Treatment of singularities and quasi-static terms in the EFIE analysis of planar structures

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 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:18:02 UTC om IEEE Xplo e. Res ic ions apply. 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 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:18:02 UTC om IEEE Xplo e. Res ic ions apply. 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) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:18:02 UTC om IEEE Xplo e. Res ic ions apply. 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 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:18:02 UTC om IEEE Xplo e. Res ic ions apply. 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) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:18:02 UTC om IEEE Xplo e. Res ic ions apply. 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. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:18:02 UTC om IEEE Xplo e. Res ic ions apply. 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. REFERENCES [1] D. C. Chang and J. X. Zheng, “Elec omagne ic modeling o passi e ci cui s elemen s in MMIC,” IEEE T ans. Mic owa e Theo y Tech., ol. 40, pp. 1741–1747, Sep . 1992. [2] J. Se cu, N. Faché, F. Libb ech , and P. Lagase, “Mixed po en ial in eg al equa ion echnique o hyb id mic os ip-slo ine mul ilaye ed ci cui s using a mixed ec angula - iangula mesh,” IEEE T ans. Mic owa e Theo y Tech., ol. 43, pp. 1162–1172, May 1995. [3] K.A. MichalskiandJ. R.Mosig,“Mul ilaye edmediaG een’s unc ions in in eg al equa ions o mula ions,” IEEE T ans. An ennas P opaga ., ol. 45, pp. 508–519, Ma . 1997. [4] J. Se cu, N. Faché, F. Libb ech , and D. De Zu e , “Full-Wa e space- domainanalysiso openmic os ipdiscon inui iesincluding hesingula cu en -edge beha io ,” IEEE T ans. Mic owa e Theo y Tech., ol. 41, pp. 1581–1588, Sep . 1993. [5] A. Toscano and L. Vegni, “Spec al dyadic G een’s unc ion o mula ion o plana in eg a ed s uc u es wi h a g ounded chi al slab,” J. Elec o- magn. Wa es Applica ., ol. 6, no. 5–6, pp. 751–769, 1992. [6] C. M. K owne, “Fou ie ans o m ma ix me hod o inding p opa ion cha ac e is ics o complex aniso opic laye ed media,” IEEE T ans. Mi- c owa e Theo y Tech., ol. MTT-32, pp. 1617–1625, Dec. 1984. [7] R. Ma qués and M. Ho no, “On he spec al dyadic G een’s unc ion o s a i ied linea media,” P oc. Ins . Elec . Eng. H, ol. 134, June 1987. [8] F. L. Mesa, R. Ma qués, and M. Ho no, “A gene al algo i hm o com- pu ing he bidimensional spec al G een’s dyad in mul ilaye ed complex bianiso opic media: he equi alen bounda y me hod,” IEEE T ans. Mi- c owa e Theo y Tech., ol. 39, pp. 1640–1649, Sep . 1991. [9] J. an Bladel, “Some ema ks on G een’s dyadic o ini in e space,” IEEE T ans. An ennas P opaga ., ol. AP-9, pp. 563–566, 1961. [10] J. D. Jackson, Classical Elec odynamics, 3 d ed. New Yo k: Wiley, 1999. [11] S. M. Rao, D. R. Wil on, and A. W. Glisson, “Elec omagne ic sca e ing by su aces o a bi a y shape,” IEEE T ans. An ennas P opaga ., ol. AP-30, pp. 409–418, May 1982. [12] M. B essan and G. Conciau o, “Singula i y ex ac ion om he elec- ic G een’s unc ion o a sphe ical esona o ,” IEEE T ans. Mic owa e Theo y Tech., ol. MTT-33, pp. 407–414, May 1985. 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. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 07,2020 a 15:18:02 UTC om IEEE Xplo e. Res ic ions apply.