Enhanced backward wave propagation in evanescent waveguides loaded with split ring resonators

J. Appl. Phys. 102, 044902 (2007); h ps://doi.o g/10.1063/1.2769141 102, 044902 © 2007 Ame ican Ins i u e o Physics. Enhanced backwa d wa e p opaga ion in e anescen wa eguides loaded wi h spli ing esona o s Ci e as: J. Appl. Phys. 102, 044902 (2007); h ps://doi.o g/10.1063/1.2769141 Submi ed: 18 Janua y 2007 . Accep ed: 28 June 2007 . Published Online: 20 Augus 2007 J. Ca bonell, L. J. Roglá, V. E. Bo ia, and R. Ma qués ARTICLES YOU MAY BE INTERESTED IN Elec ic- ield-coupled esona o s o nega i e pe mi i i y me ama e ials Applied Physics Le e s 88, 041109 (2006); h ps://doi.o g/10.1063/1.2166681 Spli ing esona o -based le -handed coplana wa eguide Applied Physics Le e s 83, 4652 (2003); h ps://doi.o g/10.1063/1.1631392 Enhanced backwa d wa e p opaga ion in e anescen wa eguides loaded wi h spli ing esona o s J. Ca bonell,a兲L. J. Roglá, and V. E. Bo iab兲 Ins i u o de Telecomunicaciones y Aplicaciones Mul imedia (iTEAM), Uni e sidad Poli écnica de Valencia, Camino de Ve a s/n, 46022 Valencia, Spain R. Ma quésc兲 Depa amen o de Elec ónica y Elec omagne ismo, Uni e sidad de Se illa, A . Reina Me cedes s/n, 41012 Se illa, Spain 共Recei ed 18 Janua y 2007; accep ed 28 June 2007; published online 20 Augus 2007兲 This pape epo s on he design and expe imen al cha ac e iza ion o e anescen wa eguides loaded wi h spli ing esona o s 共SRRs兲suppo ing enhanced backwa d wa e p opaga ion. Wi h espec o p e ious nega i e e ac i e index de ices based on me allic wa eguides, he use o a di ec coaxial p obe exci a ion inside an e anescen wa eguide d as ically imp o es ansmission and e lec ion cha ac e is ics o he pe iodic s uc u e. Mo eo e , he addi ion o me allic windows in e lea ed be ween consecu i e esona o s o e s an ex a deg ee o eedom in o de o con ol in e esona o couplings, and hence, de ice pa ame e s like o e all ansmission, e lec ion, and bandwid h o he backwa d wa e pass band. A nume ical and expe imen al analysis o di e en s uc u es o e coming he high loss issue o p e iously published esul s is de ailed. An expe imen al alida ion o he p oposed enhancemen s is done o pe iodic s uc u es, whe e no indi idual uning o he esona o sec ions is pe o med. The p oposed echnology mainly o e s he ad an age o minia u iza ion, and i could be combined in a u he s ep wi h s anda d design echniques o p o ide equency selec i e de ices. © 2007 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.2769141兴 I. INTRODUCTION In he las ew yea s, nega i e e ac i e index ma e ials ha e opened mo e possibili ies o he design o mic owa e de ices. Gene ally included in a b oade ca ego y o “me ama e ials,” hese a i icial composi es owe hei p op- e ies o subwa eleng h de ails o s uc u e a he han o hei chemical composi ion. Possible applica ions o his ype o de ices a e based on he use o hei unique ea u es, such as backwa d wa e p opaga ion,1nega i e e ac i e index,2 o e anescen wa e egene a ion.3E en i o da e he opic is s ill an eme ging esea ch line, one can ci e, among o he s, p omising po en ial applica ions such as imaging below he di ac ion limi wi h he so-called “supe - esolu ion,”4 cloaking,5,6de ice minia u iza ion,7o , e y gene ally speak- ing, any unusual p opaga ion e ec no achie able wi h na u- al “ma e ials.” In his con ex , a numbe o wa eguide-based backwa d p opaga ion de ices ha e been p oposed ecen ly, s a ing om he o iginal p oposal in Re . 8. In ha pape , le - handed o backwa d wa e p opaga ion was ob ained o a s uc u e composed o an a ay o spli ing esona o s 共SRRs兲9inside an e anescen wa eguide. This composi e s uc u e possesses a le -handed ansmission band sligh ly abo e he esonan equency o he esona o s, which is gen- e a ed h ough he combina ion o a nega i e e ec i e pe - mi i i y syn hesized by he e anescen wa eguide enclo- su e, and a nega i e e ec i e pe meabili y syn hesized h ough he a ay o SRRs. A e his pionee ing pape , a numbe o s udies ollowed using wa eguide s uc u es as he basis o hei design.10–14 Ye , he common d awback o poo ansmission pe o mance, which could only be pa - ially alle ia ed h ough he use o long ape ing sec ions, 共Re . 15兲, was acknowledged. Two enhancemen s we e heo- e ically in oduced in, Re . 16, which is a ecen con ibu- ion aiming a imp o ing he poo ansmission le els p e i- ously epo ed. These can be ci ed as he use o a coaxial exci a ion o eed he wa eguide s uc u e and he use o induc i e windows in e lea ed be ween he SRR a ay. This echnology makes he ansmission s uc u e e y compac in size wi h espec o o he con en ional wa eguide con igu- a ions. The p esen pape analyzes bo h ypes o enhance- men s no only in e ms o ansmission, bu also in e ms o ma ching, in e esona o coupling, loca ion, and wid h o he pass band and loss con ibu ions, hus, pa ing he way o hei u u e use owa d he applica ions. Fu he mo e, expe i- men al esul s con i ming he heo e ical wo k ha e been ob- ained. In his pape , Sec. II e iews he p oposed s uc u es, and he causes o pe o mance imp o emen associa ed wi h he wa eguide heigh educ ion, wi h he use o he coaxial eed- ing p obe and wi h he use o he induc i e windows. Sec ion III desc ibes he design and manu ac u ing o a p o o ype and he expe imen al alida ion o he nume ical p edic ions. Also, a measu ed esul gi ing e idence o backwa d wa e p opaga ion is p o ided, oge he wi h a sho discussion on a兲Elec onic mail: [email p o ec ed].es b兲Elec onic mail: [email p o ec ed].es c兲Elec onic mail: [email p o ec ed] JOURNAL OF APPLIED PHYSICS 102, 044902 共2007兲 0021-8979/2007/102共4兲/044902/8/$23.00 © 2007 Ame ican Ins i u e o Physics102, 044902-1 he main ea u es associa ed wi h he p o o ypes. Finally, Sec. IV summa izes he conclusions o his wo k. II. TOPOLOGIES FOR IMPROVING THE TRANSMISSION RESPONSE OF AN EVANESCENT WAVEGUIDE LOADED WITH AN SRR ARRAY All subsequen designs a e based on a common SRR geome y. SRRs a e all o he b oadside-coupled ype: one spli ing on each side o he subs a e and symme ically o ien ed wi h espec o hei axis.17 B oadside-coupled eso- na o s do no p esen bianiso opy and ha e a educed elec- ical size as compa ed o he pa icles p oposed by Pend y in Re . 9. Squa e-shaped de ices we e selec ed since pe o - mance was much be e han o ing-shaped ones 共be e ansmission and e lec ion cha ac e is ics兲, especially o he expe imen al esul s. This choice was jus i ied because o he coupling imp o emen be ween he i s esona o and he coaxial p obe and also be ween neighbo ing esona o s. Mo eo e , he ab ica ion p ocess p oduced mo e accu a e samples in he case o he squa e-shaped esona o s. The la e al dimension o his “pa icle” is l=3.2 mm and i has me allic ack wid hs o c=0.5 mm. The ing gap o sli is 0.5 mm wide. These dimensions, pa e ned on a low pe mi - i i y subs a e 共␧ =2.2兲wi h a hickness o 0.508 mm, gi e an app oxima e esonan equency o =8.5 GHz. The pe- iodic a ay has a pe iod o p=4 mm. Fo he simula ions, i losses a e included o he SRRs, ini e conduc i i y o cop- pe 共 ␴ Cu=5.8⫻107S/m兲and dielec ic loss angen o he subs a e 共 an d=0.0009兲a e aken. A. In luence o a educed heigh wa eguide S a ing om he s uc u e p oposed in Re . 12, he au- ho s ha e i s analyzed nume ically he double in luence o he use o educed heigh wa eguides ins ead o a ull heigh 共o s anda d兲wa eguide. Nume ical simula ions h oughou his wo k use he comme cial so wa e simula o Anso HFSS. S anda d wa eguide dimensions o X-band ope a ion co espond o hose o a WR-90 共22.86 mm⫻10.16 mm兲, whe eas he educed heigh wa eguide has a e ical dimen- sion sligh ly bigge han he SRR dimension 共4.1 mm兲.An analysis in e ms o Spa ame e s shows ha he use o a mo e-con ined shielding s uc u e a ound he esona o s in- c eases no only he amoun o powe being in oduced in- side he pe iodic s uc u e, bu also he in e esona o cou- pling. This is di ec ly obse able in Fig. 1, since ansmission and e lec ion le els a e bo h imp o ed. A ansmi ed back- wa d wa e band is obse able close o he esonan e- quency o he SRRs. The numbe o esonan equencies in his ansmi ed band equals heo e ically he numbe o eso- na o s in he ini e-leng h s uc u e 共4, in his case兲, howe e , he e ec o ma e ial losses in he dielec ic and conduc o s, combined wi h a high misma ch 共 ep esen ed by S11兲, e- duces he numbe o esonan peaks o dips. Mo eo e , e- ducing he heigh o he wa eguide also inc eases in e eso- na o couplings by widening he bandwid h o he backwa d wa e band. The inc eased in e ac ion be ween he esona o s shi s upwa ds he indi idual esonan equencies obse - able in he ansmi ed ange. In bo h cases, eeding o he simula ed s uc u es is done di ec ly h ough he de ini ion o access po s a bo h ends, wi h hei shapes co esponding o he wa eguide ans e se dimensions. The exci a ion o he esona o s inside he wa eguide can be in e p e ed h ough he o iginal ac ional a ea con- cep in oduced by Pend y o he SRRs in Re . 9共a ea o he SRR o e a ea o he uni cell兲. I was shown in Re . 18 ha he coupling o he ene gy ca ied ou inside he wa eguide by he dominan TE10 mode o he esona o is based on his ac ional a ea concep . Since he po ion o he educed heigh wa eguide occupied by he SRR is bigge , much mo e magne ic- ield lines will exci e he esona o s. I is ob ious ha coupling be ween consecu i e esona- o s also depends on he sepa a ion be ween hem. The e is a signi ican in e ac ion among neighbo ing SRRs, bu his only holds o small sepa a ion dis ances. Hence, i is he collec i e esponse o he a ay ha in luences he bandwid h o he de ice. Using a educed heigh wa eguide s eng hens he in e ac ion be ween consecu i e esona o s and he ma ching o exci a ion o he pe iodic s uc u e, yielding an inc eased bandwid h. B. Coaxial p obe exci a ion The use o a educed heigh wa eguide, as men ioned in he p e ious pa ag aph, p ecludes he use o a s anda d wa eguide ansi ion in o de o eed he pe iodic s uc u e, hus leading o he adop ion o an op imized exci a ion FIG. 1. 共Colo online兲Re lec ion and ansmission coe icien s o a ou spli ing esona o backwa d wa e s uc u e in WR-90 including induc i e windows. Inse s show s anda d heigh and educed heigh wa eguide p o o- ypes. These nume ical esul s om ull-wa e simula ions include me allic and dielec ic losses in he esona o s. 044902-2 Ca bonell e al. J. Appl. Phys. 102, 044902 共2007兲 sou ce o he pe iodic SRR a ay. Coaxial exci a ion o he de ice was hen selec ed as an imp o ed means o inc easing a ansmi ed signal ac oss an e anescen wa eguide. A his s age, “e anescence” can be p oduced wi h pe iodic induc- i e windows, as shown in he inse s o Fig. 1, o by di ec ly using a wa eguide wi h educed ans e se dimensions 共wid h and heigh 兲, which is he s a ing poin o his subsec- ion. The desc ip ion o he analyzed s uc u e can be ound in Re . 16, as a wa eguide app oach d awn om he plana p oblem s udied in Re . 18. Le us ecall ha he uni cell size in all he pe iodic s uc u es employed in he p esen pape is 4 mm 共in he p opaga ion di ec ion兲, and he guided wa eleng h a 8.5 GHz 共Xband兲is ␭g=55.5 mm in he WR-90. Roughly, he pe iodic s uc u es ha e a ela i e uni cell size o ␭g/14 as compa ed o he s anda d wa eguide case. Figu e 2illus a es he simula ed magne ic ield Hgen- e a ed in he longi udinal plane o a p o o ype s uc u e wi h 3 SRRs. This p o o ype includes a sho -ci cui ed coaxial p obe exci a ion eeding a SRR a ay in an e anescen wa e- guide. The e y s ong magne ic ield gene a ed is op imally o ien ed since i is pa allel o he axis o he SRRs in he a ay plane. This p o ides enhanced coupling be ween he p obe and he esona o s, which causes he educ ion o he e lec ion coe icien , and hence he imp o ed ma ching o he s uc u e. In his con igu a ion di ec p obe- o-p obe coupling may a ec he inal ansmission h ough he de ice. Fo hese pu poses, he s uc u e wi hou he me allic esona o s, al- hough including he dielec ic subs a e, has been simula ed. Resul s 共no shown兲demons a e ha o a p obe- o-p obe dis ance o 7.2 mm 共co esponding o a wo-SRR de ice兲, he ansmi ed signal is no signi ican and is below −50 dB. The e y e anescen wa eguide enclosu e p ecludes he di- ec ansmission in he absence o he SRRs. On op o he in luence o he educed heigh wa eguide, he ma ching p oblem is clea ly dependen on he dis ance om he p obes o he i s o las esona o s o he a ay. This can be no iced in Fig. 3, which shows nume ically com- pu ed Spa ame e s o a es s uc u e wi h ou SRRs. Ma- e ial losses a e no included in hese simula ions. Fo h ee alues o he dis ance om he i s and las esona o o he coaxial p obe 共d→0, d=0.4, and d=1 mm兲, he plo s show ha enhanced ma ching, wi h a educed e- lec ion le el, is achie ed when he p obe is close o he esona o s. This is appa en in he ansmission cu e whe e, o d=1 mm, e y selec i e esonan peaks a e de ined in he ansmission band. S11 shows ha only a hese indi idual equencies is he s uc u e comple ely ma ched. Again, he numbe o hese peaks co esponds o he numbe o esona- o s in he pe iodic s uc u e. As he coaxial p obe is close o he dielec ic subs a e wi h he p in ed SRR acks, and in he limi case bo h a e “ angen ” wi h d→0mm共Ohmic con ac be ween he p obe and he SRR ings, and hence, sho ci cui s a e a oided because o he ci cula shape o he coaxial p obe, see he inse in Fig. 3兲, he selec i i y o hese esonan peaks is educed wi h an inc eased o e all ansmis- sion le el. The e lec ion coe icien also shows an imp o ed beha io wi h educed le el as compa ed o he p e ious cases. Coupling o he ene gy o he de ice is enhanced since he elec omagne ic 共EM兲 ields a e e anescen inside he wa eguide o educed dimensions, and any small sepa a ion would imply a coupling loss. Only when hey a e e y close o he p obes, he i s and las esona o s s ongly in e ac wi h hem. The coaxial p obes locally gene a e high EM FIG. 2. 共Colo online兲Magne ic ield Hgene a ed om he sho -ci cui ed coaxial p obe. No e he ield o ien a ion is aligned wi h he spli ing eso- na o axis. FIG. 3. 共Colo online兲Nume ical esul s o di e en alues o he dis ance om he coaxial p obes o he i s and las esona o s o he a ay 共calcu- la ed ansmission and e lec ion coe icien s o a ou SRR s uc u e兲.In- se s show la e al and op zoomed iews o he design wi h indica ions on he dis ance om he p obe o he i s esona o . 044902-3 Ca bonell e al. J. Appl. Phys. 102, 044902 共2007兲 ields due o he sho ci cui s wi h he wa eguide enclosu e. The s ong cu en gene a ed locally by he sho ci cui is, hence, con ibu ing o he SRR exci a ion 共which is essen- ially magne ic兲. I is also possible o obse e in Fig. 3 ha he ou o band ejec ion le el is educed wi h a smoo he all o he ansmission cu e a high equencies. Ne e heless, o e all he ansmi ed bandwid h is in p ac ice no depen- den on his pa ame e . The ansmission and e lec ion cha ac e is ics o di e - en leng h s uc u es ha e been simula ed in he case whe e he p obes a e quasi in con ac 共d→0 case兲wi h he i s and las esona o s o he a ay 共see Fig. 4兲. Simula ions do in- clude losses in he conduc o s and he dielec ic subs a e. T ansmission cu es show a ela i ely high le el band cen- e ed close o he esonance equency o he SRRs wi h low losses. These losses sligh ly inc ease wi h he numbe o esona o s bu a e smalle han 4 dB e en o he case o he 6 SRRs s uc u e. An es ima e o he a enua ion cons an e alua ed om he maximum ansmi ed le el in he pass band o he di e en leng h s uc u es gi es an app oxima e alue o ␣ =1.06 dB/cm. This alue clea ly imp o es he pe o mance o he esul s p e iously ob ained by he au ho s in Re . 12, which was a ound ␣ =4 dB/cm, and is compa- able o he esul s o Re . 10, claiming a=0.8 dB/cm. I is impo an o no e ha o ob aining such low alues ex e nal wa eguide une s we e employed in his las wo k o ma ch he e anescen s uc u e. Also, an e anescen wa eguide o a much la ge c oss sec ion 共12 mm⫻12 mm agains 4.1 mm⫻4.1 mm in his case兲was used, hence, his imple- men a ion p o ides mo e compac ness 共ope a ion equencies a e in bo h cases close o 8 GHz兲. A lowe e anescence o he wa eguide in Re . 10 lowe s he a enua ion cons an , since in e esona o coupling is based on p oximi y and e a- nescen ields. The bandwid h o he ansmi ed band e- mains cons an wi h he leng h a ia ions, whe eas ou o band le els dec ease as he de ices a e longe , and hence, a e mo e “selec i e.” F ac ional bandwid h can be es ima ed a ound 5.7%共0.5 GHz bandwid h wi h 8.5 GHz cen al e- quency兲. These esul s a e also in line wi h p e iously pub- lished esul s o de ices based on spli ing esona o s 共in- insically na ow band s uc u es兲, wi h, o example, 5% bandwid h in Re . 10. Main conclusion is, he e o e, ha he good ma ching p o ided by he coaxial p obe eeding pe - mi s a he high ansmission le els e en o pe iodic s uc- u es su icien ly long o p o ide a good selec i i y. C. Induc i e windows The use o induc i e windows o idges in be ween he spli ing esona o s is a means o con olling he ene gy coupling be ween consecu i e esona o s, as i was s a ed in Sec. II A. The e o e, i is possible o include hem in he p o o ype oge he wi h he coaxial eeding p obe. This o e s an ex a deg ee o eedom in o de o de ine, wi hin a ce ain ange o alues, he e lec ion cha ac e is ics o he de ice as well as he bandwid h o he backwa d wa e pass band. The window canno ha e a la ge ape u e han he e anescen wa eguide wid h. In his case, he e would be no window and i he window we e e y igh 共small ape u e兲 he ene gy low would be supp essed. Fo compa ison pu poses, he pe- iodic a ay o SRRs o he p e ious sec ion has been eused bu including in e lea ed me allic windows inside he e a- nescen wa eguide. Fo a wo SRR de ice, wi h one window in e lea ed be ween he esona o s, Fig. 5shows he nume ical esul s in e ms o Spa ame e s and bandwid h as a unc ion o he window ape u e. I is obse able ha he e is an op imum window size ha minimizes e lec ion losses gi ing an im- p o ed ma ching. This op imum alue o he window ape - u e is sligh ly highe han 2.9 mm. Wi h espec o he ans- mission coe icien , S21 has a la cha ac e is ic wi h a high and qui e cons an le el in a la ge ange o window sizes. Fo FIG. 4. 共Colo online兲T ansmission and e lec ion cha ac e is ics o di e - en leng h pe iodic s uc u es. FIG. 5. 共Colo online兲T ansmission and e lec ion coe icien s a he cen al equency o he pass band and −3 dB bandwid h as a unc ion o he in e - lea ed window ape u e wid h o a wo SRR de ice. 044902-4 Ca bonell e al. J. Appl. Phys. 102, 044902 共2007兲 low alues o he window wid h, he S21 esponse is de- g aded wi h lowe ansmission. In e ms o bandwid h, i is easily deduced ha he bandwid h o he de ice inc eases oge he wi h he ape u e wid h. This bandwid h a ia ion is, hence, associa ed wi h he con ol o he in e esona o cou- plings, jus as in a classical design ule in il e heo y. Figu e 6shows he ansmission and e lec ion cha ac- e is ics o se e al s uc u es wi h di e en numbe o eso- na o s. The me allic idges employed ha e a hickness o 1 mm and a la e al sepa a ion o 2.7 mm; hese a e common alues o all cu es since he analysis is based on ini e-size pe iodic s uc u es. I is possible o obse e ha , by includ- ing he induc i e windows, he wo SRRs cu e shows a e y la ansmission band wi h a educed in-band e lec ion loss. Gene ally speaking o he whole se o cu es, bandwid h is educed wi h espec o he cu es in Fig. 4, and a ce ain con ol o he cha ac e is ics o he pass band can be achie ed h ough he adop ion o a speci ic window ape u e alue. The op imum alues, maximizing ansmission and minimizing e lec ion, o he window ape u es depend on he numbe o esona o s o he s uc u e. Ob aining hese op imum alues would equi e employing, o example, il e design echniques, which is no he pu pose o his wo k. III. EXPERIMENTAL VALIDATION AND DISCUSSION A p o o ype has been ab ica ed in o de o alida e he expec ed pe o mance enhancemen o he p oposed ea- u es, see Fig. 7. The e anescen wa eguide has ans e se dimensions o 4.1 mm⫻4.1 mm, and a segmen ed ap- p oach has been used in o de o es di e en leng h de ices, as in Re . 12. Two segmen s o a ull wid h e anescen wa e- guide and an in e lea ed window sec ion a e isible on he op pic u e o Fig. 7. P o o ypes ha e been ab ica ed includ- ing ei he wo o h ee esona o s in he p opaga ion di ec- ion. Also, wo coaxial p obe ansi ions ha e been imple- men ed s a ing om s anda d coaxial SMA connec o s inse ed in o he wa eguide body e mina ions. These con- nec o s ha e a long cen al conduc o in o de o assu e he con ac o he p obe wi h he op o he e anescen wa e- guide p oducing a sho . An in e lea ed window wi h an ap- e u e wid h o 2.7 mm has been inse ed be ween he eso- na o s, a alue close o he S11 op imum poin o he wo SRR s uc u e. This choice is jus i ied in o de o euse he same window elemen s o measu e a h ee spli ing esona- o line wi h mode a ely good ansmission and e lec ion esponse. All he samples ha e been mechanically milled on a dielec ic subs a e o ype Nel ec NY9220 om Nelco, wi h he p e iously men ioned elec ical and geome ical cha ac e is ics. An HP 8510C Vec o Ne wo k Analyze has been used, wi h a s anda d coaxial h ough- e lec -line 共TRL兲calib a- ion, o measu e he Spa ame e s in he magni ude and phase o he p o o ypes. FIG. 6. 共Colo online兲T ansmission and e lec ion cha ac e is ics o di e - en leng h pe iodic s uc u es including in e lea ed induc i e windows. FIG. 7. 共Colo online兲Segmen ed p o o ype composed o coaxial- o- e anescen wa eguide ansi ions, e anescen wa eguide po ions and squa e spli ing esona o a ays 共 op兲; compac wo SRR p o o ype wi h o al leng h o 12.6 mm 共bo om兲. 044902-5 Ca bonell e al. J. Appl. Phys. 102, 044902 共2007兲 Figu e 8shows a compa ison o he nume ical and mea- su ed esul s o a wo SRRs de ice in e ms o ansmission and e lec ion coe icien s. Bo h pa ame e s show a good ag eemen be ween he expec ed and he measu ed cha ac e - is ics, especially o he cen al equency and he bandwid h o he pass band. The la e is limi ed due o he in e lea ed window e ec and his ac esul s in he p esence o only one esonance in he S11 cha ac e is ic. Table Isumma izes he nume ical and expe imen al esul s ob ained a he cen- al equency o he pass band. A e y high ansmission has been measu ed, combined wi h a mode a e selec i i y o he pass band and a e y low e lec ion loss, hus alle ia ing he ma ching p oblems o p e ious implemen a ions. Nume ical esul s conside losses in he spli ing esona o s.19 In o de o p o e he le -handed o backwa d wa e na- u e o his pass band, basically ela ed o he spli ing eso- na o beha io and he e anescen shielding, he measu ed phase di e ence be ween wo di e en leng h ansmission lines 共l1and l2, wi h l2⬎l1兲is displayed in Fig. 9. I is pos- sible o analyze he sign o he e ac i e index associa ed wi h a p opaga ing s uc u e wi h espec o he phase di e - ence sign, analyzing i as aken om Re . 20, which assumes he enginee ing con en ion ej ␻ o he a ia ion o he elec- omagne ic wa e ⌬ ␾ = ␾ 2− ␾ 1=− ␤ l2+ ␤ l1 =− ␤ 共l2−l1兲 =− ␻ n共 ␻ 兲 c共l2−l1兲.共1兲 Then, i he phase di e ence is posi i e his would indica e he p esence o a nega i e e ac i e index o he p opaga- ion o he EM wa e. This is he case o he equency ange in Fig. 9, co esponding o he pass band o Fig. 8, i.e., 8.1–9 GHz, which is in u n close o he esonance equency o he esona o s. Abo e his equency ange, o he ejec ed band whe e he e is no ansmission, bo h phase cu es ha e equal alues, hence p oducing a null phase di e ence wi h he con- sequence o p ohibi ed p opaga ion. Abo e app oxima ely 10 GHz, he phase di e ence would u n nega i e, hen indica - ing a posi i e e ac i e index o a p opaga ing band in an uppe equency ange. Figu e 10 shows he measu ed cha ac e is ics 共S11 and S21兲 o a wo SRR de ice including o no he in e lea ed window. In his compa ison, he wid h o he e anescen wa eguide is 4.1 mm and he window employed has an ap- e u e o 2.7 mm. While he a ia ion in he ansmission le el in he pass band is small 共a ound 1.4 dB in he middle o he pass band兲, he in luence o he me allic window is g ea in e ms o he e lec ion coe icien and impo an a ia ions in he S11 cha ac e is ic can be achie ed. Fo in- s ance, since he e is a s ong in e ac ion be ween bo h eso- na o s in he absence o he induc i e window, wo eso- nances a e isible in he S11 cu e. In oducing he in e lea ed window limi s his in e ac ion and de-couples bo h esona o s, gi ing jus one esonance in he S11 cu e. In addi ion, pass band selec i i y is inc eased wi h he use o he in e lea ed window wi h a sha pe all a bo h sides o he pass band. FIG. 8. 共Colo online兲Re lec ion S11 and ansmission S21 coe icien s o a wo SRR p o o ype, compa ison be ween measu ed and nume ical esul s. TABLE I. Spa ame e compa ison o wo and h ee SRR de ices. S uc u e S-pa ame e Type Value 共dB兲 F equency 共GHz兲 2 SRRs S21 Measu ed −1.71 8.48 Simula ed −0.85 8.48 S11 Measu ed −44.65 8.42 Simula ed −18.64 8.49 3 SRRs S21 Measu ed −3.90 8.70 Simula ed −1.62 8.73 S11 Measu ed −17.79 8.73 Simula ed −19.70 8.73 FIG. 9. 共Colo online兲Unw apped ansmission S21 phase measu ed o wo and h ee SRR pe iodic s uc u es including in e lea ed windows. 044902-6 Ca bonell e al. J. Appl. Phys. 102, 044902 共2007兲 A h ee SRR s uc u e has also been measu ed in e ms o he Spa ame e s, wi h he esul s ob ained shown in Fig. 11. The ag eemen be ween he p edic ed alues o ull-wa e simula ions and he measu ed da a is easonably good. The window size emains cons an wi h espec o he wo SRR p o o ype, and hence, i is no an op imum alue in o de o ma ch he de ice. Ne e heless, o his pe iodic s uc u e s onge selec i i y has been ob ained 共wi h lowe ou -o - band ansmission le els兲and s ill a good ansmission has been measu ed. Resul s a e also included in Table I.I may be use ul o he applica ions ha a low numbe o esona o s allow o a sha p all o he pass-band edges. In con as , and as p edic ed in he simula ions, losses sligh ly inc ease wi h he numbe o esona o s in he lines, bu a e s ill accep able as compa ed o p e ious implemen a ions o wa eguide- based me ama e ial lines. Disc epancy be ween heo e ical and expe imen al alues may be explained because o ab i- ca ion ole ances, since SRRs a e qui e sensible o he me- chanical milling ole ances. IV. CONCLUSIONS In his wo k, he e y high losses associa ed wi h e- cen ly epo ed wa eguide-based me ama e ial ansmission lines a e pa ially o e come wi h he use o a di ec coaxial exci a ion and he use o in e lea ed induc i e windows in- side a backwa d wa e e anescen wa eguide loaded wi h SRRs. These wo ea u es imp o e he ansmission, e lec- ion, and bandwid h cha ac e is ics o p e iously published p o o ypes and p o ide a means o con olling he basic cha - ac e is ics o backwa d wa e p opaga ion in he pe iodic s uc u es. Measu ed ansmi ed le els may allow he u u e use o his echnology owa d he applica ions. Also, his s udy opens he possibili y o use s anda d syn hesis ech- niques in o de o mo e om pe iodic s uc u es o egula il e designs, based on backwa d wa e p opaga ion. The main ad an age o his ype o po en ial echnology is he e y compac size o he analyzed p o o ypes. ACKNOWLEDGMENTS This wo k was suppo ed by he Minis e io de Educación y Ciencia, Spanish Go e nmen , unde a Ramón y Cajal Con ac and unde he Resea ch P ojec s TEC-2004-04313- C02-01 and TEC-2004-04249.C02-02 and by Gene ali a Va- lenciana unde p ojec s ACOMP06/013 and GVA/2007/215. 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