IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 9, SEPTEMBER 2002 2185
On he Fas App oxima ion o G een’s Func ions in
MPIE Fo mula ions o Plana Laye ed Media
N. V. Shuley, Membe , IEEE, R. R. Boix, Membe , IEEE, F. Medina, Senio Membe , IEEE, and M. Ho no
Abs ac —The nume ical implemen a ion o he complex image
app oach o he G een’s unc ion o a mixed-po en ial in eg al-
equa ion o mula ion is examined and is ound o be limi ed o low
alues o
0
(in his con ex
0
=2
0
, whe e is he dis-
ance be ween he sou ce and he ield poin s o he G een’s unc-
ion and
0
is he ee space wa eleng h). This is a clea limi a ion
o p oblems o la ge dimension o high equency whe e his limi
is easily exceeded. This pape examines he a ious s a egies and
p oposes a hyb id me hod whe eby mos o he abo e p oblems
can be a oided. An e icien in eg al me hod ha is alid o la ge
0
is combined wi h he complex image me hod in o de o ake
ad an age o he ela i e me i s o bo h schemes. I is ound ha a
wideo e lapping egionexis sbe ween he wo echniquesallowing
a e y e icien and consis en app oach o accu a ely calcula ing
he G een’s unc ions. In his pape , he me hod de eloped o he
compu a ion o he G een’s unc ion is used o plana s uc u es
con aining bo h lossless and lossy media.
Index Te ms—An enna, complex images, G een’s unc ion,
nume ical analysis, sca e e .
I. INTRODUCTION
IN THE nume ical modeling o p in ed plana s uc u es
used in monoli hic in eg a ed mic owa e and millime e
s uc u es, i is gene ally accep ed ha he me hod o momen s
(MoM) [1] is one o he mos e icien and igo ous me hods
o he analysis o small- o-medium-sized s uc u es (up o
se e al wa eleng hs). MoM o mula ions in ei he he spa ial
o spec al domain in ol e he con e sion o an ope a o based
in eg o-di e en ial equa ion in o a ma ix equa ion ha is
subsequen ly passed o he compu e o nume ical p ocessing.
The indi idual en ies o he ma ix equi e spa ial o equency
in eg a ions in ol ing he G een’s unc ion in he app op ia e
domain and sui able basis and es ing unc ions. Mul iple
nume ical in eg a ion is usually equi ed o his s ep o he
MoM p ocedu e ha subsequen ly makes he illing ime a long
ime-consuming p ocess. This is pa icula ly ue o p oblems
ha in ol e elec ically la ge geome ies o many equencies,
such as is equi ed in g ound pene a ion ada (GPR) s udies,
which a e in e es ed in he sho pulse sca e ing om bu ied
objec s (e.g., [2] and [3]) o o he p oblems in ol ing la ge
Manusc ip ecei ed Sep embe 28, 1998. The wo k o N. V. Shuley was
suppo edinpa by heMinis yo Scienceand Educa ion,Mad id,Spainunde
G an SAB95-0424.
N. V. Shuley is wi h he School o Compu e Science and Elec ical
Enginee ing, Uni e si y o Queensland, B isbane, Qld. 4072, Aus alia.
R. R. Boix and F. Medina a e wi h he Depa men o Elec onics and
Elec omagne ism, Uni e si y o Se ille, 41012-Se ille, Spain.
M. Ho no, deceased, was wi h he Depa men o Elec onics and
Elec omagne ism, Uni e si y o Se ille, 41012-Se ille, Spain.
Publishe I em Iden i ie 10.1109/TMTT.2002.802333.
ans e se dis ances, such as in he mu ual coupling o mi-
c os ip a ays o mo e han a ew elemen s. O he ypically
la ge geome y s uc u es such as a eling-wa e an ennas o
e en ela i ely small a ays o esonan elemen s could bene i
om he as e alua ion o he G een’s unc ions ha would
conside ably speed up ma ix ill imes.
Fo p oblems in ol ing plana s a i ied media, one ecen
end ha is becoming widely accep able is o implemen a dis-
c e e complex image app oach [4]–[9]. O iginally p oposed and
o mula ed by Fang e al. [4], his me hod is essen ially an al-
e na i e means o e alua ing he ime-consuming Somme eld
in eg als ha a e in eg al ep esen a ions o he G een’s unc-
ions. The main ad an age o a complex image scheme is ha
he nume ical e alua ion o he Somme eld in eg al in con-
junc ion wi h an in e pola ion scheme is comple ely a oided.
The me hod is pa icula ly a ac i e o mixed-po en ial in e-
g al-equa ion (MPIE) o mula ions since all he G een’s unc-
ions o his o mula ion a e ypically cas in e ms o scala
unc ions in ol ing Hankel ans o ms. The ad an age o he
me hod becomes appa en when i is ealized ha he Hankel
ans o m o a sphe ical wa e can be exp essed,in closed o m,
as a summa ion o cylind ical wa es ia he Somme eld iden-
i y [11]. In addi ion, he MPIE o mula ion bene i s om he
ac ha he singula i y o bo h he scala and ec o po en-
ials a e o he o de o and, he e o e, less singula han
spa ial-domain elec ic- ield in eg al-equa ion (EFIE) o mula-
ions. Fu he mo e, he momen in eg als associa ed wi h he
singula e m a e known in closed o m o ce ain basis unc-
ions.
Un o una ely, he nume ical implemen a ion o he com-
plex image app oach is gene ally limi ed o low alues o
, whe e is he dis ance be ween he sou ce and
ield poin s o he G een’s unc ion and is he ee-space
wa eleng h. As p e iously men ioned, his is a clea limi a ion
o p oblems o la ge dimension o high equency whe e
his limi is easily exceeded. This pape examines he a ious
s a egies and p oposes a hyb id me hod whe eby mos o he
abo e p oblems can be a oided. An e icien in eg al me hod
ha is alid o la ge is combined wi h he complex image
me hod (CIM) in o de o ake ad an age o he ela i e me i s
o bo h schemes. I is ound ha a wide o e lapping egion
exis s be ween he wo echniques allowing a e y e icien
app oach o accu a ely calcula ing he G een’s unc ions.
Sec ion II e iews he CIMs cu en ly in use and explains he
di icul ies o la ge o bo h lossless and lossy s uc u es.
Sec ion III discusses he co esponding in eg al me hod and
Sec ion IV p esen s some esul s, which combine Sec ions II
and III o illus a e he echnique.
0018-9480/02$17.00 © 2002 IEEE
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2186 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 9, SEPTEMBER 2002
Fig. 1. Mul ile el plana geome y.
Wi hou loss o gene ali y, illus a ions ha e been con ined o
he scala po en ial G een’s unc ion . This componen con-
ainsall heessen ial ea u eso heSomme eldin eg al:slowly
con e gen oscilla o y beha io wi h su ace wa e poles in he
in eg and, b anch poin singula i ies, and a nonze o quasi-s a ic
limi .The wo s -casescena ioalsooccu swhensou ce and ield
poin s a e assumed o be coplana , which co esponds o he
bounda y condi ion used in o ming he MPIE. This condi ion
is assumed h oughou his pape . Again, wi hou loss o gene -
ali y, a en ion has been ocused on he mic os ip s uc u e o
illus a i e pu poses.
II. CIM
We begin by conside ing a mul ilaye ed medium, whe e
. is he sou ce
poin and is he ield poin . Le , whe e
be a unc ion ep esen ing ei he
he scala G een’s unc ion o any o he ec o po en ial en ies
o he G een’s dyad o he mul ilaye ed medium o Fig. 1. The
unc ion can be w i en in e ms o i s ze o h-o de
Hankel ans o m as ollows:
(1)
whe e he Somme eld in eg a ion pa h (SIP) is passing abo e
he poles and b anch poin a o (Fig. 2)
[6]. The unc ion can be ob ained in a ela i ely
s aigh o wa d way o he mul ilaye ed geome y o Fig. 1 by
means o i e a i e algo i hms [6]. Howe e , once
is known, he Somme eld in eg al o (1) canno be ob ained in
closed o m and, he e o e, i is no possible o ob ain an exac
analy ical exp ession o . The idea unde lying he
complex image app oach is o ob ain an accu a e app oxima-
ion o in such a way ha he Somme eld in eg als
o (1) can be de e mined in closed o m and, hus, an accu a e
analy ical app oxima ion o can be ob ained acco d-
ingly. As we shall see la e in his pape , he p oblem a ising
(a)
(b)
Fig. 2. (a) Two-le el complex in eg a ion pa hs in he complex
k
-plane. SIP
is he Somme eld in eg a ion pa h; he same pa h is also applicable o lossy
s uc u es. (b) In eg a ion con ou in he
k
-plane co esponding o (a).
om his idea is ha i is impossible o build accu a e app oxi-
ma ions o bo h and o all alues o
and , espec i ely.
In he ollowing, we will show how he CIM has been applied
in hispape o hepa icula caseo .Wewilldis inguish
be ween he wo cases.
Case 1: I is assumed ha and lie on a plane pa allel o
he in e aces inside he h laye . In his case, o e e y alue
o , we app oxima e as
whe e and
o scala po en ial
o ec o po en ial
(2)
He e, is he quasi-s a ic limi o he asymp o ic
limi o as becomes la ge. a e
he poles o and
a e he co esponding esidues
a he poles. is he unc ion o be app oxima ed
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SHULEY e al.: FAST APPROXIMATION OF GREEN’S FUNCTIONS IN MPIE FORMULATIONS FOR PLANAR LAYERED MEDIA 2187
by an exponen ial se ies and is gi en by
(3)
Case 2: and a e a he in e ace be ween he h and
h laye s. Again, . This is nea ly always he case e-
qui ed o single-laye mic os ip-based p oblems wi hou ias.
In his case, he abo e o mulas s ill hold wi h he excep ion o
he quasi-s a ic limi , which should be changed o
(4)
The complex image o mula ion as ou lined abo e is essen-
ially heme hodo Chowe al.[5] o amic os ips uc u e and
la e ex ended by Aksun and Mi a [8]. In bo h o hese pape s,
he scala and ec o po en ial G een’s unc ions in he spec al
domain we e ans o med o he spa ial domain by i s analy -
ically ex ac ing om he spec al domain, as in he abo e o -
mula ion, he su ace-wa e poles and quasi-s a ic images. The
emaining e ms a e hen handled by pe o ming a P ony, leas
squa e P ony, o ma ix pencil o unc ions (MPOF) [12] expo-
nen ial i o he emaining e ms ha cons i u e he complex im-
ages. Aksun [9] hen gene alized he CIM o mul ilaye plana
s uc u es as o mula ed he e. Howe e , in an e o o p ese e
gene ali y, whe e a p io i in o ma ion o he su ace-wa e poles
and quasi-s a ic in o ma ion may no be known explici ly, nei-
he he poles, no he quasi-s a ic e ms we e ex ac ed, bu a
me hod o p oceeding di ec ly o he en i e G een’s unc ion
was conside ed. This gene aliza ion conside ed a wo-s ep sam-
pling scheme, which add esses he p oblem o ying o simul-
aneously p ese e bo h he low beha io (p edominan ly
a - ield in o ma ion) and he slow con e gen asymp o ic be-
ha io o he spec um (p edomina ely nea - ield in o ma ion),
al hough Kipp and Chan [18] we e he i s o epo on such a
spli sampling scheme in o de o p o ide o a mo e e ec i e
u iliza ion o he exponen ial app oxima ion. Gi en ha egu-
la ly spaced da a is equi ed by any o he exponen ial i ing al-
go i hms, his s a egy ep esen s a good comp omise be ween
he con lic ing equi emen s as no ed abo e.
The applica ion o he wo-s ep exponen ial i ing algo i hm
o Aksun [9] o (3) equi es wo se s o samples o he known
unc ion . The i s se o samples is ob ained
along a de o med pa h in he -plane passing abo e
he su ace-wa e poles and b anch poin o [see
Fig. 2(a)]. Once he su ace-wa e poles and b anch poin a e
sa ely passed, he second se o samples is ob ained along a
second pa h o e he eal axis o he -plane. The com-
bined pa h is hus equi alen o he SIP as equi ed
by he Hankel ans o m. As in [9], his pape uses he MPOF
algo i hm [12] o de e mine he unknown coe icien s (am-
pli udes and exponen s) o (3). This equi es ha he samples
o a e aken in e ms o a eal a iable , which
mus be chosen in such a way ha he complex unc ion o ,
[appea ing in he exponen s o (3)] is a linea
unc ion o . Following Aksun [9], he ans o ma ions linking
and in he pa hs and o he complex -plane a e
gi en by
(5)
Full guidelines o choosing , — he poin s ha delimi he
espec i e pa hs—may be ound in [9]. Co esponding plo s o
and in he complex -plane a e shown in Fig. 2(b).
The equa ions o and used in his pape (5) a e he
same as hose o Aksun, excep o he ac ha he wa enumbe
o he open laye has been used ins ead o he wa enumbe o
he sou ce laye . This is an impo an di e ence, as discussed
by Kipp and Chan [10]. Had we used in
he exp essions o and ins ead o using
, hen and would ha e
had a b anch poin a while would s ill
ha easingleb anchpoin a [10]. Mo eo e , he eisno
concep ualdi icul yinusing he sameequa ions o ea ing he
lossy subs a e case. The pa h is he same. Whe eas i was
usedins ead o , hecomplexdielec icwould o cea change
in pa h. This di e ence ein o ces he no ion o ha ing he unc-
ion o be app oxima ed and he exac unc ion sha ing he same
b anch poin . Analy ical con inua ion equi es he unc ion o
be app oxima ed o be analy ic no only on he in eg a ion pa h
in he complex -plane, bu also in all egions o in he
neighbo hood o he pa h.The pa hs in he -and -planesa e
shown in Fig. 2. Sampling and applica ion o he MPOF algo-
i hm hen con e s he G een’s unc ion o a sum o exponen-
ials, which may be di ec ly in e ed o he spa ial domain ia
he Somme eld iden i y [11]
(6)
whe e is an a bi a y complex cons an .
I we a e dealing wi h small , he abo e p ocedu es wo k
well. Howe e , he a - ield G een’s unc ion is domina ed by
cylind ical su ace wa es and he exp ession p o ided o he
G een’s unc ion by he CIMcon ains ase ieso quasi-sphe ical
wa es o he ype shown in he le -hand-side e m o (6), which
do no ha e a clea physical meaning. Thus, we expec p ob-
lems wi h he CIM o la ge . We now illus a e some o he
abo e obse a ions wi h he canonical se o da a o [5]. He e,
weha eused he wo-s epme hodbo hwi h andwi hou ex ac-
ion o he su ace-wa e poles. We also include in he plo he
e ec o using he ee-space (called “case A” in he igu es)
as compa ed o he wa enumbe o he dielec ic laye
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2188 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 9, SEPTEMBER 2002
Fig. 3.
G
o he mic os ip da a o [5] using he wo-s ep MPOF algo i hm.

=12
:
6
,
=1
:
0
mm,
=10
GHz. Case A co esponds o using
k
0
k
in (2)–(4) and
k
in (5), whe eas case B co esponds o using
k "
0
k
ins ead o
k
0
k
in (2)–(4) and
k
p

ins ead o
k
in (5). Fo case A
cu es:
N
=35
samples on
C
,
N
=20
samples on
C
. Fo case B
cu es:
N
= 100
samples on
C
,
N
= 300
samples on
C
. Wi h/wi hou
pole ex ac ion e e s o i he su ace wa e e m(s) is/a e ex ac ed p io o
exponen ial app oxima ion.
Fig. 4. Same da a as o Fig. 3.
=50
GHz.
(called “case B”) as pe he abo e discussion (Fig. 3). The di -
e ence in sampling le els equi ed o achie e co espondence
o he cu es a low equency is shown in he cap ion. Figs. 3
and4a e o di e en equencyda ao [5].I is o beno ed ha ,
o Fig. 3, he e is only one TM su ace wa e p opaga ing, while
in Fig. 4, he e a e h ee su ace wa es. The “exac ” esul using
he me hod o a e ages [17] oge he wi h s aigh quad a u e is
shown o compa ison.
While bo h da a se s ag ee e y well wi h [5] o low
(apa om a no malizing ac o ), he e is clea ly a egion
a ound whe e he image me hod ails. The
ype o ailu e is di e en o cases A and B as e idenced by
he igu es, which seem o con i m he indings o [10]; in
pa icula , he da a o [10, Fig. 10]. Howe e , he maximum
ho izon al dis ance be o e di e gence o he wo esul s seem
o di e by an o de o magni ude compa ed o ha epo ed in
[10]. In o de o explain his, one should no e ha , whe eas in
Figs. 3 and 4, he sou ce and ield poin s a e on he bounda y
be ween he subs a e and ee space, while in [10, Fig. 10],
he sou ce and ield poin s a e inside he subs a e be ween
dielec ic laye s. Thus, he e o s esul ing om using he
w ong b anch poin ins ead o he igh one
will be ampli ied. An a emp o inc ease he numbe
o samples on so as o imp o e he a - ield esul b ough
abou inconsis en and numbe -o -samples-dependen esul s,
some imes wi h an e en wo se esul o an inc eased numbe
o samples. This is no eally su p ising when he su ace-wa e
e m is no ex ac ed om (2) and (3). Gene ally,
he exponen ial i ing p ocedu e has a p oblem wi h e ms
in ol ing a beha io , i.e., in cases whe e he su ace
wa e is domina ing, as well as he b anch-poin p oblems o
case B p e iously men ioned.
By ealizing ha he p oblem in he MPOF algo i hm exis s
when he su ace wa es a e dominan , i is ins uc i e o ex ac
he su ace-wa e con ibu ion be o e commencing he exponen-
ial app oxima ion. This p ocess makes use o he Hankel ans-
o m pai de ined by he iden i y [16]
(7)
I migh be hough ha p io ex ac ion o he su ace-wa e
con ibu ionwouldsigni ican lyimp o e hecompleximageap-
p oach in he a ield, as hese e ms a e he domina ing ones
he e. In Figs. 3 and 4, i is shown ha his is ue in case A, bu is
no ue in case B. Since case B is he case ea ed by Aksun [9],
hisseems ojus i yAksun’sapp oachina oiding heex ac ion
o he su ace-wa e e ms. De e mina ion o he su ace-wa e
con ibu ion is a ime-consuming ask (because he poles and
associa ed esidues ha e o be ound) and, in case B o Figs. 3
and 4, he bene i o ob aining he su ace-wa e con ibu ion is
los due o he ailu e o he MPOF algo i hm o la ge .
I isalsoins uc i e oexamine hedependencyon henumbe
o samples in he wo egions o Fig. 2(a). Ideally, all esul s
should be independen on he sampling scheme employed o
make he exponen ial se ies app oxima ion. Howe e , due o he
limi a ions o he nume ical algo i hm, his is a ely he case. A
la ge numbe o nume ical es s show ha he esul s a e qui e
s able wi h di e en numbe s o samples ( ) on he pa o
he spec um. This is unde s andable since once he quasi-s a ic
con ibu ion is emo ed he wo-s ep p ocess makes he spec-
um decay quickly o ze o wi h e y li le ine de ail. On he
o he hand, apid changes in he spec um can occu a low
alues o , which co espond o he a - ield egion o
he spa ial domain. I , he e o e, seems plausible o inc ease he
numbe o samples ( ) in he egion o inc eased accu acy
in he a ield. Fig. 5 shows ha good esul s o la ge can
be ob ained o a low numbe o samples. I should also be no ed
ha he numbe o samples equi ed o achie e good esul s
o when in hese igu es is subs an-
ially less han hose men ioned in [9]. This is p obably because,
in he cu en pape , he quasi-s a ic limi and he su ace-wa e
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SHULEY e al.: FAST APPROXIMATION OF GREEN’S FUNCTIONS IN MPIE FORMULATIONS FOR PLANAR LAYERED MEDIA 2189
Fig. 5. Con e gence s udy o he numbe o samples used in he a - ield
egion. Mic os ip s uc u e:

=12
:
6
,
=1
:
0
mm,
=50
GHz. Case A
wi h ex ac ion o all su ace-wa e poles.
Fig. 6. Compa ison be ween he CIM and quad a u e ( ull line) o
G
o
wo le els o loss. The CIM (do ed line) uses case A wi h all poles ex ac ed.
=10
GHz,

=4
:
4
,
=10
mm.
N
=35
samples on
C
,
N
=20
samples on
C
.
e m a e ex ac ed be o e applying he CIM. Fu he mo e, he
igh b anch poin is used in he app oxima ion o he spec al
G een’s unc ion, which a e s eps ha a e no conside ed in [9].
Howe e , when , no ma e how many samples
a e used in he applica ion o he CIM p oposed in his pape ,
he esul s o end o be e y poo . Once he s able
numbe o samples has been asce ained, he exponen ial se ies
app oxima ion is s able wi h equency da a, as shown in Fig. 6
o bo h he lossless and lossy cases. All da a o his plo using
he CIM we e calcula ed using he same numbe o samples .
Typical esul s o he CIM in ol ing lossy dielec ics ha e
no ye been ea ed in he li e a u e. As men ioned p e iously,
he -plane con ou does no need o be modi ied i he case A
o mula ion is ollowed. Fig. 6 shows some ypical da a o in-
c easing loss angen using he CIM as compa ed o quad a u e.
Again, he ailu e o he CIM o la ge and inc easing loss
angen is appa en . Once again, i appea s ha he a enua ed
su ace wa e canno be p ope ly econs uc ed. Fo he lossy
case, he su ace wa es a e decaying a while he sphe -
ical wa e decays a , he su ace wa e will hus p edomina e
a smalle o inc easing loss. Ca e needs o be exe cised
when using he CIM in high loss si ua ions whe e hese su ace
wa es con ol he quali y o he app oxima ion.
III. INTEGRAL METHOD
A. Lossless Case
I appea s om he abo e sec ion ha he only di icul y wi h
he CIM is i s inabili y o deal wi h he su ace-wa e beha io a
la ge . An in eg al echnique ha is a ac i e o la ge
beha io was p oposed in [13]. Al hough he echnique de e i-
o a es o dec easing , his exac ly sui s ou pu poses. This
app oach conside s he imagina y axis as pa o an al e na i e
pa h o he e alua ion o he Somme eld in eg al. I Cauchy’s
heo em is applied o he con ou o he -plane, shown in
Fig. 7(a), i can be shown ha he Somme eld in eg al o (1)
can be ew i en as
(8)
whe e is he modi ied Bessel unc ion o ze o h o de and
a e he esiduesa hepoles o
. Al hough he esul con ains an in ini e in eg al in
he i s e m, hemodi iedBessel unc ion decaysexponen ially
as i s a gumen inc eases, hus ensu ing a quick compu a ion o
he in ini e in eg al wi h inc easing . The second e m is an
in eg al o e he ini e in e al and is eadily compu ed.
Al hough he echniquealso equi es hepolesand hei esidues
o be e alua ed, he o al nume ical e o equi ed is ac ually
in e sely p opo ional o . As no ed by he o igina o s o
his echnique, he a ious e ms become dominan acco ding
o he dis ance om he sou ce. Thus, he e ms ha co espond
o hede iciencyo hecompleximageapp oach,may be eadily
iden i ied.
B. Lossy Case
Including losses implies ha he poles now become complex
and mo e downwa d o he eal axis (and owa d he imagina y
axis o inc easing loss). This means ha hey would no longe
be cap u ed in he lossless pa h no ed abo e. Howe e , a se
o wo con ou s and [see Fig. 7(b)] can be used o
he compu a ion o he G een’s unc ion in his case. This
new se o wo con ou s makes i possible o accoun o he
complex poles o he G een’s unc ion, as explained in [14].
E alua ion o he G een’s unc ion using he wo new con ou s
equi es ha he adi ional Somme eld b anch cu (which
di ides he complex -plane in o a p ope and an imp ope
shee ) be modi ied o a new cu . This new cu is de ined as
. O dina ily, using di e en b anch
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2190 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 9, SEPTEMBER 2002
(a)
(b)
Fig. 7. In eg a ion con ou s in he
k
-plane o he in eg al me hod o
Sec ion III. (a) Lossless case. (b) Lossy case.
cu s means ha he leaky-wa e poles, which we e all on he
lowe Riemann shee in he case o he Somme eld cu s,
can now be a p oblem i he con ou is no chosen co ec ly.
Howe e , i can be shown ha his pa icula pa h a oids he
leaky-wa e poles p o ided ha any leaky-wa e pole is no
close o he nega i e imagina y axis. Thus, wi h his choice o
in eg a ion pa h, any lossy poles p esen a e enclosed, which
was no he case o he lossless o mula ion. When Cauchy’s
heo em is applied o he wo closed con ou s and
o Fig. 7(b), i is possible o ob ain an exp ession o he
Somme eld in eg al o (1), which is a gene aliza ion o (8)
This exp ession is gi en by
(9)
Fig. 8. Compa ison o he calcula ion o
G
using he CIM (case A wi h ull
su ace-wa e pole-ex ac ion ull line) and he in eg al me hod (do ed line) o
Sec ion III,

=12
:
6
,
=1
:
0
mm,
N
=35
samples on
C
,
N
=20
samples on
C
.
IV. COMBINATION OF COMPLEX IMAGES AND
INTEGRAL METHOD
I is sugges ed ha , o all alues o , a combina ion o
he CIM and he in eg al me hod will be a e y e icien means
o calcula ing any G een’s unc ions equi ed in a mixed-po-
en ial o mula ion. As has been obse ed, he CIM can expe i-
ence p oblems in he egion o la ge . On he o he hand, he
in eg al me hod becomes e y ime consuming o small .
Clea ly he e mus be a egion o o e lap o he wo app oaches
ha will allow a ce ain amoun o lexibili y in an au oma ic se-
lec ionscheme.Twobene icial esul so hes udyo heo e lap
p ocedu e a e, i s ly, ha we a e able o check he ela i e ac-
cu acy o each o he echniques agains each o he in espec i e
egions and, secondly, since he e ms in he in eg al app oach
ha e a di ec co espondence o ce ain sec ions o he ield, we
a e able o eadily iden i y he sou ce o any inaccu acies in he
compu a ional p ocess.
Fig. 8. shows he combina ion o he wo echniques o he
lossless case. The in eg al me hod has been compu ed o an
uppe limi o 10 o he modi ied Bessel e m. I equi ed,
a highe unca ion le el gi es a mo e accu a e esul o lowe
, bu a inc eased compu a ional e o . I should be no ed
ha hepoin o depa u eo heCIMde ines heuppe endpoin
o a e y wide o e lapping egion be ween he wo me hods and
he lowe end poin can be changed by selec ion o an app o-
p ia e uppe limi o he i s in eg al in (5). As he e a e wo o
mo e su ace wa es in e ac ing in he 30- and 50-GHz plo s, he
ipple s uc u e is mo e p onounced in hose plo s.
Fig. 9 shows some new esul s o he CIM o a lossy s uc-
u e o he same pa ame e s o Fig. 6. He e, he “exac ” quad a-
u e cu e has been epea ed o con enience. Losses in he
sys em end o la en ou he cha ac e is ic “pla eau” o he
cu e and he co esponding ailu e o he MPOF algo i hm is
less d ama ic, bu as discussedea lie . The su ace wa e also ex-
pe iences a decaying mechanism and e en ually e ms ha a e
wi hou a enua iondomina e,whichlead oanin e e encephe-
nomena a la ge , shown he e a ound o he
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SHULEY e al.: FAST APPROXIMATION OF GREEN’S FUNCTIONS IN MPIE FORMULATIONS FOR PLANAR LAYERED MEDIA 2191
Fig. 9. Compa ison be ween he in eg al me hod o Sec ion III (do ed line)
and quad a u e (solid line) epea ed om Fig. 6.
=10
GHz,

=4
:
4
,
=10
mm. The unca ion le el o he i s in eg al o (9) has been se a
10
k
.
cu e. This in e ac ion has been in e p e ed as a
Zenneck–su ace-wa e in e ac ion and is due o he in e ac ion
o a pole and b anch poin [15].
Finally, i is also ins uc i e o check he claim ha he su -
ace-wa e con ibu ions indeed domina e in he a ield. We
chose a alue o co esponding o a nominal
dis ance a which CIM expe iences di icul ies. We e alua ed
he con ibu ion o he su ace-wa e e m(s) compa ed o he
o al ield, as o mula ed in (8) and(9). The pe cen agee o be-
ween he app oxima e (su ace-wa e ield) and “exac ” [com-
ple e (8)] was 4.2% o he da a o Fig. 3 and 4.8% o Fig. 4,
bo h educing o la ge . Fo he medium-loss case o Fig. 9
(), he e o inc eased o 26%, only alling o 3.5%
by . Al hough somewha equency depen-
den , such e o le els indica e ha , o high-accu acy e alua-
ion o he G een’s unc ion, he su ace-wa e e ms a e, in ac ,
no comple ely dominan when he CIM begins o ail.
The o ganiza ional s a egy in a gene alized code cons uc-
ion would, he e o e, be o compu e he G een’s unc ion using
he CIM o small and hen implemen he in eg al me hod
o in e media e dis ances. The combina ion o he CIM and in-
eg al me hod o Sec ion III o he lossless case is shown di-
ec ly in Fig. 8 and he lossy case may be in e ed om a com-
bina ion o Figs. 6 and 9. A la ge , he su ace wa es (bo h
cylind ical and Zenneck wa es) a e domina ing and he G een’s
unc ion may be se wi hou any in eg a ion a all, as explained
in[15].Theo e lapbe ween hese egionsmaybechosensome-
wha a bi a ily gi en he la ge o e lap be ween he s a egies.
Al hough hispape hasconcen a edon hescala po en ial ,
simila obse a ions may be made o he o he componen s o
he ec o po en ial dyadic .
V. CONCLUSION
Fo he compu a ion o he G een’s unc ions using an MPIE
app oach in a plana mul ilaye ed s uc u e, i is no su icien o
ely solely on he CIM o he cases whe e la ge ans e se dis-
ances a e in ol ed. This pape has demons a ed a s a egy o
combining an in eg al and he CIM o e y apid and accu a e
e alua ion o he G een’s unc ion. The me hod has also been
ex ended o he case o lossy media. Howe e , i mus be s a ed
ha he p ice paid o his combined app oach is ul ima ely an
inc ease in CPU ime o e he di ec applica ion o he CIM
alone in spi e o he e y e icien nume ical in eg a ion called
o in he app oach, as discussed in his pape .
ACKNOWLEDGMENT
This s udy was pe o med while au ho N. V. Shuley was a
Visi ing Academic wi h he Facul y o Physics, Depa men o
Elec onicsandElec omagne ics,Uni e si yo Se ille,Se ille,
Spain.
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[2] S. Vi ebskiy and L. Ca in, “Momen -me hod modeling o sho -pulse
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[4] D. G. Fang, J. J. Yang, and G. Y. Delisle, “Disc e e image heo y o
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[5] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howa d, “A closed- o m
spa ial G een’s unc ion o he hick mic os ip subs a e,” IEEE T ans.
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[6] G. Du al and M. I. Aksun, “Closed- o m G een’s unc ions o gene al
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43, pp. 1545–1551, July 1995.
[7] K. A. Michalski and J. R. Mosig,“Disc e e compleximage mixedpo en-
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[8] M.I. Aksun andR.Mi a, “De i a iono closed- o m G een’s unc ions
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[9] M. I. Aksun, “A obus app oach o he de i a ion o closed- o m
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Mic owa e Theo y Tech., ol. 44, pp. 651–658, May 1996.
[10] R. A. Kipp and C. H. Chan, “Complex image me hod o sou ces in
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magn. Wa es. Pisca away, NJ: IEEE P ess, 1995.
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aga . Mag., ol. 37, pp. 48–55, Feb. 1995.
[13] J. R. Mosig and F. E. Ga diol, “Analy ical and nume ical echniques in
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[14] J. R. Mosig and A. A. Melcon, “G een’s unc ions in laye ed media:
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AP-S Con ., Bal imo e, MD, July 1996, pp. 416–419.
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2192 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 9, SEPTEMBER 2002
N. V. Shuley (S’79–M’85) was bo n in Melbou ne,
Aus alia, in 1951. He ecei ed he B.E.Sc. and
M.E.Sc. deg ees in elec ical enginee ing om he
Uni e si y o New Sou h Wales, N.S.W., Aus alia,
in 1973 and 1975, espec i ely, and he Ph.D.
deg ee in elec ical enginee ing om he Chalme s
Uni e si y o Technology, Gö ebo g, Sweden, in
1985.
F om 1977 o 1978, he was an RF Enginee wi h
Mic owa e Associa es, Duns able, U.K., whe e he
was mainly conce ned wi h he design and de elop-
men o mic owa e solid-s a e powe gene a ion. F om 1979 o 1985, he was a
Resea ch and Teaching Assis an and la e a Pos -Doc o al Scien is , un il May
1988, wi h he Chalme s Uni e si y o Technology. In May 1988, he joined
he Depa men o Elec ical Enginee ing, Uni e si y o Queensland, B isbane,
Qld., Aus alia, as a Senio Lec u e . He has ca ied ou consul ancy wo k o
No dic indus y since 1981 and is an acc edi ed consul an o he Eu opean
Space Agency. His cu en esea ch in e es s include mic owa e an ennas,
dich oic su aces, and analy ical and compu e -aided elec omagne ics.
R. R. Boix (M’96) was bo n in Melilla, Spain,
in 1962. He ecei ed he Licenciado and Doc o
deg ees in physics om he Uni e si y o Se ille,
Se ille, Spain, in 1985 and 1990, espec i ely.
Since 1985, he has been wi h he Elec onics and
Elec omagne ics Depa men , Uni e si y o Se ille,
whe e he became an Associa e P o esso in 1994.
Du ing he summe s o 1991 and 1992, he was a
Visi ing Schola wi h he Elec ical Enginee ing
Depa men , Uni e si y o Cali o nia a Los Angeles
(UCLA). Du ing he summe o 1996, he was a
Visi ing Schola wi h he Elec ical and Compu e Enginee ing Depa men ,
Sy acuse Uni e si y, Sy acuse, NY. His cu en esea ch in e es a e ocused
on he analysis o he e ec s o complex subs a es on he pe o mance
o plana passi e mic owa e ci cui s, plana pe iodic ansmission lines,
equency-selec i e su aces, and p in ed ci cui an ennas.
D . Boix is on he Edi o ial Boa d o he IEEE TRANSACTIONS ON
MICROWAVE THEORY AND TECHNIQUES.
F. Medina (M’90–SM’01) was bo n in Pue o Real,
Cádiz, Spain, in No embe 1960. He ecei ed he
Licenciado and Doc o 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
wi h he Labo a oi e deMic oondesde l’ENSEEIHT,
Toulouse, F ance. F om 1985 o 1989, he was a P o-
eso Ayudan e(Assis an P o esso )wi h heDepa -
men o Elec onics and Elec omagne ism, Uni e -
si y o Se ille, and since 1990, he has been a P o eso
Ti ula (Associa e P o esso ) o elec omagne ism. He is also cu en ly Head o
he Mic owa es G oup, Uni e si y o Se ille. His esea ch in e es includes an-
aly ical and nume ical me hods o guidance, esonan and adia ing s uc u es,
passi e plana ci cui s, and he in luence on hese ci cui s o aniso opic ma e-
ials.
D . Medina was a membe o bo h he Technical P og am Commi ee (TPC)
o he 23 d Eu opean Mic owa e Con e ence, Mad id, Spain, 1993, and he
TPC o ISRAMT’99, Malaga, Spain. He is on he Edi o ial Boa d o he IEEE
TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He has been a e-
iewe o o he IEEE and Ins i u ion o Elec ical Enginee s (IEE), U.K., pub-
lica ions.
M. Ho no was bo n in To e del Camp, Jaén, Spain,
and died in Sep embe 1998, in Se ille, Spain. He
ecei ed he Licenciado and Doc o deg ees om
he Uni e si y o Se ille, Se ille, Spain, in 1969 and
1972, espec i ely, bo h in physics.
In Oc obe 1969, he joined he Depa men o
Elec onics and Elec omagne ism, Uni e si y o
Se ille, whe e he became an Assis an P o esso in
1970, Associa e P o esso in 1975, and P o esso in
1986. His main ields o in e es included bounda y
alue p oblems in elec omagne ic heo y, wa e
p opaga ion h ough aniso opic media, and mic owa e in eg a ed ci cui s.
Du ing his inal yea s, he was engaged in he analysis o plana ansmission
lines embedded in aniso opic ma e ials, mul iconduc o ansmission lines,
and plana an ennas. He was a membe o he Elec omagne ism Academy,
Massachuse s Ins i u e o Technology (MIT), Camb idge.
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