Demonstration of negative refraction of microwaves

Demons a ion o nega i e e ac ion o mic owa es Ma ia C. Velazquez-Ahumada, Manuel J. F ei e,a兲Jose M. Alga in, and Rica do Ma ques Depa men o Elec onics and Elec omagne ism, Uni e si y o Se ille, 41012 Se ille, Spain 共Recei ed 30 June 2010; accep ed 7 Decembe 2010兲 An expe imen al se up o demons a e nega i e e ac ion is desc ibed. A simple me hod o designing and ab ica ing a me ama e ial wi h nega i e e ac i e index a mic owa e equencies is discussed. The me ama e ial is made o a mul ilaye plana a angemen o la uni cells. A p ism was ab ica ed and used o demons a e nega i e e ac ion a he p ism-ai in e ace. The p ism is designed o demons a ions and wo ks a he equency o comme cial mic owa e ansmi e s and ecei e s. © 2011 Ame ican Associa ion o Physics Teache s. 关DOI: 10.1119/1.3537122兴 I. INTRODUCTION In he well-known law o e ac ion in op ics, he angles o incidence and e ac ion a e measu ed on di e en sides om he no mal o he bounda y be ween wo o dina y iso opic media 关see Fig. 1共a兲兴. In ecen yea s, a new class o ma e- ials, called me ama e ials,1ha e appea ed which do no obey he law o e ac ion. Ins ead, a he bounda y be ween an o dina y medium and a me ama e ial, he angles o inci- dence and e ac ion can be on he same side om he no - mal 关see Fig. 1共b兲兴. This phenomenon is known as nega i e e ac ion and is a p ope y o ma e ials o which bo h he ela i e pe mi i i y ␧ and he ela i e pe meabili y ␮ a e bo h nega i e.2Me ama e ials a e composi e ma e ials wi h e ec i e alues o ␧ and ␮ de e mined by hei s uc u e a he han by he in insic p ope ies o he ma e ial compo- nen s 共which a e con en ional conduc o s and dielec ics兲. In his pape , an expe imen al se up o demons a ing nega i e e ac ion is discussed. The demons a ion uses elec omagne ic wa es wi h a equency in he mic owa e band. A p ism o me ama e ial is designed and ab ica ed o ope a e in combina ion wi h comme cial mic owa e ans- mi e s and ecei e s commonly used in physics labo a o ies. II. THEORY Me ama e ials wi h ␧ ⬍0 and ␮ ⬍0 a e also known as le -handed ma e ials o nega i e e ac i e index ma e ials, in con as o o dina y ma e ials which a e igh handed. The eason o labeling ␧ ⬍0 and ␮ ⬍0 media as le -handed media can be b ie ly explained as ollows.2Conside he Maxwell equa ions, ⵜ⫻E=−i ␻ ␮ H,共1兲 ⵜ⫻H=i ␻ ␧E,共2兲 whe e ␧=␧ ␧0, ␮ = ␮ ␮ 0, and ␧0and ␮ 0a e he dielec ic pe mi i i y and magne ic pe meabili y in acuum. Fo he plane wa es E=E0exp共−ik· +i ␻ 兲and H=H0exp共−ik· +i ␻ 兲, Eqs. 共1兲and 共2兲 educe o k⫻E= ␻ ␮ H,共3兲 k⫻H=− ␻ ␧E.共4兲 We see ha o posi i e ␧and ␮ ,E,H, and k o m a igh - handed o hogonal sys em o ec o s 关see Fig. 2共a兲兴. How- e e , i ␧ ⬍0 and ␮ ⬍0, hen Eqs. 共3兲and 共4兲can be e- w i en as k⫻E=− ␻ 兩 ␮ 兩H,共5兲 k⫻H= ␻ 兩␧兩E,共6兲 showing ha E,H, and k o m a le -handed iple , as illus- a ed in Fig. 2共b兲. The main implica ion o his analysis is backwa d-wa e p opaga ion. The di ec ion o he ime-a e aged lux o en- e gy is de e mined by he eal pa o he Poyn ing ec o , S=1 2E⫻Hⴱ,共7兲 which is una ec ed by a simul aneous change o sign o ␧ and ␮ . Thus, E,H, and Ss ill o m a igh -handed iple in a le -handed medium. In such media, he ene gy and wa e on s a el in opposi e di ec ions. Now conside e ac ion a he in e ace be ween an o di- na y and a le -handed medium. The bounda y condi ions equi e he con inui y o he angen ial componen s o he wa e ec o along he in e ace. F om he backwa d p opa- ga ion in he le -handed egion i ollows ha , unlike in o dina y e ac ion, he angles o incidence and e ac ion mus ha e opposi e signs. This e ec is illus a ed in Fig. 3. F om he con inui y o he angen ial componen s o he wa e ec o s o he inciden and e ac ed ays, i ollows ha 共see Fig. 3兲 sin ␪ 1 sin ␪ 2 =−兩k2兩 兩k1兩⬅n2 n1 ⬍0, 共8兲 which is he well-known law o e ac ion. In Eq. 共8兲n1and n2a e he e ac i e indices o he o dina y and he le - handed medium, espec i ely. I n1⬎0, i ollows ha n2 ⬍0 om Eq. 共8兲. Tha is, he sign o he squa e oo in he e ac i e index de ini ion mus be nega i e,2 n⬅−冑␧ ␮ ⬍0. 共9兲 Fo his eason, le -handed media a e also e e ed o as “nega i e e ac i e index” o “nega i e e ac i e” media.1 III. METHODOLOGY AND RESULTS A common way o demons a e e ac ion is o o m a p ism, shine a beam h ough i , and obse e he de lec ion o he beam on he o he side. Wa es en e he p ism h ough one o he in e aces a no mal incidence and s ike he op- posi e in e ace a an oblique angle. Fo a p ism made o o dina y o igh -handed ma e ial his angle is posi i e 关see Fig. 4共a兲兴and is nega i e o a p ism o le -handed ma e ial 关see Fig. 4共b兲兴. Nega i e e ac ion o wa es in he mic o- 349 349Am. J. Phys. 79 共4兲, Ap il 2011 h p://aap .o g/ajp © 2011 Ame ican Associa ion o Physics Teache s wa e ange is demons a ed in an expe imen whe e a mic o- wa e beam is e ac ed h ough a le -handed p ism o a ab ica ed me ama e ial. Nega i e e ac ion o mic owa es was i s shown expe i- men ally by Shelby e al.3 o a p ism consis ing o a h ee- dimensional a ay o me allic wi es and me allic spli - ing esona o s. Spli - ing esona o s p o ide a s ong magne ic esponse and ha e been well s udied 共see, o example, Re . 1and e e ences he ein兲. A ays o wi es ha e also been analyzed o designing me ama e ials wi h speci ic elec ic beha io .1In he me ama e ial p ism conside ed in Re . 3 he a ay o wi es had ␧ ⬍0, in con as o he magne ic e- sponse o he spli - ing esona o s which was ␮ ⬍0. We will ab ica e a me ama e ial wi h a simple s uc u e. Because he p ism is illumina ed by linea ly pola ized plane wa es coming om a mic owa e sou ce, i is no necessa y o ha e an iso opic ma e ial wi h scala ␧ ⬍0 and ␮ ⬍0. I is su icien o ab ica e an aniso opic ma e ial wi h ␧ ⬍0in he di ec ion o he elec ic ield Eand ␮ ⬍0 in he di ec ion o he magne ic ield H. A simple way o ob ain ␧ ⬍0 in he di ec ion o Eis o pile me allic pla es which a e pa allel o he di ec ion o E, so ha each pai o closed pla es cons i- u es a pa allel pla e wa eguide.4The p opaga ion cons an ␤ o he undamen al ans e se elec ic 共TE1兲o he magne ic 共TM1兲mode in a pa allel pla e wa eguide is gi en by4 ␤ =冑k0 2− 冉 ␲ d 冊 2 ,共10兲 whe e k0is he ee-space wa e numbe , dis he dis ance be ween he pla es, and ␲ /dis he cu o wa e numbe . The e ec i e ela i e pe mi i i y o his wa eguide is4 ␧ = 冉 ␤ k0 冊 2 =1− 冉 ␲ dk0 冊 2 =1− 冉 c 2d 冊 2 ,共11兲 whe e is he ope a ing equency, cis he speed o ligh in acuum, and c/2d ep esen s he cu o equency. By choos- ing o be less han he cu o equency, i is possible o ob ain ␧ ⬍0. To ob ain ␮ ⬍0 in he di ec ion o he H ield, a wo-dimensional plana a ay o spli - ing esona o s pe - pendicula o he H ield is in oduced be ween he pla es. The ab ica ed p ism consis s o a mul ilaye ed plana a - angemen o la uni cells as shown in Fig. 5共a兲. Inside each cell ␧ ⬍0 in he di ec ion o Eand ␮ ⬍0 in he di ec ion o H. Figu e 5共b兲shows a ske ch o one la uni cell consis ing o an a ay o spli - ing esona o s placed be ween wo oam laye s 共Rohacell 51 HF兲, 6 mm o hickness, and pe mi i i y close o uni y. The wo oam laye s a e shielded by hin me allic pla es a he op and he bo om 共 he me al pla es we e made using he coppe me alliza ion o hin FR4 ci cui boa ds兲. The hickness o each la uni cell is 12 mm. The ope a ing equency o 10.5 GHz is ixed by he mic owa e ansmi e and ecei e used in he demons a ion. Fo his equency, Eq. 共11兲gi es ␧ =−0.4 o d=12 mm. The geome y and dimensions o he a ay o spli - ing esona o s a e shown in Fig. 5共c兲. We ha e de eloped models o making uni o m me ama e ials.5Wi h he help o his model, which akes in o accoun he esonance equency o he spli - ing esona o s, dimensions, mu ual couplings, and pe iodici y o he a ay, a desi ed magne ic esponse can be ob ained. As a simple ule o humb, o ob ain ␮ ⬍0a a desi ed equency, he spli - ing esona o has o be designed o esona e a a equency sligh ly below he ope a ing e- quency. In he de ice used in his pape , he esonance e- quency o he ab ica ed spli - ing esona o s has an a e age alue 共due o ole ances in he pho oe ching p ocess兲o 10.35 GHz. Taking in o accoun his esonance equency Fig. 1. Re ac ion a he bounda y be ween 共a兲 wo o dina y media o igh - handed media and be ween 共b兲an o dina y medium and a le -handed medium. E H k S E H k S R igh -handed iple Le -handed iple ( a )( b ) Fig. 2. Illus a ion o he sys em o ec o s E,H,k, and S o a plane wa e in 共a兲an o dina y medium and 共b兲a le -handed medium. S 1 k 2 q 2 q 1 k 1 S 2 Fig. 3. Demons a ion o he nega i e e ac ion be ween o dina y 共1—whi e兲and le -handed 共2—g ay兲media. The Poyn ing and wa e ec- o s a each media a e labeled as S1,S2,k1, and k2, espec i ely. Nega i e e ac ion a ises om he con inui y o he componen s o he wa e ec o s, k1and k2, pa allel o he in e ace and om he ac ha ays p opaga e along he di ec ion o ene gy low. Tha is, ays mus be pa allel o he Poyn ing ec o s S1and S2. Righ -handed p ism qiq qiq Le -handed p ism qiq (a) (b) Fig. 4. Re ac ion h ough a p ism. The angle o e ac ion is posi i e o a p ism made o 共a兲an o dina y o igh -handed ma e ial and 共b兲is nega i e o a p ism made o le -handed ma e ial o me ama e ial. 350 350Am. J. Phys., Vol. 79, No. 4, Ap il 2011 Velazquez-Ahumada e al. and he dimensions indica ed in Fig. 5共c兲, ou model gi es ␮ =−0.75 a he wo king equency o 10.5 GHz.5 The alues o ␧ =−0.4 and ␮ =−0.75 gi e a nega i e in- dex o e ac ion equal o ⫺0.55. The dimensions o he spli - ing esona o s we e chosen wi h he help o eely a ailable so wa e.6The geome y shown in Fig. 5共c兲was chosen o i s symme y,7which ensu es a pu ely magne ic esponse, so ha he e ec i e ␧ p o ided by he me al pla es is no a ec ed by he p esence o he spli - ing esona o s. The spli - ing esona o s we e pho oe ched in an ARLON dielec ic subs a e LX02003311 wi h dielec ic cons an o 2.33 and dielec ic hickness o 0.508 mm. A me ama e ial consis ing o a mul ilaye ed s uc u e o h ee uni cells was ab ica ed, and a 20° p ism was cu om i . The p ism was placed in a slo made in a poly inyl chlo- ide 共PVC兲moun ing base, and he base was placed be ween a mic owa e ansmi e and a ecei e . Figu e 6shows a pho og aph o he se up. In he se up he ansmi e is ixed and is no mal o he ou pu in e ace o he p ism; he e- cei e can be o a ed a ound he p ism by means o a goni- ome e . The mic owa e ansmi e and ecei e we e 10.5 GHz mic owa e ho ns 共PASCO兲. The me ama e ial was de- signed o wo k a his equency, bu i can be easily designed o wo k a o he equencies by changing he esonance e- quency o he spli - ing esona o s and he dis ance be ween he me al pla es. The ecei ing ho n was placed on he goni- ome e a m a a dis ance om he moun ing base la ge han en wa eleng hs 共⬇30 cm兲 o ensu e ha he wa e a he ecei ing ho n was app oxima ely plana . The PVC moun ing base was co e ed wi h adhesi e cop- pe oil, and py amidal oam abso be was placed a bo h sides. This oam abso be p e en s he o ma ion o s anding wa e pa e ns be ween he ho ns and he moun ing base. I s anding wa e pa e ns a e p esen , he signal in he ecei e would oscilla e wi h he posi ion o he ecei e along he goniome e a m. A second 20° PVC p ism was also ab i- ca ed o compa ison pu poses. Figu e 7shows he measu ed adia ion pa e n o bo h p isms. The measu emen s we e ob ained by o a ing he e- cei e be ween 50° and ⫺50° in s eps o 5° and eco ding he ol age ha was measu ed by a ol me e connec ed o his ecei e 8共see he pho og aph in Fig. 6兲. The ol age is Me al pla es Foam slabs P in ed-ci cui boa d LH P ism 2d=12 mm (d=6 mm ) =1.63 mm ex g=0.2 mm w=0.18 mm a=5 mm E H (a) (b) (c) Fig. 5. 共a兲Mul ilaye s uc u e o he ab ica ed p ism o le -handed ma e- ial. 共b兲De ails o he s uc u e o a laye consis ing o a p in ed ci cui boa d wi h esonan ings be ween a pai o oam slabs; he laye is co e ed by me al pla es. 共c兲Geome ical de ails o he esonan ings. Moun ing Base wi h Py amidal Foam Abso be P ism o PVC P ism o Nega i e- e ac i e Index 10.5 GHz Mic owa e Ho n Fig. 6. Pho og aph o he expe imen al se up. A p ism o le -handed ma e- ial is placed in a moun ing base co e ed wi h py amidal abso bing oam. The moun ing base is placed in he cen e o a goniome e , which allows a mic owa e ho n ecei e o o a e. A mic owa e ho n emi e is shown on he o he side. Bo h he emi e and he ecei e a e comme cial ho ns used o physics demons a ions. The pho og aph also shows an o dina y p ism made o PVC. Fig. 7. Radia ion pa e n ob ained wi h he se up shown in Fig. 6 o he o dina y o igh -handed p ism o PVC and he le -handed p ism. Main lobes o adia ion a e ob ained a 38° o he igh -handed p ism and a ⫺15° o he le -handed p ism. 351 351Am. J. Phys., Vol. 79, No. 4, Ap il 2011 Velazquez-Ahumada e al. g ea ly educed by he p esence o he oam abso be , bu he numbe o laye s in he me ama e ial was su icien o ob ain a measu able signal 共 he wid h o he slo in he moun ing base co esponds o he hickness o he mul ilaye ed s uc- u e o h ee la uni cells, ha is, 36 mm, which is also app oxima ely one wa eleng h a 10.5 GHz兲. The esul s in Fig. 7 o he adia ion pa e n o he PVC p ism show a main lobe o ansmission o a posi i e angle o ⬇40°. Fo an incidence angle o 20°, Eq. 共8兲leads o an index o e- ac ion o n1=1.8. This esul is e y close o he expec ed alue based on he ela i e pe mi i i y o PVC,9which is close o n⬇冑3⬇1.7. When he me ama e ial p ism is measu ed in ou se up, a main lobe is ob ained o a nega i e angle o ⫺15° 共see Fig. 7兲. This nega i e angle is close o he alue p edic ed by he calcula ed nega i e index o e ac ion o ⫺0.55. A mino lobe is also ob ained when he ansmi e and he ecei e a e aligned, co esponding o a posi i e angle o 20°. This mino lobe appea s jus in on o he emi e and can be explained by he ac ha he Ewald-Oseen ex inc ion heo em10 is no ully sa is ied in ou medium due o he medium’s disc e e and ini e na u e. The main sou ces o e o s in he expe imen s include mis- alignmen o he moun ing base wi h espec o he goniom- e e a m and ole ances in he ab ica ion p ocess o he spli - ing esona o media and in he dis ance be ween he me allic pla es. We obse ed a maximum misalignmen o 2° in he loca ion o he maxima, which s ill le s us dis inguish be- ween posi i e and nega i e e ac ions. These e o s will a y depending on he abili y o he s uden s and he quali y o he componen s. We used a pho oe ching echnique wi h a minimum esolu ion o 20⫻10−6 m, which is su icien o ou pu poses. We also ound ha he alignmen o he me al- lic pla es so ha hey a e easonably pa allel is c ucial o he success o he expe imen . We epea ed ou expe imen s in- side a small anechoic chambe wi hou signi ican changes in he esul s, which shows ha he pa ial shielding we used is su icien o he expe imen . ACKNOWLEDGMENTS This wo k was suppo ed by he Spanish Minis e io de Ciencia e Inno ación unde P ojec Consolide -EMET CSD2008-00066. a兲Elec onic mail: [email p o ec ed] 1R. Ma ques, F. Ma in, and M. So olla, Me ama e ials wi h Nega i e Pa ame e s: Theo y and Mic owa e Applica ions 共Wiley, Hoboken, NJ, 2008兲. 2V. G. Veselago, “The elec odynamics o subs ances wi h simul aneously nega i e alues o ␧and ␮ ,” So . Phys. Usp. 10, 509–514 共1968兲. 3R. A. Shelby, D. R. Smi h, and S. 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