2D ray tracing model for multilayer dielectric dome arrays with inner reflections

IEEE OPEN JOURNAL OF ANTENNAS AND PROPAGATION, VOL. 5, NO. 4, AUGUST 2024 845 Recei ed 7 Decembe 2023; e ised 16 Janua y 2024; accep ed 5 Feb ua y 2024. Da e o publica ion 12 Feb ua y 2024; da e o cu en e sion 6 Augus 2024. Digi al Objec Iden i ie 10.1109/OJAP.2024.3365039 2-D Ray-T acing Model o Mul ilaye Dielec ic Dome A ays Wi h Inne Re lec ions MARIA PUBILL-FONT 1, FRANCISCO MESA 2(Fellow, IEEE), ASTRID ALGABA-BRAZÁLEZ 3, SARAH CLENDINNING 4, MARTIN JOHANSSON 3(Senio Membe , IEEE), AND OSCAR QUEVEDO-TERUEL 4(Fellow, IEEE) 1The Global Big Da a Technologies Cen e , Uni e si y o Technology Sydney, Ul imo, NSW 2007, Aus alia 2Depa men o Applied Physics 1, Uni e sidad de Se illa, 41012 Se illa, Spain 3E icsson Resea ch, E icsson AB, 417 56 Go henbu g, Sweden 4Di ision o Elec omagne ic Enginee ing and Fusion Science, KTH Royal Ins i u e o Technology, 100 44 S ockholm, Sweden CORRESPONDING AUTHOR: O. QUEVEDO-TERUEL (e-mail: [email p o ec ed]) The wo k o F ancisco Mesa was suppo ed in pa by MCIN/AEI/10.13039/501100011033 unde G an PID2020-116739GB-I00. The wo k o As id Algaba-B azález, Ma in Johansson, and Osca Que edo-Te uel was suppo ed by he S a egic Inno a ion P og am Sma e Elec onics Sys em—a Join Ven u e o Vinno a, Fo mas, and he Swedish Ene gy Agency unde P ojec 2023-00648. ABSTRACT The applica ion o lenses combined wi h a ay an ennas (also known as dome a ays o dome an ennas) o he nex gene a ion o e es ial and sa elli e communica ion sys ems b ings a wide ange o ad an ages in e ms o imp o ed adia ion pe o mance, econ igu abili y in he use case, and educ ion in powe consump ion. To acili a e he indus ial implemen a ion o dome an ennas, highly e icien simula ion ools a e equi ed. In his pape , we p esen a s eamlined implemen a ion o ay acing o as and e icien nume ical analysis o he a - ield adia ion pe o mance o 2D mul ilaye dielec ic lenses combined wi h phased a ays. Unlike comme cial physical-op ical me hods, ou p oposed ay- acing me hod is capable o compu ing he e ec s o in e nal e lec ions in he dome in a mul ilaye con igu a ion. In addi ion, he me hod es ima es he abso p ion losses as a esul o he Joule e ec . To demons a e he e ec i eness o he p oposed app oach, we p o ide compa isons o he simula ed adia ion pa e ns using ou p oposed ay acing wi h he esul s ob ained om comme cial ull-wa e simula ion ools. INDEX TERMS A ay an enna, abso p ion loss, dielec ic lens, dome, ma ching laye s, lens a ay, ay acing, adia ion pa e n, e lec ion losses, scanning, 6G. I. INTRODUCTION THE INTRODUCTION o he six h gene a ion (6G) [1] mobile communica ions is an icipa ed o 2030, and i is expec ed o ha e a much g ea e e ec on ou socie y han 5G. Examples o 6G use cases include senso ial expe- iences (In e ne o Senses), machine- ype communica ion, augmen ed eali y and i ual eali y (AR/VR), and join communica ion and sensing (JCAS). To mee he needs o an inc easing numbe o connec ed de ices and use s, 6G mus alloca e addi ional spec um ha has no been used be o e, as well as add ess c i ical echnological issues ela ed o ha dwa e and an enna solu ions [2]. Phased a ay an ennas (PAA) cons i u e an in e es ing an enna solu ion o 5G/6G e es ial communica ions [3], [4],[5] and sa elli e sys ems [6] due o hei simplici y o design and beam s ee ing capabili y, which can be achie ed wi hou physically mo ing he an enna. Howe e , he pe o mance o elec onically scanned PAA is comp omised when s ee ing owa d ex eme angles due o a educ ion in he p ojec ed an enna ape u e size in he scanning di ec ion, esul ing in inc eased scanning losses [7],[8]. An ennas capable o achie ing wide beam scanning wi hou pe o mance deg ada ion a e highly desi able o some applica ions, such as ada s sys ems [9], ai c a -sa elli e communica ions sys ems [10],[11], and mode n wi eless communica ion sys ems [12],[13]. An a ac i e solu ion is he combina ion o phased a ays wi h quasi-op ical sys ems, which p o ides good ma ching pe o mance and c 2024 The Au ho s. This wo k is licensed unde a C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ PUBILL-FONT e al.: 2-D RAY-TRACING MODEL FOR MULTILAYER DIELECTRIC DOME ARRAYS 846 high gain o la ge scanning angles. Well-known solu- ions include pillbox an ennas [14],[15],[16],Ro man lenses [17],[18],[19], Lunebu g lenses (which can be imple- men ed in a ully me allic o m wi h me asu aces [20] o geodesic su aces [21]), shaped pa allel pla e lenses [22], and homogeneous dielec ic lenses [23],[24],[25],[26],[27]. The combina ion o dielec ic lenses and a ays, known as dielec ic dome an ennas, is a p omising an enna solu ion o he nex gene a ion o adio access sys ems and sa elli e communica ions [27],[28],[29],[30],[31],[32],[33]. These an ennas can be used o enhance ce ain p ope ies o phased a ays, such as hei ield o iew [7],[26], o modi y hei adia ing pe o mance o sui di e en scena ios[34], o o educe he powe consump ion o powe ampli ie s o achie e a speci ic Equi alen Iso opic Radia ed Powe (EIRP) alue [35]. When he goal is o inc ease he ope a ing scanning ange, he dome in oduces he phase a ia ion equi ed o de lec he beam in he desi ed di ec ion [36], and due o he e ical dimension o he lens, he e ec i e ape u e will inc ease. A dome an enna could be implemen ed wi h me asu aces [37], bu his would lead o a na ow ope a ing bandwid h caused by he na owband beha io o he employed me asu ace. The mos p ac ical app oach is he dome implemen a ion based on homogeneous dielec ics, because o i s simplici y o design and cos - e ec i eness. Fu he mo e, by choosing he lens ma e ial in a sui able way, he dome may also ac as a adome. In his way, we add lensing unc ionali y o he adome, which allows us o imp o e he scanning pe o mance o he a ay while p o iding mechanical p o ec ion om he en i onmen . The lens cu a u e is hen selec ed o manipula e he wa e on and ob ain he desi ed ocusing p ope ies o each speci ic applica ion. The e o e, he shape o he dome could be op imized by modi ying i s geome y o di e en use cases acco ding o he equi ed speci ica ions, as p e iously in oduced in [34]. Addi ional dielec ic laye s, known as ma ching laye s, can be in oduced o educe he numbe o e lec ions occu ing a he ai /dome bounda ies. The op imiza ion o lens/ adome shapes using comme cial ull-wa e simula o s is compu a ionally in ensi e and ime- consuming. To add ess his issue, ay- acing echniques ha e been de eloped o e alua e he a - ield pe o mance o la ge objec s wi h educed ime and compu a ional esou ces [38], [39],[40],[41],[42],[43],[44],[45],[46],[47],[48], [49],[50],[51]. Addi ionally, he e a e comme cial so wa e packages a ailable o his pu pose [52],[53]. Recen ly, a nume ical me hod based on ay acing o e alua e he a ield o gene alized geodesic lenses was p oposed in [54]. Al hough he gene al a ea o ay- acing echniques can be conside ed ma u e (mainly due o i s ex ensi e use in compu e g aphic ools), he oppo uni y o ad ancemen s ill emains in speci ic cases; o example, in he s udy o dome an ennas, whe e simpli ica ions can be made o signi ican ly educe simula ion ime. I we a e only in e es ed in ay- acing o a pa icula pu pose a he han using a gene alized comme cial ay- acing so wa e, he e a e a numbe o open sou ce lib a ies a ailable (e.g., [55],[56],[57]), as well as he ay- acing ool speci ically p oposed in [7], o e alua e he e ec o homogeneous dielec ic lenses in dome an ennas wi hou conside ing e lec ion o abso p ion losses. This pape p esen s an ex ended bu s ill simpli ied and e icien ay- acing ool o e alua e he adia ion cha ac e is ics o wo-dimensional (2D) mul ilaye dielec ic dome an ennas, aking in o accoun e lec ion and abso p ion losses. The simpli ied 2D model also allows us o in es iga e he e ec o ma ching laye s o mi iga e e lec ion losses. As p e iously men ioned in [32], 2D models can gi e us a good unde s anding o he pe o mance o a h ee-dimensional (3D) dome a ay an enna. In con as o [32], he H-plane adia ion pa e n o he dome a ay is compu ed he e using he s aigh o wa d Ki chho di ac ion o mula ha was al eady used in [54]. The nume ical e ec i eness and accu acy o he adia ion pa e ns compu ed wi h he epo ed simpli ied me hod ha e been demons a ed and compa ed wi h he COMSOL Mul iphysics so wa e o he 2D case and wi h he CST Mic owa e S udio o he 3D case. This ool can be used as a p ima y s ep o he design o 3D lenses and domes o 5G/6G communica ion, as i p o ides a good ini ial quali a i e and quan i a i e insigh in o he lens pe o mance. The p oposed me hod is especially ad an ageous when di e en dome s uc u es a e needed o adjus he adia ing pe o mance o an a ay an enna o di e en use cases, as desc ibed in [34], hus mee ing he cus ome ’s equi emen s in a imely manne , since he a ay can be eused and only he dome design needs o be modi ied. II. RAY-TRACING MODEL In his sec ion, we discuss se e al aspec s o he nume i- cal implemen a ion o he simpli ied ay- acing echnique p oposed o analyze dielec ic dome an ennas. Ra he han using some o he mo e comp ehensi e and igo ous asymp o ic me hods al eady epo ed in he li e a u e [27], [39],[42],[43],[46],[51], he s aigh o wa d implemen ed me hod has been inspi ed by he one epo ed in [54], al hough i has been adap ed o handle mul ilaye dielec ic 2D lenses combined wi h an a ay. The ay- acing app oach is based on h ee unde lying heo ies: 1) geome ical op ics (GO), used o de e mine he ajec o ies o he ays, 2) conse a ion o ay ube powe , applied o calcula e he ampli ude dis ibu ion a he lens ape u e, and 3) he Ki chho di ac ion o mula, employed o ob ain he a - ield pa e n. To es ou ay- acing app oach, we used a dome e e ence model om [7]. The geome y o he dome is shown in Fig. 1, and is cha ac e ized by i s dielec ic cons an , ε , wi h inne and ou e su aces s1and s2. The su aces a e o a ionally symme ic wi h espec o he e ical zaxis and ollow conic shapes. In his wo k, he dielec ic dome o ε =2.5 is e alua ed wi h an a ay o L=975mm a 13 GHz. 847 IEEE OPEN JOURNAL OF ANTENNAS AND PROPAGATION, VOL. 5, NO. 4, AUGUST 2024 FIGURE 1. Dielec ic dome an enna geome y and main pa ame e s. FIGURE 2. Examples o (a) di ec ay acing o linea phase a ay exci a ion and (b) e e se ay acing o calcula ing he phase o e he a ay. A. GEOMETRICAL OPTICS GO uses a ze o-wa eleng h app oxima ion o model he beha io o elec omagne ic p opaga ion in e ms o ays [58],[59]. The ays a e de ined as he o hogonal ajec o ies o he wa e on s, which a e he equiphase su aces o a wa e. To ace he ays h ough dielec ic lenses, he Snell-Desca es law is employed. Following he p ocedu e epo ed in [7], wo di e en app oaches o ay acing a e p esen ed he e, depending on whe e he s a ing poin s o he ays a e se . The i s , e e ed o as di ec ay acing, in ol es se ing he s a ing poin s in he a ay. This ay acing is he ini ial s ep in ob aining he ampli ude dis ibu ion and he a - ield adia ion pa e n. The second implemen a ion is e e se ay acing, in which he s a ing poin s a e speci ied in he ape u e plane. Re e se ay acing is used o ob ain he op imum phase dis ibu ion o e he a ay o maximize di ec i i y. Fo comple eness, his p ocedu e is b ie ly ou lined nex . 1) DIRECT RAY TRACING The ays a e emi ed om he a ay owa ds he ape u e plane, wi h a ixed angle o eme gence, θi, as seen in Fig. 2(a). This ini ial s ep conside s he case o a linea phased a ay exci a ion, meaning ha all he ays ha e he same inciden angle, θi, and a e pa allel o each o he a he sou ce. Howe e , since he su aces o he dome a e no pa allel, he ays will be de lec ed in di e en di ec ions, FIGURE 3. Schema ic o he k- h ay ube and a iables in ol ed in he ampli ude e alua ion. leading o a dec ease in di ec i i y. Consequen ly, an op imal phase dis ibu ion is needed o maximize pe o mance. 2) REVERSE RAY TRACING Rays a e sen ou om he ape u e plane owa ds he a ay in o de o maximize di ec i i y. We assumed ha he ays in he ou e pa o he dome a e pa allel, so hey each he su ace whe e he a ay is loca ed (a z=0)wi h a nonlinea phase dis ibu ion. The angle a which he ays impinge on he dome [θoin Fig. 2(b)] is he same o all ays. The phase dis ibu ion is ob ained by adding he e ec i e dis ances a eled inside he di e en media o each ay [7]: a=− (d1k0+d2kd+d3k0)(1) whe e he dis ances d1,2,3a e shown in Fig. 1 o a speci ic ay, and k0and kd=k0√ε a e he wa enumbe s in ee space and in he lens ma e ial, espec i ely. B. RAY TUBE POWER THEORY Since he ays a e no pa allel, he powe dis ibu ion a he ape u e is di e en om ha a he sou ce. Conse a ion o powe wi hin he ay ubes is applied o accoun o his e ec [58]. Some de ails on he calcula ion o he ampli ude a he lens ape u e we e p e iously gi en in [54]. Following he no a ion in Fig. 3, he ampli ude can be exp essed as A k=AkdLk dckcos θk (2) whe e A kis he elec ic ield ampli ude o he ay kon he dome ape u e wa e on , W.The e msAkand dLk e e o he ampli ude and wid h o he ay ube on he sou ce wa e on , de ined as W.The e mdck e e s o he leng h o he a c in he dome ape u e, and we de ine θkas he angle be ween he local no mal uni ec o o he ape u e ˆ nk and he local Poyn ing uni ec o ˆ sk. All he pa ame e s needed o he ampli ude calcula ion a e ob ained om he GO. Since he wa e on Wand he dome ape u e can ha e di e en cu a u es, he wid h o he ay ube in he dome ape u e, dL k, is aken as dckcos θk. PUBILL-FONT e al.: 2-D RAY-TRACING MODEL FOR MULTILAYER DIELECTRIC DOME ARRAYS 848 C. RADIATION PATTERN COMPUTATION The a - ield adia ion pa e n is e alua ed using he Ki chho di ac ion o mula [54], gi en by E(θ) ∝ k A k e−jk0( k+σk) kˆ nk·ˆ sk+ˆ nk·ˆ kTkdck(3) whe e E(θ) is he o al a -zone elec ic ield a a gi en obse a ion angle θ. The dome ape u e is ea ed as an a ay o adia ing dipoles, each wi h an ampli ude A k.σkis he e ec i e pa h leng h o he k- h ay om he sou ce o he lens ape u e ( his quan i y is de ined la e ), kis he dis ance om he ape u e o he obse e posi ion, ˆ k is he uni ec o in he di ec ion o he dome ape u e o he obse e poin . The e m Tkis he F esnel ansmission coe icien [60] ha accoun s o all e lec ions. III. STUDY OF LOSSES In dielec ic dome an ennas, abso p ion and e lec ion can lead o losses. We ha e modi ied he ay- acing app oach om [54] o e alua e hese phenomena. Ou app oach enables us o inco po a e a dielec ic ma e ial wi h a gi en loss angen and calcula e he associa ed ma e ial losses. Addi ionally, he o wa d and backwa d ields p opaga ing ac oss di e en in e aces a e ela ed o he ans e ma ix. We use he ma ix solu ion o calcula e he componen s o ansmi ed and e lec ed elec omagne ic wa es o a mul ilaye dielec ic s uc u e in o de o e alua e e lec ion losses [60],[61]. A. ABSORPTION LOSSES The dielec ic loss angen , an δ, is a measu e o he elec ical ene gy dissipa ed due o a ious physical p ocesses, such as dielec ic elaxa ion, dielec ic esonance, elec ical conduc ion, and nonlinea losses [62]. When he medium is no ideal, abso p ion losses mus be aken in o accoun , and he dielec ic cons an o he ma e ial becomes complex, ε=ε−jε. Then, he wa enumbe k=β−jα(βand α a e he phase and a enua ion cons an s) associa ed wi h a homogeneous medium can be w i en as k=ωμ0ε(1−j anδ)=k0ε (1−j an δ)(4) wi h ε being he ela i e pe mi i i y cons an o he medium, and anδ he a io be ween he eal and imagina y pa s o ε(iso opic magne ic lossless/lossy ma e ials can easily be aken in o accoun by changing μ0by i s co e- sponding magne ic pe meabili y). The complex pe mi i i y o each ma e ial in a laye ed dielec ic dome leads o di e en wa enumbe s. This is aken in o accoun when he a ield is e alua ed using he Ki chho di ac ion o mula (3). The o al e ec i e pa h o he k- h ay, which can be complex i an δi= 0, is hen gi en by he e ec i e pa h leng h σk= N  i=1 σ(i) k= N  i=1ε ,i(1−j an δi)i(5) whe e he index i ep esen s each medium o he N-laye ed s uc u e and iis he geome ic dis ance a eled by he ay wi hin he i- h ma e ial. B. REFLECTION LOSSES When a a eling plane wa e impinges on an in e ace be ween wo ma e ials wi h di e en pe mi i i ies, bo h e lec ion and e ac ion occu . Howe e , when he medium is laye ed, i is no enough o calcula e he e lec ions a each in e ace, as some o he ene gy will be apped inside one ma e ial as a esul o in e nal e lec ions. Acco ding o [61],[63],[64], he e lec ion and ansmission losses associa ed wi h each ay in a plana laye ed medium can be de e mined by using p opaga ion/ma ching ma ices o ans e se ields. The losses due o mul iple e lec ions can hen be calcula ed using he complex ansmission coe icien Tkin he Ki chho di ac ion o mula (3). The analysis o ansmission and e lec ion coe icien s o a mul ilaye plana dielec ic medium can be accomplished using he ollowing app oach ou lined in [61],[63],[64]: E+ 0 E− 0=N+1  i=1 1 TiejϕiRie−jϕi Riejϕie−jϕiE+ N+1 0 =A11 A12 A21 A22E+ N+1 0(6) whe e Tiand Ria e he F esnel ansmission and e lec ion coe icien s, espec i ely, a he i- h in e ace be ween he (i−1)- h and i- h laye s. The phase shi associa ed wi h each laye is gi en by ϕi=kicos θi i, whe e iis he hickness o he laye and θiis he p opaga ion angle wi h espec o he no mal o he in e ace (ϕN+1=0).The o al ansmission coe icien o each ay is ob ained a e compu ing he p oduc o he ma ices as Tk=1/A11.The abo e de i a ion assumes ha all in e aces a e plana and pa allel, so (6) mus be adap ed o ou case s udy, whe e he su aces o he dielec ic dome ha e a conical shape. To do his, he geome y o he p oblem can be simpli ied by b eaking down all he su aces ha make up he dome in o small segmen s/ ace s. The ma ix solu ion equi es h ee inpu s: he hickness o each laye , he ela i e pe mi i i y o each laye , and he angle o incidence. The con igu a ion in Fig. 4(a)isusedas an app oxima e equi alen local model o he laye ed lens when he su aces a e no pa allel, o he pa icula case o a h ee-laye dielec ic slab. The no mal o he i s su ace, ˆ nA, is aced a he in e sec ion poin o he incoming ay wi h his su ace, a poin ha will be used as a e e ence. Each ime he ay in e sec s ano he in e ace, an auxilia y line pa allel o he i s su ace is aced [blue dashed lines in Fig. 4(a)]. The “e ec i e” hickness o each laye ,  i,is aken as he dis ance be ween he pa allel auxilia y lines, hus c ea ing a locally e ec i e plana s uc u e, accep able o he e alua ion o he e lec ion losses associa ed wi h each indi idual ay. The use o his equi alen plana model o calcula e he ansmission coe icien T igno es any phase 849 IEEE OPEN JOURNAL OF ANTENNAS AND PROPAGATION, VOL. 5, NO. 4, AUGUST 2024 FIGURE 4. (a) O iginal locally non-plana s uc u e and he equi alen hickness used o e alua e e lec ions in he disc e ized dome. (b) E ec i e locally-plana model ha shows he phase compensa ion o he plane wa e model. shi o he ay due o i s o iginal non-plana na u e. Howe e , in ou app oxima e model, we mus ake in o accoun a phase-shi ac o ξk o each ay. This equi es mo ing he e e ence om poin A o he co esponding exi poin o he ay, poin B in Fig. 4(b), which makes ha ξk=e−jk0  T anθksinθk(7) whe e he angle θkbe ween he ay inside he lens and he no mal ˆ nAis de e mined by he Snell-Desca es law, which s a es ha θk=a csin(sinθ0/nlens). The o al equi alen hickness  Tin he case o Fig. 4is equal o he sum o  1,  2, and  3. The dis ance om he end-poin o he ay o poin A (solid ed line in Fig. 4) is ob ained om  T anθk. Then he e m sinθkis added o he phase-shi ac o ξk o mo e he e e ence om poin A o poin B (dashed o ange line). The accu acy o he app oxima e equi alen local plana model is demons a ed by he close ma ch be ween he esul s o ou model and ull-wa e simula ions, which is discussed in mo e de ail in he ollowing sec ions. IV. NUMERICAL RESULTS The 2D dielec ic dome a ay an enna was modeled using he ay- acing ool discussed in he p e ious sec ion. The dome shapes we e di ided in o 30 segmen s, which was enough o gua an ee good con e gence in his case. To alida e he p ecision and e iciency o ou app oach, we also simula ed FIGURE 5. Phase dis ibu ions in he a ay when a plane wa e is a i ing om θoo 0◦,20 ◦,40 ◦,60 ◦and 80◦. he 2D p o ile o he lens wi h COMSOL a 13 GHz. The dielec ic lens was illumina ed using an 84-elemen a ay o dipoles wi h a leng h o 975 mm (see Fig. 1). Two lenses we e simula ed: a h ee-laye dome wi h a bi a y e ac i e indexes and high e lec ion losses and a dielec ic dome wi h ma ching laye s. The e e se ay acing is used o ob ain he phase exci a ion o he a ay when he lens is applied. The phases a e shown in Fig. 5 o he wo lenses s udied: he lens wi h high e lec ions and he lens wi h ma ching laye s. Howe e , he adia ion pa e ns a e compu ed wi h di ec ay acing using he phase in o ma ion ob ained om he e e se model ( om Fig. 5). A. DOME WITH HIGH REFLECTIONS A conic-shaped dome wi h h ee laye s o di e en e ac i e indices is used o es he accu acy o he p oposed ay- acing ool in assessing e lec ion losses. The e ac i e indices o he laye s a e 3, 4, and 2.5, espec i ely, and he 2D p o ile is shown in Figu e 6(a). The hicknesses o he i s and hi d laye s a e 1=20 mm and 3=30 mm. The shape o he lens is de ined by ou su aces (s0,s1,s2, and s3) ha can be de ined analy ically o nume ically, esul ing in ou unc ions ha a e disc e ized. In his example, he su aces s0and s1 ha de ine he lowe dielec ic laye ha e he same shape; simila ly, s2and s3 ha bound he uppe laye a e also pa allel o each o he . When he su aces a e disc e ized, he p oblem can be locally iewed as wo nonpa allel bu plana su aces, s1and s2, wi h addi ional pa allel lowe and uppe pa allel su aces, s0and s3, as illus a ed in Fig. 4(a). Fig. 6(b) shows he adia ion pa e ns o 0◦,20 ◦,40 ◦,60 ◦ and 80◦, compa ing he esul s ob ained wi h ou algo i hm and he COMSOL so wa e. The phase dis ibu ion o he phased a ay is ob ained i s om e e se ay acing, as discussed in Sec ion II-A. The peak magni ude o he elec ic ield no malized o ha in b oadside wi hou he lens is illus a ed in Fig. 6(c). The ay- acing esul s a e compa ed o hose om COMSOL simula ions, as well as o he scan losses o he isola ed a ay (g ay line). The compa ison yields good ag eemen ac oss he en i e scanning ange, al hough some mino disc epancies a e obse ed o la ge angles, PUBILL-FONT e al.: 2-D RAY-TRACING MODEL FOR MULTILAYER DIELECTRIC DOME ARRAYS 850 FIGURE 6. (a) 2D p o ile, (b) adia ion pa e ns, and (c) peak magni ude s scanning angle o a conic-shaped dielec ic lens wi h high e lec ions. Ray- acing esul s a e compa ed wi h COMSOL. which a e mainly a ibu ed o he d awbacks o COMSOL when simula ing such ex eme angles. A simula ion o he isola ed a ay wi hou he lens was also conduc ed using COMSOL and was compa ed o he ay- acing algo i hm, which e ealed simila disc epancies a hese la ge angles. B. DOME WITH MATCHING LAYERS He e we s udy a dielec ic dome o ε =2.5de ined by wo su aces s1and s2when his dome is bounded by lowe and uppe ma ching laye s. To educe he le el o e lec ions, hese ma ching laye s ha e a hickness o a qua e wa eleng h and a ela i e pe mi i i y εML = √1·2.5=1.58. The shapes o he ou su aces ha de ine his lens a e exac ly he same as in he p e ious example, as shown in Fig. 7(a). FIGURE 7. (a) 2D p o ile, (b) adia ion pa e ns, and (c) peak magni ude s scanning angle o a conic-shaped dielec ic lens wi h ma ching laye s. Ray- acing esul s a e compa ed wi h COMSOL. In Fig. 7(b), he adia ion pa e ns compu ed wi h COMSOL and he ay- acing algo i hm a e p esen ed wi h he phase dis ibu ion o he phased a ay ob ained i s om e e se ay acing. Radia ion pa e ns a e compu ed o i e cases, each wi h a di e en poin ing di ec ion ha goes om 0◦ o 80◦. The peak magni ude o he elec ic ield no malized o ha in b oadside wi hou he lens is illus a ed in Fig. 7(c). Again, good ag eemen is achie ed along almos he en i e scanning ange be ween he ay- acing (RT) and COMSOL da a. Simila ly o he p e ious example, small disc epancies a e ound o e y la ge poin ing angles. This analysis clea ly shows ha ma ching laye s a e equi ed o educe losses. Ou model has also been es ed using CST S udio Sui e 2022, a h ee-dimensional (3D) ull-wa e simula o . Two lenses we e implemen ed in CST a 28 GHz and 851 IEEE OPEN JOURNAL OF ANTENNAS AND PROPAGATION, VOL. 5, NO. 4, AUGUST 2024 FIGURE 8. (a) Linea phased a ay,(b) cylind ical lens, (c) adia ion pa e ns conside ing CST and 2D ay- acing, and (d) peak magni ude s scanning angle o conic-shaped dielec ic lens wi h ma ching laye s. The esul s a e no malized o b oadside wi hou he lens. compa ed o he p oposed ay- acing ool. Figu e 8(a) shows a one-dimensional a ay o 24 wa eguides ha illumina es a cylind ical lens along he y-axis. The dimensions o he a ay and he 2D p o ile o he lens we e aken om [34]. The geome y o he 3D lens is depic ed in Fig. 8(b), and he adia ion pa e ns o di e en s ee ing di ec ions in he E-plane a e shown in Fig. 8(c). To assess he e ec o e lec ion losses, he esul s we e no malized o he b oadside wi hou he lens. Fo he second alida ion, a wo-dimensional a ay o 24x8 wa eguides was employed, as depic ed in Fig. 9(a). The second lens had he same p o ile as he i s , bu was o a ionally symme ic wi h espec o he z-axis, as illus a ed in Fig. 9(b). The adia ion pa e ns FIGURE 9. (a) 2D Phased a ay model included in he CST simula ion. (b) Sphe ical lens, (c) adia ion pa e ns no malized o b oadside wi h he lens o a o a ionally symme ic conic-shaped dielec ic lens. o he lens no malized o b oadside a e shown in Fig. 9(c). The adia ion pa e ns show he Ex- ield con ibu ion in he E-plane, which co esponds o he co-pola iza ion. In his simula ion, al hough i is no illus a ed he e, he c oss- pola iza ion is negligible. The esul s ob ained om he 2D ay- acing model and he CST ool o bo h he cylind ical and sphe ical lenses we e in e y good ag eemen . This shows ha he p oposed me hod can be used as a i s design s ep o 3D dome an ennas, as i p o ides a eliable ini ial unde s anding o he lens pe o mance, bo h quali a i ely and quan i a i ely. As a inal no e, he CPU ime employed by he ay- acing ool implemen ed in hese wo examples is abou 150 imes less han ha equi ed by COMSOL, 240 imes less han he equi ed in CST o he cylind ical lens, and 740 imes less o he sphe ical lens in CST. V. CONCLUSION An e icien ay- acing ool has been de eloped o e alua e he a - ield adia ion pa e n o 2D mul ilaye dielec- ic lenses. Radia ed ields compu ed wi h his app oach PUBILL-FONT e al.: 2-D RAY-TRACING MODEL FOR MULTILAYER DIELECTRIC DOME ARRAYS 852 ha e been success ully alida ed by compa ison wi h he COMSOL comme cial simula o , educing he compu a ional ime by a ac o o app oxima ely 150. Losses can be de e mined e en when ex a laye s a e added o he dielec ic dome. The imagina y pa o he pe mi i i y o he laye s is used o ake in o accoun abso p ion losses. Re lec ion losses ha e been accu a ely simula ed by u ilizing an equi alen local plana model and he p opaga ion/ma ching ma ices o he ans e se ields. The implemen a ion o he ay- acing me hod has been p o en o be a powe ul and p ecise ool o he design and analysis o mul ilaye dielec ic dome an ennas. The 2D ay- acing model has also been e i ied wi h a 3D ull-wa e comme cial simula o , CST S udio Sui e 2022, demons a ing i s sui abili y as an ini ial design ool o unde s anding how a 3D lens shape would pe o m when combined wi h an a ay. This has a signi ican indus ial impac , as di e en lens geome ies can be modeled quickly and accu a ely, sa ing compu a ional esou ces and enginee ing e o s. The ag eemen be ween ou simpli ied 2D model and CST is ema kable. Howe e , o make he ay- acing app oach sui able o all dome an enna applica ions, ou nex esea ch objec i e is o expand he ay- acing me hod o encompass 3D s uc u es. Mo eo e , he p oposed me hodology has shown signi ican e iciency and speed, making i a aluable complemen o ad anced syn hesis p ocedu es. 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She is cu en ly pu suing he Ph.D. deg ee in millime e and submillime e band an ennas wi h he Uni e si y o Technology Sydney, Sydney, Aus alia. He cu en esea ch in e es s include lens an ennas, ans o ma ion op ics, an enna a ay ech- nologies, and beam o ming ne wo ks. FRANCISCO MESA (Fellow, IEEE) ecei ed he Licenciado and Ph.D. deg ees in physics om he Uni e sidad de Se illa, Se ille, Spain, in 1989 and 1991, espec i ely, whe e he is cu en ly a P o esso wi h he Depa amen o de Física Aplicada 1. His esea ch in e es includes elec o- magne ic p opaga ion/ adia ion in mic owa e and quasi-op ical s uc u es.