Application of Huygens’ Metasurfaces to the Arbitrary Design of a Leaky-Wave Antenna

Applica ion o Huygens’ Me asu aces o he A bi a y Design o a Leaky-Wa e An enna Elena Abdo-S´ anchez(1), A iel Eps ein(2), Geo ge V. Ele he iades(3) [email p o ec ed], [email p o ec ed], [email p o ec ed] (1)Dp o. Ing. Comunicaciones, E.T.S.I. Telecomunicaci´ on, Uni e sidad de M´ alaga, Andaluc´ ıa Tech, E-29071 M´ alaga, Spain. (2)And ew and E na Vi e bi Facul y o Elec ical Enginee ing, Technion - Is ael Ins i u e o Technology, Hai a 32000, Is ael. (3)The Edwa d S. Roge s S . Depa men o Elec ical and Compu e Enginee ing, Uni e si y o To on o, To on o, Canada. Abs ac —Leaky-wa e an ennas a e guiding s uc u es ha leak powe along hei leng h. Thei adia ion is mainly cha ac- e ized by he p opaga ion cons an (leakage ac o and phase cons an ) o he a eling wa e. In his con ibu ion, a leaky- wa e an enna based on pa allel-pla e wa eguide is p oposed. A bi a y con ol o he leakage ac o and he phase con- s an is achie ed by eplacing he op pla e by an omega- ype bianiso opic Huygens’ me asu ace, which implemen s he de- si ed ield ans o ma ion. The heo e ical de i a ion and design me hodology a e b ie ly desc ibed. Se e al designs wi h di e en poin ing angles (phase cons an s) and leakage a es ha e been ca ied ou . Elec omagne ic simula ion esul s alida e he heo e ical de i a ion, highligh he capabili ies o he s uc u e and con i m he lexibili y in he design pa ame e s. I. INTRODUCTION In he ecen yea s, he e is an inc easing demand on di ec i e an ennas wi h low p o ile and cos , especially o applica ions such as au omo i e ada s o sa elli e communi- ca ions. Good pe o mance is ob ained wi h phased a ays; howe e , hei eeding ne wo ks a e complica ed, leading o high cos s. Leaky-wa e an ennas (LWAs), on he con a y, ha e e y simple eeding, since hey consis on a guiding s uc u e ha leaks powe while he wa e is being p opaga ed along i [1]. Plana LWAs ha e ecei ed an inc easing a en- ion la ely, especially a e he in oduc ion o me ama e ials and me asu aces, which allowed he enhancemen o hei cha ac e is ics (such as he mi iga ion o he b oadside e ec ) [2]–[5]. In o de o ob ain a ce ain adia ion pa e n, i is necessa y o be able o design he p opaga ion cons an o he leaky mode, γ=β−jα. The phase cons an , β, de e mines he poin ing angle, whe eas he leakage ac o , α, con ols he a e o he powe leakage, which se s he ampli ude dis ibu ion. Independen con ol o hese wo pa ame e s is a challenge in he design o LWAs. Con ol on he adia ion pa e n could be achie ed i we a e able o a bi a ily ans o m he ield inside he guiding s uc u e in o he desi ed adia ed ield. In his ega d, Huy- gens’ me asu aces ha e been ecen ly p oposed as a powe ul ool o a bi a y ield manipula ion [6]–[8]. They consis o subwa eleng h elec ically- and magne ically-pola izable pa icles and allow he ul illmen o he equi ed bounda y condi ions, so ha he desi ed ield ans o ma ion is achie ed. The e o e, by placing a Huygens’ me asu ace on he op o he guiding s uc u e, he equi ed bounda y condi ions o ans o m he guiding mode in o he desi ed leaky mode can be implemen ed. I has been ecen ly disco e ed ha by employing omega- ype bianiso opic me asu aces (O-BMSs) O-BMS PEC . x zy d .E , H - - qin .E , H + - qou z=-d z=0 L k = j y +b - a - - k = j y +b - a + + Fig. 1. T ans e sal sec ion o he p oposed LWA con igu a ion, which consis s o a pa allel-pla e wa eguide wi h he op pla e being a bianiso opic me asu ace (O-BMS). jus one condi ion in he s ipula ion o he ields mus be me o achie e a bi a y ield ans o ma ion using passi e and lossless pa icles: local powe conse a ion along he me asu ace [9]. This is possible due o he addi ional (mag- ne oelec ic) deg ee o eedom p o ided by he O-BMSs. The ac ha only one condi ion o a bi a y ield ans o ma ion is equi ed allows he con ol o he e lec ion and ansmission coe icien s om he guiding s uc u e o ai h ough he me asu ace. In his con ibu ion, a pa allel-pla e wa eguide which inco po a es an O-BMS as he op pla e is p oposed as a no el LWA wi h ou s anding lexibili y o he design pa ame e s and sys ema ic design, as highligh ed in [10]. II. THEORY The explo ed s uc u e (Fig. 1) is a pa allel-pla e wa eg- uide in which he op pla e is eplaced by an O-BMS (a z= 0). Fo simplici y, he p oblem conside ed he e is 2D (∂/∂x = 0). The O-BMS has a leng h in he y-coo dina e o Land he exci a ion o he esul ing pa allel-pla e wa eguide is loca ed a y= 0.ds ands o he wa eguide heigh . A ans e se elec ic (TE) pola ized ield is used as ield exci a ion (Ey=Ez=Hx= 0). Then, he ans e se ield componen s abo e (E+ xand H+ y) and below (E− xand H− y) he O-BMS a e ela ed h ough he bianiso opic shee ansi ion condi ions [11]: 1 2(E+ x+E− x) = −Zse(H+ y−H− y)−Kem(E+ x−E− x) 1 2(H+ y+H− y) = −Ysm(E+ x−E− x) + Kem(H+ y−H− y) (1) whe e Zse s ands o he elec ic su ace impedance, Ysm o he magne ic su ace admi ance and Kem o he magne o- elec ic coupling coe icien . The i s s ep o he heo e ical de i a ion o he p oblem is o ind an exp ession o he elec omagne ic ield inside he wa eguide ha ul ills Maxwell’s equa ions. Since he me asu ace design will o ce he bounda y condi ions a z= 0 o be me , he only es ic ion o he ield below he O-BMS is o anish a he PEC (z=−d). The e o e, he ollowing elec ic ield has been s ipula ed: E− x=|Ein|(ejk− z(z+d)−e−jk− z(z+d))e−jk− yy(2) whe e he p opaga ion cons an s a e complex, in o de o le he s uc u e adia e: k− y=β−−jα−;k− z=β− z−jα− z;k−2=k− y 2+k− z 2.(3) The desi ed ield o he egion abo e he me asu ace is s ipula ed as a leaky mode, simply as E+ x=|Eou |e−jk+ zze−jk+ yy,(4) whe e k+ y=β+−jα+;k+ z=β+ z−jα+ z;k+2=k+ y 2+k+ z 2.(5) We assume cons an βand αalong y. I can be demons a ed ha o he powe conse a ion condi ion o be me (P− z(y) = P+ z(y)[9]), he ield mus ha e he same decay along yabo e and below he O-BMS, i.e. Re[k+ y] = Re[k− y] = α, whe e αis, indeed, he leakage ac o . Mo eo e , he cons an |Eou |is gi en by he es o he pa ame e s [10]. Once we ha e exp essions o s ipula e he ields gi en desi ed αand β, we can calcula e he me asu ace pa ame e s {Kem, Ysm, Zse} o achie e he equi ed ield ans o ma- ion. The only condi ion is ha αmus be he same below and abo e he me asu ace, bu he es o he pa ame e s (β+,which de e mines θou ;β−, which de e mines θin; and he wa eguide heigh d) a e comple ely ee o be se . I can be shown ha , o cons an α, he me asu ace pa ame e s {Kem, Ysm, Zse} esul pe iodic, wi h a pe iod gi en by p=2π |β+−β−|.(6) III. REALIZATION AND SIMULATION RESULTS To be able o p o e he concep h ough elec omagne ic simula ion, he me asu ace pa ame e s mus be disc e ized along y(we ha e used a leng h o λ0/6). To ealize he O- BMS being compa ible wi h s anda d ab ica ion echniques, we use asymme ic h ee-laye s ack o impedance shee s [9], [12]. In his way, we ans o m he local {Kem, Ysm, Zse} in o he equi ed Z-ma ix o each h ee-laye uni -cell using (1) and he ela ions be ween he angen ial ields below and abo e he me asu ace [13]: E− x E+ x=Z11 Z12 Z21 Z22  H− y −H+ y.(7) Then, by applying he ansmission line model o he h ee- laye s ack o impedance shee s, we ans o m he ma ix [Z] in o he equi ed alues o he impedance shee s (which esul o be lossless) {Xbo , Xmid, X op}[9]. The esul ing s uc u e o simula e using he elec omag- ne ic so wa e HF SS is shown in Fig. 2. The O-BMS consis s, as can be obse ed, o h ee laye s o eac ance shee s (which a e simula ed using impedance bounda y condi ions) Fig. 2. Schema ic o he s uc u e o simula e. 0 2 4 6 8 10 −6 −4 −2 0 2 4 6 y/λ0 Kem Zse [η] Ysm [1/η] Fig. 3. Me asu ace pa ame e s o he Design 1 (wi h a uni -cell leng h o λ0/6 and a pe iod o 2λ0). wi h ce ain pe iodici y. To emula e in ini e pla es along he x- axis, PEC bounda y condi ions a e used on he aces a he zy- planes. The s uc u e is exci ed and e mina ed by wa epo s. To illus a e he design p ocedu e and he capabili ies o he p oposed me hodology, a i s design (Design1) has been ca ied ou , in which he wa eguide heigh dhas been se o 0.6λ0, he poin ing angle θou has been a bi a ily chosen o 20◦, he pe iod o 2λ0(which de e mines he phase cons an inside he wa eguide h ough (6) esul ing in θin = 57◦) and α o 0.02k0. The leng h o he me asu ace has been se o 10λ0in o de o adia e 90% o he powe wi h he chosen α. Fo hese pa ame e s, he esul ing me asu ace pa ame e s o implemen a e shown in Fig. 3. The 2D di ec i i y compa ison be ween heo y and simula- ion is plo ed in Fig. 4. I can be obse ed ha he s uc u e adia es a he aimed di ec ion (20◦) and excellen ag eemen is ound be ween he analy ical p edic ion and he HFSS esul . Fig. 5 shows a compa ison o he magni ude o he elec ic ield. Excellen ag eemen is obse ed no only in he p op- aga ing mode inside he wa eguide bu also in he adia ing ield. The exponen ial powe decay along he y−axis, which co esponds o a cons an α, can be no iced. As can be obse ed, he p opaga ing mode is g adually leaking powe −80 −60 −40 −20 0 20 40 60 80 −30 −20 −10 0 10 20 θ [deg] D [dB] Analy ical Simula ion Fig. 4. 2D di ec i i y o he Design 1 (wi h α= 0.02k0and θou = 20◦). (a) Analy ical (b) Simula ion Fig. 5. Elec ic ield pa e n o he Design 1 (wi h α= 0.02k0and θou = 20◦). h ough he me asu ace. In o de o illus a e ha we can se a desi ed α, ano he design (Design 2) has been ca ied ou by keeping all he pa ame e s he same as in Design 1 bu educing α o 0.014k0 and inc easing he leng h L= 14λ0 o adia e he same amoun o powe as in Design 1 bu mo e g adually. Fig. 6 shows he 2D di ec i i y compa ison o his case, in which he p edic ed inc ease in he di ec i i y and dec ease in he beamwid h is obse ed wi h espec o Design 1, wi h excellen ag eemen be ween heo y and simula ion. Fig. 7 shows he compa ison be ween heo y and simula ion o he magni ude o he elec ic ield o he Design 2. The ield has p ac ically he same pa e n as o Design 1, since he pa ame e s ha e been kep he same. The only di e ence is no iced in he mo e g adual powe leakage, as expec ed. The LWA can be also designed o adia e in o an a bi a y angle, by choosing he equi ed β+(β+≈k0sin(θou )). Then, ei he β−(θin) o he pe iod pcan be a bi a ily se as well. To illus a e his lexibili y, wo addi ional designs wi h −80 −60 −40 −20 0 20 40 60 80 −30 −20 −10 0 10 20 θ [deg] D [dB] Analy icall Simula ion Fig. 6. 2D di ec i i y o he Design 2 (wi h α= 0.014k0and θou = 20◦). (a) Analy ical (b) Simula ion Fig. 7. Elec ic ield pa e n o he Design 2 (wi h α= 0.014k0and θou = 20◦). di e en θou ha e been ca ied ou . In bo h cases, θin has been se o 30◦, so pis di e en in he wo cases, acco ding o (6). A mo e ex eme angle has been chosen o Design 3, θou =−50◦, whe eas b oadside adia ion will be illus a ed wi h Design 4, wi h esul ing pe iods o 0.8λ0and 2λ0, espec i ely. Due o he low pe iod o Design 3 wi h espec o he uni -cell leng h o λ0/6, some deg ada ion wi h espec o he analy ical esul s o his case is expec ed, due o he sampling o he me asu ace pa ame e s. In bo h designs, he leng h L, wa eguide heigh d, and leakage ac o αha e been kep o 10λ0,0.6λ0and 0.02k0, espec i ely. Fig. 8 shows he 2D di ec i i y compa ison be ween heo y and simula ion o he Design 3 and Design 4. Some dis- c epancies in he di ec i i y le els be ween he analy ical and simula ion esul s a e ound o Design 3 (θou =−50◦), which a e a ibu ed o he low pe iod (p= 0.8λ0), as p e iously men ioned. In ac , his pe iod leads o e en less han i e di e en uni -cells pe pe iod, which migh no be enough o cap u e he beha io o he con inuous me asu ace. Howe e , his example was chosen o highligh ha he ield pa e n inside he me asu ace can be main ained in designs wi h di e en poin ing angles by using a di e en pe iod (see Fig. 9). The poin ing angles om he simula ions ag ee wi h −80 −60 −40 −20 0 20 40 60 80 −30 −20 −10 0 10 20 θ [deg] D [dB] Analy ical θou =−50o Analy ical θou =0o Simula ion θou =−50o Simula ion θou =0o Fig. 8. 2D di ec i i y o Design 3 and Design 4 (wi h θou =−50◦and θou = 0◦, espec i ely). (a) θou =−50◦ (b) θou = 0◦ Fig. 9. Simula ed elec ic ield pa e n o he Design 3 and Design 4. he heo e ical p edic ion. Mo eo e , i is highligh ed ha no issue wi h b oadside adia ion is ound in his s uc u e, unlike adi ional LWAs. IV. CONCLUSIONS Applica ion o Huygens’ me asu ace o he design o a LWA has been explo ed. In o de o ha e enough deg ees o eedom o con ol he leaky mode pa ame e s, omega- ype bianiso opy mus be in oduced in o he me asu ace. The heo e ical de i a ion o calcula e he me asu ace pa ame e s o a bi a y leaky mode wi h cons an leakage ac o has been shown. Simula ion esul s ha e been ob ained by implemen - ing he O-BMS using h ee-laye s ack o eac ance shee s. The lexibili y in he design has been highligh ed h ough ou designs. I has been shown ha di e en leakage ac o s can be se while keeping an a bi a y poin ing angle. Mo e- o e , designs wi h mo e ex eme poin ing di ec ion and e en b oadside adia ion ha e been illus a ed. The e o e, indepen- den con ol o he leakage ac o and he poin ing di ec ion has been success ully achie ed. The design me hodology is e y powe ul since i allows a sys ema ic design wi h almos all possible deg ees o eedom (cons an leakage ac o , θin, θou and e en he wa eguide heigh d). Fu he esea ch will ocus on ex ending he de i a ion and design p ocedu e o a modula ed leakage ac o o ha e a be e con ol on he adia ion pa e n and on ge ing a physical ealiza ion o he me asu ace wi h expe imen al e i ica ion. Mo eo e , he scanning capabili ies o he p oposed LWA will be explo ed. ACKNOWLEDGMENTS This p ojec has ecei ed unding om he Eu opean Union’s Ho izon 2020 esea ch and inno a ion p og amme unde he Ma ie Sklodowska-Cu ie g an ag eemen No 706334. REFERENCES [1] A. A. Oline and D. R. Jackson, “Leaky-wa e an ennas”, An enna Eng. Handbook, Ch. 4, pp. 11-1–11-55, 1993. [2] C. Caloz and T. I oh, Elec omagne ics Me ama e ials: T ansmission Line Theo y and Mic owa e Applica ions. Hoboken, NJ: Wiley and IEEE P ess, 2006. [3] A. Mehdipou , J. W. Wong and G. V. Ele he iades, “Beam-Squin ing Reduc ion o Leaky-Wa e An ennas Using Huygens Me asu aces,” IEEE T ans. An ennas P opag., ol. 63, no. 3, pp. 978–992, Ma ch 2015. [4] G. 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