surface
currents J( and M s on S. If th e sources are constant or have all th e sam e periodic variation along
th e z-coordinate, th en th e problem can be seen as tw o-dim ensional in the xy-plane.
T he infinite cylinder can th en be uniquely characterized th ro u g h its generating curve C, i.e., the
intersection of S w ith th e xy-plane, which is shown in Figure 3.1. For convenience, an orthonorm al
coordinate system ( r , n ) is introduced on the curve C , f being the unit vector tangent to C at
the point p , and n th e u n it vector norm al to C a t p . ip is th e angle between
cos ip =
t
• x .
25
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t
and x , i.e.,
Figure 3.1: G eom etry of a two-dim ensional body.
3.1
T he coupled surface integral equations in two-dim ensions
It is shown in A ppendix B th a t for a tw o-dim ensional problem th e coupled integral equations
(2.37),(2.38) can be w ritten in the following form:
Ms ^ X n = Etl)(p) ~ j ' w f i o ( l - n n ) - ^ + Q-2 d { P i
p
') ' M p ' ) d C '
- ( I - n n ) - / * K°2D( p , p ' ) - M a( p ' ) d C ' ,
Jc+
Ms{p2)X—
=
+
PeC+
(3.1)
peC -
(3.2)
j u t i ( l - h h ) - £ _ G 2D( p , p ' ) . J t ( p ' ) d C ' +
+ (I-n n )-jf
& 2 D( p , p ' ) - M s { p ' ) d C ' ,
where the dyadic and scalar G reen’s functions have th e expressions:
i—
i / ~
Q.2 d { p , P )
K 2D( p , p ' )
sM
p
, p ')
n „/\ t i ^ 9 2 d (P i P )
= 92D{p,P)l-^
P
V
/o o\
(3.3)
=
V g 2D( p , p ' ) x l
(3.4)
=
-jflfw p-p'l)
(3.5)
and fJg2^(x) is the Hankel function of th e second kind. T he correspondent functions for free space
9$d> & 2D and ^ - 2V have th e same expressions as (3.3)-(3.5) w ith e = £oM — MoX> k = &o26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he equivalent surface current densities J t, M s and the tangential incident electric field E :o
on th e surface S can be w ritten in term s of their r - and ^-components:
J s(p)
=
J r ( p ) t + J z(p) z
(3.6)
M s(p)
=
M T( p ) f + M z {p) z
(3.7)
Ef{ p )
=
e
W(P)
t
+ E${p)z
(3.8)
S u b stitu tin g expressions (3.6)-(3.8) into equations (3.1),(3.2), and separating the z- and r-com ponents
yields the following four equations:
E i'H p )
—
7^ -
~ j u p o j> ^+ 9 2 d ( P i p ' ) J z { p ' ) d C ' +
dg\
2D- M T( p' ) d C ,
■/
Jcc+ dn'
E^(p)
=
M z ^pl + j u g ,o £
C+
+L
0
%
Mr(p)
=
M^
G C+
A , ( p , P V r ( 0 ' ) - i/cq8 &
d r 8 dr'
(3.10)
.
j u p <f 92d ( p , p' ) J z { p' ) dC' +
Jc+
-
4
M z (p)
2
cos (-ip —ip') d,C' +
d c ''
dg2D
+ Jc+ dn' M r ( p' ) dC' ,
0 =
(3.9)
P
\
t
1 d92D 9 Jt
i\ t (
+ JU„ f c + 9 2 D { p , p ) M p ) - ¥ o ^ r l ?
’
Jc+
dg2D
Tc+ dn'
M z {p' ) dC',
(3.11)
p G C+
cos {ip — ip') dC' +
p ec+
(3.12)
This system of surface integral equations will be solved for some sim ple cases in Sections 3.3 and
3.4. Using the results of A ppendix B, th e scattered electric field generated in a two-dim ensional
space by equivalent surface currents J s and M s on th e curve C has th e form:
E (s)(
-jupo
j
G °2 D{p, p' ) • U p 1) dC' -
c
j
K °2 D( p , p' ) ■M , { p ’) dC'
c
27
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(3.13)
Figure 3.2: D efinition of scattering problem for a two-dim ensional body.
3.2
Far-field and scattering w idth
For a two-dim ensional body, th e far field is defined as the region of space where p is large com pared
to th e wavelength A, so th a t th e following approxim ation can be used:
k0 \ p - p ' \
«
P = —
P
k o p -k o p -p ',
(3.14)
Using (3.14) together w ith th e large argum ent approxim ation of th e Hankel function [71], in the
far field the two-dim ensional scalar G reen’s function (3.5) is approxim ated as:
e ’*k °P e ^ 0/5' p
92d{p, p ' )
(3.15)
From this approxim ation, and from expressions (3.3) and (3.4), th e dyadic G reen’s functions in th e
far-field are obtained:
G.2d ( p >p ')
~
(z Z + 0 0 )
K.2 d ( p >p ')
~
~jk0 (
-
e_ ^ o P e i k°P' P
z 0 )
^
e ~ ^ p e i k° P' P
28
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(3.16)
(3.17)
Using expressions (3.16),(3.17) into (3.13) yields:
E (s)( p )
-j&o
C o / ( z z + 0 0 V J s ( p ' ) e ^ ° ^ ' P d,C' +
Jc
TT&oP
+ /
Jc
-
z 4>^j ■M s ( p' ) e ^ ° P ' P dC'
(3.18)
T h e incident electric field is assum ed to be a g-polarized plane wave of the form
E«(p)
=
q< e
fco ki ■P
(3.19)
where the unit p ropagation vector of th e incident wave is now two-dimensional:
kj = - cos fa x - sin fa y ,
(3.20)
w ith fa being th e angle of incidence, defined as shown in Figure 3.2.
Let
be th e corresponding scattered field, w ith u nit propagation vector of in the direction
cj)s given by
k s = cos 4>s x + sin <f>s y .
(3.21)
For two-dim ensional objects, a q u an tity called the scattering w idth is introduced in a sim ilar
fashion as the scatterin g cross section in th ree dim ensions. It is an equivalent w idth proportional
to th e apparent size of th e scatterer in a p a rticu la r direction, and it is defined as
apq (ks, £i)
=
JiHL 2?r P
p ^ 0°
•
IE ?S)(P) • P<
(3.22)
| Eg*) |2
In other words, if th e power density of th e incident field were m ultiplied by th a t length, it would
yield the same power as produced by th e scatterer.
T he incident field (3.19) generates equivalent currents
be evaluated using th e surface integral equations.
and
on the surface S th a t can
S u b stitu tio n of th e expression (3.18) for the
scattered field at a point p = k s in th e far zone, into (3.22) yields:
(ka,ki) =
1 h
IE,« i 2 4
Ps •
(o / ( z z Jc
+ j> ( 0 z - z 0 ) •
4> 0 ) • J ^ ( p O e ^ °
( p') eZ
■P dQi
29
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P dC' +
(3.23)
T h e scatterin g cross section per length I can be obtained from th e scattering w idth through the
relationship [72]:
o-pg(k s,k i)
3.3
a ^ ( k s, ki )
=
(3.24)
Perfect electric conducting tw o-dim ensional body
Let the two-dim ensional cylinder be m ade of a perfect electric conductor. Since the tangential
electric field is null on th e surface of a perfect electric conductor, th e equivalent surface m agnetic
current M s also vanishes. Therefore, only th e exterior problem is to be considered, and the equation
to be solved is (3.10), which assum es th e form:
E «(p)
=
j u ^ ( l - h h ) - £ Q.°2D( p , p ' ) - J s ( p ' ) d C ' ,
p e c
(3.25)
th a t corresponds to the following two equations for th e z- and r-com ponents:
E z H p ) = jupo £
p eC
92d ( p , p ' ) J z { p ' ) d C ' ,
E^(P)
92D\ P i P ) J t \ P )
k,Q d r
dr'
cos {ip - t f ) d C ' ,
peC
(3.26)
(3.27)
whose solution gives th e two com ponents J z and JT of th e induced equivalent surface current. Once
J z and J T are found, th e scatterin g w idth can be evaluated by using (3.23) w ith M s(p') = 0:
apq (ks, ki)
=
IE
3.3 .1
1
k0
(i)I2 4
Ps ■ £o j> ( z z — 4>
( p ') e
k s ■P dC'
(3.28)
T M incident wave
Let the two-dim ensional body be illum inated from an angle fa by a plane wave whose electric field
is directed along the 2-axis. Expression (2.49) reduces to:
{cos f a x + sin f a y )
EW(p) =
(3.29)
Such choice of incident field is denoted as transverse m agnetic (TM ) incident wave, and is illu strated
on the left side of Figure 3.4. A TM incident wave induces on th e perfect conductor an equivalent
30
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1/ Lt
t - A.t/2
(a)
t
tn
(b)
Figure 3.3: (a) D iscretization of curve C. (b) Basis (expansion) functions.
surface current J s th a t is also directed along the 2-axis:
(3.30)
j s(p) = Jz{p) Z
Therefore, only th e integral equation (3.26) for th e 2-com ponent is needed, which has th e form:
4
p£C
Jc
(3.31)
Once Jz is determ ined by solving th e integral equation (3.31), th e scattering w idth can be found
through expression (3.28). Since for 2-polarized incident electric field the scattered electric field is
also directed along th e 2-axis, the only non-zero scatterin g w id th is c r j f , for convenience hereafter
indicated as cr™ , which is given by:
a 2D (kSJ kj)
1
=
IE
fcoCo
£ J z ( p' ) e i k ° * s ' P' dC'
(3.32)
i
Numerical solution using the m e t h o d of m o m e n ts
T he integral equation (3.31) is solved using th e m eth o d of m om ents (MOM ). For this purpose, a
curvilinear coordinate t is defined on th e curve C , an d C is divided into N intervals by choosing
31
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N points t n = (n — 1)A t, (n = 1 , 2 , 3 , N ) on it, as shown in Figure 3.3. R ectangular pulses are
selected as basis functions:
1
Pn(t)
=
At ^
At
t n -- — < t < t n ---- —
2 2
elsewhere
for
<
0
n = 1,2,3,..., N
,
(3.33)
while the testing functions are D irac delta functions:
Xn{t)
= S { t - t k) ,
k = 1,2,3,. , . , N
(3.34)
W ith such choices, the equivalent current density can be approxim ated as follows
N
J 2(t')
=
t'zC
£ lnP n(0>
71=
(3.35)
1
where
In
Jz{tn) i
=
n = 1,2,3,..., N
(3.36)
S u b stitu tin g (3.35) into (3.31), and testing the resulting equation w ith th e functions (3.34) for
A: = 1,2, 3, . . . , N leads to th e following system of algebraic equations:
N
vk
where
Vk
=
Zk*
=
=
Y z knln,
n= 1
& = 1, 2,3,..., N
,
E ^ ( t k) ,
H ^ \ k Q\ p k - p ' \ ) d t ' ,
(3.37)
k = 1,2,3,..., N
(3.38)
k, n = 1,2,3, ...,1V
(3.39)
Jtn
4
where
tn
=
At
^ + y i
n ~ 1,2,3,..., N
(3.40)
In the case of a circular cylinder of radius a as in Figure 3.4, th e curve C can be described by the
equation
p'
=
a cos (j>' x + a sin ^6' y ,
<j>' € [0,27t]
(3.41)
and the curvilinear coordinate is related to the angle <f>as follows:
t = a<j> ,
A t — a A<f> ,
27r
A <j> = —
32
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(3.42)
TM incident wave
TE incident wave
Figure 3.4: TM and T E incident waves on a circular cylinder of radius a.
so th a t, for k ,n = 1 , 2 , 3 , IV, (3.38) and (3.39) sim plify into:
yk
=
jjW g j ko a cos [(k - 1) A <j>- fa]
i A <f>
Zk
(3.43)
H q2^ ^ko 01^2(1 — cos (k — n)A4>]j
A,
j,
.2 j
a A(f> —— <1 — j —
(3.44)
/T * o o A ^ A _ 1
V
4
/
7T
for
for
k = n
where 70 = 1.781072418....
T he values of th e current I n are calculated by solving the system of linear equations (3.37) for
k , n = 1 , 2 , 3 Fr om them , using (3.35)-(3.36) into (3.32) one can determ ine the scattering
w idth a t a scattering angle <j>s:
1
JTM ( 1 \
a 2D W s )
IK
(i)l
h (q
N
In e i
a C0S
~ (n ~
(3.45)
71=1
Results
Here, the solution of th e integral equation for a perfectly electric conducting cylinder w ith TM polarized incident wave is illu strate d and com pared w ith th e exact solution.
The incident electric field in (3.48) is chosen incom ing from th e positive z-axis
33
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= 0) and
Two-dimensional PEC Cylinder - TM case, ka = 0.1, N=12
$2
»
Theory
Method of Moments
135
180
Figure 3.5: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
ko a = 0.1 and num ber of points N = 12. N um ber of term s in analytical expression sum m ation is
M = 10.
(i)
w ith a u nit am plitude £*
= Coi i-e -i
z Co e j k0 x
E « (p )
(3.46)
T he analytical expressions for th e equivalent surface current and th e scattering w idth can be
found in [15]. T he z-com ponent of the current is a function of th e polar angle <f>as follows
M<t>)
2 E^l
y '
toirkoa ^
-m cos (mcp)
m
H ^ { k 0 a)
(3.47)
while the scattering w idth is a function of difference betw een the scattering and incident angles:
2
EH)
m
Cm
m —0
where
,
,
?2 )------------ COS [ m (<j)3 -
J m \ k o CL)
rt I
H ^' i*k( o a)
1
for m = 0
2
for r a ^ O
( p i )j
(3.48)
(3.49)
T he infinite sum m ations in (3.48) and (3.49) are lim ited to a num ber M or term s such th a t the
rem ainder is neglegible.
34
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Two-dimensional PEC Cylinder - TM case, ka = 1.0, N=20
2.5
«
Theory
Method of Moments
cr
0.5
45
135
180
<t>[cleg]
Figure 3.6: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa = 1.0 and num ber of points N = 20. N um ber of term s in an alytical expression sum m ation is
M = 15.
Two-dimensional PEC Cylinder - TM case, ka = 2st, N=130
2.5
Theory
Method of Moments
0.5
45
135
180
<t>[deg]
Figure 3.7: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa = 2-k and num ber of points N = 100. N um ber of term s in an alytical expression sum m ation is
M = 20.
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Two-dimensional PEC Cylinder - TM case, ka = at, N=500
2.5
Theory
Method of Moments
cr
45
135
180
Figure 3.8: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa = 87r and num ber of points N = 500. N um ber of term s in analytical expression sum m ation is
M = 40.
Two-dimensional PEC Cylinder - TM case
Theory
Method of Moments
x
=
180 °
O)
= 0°
1
2
3
4
5
6
7
8
9
10
Figure 3.9: Scattering w id th obtained w ith m eth o d of m om ents and exact solution vs. koa, for
scattering angles <f>= 0° (backscattering) and <j>— 90° (forward scattering).
36
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Two-dimensional PEC Cylinder - TM case, l^ a = 0.1, N=12
.....
2
------Theory
* Method of Moments .
,
'io'
0
si
;— x— x— x— x— x— x
u>
c
(0
o
w
■03O
.N
E
45
90
135
Scattering Angle <|> [deg]
180
Figure 3.10: Scattering w idth obtained w ith m ethod of m om ents an d exact solution vs. <fis, for
koa — 0.1, num ber of M OM points N = 12, a n d analytical expression tru n c a te d to M = 10 term s.
In Figures 3.5 to 3.8, th e equivalent current obtained by solving (3.37) is plotted together the
exact solution (3.47) for different values of koa.
Since for th e incident wave (3.29) the surface
equivalent current is sym m etrical w ith respect to the x-axis, th e values of <p are lim ited to the
range between 0° and 180°. In all cases considered, the relative difference betw een the analytical
and num erical solution has been found to be less th a n 1%.
Figure 3.9 shows th e backscattering and forw ard scattering w idth, <j2D((j)s = 0°) and cr2D((ps =
90°) respectively, as functions of th e size p aram eter koa.
T he bistatic scattering w idth
a 2D
as a function of the scattering angle
<f>s
is plotted in Figures
3.10 to 3.13 for different values of koa. Again, th e bistatic scatterin g w idth is sym m etrical w ith
respect to the x-axis, therefore plots are lim ited to the values <j>3 betw een 0° and 180°.
T he relative error on th e scattering w idth estim ated using th e m ethod of m om ents is defined
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Two-dimensional PEC Cylinder - TM case, l^ a = 1.0, N=20
Theory
Method of Moments
“ 7
O)
Scattering ^ngle <J)=.[deg] 135
180
Figure 3.11: Scattering w id th obtained w ith m ethod of m om ents and exact solution vs. <f>s , for
koa — 1.0, num ber of M OM points N = 20, and analytical expression tru n c a te d to M = 15 term s.
Two-dim ensional PEC Cylinder - TM case, l^a = 2n, N=130
— Theory
x Method of Moments
O)
Scattering jungle
.[d e g ]135
180
Figure 3.12: Scattering w idth obtained w ith m ethod of m om ents and exact solution vs. <f>3, for
koa = 27T, num ber of M OM points N = 130, and an alytical expression tru n c a te d to M = 20 term s.
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Two-dimensional PEC Cylinder - TM case, l^a =
871 ,
N=500
Theory
x Method of Moments
2 15
"O
§ 10
U)
Scattering i^ngle <()c.[d e g ]135
180
Figure 3.13: Scattering w idth obtained w ith m ethod of m om ents and exact solution vs. <f>s, for
koa = 87t, num ber of MOM points N = 500, and analytical expression tru n c a te d to M = 40 term s.
w ith respect to th e exact solution, for a scattering angle (f>s, as follows:
±
5 = 2 D(<f>s)
N
<4h)
=
^
—
~ <T2d (4>s )
/0
7
(3' 50)
)—
where G2 D is the exact value of the scattering w idth found th ro u g h (3.48) and &2 D is the result
(3.45) of th e MOM calculation.
Table 3.1 gives th e m axim um error emax, in %, defined as
emax = A m ax
\e((/>s)\ ,
(3.51)
0 s e [ O ,2 ir ]
for the four values of koa considered, and also provides th e num ber N of points in th e MOM
♦
calculation and the num ber of term s M to which sum m ation (3.47) has been truncated.
Note how the difference betw een the two solutions is always very sm all and less th a n 1
3 .3 .2
T E in c id e n t w a v e
Transverse Electric (TE) polarization is defined as th e case w hen th e incident electric field is parallel
to the transverse xy-plane, as depicted on th e right side of Figure 3.4. In this case, the incident
39
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koa
N
M
emai
0.1
1.0
27T
87T
12
20
130
500
10
15
20
40
0.8
0.9
0.3
0.2
Table 3.1: Accuracy of M OM com putation of TM scattering w idth for P E C circular cylinder of
various values of koa, w ith emax in %.
m agnetic field is directed along the axis of the infinite cylinder, i.e., th e z-axis, and can be w ritten
as
HW(p) = £ t f « e iMcos<fo:r + sin<&y)
One could derive th e expression of th e incident field
(352)
from (3.52), th e n solve (3.26) and (3.27)
for the equivalent current J s . In this case th e surface equivalent current has only th e r-com ponent:
3 s {p) = J T( p ) f
(3.53)
T his would require solving th e second integral equation (3.27) th a t contains a derivative of J s
and involves differentiation of th e G reen’s function g\D. A m ajor sim plification of the problem is
obtained by applying th e d u ality principle, together w ith th e fact th a t in this case the equivalent
current has only th e r-com ponent, to equation (3.26), to obtain
H {z \ p )
=
JT2P>>~ f c { ^ ^ C0S^ ' +
J r ( p ' ) d C 'i
P ^ c
(3-54)
Once J r is evaluated by solving (3.54), the scattering w id th can be found through (3.23), which
for the TE-case becomes:
rT E /C
02i?
(k s ,k i)
=
^ — 1f^-2
4 | rr(l) |
l-^O I
( 4> z - Z 4> ) ■r 'J t { p ')
k° ^ s ' p dC'
(3.55)
N um erical so lu tio n u sin g th e m eth o d o f m om en ts
Here, equation (3.54) by using th e m ethod of m om ents. R ectangular pulses as defined in (3.33) are
selected as basis functions to expand th e current J T, an d D irac d e lta functions (3.34) are employed
40
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as testing functions. W ith such choices, the following system of equations is obtained:
N
Vk
=
YhZknln,
k — 1,2, 3 ,...,1V
(3.56)
n —1
where
n — 1 ,2 ,3 ,..., N
(3.57)
k = 1 ,2 ,3 ,..., N
(3.58)
and, for k, n — 1 ,2 ,3 ,..., N , the coefficients of (3.56) are
Z *. =
A
(3. 59)
1
°J
\Pk~Pn\
w ith Skn being th e K ronecker’s delta:
Jkn
1
for
k =n
0
for
k j4 n
(3.60)
For the specific case of a circular cylinder of radius a, expression (3.59) becomes
Jkn
l + ( l - 8kn)aA(f> ^
H i 2)(kQa^J2[l - cos (k - n ) A f l)
(3.61)
4>]ei k ° a cos W*~{ n- l) A < l > ]
(3.62)
and th e scattering w id th is given by
°2D (4>s)
=
1
7
4
1
0
N
£
In cos [</>, - ( n - 1)A
71=1
R esu lts
In this paragraph, th e solution of the integral equation for a perfectly electric conducting cylinder
w ith T E -polarized incident wave is illu strated and com pared w ith th e exact solution.
T he incident m agnetic field is chosen w ith a u n it am p litu d e H q and incoming from the positive
x-axis, i.e., from an incident angle (fix = 0:
H w (p )
=
z e i k° x
41
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(3.63)
Two-dimensional PEC Cylinder - TE case,
1
r
a = 0.1, N=12
'
2c 1 .
o
11 .
O
c
CO
-i o.i
cr
Q O .i
CO
= 0.< ........
-----Theory
» Method of Moments .....................
C/3
90
[deg]
45
135
180
Figure 3.14: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa = 0.1 an d num ber of points N = 12. N um ber of term s in analytical expression sum m ation is
M = 10.
T he analytical expressions for the equivalent surface current and th e scattering w idth are given
in [15]. T he surface current is given by th e infinite sum m ation
•W )
ir koa m= 0
(3.64)
H $ \ k o a)
while the scattering w idth is
JTM fi \
a 2D W s )
:
kQ
^ i fcoa)- cos [m (cf,s - ^-)]
cm J /ov
£ ( - ! ) m em
(3.65)
{k0 a)
771=0
where em is defined in (3.49).
T he m ethod of m om ents has been applied to four cases of P E C cylinders w ith different values
of koa, for a T E incident wave given by (3.63). T he results are illu strated in Figures 3.14-3.17,
where the equivalent current obtained by solving (3.54) is p lotted together to the exact solution as
a function of the polar angle <f>. In all cases, th e relative difference betw een the two solutions has
been found to be less th a n 1%.
42
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Two-dimensional PEC Cylinder - TE case,
a = 1.0, N=20
2 1.4
•§ 0.8
c
LU
a) 0.6
0.4
—
*
Theory
Method of Moments
0.2
135
45
180
<t>[deg]
Figure 3.15: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
koa — 1-0 and num ber of points N = 20. N um ber of term s in an alytical expression sum m ation is
M = 15.
Two-dimensional PEC Cylinder - TE case, 1^a = 2k, N=130
2 1.4
• § 0.8
cr
LU
a) 0.6
0.4
Theory
Method of Moments
0.2
90
<i>[deg]
135
180
Figure 3.16: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
k o a = 2 k and num ber of points N = 130. N um ber of term s in an alytical expression sum m ation is
M = 20.
43
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Two-dimensional PEC Cylinder - TE case, 1^ a = 8k, N=500
• =
0.8
0.6
Theory
Method of Moments
45
135
180
<|>[deg]
Figure 3.17: Surface equivalent current obtained w ith m ethod of m om ents and exact solution for
k^a = 87r and num ber of points N = 500. N um ber of term s in analytical expression sum m ation is
M = 40.
Figure 3.18 shows the backscattering and forw ard scattering w idth, a 2D(<ps = 0°) and a 2D((j)3 =
90°) respectively, as functions of koa. T he b istatic scattering w id th <r2D vs. the scattering angle
< is plo tted in Figures 3.19-3.22 for different values of koa.
Table 3.2 gives th e m axim um error emax, in %, for th e four values of koa considered, and also
provides the num ber N of points in the M OM calculation a n d th e num ber of term s M to which
sum m ation (3.65) has been tru n cated .
koa
N
M
tmax
0.1
1.0
2ir
Sir
12
20
130
500
10
15
20
40
0.01
0.04
0.1
0.01
Table 3.2: Accuracy of M OM com putation of TM scattering w id th for P E C circular cylinder of
various values of koa, w ith emax in %.
44
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Tw o-dim ensional PEC Cylinder - TE case
Theory
x Method of Moments
ay
-4
Figure 3.18: C om parison betw een scattering w idth obtained w ith m ethod of moments a n d exact
solution as functions of k^a for scattering angles <j> = 0° (backscattering) and 0 = 90° (forward
scattering).
45
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Two-dimensional PEC C ylinder-TE case, l^a = 0.1, N=12
-20
S -3 0
■o
§ -4 0
-5 0
-6 0
Theory
x Method of Moments
-70.
Scattering Jingle ^ .[ d e g ] 135
180
Figure 3.19: C om parison betw een scattering w idth obtained w ith m ethod of moments and exact
solution for k^a = 0.1 and num ber of points N = 12.. N um ber of term s in analytical expression
sum m ation is M — 10.
Two-dim ensional PEC Cylinder - TE case, l^a = 1.0, N=20
Theory
Method of Moments
-4
-7
- 8.
Scattering ^ng le <|>.[d e g ]135
180
Figure 3.20: C om parison betw een scattering w idth o b tain ed w ith m ethod of moments and exact
solution for koa = 1 . 0 and num ber of points N = 20. N um ber of term s in analytical expression
sum m ation is M = 15.
46
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Two-dimensional PEC Cylinder - TE case, l^a = 2n, N=130
*o
O)
-§
-1 0
■o
(13
N -2 0
o -2 5
Theory
* Method of Moments
-30.
45
90
135
Scattering Angle <|> [deg]
180
Figure 3.21: Com parison betw een scattering w idth o b tained w ith m ethod of m om ents and exact
solution for koa = 2n and num ber of points N = 100.. N um ber of term s in analytical expression
sum m ation is M = 20.
Two-dimensional PEC Cylinder - TE case, l^a = 8n, N=500
O)
Theory
» Method of Moments
-30.
Scattering ^ngle i|>c.[d e g ]135
180
Figure 3.22: Com parison betw een scattering w idth o b tain ed w ith m ethod of m om ents and exact
solution for koa = 87r and num ber of points N = 500.. N um ber of term s in analytical expression
sum m ation is M = 40.
47
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3.4
D ielectric tw o-dim ensional body
For a dielectric body, b o th th e exterior a n d the interior integral equations have to be considered, and
th e unknow n are
b o th the surface electric an d m agnetic current. As for the PE C two-dim ensional
body, the basic cases to be considered when studying th e scattering from a two-dim ensional di­
electric body are those w ith TM a n d T E incident waves. H ereafter, only the TM case will be
considered. T he T E case can be easily derived using the duality principle.
Let the incident electric field be a plane wave directed along th e 2-axis, i.e., be TM -polarized,
w ith an expression (3.29). Such incident wave induces on th e perfect conductor an equivalent surface
electric current J 6 th a t is also directed along the 2-axis, while th e equivalent surface m agnetic
current M,s will have only th e transverse com ponent. Hence,
J s(p)
= U
M s {p)
p
)z
(3.66)
= M r (p) t
(3.67)
Therefore, only equations (3.9) and (3.11) need to be solved:
e
^ \
p
) =
—
+ j ^ p o j>c+ <?2£>(P>
+ £ +
0 _
_ M t(p ) _
+
p
') U
p
') dC' +
cos ^
j
g2D{ p , p') U
(^ d x ^ c o s^ ' +
p
M t { p ') dC',
p e C+
(3.68)
M r {p') dC',
p ec~
(3.69)
') d C ' +
s*n ^
Once Jz and M r are known, the scattering w idth can be determ ined using (3.23), th a t for this
case becomes:
where cr™ = a 1® is th e only non-zero scattering w idth.
48
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N u m erical so lu tio n u sin g th e m eth o d o f m om en ts
In addition to ap proxim ating Jz in term s of basis functions as in (3.35), the m agnetic current r com ponent is expanded as
N
t' e C
(3.71)
71=1
S ub stitu tin g (3.35) an d (3.71) into th e system of integral equations (3.69),(3.70), and testin g
the resulting equations w ith the functions (3.34) for k = 1 , 2 , . . . , N , leads to th e following linear
system of algebraic equations:
N
N
'y ^-A-kn I n 4" y
B k n M -n
n =1
~
Vk
(3.72)
=
0
(3.73)
n=l
N
y
N
] Cfcn I n "
b y ( -Dfcn M n
n—
1
where
n=l
Ffe = £ « ( * * ):
In — Jzi^n) >
(3.74)
= 1 ,2 ,3 ,..., N
— M r (tn) ,
n — 1 ,2 ,3 ,..., N
(3.75)
For the p articu lar case of a circular cylinder of rad iu s a, th e coefficients Akn an d B^n are found
to be
fcoCo u {2)(
a A(j) ----------- H q (kou)
U)
Akn
a^
oCo
iU t
A;
-f— I b
<3-76)
J\
-2
i l - , -
(3.77)
where 5kn is the K ronecker’s delta defined in (3.60), and
a y 2[1 — cos (k — n)A(j)]
(3.78)
T he coefficients Ckn an d Dkn have the same expressions as Ak n and Bk n, w ith eo, ^o> ko, and (o
replaced by e, ju, k, an d (, respectively.
49
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Once th e system (3.72),(3.73) is solved for I n an d M n , the scattering w idth is given by:
N
kQ
(Oi2
Y
{Co I n ~ M n cos
[<j>s -
(n
-
1) A 0 ]} e
jko a cos [(f)s — (n - 1)A 0]
n =1
(3.79)
R e s u lts
In this subsection, th e solution of the integral equation for a hom ogeneous dielectric cylinder w ith
T M -polarized incident wave is illu strated and com pared w ith th e exact solution.
A nalytical expressions for the induced electrom agnetic fields and th e scattering w idth can be
found in [73]. T he equivalent surface currents can be found from th e induced tangential fields on
the cylinder surface by using definitions (2.25) and (2.35), and are given by:
p(i)
-jT -
oo
Y
•
Jz(<P)
=
£m [jmiko a) + A m H $ ' (k 0 a) cos (m<f>)
(3.80)
m —0
oo
Y
3 m
[Jm(ko a) + A m H $ (kQa)
cos (m<f>)
(3.81)
m=0
and while the expression for th e scattering w idth is:
4
ko
Y
(-1)™ em am
(k0 a) cos [m(<f)s -
<f>i)}
(3.82)
771=0
T he coefficients am in (3.80)-(3.82) are given by
am
(ki/fJ-i) Jm(ko o) Jm (ki of)
(ki/ni)
(ko j Ho) Jm(ko ®) Jm(ki a)
(3.83)
(kQa) J'm (ki a) - (kQ/no) H $ (kQa) Jm (k\ a)
T he system of integral equations (3.68),(3.69) has been solved for a num ber of cylinders of
different size param eter koa and relative dielectric constant er . Figures 3.23-3.30 show the equiv­
alent currents J s and M s vs. the azim uth angle <j>for those cases. T he solid line is the analytical
solution (3.82), while th e crosses indicate the M OM values. T h e scattering w idth is p lo tted in
Figures 3.31-3.34 as a function of th e size param eter koa, and in Figures 3.35-3.35 as a function of
the scattering angle <f>s.
50
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Two-dimensional Dielectric Cylinder - TM case,s. =2.56
«
Theory
Method of Moments
k0a = 5.0
a * 1.0
135
180
<i>[deg]
Figure 3.23: E quivalent electric current, m ethod of m om ents solution and theory, vs. (f>, for a
lossless homogeneous cylinder w ith sT = 2.56, for k^a = 1.0 a n d k^a = 5.0. N um ber of points used
in the m ethod of m om ents are N = 20 and N = 100, respectively.
Two-dimensional Dielectric Cylinder - TM case,er =2.56
3.5
x
Theory
Method of Moments
2 2.5
« 1.5
45
90
135
180
<j>[deg]
Figure 3.24: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. (j), for a
lossless homogeneous cylinder w ith eT = 2.56, for k^a = 1.0 an d k^a = 5.0. N um ber of points used
in the m ethod of m om ents are IV = 20 and N = 100, respectively.
51
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Two-dimensional Dielectric Cylinder - TM case,sr=2.56 - j0.102
Theory
Method of Moments
kQa = 5.0
135
180
Figure 3.25: E quivalent electric current, m ethod of m om ents solution and theory, vs. 0, for a lossy
homogeneous cylinder w ith er — 2.56 —j'0.102, for k^a = 1.0 a n d k^a = 5.0. N um ber of points used
in the m ethod of m om ents are N = 20 and N = 100, respectively.
Two-dimensional Dielectric Cylinder - TM case,sr=2.56 - jO. 102
3.5
x
Theory
Method of Moments
2 2.5
” 1.5
cr
45
90
<t>[deg]
135
180
Figure 3.26: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. <fi, for a
lossy homogeneous cylinder w ith eT = 2.56, k^a = 1.0 and k^a = 5.0. N um ber of points used in the
m ethod of m om ents are N = 20 and N = 100, respectively.
52
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Two-dimensional Dielectric Cylinder - TM case,e = 5.0 - j 1.0
4.5
Theory
Method of Moments
2.5
=
1.0
0.5
90
<i>[deg]
135
180
Figure 3.27: E quivalent electric current, m ethod of m om ents solution a n d theory, vs. (f), for a lossy
homogeneous cylinder w ith er = 5.0 —j‘1.0, k$a = 1.0 and k^a = 5.0. N um ber of points used in the
m ethod of m om ents are N = 20 and N = 100, respectively.
Two-dimensional Dielectric Cylinder - TM case,£.= 5.0 - j 1.0
«
Theory
Method of Moments
0.2
45
135
180
(j>[deg]
Figure 3.28: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. (f>, for a
lossy homogeneous cylinder w ith sT = 5.0 —j 1.0, k^a = 1.0 and k^a — 5.0. N um ber of points used
in th e m ethod of m om ents are N = 20 and N = 100, respectively.
53
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Two-dimensional Dielectric Cylinder - TM case, s =60.0 - j59.9
—
*
Theory
Method of Moments
135
180
Figure 3.29: Equivalent electric current, m ethod of m om ents solution and theory, vs. tfi, for a lossy
homogeneous cylinder w ith er = 60 —j59.9, k0a = 1.0 and k^a = 5.0. N um ber of points used in
th e m ethod of m om ents are N = 20 and N = 100, respectively.
Two-dimensional Dielectric Cylinder - TM case, s =60.0 - j59.9
0.3
£0.25
c 0.15
Theory
Method of Moments
= .1,0
.k0a.=.S.O
0.05
135
180
Figure 3.30: Equivalent m agnetic current, m ethod of m om ents solution and theory, vs. </>, for a
lossy homogeneous cylinder w ith er = 60 —J59.9, fcoa = 1-0 an d k^a = 5.0. Num ber of points used
in the m ethod of m om ents are N = 20 and N = 100, respectively.
54
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Two-dimensional Dielectric Cylinder - TM case,£.=2.56
Theory
Method of Moments
£ 10
■a
=
—<
180'
CD
Figure 3.31: S cattering w idth, m ethod of m om ents solution and theory, size param eter k$a, for
a lossless homogeneous cylinder w ith er = 2.56, for k^a = 1.0 a n d k^a = 5.0. Solid line is
backscattering (<f>s = 0°), and dashed line is forw ard scattering (<f>s = 180°).
Two-dimensional Dielectric Cylinder - TM case,er=2.56 - j 0.1
Theory
Method of Moments
=
180'
£ 10
o>
=
0
Figure 3.32: Scattering w idth, m ethod of m om ents solution and theory, size param eter k^a, for a
lossy homogeneous cylinder w ith er = 2.56 — j'0.102, for koa = 1.0 and koa = 5.0. Solid line is
backscattering (cf>s = 0°), and dashed line is forw ard scattering (<f>s = 180°).
55
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Two-dimensional Dielectric Cylinder - TM case,£=60.0 - j59.9
Theory
Method of Moments
=
£ 10
180'
=
0'
Figure 3.33: Scattering w idth, m ethod of m om ents solution and theory, size param eter k^a, for
a lossy homogeneous cylinder w ith eT = 5.0 — j l .0 , k§a = 1.0 and k^a = 5.0. Solid line is
backscattering (<f>s = 0°), an d dashed line is forw ard scattering ((f)s = 180°).
Two-dimensional Dielectric Cylinder - TM case,er= 5.0 - j 1.0
25
Theory
» Method of Moments
20
=
0'
Figure 3.34: S cattering w idth, m ethod of m om ents solution a n d theory, size param eter koa, for
a lossy homogeneous cylinder w ith s r = 60 — j 59.9, k^a = 1.0 and k^a = 5.0. Solid line is
backscattering (cf>s = 0°), and dashed line is forw ard scatterin g (<ps = 180°).
56
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Two-dimensional Dielectric Cylinder - TM case, e =2.56
Theory
x Method of Moments
k„a = 5.0
-15
- 20.
45
90
135
Scattering Angle 0 [deg]
180
Figure 3.35: Scattering w idth, m ethod of m om ents solution an d theory, vs. scattering angle <f>s, for
a lossless homogeneous cylinder w ith eT = 2.56, for koa = 1.0 and koa = 5.0. N um ber of points
used in the m ethod of m om ents are IV = 20 and N = 100, respectively.
Two-dimensional Dielectric Cylinder - TM case, e =2.56 —j0.102
Theory
Method of Moments
D)
-5,
k.a = 5.0
-15
-20
Scattering jungle <j>e.[deg] 135
180
Figure 3.36: S cattering w idth, m ethod of m om ents solution and theory, vs. scattering angle <f>s, for
a lossy homogeneous cylinder w ith eT = 2.56 —y’0 .102, for koa = 1.0 and koa — 5.0. N um ber of
points used in the m ethod of m om ents are N — 20 and N = 100, respectively.
57
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Two-dimensional Dielectric Cylinder - TM case,er=60.0 - j59.9
Theory
x Method of Moments
■o
O)
k.a = 5.0
-6
-8
-10
Scattering ^ngle <j>s.[deg] 135
180
Figure 3.37: S cattering w idth, m ethod of m om ents solution an d theory, vs. scattering angle <j>3, for
a lossy homogeneous cylinder w ith sr = 5.0 — j 1.0, koa = 1.0 and koa = 5.0. N um ber of points
used in the m ethod of m om ents are N — 20 and N = 100, respectively.
Two-dimensional Dielectric Cylinder - TM case,e = 5.0 - j 1.0
Theory
x Method of Moments
TO
-10
= -15
-25
45
Scattering Angle <j> [deg]
135
180
Figure 3.38: S cattering w idth, m ethod of m om ents solution and theory, vs. scattering angle <f>s, for
a lossy homogeneous cylinder w ith er = 60 — y59.9, koa — 1.0 a n d koa = 5.0. N um ber of points
used in the m ethod of m om ents are N = 20 and N — 100, respectively.
58
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3.5
Conclusions
In this chapter, the m ethod of m om ents has been used to solve the coupled surface integral equa­
tions, and detailed results have been given for cylinders of circular cross section. B oth perfectly
conducting an d dielectric - lossless and lossy - cylinders have been considered. For perfectly con­
ducting cylinders, th e cases of TM and T E incident wave has been investigated, while for conducting
cylinders only th e case of TM incident wave has been studied. All results have been found to agree
very well w ith the theory. T his validates th e use of th e coupled integral equations derived in C hapter
2 as a tool to determ ine th e scattering from a rb itra ry objects located in an infinite m edium.
59
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Chapter 4
B odies of R evolution
T he previous chapter has dealt w ith infinite cylinders illum inated by an electrom agnetic source th a t
is function of the coordinate z. A lthough th e resulting problem is three-dim ensional in n atu re, a
Fourier decom position along th e cylinder axis can be used to reduce the problem to a superposition
of two-dim ensional problem s m ore su itable for num erical solution. In th is chapter another specific
class of three-dim ensional problem s will be considered, th a t can also be tre a te d as a superposition
of two-dim ensional problem s. A xisym m etric objects, often called bodies of revolution because they
can be obtained by ro ta tin g a plan ar arc around an axis, constitutes such particular class.
In
m ost cases, vegetation elem ents can be m odeled as axisym m etric objects, therefore the ability to
tre a t bodies of revolution is very im p o rta n t in the scattering from vegetation. For them , a Fourier
series in the ro tatio n angle can be employed to transform the problem into one involving uncoupled
equations for each harm onic.
4.1
G eom etry of th e problem
As shown in Figure 4.1, a b o d y of revolution is obtained by ro ta tio n of a planar arc C - called the
generating curve - ab o u t a n axis, w hich is hereafter chosen as the z-axis of a C artesian coordinate
system. T he surface 5 th u s generated represents the bou n d ary betw een free space and the m aterial
body, which is assum ed as having p e rm ittiv ity e = £q
an d perm eability /r = mo Md- Any point P
60
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Figure 4.1: B ody of revolution
on S is uniquely identified by a pair of variables (f, 0), where t is th e curvilinear coordinate along
the generating curve to which P belongs, and 0 is th e ro ta tio n or azim uth angle m easured from
the ajz-plane. A right-handed tria d ( n , 0 , r ) of orthonorm al vectors is defined on S, w ith n
and f being the norm al and the tangent u nit vectors to S in th e plane of the curve C.
All these definitions are illu strated in Figure 4.1. T he tria d ( r , n , 0 ) can be expressed in
term s of ( x , y , z ) as follows:
f
= s in 0 c o s 0 x + s in 0 s in 0 y
+ cos0z
(4.1)
n
= cos 0 cos (f>x + c o s 0 s i n ^ y
—s i n 0 z
(4.2)
0
= —s i n ^ x - f c o s ^ y
(4.3)
where 0 is the angle betw een r and z , i.e. cos 0 = r ■ z , w ith 0 = 0 if f = z and 0 = n / 2 if
61
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f
= p . The infinitesim al element of area on S is given by
d S = pdtd<j)
4.2
4 .2 .1
Let th e
(4.4)
The electric field integral equations for a b od y of revolution
F o r m u la tio n
body of revolution be
excited by an incident plane wave w ith tangential com ponent
(i)
E) on the surface S. T he induced equivalent tangential electric and m agnetic currents J 5, M s
on th e surface S satisfy th e surface integral equations (2.37)-(2.38), on th e exterior and interior
surface, S + and S ~ , respectively :
C
M s(r) x n
=
2 E$ (r) — 2ju>po (I — n h ) ■
Js +
- 2 (I - n
M ,(r) x n
=
n ) ■[
Js+
2jup (I - h
- 2 (I - n
where
r7 /-t0
&o
VG°(r, r') x M , (r') d S '
n) ■f
,
r € S+
(4.5)
J s (r ' ) d S ' +
G( r, r ' ) I — ^ttt-V
V '
k2 s
Js- L
n )•
J s(r') dS' +
G0( r , r ' ) I ------
VG (r,r') x M .( r ') d S ' ,
,
r € S~
(4.6)
I is the unit dyad,
I = x x 4 - y y + ^2 = n n + 0 0 + f f
(4.7)
,
G ° ( r ,r ') and G (r, r ') are the scalar G reen’s functions in the two m edia,
- j k |r - r '|
, ~ j k o \ r - r '|
G °( r , r ') =
G ( r ,r ') =
47r |r — r
47r |r — r '|
’
(4.8)
ko and k are the wavenum bers,
k0 =
lu^/ eopo
,
k = u ^ / e p = kQyJpded ,
(4.9)
and V's ■J s is the divergence of J s on th e surface 5 ,
X-7>
T
(r l\
V , - J .(r ) =
<K ^
r
9 3 s
62
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(4.10)
T he exterior an d interior surfaces S + and S ~ , respectively, have been introduced in C h ap ter 2: S +
is defined as a surface infinitesim ally close to th e surface S and enclosing S itself, and S ~ is defined
as a surface infinitesim ally close to the surface 5 and enclosed by S itself.
E quations (4.5),(4.6) are expressed in term s of vector quantities. T h e very first step in their
solution is to decompose th em into a pair of scalar integral equations for each of their com ponents.
(i)
T he tangential incident electric field E ) a n d th e equivalent surface currents J s, M s on th e surface
S can be w ritten in term s of their r - and ^-com ponents as follows:
E ^ ( t , c f > ) f +Ef(t,cj>)4>
E « (r) =
J s (r)
=
M s(r) =
(4.11)
JT(t,<p)
t
+
4>
(4.12)
M T(t,<j>)
t
+ M<p(t,<f>) $
(4-13)
S u b stitu tio n of these expressions into th e first integral equation (4.5) and separation th e single
com ponents yields two integral equations th a t contain only th e r - and ^-com ponents of J s, M s
and E j^ . Introducing eight integral operators a pq and f3pq, (p, q = r , (p), these equations m ay be
w ritten in com pact form as follows:
£
a ' « { J J( t V ) } +
£
= £?> (<,«
(4.14)
q=T,<t>
? = T ,0
+
£
q = T ,ip
(4.15)
q = T ,< t>
T he operators a pq and
f3vq (p,q = r , <f>) perform integrations of th e functions J q(t', <pr) and
Jq( f , </>'},respectively, in th e variables t' and (j)', which axe therefore left indicated in their notation.
These operators are defined as:
a pT{ J T{t',(f>')}
d pq{ M q{t',<4>')}
—
p • r 'G ° ( r , r ') -
=
- j u j f j ,o [
=
-ju p
=
- [ p ■V G 0( r ,r ') x q ' M g t f , # ’) dS'
Js
Js p
o
[
Js
, / _ 0,
p • 0 G (r, r ') -
k i dp dt'
[p'JT( t ' d S '
1 1 dG° d
-2 p'
rJ dp dcp_
kg
63
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(4.16)
(4.17)
(4.18)
M ore detailed expressions are given in A ppendix C.
T h e integral equation (4.6) for the interior of th e body can be decom posed into its r - and
^ —com ponents in a sim ilar m anner, and w ritten as
£
7t8
q=T,<j>
Y ^
q=T,<t>
# ' ) } - £ S'"
q=T,<j>
« }
4)} - q-r,<p
E S**
*')} +
w here the expressions for the operators 7P9 and
5pq
-
= o
(4.19)
= 0
(4.20)
(p, q = T,<fr) can be found from the expressions
(4.16)-(4.18) for a pq and (3pq by replacing G °(r, r ') , &o, £o>and
po w ith G(r, r1), k, e, and p,
respectively, and they are also given in A ppendix C.
4 .2 .2
S e p a r a tio n o f m o d e s
Expressions (4.14),(4.15) and (4.19),(4.20) are tw o-dim ensional integral equations on th e surface
5 . One could solve such system of equations using th e m ethod of m om ents w ith th e definition of a
two-dim ensional grid of points on S. However, this approach would require a large num ber of points
and th e evaluation of tw o-dim ensional integrals, a n d would result in a a heavy com putational cost.
Fortunately, the axial sym m etry of th e surface of revolution S allows a much sim pler tre a tm e n t
of th e problem because it im plies th a t all th e quantities involved in such equations have th e sam e
periodic dependence on th e azim uth angle 0, w ith a period equal to 2 tt or an integer m ultiple of it.
In particular, since Ep \ Jp and M p (p = r, <j>) are periodic functions in the variable <j>, on the
surface S they can be expanded into Fourier series as follows:
+oo
4 °M )
=
E ® {t)ein*
E
(4.21)
71 — — OO
-boo
Jp( t J ) =
Y
-boo
M p(t,cf>) =
£
M Pn( t ) e qn<»
(4.22)
n = —oo
n=—
oo
Using such Fourier m ode expansions into the expressions for oiPq and f3pq results into a series
64
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expansion for those operators themselves:
a T ,W + ' ) }
=
+00
2* £
(4.23)
71= —00
=
+00
27T £
T
n q { M qn(t')}
(4.24)
n=—
oo
a pq and /3pq are linear operators perform ing curvilinear integrals along C. For example:
< T {Jrn ( t ' ) } =
(4.25)
+ c o s , 0 o ( t i t ,, _
P Z { M r [<, # ) } =
(4.26)
= —j [ [p cos ip sin ip' —p' sin ^ cos ip' — (z — z ') sin ip sin ip']
«/c
-^n+i(A O
where p' and ip' are the values of p and ipfor t = t'. E xplicit expressions
given in A ppendix C. T he expressions (4.25) and (4.26)
^
2
for all a pq and (3pq are
for a T
nT and /3£T, aswell as those for the
other operators a pq and (3pq, contain th e functions:
e°(t, f )
=
K° ^
=
-i rZ'K
- jf
C W j e - ’W - W < ! ( # '- * )
(4.27)
(4-28>
where R is the distance betw een th e two points a t r, r':
R = |r — r '| = \ ! p 2 + p'2 — 2pf/ cos(0' — <p) + (z — z ' ) 2
(4.29)
T he integrals in (4.27)-(4.28) are defined on the surface S. T h e in teg ratio n in <p' — <p removes the
angular dependence, so th e results are functions of t a n d t' only.
Expressions (4.27) an d (4.28) can be regarded as the coefficients of th e expansion of the G reen’s
function G0(r, r') a n d its derivative into Fourier series on th e surface S :
+oo
G ° ( r ,r ')
=
J]
G0n { t , t ' ) e j n W - ®
v ,r'e S
n=—
oo
65
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(4.30)
'N+l
'N+l
’N+l
(a)
(b)
Figure 4.2: (a) Choice of discretization points {f&} on the curve C. (b) Discretized curve C
1 dG°{ r , r ')
R
dR
+00
53
r ,r 'e S
(4.31)
If (4.21) and (4.22) are su b stitu te d into (4.14)-(4.15) and (4.19)-(4.20), and th e resulting equa­
tions are integrated side-by-side in th e variable 0 over th e interval [0,27r], the following system of
linear equations is o b tained for each single harm onic m ode n = 0, ± 1 , ± 2 ,...:
« * „ (* )
9 = T ,<
E in
g~T,<f>
E
q = T ,i
(4.32)
4 5 (0
(4.33)
q=T,<,
E a? «»(<')}+ E 0t, {M,Jt)} + M tM
?=T,<
4 5 (0
<7=r,<?
- E C
- 444
(4.34)
q=T,<t>
y . ^’ {v,„(t')) + 4 4 4
=
0
q = T ,< f
66
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(4.35)
4 .2 .3
M e t h o d o f m o m e n ts s o lu t io n
A set of N + 2 points {£*}, k = 0 ,1 ,2 , . .. N + 1, is chosen on the generating curve C as illustrated
in Figure 4.2 (a), w ith to and fjv+i being th e end points. The points {£&} subdivide C into N + 1
segments, which represent an approxim ation C of the generating curve, as shown in Figure 4.2 (b).
T hen, a second set of N + l points {£& }, k = 1,2, ...N + 1, is chosen on C. T h e generic point t
is
defined as the m idpoint on the segment [tk-itk].
Following the M OM, th e tangential incident electric field and the tangential surface equivalent
currents are approxim ated by th e following sum m ations:
N
N + l
PE${t) =
E f n (t) =
>
N
P jrjt) =
J 0n(f) =
(4.37)
m= 1
iv
n - r t
E M % p W (t)
m=
(4.36)
N + l
£ j ^ T m (f) ,
m = 1
,(t) =
p£> (f)
,
M tJt) =
1
(4.38)
m = 1
where the expansion functions are chosen as
P « (t)
=
P (~
t ^ ) ,
\ A fm /
P ${t)
=
p
Tm{t)
m = 1 , 2 , 3 ,..., N
{ ^ ^ Y
= r(^ r )’
(4.39)
™= 1 , 2 , 3 , . . . , N + 1
(4.40)
rn = l , 2 , 3 , . . . , N
(4.41)
and are illustrated in Figure 4.3. T he segm ent lengths A tm and A f~ are defined, respectively, as
Atm = t S —i ~ t m ,
A tm = tm —fm-i ,
m. = 1 ,2 , . . . , N + 1
(4.42)
and P(t) and P(t) are rectangular and trian g u lar pulse functions defined as
f 1
P(t) =
I 0
for _ I < t < I
2 - C<2
elsewhere
,
T( t)
=
f 1 — |*|
' '
I 0
for - 1 < t < 1
,
u
■ (4.43)
elsewhere
67
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f
A tm
lm--^T
"
f j_ A tn
lm~I—
tm— A tn
tm~~ A.trr.
A tn
(b)
(a)
(C)
Figure 4.3: D efinition of expansion functions: (a) Pm, ^(t ), (b)
5(2) 1
and (c) Tm (t).
T he choice of tria n g u la r pulses for the expansion of th e r-com ponent of the equivalent surface
current Jt is necessary due to th e presence of a derivative of JTn{t). T he use triangular pulses T(t)
makes possible to express such derivative as a linear com bination of rectangular pulses P{t). In
particular, it is easy to see th at:
,(2)
P t t X t ) + P%1 iCO
A t m+1
Atn
N
Tn
1 = £ • £m
m= 1
dt
(4.44)
Using th e approxim ating sum m ations (4.36)-(4.38) and (4.44) into equations (4.32)-(4.35) yields:
N
E
N + l
O'
)A \ j T
m + Y , aTn { P£ ) (t' ) } j t +
P'
771=1
J
m= 1
,(1)
N
(4.45)
+
771=1
V
“
J
N
E a. — Pn[P '}J t + E
rjrr
771=1
'
{ P t (0}J t +
771=1
( p ( 1) 'I
N
771=1
N + l
+ t M l
+
771=1
P
-E
771=1
r rp
(4.47)
\ 'N
771=1
f
p (l)
)
I
P
)
Hr
i V - |- l
E ^ P #P 2}J t + E
771=1
= 0
E 7? H 2)(0} J t +
J
JV
N
(4.46)
771=1
' T m (t')
71=1
= E ( + t)
1V+1
“E
H
’m } M t -
771=1
{ p t d ') } J t +
no — 1
68
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Pm
m =T
1
i
p r 1 J,
K
m =
(4.48)
11
Using the following testing functions on equations (4.45),(4.47) and (4.46),(4.48), respectively,
xl(t)
=
X*(t)
=
S (t-tj),
k = 1 ,2 ,3 ,...,1V
(4.49)
k = 1,2,3,...,N + 1
(4.50)
leads to the following linear system of N p algebraic equations, for each polarization p =
N q
N q
Y Y
q=r,(f>
aPn ( k i m ) Xm
m—1
+Y Y
K q( k , m ) M ^
= Vpkn
(4.51)
= o
(4.52)
q—T,<t> m=l
N q
N q
Y Y
+ Y
q=r,4> m = l
q=T,4>
Y < " ( k’™ ) M i
m=l
for k = 1 , . . . , N p
where
Np
N
N + l
=
for p = t
for p = (f>
(4.53)
T he coefficients a ^ ( k , m ) a n d b%}(k,m)
k = 1,..., N p-, m = 1,..., ./V
ar ( t , m )
=
-jf
a^{k,m
=
~ [ <*P
n { P $ ( 0 } Xi (*) d t ,
Jc
bT
nT( k , m
_
r
P k
bT
J(K m
btT(k, m
b^(k,m
^
=
km
[
OTT
I
k = 1,..., A/p; 77i = 1,..., N + l
X litfdt,
(4.54)
(4.55)
k , m = 1,..., AT
(4.56)
j c f t * { P $ ( t ' ) } xl(t) dt,
k = 1,..., AT; m = 1,..., N + 1
(4.57)
jc ^ T\ ^ p ^ x W ) d t ,
k = 1,..., N + 1; m = 1,..., AT
(4.58)
Pk
A tk
k,
771
= 1,..., Af + 1
(4.59)
where Skm is the th e K ronecker’s delta. Expressions for th e coefficients bf f(k ,m ) and d ^ ( k , m )
can be obtained from (4.54)-(4.59) by replacing the G reen’s function, ko,
eq
and p,o. Defining th e
69
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i
1
vectors
■
' M {n ‘
M v2n
3
3
■ J-P n
V
Pn
_
,
l Pn =
,
=
p = r,4>
,
(4.60)
H
.........
■
• £ £ ?
1
1
i
1
i
i
1
the system of equations (4.51)-(4.52) can be w ritten in th e following convenient m atrix form:
] M Tn +
+
_
Cd
\T<t> J<t>r
y Tn
(4.61)
— y<t>n
(4.62)
i
[A ? ] I T" +
4><t>
[ A f ] l Tn + [a ^ \ J<Pn + [ B f M Tn + ±■R
Jn
[C 7 ] ITn 4- [c ;* ] J4>n - p r
M Tn -
[ c f ] l Tn + [C ^ ] I*" - [ ° f ; M Tn -
d
t<i> M <Pn =
0
(4.63)
M^n
0
(4.64)
71
=
where [A£9] is th e m a trix whose elem ent k , m is apq( k , m ) given by (4.54),(4.55), i.e.,
[A -n9]
=
.
.
_ a pq(Np, 1)
.
apq(Np, 1)
•••
.
P ,q =
r,<
(4.65)
a g ( N p, N q) .
and so on for [B£9], [B£9] , and [D£9].
4 .2 .4
I n c id e n t field
This section focuses on th e expansion into angular m odes of th e tangential electric field on the
surface 5, as given in (4.21). A n expression for th e n -th coefficients E Pl (t) will be derived based
on their definition
\
/"27T
P = T,
E ^ [t) =
(4.66)
Let the body of revolution be illum inated by a uniform plane wave
E w (r) =
In equation (4.67), ko =
uj^/ uqS q
(4.67)
is th e free space wavenum ber, and
k i = — sin 9i cos <pi x —sin di sin fa y — cos
z
70
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(4.68)
is the propagation vector of th e incident wave from th e direction (9U fa). H orizontal and vertical
polarizations of the incident wave are defined by the direction of th eir u nit polarization vectors:
A
hi =
^ ^
------ - =
|kj x z|
—sin fa x + cos fa y
(4.69)
Vj =
hi x ki =
- cos 9i cos fa x - cos 9i sin fa y — s i n ^ z
(4.70)
T he vector E t can be w ritte n as a linear com bination of hi and Vi as follows:
Ei -
E f hi +
^
(4.71)
Using the definitions (4.1)-(4.3), the r - and ^-com ponents of E ^ can be w ritten as
E^
=
|E ^ s i n '0 s i n ( 0 — 0i) —
[sin-0cos0icos (0 — 0i) + c o s0 sin 0 i]j
=
\E h^ cos (0 ~ & ) + E V5 cos 0 i sin (0 _ <f>ij\ e
(4.73)
ftokj-r = —u p c o s f i — v z
where the exponent is
u = fcosin^i,
w ith
k° ^ r
(4.72)
(4.74)
v = ko cos 9i .
(4.75)
Perform ing the integration in (4.66) yields the coefficients of th e n -th mode for the r - a n d 0com ponents of E W :
E^
= j n | E ^ sin-0 T+ (up) -
E ^ [ - j s in -0 cos 9i T ~( up ) + cos 0 sin#; Jn (up)] | e~:in^ieE z (4.76)
E«
= f
co s9iT+(up)\ e ~ ^ e ? vs
[jE ^T -(up) + E ^
where the functions
(4.77)
are defined as
T +( X) = Jn + i ( x) + Jn- lM
T ~ ( x ) = Jn+l{x) ~ Jn~l{x)
,
Li
(4.78)
Li
Of particular interest, due to its simplicity, is the case of a plane wave incident along th e 2-axis,
i.e., from an angle 9i = 0. Since u = 0, and Jn (0) ^ 0 only for n = 0, it follows the only m odes
71
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w ith non-zero com ponents are those w ith n = ± 1 , which have r - and ^-com ponents:
4 .2 .5
sin $ ( j '4 ° + 4 ° ) eT
ejk°Z
E rl
=
^
B f ±i
=
± ± ( E P + j E U ) e ^ i h e j k 0z
(4-79)
(48Q)
S c a tte r e d field in th e r a d ia tio n z o n e
T he coupled surface integral equations are solved w ith (4.67) as incident electric field. In particular,
a polarization q is considered, i.e.,
q = h,v
EW = f ? W q i ,
T he equivalent surface currents
(4.81)
th a t are found as solutions are subsequently used to
com pute the scattering am plitudes.
According to expression (2.63), the scattering am plitudes of th e object generated by th e equiv­
alent surface currents
(k , Kk td)
Jfpq{Ks
-—
and
are given by
___ 1 jkoCo
^
p- s
Co Js
£ ( 0 e + 4>
J i 9)(r')
k°
r ' dS' +
( 0 e - e 0 ) •M ^ (r ') e i k° ^ s ' r' dS'
q = h,v
(4.82)
where k 5 is the propagation vector of th e scattered wave in the direction (9S, <fis),
k s = sin
cos
x + s i n 0 s s i n ^ s y + cos0s z .
(4.83)
while the associated horizontal and vertical polarizatio n unit vectors are
A
hs =
k x z
------- =
|k s x z
s i n 0 s x — cos
vs =
hs x ks =
cos
y
=
xv
0
cos 05 x + cos 9S sin 0 S y + sin05 z
(4.84)
=
0
(4.85)
Once the scattering am plitudes fpq are known, th e b istatic scattering coefficients apq can be
calculated from them using th e relationship (2.64).
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In order to evaluate th e integrals in (4.82), the equivalent surface currents J 5, M s axe w ritten
as sum m ations o f Fourier m odes an d approxim ated in term s of basis functions used in th e M OM
solution of the integral equations (4.5)-(4.6). As a result, th e scattering am plitude will also have
the form of a sum m ation of an infinite num ber of modes, which is tru n c a te d to include only those
m odes th a t produce a significant contribution to th e scattering.
Introducing the functional operators V pw and Q pw as
V pw
=
j> ( p - w ' ) Jw(r')
r dS' ,
4>')}
=
£ ( h - w ' ) M w{ v ' ) e i k^
' r ' dS' =
Q vw { M w(t', $')}
=
Q
hw
<f ( v . w ' ) M ffi( r ' ) e j l o ^
Js
p = h , v , w = T,<j>
V vw { M w(t',4>')}
r , d5' = - 1
Co
V hw
(4.86)
(4.87)
(4.88)
the expression (4.82) for th e scattering am plitudes can be w ritten as, for p ,q — h,v,
f pq( k , k )
E
=
E q
77
+
(4.89)
w = r ,4 >
T he expansion of th e r - and qJ- com ponents of the equivalent surface currents into Fourier series
as in (4.22) and approxim ation th rough (4.37)-(4.38) leads to w rite th e operators V pw as:
+oo.
V pw { J w{t',<p')}
=
E
Nw
’
p — h , v \ w = T,(j)
(4.90)
n=—oc m=l
(n)
where the dependence of the solution term s Z™n and of th e coefficients pm, w on the p articu lar
direction and polarization of th e incident and scattered wave, respectively, have been explicitly
indicated.
Substitution of (4.90) into (4.89) and use of the relationships (4.87)-(4.88) between V pw and
Q pw allows to w rite th e scatterin g am plitudes as an infinite sum m ation of angular modes, i.e.,
+oo
f pq(k s,ki)
=
E
U ) e ? n*‘
n = —oo
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.91)
X
X
disk
w asher
sphere
cone-sphere
cylinder
Figure 4.4: G eom etry of various bodies of revolution,
whose n -th m ode term s are
/^(k a.k i) =
- j k 0C0
4 tT
Nw
q i) - i - P ^ U k ^ v , )
w = T ,< t> m = z l L
q j)
(4.92)
^
(4.93)
47t E (q^ w=T,<j>m=l
Explicit expressions for the coefficients p ^ L ( k 3, p s) (p = h ,v , w = r , <f>) are given in A ppendix
C. T he b istatic scatterin g coefficient can be found from th e scattering am plitudes using
<jp9( k s, k j )
4.3
= 47t \f pq{9s)\2 ,
p ,q = h , v
(4.94)
Validation
T he M OM procedure to evaluate the equivalent surface currents and b ista tic scattering coefficient
for a body of revolution, described earlier in this chapter, has been im plem ented and tested on a
num ber of objects of different shapes and dielectric properties. In this section, in order to validate
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
th e correctness of this m ethod, com parisons will be m ade w ith o ther results, b o th num erical and
analytical, found in the literatu re. Perfectly electric conducting (PE C ) bodies will be treated first,
a n d lossy dielectric bodies will be considered in the subsequent subsection.
4 .3 .1
P e r f e c t ly e le c t r ic c o n d u c tin g b o d ie s o f r e v o lu tio n
Bodies of revolution m ade of a perfect electric conductor are considered. The tangential electric
field is null on the surface of a perfect electric conductor, hence th e equivalent surface m agnetic
current
also vanishes on S. O nly th e surface integral equation for exterior problem needs to be
solved:
Ejl}(r) = jwpo (I - n n ) ■ f
G °(r,r')I -
Js+
J s (r/) dS' ,
r e 5+
(4.95)
Ko
a n d the m atrix form (4.61)-(4.64) of the system of algebraic equations becomes
[ a ; t ] i Tn + [ a ^ ] i*»
=
v T"
(4.96)
[ a £ t ] r™ + [a**] i*»
=
v ^
(4.97)
Using these equations, th e electric surface current J 3 has been calculated for P E C bodies or
revolution of various shapes, as illu strated in Figure 4.4. C om parisons have been m ade w ith values
from exact analytical expressions - when available - or w ith num erical results from the literature.
An im portant source of d a ta for com parisons has been th e num erical code from Glisson and
W ilton [47], also based on th e M OM solution of surface integral equations. The d a ta used for the
validation th a t is referred to as W G in the plots has been generated using this code. M autz and
H arrington have also stu d ied the num erical solution of th e surface integral equation for bodies of
revolution [42, 44, 45]. Figures 4.5-4.8 illu strate the results of th e m ethod of m om ents approach
derived in this dissertation, denoted as BO R, com pared to th e values reported in [44] (MH) and
th e Glisson and W ilton (W G) solution.
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
J - BOR
J -WG
J -MH
O"
•o
SA?a
A
0.5
A qV Jt-Ar
0.5
Figure 4.5: Surface currents vs. t /X on a P E C disk w ith radius a = 1.5A, for 6t = 0°. Com parison
betw een results of th e present m ethod (B O R ), W ilton-G lisson (W G ), and M autz-H arrington (MH)
solutions.
J - BOR
3.5
X
J - WG
J - MH
"(/>
J - BOR
C
CD
5
o
c
^
>
’3c
•
oN
J - WG
2.5
J -M H
9
1.5
CO
£
o
c
0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 4.6: Surface currents vs. t /X on a P E C washer w ith radii ai = 0.4A and oi = 1.2A, for
6i = 0°. C om parison betw een results of th e present m ethod (B O R ), W ilton-G lisson (W G), and
M autz-H arrington (MH) solutions.
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.5
J - BOR
J - WG
I
3.5
J - MH
cCD
J - BOR
♦
3O
c 2.5
o
A
CO
J.
-
WG
J - MH
•5
cr 2
d)
TJ , 0
) 1.5
N
co
E
o
c
0.5
0.2
0.4
0.6
0.8
Figure 4.7: Surface currents vs. t j A on a P E C open-ended cylinder of radius a = 0.4A and length
L = 1.2A, for 6i = 0°. C om parison betw een results of th e present m ethod (BOR), W ilton-G lisson
(W G), and M autz-H arrington (MH) solutions.
J - BOR
J - WG
—_ 2.5
J - MH
J - BOR
J - WG
J - MH
CT
0.5
0.5
t/X
Figure 4.8: Surface currents vs. t/A on a P E C cone-sphere of rad iu s a = 0.2A and length L — 1.2A,
for 0j = 0°. Com parison betw een results of th e present m ethod (B O R ), W ilton-G lisson (W G), and
M autz-H arrington (MH) solutions.
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
JT - BOR
Jt -MIE
J - BOR
<>
J - MIE
0.4
40
80
100
elevation angle 9 [deg]
120
140
160
180
Figure 4.9: Surface currents vs. t /A on a P E C sphere w ith size param eter k^a = 1.0. C om parison
betw een results of the present m ethod (BOR) and Mie series (M IE) solution.
Figure 4.5 shows th e norm alized electric surface currents as a function of the norm alized coor­
dinate t /X (defined along th e generating arc) for a P E C disk of radius a = 1.5A. T he incident field
is a plane wave illum inating th e disks from the positive z-axis, i.e., from an incident elevation angle
9l = 0° Figures 4.6 and 4.7 show th e norm alized electric surface currents along the generating arc
for a P E C washer of radii a\ — 0.4A an d a i = 1.2A, and for a P E C open-ended cylinder of radius
a = 0.16A and length L = 1.0A, respectively, also for 6t = 0°. T he thickness of the disk, washer,
and of the walls of th e open ended cylinder are assum ed infinitesim al. Theses cases can be used as
approxim ations of real bodies in which the thickness ten d to zero.
Figures 4.8 shows th e electric surface currents for a P E C cone-sphere of radius a = 0.2A and
length L = 1.15A, for a plane wave incident from 6l = 0°.
For a perfectly conducting sphere, th e exact analytical expressions of th e currents excited by a
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.8
I
I
!
1
x
1.6
X
O
3s.
—)
1.4
t
3
C/i
1
........ -o
1
<3
©
©
©
(9
X
0 .8
0.2
0
P
.© .
/
. ....
c
6
■Ar
a O o 0 O Q G ° QC
I
20
40
i
60
.
Q
0.4
JO/
Q
E
o
\x#
r ......
E 0.6
•o
05
N
CO
:
3
C
05
CO
M IC
v .0
NO
1.2
O
a>
•a
_
mie
VBOR
-------
c
a?
3
U
<D
CO
- BOR
____ jt -
1
80
100
120
elevation angle 8 [deg]
140
_i
160
180
Figure 4.10: Surface currents us. t / X on a P E C sphere w ith size p aram eter k^a = 10.0. C om parison
betw een results of th e present m ethod (BOR) and Mie series (MIE) solution.
incident plane wave exist, and can be found in [14, 15]. T hey have the form of Mie series:
J
(0 )
=
A
.
Y
koa
jn
2 n +
P nx ( CO S0)
1 ,
n ( n + 1)
sint
[(£:0a) h n \ k o a )
1
k0a ^
P n (cos 9)
n ( n + 1) | [(jfeoa) ^ ( j f e o a ) ] '
sin0
+
5 f£ (c o s0 )
89
(k0a) h n \ k o a )
dP^ (cos 9)
+
(k0a ) h ^ ( k 0a)
(4.98)
► (4.99)
99
Figures 4.9 and 4.10 show th e surface currents on the generating curve for two P E C spheres w ith
size param eters koa = 1.0 and k^a = 10.0, respectively. T he solid and do tted lines represent the
exact value calculated from expressions (4.98)-(4.99), while th e symbols denote the B O R num erical
solution, which is in excellent agreem ent w ith th e Mie solution.
Only plane wave incidence from 9i = 0°, i.e., along the z-axis, has been considered so far. Since
in this case, as noted in Section 4.2.4, only the m odes n — ± 1 are excited, the validity of th e present
approach has been proved only for those two m odes. In order to consider other modes as well, the
incident wave m ust be a t an oblique angle. For this reason, th e results for a PE C finite-length
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.5
2.5
x
Jt - BOR
O
j '- B O R
_
J - WG
Jt -W G
X
s)
C
©
©
t:
<n
o
■®
3o
c
sE
a
a'O'cro-o-s © o a d o -
■©
o
N
«
E
o
c
0.2
0.4
0.6
0.8
2
1.2
0.2
0.4
0.6
1.2
0.8
t/X
Figure 4.11: Surface currents vs. t f A on a P E C finite cylinder of radius a = 0.5A and length L = A,
for 9i = 0° (left) and 9{ = 45° (right). C om parison betw een results of th e present m ethod (BOR)
and W ilton-G lisson (W G) solution.
1.3
X'
”« 1.6
c
®
3
U
|
5
o
®
T
3
3
'c
0.8
0.8
I
0.6
0.6
CO
®
*
ia
E
o
c
Jt - BOR
J t - WG
0.2
99
0.5
1.S
2.5
o V
0.2 a ©
3.5
bor
J - WG
0.5
2.5
3.5
t/X
Figure 4.12: Surface currents vs. t / X on a P E C finite cylinder of radius a = 0.5A and length
L = 2.5A, for 6i = 0° (left) a n d 9{ = 45° (right). C om parison betw een results of th e present
m ethod (BOR) and W ilton-G lisson (WG) solution.
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.5
X '
~x "
c©
3
2 1.4
O
3
3.5
©
©
jg
©
3(A
3U>
o
c
J - BOR
C
O)
£3 0.6
7©
Jt - WG
•o
Ol
©
© 0.4
£
o
c
©
£
©
£
o
c
o V B0R
J -W G
9
2.5
0.5
t/X
2.5
Figure 4.13: Surface currents vs. t /X on a P E C finite cylinder of radius a = 0.05A and length
L = 2.5A, for 9l — 0° (left) and 9i = 45° (right). C om parison betw een results of th e present
m ethod (BOR) and W ilton-G lisson (WG) solution.
cylinder at two different incident angles 9i will now be included.
In Figures 4.11-4.13, th e results of the present m eth o d are com pared to those of th e GlissonW ilton code for incident angles 9i = 0° (left) an d 9i = 45° (right). R adius and length are a = 0.5A,
L = A for the cylinder in Figure 4.11, a — 0.5A, L — 2.5A for the cylinder in Figure 4.12, and
a = 0.05A, L = 2.5A for th e cylinder in Figure 4.13.
From the electric surface currents found by solving th e surface integral equation, th e b istatic
scattering coefficient is com puted using expression (4.91) (4.93). E xam ples of results are shown in
Figures 4.14-4.21 for th e sam e type of bodies for which th e surface currents have been calculated.
In m ost cases, the com parisons are m ade w ith the results of th e G lisson-W ilton code. T he incident
angle is , for 9{ = 0°, except for the finite P E C cylinder, w here , an oblique angle 9X = 45° is also
considered. Figure 4.14, for a P E C disk of size p aram eter koa = 10, shows a comparison betw een the
norm alized bistatic scatterin g coefficient calculated w ith th e present num erical approach, and an
analytical solution derived by Hodge [74]. Figures 4.17-4.18 show th e norm alized b istatic scattering
coefficient for the two P E C sphere whose currents are given in figures 4.9-4.10. The num erical values
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
o
HH - BOR
HH - Hodge
VV-BOR
VV - Hodge
e
m
T3
-10
-15
scattering angle 88S [deg]
60
75
Figure 4.14: N orm alized b istatic scattering coefficient vs. 9S for a P E C disk of size param eter
koa = 10, for 6l = 0°. Com parison betw een results of the present m ethod (BO R ), and Hodge
solutions.
s
CD
T3
CM
cti
-5
O
-10
HH-BOR
x HH-WG
- - VV-BOR
o VV - WG
-15
-20
20
40
80
100
120
scattering angle 0s [deg]
140
160
180
Figure 4.15: Norm alized b istatic scattering coefficient vs. 9S for a P E C open-ended cylinder of
radius a = 0.4A and length L = 1.2A, for 9i = 0°. C om parison betw een results of th e present
m ethod (BOR) and W ilton-G lisson (WG) solution.
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
x
o
HH-BOR
HH - WG
VV-BOR
VV - WG
B
CD
■O
<3
"6
es
-2
-3
-4
20
60
80
100
120
scattering angle 8a [deg]
140
160
180
Figure 4.16: N orm alized b ista tic scattering coefficient vs. 9S for a P E C cone-sphere of radius a =
0.2A and length L = 1.2A, for plane wave incident a t 9t = 0°. C om parison betw een results of the
present m ethod (BOR) and W ilton-G lisson (WG) solution.
E
co
13
-4
x
-8
o
-10
20
)
80
100
120
scattering angle 0 [deg]
140
HH-BOR
HH - MIE
VV-BOR
VV - MIE
160
180
Figure 4.17: N orm alized b ista tic scattering coefficient vs. 9S for a P E C sphere w ith size param eter
k^a — 1.0, for 0j = 0°. C om parison betw een results of the present m ethod (BOR) and Mie series
(MIE) solution.
83
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o-o-
E
CD
■
o
03
j;
B
-10
-15
HH-BOR
« HH-MIE
- -V V -BO R
o VV-MIE
-20
-25
3
80
100
120
scattering angle 9 [deg]
40
140
160
180
Figure 4.18: N orm alized b istatic scattering coefficient vs. 9S for a P E C sphere w ith size param eter
koa — 10.0, for 9%= 0°. C om parison betw een results of th e present m ethod (BOR) and Mie series
(MIE) solution.
are com pared w ith th e exact values calculated according to th e following form ulas for the scattering
am plitudes [15]:
-j
fhh(0s)
OO
n+1
=
an
p i (cos 0)
<9pi(cos0)
„
- J Pn
sin 9
89
(4.100)
OCv
dP.%(cos 9)
-3&
89
(4.101)
n =1
fv v(9s)
~
k
Y ,J n+ l
wn =1
P n (cos 9 )
sint
where the coefficients a n and /3n are
,n 2n + 1
aT
-J
n{n + l) h (n \ k 0a)
2n + l
Pn
=
f
j n {koa)
[(k0a) j n {k0a)}'
n{n + l) [(k0a ) h ^ ( k Qa)]'
(4.102)
(4.103)
and the bistatic scatterin g coefficients are
CTpq{9s)
=
4tT\ f p q { 9 s ) \ :‘
(4.104)
Given th e good agreem ent betw een the results, th e num erical m ethod derived earlier in this
chapter can be considered validated in the case of P E C bodies of revolution.
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E
CD
■o
N(0
e'l<
s
K
B
B
-15
-1 0
*
HH - BOR
H H -W G
0 W -B O R
- - VV - WG
-15
-20;
-10
20
80
100
scattering angle 0 [deg]
120
140
160
x
o
180
-25
scattering
100
120
ie 0 [deg]
140
HHHH W W -
BOR
WG
BOR
WG
160
180
Figure 4.19: N orm alized b istatic scattering coefficient vs. 9S for a P E C finite cylinder of radius
a = 0.5A and length L = A, for Oi = 0° (left) and 0t = 45° (right). C om parison betw een results of
the present m ethod (BOR) and W ilton-G lisson (WG) solution.
E
CQ
T
3
CO
£
B
e -15
-20
-1 0
X
-15
O
-20
80
100
120
scattering angle 9 [deg]
140
-25
HH - BOR
HH - WG
W -B O R
VV - WG
160
x
-30
-35
180
o
80
100
scattering angle 0 [deg]
120
140
H H -B O R
H H -W G
W - BOR
W - WG
160
180
Figure 4.20: Norm alized b istatic scattering coefficient vs. 6S for a P E C finite cylinder of radius
a = 0.5A and length L = 2.5A, for 6t = 0° (left) a n d 6t = 45° (right). C om parison betw een results
of the present m ethod (BOR) and W ilton-G lisson (W G) solution.
85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6 0 1
0
1
20
1----------- — 1--------------- 1
40
60
1
1
1
80
100
120
scattering angle 9g [deg]
1
140
160
1
180
.401---------- >— 1!------'-------------1------------1----------- '----------- 1------------1------------1------------1
0
20
40
60
80
100
120
140
160
180
scattering angle 8 [deg]
Figure 4.21: N orm alized b istatic scattering coefficient vs. 0S for a P E C finite cylinder of radius
a = 0.05A and length L = 2.5A, for 9i = 0° (left) and Oi = 45° (right). Com parison betw een results
of the present m ethod (BOR) an d W ilton-G lisson (WG) solution.
4 .3 .2
H o m o g e n e o u s d ie le c tr ic b o d y o f r e v o lu tio n
For an homogeneous dielectric body of revolution, b o th integral equations (4.5) and (4.6) need to
be solved sim ultaneously. For th e validation, only the case of th e dielectric sphere is treated. Two
different values of th e size p aram eter are considered, i.e., k^a = 1.0 a n d k^a = 10.0, in com bination
w ith three different values of th e relative dielectric constant, i.e., ec = 4, ec = 4 —j, and ec = 18—jQ.
In Figures 4.22-4:27, th e electric and m agnetic surface currents obtained w ith the present num erical
m ethod are com pared to th e results of the G lisson-W ilton code.
T he bistatic scatterin g coefficients in the hh and un-polarization are calculated from these
equivalent surface currents using (4.92)-(4.94), and are p lo tted in Figures 4.28-4.33. In th e same
plots, the exact values of th e b istatic scattering coefficients are also shown.
These values are
calculated from (4.94) and expressions (4.100)-(4.101), w ith th e coefficients a n and /3„ of the Mie
series given by:
OLji
.n 2n + l
—
—J
j n (k0a) [ka j n (ka )} '- j n (ka) [kQa j n {k0a)}'
~ T — 7T7
7T7
n ( n + 1) hb, (koa) [ ka jn (ka)]' - j n (ka)[k0a h ^ ( k 0a)]'
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( 4 .1 1 ) 5 j
0.3
1.4
S B 0.7
0.6
0.5
^
c
0.2
0.4:
*
MX~ BOR
Mt - WG
0
0.2
M ^-BO R
M - WG
0.05
0.1
0.15
0.2
0.25
t/X
0.3
0.35
0.4
0.45
0.5
0.05
0.1
0.15
0.2
0.25
t/X
0.3
0.35
0.4
0.45
0.5
Figure 4.22: Equivalent surface currents vs. t / X on a dielectric sphere w ith k0a = 1.0 and ec — 4.
C om parison betw een results of the present m ethod (BOR) and Mie series (MIE) solution.
0.8
S" 0-7
0.6
0.5
.1
0.4
© 0.6
0.4,
0.2
0.2
0.1
0.05
0.1
0.15
0.2
0.25
t/X
0.3
0.35
0.4
0.45
0.5
0.05
0.1
0.15
0.2
0.25
t/X
0.3
0.35
0.4
0.45
0.5
Figure 4.23: Equivalent surface currents vs. t / X on a dielectric sphere w ith k$a = 1.0 and ec = 4 —j.
Com parison betw een results of the present m ethod (BOR) and Mie series (MIE) solution.
87
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.8
x
Mt - BOR
M - WG
2 " 0.7
0
- BOR
M - WG
0.6
0.5
>
0.8
.1 0.4
0.6
0.3
0.4
x
0.2
- BOR
Jt - WG
o VSOR
0.2
J - WG
0.05
0.1
0.15
0.2
0.25
t/X
0.3
0.35
0.4
0.45
0.5
0.05
0.1
0.15
0.2
0.25
t/X
0.3
0.35
0.4
0.45
0.5
Figure 4.24: Equivalent surface currents vs. t /X on a dielectric sphere w ith k^a = 1.0 and ec =
18 — j 6. C om parison betw een results of the present m ethod (BOR) a n d Mie series (MIE) solution.
c
M? - BOR
hi
X'
2.5
c
©
_
- WG
M -B O R
*
2.5
_ M - WG
3
&
■
c
in
©
©
3
3
tft
c
C
3
CT
©
<0
>
3
CT
©
■©
o
3
©
•o
3
&
©
(0
>
O
O
'c
C
CO
o»
O)
E
i
■o
■o
N 0.5
cO
E
N
0.5
o
c
0.5
2.5
0.5
2.5
Figure 4.25: Equivalent surface currents vs. t /X on a dielectric sphere w ith k$a = 10.0 and ec = 4.
Com parison between results of the present m ethod (BO R) and Mie series (MIE) solution.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.5
-* J t -B O R
_____Jt - WG
o
j’
=S 0.45
- bor
_____J - WG
D
0.35
0.3
3®
0.25
.........
x x * * x £JJx
es°
x
Mt - 8 0 R
o
M - BOR
M -W G
o.
Ov
°&(S
M - WG
0.5
2.5
t/X
Figure 4.26: Equivalent surface currents vs. t /X on a dielectric sphere w ith k$a = 10.0 and ec =
4 - j . C om parison betw een results of the present m ethod (BOR) and Mie series (MIE) solution.
0.5
X'
^
c
o
0.45
0.4
3
U
0.35
0.3
c
JJ>
.1
>
3
•0
cr
0.25
o
CD
"O
3
C
cs
03
E
■o
N
E
o
c
0.5
2.5
0.5
2.5
t/X
Figure 4.27: Equivalent surface currents vs. t / X on a dielectric sphere w ith k^a — 10.0 and ec =
18 - j6 . Com parison betw een results of th e present m ethod (BOR) and Mie series (MIE) solution.
89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1 0
# -15
-20
HH x HH - - VV o VV -
-25
-30
40
80
100
120
scattering angle 9s [deg]
140
MIE
BOR
MIE
BOR
160
180
Figure 4.28: N orm alized b istatic scattering coefficient vs. 9S for a dielectric sphere w ith k^a = 1.0
and ec = 4. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE)
solution.
-1 0
-20
-HH-MIE
X HH - BOR
- - VV - MIE
O W - BOR
-25
-30
40
80
100
120
scattering angle 0g [deg]
140
160
180
Figure 4.29: Norm alized b istatic scattering coefficient vs. 9S for a dielectric sphere w ith k^a = 1.0
and ec = 4 —j. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE)
solution.
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5
T>
£
m
■a
CM
'B
-2
-3
HH - MIE
x HH - BOR
- -W -M I E
o W - BOR
-
80
100
120
scattering angle 0s [deg]
20
140
160
180
Figure 4.30: N orm alized b istatic scattering coefficient vs. 9S for a dielectric sphere w ith k^a — 1.0
and ec = 18 — j 6. C om parison betw een results of the present m eth o d (BOR) and Mie series (MIE)
solution.
HH-MIE
* HH - BOR
- - VV - MIE
o VV - BOR
E
•a
Nas
CD
'B
-5
-1 0
-15
-20
20
40
80
100
120
scattering angle 0g [deg]
140
160
180
Figure 4.31: Norm alized b istatic scattering coefficient vs. 0S for a dielectric sphere w ith k Qa = 10.0
and ec = 4. C om parison betw een results of th e present m eth o d (BOR) and Mie series (MIE)
solution.
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
»r -10
-20
HH - MIE
x HH - BOR
- - VV - MIE
O VV - BOR
-30
-40
80
100
120
scattering angle 9s [deg]
40
140
160
180
Figure 4.32: N orm alized bistatic scattering coefficient vs. 9S for a dielectric sphere w ith kqa = 10.0
and ec — 4 —j. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE)
solution.
HH HH VV O VV x
MIE
BOR
MIE
BOR
E
C
Q
T
J
cv
-5
-1 0
-15
-20
40
60
80
100
120
scattering angle 9g [deg]
140
160
180
Figure 4.33: Norm alized b ista tic scattering coefficient vs. 0S for a dielectric sphere w ith k^a = 10.0
and ec = 18 —j 6. C om parison betw een results of th e present m ethod (BOR) and Mie series (MIE)
solution.
92
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2n + 1
Pn
=
jnjkoa) [kajn (ka)]' - £rjn{ka) [kQa j n {k0a)]'
f
( 4 . 106 )
n in + !) h ^ \ k 0a) [kajn (ka)}f - erj n ( k a ) [ k o a h n \ k o a )]'
T he com parisons show th a t th e b o d y of revolution num erical m ethod gives p retty accurate
results in th e case of a dielectric object, and only a sm all degradation in accuracy for th e larger
sphere w ith higher dielectric constant can be noted.
4.4
A pplication to scattering from vegetation
T he num erical procedure developed an d tested for bodies of revolution in this chapter can be
applied to stu d y the scatterin g from vegetation com ponents.
In microwave rem ote sensing, a
vegetation canopy is usually m odeled as a layer of dielectric cylinders and disks placed over a half
space representing the ground. In th e case of a forest such cylinders and disks represent tru n k s,
branches, needles and leaves, while in th e case of sm aller vegetation such as agricultural crops, they
represent stalks and leaves. If they are circular and hom ogeneous, these scatterers can be m odeled
as bodies of revolution,
Since a com parison betw een num erical calculations a n d analy tical approxim ation has already
been done by W illis et al. [29], only cylinders will be considered here. T he cylinders are assum ed
circular, homogeneous, an d lossy, w ith dimensions a n d orientations typical of trunks, branches,
needles and stalks.
T ypical values - based on ground d a ta collected during experim ental - for
conifer forest are those given in [9] for hemlock trees in a boreal forest. C orn stalk param eters can
be found in [10].
vegetation
type
vegetation
element
radius
a
length
L
hem lock
hem lock
hem lock
hem lock
corn
secondary branch
prim ary branch
sm all tru n k
large tru n k
stalk
1.8 m m
0.6 cm
3.0 c m
10.0 c m
1.25 c m
16 cm
90 cm
2.5 m
7.5 m
62.5 m
Table 4.1: G round d ata.
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he relative dielectric constant ec of vegetation structures greatly depends on their w ater con­
tent. Tree tru n k and branches are usually drier and therefore have a lower ec th a n corn stalks. For
the sake of brevity, in th e following, only one value for the dielectric constant will be used, but
other values of ec have also been considered, leading to sim ilar results. T he value chosen here is
ec = 18 —j’6, which is consistent w ith ground d a ta m easurem ents and U laby’s em pirical m odel [75].
In ra d a r rem ote sensing, particularly when considering scattering from a layer of dielectric
objects, th e interest lies in th e m ain scattering lobes. T his is a consequence of th e large num ber of
scatterers in such vegetation media. Since the contributions from the single scatterers are added
all together, the relative weigh of th e scattering from th e side lobes becomes negligible. Therefore,
when com paring approxim ate and exact scattering coefficients, th e approxim ation is considered
acceptable if it agrees w ith the theory inside the m ain scattering lobe and does not produce high
scattering elsewhere. It has to be stressed th a t in cases where all scattering of any level m ust be
taken into account - such as antenna applications - th e conclusions draw n here do no apply.
In the following sections, after a brief description of the analytical models, th e values of the
scattering coefficient estim ated by th e approxim ate m ethods will be com pared w ith its exact values
found numerically. In particu lar, for various incident angles 6i, 0j, th e bistatic scattering coefficient
will be determ ined over a range of scattering angles Qs ,<t>s using b o th m ethods. T he illustration
of results is lim ited to th e case of scattering in th e sam e plane cj> = fa of the incident wave, b u t
sim ilar conclusions apply for scattering a t different azim uth angles <f>s.
4 .4 .1
F in it e - le n g t h d ie le c tr ic c y lin d e r a p p r o x im a tio n
Consider a finitely-long circular dielectric cylinder of radius a an d length L, w ith p erm ittiv ity
e = e0 s c (where ec = eT — j s j is th e complex relative p e rm ittiv ity or dielectric constant of the
cylinder) and perm eability y, = y,o , located in free space. A C artesian coordinate system (x, y, z)
is defined w ith its origin in the center of the cylinder, an d th e z-axis coincident w ith th e cylinder
94
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z
8 ,|A
Figure 4.34: G eom etry of the finite-length dielectric cylinder problem
axis, as illustrated in Figure 4.34.
T he cylinder is considered being illum inated by a g-polarized uniform plane wave
q = h,v.
E W (r) =
(4.107)
T he incident wave (4.107) induces an internal field E?nt inside th e cylinder. T he scattering
am plitude for the cylinder is related to the internal field by (see [20] for details):
f pq( k s S i ) = fc°2(£4C7r7—
[ p s - E?n t( r V fc°k s ' r V ' ,
p,q = h , v
(4.108)
V
Because the internal fields w ithin a finite-length cylinder are not known exactly, they are approxi­
m ated by the internal fields inside an infinite cylinder of the sam e radius, orientation and dielectric
constant. This assum ption requires th a t the cylinder length be large com pared to its radius, i.e.,
L
> 1.
95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.109)
T he internal electric field for horizontally (TE) or vertically (TM ) polarized incident wave can be
found in W ait [13]. T he integration w ithin the cylindrical volume V is carried out in [25]. T he
resulting expression for th e scattering am plitudes is
k l ( s c - 1)
47r
fpqfe-si kj)
7T ■
+ j
e
[ l | n)( q • a x ■ p ) + 4 n)( q • a 2 • p )
- j e ~ ^ s [ 4 n)( q
a! • p ) + l \ n\ q • a 2 • p ) ] +
+ e M * [ 4 n)( q - h i - p ) + 4 n)( q • b 2 • p )
+ e
where
h
=
L
+
[ 4 n)( q ■b x • p ) + jJ n)( q ■b 2 • p ) }
sin[fco(cos 9i + cos 9S) L / 2]
ko(cos9i + cos 9s) L / 2
(4.110)
(4.111)
a
r(n)
=
JnM ) Jn M )
J
P'
dp'
0
Q»[ Ai Jyi (^1&) Jn (^1&)
]
Af +
(4.112)
a
r(n)
=
f J'n { \ lP') Jn+liMp') P' dp'
0
(4.113)
a
r(n)
15
=
J
J ^ X lP ^ J n -lM p 'd p '
(4.114)
0
a
(n)
=
J
J n M ) J n + l M ) P'dp'
(4.115)
0
a
r(n)
=
J
J n M ) J n - l M ) P'dp'
(4.116)
0
and
Ai
=
A2 =
ko sin 9S
(4.117)
kQy/er — cos2 9S
(4.118)
In addition the following relationships perm it a m ore efficient calculation of the l\H .
r(«)
4
_
n
~~
Ai
A n + l)
r(n)
~~
96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.119)
=
n
r(n -l)
II
j•(—n)
J6
7?
''f2?
1
II
r(~n)
Ji
(4.121)
(4.122)
1
II
r(~ n )
3
( 4 . 120 )
'
Ai
J3,
3
(4.123)
Such expressions are valid for cylinders of any thickness, as long as th e constrain given in (4.109)
is satisfied. However, if th e cylinder is very th in , a furth er approxim ation can be applied. It is shown
in [25] th a t if the radius a of th e cylinder is sm all com pared to its internal wavelength, a quasi-static
approxim ation is utilized to reduce the com plexity of th e scattering am plitude expressions.
Here the general case of a cylinder w ith an a rb itra ry radius a and length L satisfying (4.109)
will be considered, and th e expressions for
4 .4 .2
f h v , f v h and f vv explicitly given in [25] will be used.
T a p ered c y lin d e r a p p r o x im a tio n
z3
L
Z.2
z,
v
Figure 4.35: T apered cylinder m odel
Consider a dielectric tap e red cylinder of length L a n d radii a and b a t the m ajor and m inor
base, respectively, centered at th e origin of a C artesian coordinate system , as shown in Figure 4.35.
97
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Let th e p e rm ittiv ity and perm eability of such body be e = ec£o and p = /tq, respectively, and the
axis of the cylinder be coincident w ith th e z-axis.
T he sim plest way to m odel such ta p e re d cylinder is to approxim ate it w ith a finite cylinder of
same length an d volume. However, as it will also be shown later, th is works well only when the
ta p e r is small. In general, it is m ore ap p ro p riate to divide th e tap ered cylinder into a num ber N s
of sections, an d th en approxim ate each one w ith a cylinder of like length and volume. This m odel
has been introduced by [25], following m easurem ents of tru n k diam eters at different heights. T he
approach is illu strated in Figure 4.35 for a num ber of sections N s = 3. T his approxim ation is valid
when the ta p e r is small:
R =
Li
> 1.
(4.124)
In general, th e higher th e factor R, th e larger the num ber of sections N s needed to achieve a good
approxim ation. T he scattering am plitude of the m -th cylinder, tra n sla te d by a distance z rn along
th e z-axis is:
p ,q = h , v , m = l , ..., N s
/ ^ ( k s ,k i) =
(4.125)
In (4.125), f p ^ ( k s,ki) is th e scatterin g am plitude of th e cylinder centered a t the origin, and the
exponential term accounts for th e phase shift in b o th th e incident and scattered wave produced by
th e translation of th e cylinder by z m .
Assuming no coupling exists betw een th e cylinders, th e to ta l scatterin g am plitude of the con­
figuration is the sum of N s term s fp^1' (k >3, kj) as in (4.125), i.e.,
p,q = h,v
/ p, ( k s, k i ) = X ) / m ° (
m= 1
(4.126)
The individual scatterin g am plitudes / p ^ ( k s,k j) in (4.126) are determ ined using the finite
cylinder approxim ation described in th e previous section.
T he rationale behind this approxim ation is th e following. C onsider a finite cylinder of length
L, and divide it in N s sections of a rb itra ry length. T he scattering am plitudes of all cylindrical
98
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
incident
wave
forward
scattering lobe ^
specular
scattering lobe
•• scattering
/
cone
Figure 4.36: Scattering p a tte rn of a cylinder
sections are estim ated using th e finite cylinder m odel described in th e previous section 4.4.1, and
added together according to (4.126), to yield the to ta l scatterin g am plitude of th e configuration.
Since the contributions of th e end currents from two contiguous cylindrical sections cancel w ith
each other, th e final resu lt is no interaction betw een those sections. T he only rem aining effects
in the sum (4.126) are those a t the two term inal sections. T he resulting scattering am p litu d e is
therefore the same as th a t provided by th e finite cylinder m odel for the original cylinder of length
L.
The tap ered cylinder m odel described in this section is a p e rtu rb a tio n of the situ atio n ju st
illustrated. If the difference in radius betw een contiguous sections is small, only weak interactions
exist between those adjoining cylinders, while th e two term in al sections will experience th e effect
of replacing their in tern al fields w ith those of two infinite cylinders, as explained in the previous
section.
99
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4 .4 .3
R e s u lt s a n d c o m p a r is o n s
A n exact solution is needed as a reference to evaluate the accuracy of th e finite cylinder m odel
and the tap e red cylinder m odel introduced in th e two previous sections.
Since no closed-form
exact analytical expressions are known for the scattering from finite or tap e red cylinders, th e only
available option is to use a num erical analysis approach. In particu lar, the results of the num erical
m ethod developed earlier in this chapter will be considered as the reference to evaluate the accuracy
of th e finite cylinder and th e tap ered cylinder approxim ations.
To b e tte r quantify the accuracy of the analytical model, a p aram eter epp is introduced to
m easure th e error betw een exact an d approxim ate solution over a range of scattering angles 9S
for a single incident angle 9t. Let app and app be th e values of th e b istatic scattering coefficient
obtained using the m ethod of m om ents (i.e., the exact solution) an d th e analytical m odel (i.e., the
approxim ate solution), for polarization pp = hh, vv, respectively. T he absolute error in dB, epp, of
th e finite cylinder m odel is defined as follows:
i Mpp
ePP(9i) = j z —
I&Pp(dsm\ 0 i ) ) [ d B ] ~ Vppi9^ , 8 i ) [ d B } \ ,
m vv m= 1
PP = h h , v v
(4.127)
where the bistatic scatterin g coefficient is expressed in dB:
cfpp\dB\ = 101og10 (cTpp) ,
app[dB] = 10 log10 (app) ,
pp = h h , v v
T he sum is carried out over a set of m = 1 ,2 ,.., M pv scattering angles 9 ^
(4.128)
where the scattering
coefficient is no lower th a n 10 d B below th e peak value cr™ax(9i), i.e., such th a t
aPM m\ 9i)[dB] > a™ax{9i)[dB] - 10 dB ,
pp = hh, vv
(4.129)
T he quantity epp provides an estim ate of the difference - expressed in dB - betw een the approxi­
m ation and the exact solution over a range of angles w here the level of scattering is high enough. In
other words, epp m easures th e area betw een th e two exact and approxim ate d B curves app and app
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
vs. 9i, lim ited to th e values 9 ^ of the scattering angle for which app satisfies expression (4.129).
T he error could be defined in m any other different ways. Here the definition (4.127) is chosen
because of its sim plicity and intuitive m eaning, an d will be used m erely as a tool to com pare how
the accuracy changes w ith th e incident angles and the cylinder size.
F in ite-len g th , con stan t-rad iu s cylinder
In order to investigate th e accuracy of the finite cylinder m odel, in this section a num ber of cases
of cylinders w ith different dim ensions, chosen consistently w ith th e ground d a ta in [9] and [10], is
considered.
Since the finite cylinder m odel assum es th a t th e currents inside the cylinder are the sam e as if
the cylinder were infinite, one would expect th a t th e accuracy of th e approxim ation decreases for
shorter cylinders w ith a sm aller length-to-radius ratio. However, it will be seen th a t for incident
waves sufficiently close to th e norm al to the cylinder axis, th e error is still sm all even in such cases.
T he general shape of the scattering p a tte rn of a dielectric cylinder is illu strated in Figure 4.36.
T he am plitude of the scattering has its m axim um in a conical region also referred to as the scattering
cone. In a section cj) — 4>i of the scattering p a tte rn th ere will be two relative m axim a corresponding
to the m ain scattering cone, one in the forw ard direction, th e other one in th e specular direction
w ith respect to the cylinder side.
In the following, th e norm alized bistatic scatterin g coefficient a / ( n a 2) will be p lotted as a
function of the scatterin g angle for a fixed incident angle, in b o th hh- and w -polarization. T he
scattering angle 9S in th e plots ranges between 0° an d 180°, therefore only the specular lobe at
9S = 180° —9i will be visible.
T he first case exam ined is a cylinder of length L = 10.0A and radius a = 0.04A. For a wavelength
A = 60 cm (or / = 500 M H z , in P -band), this corresponds to a tree tru n k 6 m long w ith a diam eter
of 4.8 cm. This cylinder is several wavelengths long an d its length and radius satisfy the condition
101
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HH-POLARIZATION
VV-POLARIZATION
I(n a2) [dBm]
MM
PO
MM
PO
-10
-io
-20
-2 0
-3 0
-3 0
.c
D
-4 0
60
80
100
120
140
160
180
60
scattering angle 0g [deg]
80
100
120
140
160
scattering angle 0 [deg]
Figure 4.37: N orm alized bistatic scattering coefficient vs. scattering angle for finite dielectric cylin­
der w ith L = 10.0A, a = 0.04A, and incident angle Qt = 20°.
(4.109). T he norm alized bistatic scattering coefficient
<7 /
(7ra2) is p lotted in Figure 4.37 as a function
of th e scattering angle 0S for a n incident angle 0j = 20°, in b o th hh- and wv-polarization. T he
solid curve represents the m ethod of m om ents num erical solution (MM), while the dashed curve
is obtained using the approxim ate physical optics analytical solution (PO ). T here is very good
agreem ent betw een the two solutions except for w -p o la riz a tio n a t angles 0S far from th e specular
scattering lobe located at approxim ately 9S = 160°.
As the cylinder becomes thicker, i.e., the ratio L / a decreases, one would expect the finite cylinder
m odel to lose its accuracy. Nevertheless, it will be seen th a t in such case the approxim ation is still
satisfactory for incident angles 0* close to th e norm al to th e cylinder axis. There is a possible
intuitive explanation for this behaviour. In the analytical m odel, the finite cylinder is tre a te d as
infinite and the effects of its ends are not accounted for. T he co ntribution of such end-on scattering
becomes more pronounced in thicker cylinders due to th e larger area of th e ends. However, the
ends are alm ost invisible to norm al or quasi-norm al incident waves, and therefore in th a t case their
contribution to the scattering is negligible.
102
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180
HH-POLARIZATION
VV-POLARIZATION
MM
PO
----- MM
- - PO
E
CQ
•o
"O
-1 0
-1 0
-20
-2 0
-3 0
-3 0
0
20
40
60
80
100
120
140
160
0
180
20
40
scattering angle 0 [deg]
60
80
100
120
140
160
180
140
160
180
scattering angle 9s [deg]
----- MM
----- MM
- - PO
- -P O
E
C
Q
*o
T3
-10
O
-20
-2 0
-3 0
-3 0
0
20
40
60
80
100
120
140
160
180
scattering angle 0g [deg]
0
20
40
60
80
100
120
scattering angle 0 [deg]
Figure 4.38: N orm alized b ista tic scattering coefficient vs. scattering angle for finite dielectric cylin­
der w ith L = 10.0A, a = 0.4A, eT = 18 —
a nd incident angle
= 20°(top) and
= 80° (bottom ).
103
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HH-POLARIZATION
VV-POLARIZATION
3.5
absolute error [dB]
—
L = 5.0 X
L = 3.0 X
L = 1.0 X
3.5
2.5
2.5
0.5
0.5
incident angle 0. [deg]
incident angle 0. [deg]
Figure 4.39: A bsolute error vs. incident angle for finite dielectric cylinder w ith a — 0.04A, eT =
18 —j'6, L = 5.0A, 3.0A, a n d 1.0A, for hh- and iw -polarization.
Figure 4.38 shows th e norm alized bistatic scattering coefficient vs. the scattering angle for a
cylinder w ith the sam e length and p e rm ittiv ity as in Figure 4.37, b u t w ith a radius a = 0.4A, for an
incident angle
= 20°. A lthough it is still L » a, the plots of th e exact and approxim ate a now
differ, especially for w -p o lariza tio n , even in the m ain lobe. T he actu al am plitude of the scattering
off the m ain scattering cone is m uch higher th a n w hat predicted by the analytical model. It is
reasonable to conclude th a t th e m ain co ntribution to th e scatterin g a t those angles 8S comes from
th e ends, while in the m ain lobe it comes from the sides. N onetheless, as it has been discussed
earlier, the agreem ent im proves considerably when th e incident angle approaches the norm al to
th e cylinder axis. Indeed, for a higher incident angle such as
= 80°, shown in the two plots at
th e bo tto m of Figure 4.38, th e agreem ent is very good, except for a couple of dB difference a t the
peak value of the hh-polarization. In th is case, the ends are alm ost invisible when looking a t the
cylinder from the angle 9i, which could explain why th e analytical m odel yields accurate results.
As explained previously, w hen tre a tin g rem ote sensing problem s, only the scattering w ithin 10
dB of the peak value is relevant, therefore we are concerned w ith achieving a good approxim ation
104
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HH-POLARIZATION
VV-POLARIZATION
—
L= 10.0 A
L = 5.0 A
—■»— L = 3.0 A
—
—
L= 10.0 X
L = 5.0 A
L = 3.0 A.
absolute error [dB]
CQ
•o
0._1
>_
o
03
=3
30
40
50
60
30
40
50
60
70
incident angle 0. [deg]
incident angle 0. [deg]
Figure 4.40: A bsolute error vs. incident angle for finite dielectric cylinder w ith a = 0.4A,
18 - j 6, L = 10.0A, 5.0A, and 3.0A, for hh- and w -p olarization.
e y
=
only in the region surrounding the m axim a. T his m eans th a t for th e th in n er cylinder illu strated
in Figure 4.37, there is very good agreem ent, while for the thicker cylinder in Figure 4.38, there
are some problem s a t 9i — 20° due to th e scattering from the ends th a t are not accounted for
in the approxim ate model.
Such problem s are m ore pronounced for wu-polarization, a possible
explanation for which being the discontinuity in th e perpendicular com ponent at th e cylinder ends,
b u t disappear a t &i = 80°.
To b e tte r und erstan d how the accuracy changes w ith the cylinder dim ensions as well as w ith
the incident angle, the absolute error defined by (4.127) is now com puted. Two different values for
th e radius are considered a = 0.04 A to represent a th in cylinder, a n d a = 0.4 A to represent a thick
cylinder. In b o th cases, th e absolute errors e^h an d evv are p lo tted versus the incident angle 9i for
three different values of th e length, e.g., long, m edium ,and short.
T he th in cylinder cases (a = 0.04 A) are shown in figure 4.39. T he cylinder lengths are L = A,
3 A, and 5 A, w ith th e hh- and vv- po larization on th e left and right, respectively. All these cylinders
have a -C L, in p articu lar L / a = 25, 75, and 625, respectively. If an absolute error below 2 d B is
105
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considered as acceptable, th en th e agreem ent between finite cylinder m odel and num erical solution
a t hh-polarization is excellent at m ost angles and for m ost lengths.
T he only exception is th e
shortest cylinder (L = A), perhaps due to resonances. As expected, th e error is lower for longer
cylinders because the infinite cylinder assum ption is more correct, a n d decreases as the incident
angle increases since th e end-on effect becomes less im portant. Also to be noted is the fact th a t
evv is generally higher th a n the corresponding
Figure 4.40 illustrates three thicker cylinder cases, w ith a rad iu s a = 0.4 A and lengths L = A,
5 A, and 10 A, i.e., L / a = 7.5, 12.5, and 25, respectively. T he m ost obvious comm ent is th a t th e error
in b o th polarizations is m uch higher th a n for the previous cases of Figure 4.38, which is expected
given th e lower value of L / a . T here seems to be a threshold sc atterin g angle 9° below which th e
error increases as 9{ decreases, and above which the error rem ains fairly constant. Supposedly, those
two regions - i.e, 6i < 9° an d 9i > 6° - correspond to situations w here end-on effects respectively
have or d o n ’t have a significant effect on the m ain scattering cone.
Such threshold value of 0j
decreases w ith the cylinder length, and varies betw een 15° for th e longest cylinder, and 25° for the
shortest. T he approxim ation can be considered good (i.e., w ith an error lower th an 2 dB) for angles
th a t vary between 9 i > 9° = 15° for th e longest cylinder, and 9i > 6° = 40° for the shortest cylinder
at uv-polarization. As also seen in th e case of Figure 4.38, th e analytical m odel perform ance is
slightly worse at wu-polarization th a n a t h/i-polarization, especially for shorter cylinders.
T apered cylinder
Two cylinders of length L = 10.0 A and different ta p e r have been considered to illustrate the
accuracy of the tap ered cylinder m odel described in section 4.4.2.
For each case, th e results of th e num erical code are p lo tte d together w ith the approxim ate
solutions obtained using only one cylinder (see also equation (4.108)) an d the tapered cylinder m odel
(see also, equation (4.126)) w ith two and four cylinders. T h e param eters a, b and L are defined
106
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in Figure 4.35, a n d am is the radius of the cylinder replacing th e m -th section (m = 1 , 2 , N s ).
A sum m ary of th e values of the cylinder lengths and radii used in the approxim ations is given in
Table 4.2. A relative dielectric constant ec = 18 —j 6 is assum ed, i.e., th e sam e value used in the
previous section for the analysis of th e finite cylinder model.
No. of cylinders N s
( L / N s )/A
ai /X
o2/ A
a 3/ A
a^jX
a = 0.7 A, 6 = 0.1 A
2
1
4
a = 0.5 A, b = 0.3 A
1
2
4
10.0
0.4
10.0
0.4
5.0
0.55
0.25
2.5
0.625
0.475
0.325
0.175
5.0
0.45
0.35
2.5
0.475
0.425
0.375
0.325
Table 4.2: Geom etric param eters used in tap ered cylinder approxim ations.
Figure 4.41 shows th e norm alized b istatic scattering coefficient vs. scattering angle for a tap ered
dielectric cylinder w ith L — 10.0 A, a = 0.7 A, b = 0.1 A, for a wave incident at an angle 6t — 40°.
T his cylinder has a ta p e r factor - as defined in (4.124) - R = 0.06. In these plots, a range of
scattering angles 120° < 9S < 240° has been chosen in order to show b o th the specular and the
forw ard scattering lobe.
Note how the approxim ation w ith a single cylinder is inadequate, p articularly around the spec­
ular scattering lobe, which is not located a t th e correct angle 9S. Such displacem ent occurs because
the slope of the side walls of the tap e red cylinder cannot be m odeled using only one cylinder. T he
direction of the forw ard scattering lobe, on th e other hand, is not affected by the taper, an d is
correctly estim ated by the approxim ate model. T he use of two stacked cylinders (of length and
radii given in Table 4.2) leads to a significant im provem ent. T h e m ain scattering lobe is now closer
to its correct location, b u t there is still the problem of a high side lobe around 9S — 150°. T he
approxim ation w ith four cylinders brings the m ain lobe even closer to its correct position and more im portant - lowers the side lobes. Em ploying m ore th a n four cylinders still increases the
accuracy, b u t the im provem ent is quite small.
107
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VV-POLARIZATION
HH-POLARIZATION
40
40
CM
ns
X
-1 0
-1 0
tapered cylinder
1 cylinder
—«—2 cylinders
—9—4 cylinders
-20
-30
120
140
160
180
200
220
tapered cylinder
1 cylinder
—**—2 cylinders
—9—4 cylinders
-20
-30
120
240
140
160
180
200
220
240
scattering angle 0 [deg]
Figure 4.41: N orm alized b istatic scattering coefficient vs. scatterin g angle for tap ered dielectric
cylinder w ith L = 10.0 A, a = 0.7 A, b — 0.1 A, and incident angle 9t = 40°. h h - and vv- polarization
(left and right, respectively).
HH-POLARIZATION
VV-POLARIZATION
tapered cylinder
1 cylinder
—•*—2 cylinders
—9—4 cylinders
— tapered cylinder
- -1 cylinder
-«—2 cylinders
■9— 4 cylinders
£
■O
E
CQ
CQ
-a
cj
ns
as
K
K
o's
-1 0
-1 0
-20
-2 0
-30.
120
140
160
180
200
220
240
140
160
180
200
220
scattering angle 0s [deg]
scattering angle 0 [deg]
Figure 4.42: Norm alized b istatic scattering coefficient vs. sc atterin g angle for tap ered dielectric
cylinder with L = 10.0 A, a = 0.5 A, b = 0.3 A, and incident angle 9{ — 40°. hh- and vv- polarization
(left and right, respectively)
108
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240
1 cylinder, HH
2 cylinders, HH
4 cylinders, HH
1 cylinder, VV
- •* - 2 cylinders, VV
4 cylinders, VV
1 cylinder, HH
—“— 2 cylinders, HH
4 cylinders, HH
1 cylinder, VV
- k - 2 cylinders, VV
4 cylinders, VV
absolute error [dB]
—
m
■a
I—
o
0
0
3
o
CO
.Q
0
30
45
60
incident angle 0. [deg]
incident angle 0. [deg]
Figure 4.43: E rro r vs. incident angle for tap e red dielectric cylinder w ith L = 10.0 A, (left) a — 0.7 A,
ct2 = 0.1 A, and (right) a — 0.5 A, b = 0.3A. Solid lines are hh-pol, dashed lines w -p o l.
Figure 4.42 illustrates th e case of a tap e red cylinder of th e sam e length and volume as th e
cylinder in Figure 4.41, b u t w ith less tap e r, specifically w ith a — 0.5 A, b = 0.3 A, which results
in a tap e r factor R = 0.02. T h e wave is incident from an angle 0Z = 40° as in the previous case.
As one would expect given th e sm aller tap e r, th e tap ered cylinder m odel works m uch b e tte r in
this situation, and even th e one cylinder approxim ation exhibits only a sm all displacem ent in the
m ain scattering lobe. In b o th Figure 4.41 a n d 4.42, it is evident th a t th e analytical approxim ation
works b e tte r in th e forw ard scattering direction. T his behaviour is probably due to the fact th a t
the forward scattering depends more on th e dielectric properties of th e cylinder, which are not
approxim ated, while the scattering in th e specular direction is m ore affected by the geometry,
which depends on th e m odel used.
T he error plots in Figure 4.43 help u n d e rsta n d how th e accuracy of th e various approxim ations
changes w ith the incident angle
T he definition of the errors thh, and evv is still the sam e as in
(4.127).
T he first of them is th e long tap e red cylinder of Figure 4.41, w ith L = 10.0A, a — OTA,
109
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a 2 = 0.1 A, and. R = 0.06. Its absolute d B error is plo tted at the left of Figure 4.43 as a function
of the incident angle 9i. As already noted in Figure 4.41, the single cylinder solution does not
provide a good approxim ation for th e tap e red cylinder around th e specular scattering lobe. The
curves on the left of Figure 4.43 confirm th a t by showing an error betw een 3 d B an d 10 d B a t m ost
incident angles in h/i-polarization. Such a large error is m ostly caused by th e m isplacem ent of the
specular scattering lobe by th e analytical approxim ation. For the sam e reason, using two cylinders
produces a lower - b u t still unacceptable - error a t h/i-polarization. T he approxim ation w ith four
stacked cylinders yields th e best results, th e error being below 1.5 d B at th e most angles, except
for low incident angles, i.e., 9i < 15°. T he problem of m isplacem ent of th e specular lobe in the
approxim ation does not seem to affect the w -p o lariza tio n . In reality, as obvious from Figure 4.41,
th e problem is present a t b o th polarizations, b u t since a t uw-polarization the specular scattering
lobe is more th a n 10 d B lower th a n the forw ard scattering lobe, it does not enter in the calculation
of the error as defined in (4.127).
On the right side of Figure 4.43 are the error plots for the cylinder of Figure 4.42, w ith L =
10.0 A, and a = 0.5 A, b = 0.3A, and R = 0.02. T he cylinder has a sm aller tap er, and a good
approxim ation is already achieved w ith a finite cylinder of th e same length and volume, w ith no
m ajor im provem ent using th e tap ered cylinder approxim ation.
O n the other hand, com parison
betw een the two plots in F igure 4.43 indicates th a t th e error a t low incident angles is not affected
by the taper. T he m ost obvious conclusion is th a t th e end-on effect are predom inant there and
cannot be predicted accurately by any of the approxim ate m odels here used.
C om parison o f co m p u ta tio n a l tim es
In this section, a com parison will be m ade betw een the co m putational tim es of the various m ethods,
in order to show the advantage of choosing th e analytical approxim ations over the num erical solution
approach.
In order to estim ate and com pare c o m p u tatio n tim es of th e different m ethods, the
110
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following considerations are made.
T he num ber of calculations needed to evaluate th e scattering coefficient using the finite cylinder
m odel does not depend on the cylinder length and is constant for a fixed vale of its radius. T his
happens because the expression used to com pute th e scattering am plitudes is a series sum m ation
w ith a num ber of term s th a t increases w ith the radius of the cylinder. In practice this num ber has
been chosen large enough to achieve convergence in all cases considered in this comparison. U nder
this assum ption, let T p c be defined as the com putation tim e of th e scattering coefficient by the
finite cylinder model.
T he tap e red cylinder m odel requires - as a first step - th e calculation of th e scattering am plitudes
of th e N s individual cylinders th a t approxim ate sections of th e original tap ered cylinder.
T he
scattering am plitudes are th e n used in expression (4.126) to yield th e scattering am plitude of the
entire tap e red cylinder. Therefore, if one neglects th e tim e need to perform such sum m ation, the
com putation tim e of the tap ered cylinder m odel will be
Tt c = N s T p c ,
(4.130)
which depends on the num ber N s of sections in the approxim ation, b u t not directly on the tap ered
cylinder length.
On th e other hand, th e m ethod of m om ents involves th e selection of a set of N points on the
generating curve C an d requires a num ber of calculation th a t increases as N 2. The num ber of
points N varies in p ro p o rtio n to th e length I of th e generating curve C in the representation of
the cylinder as a body of revolution, m easured in wavelengths A. For a finite cylinder of length
L and radius a, the length I of the generating curve C is I = L + 2a, while I = L + a + b for a
tapered cylinder of length L and radii a, b. Therefore, it is reasonable to expect the com putation
tim e T m m ° f the m ethod of m om ent algorithm to increase approxim ately w ith the square of the
cylinder length.
I l l
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Em pirically, th e following approxim ate relationship betw een T m m and T FF are found:
T m m = Z M (j )
Tf c = Z M
' TF C .
(4.131)
where M is th e num ber of modes used in the num erical algorithm .
Similarly, if the m ethod of m om ents is used to find the scattering coefficient of a tap ered
cylinder, th e use of (4.130)-(4.132) yields
T m m = 3A f ( j J
2m
„ M
TFC = 3 —
(L + a + b \2
------—
j Ttc .
(4.132)
In order to exam ine th e gain in com putational speed, TFc and T m m have been estim ated using
expressions (4.130) a n d (4.132) for four tap ered cylinders w ith a = 0.6A, b — 0.2A, and lengths
L = A, L = 3A, L = 5A, a n d L = 10A. As an exam ple, a com puter system is considered, on which
the finite cylinder p rogram needs a tim e TFc = 5 sec to calculate the scattering coefficients. For
all four cases, the num ber of sections in the tap ered cylinder approxim ations is chosen as N s = 4,
therefore according to (4.130) the com putation tim e is TFc — 20 sec.
T he com putational tim e T m m if th e num erical algorithm has been estim ated through expression
(4.132), and listed in Table 4.3, where th e ratio T m m / T t c is also given. A num ber of modes M — 5
has been found to achieve good accuracy in the num erical solution in these p articular cases, and
used in (4.132).
length L
A
3A
5A
10 A
Tmm
Am in
18 m i n
42 m i n
145 m i n
Tm m /T tc
12
54
126
438
Table 4.3: C om parison of com putation tim e betw een an alytical m odel and num erical algorithm for
four tapered cylinders of length L and a = 0.6A, b = 0.2A.
The values in T able 4.3 clearly show the great gain in com putational efficiency at the cost of
some loss of accuracy. Indeed, it is easy to see th a t even for a ra th e r short cylinder of length L = 3A,
112
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a com putation tim e of T m m = 18 m i n for the num erical approach com pared to T t c = 20 sec for the
analytical approxim ation makes th e la tte r preferable when dealing w ith a m edia where thousands
cylinders of different sizes are present and estim ation of the scattering coefficient from each cylinder
is needed.
4 .4 .4
C o n c lu s io n s
A nalytical approxim ations for two different types of finite cylinders have been considered, and their
accuracy evaluated. For cylinder sizes and dielectric constant sim ilar to those used to represent
vegetation elements, such approxim ations work well in th e m ain lobes, which is w hat is most
im p o rta n t w ith w hen studying rem ote sensing from vegetation.
W hile the physical optics solution for the finite cylinder works very well and is readily and
safely usable in vegetation m odels, th e tap ered cylinder approxim ation requires more a tte n tio n in
the choice of the num ber N g of cylinders, in order to reduce th e error due to the displacem ent of
the specular scattering lobe. T he results also prove th a t w hen th e ta p e r is significant, the single
cylinder approxim ation is not adequate for a tap ered cylinder.
As seen in the previous section, th e relative sm all loss in accuracy suffered by the use of such
m odels is outbalanced by a trem endous gain in com putational speed. However, more work could be
done to try to m odel th e end-on scattering. T he integration of such a m odel into the approxim ations
exam ined here could allow th eir use in a broader range of electrom agnetic problems.
113
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Chapter 5
H alf-Space Problem
T he concepts a n d techniques introduced and developed in th e previous chapters can be used to
build more realistic m odels for vegetation th a t account for th e effect of th e ground. A very simple
approach is to m odel th e ground and the air above it as two sem i-infinite m edia separated by a
plane interface. T his type of configuration will be referred to as an Half-Space Problem .
In order to s tu d y this problem , it is m ore convenient to use the G reen’s functions in their
dyadic forms. For th is reason, the Section 5.1 of this chapter is dedicated to the derivation of
explicit forms for th e com ponents of the dyadic G reen’s functions, b o th electric and m agnetic, in
the case of an infinite, homogeneous m edium. Following th e sam e steps, in Section 5.2 the dyadic
G reen’s functions for th e half-space problem are introduced, and expressions of their com ponents
are derived. Section 5.3 shows how to modify the coupled surface integral equations for the half­
space problem, an d how to solve them by th e M ethod of M om ents for the case of a B ody of
Revolution located in th e u p p er half-space. T he im plem entation of th e BOR-M OM m ethod in the
half-space often requires the evaluation of slowly converging integrals. Section 5.4 is devoted to
the investigation of some efficient num erical techniques for th eir integration. Results are given and
discussed in Section 5.5, and finally the application to electrom agnetic scattering from vegetation
is illustrated in Section 5.6.
114
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5.1
The dyadic G reen’s function for an infinite, hom ogeneous
m edium
T h e dyadic G reen’s function for an infinite, homogeneous m edium w ith perm ittiv ity e an d perm e­
ability fj, relates th e vector electrom agnetic fields to the vector current sources, and can be found
from the corresponding scalar G reen’s function using th e alternative form (2.44):
G (r,r')
=
f l + ^ w
k2
) 5 (r,r')
(5.1)
where k = y/]H is th e wavenum ber of the m edium . A m agnetic dyadic G reen’s function is defined
by expression (2.11) as:
K (r,r')
=
V j ( r ,r ') x l
(5.2)
D epending on th e geom etry of the problem under consideration, it is useful to express the
dyadic G reen’s function in different forms. One of such representations makes use of plane waves.
As shown in [76], in order to derive it, the equation for the scalar G reen’s function
V 2fir(r, r ') + k 2g(r, r ') = - 5 { r - r ')
(5.3)
is solved using Fourier transform s. T he result is an expression in term s of planes waves:
,•
r e - j k 0Z\z - A
,
9 ( r ' r ' , = “ 4 ^ / ------- 2 K , ------- 6
J
(5'4)
where k £ and r £ are tw o-dim ensional vectors defined as
k s=
and
kx ± + k y y ,
r t = j;x + y y
koz = y jk ,2 - k l - k l ,
X m { k oz} < 0 ,
(5.5)
(5.6)
Consider a plane electrom agnetic wave traveling in th e direction of a u nit vector k. A ssociated
w ith this electrom agnetic
wave
is a propagation vector k = k k and two
other u nit vectors h and
v , defined so th a t th e vector tria d (k, h, v) is orthonorm al:
k x z
h = —----------,
|k x z |
„
v =
h x k
115
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(5.7)
A
h
\
\
Figure 5.1: D efinition of (k, h , v) for waves traveling forward and backw ards in th e ^-direction.
T he u nit vector h corresponds to th e direction of the h-polarization of th e electrom agnetic wave,
while v is in the direction of th e u-polarization. Those vectors are shown in Figure 5.1 for the two
cases of plane waves traveling in the positive and negative z-direction, identified by the superscripts
+ and —, respectively.
Since
= k k*
= kx x + ky y ± koz z = k t ± koz z ,
(5.8)
th en the scalar G reen’s function (5.4) m ay also be w ritten as
^ J
- j k + • (r - r')
—j------------ dkt
for z > z'
47T
2J
g(r , r ' ) = - <
(5.9)
g - j k~ • (r - r')
2 kn
dkt
for z < z'
T he dyadic G reen’s function can be o b tained by differentiation of (5.4) th ro u g h expression (5.1).
T he procedure is illu strate d in [76] and yields th e result:
j k + ■(r - r')
J
2tr2
kl
h + h+ + v+ v+ )
~ 2 J { h - h - T v - v - ) -------- _
dkt
--------- d k t
for z > z'
for z < z'
— OO
T he dyadic G reen’s function can also be w ritte n in term s of vector cylindrical waves. Such a
116
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representation is particu larly useful when dealing w ith problem s th a t have a cylindrical or ro tatio n al
sym m etry.
T he vector cylindrical wave functions are defined as
^jn(kpi k z ,v)
=
k Z: r)
M n {kp, k z ,r)
=
V x [z ipn (kp,kz ,r)}
(5.12)
N n {kp, k z ,r)
=
i v x M ^ ^ r )
(5.13)
(5.11)
where ipn is the solution of th e scalar wave equation in cylindrical coordinates
{ } r P%
+ ^ w
w ith
+ h
k
+ k 2 ) i ’ir)
= 0
9
9
kp + kz ,
=
(5-14)
(5.15)
and is related to the Bessel function of ra-th order J n :
^ n { k p, k z , r)
=
Jn ( k p p ) e ~ 3 kzZ ~ ^
(5.16)
kx = kp cos^fc ,
ky = kp sin$fc
(5-17)
Using the transform ation
and the m athem atical id entity
e - j { k xx + kyy) = e - j k pPzos{<t>-§k) =
4-00
Y , ( - j ) n Jn(kPp ) e ~ i n (^ - $*)
(5.18)
it is possible to w rite th e vector electrom agnetic plane waves in term s of vector cylindrical waves:
+°o
h e " j k 'r
=
- —
5 3 ( - j ) U^ - n ( k p , k z , r ) e ~ i n ®k
(5.19)
P n =—
oo
i
v e-jk-r
=
_
+oo
Y
( - j ) n $ n (kP, k z , r ) e - 3 n ^
P n=—
oo
Since
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(5.20)
su b stitu tio n of (5.20), (5.21) into (5.9), followed by integration in th e variable
yields the
cylindrical wave representation of the dyadic G reen’s function:
(5.22)
koz, r ) + ]NTw(fcp, k oz, r) 1ST—
n {kp,
M-njkp, k oz, r)3Vl—n {kp,
Air
k oz, v )
koz kp
k oz, r)!VI_n{kpi koz, r ) +
~M-n{kpi
47r
fcoz,r )N _ n(fc^,A:oz,r )
koz kp
dkg
for z > z'
dkg
for z < z'
T he m agnetic dyadic G reen’s function can be obtained w ith a sim ilar procedure by su b stitu tin g
the expression (5.4)
for the scalar G reen’s function into the definition (5.2).C arrying
out the
differentiation yields:
K (r,r0
( —j k + x I g( r, r')
Z
,
=
for z > z'
„
,
(5-23)
so th a t the following expression is obtained:
k
K (r,r')
=
r
~
e - i k + • (r - r ')
h + - h + v+) -------- — ---------d k ,
z > z’
-<
(5.24)
v
. 47T2 J v ’
_
h
—h
"
e ~ 3 k ■(r _ r ')
v ) -------- —------------- dkt
'
2 koz
_
for z < z'
By expanding h and v in term s of vector cylindrical waves by m eans of (5.19) and (5.20),
the following cylindrical waves representation of the m agnetic dyadic G reen’s function is obtained:
OO
k
47T
K (r,r')
k
47T
f
/
f
/
^
A
^ n { k p , koz, r) 1V1_n {kp,
A
k0Z:T^j
A
T^/ln ^kp, k 0Z: r)N_^(A:p,
koz, v )
koz kp
koz j r)]VI_n{kpi k 0zi r )
^^-nikpi
k oz, r )N _ n(fcp, koz, r )
koz kp
dkg
for z >
dkg
for z < z
(5.25)
It is desirable to find an expansion of th e dyadic G reen’s functions into Fourier modes in the
azim uth angle <j>. T his type of expansion is indeed essential in order to use the BO R approach
developed in the previous chapter.
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S u b stitu tin g expression (5.16) into the definitions (5.12) an d (5.13), the vector cylindrical wave
functions can be w ritten explicitly as
M n {kp,kz ,v)
—
jn
LP
N n {kp, kz , r)
=
JnikpP) P "b kpJn (kpp) </>
, ~ j kz z - jn<p
(5.26)
- j k pkz J'n (kpp ) p + n k z k J n {kpp) <p - k 2Jn {kpp) z
e —J kzz — jruf>
(5 27)
In this form, the periodic behaviour in th e variable <f>is evident. Using (5.26) and (5.27) yields the
Fourier series expansion of th e dyadic G reen’s function:
+oo
G (r,r') =
Y.
(5.28)
S
T he coefficients G n are dyads them selves, and have different expression depending on th e coordi­
nate system used. Using a unprim ed-prim ed cylindrical coordinate system in which a dyad S can
be w ritten as
S =
^2
S pq p q '
(5.29)
p ,q = p,faz
the com ponents of G n have th e form:
OO
Gp/ ( P,z-,p',z') = - j -
J
-j koz\z - z'\
g pq, ( k p , k o z ] p , p ' )
2 koz
kpdkp,
p,q = p,(/),z
(5.30)
Similarly, the m agnetic dyadic G reen’s function (5.26) can be w ritten as a Fourier series
+ °°
K (r,r') =
,
K n ( / > , * ; / / , * ,) e J’n ^ ~ 0 )
(5.31)
n=—oo
and the com ponents of K „ as
. ~ j k0Z\z - z'\
K pq' { p ,z - p ' , z ' ) = ~ L
5.2
J
K,iq'{kp,koz-p,p')
2fcoz
kpdkp,
p,q = p,(f>,z
(5.32)
The dyadic G reen’s function for sem i-infinite media
Let the space be divided into two regions w ith different dielectric properties by an infinite plane,
which for convenience will be considered coincident w ith th e xy-plane. Each region is referred to
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as an half-space. T he two m edia are characterized by p e rm ittiv ity and perm eability respectively
given by:
ea
£b
for z > 0
Ha
f
A4 =
for z < 0
|
lu a
z > 0
,
n
for £ < 0
for
,
(5.33)
Let b o th the sources and the observation points be located only in th e upper half-space z >
0. T he dyadic G reen’s functions in th e u p p er half-space, i.e., for z, z' > 0 have different term s
corresponding to th e direct (incident) and reflected electrom agnetic waves:
G (* ;„$ * ; z,z')'
=
& D)(kp, $ k-,z,z') + £ W { k p, $ k]z,z')
(5.34)
K ( k p, $ k -,z,z')
=
K ^ ( k p, ^ k - z,z ') + K W ( k p, $ k -,z,z')
(5.35)
T he direct term s have the sam e expression as for the dyadic G reen’s functions in an infinite m edium
(5.10) and (5.24):
i
G ^ (r,r')
r - e _ i k + ' (r _ r 0
' ''h + h + + v + v + ) ---------—------------ d k t
2 koz
47T2
=
j
, 47r2
K ('D)(r, r ;)
=
-<
-
e
’ (r — r ' )
for z > z'
(5.36)
v ) ---------—------------ dkt
2 rCor
for z < z'
j
k
„ , . v + h + — h + v + ) ------- —------------- dkt
47T2 J
2 ka
for z > z'
h
h
+ v
**
f0r2f<i!
Using the cylindrical wave expansion of th e plane waves h an d v , these expressions take the
following form:
/
OO ~
4 tt
r
/
A
]&ln (kp,
K U U ( r , r ') = <
f
a
a
f
u u
/
A
dkD
A
for z < z'
a
z
z1
p
koz kp
-~koz, r)]Vl—n
for z > z'
(5.3?
koz, r ) N _ n (fcp, —k0Z: r ) ^
fepz, r ) —
J
dk0
A
a
a
j4n (kp, k oz, r ) M _ n (fcp,
OQ A
&
47T
x
k oz, r ) M ~ n (kp, k oz, r ) + N n (A;p, —A;oz, r ) N _ n (fcp, k oz, r )
koz kp
Ca J
k
47t
/s.
koz kp
QQ
47T
^
k oz, r ) -I- ~Nn (kp, koz, r)N _ ri(A:j0, —koz, r )
J
G ^ }( r , r ') =
j
^
F WLn {kp, k 0z-, r)JVl_rj(A:p,
j
(5.31
A
k0Zj r ) — 3VIn (fcpT~ k oz, r ) N _ 7i(A^, k0zi r )
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dkp
for z < z'
T he reflected term s have th e following expression:
.
i
f i W (r . O = - ^
/
b
K ^ ( r , r ' ) = —^~2 /
r T h h + e ~3 k + r h~ e l k_ ' ^ +T„ v + e~^' k + ' r v"
--------------- —
^
k_ ' ^
dki
(5'40)
r r , o+ « —j k+ • r it- „ j k~ ■r' r fl+ ~3 k+ ■r <>- „ j k~ • r'
6
—
---------2 ^ " ------------ ----------- —
**
(5.41)
where IT and r „ take into account the reflection of the horizontal and vertical plane waves, respec­
tively. T hey depend on th e dielectric properties of the two m edia, as well as on kp, as follows:
_
h
Mfr K z - Ma K z
e b K z - £a K z
r
lib K z + Mo K z ’
V
^
£b K z + £a K z
,----------
w ith
=
y /K ~ K
=
k oz
(5.43)
Kz =
\jK ~ K
=
\ ! k oz + ( K - kl)
(5.44)
Kz
and the wavenum ber in th e two m edia given by
= uj^JiJ,aea
for 2 > 0
kb = ujy/JIb£b
for 2: < 0
ka
k = {
(5.45)
In term s of cylindrical waves:
ri(R)r
-
i\
3
47T
J0
OC
f
h ■hdn(^p) k oz, r)M _ n (fc^, k oz, r ) 4 -1\, N n (kp, k oz, r)N _ n(&p, k oz, r ) J7
k^ kp
9
/e
OO
~
f 'E'fl i'fn ( kp, koz,v)ls/L—n (kp, k o z i Y ) — T'v 7S/Ln(kp, k oz,ic')N~n (kp, k oz, Y )
-------------------------------------------- k j , ---------------------------------------------dk> (5-47)
A
is-(R)f
!\
k
{T' r ) = ^ J
0
As already m entioned in th e previous section for th e infinite-m edium dyadic G reen’s functions,
the half-space dyadic G reen ’s functions (5.38)-(5.41) can also be w ritten as infinite sum m ation of
angular modes
S l ‘>(r,r') =
£
=
n=—00
£
[ £
S * p q '] e 3
n=—oo \p,q—p,(ji,z
)
for SW = £ W , K W ,
t = D,R
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- «
(5.48)
where th e dyadic com ponents of the n th m ode coefficients have th e following integral forms:
Sd^{p^-,p',z')
, - j k 0Z\ z - z '|
= -zL j
S $ n {kp, k oz;p,p')
2k„
o
OO
J
kp d k p
,
p, q = p,4>,z
(5.49)
■k p d k p
,
P , q = P,<f>,z
(5.50)
, - j k 0z { z + z ' )
S ^ n {kp,k o z \P ,p ')
-
2 k oz
Explicit expressions for Qv^ n , fcP
r jn , Qv^ n -, and KF^n are given in A ppendix D.
5 .2 .1
T h e d y a d ic G r e e n ’s fu n c tio n in t h e far field
In scattering problem s it is very im p o rta n t to be able to evaluate th e electric and m agnetic fields,
generated by electrom agnetic sources, a t large distances from th e sources themselves.
For this
purpose, it is essential to derive the far-field expressions of th e dyadic G reen’s functions derived in
th e previous section.
" .///
Figure 5.2: Far-field in half-space m edium
W ith reference to F igure 5.2 (left), a point in th e far-field - or rad iatio n zone - is defined by a
position vector r such th a t k^r
1 a n d r » r ', w here r ' identifies an a rb itrary source point. If
122
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the source has a m axim um overall dim ension D th a t is large com pared to the wavelength A, the
far-field region is comm only taken to exist at distances |r —r '| greater th a n 2 D 2/ X from the source.
B oth direct an d reflected term s of th e half-space dyadic G reen’s functions need to be evaluated in
the far-field. T he direct term has two different expressions, depending on w hether z > z'
01
z < z 1.
However, only th e first case needs to be considered, and Figure 5.2 (right) illustrates why.
An observation point r has z > z' if it is not inside an infinite slab of thickness h (equal to the
height of th e m ost d istan t source point from the half-space interface). T he m axim um allowed value
of the elevation angle for th e observation point not to fall inside th a t slab is 6 — arctan (p/h). In
the rad iatio n zone, p becomes very large and 6 approaches
tt/2.
T his m eans th a t for observations
angles 6 < 7r/2, which is th e range of elevation angles considered in half-space problems w ith plane
wave coming from th e u p p er m edium , only the case z > z' needs to be considered in the evaluation
of the far-field dyadic G reen’s functions.
Hence, th e problem is now reduced to the evaluation of expressions (5.36)-(5.37) for z > z' and
(5.46)-(5.47). All these expressions have the general form
F ( r , r ')
j
=
l ( k p, k oz-v') 6
dkt
(5.51)
An approxim ation for r > r' for th is generic F ( r , r ') can be found by employing the so-called
stationary-phase m ethod. T he exponential
k+ •r
=
kx x + ky y + ^Jk2 — k 2 — k 2 z
(5.52)
is a rapidly oscillating function of k t = kx x + ky y for very large values of r. T hus the contributions
to the integral from various points in th e kx ky plane ten d to cancel each other because th ere is
a lack of in-phase add itio n from th e various regions. A n exception is a point where the exponent
does not vary w ith sm all changes in k x , ky . Such point is a called a stationary-phase point and is
characterized by th e vanishing of th e first derivative of the exponent w ith respect w ith kx and ky ,
i.e.,
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At a stationary-phase point, th e phase of (5.51) does not vary rapidly, and a nonzero contribution
would be obtained from this region of th e kx ky plane. In a sm all region surrounding the stationaryphase point k x = k x , k y = k y , th e rem aining, slowly-varying p a rt of the integrand of (5.51) is set
equal to its value at kx , ky . T he integral th a t rem ains th en involves only the exponential function,
and can be easily evaluated.
T h e stationary-phase point is at
k + = kx x + ky y + k z z
w ith
k x = k sin 9 c o s^ > ,
ky = k s in 9 sin <fr ,
(5.54)
k z = k cos 9
(5.55)
T h e result of the integration around th is point is shown [77] to be
F ( r , r ')
~
F (7c sin
(9 ,
& co s# ;r')
-r
(5.56)
At the stationary-phase point the plane waves k , h and v become
k+ = r ,
k~ = r — 2 cos 9 z
(5.57)
h + = h " = - 4>
v + = —0 ,
(5.58)
= — 0 + 2 cos 9 p ,
(5.59)
Using the general result (5.56) and expressions (5.57)-(5.59), th e far-field expressions of th e direct
term s of the dyadic G reen’s functions are found - as expected - to be identical to the ones for an
infinite m edium, i.e,
G ^ D^ ( r ,r ')
~
( d 0 + (j) <fr^) e ^ r
K ^ ^ r,^ )
~
—j k ( <j >0 v
0 <^)
/
k
4tt r
r
(5.60)
•r
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(5.61)
while the reflected term s of the dyadic G reen’s functions in th e far field are
J
~ (fv 9
\
G ^ (r,r')
K ^ (r,r')
where
~
0
/
rp
\
/
A
^
e i fck“ ' r '
47v r
- j k ( T v 4 > d - T h e 4 > - 2 TV c o s e 4 >p )
a n d r „ are the values of
jjj
+ T h 4 > 4> - 2 TV c o s d 9 p ) — A
__j T
^
------- e j k k - - r '
47r r
(5.62)
(5 ^
an d r „ , defined in (5.42), calculated a t the stationary-phase
point given by (5.54).
5.3
Scattering from a B o d y in a Sem i-Infinite M edium
5 .3 .1
E x p r e s s io n o f S u rfa ce I n te g r a l E q u a tio n s fo r S e m i-I n fin ite M e d iu m
Figure 5.3: M aterial body in half-space m edium .
T he coupled surface integral equations for a m aterial body in an homogeneous, infinite m edium
have been derived in C hapter 2. T h eir expressions are (2.37) a n d (2.38). These equations can be
m odified and applied to an half-space problem .
T he problem is defined in Figure 5.3. A n homogeneous m aterial body, w ith perm ittiv ity
£4
and
perm eability jj,d inside its volume V, is located in th e upper half-space, which has perm ittiv ity £0
and perm eability /j,q. T he lower half-space has p erm ittiv ity eg a n d perm eability n g.
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T h e m aterial body can be regarded as being located in a n equivalent homogeneous, infinite
m edium w ith th e same p e rm ittiv ity £q, perm eability /xq as the u p p er half-space but w ith th e dyadic
G reen’s functions G and K previously derived for the half-space problem . A similar approach has
been followed by [78]. T he derivation of the coupled surface integral equations can be carried out
for th is equivalent m edium in a sim ilar way as done for th e hom ogeneous, infinite m edium case.
T h e exterior problem is illu strated in Figure 5.4. If th e surface S 2 is entirely inside th e upper
m edium , as in Figure 5.4(a), th e n (2.27) can be w ritten as
f
—jui/jiQ [ G HS ( v , r ' ) - 3 { r ' ) d V ' K H5 (r, r ') ■M ( r ' ) d V ' +
Jv
Jv
=
2
-jufxo
-joJH0
f
G HS (r, r') ■ [ n ' x H (r')] dS' -
f
K HS (r,r' ) ■ [E (r') x n '] dS' +
G HS,(r, r') • [ n ' x H(r')] dS' - <£ K fflS(r, r ') ■ [E (r') x n '] d S ' , r G Si U S 2 (5.64)
T he dyadic G reen’s functions G HS and ~KHS for th e half-space m edia consist of direct and reflected
term s in the upper m edium , an d a tra n sm itte d term in th e lower m edium :
for z > 0
for 2 < 0
for 2 > 0
for 2 < 0
(5.65)
(5.66)
’
T he direct and reflected term s are given by expressions (5.37),(5.38) an d (5.47),(5.48), respectively,
while the expressions of th e tra n sm itte d term s are not given here because they will not be used in
this dissertation.
As th e surface S 2 expands, p a rt of it sta rts entering th e lower m edium , as shown in Figure
5.4(b). Considering th e surfaces Si U S ^ , w ith 5 ^ = S a U £a> a n d 5 ^ = S b U Eg in Figure 5.4(c),
it is possible to w rite the following two expressions:
^
2
=
- j uf i Q [ G H5 ( r , r ' ) • J ( r ') d V ' - [ K K5 (r, r ') • M ( r ') dV' +
Jv
Jv
-jwn0
f
G H S {r,v') • [ n ' x H (r')] dS' -
f
K H S {r,v') ■ [E (r') x n '] dS' +
/
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
J,M * 0
J,M * 0
(b)
J,M * 0
Figure 5.4: E xterior problem for half-space m edium .
127
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-juii o
f
G H S (v, r') ■[ n ' x H (r')] dS' -
f
G HS {r,r') • [ n ' x H (r')] dS' -
JSa
-jojfj, 0
f
K HS (r,r' ) • [E (r') x f i '] dS'
f
K ff5 ( r, r ') • [E (r') x n '] dS'
j Sa
JH
Ha
JHa
for r e S i U S i
E (r)
j —r~y r
r,
JS b
2
f
JzB
G h s {y,y')
•
[ n ' x H (r')] dS' - f
K HS (v,r')
J Sb
f
G H S (r,r' ) • [ n ' x H (r')] dS' -
•'S b
■
(5.67)
[E (r') x n '] dS' +
K HS {r,r') ■ [E (r') x n '] dS'
for r e S i
(5.68)
Note th a t there are no volum e integrals in (5.68) because there are no sources in th e lower m edium .
W hen the surfaces E a a n d E # approach the interface z = 0, due to continuity a n d to the opposite
direction of th e surface norm al vectors,
Vo
= —n g (f
/ G ff5 ( r , r ' ) ' [ n ' x H ( r ' ) ]
f K HS (r,r' ) • [E (r') x n '] dS'
JT,a
=
G ffS( r, r ') ■[ n ' x H (r')] d 5 '
- f K HS {r,r') ■[E (r')
■'2s
x n '] dS'
(5.69)
(5.70)
while the integrals over th e surfaces S a , S b vanish at infinity. Therefore, in th e lim it of S a , S b
expanding to infinity, adding equations (5.68)-(5.68) side by side yields th e following result:
=
2
—juifiQ
-jufio [
Jv
Js+
G H S ( r , r ' ) - J ( r ' ) d F ' - / K F 5 (r, r ') • M ( r ' ) d V ' +
Jv
G H S (r,r' ) • [ n ' x H (r')] dS' —
Js+
K H S (r,r' ) ■ [E (r') x n '] d S ' ,
r e S + (5.71)
O n the other hand, th e in terio r equation depends only on the properties of the m edium inside the
surface 5 of the m aterial body, therefore its form is th e sam e as for free-space, i.e.,
^ 7^2
=
jwfJ-o /
Js~
G ( r , r ' ) • [ n ' x H (r')] dS' + <£
Js~
K ( r , r ') • [E (r') x n '] d S r ,
r 6 S ~ (5.72)
Taking the tangential com ponent of b o th sides of equations (5.71)-(5.72), and using th e definitions
(2.25) and (2.35) of equivalent surface currents, yields
2
=
E^(r) -
Js+
G ff5 ( r , r ' ) • J s (r') dS' +
128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- ( I - n n ) - < T K HS {r,r') • M s ( r' ) dS' ,
7s+
=
r 6 S+
(5.73)
r € 5"
(5.74)
& H S {*,*')■ M { r ' ) d V '
(5.75)
- W < i ( I - n n ) ' | _ G ( r , r ;) • J s(r')dS" +
- ( I - n n ) . j f * K ( r , r ' ) . M s (r')(l5'
where th e ’’incident field” is defined as
=
5 .3 .2
-juM J
& HS ( r , r ' ) - 3 { r ,) W ' - J
S o lu tio n o f t h e S u rfa ce In te g r a l E q u a tio n s for a B O R in H a lf-S p a c e
Let th e surface of th e m aterial body have a ro tatio n al sym m etry around the z-axis, so th a t th e
B O R approach discussed in C hapter 4 can be applied.
D ividing the half-space dyadic G reen’s functions into th eir direct and reflected term s as in
(5.65)-(5.66), the exterior integral equation (5.73) can be w ritte n as
M *( 3)X n
=
E ? \ r ) - jujfxQ( 1 - n h ) - £ + G W ( r , r ' ) ■J s (r’) d S ’ +
- ( I - n n ) - < T K ^ ( r , r ' ) - M s (r' ) d S ' +
Js+
- j u ^ o ( I - n n)- <f
Js+
G p ) ( r , r ') ■J s (r') dS' +
- ( I - n n ) i * K (jR)( r ,r ' ) ■ M s {r') d S \
Js+
r GS+
(5.76)
D iscretization of the surface integral equations (5.76) a n d (5.74) by m eans of the m ethod of m om ents
yields a linear system of equations of th e form:
E ^ { ( [ A ^ ] (D) + [ A ^ p ) ) i?« + ( [ B W p ) + [Bpp R)) M «"} = V Pn
(5.77)
E {[CW ]F»-[DM ]M 9»} = 0
q=T,<f>
(5.78)
T he elements of th e m atrices [CP9] and [DP9] can be evaluated by th e same procedure and expres­
sions derived
in C h ap ter 4. T he direct term s G p ) and G P
have the same expressions as th e
129
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dyadic G reen’s functions for the half-space. However, the calculation of th e elem ents of [A^9] ^
and [B?9] ^
depends on th e particu lar form of the dyadic G reen’s functions used. The operator
form is preferred here because the direct term s of dyadic G reen’s functions are highly singular. T he
rem aining term s in (4.70), i.e.,
E
q= t ,4>
{ [A£9](r) I 9n + [B£9](k) M 9" }
(5.79)
J
1
are obtained by discretizing th e surface integrals
Ls(r)
=
- j u p 0 (I - n n)- <fi G (iJ)(r,r') ■J s(r') dS' +
Js+
- ( I - n n ) i * K ^ )(r ,r ') • M s(r')dS',
J s+
r e S+
(5.80)
As seen in Section 5.2, the reflected term s of th e dyadic G reen’s functions can expressed in
cylindrical coordinates as follows
G ^ ( r ,r ') -
£
^ (r.O p q ',
K^(r,r')=
p ,q= p,4> ,z
£
K $ ( r , r') p q '
(5.81)
p,q=p,<t>,z
Therefore, use of (5.81) and separation of th e r - and ^-com ponents of the incident electric field
and surface equivalent currents, i.e.,
J s(r')
=
M s {r')
=
+
M T(f,<f>')
t
4>'
(5.82)
' + M <p{t',<j>') 4>'
(5.83)
allows to transform th e vector term (5.80) into th e following two scalar term s for p = r , (p
L p{t, <p) = -jcuMo
£
f G $ ( r , r ' ) J qtf,<l>')dS' + £
a=T,< PJ s
[ K ^ ) { v y ) M q{ t ' A ' ) d S '
(5.84)
q=r,<t>J S
where,
G ^ (r ,r ')
= f ■ G ^ )(r ,r ')- f '
= sin ip ( p ^ J sin ip' + G ^ ) cos ip'^ + cos ip [ G ^ sin ip' + g [^) cos i
G%}{ r,r')
= f - G ^ ( r , r ' ) - 4>' = sin ip G ^ j + cos ip g [^!
130
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
p
(5.85)
(5.86)
^ 5 ( r , r ')
=
0 • G (E)(r,r')- f ' = G^p! s in 0 ' + G ^ l c o s 0 '
(5.87)
=
$ ■ G ^ ( r , r') ■ 0 ' = g JJ)
(5.88)
where G ^ ? (p, q = p, </>,z) are th e com ponents of th e dyadic G reen’s functions G
coordinates, as defined in (5.81). T he expressions for the
in cylindrical
(p ,q = r, 0) can be obtained from
pq'
w ith K ^ ) , an d G ^ ) w ith K ^ ) for p, q = p,<f>,z.
(5.86)-(5.88) by su b stitu tin g
T he cylindrical com ponents of G ^
have the form of Fourier series, for p, q = p, 0, z,
and
+oo
+oo
G ^ i r . r 1) = Z
= £
n = —oo
•
n=
K ^ ( p , z ; / , z')
(5.89)
—00
where, from (5.50),
7 r00
GP£ n ( P ’ z ' >p' i z ')
=
J
,
e - j M * + z ')
— ----------------------------------------------- (5.90)
O^
p n {k p ,k oz- p , p ' )
o
02
r°
^■^n(P-> Z i P J ^ ) = _ 4^ J
o
and th e expressions for Qp^ n an d
p —j k o z { z J r z ')
^R,n^p->k0Z\P, p )
k p dkp
^
(5.91)
02
are given in A ppendix D.
Since the dyadic G reen’s functions G and K th a t are already in the form of a Fourier series, a
m ore straightforw ard approach th a n th e one followed in th e infinite m edium solution of th e integral
equations can be employed to decouple the angular m odes in th e term (5.84).
E xpanding,the equivalent? surface currents into Fourier serieSnfbr p = r, 0 , ,
%(f,0) = E
J pn( t ) e Jn<P,
M p( t,0 ) = Y
M Vn{ t W n *
71 = — OO
(5.92)
71 = — OO
the term (5.84) can be w ritte n as
+00
Y
Lp (f,0) =
C
w h G re
L Pn(t)
=
L Pn(t )ejn<P
i
(5-93)
r
/
-jujpo E
/ A%n ( t , t ' ) J qn(t') d l ' + E
/
q=T,(j>
q=T,<pJC
dl'
(5.94)
and the integral kernels in (5.94) are
4 ^ n ( i, t')
=
sin 0 (G p^ n sin 0 ' + G£?n cos 0 ') + cos 0 (G z£ n sin 0 ' + G f£ cos 0 ')
131
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.95)
At M
=
Axi^ G R,n
Rn + COsV’ ^ R,n
n
(5.96)
=
G%n s m $ + G%n co& tf
(5.97)
=
G *
(5.98)
T he expressions for the term s Bp*n (p ,q = p, <j>,z) can be obtained from (5.95)-(5.98) by replacing
GZ
w ith K RnApplying th e m ethod of m om ents as in th e unbounded m edium case, the incident tangential
electric field and th e tangential equivalent currents are approxim ated by th e sum m ations (4.37)(4.38). Then, th e use of the testing functions (4.49)-(4.50) leads to the two term s (for p = r,(p):
N q
N q
-jun 0
m)M m >
S
q=T,<f> m =
q=T,(j> m =
1
k= l,...,N p
(5.99)
1
T he coefficients ap^ n (k ,m ) are given by The
+
Ir t m
oKn(fe>m )
=
-
/
.
A R,n{t k , t ) d t
''t - r n
=
sini
sini/>k [ G ^ ( t fc; i " , f+ ) sim pm + Gp^ n (tk ; i “ , t+ ) cos ipv
+ c o sijjk [ G ^ ( t A:; t “ , t + ) s i n ^ m + G!^ ( t f c ; t ' , t + ) c o s ^ m]
(5.100)
i tm~\
— sin 'lf/icG ^ n {tk '1fm_i, fm) + COS IpkGQn (tk \ tm —1>tm)
at n ( k ’ m )
at n ( k ’ m )
=
/
A Z n ^ k ^ ' ) dt '
J t-m
=
Gt n ( f k ; *m» *m) s i n + <?££(**; f~ , t+ ) COS
=
[
A t n ^ k J ) dt'
^771—
*1
=
G Z ^ k >^m-li ^m)
(5.101)
(5.102)
(5.103)
where
132
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G % ( t 0-tA , t B )
T he coefficients b ^ n {k, to) , for p =
t ,4>
=
[ tB Gp*n (t0,t') d£
Jtj4
(5.104)
can be found from the expressions for aR,n(k ’m ) by
replacing Gp£ n w ith Gp^ n , a n d G ^ n w ith
=
[ tB K r t ( t 0, t ' ) d t '
(5.105)
Therefore, th e elem ents of th e m atrices [A^9] ^ and [B£9] ^ in (5.77) are, for p = T,<j>,
5 .3 .3
[A " ] S
=
[BJ’l S
=
- i wf*o
(5.106)
<5-107)
E x p r e s s io n o f “in c id e n t ” field o n t h e su r fa c e S
A .
A.
Figure 5.5: P lan e waves incident on a body in a half-space m edium.
As in the previous cases of two-dim ensional bodies a n d bodies of revolution in free space, the
coupled surface integral equations are solved considering an excitation in the form of a plane wave
incident on the m aterial body. C om parison of the expression for the “incident” electric field in
half-space (5.75) w ith the expression for the “incident” electric field in free-space (2.29) suggests
th a t, in the half-space problem , th e former needs to take into account th e presence of th e interface
betw een the two m edia located a t z — 0. Since the effect of th is plane interface is to produce a
reflected field in response to th e electric field incident on it, as illu strated in Figure 5.5, it m eans
133
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th a t a reflected term m ust be added to th e direct term given by expression (2.49). Hence, the to ta l
“incident” electric field on the surface S of the m aterial body will have th e form:
-3 h K - r + R qe ~ j h
E (0 = £ (0 q j
T h e expressions for th e reflection coefficients
Rh
and
R v
-r ^
^5>108j
in the horizontal and vertical polarization,
respectively, will now be briefly derived.
R e f le c te d a n d t r a n s m i t t e d fields a t t h e in te r f a c e b e tw e e n tw o lo ssy m e d ia
A_
A_
Figure 5.6: Reflection and transm ission of plane wave across a plane boundary.
As shown in Figure 5.6, let the plane z = 0 be th e bou n d ary betw een two lossy m edia w ith
perm ittiv ity e0, e* and perm eability p a , p r e s p e c t i v e l y . T he incident electric field be a ^-polarized
plane wave of the form
Ei
=
q i E \ e ~ ^ ka^
v
134
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(5.109)
and the reflected and tra n sm itte d electric field also have the form of ^-polarized plane waves:
Er
=
q r ^ e - ^ a ^ +-r
(5.110)
E*
=
^ t E tq e ~ ^ k b ^ ~ ' T'
(5.111)
T he incident, reflected a n d tra n sm itte d m agnetic fields can be found through th e M axwell’s equation
(2.3), which in the case of plane waves becomes:
jk x E
=
—jujfiE
(5.112)
T he problem of finding th e reflected an d tra n sm itte d waves associated w ith th e incident wave
(5.109) is solved by requiring the continuity of th e electric and m agnetic field across the b oundary
betw een th e two m edia. In p articular, since th e b o u n d ary surface corresponds to th e xy-plane, the
conditions to be enforced are:
z xEt
=
i x ( e * + E r)
(5.113)
z x H*
=
z x (H ‘ + H r)
(5.114)
Two cases are considered, horizontal an d vertical incident electric field. In b o th cases, Snell’s law
provides th e reflection a n d transm ission angles:
9r =
H orizontal p olarization .
9i ,
ki, cos Qt = k a cos 9{
(5.115)
In the case of ^-polarized incident electric field, th e reflected and
tra n sm itte d electric fields are also h-polarized, and th e b oundary conditions (5.113)-(5.114) become:
Ei
=
E^
—r- cos 6t
=
Ei + El
EP1
Er
- r 1 cos 9{ — —- cos 9r
Qb
sa
(5.116)
(5.117)
Sa
Using Snell’s law, equations (5.116)-(5.117) can be solved to yield
K
=
R h E{
(5.118)
El
=
(1 + R h) E l
(5.119)
135
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where th e reflection coefficient in th e h-polarization is
_ Qcosej-C'Cost),
(ft COS 6 i
V e r tic a l p o la r iz a tio n .
+
COS Qt
( a
In th e case of u-polarized incident electric field, the reflected a n d tra n s ­
m itted electric fields are also u-polarized, while th e m agnetic fields axe /i-polarized. T he bou n d ary
conditions (5.113)-(5.114) become:
E lh cos Qt
= E\
jp t
cos Qi - E rh cos 9r
jr i
=
Sb
(5.121)
jp r
+
(5.122)
S>a
Qa
Using Snell’s law, equations (5.121)-(5.122) can be solved to yield
K
=
El =
RvK
(5.123)
(l + R u ) E i
(5.124)
where the reflection coefficient in th e u-polarization is
=
C . 008 ft
Ca
-
COS Qi
a
COS 9 ,
+ (& cos Qt
S tartin g from expression (5.108) for the “incident” electric field, a n d following a derivation
sim ilar to th e one employed in Section 4.2.4, th e p-com ponents of th e “incident” electric field on
the surface S can be w ritte n as
+oo
E?{tA)
=
+oo
£
=
n=—
oo
£
[ 4 ? ( f ) + 4 ? ( f ) ] e jn*
(5-126)
7i—
—
oo
where the E p „ \ t ) have th e sam e expressions as the E p } (t ) in (4.76)-(4.77), while thep-com ponents
of the reflected term for th e n -th m ode are
E ?
= j n { Rh E ^ sin ip
(up) +
- Ry E ^ [—j sin if; cos Qi T ~ (up) + cos ip sinfli J n ( up)] } e
= j n [ j R h 4 ° T~ ( u p ) + Ry E $ cos Q, T + ( u p )] e
vz e
vz e
136
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n(^>i
(5.127)
(5.128)
w ith u and v given by (4.75), and T * defined in (4.78).
5 .3 .4
C o m p u t a t io n o f t h e s c a tte r in g c o e ffic ie n t
Let the body o f revolution be illum inated by a g-polarized pane wave of th e form
=
q i e ~-7’fc° k ' r
(5.129)
T his incident electric field generates equivalent surface currents j i 9'1 and
th a t evaluated by
solving the surface integral equations for the half-space problem w ith the m ethod of m om ents. Using
(5.35)-(5.36) an d expressions (5.61)-(5.64) for th e far-field half-space dyadic G reen’s functions, the
scattering am plitudes associated w ith these surface currents can be w ritten as the sum of two term s:
(5.130)
f pq(k s , ki ) = f g \ t a& ) + f W ( k , k )
T he first term is th e direct term corresponding to th e far-field generated by the equivalent currents
; absence of th e lower half-space, an d is given by
and Mr(9)
[sq> in
/W (k„ki)
=
+ ^ ^ ) . j W ( r O e ^ o k s+ T , d 5 / +
/?(*) 47r
j
(<£ d -
9 0 ) ■ M ^ ( r ') e i k^ t - r' d S '
q = h , v (5.131)
,
while the second te rm is th e reflected p a rt, which accounts for the presence of the lower half-space,
has th e following form
1 jkoCo ~
pO)
An
&q
-T -ffiv
fs (rv 0
e
+rh
4>
-
2
FV cos9s d p ) - j ^ ( r ' )
4> 0 - T h 6 4> - 2 f v cos 9S 4> p ) - M ^ ( r ' ) e ^ k ° ^ ' r ' dS'
,
, j k o k s • r>dS' -f
q = h,v
(5.132)
In expressions (5.131) and (5.132), k j and k j are the u nit propagation vectors of th e scattered
plane waves traveling in th e positive and negative ^-direction, respectively:
sin#,, cos fa x + s in # s sin<jf>s y ± cosds z
137
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(5.133)
It is shown in A ppendix D th a t, using (5.131) and (5.132) in (5.130), th e scattering am plitudes
can be w ritten as infinite sum m ations of angular modes,
+00
/ p?(k s,£ i)
=
£
(5.134)
whose n -th m ode term s are, for q = h , v ,
N,
/ £ ’( + + )
4”>(ks,k,.)
=
-7 ^ § 7
E
(5.135)
47T E g
w=Tj<p m = l
47T E g
w~Tt 0
E
=
t
so
J
SO
J
+
E
m =i
1
(5.136)
where the currents depend on th e direction and polarization of th e incident wave:
= M Z n(ki, qi)
q<) ,
5.4
(5.137)
Evaluation of spectral integrals
T h e integrals th a t need to be com puted when solving th e coupled surface integral equations all
have the general form
I ( p, z - , p' , z ' ) =
00
—j y / k 2 - k 2 Z
f f ( k p;p,p') ... kpdkp
o
V^ —
(5.138)
T he factor f k p, p,p') contains Bessel function term s of the type Ju {kpR ),w ith R > 0 and u being
an integer. T he w avenum ber k is generally a complex num ber:
k = kT — j k j
(5.139)
A n integral over a sem i-infinite range of th e kind (5.138) is called a Sommerfeld-like integral.
A lthough the function / has no singularities in kp and is finite for k p —> oo, perform ing the
integration of (5.138) presents some problem s due to the presence of two branch points a t kp =
± k — ±A+ =F j kj. A sim ple way to avoid this issue is to tra n sfo rm th e integral I using th e change
138
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a Im[kz}
Figure 5.7: Integration p a th of I in th e complex fc,-plane.
of variable
hp dkp
—- - -
\J k 2 - k j = k z ,
= -dkz
(5.140)
T he integral I th u s becomes
I{p,z\p',z') =
k 2 ~ k f , p , p ' ) e °kzZ dkz
(5.141)
and the integration is perform ed in the com plex fcz-plane instead of th e /cp-pla:ne. F igure 5.7 shows
the integration p a th 7 , which follows an hyperbole from th e point kz = k = fcr —j kj to infinity as
kp increases. Since th e integrand is an analytical function in kz , th e integral I can also be evaluated
along the p a th 71 U 72, which corresponds to breaking it into two parts:
I { p , z \ ( J , z ' ) = j f { y k 2 - kj; p, p') e jkzZ dkz + J f (y /k 2 - k 2z ] p, p') e jk*z dkz
71
(5.142)
72
On 7 x, kz = a — j k j , w ith a G [0, fcr ], so th a t th e first integral on the right side of (5.142) can be
w ritten as
I a = J f { \ ] k 2 - k 2] p, p') e~jkzZ dkz = e~kjZ J
71
f { ^ k 2 - {a - j k j ) 2-,p,p') e~jaZ d a
0
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.143)
while settin g k z = —j/3, w ith /3 G [kr , oo], th e integral on 72 becomes
Ib =
f { \ / k 2 - % p , p ' ) e jkzZ dkz = j
f { J k 2 + P2-,p,p')e 132 d/3
(5.144)
kj
72
W hile th e integral I a is relatively sim ple to evaluate num erically, I s can be difficult to com pute
for some values of p, p \ and Z. P articu larly for values of Z close or equal to zero, the integrand of
I b m ay exhibit an highly oscillatory behaviour and decay extrem ely slowly, rendering its num erical
integration very dem anding in term s of com putational tim e a n d m em ory requirem ents. Several
m ethods have been proposed [79, 80, 81] to overcome this problem . Here, two of those approaches,
the Tanh T ransform ation and th e W eighted-Averages m ethod, will be considered and im plem ented,
and their perform ance com pared to th a t of a sim pler num erical integration routine based on the
trapeziodal rule.
5 .4 .1
I n te g r a tio n b y T r a p e z o id a l R u le
A very sim ple way to in teg rate I b is by using th e com posite trapezoidal rule. D espite its simplicity,
it will be shown th a t th is m ethod provides fairly accurate results. Since the integral is over an
semi-infinite interval, it first needs to be tru n c a te d as follows:
Pmax
Ib ~ j
J
f { \ J k 2 + /32] p , p ' ) e ~ 0 z dp
(5.145)
kj
Choosing a suitable value of j3max can be difficult and tim e-consum ing, and represents one of the
problem s of using this procedure.
A pplication of th e com posite trapezoidal rule th en yields:
I b « j A/3 /1 + 2 X ] fp + /ivp
(5.146)
p= 2
where
fp =
(5.147)
140
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5 .4 .2
T a n h T r a n s fo r m a tio n
T he Tanh T ransform ation (T T ) was first introduced by [81] and has been further investigated by
Squire [82, 83], Evans et al. [84], and Singh et al. [85]. It is used to evaluate integrals of the general
form
b
I = J f(x)dx
(5.148)
a
where th e integrand f ( x ) m ay be singular in one or b o th of th e end points x = a, x = b.
Let g(t) be a function such th a t g(t ) —>■± oo as t —> ±oo. W ith th e su b stitu tio n
x = i (b + a) + i (b - a) ta n h [g{t)]
z
z
(5.149)
the integral (5.148) m ay be w ritten
b
I
I
a
1
\ ( b + a) + \ ( b ~ a) ta n h [g(t)] sech2[g(t)]g'(t) dx
(5.150)
W ith a suitable choice of g(t), the sech2[g(t)] factor falls off extrem ely rapidly as t —> ± o o and
controls th e behaviour of the transform ed integrand even when th e singular values of f ( x ) a t x = a
and x = b are being approached.
T he integral (5.148) th e n can be evaluated by th e m idpoint rule by setting
t = tk = k h ,
k = 0, ± 1, ± 2 , ± 3 , . . .
(5.151)
and tru n catin g the infinite sum m ation a t k = ± N . T his yields
I = 2 h ( b — a)
N
£
-2 g{kh)
+ e - 2gM p 3>(<k h )
(5-152)
where
1
i
i _ e-2g(kh)
x k = x{t k ) = - ( 6 + 0) + - ( 6 - o ) 1 + e_ 2g(^
(5.153)
T he selection of grid spacing h and num ber of term s N depends on th e particu lar function g(t)
used. A choice th a t has shown a very high level of perform ance [84], and therefore has been used
in the present d issertation work, is
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
g(t) = sinh(t) .
(5.154)
For th is choice, th e optim al grid spacing for a given N is [84];
L — ln(L —In L)
L = In (7r2iV)
(5.155)
T he algorithm th a t perform s the evaluation is iterative. It sta rts w ith an a rb itra ry initial value
of N and a corresponding value of grid spacing h
found using (5.155). T he initial value
of the integral is calculated based on (5.152)-(5.153). T he value of N is then increased, usually
logarithm ically, a n d new values of h =
and I = /W are com puted. T h e procedure continues
until the difference betw een th e results at two consecutive steps, i.e.,
j { k ) _ j(A-l)
i=
W)
(5.156)
reaches a required value.
5 .4 .3
W e ig h te d -A v e r a g e s M e th o d
T he W eighted-Averages (WA) m ethod is an extrap o latio n m ethod for accelerating th e convergence
of Sommerfeld-like integrals. A dditional general inform ation ab o u t extrapolation m ethods can be
found in [79], while th e WA procedure is discussed in d etail in [80]. W hen using ex trap o latio n
m ethods, the integrals are evaluated as sums of a series of p a rtia l integrals over finite subintervals
whose speed of convergence is increased th rough an e x trap o latio n procedure.
Let
OO
be the integral th a t needs to be evaluated, and its integrand f(/3) have, for (3 —> 00, an asym ptotic
form
(5.158)
142
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where p(P) is a 2g-periodic function such th a t
p(P) = ~P(P + q)
(5.159)
T he integral I can be evaluated using a n “integration-then-sum m ation” procedure, in which I is
expressed as an infinite sum of p a rtial integrals over finite subintervals. Defining the m -th partial
sum as
m
Sm = £
[
f ( P) d p ,
(5.160)
<=0 A -i
th e integral I can be regarded as equivalent to the lim it
I =
lim Sm ,
(5.161)
m —>oo
a n d estim ated by a p a rtial sum S m , w ith M chosen large enough so th a t th e rem ainder
-
'm —
OO
J
f(P)dP,
(5.162)
is negligible. However, usually th e sequence { S m} approaches I slowly because r m does not decay
rapidly w ith increasing m. A solution to th is problem is to transform {5m } into another sequence
{S ^ } th a t converges rapidly to I. Such transform ation is said to “accelerate”the convergence and
is referred to as an extrap o latio n m ethod since the underlying principle is
to ob tain an improved
estim ate from th e sequence of approxim ated values.
In particular, th e W eighted-Averages m ethod employs weighted m eans of consecutive p a rtial
sums, w ith weights selected based on rem ainder estim ates. T his tran sfo rm atio n is linear and defined
by
5” =
S m "h Vm Sm+l
i + ^
/r i eo\
( 5 ' 1 6 3 )
where
rim =
r m+ 1
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.164)
T he tran sfo rm atio n is applied repeatedly, leading to a recursive scheme where th e m -th p a rtia l
(k)
sum s at the A;-th step Sm are used to find the p a rtial sums a t th e (k + l)- th step,
s S + 1 , = S m +i <«,(*)
b n Sq(k)
_
(5 165)
1 +Vm
(k)
and rjm has th e following expression:
V(m } = e9C
+
(5-166)
T he algorithm sta rts by choosing a num ber M of subintervals and com puting th e integrals over
them . Then, th e in itial sequence {Sm'1} is set equal to th e sequence {5m} of p a rtial sum s (5.163) of
th e previously com puted m = 1,2, • • •, M sub-integrals. T h e iterative com putation of the sequences
{Sm}} for k = 1,2, ■• •, M using (5.165) and (5.166) follows. Every sequence { S ^ } has one less
term th a n the sequence { S m ~ 1'>}. Therefore, a t th e M -th step the sequence {Sm}} only has one
term , which represents th e best estim ate for the integral (5.157), i.e., I ~ S ^ f \
5 .4 .4
V a lid a tio n o f I n te g r a tio n M e th o d s
A simple case is considered in this section in order to validate th e im plem entation of the Tanh
T ransform ation an d W eighted-Averages m ethods and to investigate th eir performance. T he results
are also com pared w ith those obtained by the com posite trapezoidal rule.
T he following identity [71] is used for validation
e —j k ( R 2 + Z 2)
VR2
/
+ Z2
z
e ~ jj ki v oozZ
Jo(kpR) — —------- k p d k p ,
R > 0,
Z> 0
(5.167)
where
k — kT — j k j ,
k oz = ^ k 2 — k 2p
(5.168)
T he integral on th e right side is a Sommerfeld-like integral th a t can be evaluated over the two p ath s
7 i and 72 as in (5.142). Hence, the identity (5.167) can be w ritten as
J
r
Ib =
1—
e ~ j k ( R 2 + z 2)
M R ^ k * + f 3 i ) e - P z dS ■
Ia
kj
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.169)
where
I A = - j e ~ ki z
j
J0 R y / k P - i a - j k j ) *
e i aZ da
(5.170)
can be easily num erically integrated, while I b is the slowly converging integral th a t will be evaluated
using th e m ethods discussed earlier.
In th e integration w ith the com posite trapezoidal rule, (3max is chosen as any value where
3maxZ < \ M k R ) \
1000
J o { R ^ k 2 + (32m ax)
(5.171)
Using th e asym ptotic form of the Bessel function J q, this inequality corresponds to
-a™ z < \ M m
(5.172)
1000
7T p /3 n
Once (3-max is found, a num ber N p of po in ts is selected such th a t
M
> 10 p
(3max "F kj
(5.173)
7T
which assures th a t there are a t least 20 subintervals of integration for each period of oscillation
Tp
2tt
—
(5.174)
,
of the Bessel function.
In order to use the Tanh T ransform ation, th e following change of variable is made:
x = e P,
d(3 = —
dx
(5.175)
T his leads to write I b as
e
IB =
J
j
Jq R y J k 2 + {ln x )2 x z 1 dx
(5.176)
which is in the form (5.148), w ith a = 0, b = e hi , and
f{x) -
Jo R ^ J k 2 + (l n z ) 2
.z
-
1
145
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(5.177)
Finally, th e W eighted-Averages m ethod has been applied to th e evaluation of the integral I s Its integrand has an asym ptotic form
m
~
\ f ^ p cos( R l i - i ) e
2/5 •
(5-178)
therefore com parison w ith expressions (5.158) and (5.159) yields:
C = Z,
« = I »
a = y
(5.179)
I b has been com puted for several values of Z and R, using all th e integration m ethods pre­
viously discussed. A value k = 20 — 3 j has been chosen, which approxim ately corresponds to
the wavenum ber for a n homogeneous m edium w ith relative p e rm ittiv ity ey = 10 — 3 j . T he case
R = 0 has not been considered because I b reduces to a very sim ple form th a t can be integrated
analytically. In addition, for Z < 0.001 the in teg ratio n has not been perform ed by the com posite
trapezoidal rule because it required too m any points.
z
R
Ib
1.000
2.00
- 0.718 + 0.737;
tT T
*B
I'C
BT
- 0 .7 1 8 + 0.737;
—0.720 + 0.740/
0.010 + 0.01%'
<
1 0 -2
<
1 0~ 2
0.40
0.010 + 0.019/
0.010 + 0.019/
1.000
0.08
0.022 + 0.007;
0.022 + 0.007;
0.022 + 0.007;
0.022 + 0.007;
0.100
2.00
-3 6 .4 2 4 + 6.905/
- 36.424 + 6.905/
-3 6 .4 0 1 + 6.897;
-3 6 .4 1 6 + 6.914;
0.100
0.40
- 0 .2 0 7 + 1.546/
- 0 .2 0 7 + 1 .5 4 6 /
- 0.208 + 1 .5 4 6 /
- 0.206 + 1 .5 4 7 ;
< 10- 2
<
0.100
0.08
1.908 + 0.833;
1.908 + 0.833/
1.905 + 0.834;
1.915 + 0.836/
0.010
2.00
- 5 3 .3 7 8 + 3.211;
- 5 3 .3 7 8 + 3.211;
- 5 3 .3 5 8 + 3.204;
-5 3 .5 3 8 + 2.793;
0.010
0.40
- 1 .2 6 5 + 2.528 j
- 1 .2 6 5 + 2.528/
- 1 .2 5 0 + 2.527;
0.010
0.08
- 0 .4 3 6 + 2.346/
- 0.436 + 2.346/
- 0 .5 5 9 + 2.342/
0.001
2.00
- 5 5 .3 9 8 + 2.570/
- 55.398 + 2.570/
- 5 5 .3 6 1 + 2.543/
-
<
0.001
0.40
—1.447 + 2.635/
— 1.447 + 2.635/
- 1 .3 5 8 + 2.621;
-
<
<N
1
O
i—I
1.000
0.010 + 0.019/
0.01
< 10-2
«TT
ecr
Tw a
TTt
Tct
< 10~2
0.27
1.0
1.1
0.2
1 0 -2
0.24
0.5
0.7
0.2
0.5
0.6
0.1
<
N
1
O
t-H
V
fW A
- 0.718 + 0.738/
0.28
0.07
0.03
1.4
3.7
0.8
0.04
0.13
0.7
1.3
0.2
0.11
0.40
0.7
1.1
0.2
0.04
0.84
2.1
18.1
0.5
10~2
- 1 .2 1 4 + 2.527;
<
1 0 -2
0.53
1.80
1.6
18.0
0.5
—0.421 + 2.34%
<
0.62
1.3
21.2
0.6
5.17
10“ 2
0.08
-
3.8
22.7
-
-
3.4
34.6
-
3.01
0.001
0.08
- 1 .5 9 5 + 2.588 j
- 1.595 + 2.588/
- 2 .5 4 7 + 2.558/
-
<
10~2
31.33
-
4.0
34.7
0.000
2.00
—55.626 + 2.495/
- 5 5 .6 2 6 + 2.495/
- 55.579 + 2.460/
-
< 1 0 -2
0.10
-
7.5
26.8
-
0.000
0.40
- 1 .4 6 9 + 2.647;
- 1.469 + 2.647;
— 1.355 + 2.629/
-
< 1 0 -2
3.80
-
7.7
40.0
-
0.000
0.08
—1.740 + 2.616/
- 1 .7 4 0 + 2.616/
- 2 .9 2 4 + 2.577;
-
<
10~2
37.72
-
14.3
44.9
-
o
(M
1
1
O
H
<
Table 5.1: Results of the evaluation of I b using the Weighted-Averages (WA) method, Tanh Transformation
(TT), and Composite Trapezoidal (CT) integration, for various values of Z, R, and k = 18 —6j. I b is the
exact value, e is the absolute accuracy in %, and T is the computational time in sec.
T he results are sum m arized in Table 5.1.
For each m ethod, th e estim ated value I s of the
146
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
integral is given, together w ith the absolute accuracy
e =
Ib ~ Ib
Ib
(5.180)
in %, and th eir com putational tim e T in sec. T he exact value of I b calculated from the right side
of eq u ation (5.169).
T h e following considerations can be m ade. T he W eighted-Averages m ethod shows the best
accuracy, followed by the Tanh T ransform ation and the Com posite Trapezoidal rule integration.
T he C om posite Trapezoidal rule integration is faster but for sm all values of Z cannot be used due to
the large num ber of points needed. T he Tanh T ransform ation perform ance degrades considerably
for sm all values of R and Z , w ith th e error and th e com putational tim e b o th increasing. Figure
5.8 b e tte r illustrates this issue. T he plot shows the absolute error as a function of the num ber
of integration points N , for Z = 0 and four different values of R . It is easy to notice th a t as
R decreases, th e error tends to reach a constant value th a t does not decrease even w hen N is
increased. Therefore, the Tanh T ransform ation m ethod does not seem to be of any p articular value
for the num erical evaluation of th e integrals here considered. Indeed the algorithm is outperform ed
by th e Com posite Trapezoidal rule in term s of speed, and - for sm all values of R and Z - by the
W eighted-Averages in term s of accuracy.
5.5
Validation
T his section describes a few results of th e application of the num erical m ethod derived earlier in
this chapter. The goal is to validate the procedure and prove th a t it provides correct results. As
done in the previous chapters, first th e case of P E C bodies is considered, followed by th e case of
homogeneous dielectric scatterers.
147
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VP
'
'
k-
o
i_
CD
D
■4<
—
•
a
ocn
.a
cc
IQ-
50
100
200
500
1000
2000
N
Figure 5.8: A bsolute error [%} for th e Tanh T ransform ation m ethod vs. num ber of integration
points N .
5 .5 .1
P e r f e c t ly c o n d u c tin g b o d y o f r e v o lu tio n
In the case of perfectly conducting bodies of revolution, there is only one surface integral equation
to solve, i.e., the one for th e exterior problem . In order to test th e correctness of th e solution, the
sim pler situation of a perfectly conducting half-space is considered first, and the surface integral
equations are solved for a P E C open-ended cylinder located on th e half-space interface, as shown
on the left of Figure 5.9. T his configuration is equivalent to th e one shown on the right in the same
figure. Here, the lower half-space is rem oved and its effect is tak en into account by introducing the
image of the P E C open-ended cylinder for z < 0. Therefore, th e equivalent configuration consists of
an PE C open-ended cylinder twice as long, centered at th e origin, and excited by th e com bination
of the incident field and its ” im age” . T h e currents on th e surface of the open-ended cylinder in the
half-space problem m ust be th e sam e as those on th e surface of th e equivalent open-ended cylinder
148
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PEC
-L
Figure 5.9: S cattering from a vertical P E C open-ended cylinder on a P E C half-space (left), and
equivalent problem (right) in free space.
for z > 0.
T his can be seen in Figure 5.10, left side, where th e currents on an open-ended cylinder of
length L — 0.5A and radius a = 0.1A are p lo tted as a function of z / A, together w ith the currents
on th e open-ended cylinder of length L = A and radius a = 0.1A in the equivalent configuration.
T h e incident angle is Qi = 60°. T he two curves m atch very well, except for points very close to
th e interface.
The scatterin g coefficients of th e two configurations should also be the sam e for
scattering angles in th e u p p e r half-space. O n the right side of Figure 5.10 the norm alized b istatic
scattering coefficient is p lo tted for scattering angles Qs betw een 0° and 90°. Also in this case, there
is good agreement betw een th e half-space an d th e equivalent im age solutions.
To validate the results for P E C bodies of revolution over a dielectric half-space, a case studied
by Abdelmageed an d M ichalski [86] is considered. T h e body is a perfectly conducting cylinder of
length L — A and k^a = 1.0 located at a distance d = 0.2A above a homogeneous, lossy half-space
149
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20
— ■
— t h a lf- s p a c e
4.5
- -
r
- x e q u iv a le n t
— * — 4 h a lf- s p a c e
- * - ( ) > e q u iv a le n t
3.5
i 25
/0
-1 0
-15,
-20
-25
0.5
-30,
-0 .5
—1*—HH half-space
HH equivalent
- e - VV half-space
VV equivalent
-0 .2 5
0.25
0.5
20
30
40
50
6(
scattering angle 0g [deg]
70
90
Figure 5.10: C om parison betw een P E C open-ended cylinder of length L = 0.5A and radius a = 0.1 A
on a P E C half-space, an d equivalent image problem , for incident angle 6t = 60°. Left: surface
electric currents vs. z /X . Right: norm alized b ista tic scattering coefficient vs. 6S.
w ith relative dielectric constant 16—jlQ . T he incident wave is coming from an angle 6t = 0°. Figure
5.11 shows the com parison betw een th e present half-space m ethod (BOR) and th e A bdelm ageed
and M ichalski results (AM ), which axe in good agreem ent w ith each other.
5 .5 .2
D ie le c t r ic b o d y o f r e v o lu tio n
In order th e test the validity of the half-space procedure, th e case of a dielectric cylinder on a P E C
half-space is considered. T h e geom etry of the problem , illu strated on the left of Figure 5.12, is
sim ilar to th e one for th e P E C open-ended cylinder on a P E C half-space, except th a t a homogeneous,
lossy cylinder is now considered. Therefore, the sam e considerations m ade for the P E C open-ended
cylinder case apply. T he cylinder has length L = 0.6A, radius a — 0.1A and a relative dielectric
constant 4 —j , and its b o tto m end is on th e interface. C om parisons between equivalent surface
currents and between b ista tic scattering coefficients are given in Figures 5.13-5.14, and show a good
agreem ent between half-space num erical solution a n d equivalent configuration num erical solution.
150
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2.5
X
0.5
-0-0- e -9 -g rg ^ g r
0.2
0.4
0.6
VX
0.8
Figure 5.11: Surface electric currents vs. t/X on a perfectly conducting cylinder of length L — A
and koa = 1.0 located a t a distance d = 0.2A above a hom ogeneous, lossy half-space w ith relative
dielectric constant 16 —j l § . Incident angle is
= 0°.
5.6
A pplication to scattering from vegetation
T he m ethod derived earlier in this chapter can be used to m odel vegetation more realistically. In
particular, in this section th e problem of a vertical tree tru n k over a fiat ground is considered. A
comm on approach is to m odel the tru n k as a finite-length lossy dielectric cylinder of circular cross
section, and the ground as a flat interface separating th e free space from a lossy dielectric m edium
representing the soil.
T his problem can be easily studied by solving th e surface integral equations for the half-space
problem w ith the b o d y of revolution being the dielectric cylinder. T his num erical solution can be
used to evaluate th e accuracy of approxim ate analytical m ethods applied to the same problem .
A n analytical analy tical approxim ation is considered here, bases upon two assum ptions, i.e.,
th a t
1. the length of th e cylinder is large com pared to its radius;
151
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PEC
-L
Figure 5.12: Scattering from a vertical dielectric cylinder over a P E C half-space (left), and equiv­
alent problem (right) in free space.
2. th e cylinder and th e region on th e ground w here induced currents exist are in th e radiation
zone of each other.
T he first assum ption allows th e use of the finite-length cylinder m odel discussed in C hapter 4.
In addition, assum ing th a t the cylinder and th e th e region on th e ground where induced currents
exist are in the far-field region of each other, th e effect of th e ground plane interface is taken into
account only by considering the contributions from the m irror image. U nder this assum ption, the
incident field illum inating the cylinder is a plane wave of th e form
E « = q
+
(5.181)
where the reflection coefficients R q have been derived in Section 5.3.3. T h e to ta l scattering ampli152
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2.5
400
1 half-space
equivalent surface electric current
- e-
half-space
equivalent
i)i half-space
d equivalent
t
t equivalent
- e-
350
$ half-space
—>*— (j>equivalent
—
tz
x
300
S 250
» 200
" 150
•5 100
0.5
0.2
0.4
0.3
0.5
0.6
0.7
0.2
0.3
0.4
0.5
0.7
t/X
Figure 5.13: C om parison betw een dielectric cylinder of length L = 0.6A, radius a = 0.1A and a
relative dielectric constant 4 —j on a P E C half-space, and equivalent image problem , for incident
angle 0* = 45°. E quivalent surface electric currents vs. z /X . R ight: norm alized bistatic scattering
coefficient vs. 9S.
/(it a2) [dBm]
half-space
— equivalent
E
C
TQ
J
b
O'
-4
-4
half-space
equivalent
-10
-1 0
scattering angle 9 [deg]
10
40
,50 rJ 6(
scattering angle 8 [deg]
Figure 5.14: C om parison betw een dielectric cylinder of len g th L = 0.6A, radius a — 0.1A and a
relative dielectric constant 4 — j on a P E C half-space, and equivalent image problem , for incident
angle 0, = 45°. N orm alized b ista tic scattering coefficient vs. 9S.
153
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e,
direct
direct-reflected
A -
A_
e,
reflected-direct
reflected-reflected
tudes fpq of the configuration are the sum of four com ponents:
= I > « (£ ,+ ,£ - )
(5.182)
1= 1
As illustrated in Figure 5.6, these com ponents are:
1. direct scattering from th e cylinder,
e£ W > : k O
= f $ l){k + ,k O
(5.183)
2. scattering from th e cylinder, followed by reflection from th e ground
e g )(k + k -) = R P(9S) 4 f ) ( k + , k + ) e - ^
154
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.184)
3. reflection from th e ground, followed by scattering from th e cylinder
eg ( k + , k n
= / j f ) ( k s- , k r ) R q(ei ) e ~ ^
(5.185)
4. reflection from th e ground, followed by scattering from th e cylinder, and by reflection from
the ground
eg ) ( k + k - )
=
R pids)
/ i f }(ks- , k + )
(5.186)
Rqifii)
In expressions (5.183)-(5.186), fpqjl> are th e scattering am plitudes of th e cylinder in free space.
T h e u nit propagation vectors of the incident and scattered plane wave are:
k^
= —sin 9i cos (pi x —sin 9i sin (pi y ± cosOi i
(5.187)
k^
= sin 0 s cos ^ i c + sin 0 s sin(ps y ± cosOs z
(5.188)
T he additional phase shifts i/'i and ips have been introduced to account for th e difference in
length between the direct p a th and th e p a th reflected from th e ground.
T hey depend on the
incidence and scattering angles 6i and 0S, an d are given by
ipi
=
koL cosdi
(5.189)
ips
=
k o L c o s 9 s
(5.190)
T h e bistatic scattering coefficient can be found from th e scattering am plitudes as follows:
f W k ^ . k " ) = 4 7r|/pg( k + , k - ) |2
(5.191)
R e s u lts a n d c o m p a r is o n s
In this subsection, results of th e application of th e previously described approxim ation on th ree
cylinders of different dim ensions will be given, a n d com pared to the num erical
solutions.
An
interm ediate approach will also be considered, in which th e free space scattering am plitudes fpqy^
of the cylinders are determ in ed num erically by th e M O M -B O R approach rath er th a n by analytical
approxim ation.
155
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T hree hom ogeneous, lossy dielectric cylinders are considered. All th e cylinders have th e same
relative dielectric constant 4 — j, and the relative dielectric constant of th e ground is 10 — j 5 for
all cases. These values are th e sam e used by Lin and Sarabandi in [87]. Two values have been
chosen for the cylinder length, i.e., L = 2.5A and L — 4.0A, and two values for the radius, i.e.,
a = 0.025A and a = 0.25A. T he results are illustrated in Figures 5.15-5.26, for two incident angles,
i.e., 9i = 30° and 9t = 60°. As m entioned above, three separate solutions are plotted. In th e legend
and in the following, th ey are referred to as:
M O M half-space indicates the result of the application of th e half-space m ethod of m om ents
algorithm , considered as the exact reference solution;
PO - f approx is th e m odel described by (5.181)-(5.186), w ith f p q ^ given by the approxim ate
analytical m odel outlined in Section 4.4.1;
M O M + approx is th e m odel described by (5.181)-(5.186), w ith fp q ^ determ ined num erically
by th e m ethod of m om ents.
In particular, the results in Figures 5.15-5.18 are for th e th in n e r (a = 0.025A), b u t shorter
(L = 2.5A) cylinder.
Especially when considering the m ain scatterin g lobe, the PO + approx
solution is in good agreem ent w ith th e M OM half-space results in th e /i/i-polarization, b u t there
are some discrepancies in th e uu-polarization. T hese differences alm ost disappear in the M OM +
approx solution.
If the cylinder length is increased from L = 2.5A to L = 4.0A, as in Figures 5.19-5.22, the
agreement of the P O + approx w ith th e MOM half-space results substantially improves for hhpolarization, even in th e side lobes, while for w -p o la riz a tio n degrades a t sm aller incident angles,
such as 9i = 30°.
If the length of th e cylinder is kept a t L = 2.5A an d its radius increased by a factor 10 to
a = 0.25A, there is no m ore agreem ent betw een th e P O -t- approx a n d th e M OM half-space solutions,
156
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to
S
e
£ -20
-3 0
MOM half-space
o PO + approx
- « - MOM + approx
-4 0
scattering angle 9 [deg]
70
Figure 5.15: N orm alized scattering coefficient cr^h vs. 9S. C om parison betw een analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A.
Relative dielectric constant is 4 —j for the cylinder, and 10 —jb for the ground. Incident angle is
9i = 30°.
I
-1 0
-a
CVJ
<s
S
o'g
-20
-3 0
MOM half-space
o PO + approx
- * - MOM + approx
-4 0
10
40
50
60
scattering angle 9 [deg]
70
Figure 5.16: Norm alized scattering coefficient a vv vs. 6S. C om parison betw een analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A.
Relative dielectric constant is 4 —j for th e cylinder, a n d 10 — j5 for the ground. Incident angle is
9i = 30°.
157
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I -10'
;o
to
-3 0
— MOM half-space
— PO + approx
—MOM + approx
-4 0
SO
40
50
60
scattering angle 0s [deg]
Figure 5.17: N orm alized scattering coefficient c r^ vs. 6S. C om parison betw een analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a — 0.025A.
Relative dielectric constant is 4 —j for the cylinder, and 10 —j 5 for th e ground. Incident angle is
Qi = 60°.
MOM half-space
o— PO + approx
- « - MOM + approx
& -10
73
S -20
-3 0
-4 0
10
40
50
6<
scattering angle 0 [deg]
70
Figure 5.18: Norm alized scattering coefficient avv vs. 0S. C om parison betw een analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.025A.
Relative dielectric constant is 4 — j for th e cylinder, and 10 —j5 for th e ground. Incident angle is
Qi = 60°.
158
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MOM half-space
—e— PO + approx
- « - MOM + approx
CM
o
-20
-3 0
-4 0
10
40
50
6<
scattering angle 9s [deg]
70
Figure 5.19: N orm alized scattering coefficient a^h vs. 9S. C om parison between analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A.
R elative dielectric constant is 4 —j for the cylinder, and 10 —j 5 for th e ground. Incident angle is
di = 30°.
— MOM half-space
— PO + approx
—MOM + approx
E
co
■o
O'
-2 0
-3 0
0
10
20
30
40
50
60
scattering angle 9 [deg]
70
80
90
Figure 5.20: N orm alized scattering coefficient a vv vs. 9S. C om parison betw een analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A.
Relative dielectric constant is 4 —j for the cylinder, an d 10 — y*5 for the ground. Incident angle is
Qi = 30°.
159
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MOM half-space
o PO + approx
- * - MOM ■+■approx
E
m
-o
"co -10
e
b
-20
-30.
0
10
20
40
50
30
60
scattering angle 9 [deg]
70
80
90
Figure 5.21: N orm alized scattering coefficient
v s . 9s. C om parison betw een analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A.
R elative dielectric constant is 4 — j for the cylinder, and 10 —j 5 for th e ground. Incident angle is
9i = 60°.
even in th e m ain lobe, as can be seen in Figures 5.23-5.26. In this case, the results of th e P O -tapprox become acceptable only for hh-polarization a t higher incident angles such as 9{ = 60° in
Figure 5.25.
T he fact th a t, in m ost cases here considered, th e M OM + approx shows a good agreem ent w ith
th e M OM half-space solution suggests th a t th e sim plification based on expressions (5.181)-(5.186)
is valid, and th a t th e differences are likely due to th e P O analytical approxim ation of the scattering
am plitudes f p q l\
5.7
Conclusions
A num erical procedure has been developed to tre a t th e scattering from P E C and homogeneous
dielectric objects located in free space above a P E C or hom ogeneous dielectric semi-infinite m edium .
T he procedure has been validated, and subsequently applied to investigate a simplified analytical
m odel for the scattering from a vertical dielectric cylinder over a dielectric ground.
160
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
— MOM half-space
— PO + approx
—MOM + approx
E
m
-o
c\j
-10
O'
-20
-3 0
0
10
20
30
40
50
60
scattering angle 9 [deg]
70
80
SO
Figure 5.22: N orm alized scattering coefficient avv vs. Qs. C om parison betw een analytical approxi­
m ation and num erical solution for a dielectric cylinder of length L = 4.0A and radius a = 0.025A.
Relative dielectric constant is 4 —j for th e cylinder, and 10 —jb for th e ground. Incident angle is
Qi = 60°.
T he com parison betw een th e results of such analytical approxim ation and those of the num erical
m ethod th a t m odels th e half-space exactly indicates th a t the sim plified analytical model can be
successfully used to take into account th e effect of the ground in th e scattering from the elem ents
of a vegetation layer.
161
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E
m
T
J
K
-10
-20
— MOM half-space
— PO + approx
—MOM + approx
-3 0
0
10
20
30
60
40
50
scattering angle 9 [deg]
70
80
90
Figure 5.23: Norm alized scattering coefficient ahh, vs. 9a. C om parison between analytical approx­
im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A.
Relative dielectric constant is 4 — j for th e cylinder, and 10 —j 5 for the ground. Incident angle is
0i = 30°.
E
m
-a
-20
MOM half-space
—e— PO + approx
- ■» - MOM + approx
-3 0
0
10
20
30
40
50
60
scattering angle 9 [deg]
70
80
90
Figure 5.24: Norm alized scattering coefficient a vv vs. 9S. C om parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A.
Relative dielectric constant is 4 —j for the cylinder, an d 10 —j 5 for the ground. Incident angle is
Qi = 30°.
162
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
E
m
T3
-20
— MOM half-space
— PO + approx
—MOM + approx
-3 0
0
10
20
40
30
60
.50
scattering angle 80 [deg]
70
80
90
Figure 5.25: Norm alized scattering coefficient ahh vs. 9S. C om parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A.
R elative dielectric constant is 4 - j for th e cylinder, and 10 —j 5 for th e ground. Incident angle is
Oi = 60°.
MOM half-space
—e— PO + approx
- ■» - MOM + approx
E
co
-o
“a -1 0
•£.
<
e
-20
-3 0
0
10
20
30
40
50
60
scattering angle 9 [deg]
70
80
90
Figure 5.26: Norm alized scattering coefficient a vv vs. Qs. C om parison betw een analytical approx­
im ation and num erical solution for a dielectric cylinder of length L = 2.5A and radius a = 0.25A.
Relative dielectric constant is 4 —j for th e cylinder, and 10 — j5 for th e ground. Incident angle is
9i = 60°.
163
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Chapter 6
C onclusions
In this dissertation, the problem of solving surface integral equations for electrom agnetic scattering
applications has been studied. T his investigation has focused on th e scatterin g properties of objects
used 'to represent individual constituents of vegetation layer m odels.
T he approach considered
for solving the electrom agnetic scattering problem is num erical an d based on a surface integral
equation. N um erical procedures have been developed and applied to stu d y bodies of increasing
complexity. Once these num erical m ethods have been validated, th ey have been used to investigate
th e accuracy of approxim ate analytical solutions.
S tartin g for M axwell’s equation, a pair of tan gential electric field integral equations, valid on the
boundary surface of th e body of interest, is derived for the case of bodies located in an unbounded
medium.
Following this in tro d u ctio n of th e tangential electric field integral equations, the m ethod of mo­
m ents is used to solve these integral equations in th e case of tw o-dim ensional bodies. R esults and
com parison w ith th e exact theoretical expressions are given for th e case of circular perfectly con­
ducting and lossy hom ogeneous dielectric cylinders. T he good agreem ent w ith the theory confirms
the validity of the surface electric field integral equations form ulation.
Next, th e case of bodies w ith ro tatio n al sym m etry is considered. Such representation is very
useful because m any vegetation elem ents exhibit axial sym m etry, a t least in an ideal case, an d can
164
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be therefore be m odeled as bodies of revolution. A num erical solution to the tan gential electric
field integral equations for this class of objects is presented. T his solution is based on the m eth o d of
m om ents an d exploits th e ro tatio n al sym m etry to reduce the surface integral equations to integral
equations along th e generating arc of the body of revolution.
Results are given for perfectly
conducting and lossy homogeneous dielectric objects of several different shapes. C om parisons are
m ade w ith the literatu re and, w hen available, w ith the exact solutions, and show th e validity of
th e algorithm .
T he body of revolution num erical procedure is th en used to stu d y th e accuracy of two analytical
models for cylindrical structures, such as tree tru n k s, based on a physical optics approxim ation.
Plane wave scattering by dielectric cylinders of finite length and circular cross section is exam ined.
T his finite cylinder m odel is based on th e assum ption th a t th e currents inside the cylinder are
the sam e as if th e cylinder were infinite. C ylinders w ith a radius th a t varies linearly along the
cylinder length - hereafter referred to as tap ered cylinders - are also considered. T apered cylinders
are m odelled by a num ber of coaxial finite cylinders stacked on top of each other. B oth these types
of objects can be used to m odel tree tru n k s, branches, and stalks.
T he results show a good agreem ent of th e analytical approxim ation w ith the num erical solution,
here being considered as the exact reference, for rem ote sensing applications where th e low level
side-lobe scattering is neglected. For b o th constant-radius and tap ered cylindrical stru ctu res, a
good agreem ent w ith th e num erical solution is found in th e region of th e m ain scattering lobe,
which is the one of interest w hen considering com plex m edia such as vegetation canopies. However,
the accuracy of th e approxim ate solutions decreases as th e angle of the incident wave approaches
the end-on angle, and is generally b e tte r for hfi-polarization th a n for un-polarization. W hile the
physical optics solution for the finite cylinder works very well and is readily and safely usable in
vegetation models, th e tap ered cylinder approxim ation requires m ore a tten tio n in the choice of
the num ber N s of cylinders, in order to reduce th e error due to th e displacem ent of the specular
165
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scattering lobe.
T he results also prove th a t when th e tap e r is significant, the single cylinder
approxim ation is not adequate for a tap e red cylinder.
A brief discussion of th e com putational tim es for th e two approaches is also provided, which
shows th a t the com putational cost of th e num erical algorithm increases quadratically w ith the
cylinder length, while it rem ains alm ost constant for th e approxim ate solutions. T he im p o rtan t
conclusion of th is stu d y is therefore th e following. T he analytical m odels should be preferred to
th e m ethod of m om ents approach because they are accurate enough an d have shorter com putation
tim e, while the num erical com putation is too heavy for practical purposes but represents a powerful
tool to validate analytical models.
Finally, the num erical m ethod for tre a tin g axisym m etric bodies in an unbounded m edium has
been m odified to stu d y th e sam e type of bodies located above a sem i-infinite m edium . T his con­
figuration is referred to as th e half-space problem . T he tan g en tial electric field integral equations
derived for an object in an unbounded m edium are reform ulated to take into account th e presence
of the second m edium .
T his derivation implies th e definition of a n equivalent infinite m edium
whose properties are described by half-space G reen’s functions. T he num erical approach is based
on a series expansion of th e dyadic G reen’s functions in term s of cylindrical wave functions. Such
representation of the dyadic G reen’s functions allows a sim plification of tangential electric field
integral equations for th e half-space problem in th e case of bodies of revolution. However, this
approach requires the evaluation of slowly converging Som merfeld-like integrals, and section is ded­
icated to the analysis a n d discussion of some m ethods to estim ate these kinds of integral correctly
and efficiently. Following th e description of the num erical m ethod to solve the surface equations
for the half-space problem , some results are given and discussed.
As an exam ple of application of th e half-space body of revolution algorithm , the case of sc at­
tering from a vertical tree tru n k located on a flat ground is studied. A simplified analytical m odel
is introduced, which uses a physical optics approxim ation an d also neglects the near-field interac166
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
tions betw een the tree tru n k located on the ground. The num erical procedure is applied to the
sam e configuration, and com parisons are m ade w ith the simplified m odel. T he results analytical
approach are found to m atch those given by the num erical m ethod around the m ain scattering
direction. M ost of th e differences betw een analytical and num erical solution can be a ttrib u te d to
th e use of the physical optics approxim ation, while the near-field interactions are very small.
In sum m ary, the end result of th is dissertation has been th e analysis and development of num er­
ical procedures th a t can be applied to stu d y the electrom agnetic scatterin g from individual vegeta­
tion elem ents. W hile th eir use for large bodies can be in some cases lim ited by com putational costs,
they nevertheless co n stitu te a powerful tool to evaluate approxim ate analytical solutions th a t are
faster and more com m only used in electrom agnetic m odeling of vegetation. T hey axe also essential
if one wants to study situations in w hich approxim ate analytical m odels lose their validity, such as
in th e near-field. Some exam ples of application of these num erical m ethods have been provided in
this dissertation, especially for electrom agnetic scattering from cylindrical structures such as tree
tru n k s and branches, a n d plant stalks. In particular, th e ap p lication of th e half-space num erical
m ethod has shown prom ising results, b u t more work is needed b o th to stu d y its full poten tial and
to improve its com putational efficiency.
167
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B ibliography
[1] A. Fung, M icrowave Scattering and E m ission Models and T heir Applications. IEEE, 1994.
[2] F. Ulaby, K. Sarabandi, K. M cdonald, and M. Dobson, “M ichigan microwave canopy scattering
m odel,” International Journal o f Rem ote Sensing, vol. 11, pp. 1223-1253, 1990.
[3] N. E ngheta and C. Elachi, “T he fast m ultipole m ethod (fmm) for electrom agnetic scattering
problem s,” IE E E Transactions on Geoscience and Rem ote Sensing, vol. 20, no. 2, pp. 212-215,
1982.
[4] R. Lang and J. Sidhu, “Electrom agnetic backscattering from a layer of vegetation: a discrete
approach,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 21, pp. 62-71, 1983.
[5] F. Ulaby, A. Tavakoli, and T. Senior, “Microwave propagation constant for a vegetation canopy
w ith vertical stalk s,” IE E E Transactions on Geoscience and Rem ote Sensing, vol. 25, pp. 550557, 1987.
[6] A. Tavakoli, K. Sarabandi, and F. Ulaby, “H orizontal propagation through periodic vegetation
canopies,” IE E E Transactions on A ntennas and Propagation, vol. 39, no. 7, pp. 1014-1023,
1991.
[7] M. K aram , A. Fung, R. Lang, and N. C hauhan, “A microwave scattering model for layered
vegetation,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 30, pp. 767-784,
1992.
168
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[8] R. Lang, S. C hauhan, K. R anson, and O. Kilik, “M odeling p-band sar retu rn s from a red pine
s ta n d ,” Rem ote Sensing o f the Environm ent, vol. 47, pp. 132-141, 1994.
[9] S. C hauhan, R. Lang, and K. R anson, “R adar m odeling of a boreal forest,” IE E E Transactions
on Geoscience and Rem ote Sensing, vol. 29, pp. 627-638, 1991.
[10] S. C hauhan, D. L. Vine, and R. Lang, “D iscrete scatter m odel for microwave rad a r and ra ­
diom eter response to corn: com parison of theory and d a ta ,” IE E E Transactions on Geoscience
and R em ote Sensing, vol. 32, pp. 416-426, 1994.
[11] I.-S. Koh, F . W ang, and K. Sarabandi, “estim ation of coherent field a tten u atio n th rough dense
foliage including m ultiple scatterin g ,” IE E E Transactions on Geoscience and Rem ote Sensing,
vol. 41, no. 5, pp. 1132-1135, 2003.
[12] J. W ait, “S cattering of a plane wave from a right circular dielectric cylinder a t oblique inci­
dence,” Canadian Journal of Physics, vol. 33, pp. 189-195, 1955.
[13] J. W ait, Electromagnetic Radiation from Cylindrical Structures. Pergam on Press, 1959.
[14] G. Ruck, D. Barrick, W . S tu a rt, and C. K richbaum , Radar Cross Section Handbook. Jo h n
Wiley, 1970.
[15] J. Bowman, T . Senior, and P. Uslenghi, Electromagnetic and Acoustic Scattering by Sim ple
Shapes. N orth-H olland P ublishing Company, 1969.
[16] P. B arber and S. Hill, Light Scattering by Particles: Com putational Methods. W orld Scientific,
1990.
[17] R. Gans, “R ad iatio n diagram s of ultram icroscopic p articles,” A nnalen der Physik, vol. 76,
no. 1, pp. 29-38, 1925.
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[18] C. A cquista, “Light scattering by tenuous particles: a generalization of th e rayleigh-gansrocard approach,” Applied Optics, vol. 15, pp. 2932-2936, 1976.
[19] L. Cohen, R. H aracz, A. Cohen, a n d C. A cquista, “Scattering of light from arb itrarily oriented
finite cylinders,” Applied Optics, vol. 22, pp. 742-748, 1983.
[20] R. Schiffer and 0 . Thielheim , “Light scattering by dielectric needles and disks,” Journal of
Applied Physics, vol. 50, pp. 2476-2483, 1979.
[21] J. Shepherd and A. Holt, “T he scattering of electrom agnetic rad iatio n from finite dielectric
circular cylinders,” Journal o f Physics A : M athematical and General, vol. 16, pp. 651-662,
1983.
[22] M. K aram a n d A. Fung, “Electrom agnetic scattering from a layer of finite-length, random ly
oriented dielectric circular cylinders over a rough interface w ith application to vegetation,”
International Journal o f Rem ote Sensing, vol. 9, pp. 1109-1134, 1988.
[23] M. K aram , A. Fung, and A. Y.M .M ., “E lectrom agnetic wave scattering from some vegetation
sam ples,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 26, pp. 799-808, 1988.
[24] J. Stiles and K. Sarabandi, “A scattering m odel for th in dielectric cylinders of a rb itra ry cross
section and electrical length,” IE E E Transactions on A n tennas and Propagation, vol. 44,
p p . 2 6 0 -2 6 6 , 1996.
[25] S. Seker and A. Schneider, “Electrom agnetic scattering from a dielectric cylinder of finite
length,” IE E E Transactions on A ntennas and Propagation, vol. 36, pp. 303-307, 1998.
[26] D. L. Vine, R. M eneghini, R. Lang, and S. Seker, “S cattering m odel from a rb itra ry oriented
dielectric disks in th e physical optics regim e,” Journal o f the Optical Society o f Am erica,
vol. 73, pp. 1255-1262, 1983.
170
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[27] D. L. Vine, “T he ra d a r cross section of dielectric disks,” IE E E Transactions on A ntennas and
Propagation, vol. 32, no. 1, pp. 6-12, 1984.
[28] D. L. Vine, A. Schneider, R. H. Lang, and H. C arter, “S cattering from thin dielectric disks,”
IE E E Transactions on A ntennas and Propagation, vol. 33, no. 12, pp. 1410-1413, 1985.
[29] T . W illis, H. Weil, and D. Le Vine, “A pplicability of physical optics th in plate scattering
form ulas for rem ote sensing,” IE E E Transactions on Geoscience and Remote Sensing, vol. 26,
no. 2, pp. 153-160, 1988.
[30] R. H arrington, Field Com putation by M om ent Methods. IE E E Press, 1993.
[31] J. H. Richm ond, “Scattering by a dielectric cylinder of a rb itra ry cross section shape,” IE E E
Transactions on A ntennas and Propagation, vol. 13, pp. 334-341, 1965.
[32] A. Peterson, S. Ray, an d R. M ittra, Com putational M ethods fo r Electromagnetics.
Oxford
University Press, 1998.
[33] S. Raz and J. Lewinsohn, “Scattering and absorption by a th in , finite dielectric cylinder,”
Applied Physics, vol. 22, pp. 61-69, 1980.
[34] A. Papagiannakis, “A pplication of a point-m atching m om reduced scheme to scattering from
finite cylinders,” IE E E Transactions on M icrowave Theory and Techniques, vol. 45, pp. 15451553, 1997.
[35] A. Papagiannakis an d E. Keiezis, “Scattering from a dielectric cylinder of finite length,” IE E E
Transactions on A n ten n a s and Propagation, vol. 31, pp. 725-731, 1983.
[36] C.-M. Chu and H. Weil, “Integral equation m eth o d for scatterin g and absorption of electro­
m agnetic rad iatio n by th in lossy dielectric discs,” Journal of Computational Physics, vol. 22,
pp. 11-124, 1976.
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[37] H. Weil an d C.-M. Chu, “S cattering and absorption of electrom agnetic rad iatio n by th in lossy
dielectric discs,” Applied Optics, vol. 15, pp. 1832-1836, 1976.
[38] A. Poggio and E. M iller, “Integral equation solutions of three-dim ensional scattering prob­
lem s,” C om puter Techniques fo r Electromagnetics, pp. 159-274, 1973.
[39] A. Glisson, “An integral equation for electrom agnetic scattering from homogeneous dielectric
bodies,” IE E E Transactions on A ntennas and Propagation, vol. 32, no. 2, pp. 173-175, 1984.
[40] Y. Leviatan, A. Boag, a n d A. Boag, “Generalized form ulations for electrom agnetic scattering
from perfectly conducting and homogeneous m aterial bodies-theory and num erical solution,”
IE E E Transactions on A ntennas and Propagation, vol. 36, pp. 1722-1734, 1988.
[41] M. A ndreasen, “S cattering from bodies of revolution,” IE E E Transactions on A ntennas and
Propagation, vol. 13, pp. 303-310, 1965.
[42] J. M autz and R. H arrington, “A com bined-source solution for rad iatio n and scattering from
a perfectly conducting body,” IE E E Transactions on A n ten n a s and Propagation, vol. 27,
pp.
445-454, 1979.
[43] J. M autz and R. H arrington, “Electrom agnetic scatterin g from a homogeneous m aterial body
of revolution,” Archiv fu r Elektronik und Ubertragungstechnik, vol. 33, pp. 71-78, 1979.
[44] J. M autz and R. H arrington, “An im proved E-field solution for a conducting body of revolu­
tion,” Phase R eport RA DC-TR-80-194, Syracuse University, D ept, of Electrical and C om puter
Engineering, Syracuse, New York, June 1980.
[45] J. M autz and R. H arrington, “An H-field solution for a conducting body of revolution,” P hase
R eport RADC-TR-80-362, Syracuse University, D ept, of E lectrical and C om puter Engineering,
Syracuse, New York, Nov. 1989.
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[46] A. Glisson and D. W ilton, “Simple and efficient num erical m ethods for problem s of elec­
trom agnetic rad iatio n a n d scattering from surfaces,” IE E E Transactions on A ntennas and
Propagation, vol. 28, no. 5, pp. 593-603, 1980.
[47] A. Glisson an d D. W ilton, “Simple and efficient num erical techniques for treatin g bodies of
revolution,” P hase R eport RADC-TR-79-22, T he U niversity of M ississippi, Syracuse, New
York, M ar. 1979.
[48] C. B utler, “C urrent induced on a conducting strip which resides on th e planar interface be­
tween two sem i-infinite half-spaces,” IE E E Transactions on A n tennas and Propagation, vol. 32,
pp. 226-231, 1984.
[49] C. B utler, X. Xu, an d A. Glisson, “C urrent induced on a conducting cylinder located near the
p lanar interface betw een two semi-infinite half-spaces,” IE E E Transactions on A ntennas and
Propagation, vol. 33, pp. 616-24, 1985.
[50] C. B utler and X. X u, “T E scattering by p a rtially b uried and coupled cylinders at th e interface
betw een two m edia,” IE E E Transactions on A nten n a s and Propagation, vol. 38, pp. 1829-1834,
1990.
[51] X. X u and C. B utler, “Scattering of TM excitation by coupled and partially buried cylinders at
the interface betw een two m edia,” IE E E Transactions on A n tennas and Propagation, vol. 35,
pp. 529-538, 1987.
[52] X. Xu and C. B utler, “C urrent indued by T E excitation on coupled and p artially buried cylin­
ders a t the interface betw een two m edia,” IE E E Transactions on A ntennas and Propagation,
vol. 38, pp. 1823-1828, 1990.
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[53] K. M ichalski and D. Zheng, “Electrom agnetic scattering an d rad iatio n by surfaces of a rb itra ry
shape in layered m edia, P a rt I: Theory,” IE E E Transactions on A ntennas and Propagation,
vol. 38, no. 3, pp. 335-344, 1990.
[54] K. M ichalski a n d D. Zheng, “E lectrom agnetic scattering a n d rad iatio n by surfaces of arb itra ry
shape in layered m edia, P a rt II: Im plem entation and results for contiguous half-spaces,” IE E E
Transactions on A ntennas and Propagation, vol. 38, no. 3, pp. 345-351, 1990.
[55] T. J. Cui and W . W iesbeck, “TM wave scattering by m ultiple two-dim ensional scatterers buried
under one-dim ensional m ulti-layered m edia,” in IG A R S S ’96: Proceedings o f the International
Geoscience and Rem ote Sensing Symposium, vol. 1, pp. 763-765, May 1996.
[56] T . J. Cui and W . W iesbeck, “T E wave scattering by m ultiple two-dim ensional scatterers buried
under one-dim ensional m ulti-layered m edia,” in IG A R S S ’96: Proceedings o f the International
Geoscience and R em ote Sensing Symposium, vol. 1, pp. 766-768, May 1996.
[57] T . J. Cui, W . W iesbeck, and A. Herschlein, “Electrom agnetic scattering by m ultiple threedim ensional scatterers buried under m ultilayered m edia - Paxt I: Theory,” IE E E Transactions
on Geoscience and R em ote Sensing, vol. 36, no. 3, pp. 526-534, 1998.
[58] T. J. Cui, W. W iesbeck, and A. Herschlein, “Electrom agnetic scattering by m ultiple threedim ensional scatterers buried under m ultilayered m edia - p a rt II: Num erical im plem entations
and results,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 36, no. 3, pp. 535—
546, 1998.
[59] S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried p e r­
fectly conducting bodies of revolution,” IE E E Transactions on A ntennas and Propagation,
vol. 44, pp. 143-151, 1996.
174
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[60] S. V itebskiy and L. Carin, “Resonances of perfectly conducting wires and bodies of revolution
buried in a lossy dispersive half-space,” IE E E Transactions on A ntennas and Propagation,
vol. 44, pp. 1575-1583, 1996.
[61] S. Vitebskiy, L. Carin, M. Ressler, and F . Le, “U ltra-w ideband, short-pulse ground-penetrating
radar: Sim ulation and m easurem ent,” IE E E Transactions on Geoscience and Rem ote Sensing,
vol. 35, pp. 762-772, 1997.
[62] N. Geng, C. Baum , and L. C arin, “On th e low-frequency n a tu ra l response of conducting and
perm eable targ e ts,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 37, pp. 347359, 1999.
[63] N. Geng and L. Carin, “W ide-band electrom agnetic scattering from a dielectric bor buried in
a layered lossy dispersive m edium ,” IE E E Transactions on A ntennas and Propagation, vol. 47,
pp. 610-619, 1999.
[64] N. Geng, A. Sullivan, and L. C arin, “Fast m ultiple m ethod for scattering from an a rb itra ry pec
targ et above or buried in a lossy h a lf space,” IE E E Transactions on A ntennas and Propagation,
vol. 49, pp. 740-748, 2001.
[65] J. He, T. Yu, N. Geng, an d L. C arin, “M ethod of m om ents analysis of electrom agnetic scatter­
ing from a general three-dim ensional dielectric targ et em bedded in a m ultilayered m edium ,”
Radio Science, vol. 35, pp. 305-313, 2000.
[66] J. He, N. Geng, L. Nguyen, a n d L. C arin, “Rigorous m odeling of ultraw ideband vhf scattering
from tree trunks over flat and sloped terrain ,” IE E E Transactions on Geoscience and Rem ote
Sensing, vol. 39, pp. 2182-2193, 2001.
[67] J. He, A. Sullivan, and L. C arin, “M ultilevel fast m ultiple algorhithm for general dielectric
targets in the presence of a lossy half-space,” Radio Science, vol. 36, pp. 1271-1285, 2001.
175
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[68] J.-M . Jin, The F inite E lem ent M ethod in electromagnetics. Wiley, 1993.
[69] J. J. W ang, Generalized m om ent methods in Electromagnetics. Jo h n Wiley, 1991.
[70] W. D. M urphy, M. S. Vassiliou, and V. Rokhlin, “Solving electrom agnetic scattering problem s
a t resonance frequencies,” Journal o f Applied Physics, vol. 67, pp. 6061-6065, M ay 1990.
[71] M. A bram ow itz and I. Stegun, Handbook o f M athem atical Functions with Formulas, Graphs,
and M athem atical Tables. U.S. D epartm ent of Commerce, 1972.
[72] C. A. Balanis, Advanced Engineering Electromagnetics. Wiley, 1989.
[73] T.-K . Wu, Electrom agnetic Scattering from Arbitrarily-Shaped Lossy Dielectric Bodies. P hD
thesis, T he U niversity of M ississippi, 1976.
[74] D. Hodge, “S cattering by circular m etallic disks,” IE E E Transactions on A ntennas and Prop­
agation, vol. 28, no. 5, pp. 707-712, 1980.
[75] F. Ulaby a n d M. El-Rayes, “Microwave dielectric sp ectru m of vegetation - p a rt II: D ual dis­
persion m odel,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 25, pp. 550-557,
1987.
[76] L. Tsang, J. Kong, and K. Ding, Scattering o f Electromagnetic Waves. Theories and Applica­
tions. Jo h n Wiley, 2000.
[77] R. E. Collin, A n ten n a s and Radiowave propagation. M cGraw-Hill, 1985.
[78] J. He, M O M and M L F M A fo r Scattering from Dielectric Target in Layered-M edium E nviron­
ment. PhD thesis, Duke University, 2000.
[79] K. Michalski, “O n th e efficient evaluation of integrals arising in th e somm erfeld half-space p rob­
lem ,” IE E Proceedings H - Microwaves, A n ten n a s and Propagation, vol. 132, no. 5, pp. 312-317,
1985.
176
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[80] A. M ohsen, “O n th e evaluation of somm erfeld integrals,” IE E Proceedings, vol. 129, no. 4,
pp. 177-182, 1982.
[81] S. H aber, “T he T a n h rule for num erical integration,” S IA M Journal on Num erical Analysis,
vol. 14, pp. 668-685, 1977.
[82] W. Squire, “A q u a d ra tu re m ethod for finite intervals,” International Journal for Num erical
M ethods in Engineering, vol. 10, pp. 708-712, 1976.
[83] W . Squire, “A com m ent on q u a d ra tu re in the presence of end p oint singularities,” International
Journal o f Com puter M athem atics, vol. 7, pp. 239-241, 1979.
[84] G. Evans, R. Forbes, and J. Hyslop, “T he T a n h tran sfo rm atio n for singular integrals,” In te r ­
national Journal o f Com puter M athem atics, vol. 15, pp. 339-358, 1984.
[85] R. Singh and S. Singh, “Efficient evaluation of singular and infinite integrals using the ta n h
transform ation,” IE E Proceedings - M icrowaves, A ntennas and Propagation, vol. 141, pp. 464466, dec 1994.
[86] A. A bdelm ageed a n d K. M ichalski, “Analysis of EM scatterin g by conducting bodies of revolu­
tion in layered m edia using the discrete com plex image m eth o d ,” in A ntennas and Propagation
Society International Sym posium 1996, Digest, vol. 1, pp. 402-405, June 1995.
[87] Y. C. Lin and K. Sarabandi, “E lectrom agnetic scattering m odel for a tree tru n k above a tilte d
ground plane,” IE E E Transactions on Geoscience and R em ote Sensing, vol. 33, pp. 1063-1070,
1995.
[88] C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory. In tex t Educational Publishers,
1900.
[89] J. V. Bladel, Electrom agnetic Fields. M cGraw-Hill, 1964.
177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix A
Electrom agnetic fields in unbounded media
A .l
E lectrom agnetic fields generated by electrom agnetic sources
T he M axwell’s equations for a homogeneous, isotropic dielectric m edium w ith p e rm ittiv ity e and
perm eability fj. are:
V xE
=
—jc jju H — M
(A .l)
V x H
=
jue E + J
(A.2)
F irst, the case w hen only J ^ 0 and M = 0 is considered, so th a t:
V xE
=
—jwfj,H
(A.3)
V x H
=
juje E + J
(A.4)
In order to solve (A .3),(A .4), an electric vector p o ten tia l A is defined such th a t
nH = V x A
(A.5)
V x (E + j u A ) = 0
(A.6)
S ubstitu tio n of (A.5) into (A.3) yields:
For this equation to be tru e, it m ust be:
E + j'wA = —V $
178
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A .7)
Once A an d 4> are known, the electric and m agnetic fields can be determ ined from (A.5) and (A .7)
as follows:
E
=
- j u A + V4>
(A .8)
H
-
—V x A
H
(A .9)
Using (A .9) into (A.4) yields (k = u^fjle):
V 2A + A;2 A = V V • A + jfw/je V4> E — fi 3
(A.10)
T he vector potential can be chosen such th a t
V • A = —j u j g e Q E
(A .11)
which leads to a sim plification of equation (A.4):
V 2A + A;2 A = —fj, J
(A .12)
O n th e other hand, tak in g th e divergence of ( A .ll) and using M axwell’s equation:
V •E = — =
e
jue
(A.13)
yields th e following differential equation:
V 2$ + fc2 $ =
(A. 14)
jus
T he scalar G reen’s function g( r, r') is th en defined as th e solution of:
V 2 g{ r, r') + k2 g{ r, r') = - 5 { r - r')
(A.15)
It can be shown [88] th at:
g(r ’r ' ) =
e- j k \ r - r'|
47r|r _ r / |'~ ’
k =
179
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.16)
Since the vector and scalar p o ten tial are solution of (A. 12) and (A. 14), respectively, th ey can be
w ritte n as:
A (r)
=
g f < K r ,r ') J ( r ') d y '
$(*)
=
I f
~
(A.17)
Jv
e Jv
, V '-•J ( r ')
dV'
ff(r >r ) — 37we
(A .18)
3
V being the volume where th e sources J and M are located.
T h e expressions for th e electric a n d m agnetic fields are obtained su b stitu tin g (A.17), (A. 18) in
(A.8), (A.9):
E ( r)
H (r)
= - j u g [ g ( r , r ' ) J ( r ,) d V ' +
Jv
=
-jug-
=
-ju g
Jv
Jv
g{r, r') J ( r ') + ^
g{r,r')l + ^
— V [ g ( r , r ') V '- J ( r ') dV'
3 ue
Jv
Vg{r, r ') V ' • J ( r ')
V s ( r ,r ') V '
dV'
J ( r ') dV'
(A.19)
= V x f g { r ,r ’) J ( r ') d V 1
Jv
=
=
f V x g(r,r') J(r') d V 1
Jv
[ [V £r(r,r') x I] • J ( r ') dV'
JV
(A.20)
where I is the unit dyadic.
Introducing th e electric an d m agnetic dyadic G reen’s functions as, respectively,
G ( r , r ')
=
s(r,r')I+ ^V j(r,r')V '
(A.21)
K(r,r')
=
V<7(r,r')xl
(A.22)
the expressions (A.17), (A.18) for th e electric and m agnetic fields can be w ritten as follows:
E (r)
=
H (r)
-
-ju g [
Jv
G ( r , r ' ) • J ( r ') dV'
[ K ( r , r ' ) • J ( r ') dV'
Jv
180
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.23)
(A.24)
Expressions (A.23), (A.24) are valid if only electric sources J are present. In th e case w hen only
m agnetic sources exist, i.e. J = 0 and M / 0 , M axwell’s equations (A .l), (A.2) become:
V xE
V
=
x H =
-jufx H - M
(A .25)
jue E
(A .26)
T h e solution to equations (A.25), (A.26) can be found by duality from th e solution of (A.3), (A.4)
w ith the substitutions:
E
J
-> H
-> M
H
M
->■
->■
-E
—J
£
—^ j-l
fJ*
—y
£
(A .27)
T hus, the expressions for th e electrom agnetic fields generated by m agnetic sources M w hen J = 0
are:
E (r)
- [ K ( r , v ' ) - M ( r ' ) dA'
=
(A.28)
JA
=-ju e
H (r)
fJ A G ( r , r ' )
• M (r') dA '
(A.29)
W hen b o th electric and m agnetic sorces are present, th e electrom agnetic fields can be obtained
as superposition of A.23), (A.24) an d (A.28), (A.29):
E (r)
=
—jojjj, [ G( r , r ' ) • J ( r ') dA! — [ K ( r , r ' ) - M ( r ') d A 1
JA
H (r)
=
[ K ( r , r ' ) - J ( r ') d A ' - j u e
JA
A.2
(A.30)
JA
[ G(r,r') •
M (r ') dA'
(A.31)
JA
G reen’s theorem for a m ultiply connected region
Let V be a volume corresponding to a m ultiply connected region bounded by an inner closed surface
<Si and by an outer closed surface S 2 , as shown in Figure A .I. T he u n it vector n norm al to the
surfaces Si and S2 is pointing into V If P , and Q are any two twice-differentiable vector fields,
th en the following relations apply:
V • [P x (V x Q)]
=
(V x P ) ■(V x Q ) - P • [V x (V x Q)]
(A.32)
V • [Q x (V x P )]
-
(V x Q) • (V x P ) - Q • [V x (V x P)]
(A.33)
181
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e,p
s ,|x
Figure A .l: D efinition of volume V and surfaces Si an d S 2 for th e G reen’s theorem .
S ubtracting (A.32) from (A.33) yields:
V • [Q x (V
x P ) - P x (V x Q)] = P • [V x (V x
Integrating b o th sides
of (A .34) over the volume V yields:
Q)] - Q ■[Vx (V x P)] (A .34)
[ V ■[Q x (V x P ) - P x (V x Q)1 d V =
Jv
= [ { P • [V x (V x Q)] - Q ■[V x (V x P )l} dV
Jv
(A.35)
Applying the divergence theorem , the left side of (A.35) S becomes:
[ V dQ x(V xP)-Px(V xQ )]dF
Jv
=
= £ [ P x (V x Q ) - Q x (V x P )] • n dS
(A.36)
S u bstitution of (A.36) into (A.35) gives th e G reen’s theorem :
j f [P x
(V x Q) - Q x (V x P)] • n d S =
=
[ {P • [V x (V x Q)] - Q ■[V x (V x P)]} dV
Jv
182
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.37)
A .3
D erivation of expression (2.17)
Let V be a volume enclosed by a surface S = Si U S 2 , containing all th e sources J and
sources on S, a n d
n
M
,
w ith no
be the u nit vector norm al to 5 pointing into V as shown in Figure A .I. Using
th e G reen’s theorem (A.37) w ith
P
where
a
=
E
,Q
=
a
g+ ,
g+(
r ,
= g
r ' )
( r ' ,
(A.38)
r )
is an arbitrary, constant vector, yields:
f
Jv
{
• [V' x (V ' x &g+)] -
E
a
= j[
g+ ■[V' x (V ' x
dV' =
E ) ] }
x (V ' x a g+) - a.g+ x (V ' x
E
E ) ]
• n ' dS'
(A.39)
In equation (A.39) and hereafter, th e prim ed sign is used indicate th e variables of integration. Since
there are no sources on th e surface S, then:
V' x
= - j u g H,
E
on 5
(A.40)
so th a t
[ a
x (V ' x
E ) ]
■
' = —jug,
n
( a
x H) •
' = —j u g
n
• (H x
a
n
')
on 5
(A.41)
while vector m anipulations yield
[ E
x (V ' x
a
5
+ ) ]
•
n
'
=
[ E
x Vg+ x
(
a ) ]
■
n
'
=
a
■ [ (
n
'
x
E
)
x V'^+j
on S
(A.42)
Using (A.41)and (A.42), th e right side of equation (A.39) can be w ritte n as:
£ [ E x ( V ' x a j + - a s + x (V'xE))] ■ n 'd S ' =
=
Let g+ — g{
r ' , r )
a
• j [ j u g (H x
) g+ +
n
( n
'x
E
)
x V'g+} dS'
(A.43)
be the scalar G reen’s function, i.e, th e solution of
V ' 2 g(
r ' ,
r )
+ k 2 g(
r ' ,
r )
=
—5( r ' —
r )
183
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.44)
subject to th e condition th a t g be an outgoing function of r'.
V ' 2 g{r', r) = - k 2 g(r', r) - <5(r' - r)
T hen
(A.45)
and
V ' x ( V ' x a f f +)
=
V '[V ' • (a<7+)] - V ,2(a5+)
= V'[V'■(a<7+)] - a V V
V '[V ' • (ac/+ )] - a.[-k 2 g(v', r) - <J(r' - r)]
=
(A.46)
Using th e wave equation (2.14), hereafter rep eated for convenience, in th e volume V
V x ( V ' x E ) = fc2 E - jug, J - V ' x M ,
in V
(A.47)
together w ith (A.46), we can w rite th e left side of equation (A.39) as
f {E • [V' x (V ' x a 0+)] - a g+ ■[ V x (V ' x E)]} dV' = [ E ■V '[V ' • (a5+)] dV' +
Jv
Jv
- a ■f j _ E ( r ') { - k 2 g+ - S{r ' - r)] + g +[k2 E - jug, J - V ' x M ]} dV ' =
= [ E -V '[V -(a 5+ )] d V ' + a- f E( r ') 5 ( r' — r ) d V ' + a . - [ ( j u g J + V ' x M ) g + dV'
Jv
Jv
Jv
(A.48)
If we now consider th e following identity:
V ' • (u F ) = (V ' ■F ) u + F • V 'u
u = V'<7+ • a ; = V ' • (ag+),
w ith
(A.49)
F = E
(A.50)
we can write
V ' ■[(a • V'c/+ )E] = (V ' ■E ) V'<?+ • a + E ■[ V V ■(a ff+ )]
(A.51)
Applying the divergence theorem along w ith (A.52) and expression (A.51), yields
(a- V g +) E - h ' d S '
512
=
[ V ' ■[(a • V(gf+ ) E ] dV ' ,
Jv
Su - Sx U S2
184
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.52)
Now using (A.51) on the right hand side of (A.52) yields
-<£
JSi 2
(a •V g +) E ■ n ' d S '
=
=
[ {(V ' •E) V g • a + E • [V 'V ' • (a g+)}} d V
JV
-a • [
Jv jue
V'g+ d V ' + f E • [V 'V ' • (a.g+)] dV' (A.53)
Jv
so th a t
[ E ■[V 'V ' • (a5+ )l d V = a ■ /
V'g+ (E • n ') dS' + a • [ ? - ^ V ' ff+ dV'
Jv
Jsiz
Jv jue
(A.54)
Also, since r is in th e volume V, th en
f E(r') <5(r' - r) d V ' = E(r),
Jv
r
G
V
(A.55)
S u bstitution of (A.54) and (A.55) into (A.48) yields:
[ {E ■[V' x (V ' x a.g+)] - a g + • [V' x (V ' x E)]} dV ' = - a • I
V'g+ (E ■ n ) dS' +
Jv
Js 12
+ a• [
Jv ju s
V'g+ dV' + a • E(r) + a • [ (jug, 3 + V ' x M) g+ dV'
Jv
(A.56)
S ubstitution of (A.43) a n d (A.56) into (A.39) yields:
a - E( r ) = - a - [ \(j oj gJ + V' x M) g+ + —r— V'^-1"] d V ' +
Jv 1
jue
+ a - / [ j u g ( H x n ') g+ + ( n ' x E ) x V g + + V'5+ ( E • h ' ) \ d S '
Jsu
A .4
(A.57)
Evaluation of surface integral in (2.22) at singularity point
Let V be the volume and S 12 = S i U 52 the surface defined in Figure 2.1. T he surface integral th a t
appears in (2.22), i.e.,
J Su
[jug (H x n ') g( r, r') + ( n ' x E) x V'g(r, r') + V' g(r, r') (E • n ')] dS'
has a singular integrand w hen th e observation point r is also on th e
(A.58)
surface S and corresponds
w ith the source point r', i.e., r = r'. T he purpose of th is section is to evaluate expression (A.58)
185
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure A.2: D efinition of integration dom ain.
in such p articu lar situation.
Since
1 + jkR
47TR2
= 1 + J'fek —r7! _—jk\r —r'l r ~ r'
r —r '|
V '9 ( r ' r ') =
where
R = r —r '
R =
-jkR f.
(A.59)
R
R
then, the integrand of (A.58) can be w ritten as
j u g (H x n ') g(r, r ') + ( h ' x E ) x V' g { r, r') + V g { r, r') (E • n ') =
-jkR
= jw/i(Hx n')
47T.fl
[( n ' x E) x R + R ( E • n ')
L
1 + jkR
47Ti?2
jkR
e- j k R
= { j u g (H x n ') R + [ e ( n '-R ) -
n ' (E • R ) + R (E ■n ')] (1 + j k R ) }
Let us now evaluate th e surface integral (A.58) for r = r ' as th e lim it for
4 irR 2
8
(A.60)
—>■ 0 of th e integral
over the surface Ss U F d- illu strated in Figure A.2. Using (A.60),
[jug (H x n ') g(r, r') + ( n ' x E) x V g { r, r ') + V 'g (r, r ') (E • n ')] dS' =
Sl2
= lim <£[j u g (H x n ') g(r, r ') + ( n ' x E) x V ' g ( r, r ') + V ' g ( r, r ') (E • n ')] dS' +
5-40 J
Ss
+ lim I [jug (H x n ') g{r, r') + ( n ' x E) x V'<?(r, r ') + V ' g ( r , r') (E • n ')] dS'
5-4o J
r j
186
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
= f [jufi (H x n ') g{r, r;) + ( n ' x E ) x V g { r, r ') + V '5 (r, r ') (E • n ')] dS' +
JSU
f t
r
^
^
^
-i
^
fs~jkR
n (E ■R ) + R ( E • n ')] (1 + j k R ) } J ^ j g d S '
+ lim j [ j u n { H x n ) R + [ E ( n '-R ) -
(A.61)
r4
where
£
f{r,r')dS'
denotes the principal value of th e integral of / ( r , r') over the surface S, i.e., the integral is com puted
excluding th e singularity in r = r'.
Since
dS' = R 2 dn ' = |r - r ' | 2 dft'
(A.62)
lim R = lim (r — r ') = n '
$->o
5—
>o
and
(A.63)
v
’
th e n the lim it in (A.61) can be evaluated as follows:
lim j> jjWjU ( H x n ' ) R + [ E ( n ' - R ) - h ' ( E - R ) + R ( E - n ')] (1 + j k R ) }
dS' =
r4
=
L[
& '■ & ' E ( r ) I
*2
d n ' -
L
E ( r )
i
^
M
r
e
<A-64)
Therefore, for r £ S n = S i U 52, we have
<fi[jug (H x n ') g(r , r ') + ( n ' x E ) x V g { r, r ') + V g { r, r ') (E ■ n ')] dS' -
J S12
=
A.5
^7^ + f
2
J S12
[jwyu (H x n ') g{r, r') + ( n ' x E ) x V g + V g (E • n ')] dS'
(A.65)
Equivalence o f alternative dyadic G reen ’s function
From the divergence theorem , since th ere are no sources F = J , M on th e boundary surface S,
Jv V
• [5 (r, r ') F (r')] dV'
= ^ g(r, r ') F ( r ')
'
• f i dS'
= 0,
(A.66)
therefore
0 =
V ' • [<?(r, r') F ( r ') ] d V ' = J ’ V ' g ( r, r ') • F ( r ') d V ' + j ' g{r, r ') V ' • F d V 1
187
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.67)
th a t is
[ V'g(r, r') ■F ( r ') d V ' = -
Jv
f
Jv
g(r,r')V -F d V '
(A.68)
It follows th a t
/
Jv
F ( r ') d V '
G (r, r ') • F ( r ') dV'
IV
g{r, r ') F ( r ') d V ' - ~
=
[ V V '^ r , r')] • F ( r ') dV'
=
J v g{r, r ') F ( r ') dV ' - 1 V ^ V '5 (r, r') • F ( r ') d 7 '
=
^ ff(r, r ') F ( r ') d V ' + ^ V ^
=
J
=
[g(r, r ') I + ^ ^ V
']
g( r, r') V ' ■F d y '
.F(r') dy'
[ G (r, r ') ■F ( r ') dV'
(A.69)
Jv
In addition, since for a closed surface S and a J t tan g en tial to 5 , th e following integral theorem
(in [89], page 503, no. 42) holds:
f V' • [3(rir') Jt(r')] dS ' = 0
(A.70)
then
J V ’g M - J t W d S '
and
r —
] s G(T,T')-Jt(r')dV'
r
=
IY
I -
=
- J s g ( v , r ' ) V - J t dS'
VV'N
J t (r ) d V
=
^ ff( r , r ') J t ( r ') d V ' - ~ ^
=
j
g(r, r ') J t ( r ') dV ' - ^
=
j
g(r, r ') J t ( r ') d F ' + ^ V ^ 5 (r, r ') V ' • J t d V 1
=
Js [< ?(r,r')I + V ^
=
(A.71)
[V V ,j ( r , r ') ] ■J t (r') dV '
V 'S(r, r') ■J t (r') dV'
V-] • J t (r') dV'
[ G ( r , r ' ) - J t ( r ') d y '
Js
188
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.72)
Appendix B
D erivation of tw o-dim ensional G reen’s function
Let S be th e surface of a b o d y of infinite length in the z-direction, and C be its generating curve,
as depicted in Figure 3.1. If there is no variation of th e electrom agnetic quantities along the z-axis,
th en the coupled surface integral equations on S can be reduced to integral equations on C by
defining a tw o-dim ensional G reen’s function.
Consider th e interior surface integral equation (2.38). Since
dS' = dz' dC' ,
(B .l)
th e two surface integrals in (2.38) can be w ritten as follows:
j
V g V '] ■J t (r') dS'
[<?(r, r') I + p
=
JJ
g(r , r ' ) I
V5 V'
J t ( r ) dz dC
(B.2)
C —oo
OO
J
[V 'g(r, r') x I] • M t (r') dS' =
5
j
J
C
—oo
[V'g x I] ■M t (r') dz' dC'
Since J t( r ') does not depend on z’, it is a function of only th e two-dim ensional vector
(B.3)
p ' , i.e.,
J t( r ') = J t ( p ;)- T his allows to write
OO
J
OO
5 ( r , r ' ) I • J t( r ') dz'
=
J
J t {p')
OO
p-
J
(B.4)
g(r,r') dz'
OO
V g V ' ■J t {r') dz'
=
^ V
—oo
/
V'-Jt(p')
(B.5)
g(r,v') dz'
x I • M t ( p ' ) dC'
(B.6)
L —oo
r
[V'ff(r, r') x I] ■M t (r') dS'
g(*,r') dz'
=
j V'
C
oo
j
L —oo
189
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he scalar G reen’s function can be w ritten as [72]:
OO
j
ff(r,r') =
9
(B.7)
( p , p ' \ k z ) e i k z (z ~ z ) dkz
—OO
where
.•
g { p , p ’-,kz) = - - f f j ) (fep| p - p '|) ,
,—
kp = y k 2 ■
(B.8)
Using (B.7) and exchanging th e order of integration, yields:
OO
J
OOr
g(r,r')dz'
J
=
—OO
OO
J
—OO L
g { p , p ' \ k z ) e ^ k z ^z ~
oo
=
/
r
9
dk
—OO
(p ,p '-,k ,)e ik
—oo
oo
T
J
L
—oo
e - j k . z ’ dz'
dkz
oo
{p, p ' ] k z ) e i kzZ 8 {kz ) dkz
=
J
—oo
=
9 ( p , p ' \ °)
9
=
(B.9)
-p'|)
Hence, defining a tw o-dim ensional scalar G reen’s function as
92 d { p ,
p
') =
- 3- H Q
{ ]{k\ p
(B.10)
- p ' l )
th e surface integral (B.2) becomes
/
p(r,r')I + ^ V<?V' ■J ;(r') dS" = j
5
~
92 d { p , p ' ) l + ^2
V'
3t(p')dC'
( B .ll)
C
while (B.3) can be w ritte n as the following line integral
j [Vg( r , r ' ) x I] • M t (r') dS'
5
J [V,g2 D( p , p ' )
=
’
x I] ■M j ( p ' ) dC'
(B.12)
C
By defining a a tw o-dim ensional scalar G reen’s function for th e free space as
52d(p>
p
') = - 7 H o2\ ko\p - P'l) .
(B.13)
a sim ilar procedure can be applied to the surface integral equation (2.37) to obtain the two dim en­
sional integral equations for the exterior problem .
190
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Appendix C
B odies of R evolution
C .l
E xplicit form o f operators
a ■PQ
T he operators a pT (p, q = r , 0) are defined by (4.16)-(4.17):
a pr{ J T(t',(p')}
=
- j a p .o [
Js
, 0')}
=
- jt up o Js
p ■r ' G ( r , r ' ) J T(t', (j)')
1 1 5G° d{p'Jr )
kl p' dp
1 1 dG°dJ^
pJ dp d<j>'
Using expressions (4.1)-(4.3), the dot products betw een n. 0 , f
f • f '
t
• n
dt'
and n
0
dS'
dS'
(C .l)
(C.2)
f ' are found to be:
=
s in ip simp' cos ((f)1 — 0) + cos 0 cos 0 '
(C.3)
=
sin 0 cos ip' cos (0 ' — 0) — cos 0 sin 0 '
(C.4)
=
—sin 0sin(< // — 0)
(C.5)
n •r
=
cos 0 sin 0 ' cos (0 ' —0) —sin 0 cos ip'
(C.6)
n • n
=
cos 0 cos 0 ' cos (0 ' —0) + sin 0 sin 0 '
(C.7)
n ■0
=
—cos 0 sin (0 ' — 0)
(C.8)
0 ■f
=
s in 0 ; sin ( 0 ' — 0)
(C.9)
0 • n
=
cos ip' sin (0' — 0)
(c.io)
0
=
cos (0' — 0)
( C. l l )
■
0
191
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S u b stitu tio n of expressions (C .3 )-(C .ll) into definitions (C.1)-(C.2) yields:
a TT{ J r ( t ' j ' ) } =
-jojfj .0 Js
| G °( r , r') [sin ip sin ip' cos (p'
1
- p) + cos ip cos ip'] Jr (t',p') -
,<p')} = j u p QJ^i^G0{r,r')smpsin(p' - p)J${t',p') +
dS' (C.12)
dS'
(C.13)
.. .. ..
f \nOt^
^ t u> a'i
1 1 dG°d(p'JT)
r{ J T(t',p')} = -jojfj
.0 [ \ G° (r,r') sin ip'sin {p' - p ) J T{t',p') - - j —
dt'
JS I
Kq p dp
dS' (C.14)
a « r / 7 / V ^ M _
a ', { J tj >{t',p')} = - j u n o
C.2
J
,
I G ° ( r , r ' ) c o s ( 0 ' - p)J<fi{t',p')
E xplicit form of operators
1 1 SG° <9J<»
p'
dp dp'
dS'
(C.15)
/3pq
T he operators f3pT (p , q = r,p) are defined by (4.18):
/3™{M9(f',0 ')}
=
- / p .V G °(r,r')x q 'M ,(^ V S '
Js
(C.16)
T he gradient V G °(r, r') can be expressed as
o/
_ dG° A , dG° .
V G ( r ’r ) = f t T f t +
ldG° «
(C.17)
Since
R — |r — r '| = \ J P2 + p '2 — 2pp' cos (p 1 — p) + (z — z ' ) 2
(C.18)
the p a rtial derivatives of G° can be w ritten as
dG°_ _ dGP_dR _ dCP_ „
dn
dR dn
dR
dG°
dr
dG° d R
dR dr
dG°
dG°dR
d R dp
dp
dG°
dR
t
(C.19)
-VR
(C.20)
(C.21)
where, using the expression (4.8) of the free space G reen’s function,
M±_ =
dR
d
( e - i k°R \ =
d R y AirR J
1 -jkp R
AirR2
j kn R
192
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.22)
T he gradient V i? can be calculated in cylindrical coordinates as
„ „
dR .
1 dR 2
dR .
V R - ~ a ^ p + p a f 4' + & a
x
(a2 3 )
w ith the p a rtial derivatives of i? being
^
^
( V ^ 2 + P'2 “ ZPP1 cos((t>' - (f’) + (z -
^
^
( V P2 + P'2 - W
z 0 2)
cos(^ ' - 0) + (* - ^ ) 2)
( V p 2 + P'2 “ 2PP' cos(^ ' - ^) + (« -
* ')2)
= - — P ° ° ^ --------—
=
~ PP
=
-----—
(C.24)
(C.25)
(C.26)
Therefore:
_
n -V i?
dR ^
^
IdR
= — n •p +
ap
p o<p
dR
, dR .
= -7—cos ip —— sin
op
oz
=
. „ „
T -V i?
R
=
=
—
.
3 S i? _
n • 0 + — n ■z
oz
,
ip
{[p - p' cos(^' — (j ) ) \ cos ip — (z — z') s in ^ }
(C.27)
&R _
Id R ^ 2 dR „.
— T •p +
T • 0 + — T ■Z
op
p o<p
oz
dR .
, dR
sm ip + ——cos ip
op
dz
"4 {[P — p/ cos(0' — <p)] simp + (z — z') cos ip)
R
(C.28)
S ubstitution of (C.27)-(C.28) and (C.25) into expressions (C.19)-(C.20) yields:
dG°
dn
dG°
dr
1 dG°
P
~ ~ai
_
=
1 dG°
{[p — p' cos(<p' — (p)\ cos ip — (z — z') sin ip}
R dR
(C.29)
1 dG°
{[p — p' cos(<p' — 0)] s i n ^ + (z — z') cos ip}
R dR
(C.30)
1 dG° , . f±,
- R l R psm {* - V
(CU1)
Hence, the use of (C.29)-(C.31) into (C.17) yields th e following expression for the gradient VG°:
1 dC®
VG° = - — (An n + Ht f + A0 4>)
where
193
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.32)
An
=
\p ~ p' cos(<f)' —4)} cos ip - (z — z') sinip
(C.33)
Ar
=
[p - p' cos(0' - 0)] sin ip + (z - z') cos ip
(C.34)
^
=
—p'sin(<p'— (p)
(C.35)
Therefore, for q ' — f ', 0 ',
f
•
VG° x
cp • V G ° x
q' =
q
-g
{^ n[r
• ( n X q ')] + A - [ t
=
{^n[(T x
=
[M 4>-
=
=
X q ')] + -A*[ f
q
+ A T[ ( f x t ) -
'] + A
^ t
• ( 0 X
q')]}
x
q ']}
$)■
q')]
{ A n [ 4> ■(
' =
q']
n)-
•( f
n Xq
(C.36)
')]+ A r [4> -(f
X «)■
x
1 PC 0
R ~ d R [-A n ( * ■ q ') +
X q
f).
')] + A^[ (p • (
q'] + 4 4 ( ( ^
0 X
q
')]}
X (^). q ' ] }
fi ■ q ')]
(C.37)
where, using (C .3 )-(C .ll) a n d (C.36)-(C.37),
A n (4> ■f ' ) — A<f,(n ■f ') =
{[p — p 1 cos(<p' - (j))] cosip — (z - z') s in ^ } sin ip' sin (cp' - <p) +
+p' sin (<p' — 0)[cos ip sin ip' cos (cp1 — <p) - sin ip cos ip']
=
A n (4>
• <p') -
[p cos ip simp' — p' sin ip cos ip' — (z — z') sin ip sin ip'] sin((p' — cp) (C.38)
A ^ { n •ip')
=
{[p - p'cos(<p' - cp)] cos ip - (z - z') s in ^ } cos (cp' - (p)+
—p' sin (<p' — <p) cos ip sin (<p' — (p)
— - p ' c o s i p + pcosipcos{(p' — <p) — (z — z')c o s (( p '- ( p )
- A n( f ■ f ' ) + A r ( n ■ ? ')
(C.39)
=
—{[p — p' cos {<p' — ip)\ cos ip — (z — z') sin-i/’} [sin ip sin ip' cos (cp' — <p) + cosi^cos ip'] +
+ {[p —
p' cos (<p' — (p)] sin ip + (z — z') cos V’} [cos ip sin ip' cos (<p' — <p) — sin ip cosip']
=
—[p — p' cos (cp' - <p)] cos ip' + (z — z') sin ip' cos (<p' - <p)
194
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.40)
—A n (
■ (p>) + A t ( n • ((> ) =
t
{[p —p, cos(0' — ^ j c o s V ' — (z — z ^ s i n i p } s i n ^ s i n (<f>' — <p) +
— {[p — p' cos ((j) 1 — (p)] smip + {z — z') cost/)} cos ip sin (</>' — cp)
=
- ( z - z ') sin(</>' - (p)
(C.41)
S u b stitu tin g (C.38)-(C.41) into (C.37)-(C.38), th en (C.37)-(C.38) into (C.19), the final form of the
operators f3pg (p , q = r,cp) is obtained:
r { M r {t',<p')} =
f 1 dG0
- / — -j-r-[pcos ip simp' - p' s i n c o s tp' — (z - z') sm'ipsimp'] sin (<p' — cp)MT(t', <p') dS'
JS
(C.42)
wit
^ { M
f
0
,p ') } =
1dG®
- / z z —fFr[-p'cosip + p c o s tp cos (</>'- <p) - (z - z') cos ( </ / - 0)]M ?i(t/, cp1) dS'
J 5 il C b ih
(C.43)
^ { M T(t',cP')} =
- /
J ,5 it CLa x ,
- p! cos(<p' - (p)] cos ip' + (z - 2:') sin 'ip' cos{<p' - <p)}MT(t', <p') dS'
r 1 dG®
{Af*(t', </>')} = J s k -j k (z - z ') sin (<£' - W t t f , <t>') dS'
C.3
D erivation of operators
(C.44)
(C.45)
a,m
n
From expressions (C.12)-(C.15) and th e fact th a t, from (C.17),
S 0 ° - * . V G° = ^
dr
dt
(C.46,
it follows th a t
27T
a rT{ JTn {t')e^n^ } = —jcjpo
J J|
[sin ip sin ip' cos {cp' — <p) + cos ip cos ip'] G °(r, r ') JTn(t')egn^' +
k l dt
= - j o j p o ejn ^
dt'
2tt
J | ^ sin ip sin ip' J
y
G°(r , r ' ) cos (</>' - <p) e?n^ '
o
195
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
+
+ cos ijj cos ip' J G° (r, r') e?n^
o
}_ d _
kg dt
J
^ dp' p' J T„ ( 0 +
t
} dt'
G° (r, r ') ej n W~®d(P']
(C.47)
2tt
aT<l>{ J K ^ t ')ejn4>'} = i UJP o j J
o c
=
sin
sin
— 0 )G °(r, r')
jup,o e-7”^ / [ s i n * J
c
+
1 jndG °i
Jctn {t')ejn(p,p'dt'd(P'
kn p' dt ■
2ir
G°{r, r') sin (cp1 — <fi) e^n^ ' ® d(p' +
o
27r
/
G
"
(
r
,
r
( ( ' ) *’# '
277
a
-
=-ju;poJ J
,<pT
(C.48)
{G°{r,r ')s mT p' sin {( p' - (p)JTn(t')ejn*
'
d(p
d^Pdt'Jt^
ijn*')p'dt'd(P'
2tt
= —j u p o e ^ J j s i n V / [ J
G ° ( r ,r ') sin((p' — </>) e^n^ ' ^d(p' p'JTn(t') +
c
2tt
±d_
[ J G °(r, r')
kl d(p
o
5^
Tn)- } dt'
(C.49)
O-TT-
0
c
277
j
= -jw /io
C
[ J
G°(r, r ') cos (<£' - <£) e?n{# d #
+
0
277
1 jn 3
P' d</>
J
G°{r, r ') ejn(^ '- ^ d 0 '] p' J K {t') dt'd<p'
(C.50)
Using the definition (4.27), th en
277
J G °(r, r ') cos (0' - <p) e>n^ ~ ^d < p '
=
2tt - ^ ^ ±
9
1 + ^1^).
(C.51)
=
2iry g ° - i( M O ~ G n + itM ')
(C.52)
=
2 7 r ^ G ° ( M ' ) = 2 7 r ; n G° ( t , t ' )
(C.53)
0
27T
1 G ° (v ,r ')s m{ < p '- (P )e j n W-®d<P'
277
d<j>
I
Go( r , r V n(0'■“^
,
196
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
it is possible to w rite
a ^ { J qn( t ' ) e ^ ' } = 2 i r a ? c>{Jqn( t ' ) } e ^ ,
P,q = r,<p,
n = 0, ± 1, ± 2 , . . .
(C.54)
where the a^q operators for th e n -th angular m ode are:
a nT{ J r n { t') } = - j w f i
o J { [ sin ip sin ip' gn-1
_ T(b r t
f
an
0 r /
/ 1/ \ 1
.
. . . r
[ •
$ti v J j — J ^ p o J
c
T (p\\
_
-i/in
/~ f -
.//
simp
jtjt)
~
^
^ n + 1 ft)
= - j ^ o / [ C ° - l ( f ' t ' ) ^ G° + l ( ‘' l ' )
c
C.4
G n+l 1 ^
- G ° +1( i , 0
^
•
j simp
a n {•'Tnft)} — JU(j,qJ
C
^
+ cos ip cos ip' G Q
n (t , f') p'jTn(t') +
1
k? d
0
t ) j T
,
P Jrn { t )
j T
dt
jtl
P d(j>n { t ) dt dcp (C.56)
u
^ 2 Crn (i,f J
0
^ (P
^
^Tn )
~1
j±i ( ri
g '7 ^
j at (C.57)
(C.58)
0
Derivation o f operators /^9
From (C.42)-(C.45), it follows th a t
/3TT{M Tn( O e 7‘^ } =
/’27r/>1 dC®
/ / ———■[p cos ^ sin
J J R dR
o c
-
= —
0
c
i
p'dt'd<p' (C.59)
/■2vr r l dC®
I — - ^ - [ —p 1 cosip + p cos ip cos (<p'- cp) — (z - z') simp cos (cp' - c p ^ M ^ ^ ^ e ^ p'dt'dcp'
0
27r
— p 1 simp cos ip1 —[z — z') simp simp'] sin (cp1 — <p)MTn(t')
c
f
^ ^
Up' ~ p) cos ip — 2 p cos ip sin2
R dR
^
^ + (z — z') simp cos (cp1 — cp)]M^n (t')e^n(t p'dt'dcp1 (C.60)
P*T { M Tn(t')ejn<t>'} =
27T
q
= — [ [ ^ ~ 77r{[/° — p ' cos [cp' — cp)} cos ip' + (z — z') sin ip' cos(<p' — <p)}MTn (t ' ) e ^ p'dt'dcp'
J J R dR
0
c
197
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
//
0
IdT T o ^
R dR
~ P) cos ^ ~~ ^P' cos ^ s^n2 ( ^ o ~ l ~ (z ~ z ') sin V,/ c o s (^ — 4>)\Mrn ( t ' ) e ^ ' p'dt'dcp'
V 2 /
(C.61)
c
2 ir
^
= J j I^(z_*')sin (P' 0
fp)M<t>n( t ' ) e ^ ' p'dt'd<P'
(C.62)
c
Using (4.28), an d defining
„o,
S" ( M )
1
to I
=
f 2*
1 dG °
, /
R l R Sm
(
(C.63)
d W ~ <P)
it is possible to w rite
Ppg{ M gn( t ' ) e ^ ' } = 2 t t (3pg{ M qn( t ' ) } e ^ ,
p , q = T,<t>,
n = 0, ± 1, ± 2 , . . .
(C.64)
where the /3pg operators for th e n -th angular m ode are:
PT
n T { M Tn( t ' ) }
=
pr®
A
p cos ip simp' — p' sin i/; cos
/
(f
—(z — z') simp simp1}—
f 1) _
jrO
—’■----- -—
(p; - P) cos ipK°{t, t') - 2pcos ipS°{t, t') + { z - z') sin ip Kjl 1
^ ^ ^ -+1
(f
f r)
—-— M Tn(t')p'dt' (C.65)
- )■
P t T {Mrn ( 0 }
/
{p1 —p) cosip'K®(t, t ' ) - 2 p ' cos ip'Sn(t, t') — (z —z 1) simp‘
M Tn{t')p'dt' (C.67)
p 'd t'd tf
C.5
(C.68)
D erivation of scattering am plitudes
T h e functional operators V pw are defined in (4.86) as
V pw {Jw {t',<p')}
=
j
( p - w ' ) Jw(r') e i k°
r dS' ,
p = h , v \ w = r,cp
198
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.69)
T h e use of the Fourier series expansion (4.22) for the equivalent currents Jw(t', <j>') leads to w rite
expression (C.69) itself as am infinite sum m ation:
{
OO
'j
OO
£
JWn(t)ejn(t,\ = £
V pw{ j Wn( t ) e ^ }
n= —OO
77,——CO
(C.70)
J
Since
k s - r ' = p sin 9S cos (f>s + z cos 9S
(C.71)
th en the n -th term of th e series at th e right-hand side of (C.70) is
27r
j
v pw { JWnIt') e?** } =
o c
J ( p •w') JWn (t') e j n $ e j
sin 9*cos $ + * cos 9*) p'dt'dtf
(C.72)
T h e dot products in (C.72) are:
h
— 4> ■f ' — —simp 1 sin {(j)' — <f>s)
r ' =
v ■ t '— —6
h ■
=
v •
4>' =
t'
(C.73)
= —cos 0, sin i//co s (<f>' — <j>s) + s in 9a cos ip'
(C.74)
— <p ■0 ' = —cos {(j)'- 4>s)
(C.75)
— 0 ■ 4>' = cos 9S sin (</>' — <j>s)
(C.76)
Using (C.73)-(C.76) together w ith th e integral representation of th e Bessel function, expressions
(C.72) become:
V hT{ j Tn{ t ' ) e ^ } = - 2 t r j V n*‘ j sin ^ ' T + ( u sp') JTn(t') e ^ z ' p' dt'
(C.77)
c
'PVT{ j r n{'t') ejn<t>] = 2 n j n ejn(l>3J [ - j cos9s s m i p ' T ~ ( u sp') + sin 0 s cos ip'Jn (usp')\ JTn(t') e ^ VsZ p'dt' (C.78)
c
P fc* { j* n ( t V n*} -
-2 ttj ne ^
j
e
^
'
p
'
d
t
1
(C.79)
c
( 0 ejn*} = 2
where
cos 9se ^
j T + ( u sp')
(t') e ^ z ' p'dt'
Us = k 0 sin 9S ,
v 8 = k 0 cos 9S .
199
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.80)
(C.81)
S u b stitu tin g the approxim ations (4.37) into (C.77)-(C.81), and evaluating the resulting integral as
I
m
dt' = (tB - t A) f
,
( C . 8 2 )
tA
th e expressions (C.77)-(C.81) can be w ritten as
V pw { j Wn{t') e ^ }
=
^ p $ w{ks, p s) l % n { k i,q .i )e jn^ ,
p = h , v ■ w = t , 4>
(C.83)
m = l
w ith the coefficients p ^ L ( k s, p 5) given by
p % l(k 3, h 3)
= -27Tjne ^ A t m s m ^ m T ^ ( u sPm) e ^ zm
p ^ ( k s, v s)
= 2Trjnejn<f>3 A t m [ - j c o s & 3 sinipmT ~ ( u spm) + s i n 8 3 cos'ipmJn{u 3 Pm)] e i VsZrn (C.85)
p < $ (k s, h s)
= —2-Kjne^n^s A t m pm j T ~ ( u spm ) e ^ VsZm
? £ * ( =
27rjn cosd3 e ^
A t m P m T+ {u sPm ) e ^ Zm
200
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.84)
{CM )
(C.87)
Appendix D
Half-Space Problem
D .l
D .l.l
Expression o f dyadic G reen’s function com ponents
E le c tr ic d y a d ic G r e e n ’s fu n c tio n s
T he electric dyadic G reen’s functions can be w ritten as infinite sum m ation of azim uth modes:
+00
G ^ fr .r ')
E
n=
—00
GPi(p->z i p ' i z ') p q '
D
, j n(<f> — <j>)
for t = D , R
(D .l)
_p,q=p,<bz
For the direct term , th e dyadic com ponents of th e n -th m ode coefficients have th e integral form
OU
GVD,n(Pi z i P'■>z ') ~ _ 4^r J
, - j k 0 z\z - z'\
GP
f yn {kP, k 0Z-, p, p )
kp d k p ,
2 knz
p,q = P,<f>,z
(D.2)
where the kernels Q ^ n {kp, k oz; p, p') are given by
Q%n {kp, koz] p, p')
=
T + ( k pP) T + ( k pP') +
g ^ ( k p , k o z - , p, p' )
=
- j T ^ ( k p p ) T - ( k pp ' ) - j ,^ T - ( k p p ) T + ( k p p ' )
kh
k np^oz
k
^D, n(k P i k oz', P tp )
g*D%( k p , k oz -p, p')
k2
knk.
~
=
T - ( k pp) T ~ ( k p p 1)
T n (k pp) Jn(kpp')
G%n{kp,koz-,p,(/)
kL
for Z > Z1
for z < z '
T + { k pP) T + ( k pp f)
J n { k pp) T + {kpp')
%pkpz
k2
—
^2
(D.6)
for z > z'
(D.7)
J n (kpP ) T + {kpp' )
for z < z '
kn
@D,n{kpi k 0 z] Pi P )
(D.4)
(D.5)
Tn (kpP) Jn (kpp')
T - { k pp ) T - ( k p p ' ) +
k„k.
p^oz
(D.3)
Jn{kpp) JnikpP )
201
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(D.8)
and the functions T * are defined as
T^ x)
=
Jn+l ( x) ± J n - l {x)_ '
T_ ^
=
Jn + l (x)
2
Jn-l(x)
^
2
T he rem aining th e kernels Q^fn {kp , k oz; p, p') can be found by sym m etry relationships:
g i i ( k P, k 0 2 ;p,p')
= -g p
D^ ( k p, k 0Z- p \ p )
(D.10)
&D,n(^P’ ^oz't
P)
= &D,nftpik 0 z', P , p)
(D-l l )
g^
z { k P, k 0Z-,p,p')
= g % n (kP, k 0z-,p',p)
(D . 12 )
T he dyadic com ponents of the n -th mode coefficients of the reflected term of G have th e form
j
r°
,
e - j k o z { z + z')
J S ^ n {kp, k oz; p , p )
^
kp dkp ,
o
°z
Gp^ n (P>z ; p'iz ') =
p,q = p,(j>,z
(D.13)
where the kernels g™n (kp, koz; p, p') are given by
'
g i ( k p , k 0Z-,p,P')
g & ( k p , k oz; p, p')
= r ftT + M T n+ M
=
- |r „ T ; ( v ) T ; M
r„ T+( V ) T ~( V ' ) + j
r„ T " ( V ) r + ( V )
(d .u )
(D.15)
G&( k p >k o z - , P, f / )
=
(D.16)
g Z ( k P, k oz; p , p ' )
=
(D-17)
g %n (kp , k oz] p, pf )
= -
g g n {kp , k oz ] P, t J)
= % T v Jn (kpP) J n (k pP')
^
r
v j n (kpP) T + ( k pP')
(D .i8 )
(D.19)
T he rem aining kernels GV
^ n {kpi k oz] p, p') can be found by sym m etry relationships:
g ^ n (kP, k 0Z-,p,pf)
= - g ^
p n {kp , k oz- p \ p )
(D.20)
g J n ( k P, k o z ; p , p )
= g ^ n (kp, k oz; p , p)
(D-21)
g & ( k P, k oz ] p, p' )
= g ^ { k p , k oz]P' , p )
(D.22)
202
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D .1 .2
M a g n e tic d y a d ic G r e e n ’s fu n c tio n s
Sim ilarly to G , th e m agnetic dyadic G reen’s functions can also be w ritten in the form:
+00
K « ( r ,r ')
=
for t = D , R (D.23)
£
.p,q=p,<bz
For th e direct term , th e dyadic com ponents of th e n -th m ode coefficients have th e integral form
j
1
K pJ n {p, z ■,p', z') =
r°
e
1
3 ^oz\z z \
J K,p£ n {kp,koz; p , p ' ) ---------—---------- kp dkp ,
p,q = p,(j>,z
(D.24)
where th e kernels K ^ { k pi k oz\ p, p') are given by
f - j k oz [T~ {kpp) T+ {kpp') - T + [kpp) T ~ {kpp')]
for z > z'
j koz [T~ (kpp) T + ( k pP') - T + ( k pp) T - ( k pP')\
for * < z>
lCpDPtn(kp, k oz]p,p')
(D.25)
koz [Tn {kpP) Tn {k pp ) + T + (kpp)
k °z 'i Pi P)
(kpp )]
I - k 0z [Tn {kpp) Tn {k pp') + T + (kpp) T + {kpp')]
kp
^ D , n { k pi k °Ti Pi P )
for z > z
,
for z < z ‘
(D.26)
(D.27)
(kpp) Jn {kpp )
j koz [T+ {kpp) Tn (kpp') + Tn (kpp) T + [kpp')]
I C i i ( k p, k oz]p,p')
I - j koz [T+ {kpp) Tn (kpp') + Tn (kpp) T+ {kpp')\
for z > z'
,
for z < z'
(D.28)
^n{kpikoziP j)
=
- j k p T - { k p p ) J n {kpp')
(D.29)
&D, n { k p i k o z ; P , p ' )
=
0
(D.30)
T he rem aining kernels IC^n {kp, k oz: p, p ' ) can be found by sym m etry relationships:
T ^ t n { k p i k oz 'i Pi P )
=
— K-P
t y n { k p i k oz'i P
T' Dln i k P i k oz'i Pi P )
=
f c f y n ^ k p i k ° z 'i P
^ D , n { k Pi k ° z 'i Pi P )
=
—T i ^ n i k p i k o z ' i P i p )
1
1
p)
(D.31)
(D.32)
P)
(D.33)
T he dyadic com ponents of th e n -th m ode coefficients of th e reflected term of K have th e form
IC^iikp, koz] p, p')
=
- j koz [ r h T - { k pp) T + { k pp') - r v
{kpp) T~{kpp1)]
203
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(D.34)
D .2
IC&ikp, kog;p, p')
= koz [r„ T ~ ( k pp) T - { k pP') + Tv T + ( k pP) T+{kpp')}
1Cg n (kp,koz-,p,p')
=
K%n [kp, k oz; p, p')
= - k oz [ r h T + ( k pp) T + ( k pP>) + T v T ~ ( k pP) T - { k pP')\
(D.37)
S C ^ ( k p, k 0 Z-,p,p')
= j k o z [rh T+{ kpp ) T - ( k pp') + T v T ^ ( k pP) T + ( k pp')]
(D.38)
£R,n {k P’k 0z-,P,p)
= j T v k p Tn {kpp ) J n {kpp ) J n {kp P )
(D.39)
JC^n (kp, k 0 Z-,p,p')
= T h kp Jn (kpp ) T + ( k pP')
(D.40)
lCz^ ( k p, k oz]p,p')
= j T h kp Jn {kpp ) T - { k pp')
(D.41)
}C^n (kp, k 0 Z;p,pf)
= 0
(D.42)
- r v kp T + ( k pP) J n (kpP')
(D.35)
(D.36)
D erivation o f scattering am plitudes
Com bining (5.131)-(5.131) in (5.130), th e expressions (4.82) for th e scattering am plitudes can be
w ritten as, for q = h, v ,
f pq( k s , ki)
=
- 4 ) ^ r
Eq
^
^ 1 ^ ( i ' i ' ) } + ^ “ { M W ( ^ ') } ]
(D.43)
w = r , <f>
where the operators 'Hpw are defined as
=
an d
i P ^ { j ^ ( t ' , 0 ,)} + 5 pw{ j ^ ( t ' ) 0')}
C hw{ M w(t', </»')}
=
j - 7-Lvw{ M w(t', ( j ) ' ) }
C vw{ M w{ t ' ^ ' ) }
=
- 1
Co
Co
U hw{ M w{t',<j>')}
(D.44)
(D.45)
(D.46)
T he operators S pw are defined, for w — T,(j>, as
S hw{ J w{t\<t>')} = < f T h ( p - w ^ J ^ e i ^ P ' ^ ^ e - ^ ^ ' d S '
(j)')} ~
r „ [( p -w ') - 2 cos0s ( p -w ')] Jw (r') e i Usp C0S(^ e i VsZ dS'
204
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(D.47)
(D.48)
where
u s — ko sin#,, ,
vs = ko cos 9S
.
(D.49)
and the dot p ro d u cts are
p
f'
p .
=
s i n ^ cos (<p' — cps)
(D.50)
=
- s i n ( ^ / - 0 s)
(D.51)
Using the angular m ode expansion of the currents J w, th e operators S pw {Jw(t', cp')} can be w ritten
as
OC
S ? w { J w( t ' , </>')} = E
S ^ { j Wn{ t ) e ^ }
(D.52)
where
S h r{ j rn ( f ) ejn<t>}
=
J T h sin ip' T + ( u sp') JTri (t') e ~
2 t r j V n*
^ z 'p'dt'
(D.53)
c
s VT { Jrn ({) ei"*}
J
= 2ttre? * * '
T v [j cos 9S s in i//T~ {usp') +
c
+ sin # s cos ip1J n (usp')] J T„(t') e ~ 3 VsZ p'dt'
(D.54)
j Y h j T - { u sp ' ) J ^ { t ' ) e ~ jVsZ' p'dt'
(D.55)
c
$ v*{j<t>n ( 0 e ^ }
=
-
2 t t f cos 9sejn^
J T v T + ( u sp') J K (t') e ~
^ z ' p'dt'
(D.56)
c
Com bining (D.52)-(D.56) w ith the expressions of th e operators V pw from A ppendix 4 yields
U hT{ j Tn{t')ejn<t>}
=
2Kjne ^
J r i ( v s,z') s m i ; ' T + ( u sp ' ) J Tn(t')p'dt'
(D.57)
c
'H VT{ j r n {t')e:!n<t>} = 2x j ne3n^
J
[ - - f - { v s,z') j cos9s sinip'T~{usp')+
c
+ 7 +{vs ,z') sin # s cos ip' Jn (usp')] JTn(t') p'dt'
=
- 2i r f e ^ J j ^ v ^ z ^ j T - M J ^ p ' d t '
(D.58).
(D.59)
c
=
2 -K f
cosdse ^
J
'y -( v s , z ' ) T + ( u sp ' ) J (l>n(t') p'dt'
(D.60)
c
where
7p { v , z )
=
.
_
e3vz ± r p e ~ 3 vz ,
p = h,v.
205
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(D.61)
Finally, discretization of the (D.57)-(D.60) using (4.37)-(4.38) yields the expressions for th e coeffi­
cients hrrjw (k 5, P 5) ■
h g . ( k s , h s)
= - 2 -Kjn ejn,p3 A t m ^ { v s, z m ) sini;m T + { u spm )
f t ^ ( k 4, v s)
= 2 -Kjn ejn<psA t m [ - j 7 v {vs, z m) cos9s sintpmT - ( u spm)+
+ J v ( v s , z m) sin&s cos'4>m Jn (uspm)]
(D.62)
(D.63)
/ i ^ ( k s , h s)
= 2'Kjn e3n<t>s &t Tnpmj'Y £{v s, z m ) T - ( u spm )
(D.64)
^
= —27Tjin cos 9S ejn^s A t m pm % (vs,zm ) T + {uspm )
(D.65)
4,( ^ 3 , ’Vs)
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