Efficient Spectral Domain MoM for the Design of Circularly Polarized Reflectarray Antennas Made of Split Rings

1 E icien spec al domain MoM o he design o ci cula ly pola ized e lec a ay an ennas made o spli ings Ra ael Flo encio, Ra ael R. Boix, and José Encina , Abs ac —The Me hod o Momen s (MoM) in he spec al domain is used o he analysis o he sca e ing o a plane wa e by a mul ilaye ed pe iodic s uc u e con aining conduc ing concen ic spli ings in he uni cell. Basis unc ions accoun ing o edge singula i ies a e used in he app oxima ion o he cu en densi y on he spli ings, which makes i possible a as con e gence o MoM wi h espec o he numbe o basis unc ions. Since he 2-D Fou ie ans o ms o he basis unc ions canno be ob ained in closed- o m, judicious icks (con olled unca ion o in ini e summa ions, in e pola ions, e c.) a e used o he e icien nume ical de e mina ion o hese Fou ie ans o ms. The implemen ed spec al domain MoM so wa e has been used in he design o a ci cula ly pola ized e lec a ay an enna based on spli ings unde he local pe iodici y condi ion. The an enna has been analyzed wi h ou spec al domain MoM so wa e, wi h CST and wi h HFSS, and good ag eemen has been ound among all se s o esul s. Ou so wa e has p o en o be a ound 27 imes as e han CST and HFSS. Index Te ms—Mul ilaye ed media, momen me hods, pe iodic s uc u es, e lec a ays, ci cula pola iza ion. I. In oduc ion REFLECTARRAY an ennas a e an in e es ing al e na i e o e lec o an ennas and mic os ip a ays owing o hei e sa ile adia ion pe o mance, low p o ile, ligh weigh , ease o ab ica ion, simpli ied eeding sys em, e c. [1]. In he pa icula case o ci cula pola iza ion (CP) e lec a ays, he a iable o a ion echnique (VRT) in oduced by Huang and Pogo zelski is a design app oach ha has p o en o be e y success ul [2]. In he ame o his echnique, each elemen o he e lec a ay is o a ed a di e en angle o achie e he adequa e phase shi in he impinging ci cula ly pola ized wa e ha makes i possible o gene a e he p esc ibed adia ion pa e n a e e lec ion. The VRT equi es ha he e lec ion coe icien s o he wo o hogonal linea componen s o he impiging wa e elec ic ield a e o equal magni ude and 180◦ ou o phase in o de o keep he sense o ci cula pola iza ion (LHCP o LHCP o RHCP o RHCP) a e e lec ion [2], [3]. One e lec a ay elemen ha has u ned ou o be especially sui able o CP applica ions is ha based on spli ings. Han e al. used he VRT and single spli ings o design a Ka- band RHCP e lec a ay a 31.75 GHz wi h measu ed c oss- pola iza ion le el below 40 dB a b oadside [4]. S acked spli ings o di e en size we e used in [5] o design a dual equency RHCP e lec a ay ope a ing in he C and Ka bands. Smi h e al. [6] designed a single laye dual equency CP e lec a ay by using an elemen wi h wo concen ic spli ings. The inne ings and he VRT we e used o adjus he elemen s phase a 29.75 GHz o RHCP adia ion, while he ou e ings and he VRT we e used a 19.95 GHz o LHCP adia ion. An ex ension o his wo k was p esen ed in [7] whe e he dual equency CP e lec a ay was combined wi h an FSS made o non-spli ings o combine he ope a ion o he e lec a ay wi h ha o a second CP mic os ip a ay an enna wo king a L band. Zhao e al. used he single laye elemen based on wo concen ic ings and he VRT o design a b oadband CP e lec a ay a a cen e equency o 10 GHz wi h a bandwid h la ge han 30% o 1-dB gain a ia ion, and a bandwid h la ge han 40% o an axial a io smalle han 3-dB [8]. In all p e ious pape s dealing wi h CP applica ions, he elemen s o he e lec a ays consis o symme ic spli ings wi h wo gaps and one axis o mi o symme y. Zhang e al. p oposed in [9] he use o non-symme ic single gap spli ings which can be used as phasing elemen s o bo h linea pola iza ion (LP) and CP e lec a ay ope a ion. By using his elemen , a LP e lec a ay was designed a 20 GHz wi h a bandwid h o 23% o a 1-dB gain a ia ion and an ape u e e iciency o 60%. In he design o a CP e lec a ay an enna made o spli ings, he use o he VRT equi es o adjus he size o he gaps and hei o ien a ion o e e y single elemen in o de o achie e he equi ed phase shi in he impinging ci cula ly pola ized wa e, and in o de o ensu e a phase di e ence o 180◦in he e lec ion coe icien s o i s wo o hogonal linea componen s. When es ima ing he gap size and hei o ien a ion in each elemen , i is cus oma y o assume ha he elemen s a e loca ed in a pe iodic en i onmen , which is called he local pe iodici y assump ion [10], [11]. The alidi y o his assump ion is jus i ied by he ac ha leads o e lec a ay designs in which he simula ed pe o mance ag ees e y well wi h he measu ed pe o mance [5], [7], [8]. The design o a midsize CP e lec a ay an enna made o spli ings unde he local pe iodici y assump ion equi es he analysis o a la ge numbe o di e en mul ilaye ed pe iodic s uc u es owing o he wide a ie y o gap sizes and o ien a ions ha ha e o be adjus ed in he spli ings o he di e en e lec a ay elemen s o phase adjus men [7]. Owing o his, a powe ul nume ical ool is needed o he analysis o hese mul ilaye ed pe iodic s uc u es. In his pape we apply he Me hod o Momen s (MoM) in 2 he spec al domain o he analysis o he sca e ing o a plane wa e by a mul ilaye ed pe iodic s uc u e including concen ic spli ings in he uni cell. The implemen ed MoM so wa e is subsequen ly used in he design o a ci cula ly pola ized e lec a ay an enna made o spli ings. When applying he spec al domain MoM, he 2-D Fou ie ans o ms o he basis unc ions a e exp essed as in ini e summa ions o Hankel ans o ms. This Hankel ans o m app oach was used in he pas in conjunc ion wi h he spec al domain MoM o he de e mina ion o he esonan equencies o mic os ip ing esona o s [12], he inpu impedance and adia ion pa e ns o single and s acked annula - ing mic os ip an ennas [13], [14], he esonan equencies o shielded single and coupled mic os ip ing esona o s [15], [16], and he ada c oss- sec ion o concen ic annula mic os ip ings [17]. Howe e , o he au ho s’ knowledge, i has ne e been used o mi- c os ip spli ings and o mul ilaye ed s uc u es in pe iodic en i onmen s. We ha e only ound one pape dealing wi h he analysis o mic os ip spli ing esona o s in which a magne ic wall app oxima e model is used o he de e mina ion o he esonan equencies, and whe e e ec i e dimensions a e in oduced o accoun o edge e ec s [18]. Whe eas mos pape s dealing wi h he analysis o mic os ip ing s uc u es use ei he magne ic wall ca i y mode unc ions [13], [19] o subsec ional piecewise unc ions [15], [17] as basis unc ions o he elec ic cu en densi y on he ings, in his pape we ha e used basis unc ions ha accoun o edge singula i ies [16], [19] since hese basis unc ions p o ide a as con e gence o MoM wi h espec o he numbe o basis unc ions, which implies ha only small MoM ma ices ha e o be in e ed. Since he Hankel ans o ms o he edge singula i y basis unc ions canno be ob ained in closed o m, we ha e used especially ailo ed quad a u e ules o hei nume ical compu a ion, which p o ide e y accu a e esul s wi h a small numbe o quad a u e poin s. Also, we ha e in oduced judicious s op c i e ia o he summa ion o he in ini e se ies leading o he 2-D Fou ie ans o ms o he basis unc ions, and we ha e inally ca ied ou Chebyshe in e pola ions o hese Fou ie ans o ms as a unc ion o he spec al adial a iable. As a esul o all hese s a egies, we ha e de eloped a e y e icien spec al domain MoM app oach o he analysis o mul ilaye ed pe iodic s uc u es con aining spli ings. The spec al domain MoM so wa e has been used in he design o a ci cula ly pola ized e lec a ay an enna and he esul s ob ained ha e been compa ed wi h esul s p o ided by comme cial so wa es CST and HFSS . Good ag eemen has been ound be ween he h ee se s o esul s, ou so wa e being be ween one and wo o de s o magni ude as e han CST and HFSS . II. Nume ical p ocedu e Figs. 1(a) and (b) show a mul ilaye ed pe iodic s uc u e backed by a g ound plane. Concen ic conduc ing spli ings a e loca ed in each uni cell, and a e p in ed on he uppe laye . The cen e o he ings has been chosen o be coinciden wi h he geome ical cen e o he uni cell in he z=0 plane. The conduc ing ings and he g ound plane will be assumed o be pe ec elec ic conduc o s (PEC). In a gene ic case, he e would be Rconcen ic ings in he uni cell (R=2 in Fig.1(b)). Also, each ing would be spli in o Sa cs (S=2 in Fig.1(b)) so ha he o al numbe o a cs in he uni cell would be L=RS (L=4 in Fig. 1(b)). Le ρ1land ρ2lbe he inne and ou e adius o he l- h a c espec i ely (l=1,...,L), and le wl=ρ2l−ρ1lbe he wid h o he l- h a c. The wo ends o he l- h a c a e cha ac e ized by he angles ϕ1land ϕ2l, such ha ϕ1l<ϕ2l. The angle ϕkl (k=1,2; l=1,...,L) is he angle sub ended be ween he k− h end o he l- h a c and he semi- in ini e line di ec ed along he posi i e xaxis wi h o igin a he cen e o he ing. The angle ϕkl is aken as posi i e i , s a ing om he x-di ec ed semi-in ini e line, i is d awn in he coun e clockwise sense, and is aken as nega i e i i is d awn in he clockwise sense (e.g., in Fig. 1(b), ϕ21,ϕ12,ϕ22, ϕ13 and ϕ23 a e posi i e, bu ϕ11,ϕ14 and ϕ24 a e nega i e). z xy qinc em 10 em 20 , ,h2 h1 b bbbbb (a) b a y zx a j 22 2 23 2 j 13 j 23 j 12 a2 a1 21 j 21 j 11 j 24 j 14 (b) Fig. 1. Two-laye ed pe iodic s uc u e. The uni cell con ains wo concen ic spli ings wi h di e en axes o mi o symme y. A plane wa e impinges on he mul ilaye ed pe iodic s uc u e. (a) Side iew. (b) Top iew. In he pa icula cases ea ed in his pape , we will assume ha each indi idual spli ing o he pe iodic s uc u e has an axis o mi o symme y going h ough he cen e o he ing (in he case o Fig. 1(b), his means ha ϕ21 −ϕ11=ϕ22 −ϕ12 and ϕ12 −ϕ21=360◦+ϕ11 −ϕ22, and ha ϕ23 −ϕ13=ϕ24 −ϕ14 and 360◦+ϕ14 −ϕ23=ϕ13 −ϕ24). Fo he - h ing ( =1,...,R), his mi o symme y axis makes an angle α (−180◦≤α <180◦) wi h he semi-in ini e line di ec ed along he posi i e yaxis wi h o igin a he cen e o he ing (see Fig. 1(b)). The angle α is aken as posi i e i , s a ing om he y-di ec ed semi-in ini e line, i is d awn in he coun e clockwise sense, and is aken as nega i e i i is d awn in he clockwise sense (e.g., in Fig. 1(b), α1and α2a e bo h posi i e). Please no e ha he es ic ion in oduced by hese mi o symme y axes is no necessa y in he ma hema ical de i a ions p esen ed in he es o his sec ion. The concen ic spli ings o he pe iodic s uc u e a e p in ed on a mul ilaye ed subs a e con aining Nllaye s (Nl= 3 2 in Fig.1(a)). The i- h laye has a hickness hiand a complex pe mi i i y εi=ε0ε ,i(1 −j an δi) (i=1,...,Nl). In he ollowing, a ime dependence o he ype ejω will be assumed and supp essed h oughou . As commen ed in he in oduc ion, he uni cell o he mul ilaye ed pe iodic s uc u e o Figs. 1(a) and (b) has been used as cons i uen elemen o se e al CP e lec a ay an ennas [6]–[8]. In he design o a CP e lec a ay an enna unde he local pe iodici y assump ion, each cell is cha ac e ized by means o a CP 2 ×2 complex e lec ion ma ix, RCP, which ela es he RHCP and LHCP componen s o he e lec ed and inciden elec ic ields when a plane wa e impinges on he cell su ounded by a pe iodic en i onmen . In his pape , his impinging plane wa e is assumed o p opaga e in an a bi a y incidence di ec ion gi en by he angula sphe ical coo dina es θinc and ϕinc (see Fig.1(a)). I E e RHCP and E e LHCP a e he RHCP and LHCP complex componen s o he e lec ed elec ic ield, and Einc RHCP and Einc LHCP a e he RHCP and LHCP complex componen s o he inciden elec ic ield, hen E e RHCP E e LHCP !=RCP · Einc RHCP Einc LHCP !(1) whe e RCP = RRHCP,RHCP RRHCP,LHCP RLHCP,RHCP RLHCP,LHCP !.(2) When w i ing (1), we assume ha he plane wa e impinging on he pe iodic s uc u e does no gene a e g a ing lobes a e e lec ion, which equi es ha he condi ion max(a,b)< λ0/(1 +sin θinc) (λ0=2π/ω √µ0ε0) is ul illed, aand bbeing he pe iods o he uni cell in he xand ydi ec ions espec i ely (see Fig. 1(b)). A. Spec al domain MoM o mula ion In o de o ob ain he ma ix RCP o (2) o he mul ilaye ed pe iodic s uc u e o Figs. 1(a) and (b), we need o ob ain he sca e ed elec ic ield o impinging plane wa es wi h bo h RHCP and LHCP pola iza ions. These sca e ed elec ic ields can be ob ained in e ms o cu en densi y exci ed a he me allized in e ace z=0, J(x,y), by he impinging wa es. Le Ems(x,y,z) be he elec ic ield gene a ed in all space by a plane wa e impinging on he mul ilaye ed subs a e o Figs. 1(a) and (b) in he absence o he conduc ing spli ings. The cu en densi y J(x,y) induced on he spli ings will be he solu ion o he ollowing elec ic ield in eg al equa ion (EFIE) [20] ˆ z×"Ems(x,y,z=0) + +∞ X m=−∞ +∞ X n=−∞ZSmn GE(x−x0,y−y0,z=0,z0=0) ·J(x0,y0)dx0dy0#=0(3) (x,y)∈S00 whe e Smn (m,n=. . . , −1,0,1, . . .) is he me allized po ion o he z=0 plane wi hin he mn- h pe iodic uni cell, and GEis he non–pe iodic dyadic G een’s unc ion o he mul ilaye ed subs a e [21]. Since J(x,y) is a Floque -pe iodic unc ion o xand y, in o de o sol e he EFIE o (3), we only need o de e mine J(x,y) wi hin one uni cell, e. g., he cell C00 co e ing he ec angula domain {0≤x≤a; 0 ≤y≤b}. Fo ha pu pose, we expand J(x,y) in C00 in e ms o known basis unc ions Jlq(x,y) (l=1,...,L;q=1,...,Nb) as shown below J(x,y)= L X l=1 Nb X q=1 clqJlq(x,y) (4) whe e Jlq(x,y) (q=1,...,Nb) is he se o Nbbasis unc ions used o app oxima e he cu en densi y on he l- h a c o he C00 uni cell. When (4) is in oduced in (3) and Gale kin’s e sion o MoM is applied, he ollowing sys em o linea equa ions is ob ained o he unknown coe icien s clq [20] L X l=1 Nb X q=1 Γkl,pqclq =ekp (k=1,...,L;p=1,...,Nb).(5) I we in oke Pa se al’s iden i y o 2-D Fou ie ans o ms, he MoM ma ix en ies Γkl,pq o (5) can be exp essed in he spec al domain as double in ini e summa ions gi en by [22] Γkl,pq =ab +∞ X m=−∞ +∞ X n=−∞he Jd kp(kxm,kyn)∗i ·e G E,c (kx=kxm,ky=kyn,z=0,z0=0) ·e Jd lq(kxm,kyn) (6) whe e e G E,c (kx,ky,z=0,z0=0) is he con inuous 2-D Fou ie ans o m o GE(x,y,z=0,z0=0) [21], kxm = k0sin θinc cos ϕinc +2πm/a,kyn =k0sin θinc sin ϕinc +2πn/b, k0=2π/λ0and e Jd lq(kxm,kyn) is he disc e e 2-D Fou ie ans o m o Jlq(x,y), which is gi en by e Jd lq(kxm,kyn)=1 ab ZS00 Jlq(x,y) e−j(kxm x+kyny)dxdy (7) Finally, he coe icien s ekp o he sys em o equa ions (5) can be ob ained in he spec al domain as ekp =−ab he Jd kp(kx0,ky0)∗i ·Ems(x,y,z=0) ×e−jk0(sin θinc cosϕinc x+sin θinc sinϕincy)(8) whe e he ac o e−jk0(sin θinc cosϕinc x+sin θinc sinϕincy)has been explic- i ly included in (8) o abso b he dependence o Ems(x,y,z=0) on xand y. Equa ions (5) o (8) p o ide he spec al domain MoM o mula ion o he de e mina ion o he ec o unc ion J(x,y) o (3) and (4). The p oblem wi h his o mula ion is ha he disc e e 2-D Fou ie ans o ms e Jd lq(kxm,kyn) o (6) (de ined in (7)) canno be ob ained in closed o m, e en o he simples choice o basis unc ions in (4) (i.e., a cons an alue o he wo componen s o Jlq(x,y) on he su ace o he spli ings o Figs. 1(a) and (b)). In he ollowing subsec ion, we will see how o ackle his p oblem. B. Basis unc ions and 2-D Fou ie ans o ms Fig. 2 shows he l- h a c (l=1,...,L=4) o he uni cell C00 o Fig. 1(b). In Fig. 2 we ha e de ined a shi ed sys em o coo dina es {x0,y0,z0}wi h o igin a he cen e o he uni 4 b a y z x 1 l z’ y’ x’ 2l a/2 b/2 j 1l j 2l Fig. 2. Top iew o he l- h a c in he le uni cell o Fig. 1(b). Radial and angula dimensions o he l- h a c a e shown. cell in he plane z=0. We a e going o in oduce shi ed pola coo dina es ρ0and ϕ0, which a e ela ed wi h he o iginal coo dina es xand yas x=a 2+ρ0cos ϕ0(9) y=b 2+ρ0sin ϕ0(10) The uni ec o s linked o he shi ed pola coo dina es a e gi en by ˆρ0=cos ϕ0ˆ x+sin ϕ0ˆ y(11) ˆϕ0=−sin ϕ0ˆ x+cos ϕ0ˆ y(12) In he spli ings used in CP e lec a ay an ennas, he condi ion wl=ρ2l−ρ1l(ρ1l+ρ2l)/2 (see Fig. 2) is usually ul illed (i.e., he wid h o he ings is usually much smalle han hei mean adius) [6]–[8]. Assuming his condi ion holds, we a e going o neglec he adial componen o he basis unc ions Jlq(x,y) o (4), Jlq(x,y)·ˆρ0, by compa ison wi h hei azimu hal componen , Jlq(x,y)·ˆϕ0. Also, we will assume ha he azimu hal componen o Jlq(x,y) can be ac o ed in e ms o independen unc ions o ρ0and ϕ0, i.e., we will assume ha he unc ions Jlq(x,y) can all be ma hema ically w i en as Jlq(x,y)= l(ρ0)glq(ϕ0)ˆϕ0(13) (ρ1l<ρ0<ρ2l;ϕ1l<ϕ0<ϕ2l) As we will see in Sec ion III, he assump ion shown in (13) is jus i ied by he ac ha leads o alues o he ma ix RCP in (2) ha ma ch he alues p o ided by he comme cial so wa e CST . Al hough he adial componen , Jlq(x,y)·ˆρ0, o Jlq(x,y) has been neglec ed in (13), we would like o poin ou ha his componen can be easily accommoda ed in he ma hema ical o mula ion p esen ed in he es o his subsec ion. Since he unc ions glq(ϕ0)ˆϕ0o (13) a e pe iodic unc ions o ϕ0wi h pe iod 2π, hese unc ions can be expanded as Fou ie se ies o ϕ0as shown below glq(ϕ0)ˆϕ0= +∞ X i=−∞e gi lq ejiϕ0(ϕ1l<ϕ0<ϕ2l) (14) whe e e gi lq =1 2πZϕ2l ϕ1l glq(ϕ0)ˆϕ0e−jiϕ0dϕ0(15) Now, le us in oduce he pola disc e e spec al coo dina es kρ,mn and kϕ,mn gi en by kρ,mn =q(kxm)2+(kyn)2(16) kϕ,mn =a c an kyn kxm !(17) I (9) o (14), (16) and (17) a e in oduced in (7), a e some manipula ions, i is possible o exp ess e Jd lq(kxm,kyn) as e Jd lq(kxm,kyn)=2πe−jkxma+kynb 2 ab ×(e g0 lq ˜ 0 l(kρ=kρ,mn)+ +∞ X i=1 e−jiπ/2˜ i l(kρ=kρ,mn) ejikϕ,mne gi lq +e−jikϕ,mn e gi lq∗)(18) whe e ˜ i l(kρ) (i=0,1, . . .) a e Hankel ans o ms o o de io he unc ions l(ρ0) o (13), which can be exp essed as ˜ i l(kρ)=Zρ2l ρ1l Ji(kρρ0)ρ0 l(ρ0)dρ0,(19) and whe e he unc ion Ji(·) o (19) is a Bessel unc ions o i s kind and o de i. Al hough he exp ession (18) o he compu a ion o e Jd lq(kxm,kyn) looks simple han (7), i p esen s wo d awbacks. Fi s , he unc ions ˜ i l(kρ) canno be ob ained in closed o m, e en o he simples choice o l(ρ0) (e.g., a cons an alue in he in e al ρ1l<ρ0<ρ2l). Second, (18) in ol es he de e mi- na ion o an in ini e se ies, which has a dele e ious impac on he compu a ion o e Jd lq(kxm,kyn) by means o (18). In his pape , he unc ions chosen o Jlq(x,y) a e en i e domain basis unc ions which accoun o he singula i ies o J(x,y) a he edges o he a cs o Fig. 1(b). I is well known ha hese unc ions ensu e a as con e gence o he spec al domain MoM wi h espec o he numbe o basis unc ions (LNbin (4)) [16], [19], [20], as will be demons a ed in Sec ion III. The pa icula unc ions chosen o l(ρ0) and glq(ϕ0) can be w i en as l(ρ0)=1 1−2 ρ2l−ρ1lhρ0−ρ2l+ρ1l 2i2 (20) (l=1,...,L) glq(ϕ0)=s1− 2 ϕ2l−ϕ1lϕ0−ϕ2l+ϕ1l 2!2 ×Uq−1 2 ϕ2l−ϕ1lϕ0−ϕ2l+ϕ1l 2!(21) (l=1,...,L;q=1,...,Nb) whe e Uq−1(·) is a Chebyshe polynomial o second kind and deg ee q−1. The in eg als o (15) can be ob ained in closed o m o he unc ions glq(ϕ0) o (21), and hei exp ession is gi en 5 in he Appendix A. Howe e , he in eg als o (19) canno be ob ained in closed o m o he unc ions l(ρ0) o (20). Fo una ely, we ha e checked hese la e in eg als can be nume ically ob ained wi hin e y easonable CPU imes by means o Gauss-Chebyshe quad a u e ules. In pa icula , when he unc ions l(ρ0) o (20) a e in oduced in (19), he in eg als can be ew i en as ˜ i l(kρ)=Z+1 −1 hi l(kρ, ) √1− 2d (22) whe e hi l(kρ, )=ρ2l−ρ1l 2Jikρρ2l−ρ1l 2 +ρ2l+ρ1l 2 ×ρ2l−ρ1l 2 +ρ2l+ρ1l 2 (23) The in eg als o (22) a e amenable o be compu ed by means o Gauss-Chebyshe quad a u e ules [23, Eqn. 25.4.38]. In pa icula , he in eg als should be app oxima ely ob ained by means o he closed- o m o mula ˜ i l(kρ)≈ Nqp X j=1 ηjhi l(kρ, = j) (24) whe e ηj=π/Nqp and j=cos (2 j−1)π/2Nqp(j= 1,...,Nqp) ( j=1,...,Nqp). Since he unc ions hi l(kρ, ) a e non-singula smoo h unc ions o in he in e al −1≤ ≤+1, a low numbe o quad a u e poin s, Nqp, should su ice o ob ain ˜ i l(kρ) wi h a la ge accu acy. This has been con i med by nume ical simula ions, as will be shown in Sec ion III. Conce ning he in ini e se ies o (18), i s con e gence is s ongly dependen on he alue o kρ,mn. The la ge he alue o kρ,mn, he la ge he numbe o e ms ha has o be e ained in he se ies o (18) o an accu a e de e mina ion o e Jd lq(kxm,kyn). Le us see how he con e gence o his in ini e se ies depends on kρ,mn. I we assume ha a≈b(which holds in p ac ical cases o mos e lec a ay an ennas), by i ue o he mean alue heo em o de ini e in eg als, we can w i e ha (see [23, Eqn. 9.3.1]) |˜ i l(kρ=kρ,mn)||i| ∝Jikρ,mn ρ2l+ρ1l 2|i| ≈Ji kρ,mna 2!|i|∝ ekρ,mna 4i!i =eiln(ekρ,mna/4i)(25) which indica es ha he se ies o (18) has an exponen- ial con e gence p o ided ekρ,mna/4i≤1, i. e., p o ided i≥ekρ,mna/4. Nume ical simula ions ha e shown ha he in ini e se ies o (18) has o be added in he in e al 1 ≤ i≤ekρ,mna/4+8 o an accu a e es ima ion o e Jd lq(kxm,kyn). The e o e, he la ge kρ,mn, he la ge he numbe o e ms o be e ained in he se ies. In p ac ice, he compu a ion o Γkl,pq equi es o unca e he in ini e summa ions o (6) wi hin he in e als −Nmax ⩽ m,n⩽+Nmax. I we assume ha a≈b, he maximum alue o kρ,mn we will ha e o use in (6) a e he unca ion will be (see (16)) kmax ρ,mn =q(kmax xm )2+(kmax yn )2≈√22πNmax a(26) which means he maximum numbe o e ms ha will be necessa y o e ain in he in ini e se ies o (18) will be imax =ekmax ρ,mna 4+8≈πeNmax √2 +8 (27) Fo he cases ea ed in he esul s sec ion, we ha e used Nmax =50, which leads o imax ≈310 when using (27). Fo ou pu poses, his is he wo s case scena io in he e alua ion o (18), i.e., any e alua ion o e Jd lq(kxm,kyn) wi hin −Nmax ⩽ m,n⩽+Nmax will equi e o e ain a mos 310 e ms in he compu a ion o he in ini e se ies o (18) when Nmax =50. Apa om he analy ical s op c i e ium es ablished abo e o he summa ion o he se ies o (18) (1 ≤i≤ekρ,mna/4+8), one addi ional ac ha helps o educe he CPU ime equi ed in he compu a ion o his se ies is a well known p ope y o Bessel unc ions, acco ding o which Ji(x)≈0 i x<x h =C hi when i>N h. Nume ical simula ions show his p ope y holds o N h a ound 10 and C h a ound 1. Acco ding o (19), his p ope y o Bessel unc ions implies ha ˜ i l(kρ=kρ,mn)≈0 in (18) when i>N h and (kρ,mna)/2<C hi, which makes i unnecessa y he e alua ion o a la ge numbe o summa ion e ms in he nume ical compu a ion o he se ies leading o e Jd lq(kxm,kyn) by means o (18). In Sec ion III we will p esen a nume ical me hod ha makes i possible o easily es ima e wo new quan i ies d h(l) and b h(l) (which a e a unc ion o ρ1land ρ2l) o which ˜ i l(kρ)≈0 i kρa<d h(l)(i−b h(l)) and i≥b h(l). Acco ding o his, i will only be necessa y o compu e ˜ i l(kρ=kρ,mn) in he in e al kmin ρ,mn(i)=d h(l)(i−b h(l)) a≤kρ≤kmax ρ,mn (28) when i≥b h(l). Fo i<b h(l), we will ha e o compu e ˜ i l(kρ= kρ,mn) in he in e al kmin ρ,mn(i)=0≤kρ≤kmax ρ,mn. The in eg als ˜ i l(kρ) o (19) a e a unc ion o kρ,ρ1land ρ2l o each alue o i. In mos applica ions o e lec a ay an ennas he dis ances ρ1land ρ2l(l=1,...,L) emain ixed om elemen o elemen and he angles ϕkl (k=1,...,S;l= 1,...,L) and α ( =1,...,R) a e a ied o ob ain he equi ed phase shi s. So, a la ge amoun o CPU ime can be sa ed in he design o a la ge e lec a ay an enna i he unc ions ˜ i l(kρ) (l=1,...,L;i=0,...,imax) a e in e pola ed as a unc ion o kρ o ixed alues o ρ1land ρ2l, and he in e pola ions a e subsequen ly used when adjus ing he angles ϕkl and α in each elemen . The in e pola ions ha e o be pe o med in he in e al kmin ρ,mn(i)≤kρ≤kmax ρ,mn o each alue o l and i. In p ac ice, we ha e di ided he equi ed in e pola ion in e als in o Nin subin e als, and we ha e used Chebyshe polynomials up o se en h-deg ee in he in e pola ion in each subin e al. In Sec ion III we will see ha a maximum numbe o Nin =80 subin e als su ices o ob ain in e pola ed alues o ˜ i l(kρ) wi hin ou signi ican igu es, and ha he numbe o equi ed subin e als Nin dec eases as iinc eases, which is a consequence o he inc ease o kmin ρ,mn(i) (and he e o e, o he educ ion o he wid h o he o iginal in e pola ion in e al) as iinc eases. III. Nume ical esul s and alida ions Figs. 3(a) and (b) show he ela i e e o s made in he e al- ua ion o ˜ 0 l(kρ) and ˜ 100 l(kρ) by means o (24) as a unc ion 6 (a) (b) Fig. 3. Magni ude o he ela i e e o s in he compu a ion o (a) ˜ 0 l(kρ) and (b) ˜ 100 l(kρ) by means o (24). The ela i e e o s a e plo ed as a unc ion o he numbe o quad a u e poin s Nqp employed in (24). Pa ame e s: a=b=5 mm; ρ1l=1.85 mm; ρ2l=2.05 mm. o kρ, and as a unc ion o he numbe o Gauss-Chebyshe quad a u e poin s Nqp employed in (24). The e e ence alues used o ˜ 0 l(kρ) and ˜ 100 l(kρ) ha e been compu ed by means o he double exponen ial quad a u e ule wi h 203 in eg a ion poin s (le el o he quad a u e ule M=5), since his is a nume ical p ocedu e which is pa icula ly accu a e when used in he in eg a ion o unc ions wi h in eg able singula i ies a he end poin s o he in eg a ion in e al [24]. Please no e ha a alue o Nqp equal o 10 su ices o p o ide 4 signi ican igu es in he alues o ˜ 0 l(kρ) and ˜ 100 l(kρ) in nea ly all cases, which indica es ha (24) p o ides alues ha a e accu a e enough o ˜ i l(kρ) wi hin a e y low CPU ime consump ion. In he es o he esul s o his Sec ion, we will always use Nqp =10 in (24). In Figs. 4(a) and (b) we plo he h eshold alues k h ρ(i) as a unc ion o i, whe e k h ρ(i) a e he alues o kρ o which |˜ i l(kρ)/˜ 0 l(kρ)|<10−4i kρ<k h ρ(i) and |˜ i l(kρ)/˜ 0 l(kρ)|>10−4 i kρ>k h ρ(i). The e o e, o e e y alue o i, he h eshold alue k h ρ(i) indica es he alue o kρbelow which ˜ i l(kρ) can be conside ed o be anishing. Please no e he plo o k h ρ(i) as a unc ion o i i s e y well o a s aigh line. So we ha e ca ied ou a linea leas squa es i ing o k h ρ(i) as a unc ion o igi en by k h ρ(i)a≈d h(l)(i−b h(l)) (29) which should be alid o i≥b h(l). Fo he leas squa es i ing, we ha e used Nsam samples o iin ascending o de , ik(k=1,...,Nsam) he i s sample being i1=20. In i 0 20 40 60 80 100 120 140 160 180 200 kρa 0 100 200 300 400 500 kρ h(i)a (a) i 0 20 40 60 80 100 120 140 kρa 0 100 200 300 400 500 kρ h(i)a (b) Fig. 4. The solid line k h ρ(i)as ands o he h eshold alues o kρa o which ˜ i l(kρ) is negligible i kρa<k h ρ(i)a, and ˜ i l(kρ) is non-negligible i kρa>k h ρ(i)a.k h ρ(i)ais plo ed as a unc ion o i. Pa ame e s: a=b=5 mm; ρ1l=1.85 mm and ρ2l=2.05 mm in (a); ρ1l=1.20 mm and ρ2l=1.40 mm in (b). he case o Fig. 4(a), i u ns ou ha d h(l)≈2.343 and b h(l)≈9.484 wi h a coe icien o de e mina ion 2=0.9999. In he case o Fig. 4(b), i u ns ou ha d h(l)≈3.388 and b h(l)≈8.371 wi h a coe icien o de e mina ion 2=0.9998. These esul s indica e ha he alues o d h(l) and b h(l) a e sligh ly dependen on he alues o he inne and ou e adii o he ings, ρ1land ρ2l, which a e he endpoin s o he in eg a ion in e al o (19). Since he i ing o k h ρ(i) o a s aigh line o i≥b h(l) is e y good, in p ac ice a small alue o he numbe o samples o k h ρ(i), Nsam, su ices o he de e mina ion o d h(l) and b h(l). Once hese alues o d h(l) and b h(l) a e known, one can use (28) o es ima e he in e al o alues o kρin which he unc ions ˜ i l(kρ) p esen non-negligible alues wi h a iew o compu ing e Jd lq(kxm,kyn) ia (18). Figs. 5(a) and (b) show he e o s made in he in e pola ion o ˜ 0 l(kρ) and ˜ 100 l(kρ) in he in e al kmin ρ,mn(i)≤kρ≤kmax ρ,mn when his in e al is di ided in o Nin subin e als o equal wid h, and Chebyshe polynomials up o se en h-deg ee a e subsequen ly used o ca y ou he in e pola ion in each subin- e al. Please no e ha whe eas 80 subin e als a e equi ed o he de e mina ion o ˜ 0 l(kρ) wi h an accu acy o 4 signi ican igu es in he whole in e al kmin ρ,mn(i)≤kρ≤kmax ρ,mn, only 40 subin e als a e equi ed o he de e mina ion o ˜ 100 l(kρ) wi h he same accu acy. The explana ion o his is ha whe eas he whole in e pola ion in e al o ˜ 0 l(kρ) is oughly 0≤kρa.450, he in e pola ion in e al o ˜ 100 l(kρ) is much smalle , 200 .kρa.450, which is a consequence o he ac ha ˜ 100 l(kρ) is negligible o kρa.200 as shown in Fig.4(a). Fig. 6(b) shows he phase cu es equi ed o he design 7 (a) (b) Fig. 5. Magni ude o he ela i e e o s in he in e pola ion o (a) ˜ 0 l(kρ) and (b) ˜ 100 l(kρ) as a unc ion o he numbe o subin e als Nin used in he in e pola ion. Pa ame e s: a=b=5 mm; ρ1l=1.85 mm; ρ2l=2.05 mm. o a CP e lec a ay an enna (see (2)) made o single spli ings (see Fig.6(a)) a a equency o 19.95 GHz. By i ue o he VRT o [2], a linea phase a ia ion is ob ained o bo h ∠RRHCP,RHCP and ∠RLHCP,LHCP as a unc ion o α2, which is he o a ion angle o he mi o symme y axis o he spli ing. In o de o keep he sense o CP a e e lec ion in a e lec a ay an enna, we need a phase di e ence o 180◦ be ween he e lec ion coe icien o he componen o he elec ic ield along he mi o symme y axis o he spli ing and he e lec ion coe icien o i s o hogonal componen [2], [3]. To ensu e his phase di e ence is main ained, he leng h o he a cs in he e lec a ay elemen has o be sligh ly adjus ed as α2is a ied [6]. This adjus men o he leng h o he a cs as a unc ion o α2is shown in Fig. 6(c). In Fig. 6(b) we show he con e gence o ∠RRHCP,RHCP and ∠RLHCP,LHCP as a unc ion o he numbe o basis unc ions Nbused in he wo a cs o he spli ings. I can be seen ha jus h ee basis unc ions pe a c su ice o achie e con e gence (in ac , he esul s ob ained wi h he h ee basis unc ions co esponding o q=1,2,3 in (21) o e lap hose ob ained when Nb=7 and 1≤q≤7), which means ha only 6×6 MoM ma ices ha e o be in e ed when analyzing he mul ilaye ed pe iodic s uc u e con aining single spli ings in he uni cell. In o de o alida e he esul s ob ained wi h he spec al MoM so wa e desc ibed in Sec ion II, hese esul s a e compa ed wi h esul s p o ided by he comme cial so wa e CST . Excellen ag eemen is ound be ween bo h se s o esul s. In Fig. 7(b) we plo he phase cu es o a dual- equency CP e lec a ay an enna made o concen ic spli ings (see b a y z x y’ x’ 23 j 23 j 13 a 2 13 (a) α2 (deg) 20 40 60 80 100 120 140 160 180 RRHCP,RHCP and RLHCP,LHCP (deg) -300 -200 -100 0 100 200 300 RLHCP,LHCP q=1,2 RLHCP,LHCP q=1,2,3 RLHCP,LHCP q=1,...,7 RLHCP,LHCP CST RRHCP,RHCP q=1,2 RRHCP,RHCP q=1,2,3 RRHCP,RHCP q=1,...,7 RRHCP,RHCP CST (b) α2 (deg) 20 40 60 80 100 120 140 160 180 ϕ23 −ϕ13(deg) 147 147.5 148 148.5 149 149.5 150 150.5 (c) Fig. 6. (a) Single spli ing. (b) Phase o RRHCP,RHCP and RLHCP,LHCP o he e lec a ay elemen o (a) as a unc ion o he o a ion angle o he ing α2. Con e gence is s udied wi h espec o he numbe o basis unc ions (q=1,...,7) used in he wo a cs o he spli ings. Ou esul s ob ained wi h he spec al domain MoM a e compa ed wi h esul s ob ained wi h CST . (c) Va ia ion o he leng h o he a cs o he spli ing wi h he o a ion angle α2 o ensu e a phase di e ence o 180◦in he e lec ion coe icien s o he wo o hogonal linea componen s o he CP impinging wa es. Pa ame e s: a=b=5 mm; ρ13=1.85 mm; ρ23=2.05 mm; h1=0.787 mm, ε ,1=2.2, an δ1=0.0009; h2=0 mm; θinc =30◦,ϕinc=0◦; =19.95 GHz. Fig. 7(a)). While he ou e ings a e in ended o adjus he phase a 19.95 GHz, he inne ings a e in ended o adjus he phase a 29.75 GHz as in [6]. In Fig.7(b) we plo he cu es o phase adjus men a 29.75 GHz. Since his phase adjus men is con olled by he inne ings, in he applica ion o he VRT ∠RRHCP,RHCP and ∠RLHCP,LHCP a e a ied by o a ing he mi o symme y angle o he inne spli ing, α1, while keeping ixed he mi o symme y axis o he ou e ing α2. No e ha whe eas ∠RRHCP,RHCP and ∠RLHCP,LHCP a e linea unc ions o α2in Fig. 6(b), ∠RRHCP,RHCP and ∠RLHCP,LHCP a e no any longe linea unc ions o α1in Fig. 7(b). This is due o he ac ha whe eas in Fig.6(b) he whole e lec a ay elemen is 8 b a y z x 13 y’ x’ j 13 j 21 j 11 21 11 a1 a 2 j 23 23 (a) α1 (deg) 20 40 60 80 100 120 140 160 180 RRHCP,RHCP and RLHCP,LHCP (deg) -300 -200 -100 0 100 200 RLHCP,LHCP q=1,2 RLHCP,LHCP q=1,2,3 RLHCP,LHCP q=1,...,7 RLHCP,LHCP CST RRHCP,RHCP q=1,2 RRHCP,RHCP q=1,2,3 RRHCP,RHCP q=1,...,7 RRHCP,RHCP CST (b) α1 (deg) 20 40 60 80 100 120 140 160 180 ϕ21 −ϕ11(deg) 145 150 155 160 165 (c) Fig. 7. (a) Two concen ic spli ings. (b) Phase o RRHCP,RHCP and RLHCP,LHCP o he e lec a ay elemen o (a) as a unc ion o he o a ion angle o he inne ing α1. Con e gence is s udied wi h espec o he numbe o basis unc ions (q=1,...,7) used in he ou a cs o he wo spli ings. Ou esul s ob ained wi h he spec al domain MoM a e compa ed wi h esul s ob ained wi h CST . (c) Va ia ion o he leng h o he a cs o he inne ing wi h he o a ion angle α1 o ensu e a phase di e ence o 180◦in he e lec ion coe icien s o he wo o hogonal linea componen s o he CP impinging wa es. Pa ame e s: a=b=5 mm; ρ13=1.85 mm; ρ23=2.05 mm; ϕ13=−75.2◦mm; ϕ23=75.2◦mm; α2=0◦;ρ11=1.20 mm; ρ21=1.40 mm; h1=0.787 mm, ε ,1=2.2, an δ1=0.0009; h2=0 mm; θinc =30◦,ϕinc=0◦; =29.75 GHz. o a ed o phase adjus men , in Fig. 7(b) only he inne pa o he elemen is o a ed o phase adjus men , and he e o e he p opo ionali y be ween phase and o a ion angle p o en in [2] is los . Fo una ely, he o a ion o α1in Fig. 7(b) p o ides enough phase ange o e lec a ay design. In Fig.7(c) we plo he adjus men s in he leng h o he inne a cs ha a e needed o keep a phase di e ence o 180◦be ween he e lec ion coe icien s o he wo o hogonal componen s o he elec ic ield o he impinging CP wa es. These adjus men s a e la ge han in he case o Fig. 7(b), which is a ibu ed o he ac TABLE I CPU imes (seconds) equi ed o he gene a ion o 31 poin s in he cu es o Figs.6(b)and 7(b). The CPU imes a e o ou MoMso wa e and CST . Figu e TMoM CPU TCST CPU Fig. 6(b) 89.8 2383 Fig. 7(b) 152.9 2486 TABLE II CPU imes (seconds)con ibu ions (compu a ion o Hankel ans o ms, in e pola ion o Hankel ans o ms,calcula ion o 2-D Fou ie ans o ms o he basis unc ions and e alua ion o ma ix en ies) o he o al CPU ime equi ed by ou MoMso wa e o he gene a ion o one poin in Figs.6(b)and 7(b). Figu e THT TINT T2DFT TME TMoM TOTAL Fig. 6(b) 0.398 0.0005 2.868 0.007 3.290 Fig. 7(b) 0.570 0.0008 4.871 0.020 5.477 ha he ou e a cs o he elemen a e no modi ied du ing he p ocess o phase adjus men . As in Fig. 6(b), in Fig. 7(b) con e gence o he spec al domain MoM so wa e is achie ed wi h jus h ee basis unc ions pe a c (which equi es he in e sion o 12×12 MoM ma ices), and excellen ag eemen is ound be ween he spec al domain MoM so wa e and CST . We ha e compu ed he CPU ime equi ed by CST o gene a e 31 poin s o he cu es o Figs. 6(b) and 7(b), TCST CPU, and he CPU ime equi ed by he spec al domain MoM so wa e when 3 basis unc ions pe a c a e employed, TMoM CPU . When unning he spec al MoM so wa e, we ha e used he in e pola ed e sions o ˜ i l(kρ), and we ha e included in TMoM CPU he CPU ime equi ed o compu e k h ρ(i) and in e pola e he unc ions ˜ i l(kρ). The esul s ob ained o TCST CPU and TMoM CPU a e shown in Table I. These CPU imes ha e been ob ained in a lap op compu e wi h p ocesso In el Co e i7-6700HQ a 2.6 GHz wi h ou co es and 32 GB o RAM memo y. The MoM code has been w i en in FORTRAN language. Acco ding o he esul s o Table I, he spec al MoM so wa e is ypically 27 imes as e han CST in he analysis o he pe iodic s uc u es s udied in Fig. 6(b), and a ound 16 imes as e han CST in he analysis o he pe iodic s uc u es o Fig. 7(b). When he spec al domain MoM desc ibed in Sec ion II is applied o he analysis o one single pe iodic s uc u e con aining spli ings, he e a e di e en nume ical s eps ha ing di e en CPU ime con ibu ions o he o al CPU ime equi ed by MoM, TMoM TOTAL. These CPU ime con ibu ions include he CPU ime equi ed o compu e he Hankel ans- o ms ˜ i l(kρ) ha a e needed o he in e pola ions by means o (24), THT, he CPU ime equi ed o he in e pola ions o he Hankel ans o ms by means o Chebyshe polynomials, TINT, he CPU ime equi ed o he e alua ion o he 2-D Fou ie ans o ms o he basis unc ions by means o (18), T2DFT, and he CPU ime equi ed o he de e mina ion o he MoM 9 ma ix en ies by means o (6), TME. Table II shows hese di e en CPU ime con ibu ions o TMoM TOTAL ( he CPU imes ha e been ob ained wi h he same compu e used o Table I). In pa icula , he CPU imes p esen ed in Table II a e o he pe iodic s uc u e analyzed in Fig. 6(b) when α2=92.9◦ (ϕ23 −ϕ13=150.4◦in Fig. 6(c)), and o he pe iodic s uc u e analyzed in Fig. 7(b) when α1=92.9◦(ϕ21 −ϕ11=150.0◦ in Fig. 7(c)). No e ha he mos impo an con ibu ion o he o al MoM CPU ime is T2DFT, which is a ound 88% o TMoM TOTAL. This jus i ies he e o s ca ied ou in Sec ion II o compu e he Hankel ans o ms ˜ i l(kρ) appea ing in (18) in an e icien way. I we had no op imized he compu a ion o ˜ i l(kρ), he me hod p esen ed in his pape would no ha e been compe i i e by compa ison wi h comme cial so wa e. Once he unc ions e Jd lq(kxm,kyn) o (18) ha e all been compu ed, he CPU imes equi ed o compu e he double summa ions o (6) and o sol e he sys em o equa ions o (5) a e negligible (no e ha he maximum size o he MoM ma ices o be in e ed is 12 ×12). The nex impo an con ibu ion o he o al MoM CPU ime is THT, which ep esen s a ound 11% o TMoM TOTAL. Table II shows he con ibu ion o he ime equi ed o he in e pola ion o he Hankel ans o ms o TMoM TOTAL is negligible. A his poin , we should emembe ha he compu a ion and in e pola ion o he Hankel an o ms has o be ca ied ou only once in he design o a e lec a ay an enna whe e he inne and ou e adii o he ings ρ1land ρ2l(l=1,...,L) emain cons an in all he elemen s o he an enna. The e o e, we can neglec he con ibu ion o THT o he CPU ime equi ed in he design o a whole e lec a ay an enna. In ou s udy o he compu a ional pe o mance o he spec al domain MoM so wa e as a unc ion o he di e en inpu a iables in ol ed, we ha e inally analyzed he compu a ional complexi y o TMoM TOTAL as a unc ion o he numbe o quad a u e poin s used in (24), Nqp, as a unc ion o he numbe o e ms e ained in in ini e se ies o (18), imax, and as a unc ion o he numbe o basis unc ions used in he applica ion o MoM, Nb. We ha e ound ha he e ec o Nqp on TMoM TOTAL is i ele an o 3≤Nqp ≤15, ha he spec al MoM is oughly o complexi y O((imax)2) o 48 ≤imax ≤200, and ha he spec al MoM is oughly o complexi y O(Nb) o 1 ≤Nb≤7. Based on he cu es o Figs. 6(b), 6(c), 7(b), and 7(c), a dual- equency pencil beam CP e lec a ay an enna has been designed. The an enna adia es LHCP a 19.95 GHz and RHCP a 29.75 GHz as in he case o he an enna designed in [6]. The designed an enna is ci cula and consis s o 5024 elemen s a anged in a 80 ×80 g id wi h cell size 5 mm ×5 mm ( he diame e o he an enna is 400 mm). The an enna is in ended o p oduce a ocused beam in he di ec ion θb=30◦and ϕb=0◦ (see [25, Fig. 5.a]). I is illumina ed by a co uga ed ci cula eed-ho n wi h i s phase cen e loca ed a he coo dina es x=−150 mm, y=0 mm, z=259.8 mm wi h espec o a coo dina e sys em wi h o igin a he cen e o he e lec a ay (see [25, Fig. 5.a]). The ho n is assumed o adia e LHCP wa es a 19.95 GHz and RHCP wa es a 29.75 GHz. The adia ion pa e n o he ho n is modelled as a unc ion cos7(θ), which p o ides an illumina ion le el a he e lec a ay edges 12 dB below he maximum. The an enna elemen s a e dual concen ic spli ings as hose shown in Fig.7(a). In he design Azimu h angle (deg) -50 -40 -30 -20 -10 0 10 20 30 40 50 Gain (dB) -20 -10 0 10 20 30 40 Copola LHCP q=1,2,3 Copola LHCP CST Copola LHCP HFSS C osspola LHCP q=1,2,3 C osspola LHCP CST C osspola LHCP HFSS (a) Ele a ion angle (deg) -50 -40 -30 -20 -10 0 10 20 30 40 50 Gain (dB) -20 -10 0 10 20 30 40 Copola LHCP q=1,2,3 Copola LHCP CST Copola LHCP HFSS C osspola LHCP q=1,2,3 C osspola LHCP CST C osspola LHCP HFSS (b) Fig. 8. Radia ion pa e ns in (a) he azimu h plane and (b) he ele a ion plane o a dual- equency CP e lec a ay based on an elemen wi h wo concen ic spli ings. Plo s a e p esen ed o he analysis o he e lec a ay unde he local pe iodici y assump ion wi h bo h ou spec al domain MoM so wa e (solid and dash-do ed lines), CST (dashed and do ed lines) and HFSS (× and +). The esul s plo ed a e o LHCP adia ion a 19.95 GHz. o he an enna a 19.95 GHz, he dimensions and o ien a ions o he ou e ings ha e been adjus ed in acco dance wi h Figs. 6(b) and 6(c), while assuming he inne ings we e no p esen . Howe e , when he an enna has been designed a 29.75 GHz, bo h he ou e and he inne ings ha e been aken in o accoun . The dimensions and o ien a ions o he ou e ings ha e been ixed a he alues ob ained o he design a 19.95 GHz, and he dimensions and o ien a ions o he inne ings ha e been adjus ed in acco dance wi h Figs. 7(b) and 7(c). The design o he an enna has been ca ied ou by means o he spec al domain MoM so wa e desc ibed in Sec ion II unde he local pe iodici y condi ion, while using he exac incidence angle in each e lec a ay elemen . Once he e lec a ay an enna has been designed, i has been analyzed wi h he spec al domain MoM so wa e (using h ee basis unc ions pe a c wi h q=1,2,3 in (21)), and wi h he wo comme cial so wa es CST and HFSS , unde he local pe iodici y assump ion. In he analysis, he exac incidence angle has been conside ed in each e lec a ay elemen . Figs.8 and 9 show he adia ion pa e ns ob ained in he azimu h and ele a ion planes a 19.95 GHz and 29.75 GHz espec i ely. The esul ing gain is 35.88 dBi a 19.95 GHz and 39.42 dBi a 29.75 GHz. And he an enna e iciency is o 56% a 19.95 GHz and 57% a 29.75 GHz. In Fig. 8 he ag eemen be ween ou MoM esul s, CST and HFSS is excellen . In Fig. 9 he ag eemen is excellen o he copola componen o he