On the behavior of the Sierpinski multiband fractal antenna

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4, APRIL 1998 517 On he Beha io o he Sie pinski Mul iband F ac al An enna Ca les Puen e-Balia da, Membe , IEEE, Jo di Romeu, Membe , IEEE, Ra ael Pous, Membe , IEEE, Angel Ca dama, Membe , IEEE Abs ac — The mul iband beha io o he ac al Sie pinski an enna is desc ibed in his pape . Due o i s mainly iangula shape, he an enna is compa ed o he well-known single-band bow- ie an enna. Bo h expe imen al and nume ical esul s show ha he sel -simila i y p ope ies o he ac al shape a e ans- la ed in o i s elec omagne ic beha io . A deepe physical insigh on such a beha io is achie ed by means o he compu ed cu en densi ies o e he an enna su ace, which also display some simila i y p ope ies h ough he bands. Index Te ms—An ennas, ac als, mul i equency an ennas. I. INTRODUCTION THE in e ac ion o elec omagne ic wa es wi h ac al bod- ies has been ecen ly s udied [1]–[7]. Mos ac al objec s ha e sel -simila shapes, which means ha some o hei pa s ha e he same shape as he whole objec bu a a di e en scale [8]–[11]. The cons uc ion o many ideal ac al shapes is usually ca ied ou by applying an in ini e numbe o imes an i e a i e algo i hm such as he mul iple educ ion copy machine (MRCM) algo i hm [8]. In such i e a i e p ocedu e, an ini ial s uc u e called gene a o is eplica ed many imes a di e en scales, posi ions and di ec ions, o g ow he inal ac- al s uc u e. D. L. Jagga d e al. [3] showed ha he same kind o geome ical simila i y ela ions a se e al g ow h s ages we e ound in he elec omagne ic beha io o he ac al body. The di ac ion o ac ally se a ed ape u es [5], [6] he e lec ion and ansmission coe icien s o ac al mul ilaye s [7] and he sidelobe p ope ies o some ac al a ays [2], [12], [13] a e o he examples o s udies cu en ly a ailable in he li e a u e ha ela e ac als and elec omagne ics. A i s a emp o explo e he mul i equency p ope ies o ac als as adia ing s uc u es was done in [13]. The aim in ha pape was o essay a new se o shapes o he design o mul i equency an enna a ays. Wide-band and equency- independen an ennas we e de eloped and ho oughly analyzed in he ea ly six ies [14]–[19] and some con olu ed shapes we e in es iga ed o y o elude he p inciple o he an enna adia ion pa ame e s dependence on i s physical size ela i e Manusc ip ecei ed Sep embe 18, 1996; e ised Sep embe 8, 1997. This wo k was suppo ed in pa by G an TIC-96-0724-C06-04 om he Spanish Go e nmen and by he company FRACTUS S.A. C. Puen e-Balia da is wi h he Depa men o Signal Theo y and Commu- nica ions (TSC), Poly echnic Uni e si y o Ca alonia, Ba celona, Spain. J. Romeu and R. Pous a e wi h he Poly echnic Uni e si y o Ca alonia, Ba celona, Spain. A. Ca dama is wi h he Telecommunica ion Enginee ing School, Poly ech- nic Uni e si y o Ca alonia, Ba celona, Spain. Publishe I em Iden i ie S 0018-926X(98)02678-7. o wa eleng h. Spi als and log-pe iodic s uc u es a e some examples o success ul s uc u es used o design equency- independen an ennas. F ac als migh join some o hose ea ly designs due o hei sel -scaling p ope ies [8]. Conce ning ha pa icula issue, Puen e e al. desc ibed in [20] he beha io o , a he ex en known by he au ho s, he i s ac al mul iband an enna— he Sie pinski monopole. Such a monopole displayed a simila beha io a se e al bands om bo h he inpu e u n loss and adia ion pa e ns poin s o iew. Some s eps u he in he ield o mul iband ac al an ennas we e done la e in [21]–[23]. Fu he mo e, o he in e es ing con ibu ions ega ding small [24] and equency-independen [25] ac al an ennas ha e been in oduced by Cohen e al., espec i ely. In his pape , he beha io o bo h he Sie pinski monopole and dipole is desc ibed by means o expe imen al and compu- a ional esul s and he compa ison wi h he iangula (bow- ie) an enna is done. The adia ion pa e n o he measu ed ac al dipole clea ly shows a be e esemblance a di e en equencies han hose o he monopole con i ming, hus, ha ea ly disag eemen s among pa e ns we e due o he ini e size o he conduc o g ound plane. Also, he elec ic cu en densi y dis ibu ion o e he ac al s uc u e has been compu ed, gi ing some insigh on he mul i equency beha io o he an enna. II. THE SIERPINSKI MONOPOLE A. An enna Desc ip ion The Sie pinski gaske is named a e he Polish ma hema i- cian Sie pinski who desc ibed some o he main p ope ies o his ac al shape in 1916 [8], [26]. The o iginal gaske is cons uc ed by sub ac ing a cen al in e ed iangle om a main iangle shape (Fig. 1). A e he sub ac ion, h ee equal iangles emain on he s uc u e, each one being hal o he size o he o iginal one. One can i e a e he same sub ac ion p ocedu e on he emaining iangles and i he i e a ion is ca ied ou an in ini e numbe o imes, he ideal ac al Sie pinski gaske is ob ained. In such an ideal s uc u e, each one o i s h ee main pa s is exac ly equal o he whole objec , bu scaled by a ac o o wo and so a e each o he h ee gaske s ha compose any o hose pa s. Due o his pa icula simila i y p ope ies, sha ed wi h many o he ac al shapes, i is said ha he Sie pinski gaske is a sel -simila s uc u e [23]. 0018–926X/98$10.00 1998 IEEE 518 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4, APRIL 1998 Fig. 1. The Sie pinski monopole. Each subgaske (ci cled) is a scaled e sion o he whole s uc u e. The Sie pinski gaske (also Sie pinski iangle) was cho- sen as he i s candida e o a ac al an enna due o i s esemblance o he iangula o bow- ie an enna. As shown in Fig. 1, he gaske was p in ed on a 1.588-mm- hick Cuclad 250 subs a e ( ) and moun ed o e a 800 800 mm g ound plane. The s uc u e was ed h ough a 1.5-mm diame e , 50- coaxial p obe wi h an SMA connec o on he bo om side o he plane. The an enna is a scaled e sion o he an enna desc ibed in [20] wi h a hicke subs a e o p o ide he p in ed ac al a s i e suppo . The gaske has been cons uc ed h ough i e i e a ions in his pa icula case, so i e-scaled e sions o he Sie pinski gaske a e ound on he an enna (ci cled egions in Fig. 1), he smalles one being a single iangle. I one neglec s he con ibu ion o he cen e holes o he an enna pe o mance and admi s ha he cu en lowing om he eede should concen a e o e a egion ha is compa able in size o he wa eleng h, a beha io simila o i e-scaled bow- ie an en- nas (each one ope a ing a i s esonan equency) could be expec ed. The scale ac o among he i e gaske s is , he e o e, one should look o simila i ies a equencies also spaced by a ac o o wo. B. Inpu Re u n Loss The e u n loss o he Sie pinski monopole was measu ed in an HP-8510B ne wo k analyze in he 0.05–16-GHz e- quency ange. The an enna was also simula ed on a connec ion machine CM-5 using an FDTD algo i hm. Fu he mo e, i e bow ies o he same sizes as each one o he gaske s on he Sie pinski an enna (89, 44.5, 22.3, 11.1, and 5.5 mm) we e measu ed o compa e he esul s. Fig. 2 shows he inpu TABLE I MAIN PARAMETERS OF THE MEASURED SIERPINSKI MONOPOLE e lec ion coe icien ela i e o 50 o he i e-monopole bow ies oge he wi h he Sie pinski monopole ( he plo co esponding o he Sie pinski an enna appea s a he bo om o he igu e); Table I summa izes he main pa ame e s de i ed om ha plo . The i e bands on Table I co espond o he i e ol age s anding wa e a io (VSWR) minimums o he an enna. The equencies co esponding o such minimums appea in he second column ( ). The hi d column desc ibes he ela i e bandwid h a each band o VSWR 2, he ou h one he inpu e u n loss ( ), he i h one ep esen s he equency a io be ween wo adjacen bands, and he six h one he a io be ween he heigh o each o he i e subgaske and he co esponding band equency. As i was desc ibed in [20], he Sie pinski monopole p esen s a log-pe iodic beha io wi h i e bands app oxima ely spaced by a ac o ; he an enna keeps a no able deg ee o simila i y h ough he bands, wi h a mode a e bandwid h ( 21%) a each one. To ge a be e insigh on he log-pe iodic beha io o he an enna, he inpu impedance is also shown in a semiloga i hmic scale (Fig. 3). I can be seen ha he an enna is ma ched app oxima ely a equencies (1) whe e is he speed o ligh in acuum, is he heigh o he la ges gaske , he log pe iod ( ), and a na u al numbe . The bands a e sligh ly de ia ed om hose in [20], whe e . This is ela ed o he hicke subs a e (mm as opposed o mm) and i s highe pe mi i i y ( as opposed o ), which makes he whole s uc u e appea sligh ly elec ically longe [27] han he one wi h he hinne , lowe pe mi i i y subs a e. Anyway, such a beha io is clea ly di e en o ha o he bow- ie monopole, which has he i s minimum VSWR a and he co esponding highe o de modes pe iodically spaced by a equency gap o Hz; ha is (2) This is a compa able esul o he classical one om B own and Woodwa d [28] who measu ed ( om a hinne bow ie) a i s ma ch a and a second one a om he i s . The second ma ch o he bow- ie an enna is always be e han he i s one ( dB as opposed o dB), which migh sugges ha i can ha e a mo e signi ican e ec on he Sie pinski beha io . Thus, i we assimila e each o PUENTE-BALIARDA e al.: BEHAVIOR OF THE SIERPINSKI MULTIBAND FRACTAL ANTENNA 519 Fig. 2. Inpu e lec ion coe icien ( e e ed o 50  ) o i e bow ies a scaled as each one o he i e subgaske s on he Sie pinski monopole (89, 44.5, 22.3, 11.1, and 5.5 mm). The bo om plo co esponds o he inpu e lec ion coe icien o he Sie pinski monopole. he lowes subgaske s (ci cled in dashed lines in Fig. 1) o a bow ie, he Sie pinski bands could co espond o he second one o he iangula an ennas. The equency shi owa d he o igin expe imen ed by he ac al an enna, wi h espec o he iangula , can be ela ed o he capaci i e loading o he uppe subgaske s. I is also in e es ing o no ice ha he simila i y and pe iodici y a e los in he lowe bands whe e he inpu e u n loss and a io a e close o hose o he bow ie. This ac can be ela ed o he an enna unca ion since he s uc u e is no an ideal ac al cons uc ed a e an in ini e numbe o i e a ions. Al hough an ideal ac al shape is sel simila [8] a an in ini e numbe o scales, a easible implemen a ion o he s uc u e only keeps a ce ain deg ee o simila i y o e a ini e numbe o scales, which limi s he numbe o ope a ing bands. The e o e, in he low- equency egion, he lowes ma ched equency is shi ed close o ha o he i s one o a bow ie he same size as he whole Sie pinski an enna. Such a unca ion e ec becomes mo e appa en when one looks a he cu en dis ibu ion o e he an enna (Sec ion IV). C. Radia ion Pa e ns The main cu s ( ) o he ac al monopole adia ion pa e ns whe e measu ed in a 10 7.5 7.5 m anechoic chambe . The cu s whe e measu ed a he ou uppe bands (Fig. 4), whe e simila pa e ns among bands should be expec ed. As i was shown1in [20], he pa e ns do keep a ce ain deg ee o simila i y: he pa e ns o he 1The e was a misp in in he esul s published in [20]: he ' =0 and ' =90 cu s should be swapped. 520 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4, APRIL 1998 Fig. 3. Inpu esis ance ( op) and eac ance (bo om) o he i e-i e a ion Sie pinski monopole. F equency is shown in loga i hmic scale o s ess he log-pe iodic beha io o he an enna. componen a e cha ac e ized by a wo-lobe s uc u e wi h a dip loca ed app oxima ely a an ele a ion angle o 30 ; he pa e ns display a monopole-like pa e n wi h a dip app oxima ely a he same ele a ion angle, and he azimu h cu has an ellip ic shape wi h a s onge adia ion componen owa d he axis. The pa e ns o he componen looked simila as well [20], al hough his componen is less impo an since i is usually mo e han 20 dB below he o he one. These esul s a e clea ly di e en om hose o a ypical single-band an enna such as a monopole o a bow- ie an- enna since he Sie pinski an enna has an elec ical leng h sligh ly longe han ou wa eleng hs a he uppe band and a monopole o a bow ie would ha e se e al g a ing lobes a such a high equency. Howe e , i should be no iced ha he e ec o he ini e size o he g ound plane mus be aken in o accoun when analyzing he pa e ns on Fig. 4 [19]. Fo ins ance, hose a he uppe bands show a cha ac e is ic ipple, which is due o di ac ion a he edges o he plane. The a ia ions on he ipple a e as e when equency is inc eased since he squa ed plane is ob iously no sel -scalable and he edges a e spaced a longe dis ance in e ms o he co esponding wa eleng h. Also, he expec ed null in he -axis di ec ion is hidden by he con ibu ion o he an isymme ical mode o he g ound plane o he o e all adia ed powe [19]. III. THE SIERPINSKI DIPOLE A. An enna Desc ip ion In o de o p ope ly dis inguish he eal in luence o he Sie - pinski s uc u e on he an enna adia ion pa e ns, a Sie pinski dipole was cons uc ed and measu ed. The an enna was ed by means o a coaxial ape ed balun simila o ha desc ibed in [29] o balance and ma ch he dipole h ough he whole 1:16- GHz equency ange. Basically, he balun was buil by cu ing open he ou e wall o he coax so ha a c oss-sec ional iew showed a sec o o he ou e conduc o emo ed and he ound s uc u e smoo hly yielded a wo-wi e ansmission line. Bo h a ms o he an enna we e p in ed on he same subs a e used in he monopole. The measu emen s we e ca ied ou in a oll o e azimu h con igu a ion wi h he balun moun ed along he and axis al e na i ely o minimize he e ec o bo h he o a ion axis and he balun on he and pa e ns, espec i ely. The coo dina e e e ence sys em is he same o Fig. 1. An iden ical scheme was used o measu e an equila e al bow- ie an enna he same size as he Sie pinski one. B. Radia ion Pa e ns The pa e ns o bo h he Sie pinski dipole and a bow- ie an enna o he same size (89 mm was he heigh o each a m) can be compa ed on Figs. 5 and 6. Each ow displays h ee pa e ns wi hin each one o he ou uppe bands o he Sie pinski an enna. The cu s co esponding o he Sie pinski an enna a e ep esen ed on he le -hand side o he igu e, while he same cu s co esponding o he bow ie an enna a e plo ed on he igh . Each one o he wo igu es displays a cu h ough he main planes ( , ) o bo h an ennas. The cu is no signi ican in his case due o he oll o e azimu h measu emen scheme. I is in e es ing o no ice he s ong simila i y be ween pa e ns h ough he ou bands ( ows) on he Sie pinski an enna. These simila i ies a e specially ema kable a 2, 4, 8, and 16 GHz. Again, hese esul s a e clea ly di e en o hose o he bow- ie an enna. Al hough he bow- ie an enna has many ma ched equencies as desc ibed in (2), each esonan mode has a di e en cu en densi y dis ibu ion, which is ansla ed in o a di e en pa e n o each equency. The beha io is, hus, mo e simila o ha o a single-band dipole in which inc easing he ope a ing equency esul s in an inc emen on he numbe o g a ing lobes o he pa e n. Hence, we canno ei he alk abou a dipole o a bow- ie an enna as a mul iband an enna because he pa e ns a e no held simila h ough he ma ched equencies. On he con a y, we mus hink o he Sie pinski an enna as a mul iband one since bo h he inpu e u n loss and he adia ion pa e ns a e held simila h ough he bands. Ne e heless, he an enna is no equency independen since he e is a ema kable a ia ion o bo h he pa e ns and he e u n loss h ough each log pe iod. When compa ing he dipole pa e ns o he monopole ones, he nonideali ies o bo h he balun and he g ound plane can be de ec ed. The balun does no pe ec ly balance he cu en be ween bo h a ms o he dipole, consequen ly, he cu s o e he planes o hogonal o he plane display a sligh asymme y. Such an asymme y enhances adia ion owa d he axis (see cu a 3.6 and 7.2 GHz) and ends o hide some nulls o he pa e n. Ne e heless, he lobe s uc u e becomes appa en and he simila i y among bands appea s clea ly. Fu he mo e, he expec ed nulls along he axis can be dis inguished now, which suppo s he idea ha PUENTE-BALIARDA e al.: BEHAVIOR OF THE SIERPINSKI MULTIBAND FRACTAL ANTENNA 521 Fig. 4. Main cu s o he Sie pinski monopole adia ion pa e n ( E  componen ). F om le o igh , each column co esponds o one cu ( ' = 0  ;' =90  ; =90  ) a each one o he ou uppe bands ( om op o bo om =1 : 74 GHz, =3 : 51 GHz, =6 : 95 GHz, and =13 : 89 GHz). Each pa e n is no malized wi h espec o i s own maximum. Fig. 5. F on o back cu ( ' =0  ) o bo h he (a) Sie pinski dipole adia ion pa e n and he (b) bow- ie an enna. The pa e ns co espond o he E  componen . Each ow displays h ee cu s wi hin one band. No ice he simila i y be ween ows (mul iband beha io ) as opposed o he di e ences be ween columns (no a equency-independen beha io ) on he Sie pinski an enna. he o me ipple and lobes appea ed in he monopole we e due o he g ound plane. Ano he in e es ing ea u e o he Sie pinski pa e ns is he cha ac e is ic h ee-lobe s uc u e on he cu . Such a pa e n is compa able o ha on he hi d ma ched equency o he bow- ie an enna ( GHz o mm); i we compa e such equencies o he i e bow ies [(1), Fig. 2] o hose o he Sie pinski one (2), we can see ha he hi d ma ched one on each bow ie is e y close o a Sie pinski band. Hence, i seems ha he ac al an enna could be ope a ing a 522 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4, APRIL 1998 Fig. 6. Le o igh cu ( ' = 90  ) o bo h he (a) Sie pinski dipole adia ion pa e n and he (b) bow- ie an enna. The pa e ns co espond o he E  componen . Each ow displays h ee cu s wi hin one band. No ice he simila i y be ween ows (mul iband beha io ) as opposed o he di e ences be ween columns (no a equency-independen beha io ) on he Sie pinski an enna. each band in a combina ion o wo modes co esponding o wo bow ies a di e en scales, he la ge wo king on i s hi d esonance, and he smalle on i s second one. IV. CURRENT DENSITY OVER THE SIERPINSKI ANTENNA In o de o ge a deepe physical insigh on he beha io o he Sie pinski an enna, an FDTD algo i hm was de eloped o calcula e he cu en densi y dis ibu ion o e he ac al su ace. The algo i hm was un on a connec ion machine CM- 5 using a 32 256 256 cell space. The Yee cell dimensions we e mm, mm, mm, and second-o de Mu abso bing bounda y condi ions [30] we e used. A sinusoidal signal a each o he ou uppe bands was used o exci e he an enna. A e he ansien pe iod, he cu en densi y maximum a each poin was de ec ed. On he le column o Fig. 7, he magni ude o he densi y cu en dis ibu ion a each band o e he whole an enna su ace is gi en (linea scale, a bi a y uni s). The igh column p esen s an expanded iew o he egion whe e mos o he cu en is concen a ed a each equency. All o hem a e scaled by a ac o o wo, he same scale ac o exis ing among bands. I is in e es ing o no ice ha he cu en densi y dis ibu ions on he igh column a e e y simila among hem; ha is especially ue i he e ec o he smalle wholes a he lowe bands a e neglec ed. This is a easonable app oxima ion since such wholes a e small compa ed o he wa eleng h and he cu en densi y is lowe a he egions whe e he wholes a e loca ed. The e o e, he simila i y among he pa e ns shown in he p e ious sec ion can now be explained: a each band he cu en concen a es o e a p ope ly scaled subs uc u e on he an enna which has he main con ibu ion o he o e all adia ion pa e n. I should be no iced ha his egion is smalle when equency is inc eased and ha he cu en does no each he op o he an enna a he highes bands and a la ge a ea o he s uc u e becomes his way e ec i ely disconnec ed. This phenomena is equi alen o he ac i e egion ound in he log pe iodic a ays desc ibed in he ea ly six ies by Ca el and Mayes [14], [18]. Now, an addi ional explana ion on he beha io o he ac al an enna can be gi en: when an elec omagne ic wa e is ed h ough he ip o he an enna i s a s o p opaga e o e he s uc u e owa d he la end o he gaske . When he wa e inds a clus e compa able in size o i s wa eleng h i becomes adia ed; his way he powe o he a eling wa e is los and no cu en eaches he end o he an enna. The s uc u e has many discon inui ies ha enhance adia ion and emphasizes such a p ocess. I he wa eleng h is long, he small iangles o he shape con ibu e less o he o e all adia ion and he cu en can a el u he o e i ; hen a la ge ac i e egion can be eached (which will ha e a simila shape o ha o a sho e wa eleng h) and a simila adia ion pa e n o such a lowe band will be ob ained. Finally, he plo s on Fig. 7 can explain he unca ion e ec desc ibed in Sec ion II. Al hough i is di icul o p ecisely pu bounds o he ac i e egion, i ac ually co e s a wide a ea han he enci cled egions o Fig. 1; a leas an addi ional scale le el should be aken in o accoun o p ope ly model he an enna beha io . Thus, a he lowes band he whole an enna is no la ge enough, he ac i e egion is dis o ed and he an enna looses he sel -simila i y p ope ies wi h espec he highe bands. Also, i is in e es ing o no ice ha he ac i e egion co e s an a ea longe ( di ec ion) han he ope a ing wa eleng h which can explain he simila i y o he pa e n o ha o a 1.5- dipole [19]. PUENTE-BALIARDA e al.: BEHAVIOR OF THE SIERPINSKI MULTIBAND FRACTAL ANTENNA 523 Fig. 7. Cu en densi y (magni ude) o e he Sie pinski dipole compu ed by means o he FDTD algo i hm. The le column displays he whole an enna a he ou uppe bands while he igh column shows a zoomed iew o he ac i e egion a each band. No ice he simila i y be ween he dis ibu ions o e he ac i e egion a each band. V. CONCLUSIONS Bo h expe imen al and nume ical esul s on he Sie pinski an enna ha e been p esen ed. All o hem desc ibe a mul iband beha io o he ac al an enna. This beha io is consis en om he inpu e u n loss and he adia ion pa e ns poin s o iew. The bands a e log-pe iodically spaced by a ac o o wo, he same scale ac o exis ing among simila s uc u es on he ac al shape. Thus, i can be concluded ha he sel -simila i y p ope ies o he ac al s uc u e a e ansla ed in o i s elec o- magne ic beha io . The cu en densi y o e he an enna has been also calcula ed by means o an FDTD algo i hm. The inpu e u n loss calcula ed using his algo i hm is consis en wi h he expe imen al one. The densi y cu en dis ibu ions ha e a simila shape among bands as well, hus explaining he simila i y among he pa e ns ound expe imen ally. Such dis ibu ions desc ibe an ac i e egion o e he an enna ha sel scales wi h equency and allows a simila beha io o he an enna h ough he bands. ACKNOWLEDGMENT The au ho s would like o hank D . P. E. Mayes o he Uni e si y o Illinois a U bana-Champaign and D . D. L. Jagga d o he Uni e si y o Pennsyl ania, Philadelphia, o encou aging his wo k. They would also like o hank R. Ba olom´e, A. Hijazo, F. Ben´ i ez, and X. Ga cia o con ibu ions o he measu emen s and nume ical esul s. The nume ical esul s we e ob ained a he Cen e Na ional de Calcul Pa allele en Sciences de la Te e, Pa is, F ance. REFERENCES [1] D. L. Jagga d, “On ac al elec odynamics,” in Recen Ad ances in Elec omagne ic Theo y, H. N. K i ikos and D. L. Jagga d, Eds. 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Woodwa d, “Expe imen ally de e mined adi- a ion cha ac e is ics o conical and iangula an ennas,” RCA Re ., pp. 425–452, Dec. 1952. [29] J. W. Duncan and V. P. Mine a, “100:1 bandwid h balun ans o me ,” in P oc. IRE, ol. 48, pp. 156–164, Feb. 1960. [30] G. Mu , “Abso bing bounda y condi ions o he ini e-di e ence ap- p oxima ion o he ime-domain elec omagne ic- ield equa ions,” IEEE T ans. Elec omagn. Compa ., ol. EMC-23, pp. 377–382, No . 1981. Ca les Puen e-Balia da (S’91–M’93) was bo n in Badalona, Spain, In 1968. He ecei ed he Ingenie o deg ee in elecommunica ions enginee ing om he Poly echnic Uni e si y o Ca alonia (UPC), Ba celona, Spain, in 1992, he M.S. deg ee om he Uni e si y o Illinois a U bana-Champaign (UIUC), in 1994, and he Ph.D. deg ee om he UPC, in 1997. In 1992, he ecei ed a g an om he In e depa men al Commission o Resea ch and Technological Inno a ion (CIRIT) o he Ca alan Go e nmen and joined he Elec o Op ic Sys ems Labo a o y (EOSL) a he UIUC, whe e he wo ked in lase elocime y and lida echniques un il Decembe 1993. Since 1994, he has been an Assis an P o esso a he Depa men o Signal Theo y and Communica ions a he Poly echnic Uni e si y o Ca alonia. His cu en esea ch in e es s a e ac al an ennas, lida , lase sys ems, and op ical communica ions. Jo di Romeu (S’89–M’91) was bo n in Ba celona, Spain, in 1962. He ecei ed he Ingenie o and Doc o Ingenie o deg ees in elecommunica ion en- ginee ing, bo h om he Poly echnic Uni e si y o Ca alonia (UPC), Ba celona, Spain, in 1986 and 1991, espec i ely. In 1985, he joined he Elec omagne ic and Pho- onic Enginee ing G oup o he Signal Theo y and Communica ions Depa men a he UPC. Cu en ly, he is an Associa e P o esso a he same uni e si y, whe e he is engaged in esea ch in an enna nea - ield measu emen s, an enna diagnos ics, and an enna design. Ra ael Pous (S’89–M’93) was bo n in Ba celona, Spain, in 1964. He ecei ed he Ingenie o deg ee in elecommunica ions enginee ing and he Licenciado deg ee in compu e science om he Poly echnic Uni e si y o Ca alonia (Ba celona), in 1988, he M.S. deg ee in elec ical enginee ing om he Uni- e si y o Massachuse s, Amhe s , in 1989, and he Ph.D. deg ee in elec ical enginee ing om he Uni e si y o Cali o nia, Be keley, in 1992. In 1993, he joined he acul y o he Depa men o Telecommunica ions Enginee ing a he Poly ech- nic Uni e si y o Ca alonia, Ba celona, Spain, whe e he is cu en ly employed. His esea ch in e es s a e in he a ea o compu a ional elec omagne ics and applied supe conduc i i y. D . Pous was awa ded he Fulb igh Schola ship and he Schlumbe ge Fellowship du ing his s udies. Angel Ca dama (S’67–M’73) was bo n in San iago de Compos ela, Spain, on May 13, 1944. He e- cei ed he Ingenie o de Telecommunicaci´ on deg ee om he Uni e sidad Poli ´ecnica de Mad id, in 1968, and he Sc.M. and Ph.D. deg ees in elec ical enginee ing om B own Uni e si y, P o idence, RI, in 1970 and 1973, espec i ely. In 1972, he joined he acul y o he Telecom- munica ion Enginee ing School a he Poly echnic Uni e si y o Ca alonia, Ba celona, Spain, whe e he holds he posi ion o P o esso . His esea ch in e es s ange om he de elopmen o analy ical and nume ical echniques in elec omagne ics o he design o mic owa e imaging sys ems and ada and communica ion an ennas.