IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 4, APRIL 2002 485
T ea men o Singula i ies and Quasi-S a ic Te ms in
he EFIE Analysis o Plana S uc u es
Gonzalo Plaza, F ancisco Mesa, Membe , IEEE, and F ancisco Medina, Senio Membe , IEEE
Abs ac —This pape epo s on some ma hema ical and nu-
me ical de ails o he applica ion o he elec ic ield in eg al equa-
ion (EFIE) me hod o he analysis o plana s uc u es p in ed
on mul ilaye ed subs a es. Closed- o m exp essions o singula
and hype singula e ms o he ans e se elec ic ield G een’s
dyadic (TEFGD) a e iden i ied so ha hey can be explici ly ex-
ac ed ou be o e sol ing he EFIE. The p oblems due o he p es-
ence o he hype singula con ibu ion, usually a gued o p eclude
he applica ion o he EFIE o plana s uc u es, a e sol ed. In ad-
di ion, a low- equency expansion o he TEFGD co esponding o
asinglelaye subs a eis ca iedou o ecognizenonsingula elec-
os a ic- ype con ibu ions ha can e en ually become e y im-
po an o compu a ional pu poses.
Index Te ms—Dyadic G een’s unc ion, elec ic ield in eg al
equa ion (EFIE), plana s uc u es.
I. INTRODUCTION
THE APPLICATION o he mixed-po en ial in eg al equa-
ion (MPIE) me hod o sol ing elec omagne ic p oblems
in plana laye ed s uc u es is o en desc ibed in he li e a u e as
conside ably mo e con enien han he use o he elec ic ield
in eg al equa ion (EFIE) echnique [1]–[3]. Two main easons
a e usually adduced. The i s eason is ha he G een’s unc-
ions in ol ed in he ke nels o he MPIE a e scala unc ions
ha can be ep esen ed by one-dimensional (1-D) Somme eld
in eg als. The second eason is ha he G een’s unc ion in he
EFIE in ol es singula i ieso highe o de ( ) han hose ap-
pea ing in he MPIE ( ). In his way, al hough a ew wo ks
ollow an EFIE o mula ion [4], [5], his me hod is no com-
monly used and i is gene ally accep ed he mani es supe io i y
o MPIE-based echniques. Ne e heless, he p esen wo k will
y oshow ha ca yingou anapp op ia e ea men o hesca -
e ed elec ic ield, he EFIE echnique can be posed in a o m
e y simila o ha ound o he MPIE, namely, he ke nel o
he EFIE can be educed o 1-D Somme eld in eg als o scala
unc ions and only - ype singula i ies ha e o be ea ed.
A e doing his, he MPIE and he EFIE me hods a e ound
compa able and may be ega ded as possible al e na i e and
compe en echniques o deal wi h plana laye ed s uc u es.
Manusc ip ecei ed Oc obe 3, 2000; e ised Ma ch 29, 2001. This wo k
was suppo ed by he Comisíón In e minis e ial de Ciencia yTecnología, Spain,
unde P ojec TIC98-0630.
G. Plaza and F. Mesa a e wi h G upo de Mic oondas, Depa amen o de Física
Aplicada I, Facul ad de In o má ica, Uni e sidad de Se illa, 41012-Se illa,
Spain (e-mail: [email p o ec ed]; [email p o ec ed]).
F. Medina is wi h G upo de Mic oondas, Depa amen o de Elec ónica y
Elec omagne ismo, Facul ad de Física, Uni e sidad de Se illa, 41012-Se illa,
Spain (e-mail: [email p o ec ed]).
Publishe I em Iden i ie S 0018-926X(02)04955-4.
This wo k will ocus on he applica ion o he EFIE o com-
pu e he sca e ed cu en s on a pe ec conduc o p in ed on
a mul ilaye ed dielec ic subs a e, which may also show uni-
axial aniso opy wi h i s op ical axis no mal o he in e ace.
In he de elopmen o he EFIE p oblem o his s uc u e, i
will be i s shown ha a con enien analy ic p ep ocessing o
he singula i ies o he ans e se elec ic ield G een’s dyadic
(TEFGD) is he key poin o o e come he appa en disad an-
ages o he EFIE. The s udy o he singula i ies in he TEFGD
will also make possible an addi ional unde s anding o he phys-
ical meaning o he singula - e ms con ibu ion o he sca e ed
elec ic ield. In a second s ep, he s udy will be comple ed
wi h he ob aining o he mos ele an nonsingula e ms in he
TEFGD o a pa icula (al hough ep esen a i e) s uc u e con-
sis ing o a pe ec conduc o p in ed on a g ounded aniso opic
dielec ic laye . Special emphasis will be pu on hose nonsin-
gula e ms ha a e o elec os a ic ype, whose con ibu ion
may become specially signi ican .
II. TRANSVERSE ELECTRIC FIELD GREEN’SDYADIC (TEFGD)
I is usually accep ed ha he mos impo an d awback o he
EFIE echnique when dealing wi h plana p in ed s uc u es is
he high-o de singula i ies exhibi ed by he ke nel o he co -
esponding in eg al equa ion, ha is, by he TEFGD. In his
way, any a emp o imp o ing he e iciencyo he EFIE should
ocus on gi ing an app op ia e ea men o he singula i ies
o his dyadic. This ea men can be g ea ly simpli ied i he
TEFGD o he mul ilaye ed s uc u e, as ha shown in Fig. 1,
is exp essed in a con enien way. Fig. 1 shows a su ace cu en
densi y suppo ed by a pe ec ly conduc ing su ace p in ed
on a g ounded dielec ic laye ed subs a e. Each laye o he
subs a e is la e ally unbounded and can be aniso opic wi h i s
op ical axis di ec ed along he axis, namely i s pe mi i i y
dyadic is gi en by
(1)
whe e symbol indica es uni ec o , is he
uni ans e se dyadic (subsc ip will indica e in he ollowing
ans e se o axis), and subsc ip e e s o he h laye . The
cons i u i ep ope ieso his ypeo subs a eexhibi ans e se
( o axis) homogenei y as well as ans e se iso opy.
Le us assume an inciden (o imposed) elec ic ield, ,
wi h an implied ime dependence ha will no be ex-
plici ly w i en hence o h, ha gi es ise o a sca e ed ield
. The EFIE o be sol ed o compu ing he sca e ed su ace
0018-926X/02$17.00 © 2002 IEEE
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486 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 4, APRIL 2002
Fig. 1. Plana pe ec conduc o p in ed on a uniaxial aniso opic dielec ic
mul ilaye ed subs a e wi h he op ical axis along
y
axis.
cu en densi y on he me allized su ace, , can be w i en
as
(2)
o , equi alen ly
(3)
whe e is he ans e se elec ic ield G een’s dyadic
and and a e he obse a ion and sou ce poin s espec i ely,
bo h on he me allized su ace . The explici dependence on
exhibi ed by he G een’s dyadic in he p e ious equa-
ionis animmedia econsequenceo he ans e sehomogenei y
o he dielec ic s uc u e. As is well known, he di ec compu-
a ion o TEFGD is a he an imp ac icable ask, which causes
o be nume ically compu ed as he in e se ans-
o m o i s co esponding spec al exp ession. The ob aining o
he spec al exp ession o he TEFGD o a mul ilaye ed and/o
aniso opic s uc u e is adequa ely documen ed in he open li -
e a u e, o example [3], [6]–[8]. When compu ing he in e se
Fou ie ans o m o he TEFGD, noncon e gen in eg als ap-
pea because o he asymp o ic linea g owing o he TEFGD
wi h he spec al a iable. This asymp o ic beha io is associ-
a ed wi h a hype singula e m (o he ype ) appea ing in
he spa ial exp ession o he TEFGD o . The exis ence
o his kind o hype singula i y in he elec ic G een’s dyadic,
which co esponds o he nea ield o an elemen a y dipole,
is a gene al ea u e o his dyadic ha does no depend on he
pa icula dielec ic p ope ies o he medium su ounding he
sou ce dipole. Since he p esence o he hype singula e m in
he spa ial TEFGD p ecludes di ec nume ical compu a ion o
he sca e ed elec ic ield in (3), he p ac ical implemen a ion
o he EFIE basically lies on he abili y o p ope ly deal wi h
he hype singula e m. I would be hen equi ed an adequa e
ex ac ion o he asymp o ic beha io o he spec al exp ession
o he TEFGD as well as he ob aining o a closed o m exp es-
sion o i s co esponding singula spa ial coun e pa .
In o de o ca y ou he a o emen ioned ea men , he spec-
al exp ession o he TEFGD will be i s ob ained and w i en
in a compac o m o simpli y he u he analysis. Then, closed
o m exp essions o he singula e ms in he spa ial TEFGD
will be ob ained by Fou ie –Bessel ans o ming he asymp o ic
beha io o he spec al exp ession o he TEFGD. Finally, a
simple one-laye aniso opic dielec ic g ounded subs a e will
be analyzed in o de o ob ain he nonsingula elec os a ic- ype
e ms in he TEFGD. These e ms will be ob ained a e ex-
panding he TEFGD in o a se ies o powe s o he equency.
As i will be explained, hey can become as ele an as he sin-
gula ones in his p ac ical s uc u e.
A. Spec al TEFGD
To compu e he spec al TEFGD o uniaxial subs a es wi h
he op ical axis no mal o he in e ace, i is con enien o use
pola spec al a iables and
(4)
whe e and a e he spec al a iables o he Fou ie ans-
o ms along and di ec ions espec i ely. Thus, he spec al
TEFGD can be con enien ly w i en as
(5)
whe e and a e adial spec al unc ions which,
a e some algeb a, can be compu ed s a ing om any o he
algo i hms abo e men ioned. (In he ollowing, he spec al/spa-
ial na u e o he di e en quan i ies will be appa en looking a
he co esponding a iables.)
A o mal exp ession o he co esponding spa ial G eens’s
dyadic can be now w i en as he in e se ans o m o (5), ha
is
(6)
whe e and
(7)
s ands o he in e se Fou ie –Bessel ans o m o in ege o de
.Conside ing now he ollowing iden i y:
(8)
he spa ial TEFGD can be w i en as
(9)
I is in e es ing o poin ou ha he TEFGD has been exp essed
in e ms o wo 1-D Somme eld- ype in eg als o wo scala
unc ions. Simila ly, he applica ion o he MPIE o mula ion
also in ol es wo 1-D Somme eld- ype in eg als o wo scala
unc ions. In his way and a e a p ope ea men o he sin-
gula i ies in he EFIE, he inal compu a ional e o becomes
qui e simila in bo h schemes.
B. Spa ial Exp essions o he Singula Te ms o he TEFGD
As is well known, he asymp o ic beha io o he spec al
TEFGD o ( is he ee-space wa enumbe ) is as-
socia ed wi h he in e se Fou ie –Bessel ans o m o he elec-
ic- ieldnea hedipolesou ce( ha is, o ,whe e is
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PLAZA e al.: TREATMENT OF SINGULARITIES AND QUASI-STATIC TERMS 487
he ee-space wa eleng h). Thus, he singula e ms in he spa-
ial TEFGD (i.e., he spec al dominan e ms in he asymp o ic
limi ) a e only ela ed o he laye ha he conduc o is p in ed
on, namely, he uppe one. In his sense, he singula e ms in
he spa ial TEFDG can be ob ained om he speci ic exp es-
sion o he TEFGD o a single laye s uc u e wi h he same di-
elec ic p ope ies as he uppe laye in he o iginal laye ed sub-
s a e. Fo his simple case, and assuming ha he single laye
is g ounded ( his pa icula s uc u e will be conside ed la e in
Sec ion IV because o i s p ac ical in e es ), he exp ession o
and is ound o be
(10)
(11)
whe e
(12)
(13)
(14)
(15)
wi h being he laye heigh . The asymp o ic beha io o
can be now ob ained by expanding unc ions
and in o a se ies o powe s wi h . This
expansion is ca ied ou a e assuming ha he hype bolic
unc ions in (12) and (13) ha e eached i s asymp o ic alues
(). This assump ion has no e ec on he singula
beha io o he spa ial unc ions o since he hype -
bolic unc ions in (10) and (11) jus accoun o he p esence o
he g ound plane and i is expec ed ha he g ound plane will
no a ec he singula beha io o he ield.
A e some algeb a, he dominan e ms in he asymp o ic ex-
pansion o and accoun ing o he singula spa ial
beha io o he TEFGD a e ound o be
(16)
(17)
whe e
(18)
(19)
The subsc ip “e ” in (18) s ands o e ec i e, since co -
esponds o he e ec i e ela i e pe mi i i y appea ing in he
compu a ion o he elec ic ield (o po en ial) on he subs a e
unde elec os a ic condi ions.
The co esponding spa ial exp ession o he asymp o ic a-
dial unc ions (16) and (17) a e now ob ained as he ollowing
in e se Fou ie –Bessel ans o ms:
(20)
(21)
Subs i u ing (20) and (21) in o (9) and a e s aigh o wa d ma-
nipula ions, he spa ial exp ession o he singula pa in he
TEFGD can be exp essed as
(22)
whe e
(23)
(24)
The comple e spa ial TEFGD can be hen decomposed as ol-
lows:
(25)
whe e s ands o he egula pa o hespa ialTEFGD.
This egula pa can be eadily compu ed by nume ically e-
e sing o he spa ial domain he spec al unc ions
and o a ew alues o he pola dis-
ance and hen p ope ly in e pola ing o ge an accu a e ap-
p oxima ion o any alue o . In p ac ice, he alues ob ained
o a e s o ed o u he analysis o di e en de ices
buil on he same subs a e and ope a ing a he same equency.
Finally, i is in e es ing o men ion ha he exp essions o
he singula e ms in he TEGFD (23) and (24) apply o any
mul ilaye ed subs a e wi h no es ic ion abou he elec ic o
magne ic p ope ies o i s laye s excep o he uppe one, which
mus be an uniaxial aniso opic dielec ic wi h he op ical axis
no mal o he in e ace.
III. SCATTERED ELECTRIC FIELD DUE TO THE SINGULAR
TERMS IN TEFGD
Once he exac exp ession o he singula e ms in he
TEFGD ha e been ob ained, his sec ion will ocus on he
compu a ion o he sca e ed ans e se elec ic ield due o
hese singula e ms. Fi s , he comple e sca e ed ans e se
elec ic ield will be spli in o wo pa s: one ela ed o he
singula e ms in he TEFGD and he o he o he egula e ms
in he TEFGD. Nex , and undamen al o u he analysis
and compu a ion o eac ion- ype in eg als, a simple ea men
will show ha he con ibu ion o he ans e se elec ic ield
associa ed wi h he singula i y in he TEFGD can be
con enien ly exp essed as a con e gen in eg al.
Using he decomposi ion (25) o he TEFGD, he comple e
sca e ed ans e se elec ic ield will be w i en as
(26)
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488 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 4, APRIL 2002
whe e
(27)
(28)
and
(29)
The indexes ( 1) and ( 3) do no mean ha he co esponding
ields a e singula wi h his adial dependence. They simply
s and o he e m o he TEFGD ha hey come om. In ac ,
he elec ic ield c ea ed by a su ace cu en densi y is egula
a any poin .
Nex , he wo e ms con ibu ing o he elec ic ield ela ed o
he singula e ms o he TEFGD will be discussed. Bo h i s nu-
me ical compu a ion and i s physical meaning will be analyzed
in de ail.
A. Elec ic Field
The in eg and in (27) has a singula i y o .How-
e e , p o ided is a con inuous unc ion [as is expec ed o
occu i is a physically alid cu en densi y], his imp ope
in eg al con e ges. Thus, he p ac ical compu a ion o (27) does
no pose any pa icula nume ical p oblem.
Wi h espec o he physical meaning o his ield, i should
be no iced ha (27) only p o ides he ans e se componen o
he comple e elec ic ield. Ne e heless, only a qual-
i a i e desc ip ion o i s na u e will be ca ied ou since a com-
ple e discussion would equi e o know he whole exp ession o
in o de o ob ain he sou ces o his ield. Following a
physical a ionale, he solenoidal pa o can be asso-
cia ed wi h he ime a ia ions o he nea magne ic lux densi y
ield (a Bio –Sa a ype ield). This magne ic ield is
due o he su ace cu en s on he conduc o s and he displace-
men cu en sassocia edwi h heelec os a ic- ypeelec ic ield
[ he elec os a ic- ype na u e o will be ex-
plici ly shown la e ]. Analogously, he i o a ional pa can be
ela ed o he su ace pola iza ion cha ge on he subs a e in-
e ace [no e ha in Elec os a ics, namely, i
]. This su ace pola iza ion cha ge gi es ise o he dis-
con inui y o he componen o no mal o he in e -
ace; i iswo h omen ion ha , ob iously, his su ace pola iza-
ion cha ge does no appea i is embedded in an homogenous
laye .
B. Elec ic Field
The in eg and o he con ibu ion (28) o he comple e ans-
e se elec ic ield in ol es a singula i y, which, in p in-
ciple, causes his imp ope in eg al no o con e ge (a leas in
he classical sense). Thus, exp ession (28) should be a he con-
side ed as a symbolic exp ession o . When olume cu -
en densi ies a e p esen , he singula i y can be ea ed
by using he me hod o ci cum en in eg a ing o e he singu-
la i y, ha is, excluding he olume su ounding he singula i y,
he so-called exclusion olume. The con ibu ion o his exclu-
sion olume is hen accoun ed o by an app op ia e dyadic [9].
Ne e heless, o he au ho s’ knowledge, his echnique has no
been ex ended o deal wi h su ace cu en densi ies as hose
appea ing in he p esen analysis. Thus, a di e en echnique o
ea he singula i y o (28) will be p oposed. This ech-
nique is based on he o mal equi alence o (28) o an elec o-
s a ic p oblem. Mo e speci ically, he dyadic in he in eg and
o (28) can be ecognized as ha co esponding o he elec o-
s a ic ield p oduced by an elemen a y dipole in an homoge-
neous medium o ela i e pe mi i i y [10]. Then, exp es-
sion (28) is o mally equi alen o he exp ession o he elec o-
s a ic ield due o a su ace dipole dis ibu ion whose pola iza-
ion, , is gi en by
(30)
This o mal equi alence o an elec os a ic p oblem makes i
possible oexp ess(28) as a con e gen imp ope in eg al.Thus,
ollowing he usual p ocedu e in elec os a ics, can be
exp essed in e ms o an equi alen linea cha ge densi y and
an equi alen su ace cha ge densi y associa ed wi h he su -
ace dipole dis ibu ion (30), ha is
(31)
whe e
wi h being he ans e se nabla ope -
a o and heno mal ec o poin ingou wa d o (seeFig.1).
P o ided ha he sca e ed su ace cu en s (o , equi alen ly, he
p oposedbasis unc ions o hesu acecu en swhenaMe hod
o Momen s is employed o sol e he p oblem [11]) ha e con-
inuous no mal componen s, no linea cha ge dis ibu ion will
be p esen in he p oblem, i.e., . In consequence, he
elec ic ield accoun ing o he - ype singula i y can be i-
nally w i en as
(32)
namely, is a pu ely i o a ional ield ha can be ob ained
as he ans e se g adien o a scala po en ial, , gi en by
(33)
A e he applica ion o he ans o ma ion in he p e ious
equa ion, he con ibu ion o he comple e ans e se elec ic
ield coming om he singula e ms in he TEFGD ha e
beenexp essedas in eg als ha onlyin ol e - ype singula -
i ies. In his sense, he ea men epo ed he e o he singula i-
ieso heTEFGDhaso e come hemajo disad an ageusually
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PLAZA e al.: TREATMENT OF SINGULARITIES AND QUASI-STATIC TERMS 489
ela ed o he use o he EFIE me hod. Thus, he EFIE shows a
well-condi ioned nume ical beha io e y simila o ha ound
when using an MPIE-based echnique. I is in e es ing o poin
ou ha he po en ial echnique has been only applied o accoun
o he hype singula con ibu ion o he elec ic ield, whe eas
he emaining pa s a e exp essed in e ms o he TEFGD, ha
is
(34)
Inligh o hisequa ion, hep esen analysiscouldbein e p e ed
as a hyb id EFIE/MPIE o mula ion.
Finally, i should be men ioned ha a simila ea men o he
- ype singula i y was epo ed by B essan and Conciau o
in [12] o deal wi h olume cu en s in a homogeneous iso opic
medium. In such homogeneous cases, accoun s o he
comple e con ibu ion o he i o a ional pa o he elec ic ield
due o he singula e ms in he G een’s dyadic. Howe e , in
non homogeneous s uc u es as hose conside ed in his wo k,
he al eady men ioned pola iza ion su ace cha ge densi y a he
dielec ic in e ace o [i.e., he scala sou ces o ]
appea s as an addi ional sou ce o he comple e i o a ional pa
o he elec ic ield due o he singula e ms in TEFGD.
IV. ELECTROSTATIC-TYPE NONSINGULAR TERMS IN TEFGD
Al hough he con ibu ion o he elec ic ield due o he sin-
gula e ms in he TEFGD is he mos ele an one a sho
dis ances o he sou ce, he e a e many p ac ical s uc u es in
which o he e ms can be simila ly meaning ul. This si ua ion is
ound, o example, in a single laye g ounded subs a e whose
heigh is signi ican ly smalle han he ee-space wa eleng h
. In such s uc u e i is ound ha he con ibu ion o he eg-
ula ans e seelec ic ield coming om henonsingula
elec os a ic- ype e ms in ol ed in he TEFGD can be as ele-
an as he ield. This sec ion will ocus on he s udy
o he a o emen ioned nonsingula elec os a ic- ype e ms o
his pa icula and p ac ical s uc u e. As in p e ious sec ions,
he laye is assumed o be dielec ic ei he iso opic o uniaxial
aniso opic wi h i s op ical axis di ec ed no mal o he in e ace.
Assuming ha hese nonsingula elec os a ic- ype e ms in
he TEFGD come om he in ini e elec os a ic images o an
elemen a y dipole placed a he dielec ic/ai in e ace, he sub-
sequen image e ms will show a dependence o he ype
, whe e is he co esponding e ical dis-
ance o he conside ed image wi h espec o he in e ace ( he
speci ic alues o will be ound la e ). Due o he as
decay o hese e ms as inc eases, i is easonable o expec
hei con ibu ion o o be signi ican only i he ypical
linea dimension o he me allized su ace is small compa ed
wi h ( he emaining nonsingula nonelec os a ic e ms a e
only expec ed o be mo e ele an han he elec os a ic e ms a
dis ances ).
In he o hcoming analysis, he mos ele an nonsingula
e ms in TEFGD will be ob ained by expanding he TEFGD
in e ms o a se ies o he ope a ing equency . This expan-
sion will also gi e he singula e ms al eady ob ained in a sim-
ple way in Sec ion II-B. The equency expansion would hen
p o ide an al e na i e app oach o ob ain he singula and o he
ep esen a i e e ms in he TEFGD. The ob aining o all hese
e ms and hei co esponding analy ical p ep ocessing can be
e y use ul o he e icien compu a ion o he eac ion in e-
g als in ol ing .
Looking a exp essions (10) and (11) o he adial spec al
unc ions, i can be obse ed ha bo h and
a e odd unc ions o he a iable . The e o e, hey can be ex-
panded in o se ies o odd powe s o as ollows:
(35)
(36)
whe e he dominan e ms a e ound o be
(37)
(38)
(39)
wi h
(40)
(41)
(subsc ip “eq” s ands o equi alen ). The in oduc ion o he
equi alen pe mi i i y and heigh o mally educes he compu-
a ion o ei he o o he possible aniso opic
uniaxial p oblem o a simple iso opic p oblem [no e ha o
an iso opic subs a e o ela i e pe mi i i y , i is ound om
(40) and (41) ha and ].
As is appa en in ligh o expansions (35) and (36),
he and e ms accoun o he
elec os a ic- ype con ibu ion, since he emaining e ms
anish as . Then, he elec os a ic- ype pa in
he TEFGD, , can be eadily ob ained by aking
in (9), ha is
(42)
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490 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 4, APRIL 2002
In o de o pe o m he in e se Fou ie –Bessel ans o ma ion
in (42), i is con enien o w i e p e iously as he ol-
lowing se ies o exponen ial unc ions o :
(43)
whe e . Now subs i u ing (43) in o (42) and a e
aking he e m by e m in e se Fou ie –Bessel ans o ma ion,
can be inally w i en as
(44)
whe e
(45)
(46)
The i s ze o-o de e m in (45) is ecognized as he
singula e m p e iously ound in (23), ha is,
. The e o e, as i was explained in
Sec ion III, his singula e m can be seen as he G een’s dyadic
associa ed wi h he ans e se elec os a ic ield due o an
elemen a y dipole in an homogeneous iso opic medium whose
pola iza ion is . The emaining
nonsingula elec os a ic- ype e ms, (46), can be seen as he
G een’s dyadics ela ed o he ans e se elec ic ield due o
he in ini e elec os a ic image dipole dis ibu ions o
he sou ce dipole dis ibu ion
(47)
Unlike he hype singula dyadic, he dyadic e ms in
(46) a e egula . Thus, hei con ibu ion o he ans e se elec-
ic ield, which can be exp essed as
(48)
can be di ec ly ob ained by nume ically compu ing he in eg als
appea ing in (46).
Taking in o accoun he e m-by- e m ela ionship be ween
exp essions (46) and (47), i is possible o ind an al e na i e
way o exp essing in e ms o elec os a ic- ype po en-
ials due o he image se ies o he equi alen su ace dipole dis-
ibu ion (30), as i was made in (32) o he hype singula case.
Thus, each e m o can be w i en as
(49)
ha is, as he ans e se g adien o he ollowing scala po en-
ial:
(50)
whe e
(51)
and
(52)
The p e ious expansion o in e ms o he ans e se
g adien o ce ain scala po en ials could be e y use ul when
compu ing he eac ion in eg als appea ing in he applica ion
o he me hod o momen s (MoM). Following he usual ea -
men epo ed in he li e a u e [12], [11], he use o he epo ed
decomposi ion oge he wi h i s analy ical p ep ocessing would
allow o ob ain possible closed o m exp essions o he co e-
sponding eac ion in eg als.
Finally, i will be b ie ly conside ed he nonelec os a ic
dominan e ms a low equencies in expansions (35) and (36),
namely and . Expanding he hype bolic unc ions
in (38) and (39) in o exponen ial e ms, i can be w i en ha
exponen ial e ms in ol ing (53)
exponen ial e ms in ol ing (54)
Now ecognizing he -independen e ms in he p e ious ex-
p essions as he - e ms in exp essions (16) and (17), he
spa ial exp ession o in (24) can be again ob ained
a e pe o ming in e se ans o ms. I is in e es ing o no e ha
he esul s ob ained in he p esen sec ion ela ed o he singula
e ms in he TEFGD plainly jus i y he app oxima ion assumed
in Sec ion II-B [i.e., ] o compu e hese e ms. Con-
ce ning he emaining nonsingula -dependen e ms in (53)
and(54),i shouldbemen ioned ha , o heau ho s’knowledge,
hei spa ial coun e pa canno be exp essed in closed o m
by analy ically pe o ming he co esponding Fou ie –Bessel
ans o m. Thus, conside ing ha in mos cases hese e ms a e
less ele an han hesingula e mso han henonsingula elec-
os a ic ones, hey ha e no been ex ac ed ou explici ly.
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PLAZA e al.: TREATMENT OF SINGULARITIES AND QUASI-STATIC TERMS 491
V. CONCLUSION
Some impo an de ails conce ning he applica ion o he
EFIE o he compu a ion o he cu en densi y sca e ed by
p in ed pe ec conduc o s on iso/aniso opic uniaxial dielec ic
mul ilaye ed subs a es ha e been examined. The singula con-
ibu ions o he ans e se elec ic ield G een’s dyadic ha e
been iden i ied and ex ac ed ou om he comple e dyadic.
I has been shown ha he majo d awback o he applica ion
o EFIE o he a o emen ioned ype o p oblems, namely, he
exis ence o a hype singula i y in he TEFGD, can be
o e come by means o a sui able analy ical ea men . This
ea men is based on Fou ie –Bessel in e se ans o ma ion
o he asymp o ic beha io o he spec al G een’s unc ion.
By exploi ing a ce ain o mal equi alence wi h an elec o-
s a ic- ype p oblem, he o iginal EFIE is modi ied in such a
way ha he new in eg al equa ion only in ol es a ke nel wi h
- ype singula i ies. Al hough singula e ms a e dominan
a e y sho dis ances o he sou ce poin , non singula elec o-
s a ic- ype e ms in he TEFGD can become signi ican unde
ce ain condi ions (namely, when , wi h being he
heigh o he uppe laye ). Fo his eason, hese e ms ha e
been also ecognized in he spa ial domain and ex ac ed ou
o a simple, al hough p ac ical, one laye s uc u e.
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Gonzalo Plaza was bo n in Cádiz, Spain, on
No embe 26, 1960. He ecei ed he Lic. and D .
deg ees in physics om he Uni e si y o Se ille,
Se ille, Spain, in 1986 and 1995, espec i ely.
He is cu en ly an Associa e P o esso in he
Depa men o Applied Physics I, Uni e si y o
Se ille. P esen ly, his esea ch in e es ocuses on
elec omagne ic p opaga ion along plana lines wi h
gene al ma e ials.
F anciscoMesa(M’94)wasbo ninCádiz,Spain,on
Ap il 15, 1965. He ecei ed he Lic. and D . deg ees
om heUni e si yo Se ille, Se ille,Spain, in1989
and 1991, espec i ely, bo h in physics.
He is cu en ly an Associa e P o esso in he De-
pa men o Applied Physics I, Uni e si y o Se ille.
His esea ch in e es ocuses on elec omagne ic
p opaga ion/ adia ion in plana lines wi h gene al
aniso opic ma e ials.
F ancisco Medina (M’90–SM’01) was bo n in Pue o Real, Cádiz, Spain, in
No embe 1960. He ecei ed he Lic. and D . deg ees om he Uni e si y o
Se ille, Se ille, Spain, in 1983 and 1987, espec i ely, bo h in physics.
F om 1986 o 1987, he spen he academic yea a he Labo a oi e de Mi-
c oondes de l’ENSEEIHT, Toulouse, F ance. F om 1985 o 1989, he was a P o-
eso Ayudan e wi h he Depa men o Elec onics and Elec omagne ism, Uni-
e si y o Se ille, and since 1990, he has been P o eso Ti ula o Elec omag-
ne ism.Heiscu en ly heHeado heMic owa esG oup,Uni e si yo Se ille.
His esea chin e es s includeanaly icaland nume icalme hods o plana s uc-
u es and ci cui s and he in luence on hese ci cui s o aniso opic ma e ials.
D . Medina was a membe o he TPC o he 23 d Eu opean Mic owa e Con-
e ence, Mad id, Spain (1993) and a membe o he Spanish TPC o he IS-
RAMT Con e ence, Málaga, Spain (1999). He is on he edi o ial boa d o he
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and ac s as
Re iewe o o he IEEE and IEE jou nals. He was he ecipien o a Minis e io
de Educación y Ciencia/Minis e e de la Reche che e la Technologie Schola -
ship.
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