Anomalous reflection of visible light by all-dielectric gradient metasurfaces

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1 Reseach Aicle Vol. 34, No. 7 / July 7 / Jounal of he Opical Sociey of Ameica B D Anomalous eflecion of visible ligh by all-dielecic gadien measufaces NIKOLAOS L. TSITSAS, * AND CONSTANTINOS A. VALAGIANNOPOULOS Depamen of Infomaics, Aisole Univesiy of Thessaloniki, Thessaloniki GR-544, Geece Depamen of Physics, School of Science and Technology, Nazabayev Univesiy, 53 Qabanbay Bay Ave., Asana KZ-, Kazakhsan *Coesponding auho: nsisas@csd.auh.g Received 3 Januay 7; acceped Febuay 7; posed 6 Febuay 7 (Doc. ID 85933); published 5 Mach 7 Plane wave scaeing by a plana measuface composed of wo peiodically alenaing ecangula dielecic ods is consideed. A igoous inegal equaion mehodology is employed fo he analysis and he accuae deeminaion of he efleced and ansmied fields. Sysemaic opimizaions wih espec o he configuaion s paamees ae pefomed, which eveal ha i is possible o obain significanly enhanced anomalous eflecion (wih simulaneously suppessed odinay eflecion pediced by Snell s law) wih powe vaying fom 9% o almos % of he inpu one, depending on he colo of he inciden ligh. I is shown ha hese eflecion popeies ae suppoed by measufaces easily ealizable wih specific low-loss dielecic maeials. In his way, seveal all-dielecic opimal designs ae epoed ha can be used in numeous applicaions demanding anomalous eflecion in he visible ange. 7 Opical Sociey of Ameica OCIS codes: (5.5) Diffacion and gaings; (6.398) Meamaeials; (4.3695) Linea and nonlinea ligh scaeing fom sufaces. hps://doi.og/.364/josab.34.d. INTRODUCTION Gadien measufaces have emakable popeies wih espec o manipulaing elecomagneic waves including conolling he popagaion diecion of an inciden wave ove subwavelengh disances, modifying he opical wavefon, bending and focusing ligh, geneaing anomalous eflecion and efacion phenomena as well as desied disibuions of elecomagneic waves [ ]. Meamaeial gadien index diffacion gaings, composed of meallic and dielecic pas, ae uilized in gadien index opics applicaions [], in muliband elecomagneic absobes [], and in he impovemen of nea-field opical enhancemen [3]. Recenly, all-dielecic gadien meamaeials have eceived paicula aenion and have been shown o assis significanly in subdiffacion confinemen and guiding of ligh wihou meals [4], in achieving vey high ansmission efficiency [5,6], in elecomagneic mode convesion [7], and in conollable coloing [8]. Such aificial sucues ae ealized by using low-loss dielecic maeials in building alenaing blocks of anspaen and high-efaciveindex media. Among all he afoemenioned ineesing effecs ecoded wih use of gadien measufaces, special aenion will be paid o anomalous eflecion. Such eminology is used o descibe unusual and emakable seeing of he inciden illuminaion owad diecions no pediced by Snell s law. By popely designing he exue of he measuface, one can achieve eflecion of he inciden elecomagneic field ino fee space in couneinuiive ways being vey useful in numeous applicaions. In paicula, anomalous eflecion is employed in fabicaion of ulaefficien anieflecion coaings [9], while he consucion of polaizaion and hee-dimensional beam splies becomes easie by obaining conollable ou-of-plane efacion []. The significance of anomalous eflecion is no confined only in componens opeaing a he visible o he infaed pa of he fequency specum; i has been uilized fo micowave suface plasmon coupling [] o even fo acousic wavefon engineeing []. Puing i o a moe geneal basis, anomalous eflecion is a fundamenal mechanism ha decisively assiss in uning [8], diecing [], and seeing [3] he elecomagneic waves. Theefoe, i is wohy of fuhe examinaion and sudy. In his wok, we invesigae plane wave scaeing by a wodimensional, plana sucue conaining peiodically alenaing ecangula ods of wo diffeen dielecics. A igoous and highly accuae enie-domain inegal equaion mehodology is employed fo he analysis of he associaed scaeing and diffacion phenomena by such a measuface. The mehodology was iniially inoduced and developed in [4] fo he analysis of dielecic gaing sucues and was subsequenly exended o he scaeing analysis of measufaces in [5]. We focus on he deeminaion of suiable geomeical and elecomagneic paamees of he measuface in ode fo i o exhibi 74-34/7/7D-8 Jounal 7 Opical Sociey of Ameica

2 D Vol. 34, No. 7 / July 7 / Jounal of he Opical Sociey of Ameica B Reseach Aicle significan anomalous eflecion. In paicula, i is possible, due o he inhomogeneous naue of he sucue (gadien index), o guide he majo pa of he diffaced powe owad he same egion of he plane ha he measuface is excied (diecions of he ode in eflecion and ansmission), unlike wha is happening is he case of a homogeneous slab (popagaion of only he zeo ode dicaed by Snell s law). Ou pimay aim is o simulae fuhe expeimenal as well as heoeical woks by epoing vaious simple and ealisic gaing configuaions compised of odinay dielecic media, which a specific visible fequency bands give subsanial anomalous eflecion wih simulaneously suppessed and annihilaed zeo-ode esponses. Following a ceain sysemaic opimizaion scheme, we show ha measufaces composed of convenional low-loss dielecic maeials wih specifically designed geomeical chaaceisics can eflec fo all he examined fequencies in he visible ange moe han 9% of he inciden field s powe back o he diecion whee he inciden field popagaes. I is woh noing ha fo ceain colos, he efleced powes of he ode each almos %. Fo each opimized configuaion, we also examine in deail he vaiaions of he efleced and ansmied powes of he ode wih espec o he angle of incidence as well as he opeaing wavelengh. In he subsequen analysis, an exp iω ime dependence of he field quaniies is assumed and suppessed, wih pffiffiffiffiffiffiffiffiffi ω π λ ε μ as he angula fequency, as ime, and p i ffiffiffiffiffi. The pemiiviy, pemeabiliy, and wavelengh of fee space ae denoed by ε, μ, and λ, especively.. MATHEMATICAL ANALYSIS OF THE SCATTERING PROBLEM Conside he gadien dielecic measuface, depiced in Fig., which is composed of a dielecic slab wih efacive index n and hickness w conaining Λ-peiodic ecangula inclusions of efacive index n, widh sλ, and same hickness w. Thus, he measuface unde invesigaion is a Λ-peiodic sucue wih wo dielecic maeials alenaing inside each uni cell. The semi-infinie cove and subsae plane egions, above and below he slab, ae chaaceized by efacive index n (fo vacuum, we have n ). The enie sucue has pemeabiliy μ and is assumed unifom along he diecion ŷ. Fig.. The consideed all-dielecic Λ-peiodic measuface composed of wo dielecic maeials wih efacive indices n and n, widhs sλ and s Λ, and common hickness w. The measuface is excied by a uni ampliude TE-polaized plane wave wih angle of incidence θ. The measuface is excied by a ansvese-elecic- (TE) polaized plane wave wih angle of incidence θ (see Fig. ) and elecic field given by E i x;z Ψ i x;z ŷ exp ik n sin θx cos θz ŷ; () p whee k ω ffiffiffiffiffiffiffiffiffi ε μ denoes he fee-space wavenumbe. The oal elecic field induced in evey egion of he scaeing poblem is of he fom E x;z Ψ x;z ŷ, and he unknown scala elecic field faco Ψ admis he inegal epesenaion [4 7] ZZ Ψ x;z Ψ x;z k n n G x;z; x ;z Ψ x ;z dx dz ; x;z R ; () whee Ψ and G ae he elecic field faco and he Geen s funcion boh induced on he especive sucue in he absence of he inclusions due o he plane inciden wave of Eq. () and a line-souce exciaion inside he dielecic slab, especively. Fuhemoe, S denoes he oal ansvese coss secion of he ecangula inclusions. Funcions Ψ and G have been analyically deemined (fo moe complicaed geomeies) in [4,5]. Accoding o he Floque Bloch heoem [8], he elecic field faco is expessed as Ψ x;z exp ik n cos θz u x;z ; (3) whee u x;z is a Λ-peiodic funcion of z. By educing he double inegals of Eq. () o inegals on he inclusion s coss secion on he basic uni cell ;sλ w ; w and by using he Poisson s summaion fomula fo he Diac funcion [9], we efomulae he inegal epesenaion in Eq. () wih espec o funcion u as u x;z Ψ x;z exp ik n cos θz k n n X exp i πp Λ Λ z p Z sλ Z w u x ; ζ exp i πp γ w Λ ζ p x;x dx dζ ; (4) whee funcion γ p denoes he kenel of he deemined Fouie inegal expession of he Geen s funcion G (see Appendix A of [5]). Nex, we apply o Eq. (4) a highly efficien enie-domain Galekin echnique. Funcion u x;z is expanded on he coss secion of he inclusion inside he basic uni cell in he following Fouie seies wih espec o z: u x;z X c n exp g n x c n exp g n x n exp i πn Λ z ; (5) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi whee g n k n cos θ πn Λ k n, while c n ae deeminable coefficiens. The x-dependen Fouie coefficiens in Eq. (5) ae efeed o, in he eminology of he mehod of momens, as he elecic field enie-domain expansion (basis) S

3 Reseach Aicle Vol. 34, No. 7 / July 7 / Jounal of he Opical Sociey of Ameica B D3 funcions. Then, we esic he obsevaion veco x;z in Eq. (4) on he domain of he inclusion, and hence funcion u in Eq. (4) is expanded in he Fouie seies of Eq. (5). Fuhemoe, by consideing he inne poducs of boh sides of he esuling equaion wih he es funcions [which ae he conjugaes of he expansion funcions in Eq. (5)] and caying ou he inegaions analyically, we fomulae an algebaic infinie nonhomogeneous squae linea sysem of equaions wih espec o he unknown coefficiens c n ;n Z. This infinie sysem is solved numeically by uncaion as follows: we ake ino accoun he enie-domain expansion as well as he es funcions wih maximum absolue ode N and educe he infinie sysem o a uncaed squae linea sysem of ode 4N of he fom A A A A c b c b ; (6) whee c ae he N column vecos conaining he unknown coefficiens c n, while A and b ae squae maices of ode N and N column vecos, especively; hei elemens ae given in [4] and [5]. Afe having deemined he coefficiens c n by solving he sysem in Eq. (6), he efleced and ansmied elecic field facos ae, especively, expessed by Ψ x;z X p exp i k x;p x k z;p z ; x > w ;z R; p (7) Ψ x;z X p exp i k x;p x k z;p z ; x < w ;z R; p (8) wih p and p denoing, especively, he complex eflecion and ansmission coefficiens, which ae explicily deemined by means of c n. Moeove, k x;p and k z;p denoe he componens of he efleced and ansmied wave vecos and ae given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k x;p i k n cos θ πp k Λ n ; (9) k z;p k n cos θ πp Λ : () Each em in he field seies in Eqs. (7) and (8), indexed by p Z, epesens a specific field componen and will be efeed o as he p-efleced o p-ansmied diffaced ode. Accoding o Eq. (9), he diffaced odes, which popagae along he x axis, depend on he peiod Λ, he fee-space wavelengh λ, he efacive index n, and he angle of incidence θ. I is impoan o analyze he condiions fo which a diffaced ode p is popagaing o becomes evanescen. The zeo-ode efleced and ansmied fields ae always popagaing (Snell s law). Fo odes p, we define he hesholds p Λ λ n cos θ () fo which he diffaced fields swich fom popagaion o cuoff. Moe pecisely, when p <p<p, he wave veco componen k x;p is eal, and hus he p-ode efleced and ansmied fields ae popagaing along he x axis. On he ohe hand, when p>p o p<p, he componen k x;p becomes puely imaginay and he p-ode fields ae evanescen. The pesened enie-domain inegal equaion mehodology is igoous and povides semi-analyical soluions wih high numeical sabiliy, conollable accuacy, and high efficiency. The Geen s funcion is analyically expessed as a Fouie inegal and all he involved inegaions ae analyically caied ou. Theefoe, he mehod is vey efficien in ems of CPU ime and compue memoy and he sole appoximaion in he soluion is he final uncaion of he expansion funcions ses. All hese advanages ende he poposed echnique vey suiable fo opimizaion of mulipaameic sucues equiing deailed scanning of he paamee space, like he one analyzed below. The equied uncaion ode N is deemined by applying a convegence conol o he soluions fo inceasing N accoding o he following wo cieia, in consisency wih he lieaue [3,3]: (i) an enegy consevaion condiion (efleced and ansmied fields mus conseve powe wihin pa in 8 ) and (ii) convegence o he soluion wih inceasing N fo all he inclusion s and he inciden wave s paamees. Repesenaive convegence paens of he zeo-ode eflecion coefficien, depiced in [4] and [5], show ha small values of N povide sufficien convegence, a fac which consiues a basic advanage of he mehod. Fo all he esuls pesened heeafe, i has been checked ha he choice N 5 is sufficien so ha boh cieia (i) and (ii) ae saisfied. This supeio numeical efficiency of he pesen mehod is mainly jusified due o ha he unknown elecic field and he enie-domain expansion funcions saisfy he same physical law, i.e., he Helmholz equaion, and ha he uilized backgound s Geen s funcion saisfies inheenly he bounday condiions in he inclusion-fee sucue. A deailed sudy concening convegence and numeical efficiency aspecs of he inegal equaion mehodology is included in [4], whee he mehod is also validaed by obaining coinciden esuls wih diffeen mehodologies which howeve equie a significanly lage numbe of basis funcions. 3. PARAMETER SELECTION AND OPTIMIZATION APPROACH Ou majo pupose in his wok (as aleady menioned in Secion above) is o invesigae suiable selecions of paamees of he measuface in ode fo i o geneae he so-called anomalous eflecion and ansmission phenomena in which cases he main conibuion in he efleced and ansmied fields comes fom waves aveling along diecions no pediced by Snell s law. In such cases, mos of he elecomagneic esponse of he sucue is confined o diecions lying in he same egion of space wih he incoming field, heeby focing ligh o popagae owad he side of incidence. The measuface in Fig. is consideed as fee-sanding. Theefoe, he efacive index of he backgound medium (vacuum) is aken as equal o uniy: n. A. Paamee Selecion We equie ha he examined measufaces admi only he zeoand he -eflecion and ansmission odes o popagae.

4 D4 Vol. 34, No. 7 / July 7 / Jounal of he Opical Sociey of Ameica B Reseach Aicle This implies ha p < and <p <, which lead by Eq. () o he following condiions: cos θ max cos θ; < λ < cos θ: () Λ Moeove, in ode fo he ode o be efleced back in he same (second) quadan of he plane wih ha whee he inciden field popagaes and impinges on he measuface, as well as o be ansmied back in he coesponding (hid) quadan, i mus hold ha k z; <, which by Eq. () implies cos θ < λ Λ : (3) The lae is saisfied fo all θ when λ > Λ and fo essenially all θ when λ Λ. Since cos θ > cos θ, he condiion in Eq. () implies he condiion in Eq. (3) and accodingly we poceed only wih Eq. (). The angle beween he negaive z axis and he wave veco of he -efleced ode (measued clockwise as he angle θ in Fig. ) is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos θ λ θ Λ cos θ λ C A : (4) Λ The especive angle coesponding o he -ansmied ode has he same value wih θ and is measued couneclockwise fom he negaive z axis. Figue depics he conou plo of he absolue value of he diffeence beween he angle θ of he -efleced ode and he angle of incidence θ as funcion of θ and he gaing s elecical peiod Λ λ. The especive infomaion fo he -ansmied ode is also depiced in Fig. since, as menioned above, he popagaion diecion of he -ansmission ode is symmeic wih espec o he z axis o ha of he -efleced ode. Blank (whie) egions in Fig. coespond o pais of θ and Λ λ ha do no fulfill any of he condiions in Eqs. () and (3) (namely, popagaion of only wo diffaced odes and eflecion of he ode o he same quadan whee he inciden field popagaes). Fom Fig., we can conclude he Λ/λ.5 θ θ (degees) Fig.. Conou plo of jθ θj as a funcion of θ and Λ λ ange of values of he incidence angle θ, which is meaningful and useful o selec in ode o popely invesigae anomalous eflecion and ansmission effecs. This ange coesponds o he dak blue egion of Fig., whee he diffeence beween θ and θ is elaively small (pecisely does no exceed 8 ). Moeove, in he viciniy of he angle θ ha we will selec, we need o have a sufficien ange of values of Λ λ, saisfying Eq. (), in ode o be able o have a lage paameic space fo he subsequen opimizaions. I is hus meaningful o make an indicaive selecion of θ 6 ; simila invesigaions and opimizaions wih hose epoed below can be pefomed fo alenaive choices of incidence angles θ. Up o his poin, we have consideed a measuface admiing only he zeo- and he -diffaced odes, explained he ange of λ Λ ha he lae consideaion implies, and have jusified he choice of he angle of incidence. As he nex sep, we elaboae on pope and elevan selecions fo he emaining paamees of he poblem, namely, he opeaing wavelengh λ, he efacive indices n and n, he hickness w of he dielecic slab, and he duy cycle s of he inclusions. We conside five diffeen opeaing wavelenghs λ, each one coesponding o a diffeen colo of he visible specum; moe pecisely, we ake he value of λ a he cenal wavelengh in he egime of each colo (as defined by [3]). Ohe values of λ migh also be consideed. Fo each λ and fo consan θ 6, he ange of he peiod Λ of he measuface is ha dicaed by Eq. (). Fo he efacive index of he dielecic slab, we conside ha n.35. Repesenaive maeials wih such efacive index values in almos he enie visible ange ae Teflon AF fluoopolymes [33]. Fo he inclusion, we assume ha i is occupied by lossless silicon and se he values of he inclusion s efacive index n fo each colo in he visible ange accoding o he classical fequency-vaying models of [34,35]. Concening he hickness w of he dielecic slab, he geneal equiemen is o be suiably small, pemiing he descipion as a measuface. In he subsequen esuls, he opimal values of w will acually un ou o be of he ode of a few hunded nanomees, which is much smalle han a single effecive wavelengh ino he employed media. The duy cycle s of he inclusions is kep consan, heeafe, a s. Thus, he measuface has he fom of a classical binay gaing [36]; fabicaing a binay gaing is easie han ohe ypes of gaings like mulilevel o gaded-index ones [37]. Changing he duy cycle, he numbe of inclusions pe uni cell o even hei shape migh also be wohy of fuhe invesigaion; he lae chaaceisics have been shown o play significan oles in he scaeing behavios of such sucues (see, e.g., [4] and [38] in he conex of shap esonances exploied in fequency selecive files). B. Opimizaion Appoach The basic opimizaion scheme and he meics uilized ae descibed in his subsecion. The main aim is o selec suiable measuface paamees such ha he powes caied fom he ode ae significanly enhanced while simulaneously he powes caied by he zeo ode ae significanly suppessed. The elecomagneic fields powes coesponding o he diffeen efleced and ansmied odes ae defined in

5 Reseach Aicle Vol. 34, No. 7 / July 7 / Jounal of he Opical Sociey of Ameica B D5 he sandad way, following [39] (see also he discussions in [4]), as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P p j pj cos θ p λ ; (5) sin θ Λ 9 m 5 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P p j pj cos θ p λ : (6) sin θ Λ The lae quaniies ae usually called he efficiencies of he diffaced odes, which ae defined as he aios of he flux of he Poyning veco associaed wih he p-diffaced ode ove he coesponding flux of he inciden field [39]. I is also eviden ha P j j and P j j. Hence, he enegy balance cieion akes he fom X P p P p ; (7) p P whee P is he subse of Z including only he diffaced odes coesponding o popagaing waves; in ou case P f ; g. Equaion (7) saes ha he sum of all efficiencies is equal o uniy and has he physical inepeaion ha he inciden enegy is equal o he diffaced enegy (fo lossless media as he ones consideed in he pesened examples). Noe ha he evanescen waves ae no aken ino accoun since hey do no cay enegy. In all he numeical esuls pesened heeinafe, and as explained in Secion above, he enegy balance cieion in Eq. (7) was esed o be fulfilled up o he ode of 8. As a meic of he degee of anomalous eflecion and ansmission, we devise he quaniy m P P P : (8) P Lage values of m indicae an enhanced abiliy by he consideed sucue o see he efleced/ansmied waves in an unusual diecion, oally diffeen fom ha dicaed by Snell s law. In ohe wods, m denoes how significan ae he powes caied by he waves of he -diffaced ode as compaed o he ones of zeo-diffaced ode. Accoding o he discussions of Secion 3.A, we fix he paamees as follows: λ o he cenal wavelenghs of he five colos in he visible ange, θ 6, n, n.35, n o he efacive index of lossless silicon a he examined wavelenghs, and s. The emaining paamees ae he hickness w of he slab and he peiod Λ of he inclusions. The wo lae paamees w and Λ ae he ones wih espec o which we pefom he opimizaion pocedue in ode o maximize he meic m. The opimizaion is geedy and is based on successive compuaions of he eflecion and ansmission coefficiens by means of he above-descibed semi-analyical inegal equaion mehodology. The beneficial chaaceisics of he mehodology, descibed in Secion, namely, is supeio numeical sabiliy, conollable accuacy, and apid convegence aid subsanially he fas and efficien implemenaion of he opimizaions equied in ode o deemine he desied values of w and Λ yielding enhanced anomalous eflecion and ansmission. A epesenaive conou plo of he meic m in he plane of w and Λ fo he geen colo, i.e., λ 53 nm, is depiced w (nm) Fig. 3. Conou plo of meic m as funcion of w and Λ fo λ 53 nm (geen). in Fig. 3. Afe geneaing fo each colo he especive conou plo, like he indicaive one in Fig. 3, we poceed o a seconday opimizaion whee we selec a naow ineval of Λ and hen seach fo he specific value of w in he egion of lage m (as pediced, e.g., by Fig. 3) so ha he values of P and P ae significanly lage han hose of P and P. The esuls of his pocedue ae pesened sysemaically in he nex secion. 4. NUMERICAL RESULTS AND DISCUSSION Figue 4 depics he vaiaions of he powes P, P, P, and P vesus he inclusion s peiod Λ fo he five examined colos in he visible ange. Fo each colo, he opimal value of he slab s hickness w was deemined accoding o he seconday opimizaion scheme descibed above. The opimal pais w and Λ yielding he maximum efleced powe in he ode as well as he especive angles θ ae epoed fo each wavelengh in Table. Significanly lage values of P ae obseved fo evey colo, a feaue which makes he consideed designs vey ineesing in ems of anomalous eflecion. These values sa fom.9 fo blue and geen ligh and become almos fo yellow, oange, and ed ligh. Moeove, fom Table i is noiced ha he values of he angle θ ae close o ha of he angle of incidence θ 6 ; hey have a and 7 deviaion fo blue and geen, especively, and hey ae vey close o 6 fo yellow, oange, and ed. The lae wo facs mean ha fo he seleced paamees values, he measuface exhibis songly anomalous seeing of he inciden beam, namely, i eflecs back vey close o he diecion of illuminaion nealy he enie amoun of inciden powe. The ineval of lage values of P is wide fo blue and geen and naowe fo yellow, oange, and ed, which makes ou design immune o fabicaion impefecions of he peiod Λ. Howeve, as menioned above, in he cases of he lae hee colos, nealy % of he inciden field is efleced in he ode. A schemaic of he enegy balance cieion in Eq. (7) on he map of P P and P P is depiced in Fig. 5(a). Fo a measuface composed of lossless maeials, feasible values of he obained powes lie on he line segmen joining he poins ; and ; on his map. Fo significan anomalous diffacion, we mus obain a value close o he poin 5

6 D6 Vol. 34, No. 7 / July 7 / Jounal of he Opical Sociey of Ameica B Reseach Aicle (a) (b) (c) (d) (e) P P P blue P P P P P P P P geen yellow oange ed P P P P P P P P Fig. 4. Powes of - and zeo-diffaced odes vesus he inclusion s peiod Λ fo (a) λ 47 nm (blue), (b) λ 53 nm (geen), (c) λ 58 nm (yellow), (d) λ nm (oange), and (e) λ 685 nm (ed). P Table. Opimal Pais of w and Λ Yielding Maximum P and Angles θ fo Each Consideed Wavelengh Colo λ (nm) w (nm) P θ Blue Geen Yellow Oange Red P P ;P P ;. The lae equiemen is achieved fo all examined colos, as shown in Fig. 5(b), which depics he aained values of he diffaced odes powes fo he measuface unde consideaion afe implemening he opimizaion pocedue analyzed in Secion 3 above. A visual Fig. 5. (a) Schemaic of he enegy balance cieion fo a lossless measuface suppoing wo popagaing diffaced odes. (b) Opimized values of he diffaced ode powes fom he measuface unde consideaion fo each colo of visible ligh. (c) Visual epesenaion of he diffacion pefomance fo he diffeen colos, accoding o he daa of Table.

7 Reseach Aicle Vol. 34, No. 7 / July 7 / Jounal of he Opical Sociey of Ameica B D7 epesenaion of he daa conained in Table, exhibiing he obained opimized diffacion pefomance fo he diffeen colos, is depiced in Fig. 5(c). In he same figue we show he (a) (b) (c) (d) (e) 7 6 m (db) blue colo λ (nm) 7 6 m (db) geen colo λ (nm) λ (nm) 7 6 m (db) yellow colo λ (nm) 7 6 m (db) oange colo m (db) ed colo λ (nm) Fig. 6. Conou plos of meic m in db as a funcion of he wavelengh λ and he angle of incidence θ fo he opimal values of w and Λ of Table diffeen angles of popagaion of he diffaced odes as well as ha of odinay eflecion, which is dicaed by Snell s law. Fuhemoe, fo each of he five opimal designs pesened above, we epesen in Fig. 6 on he λ ; θ map he vaiaions of he meic m in he viciniy of he cenal opeaing wavelengh ( nm ange) and he angle of incidence, which is kep fixed a 6 ( ange). The values of w and Λ wee fixed o hose of Table, coesponding o he maximizaion of P. Exemely lage values of m ae highly concenaed fo almos all examined cases in elaively naow egions in he plane of λ and θ. The decibel (db) scale uilized in Fig. 6 emphasizes he lage magniudes of hese values; noice ha he epesenaion of m in Fig. 6 is scaled diffeenly fom he indicaive plo of opimizaion of m in Fig. 3. Theefoe, mos poposed devices give highly selecive anomalous eflecion boh wih espec o he fequency and he diecion of exciaion, a popey vey useful in swiching applicaions. On he conay, ou opimal configuaion fo geen ligh (which exhibis he pooe pefomance fom all poposed designs) eains a subsanial m fo a wide egion on λ ; θ plane, which means ha i can eflec in an anomalous way wideband and spaially modulaed signals. Besides, fo he illuminaion by blue and geen ligh in Figs. 6(a) and 6(b), especively, we obseve ha he locaions of maximum m appea o be off-ceneed. In ohe wods, alhough he paamees of he measuface wee seleced as o maximize P fo θ 6, λ 47 nm, and λ 53 nm, especively, he maximizaion of m is obained aound 4 and a few nanomees away fom hese values. On he ohe hand, he maximizaion of m fo yellow, oange, and ed is achieved exacly a he specific values of λ and θ, which wee oiginally used o opimize he anomalous eflecion pefomance of he measuface [cf. Figs. 6(c) 6(e) and Table ]. 5. CONCLUDING REMARKS Anomalous eflecion phenomena by a gadien dielecic measuface composed of wo peiodically alenaing lossless maeials wee invesigaed in he visible fequency ange. The diffacion poblem was solved by a highly efficien enie-domain inegal equaion mehodology. The esuls of his mehodology ae compaible wih enegy consevaion and physical inuiion and have aleady been checked agains he ones obained wih diffeen echniques. Sysemaic opimizaions wee pefomed leading o he deeminaion of he measuface s widh and peiod such ha he powe of he -efleced ode is significanly enhanced, while he coesponding powes of he zeo-efleced and -ansmied odes ae annihilaed. Fo ceain fequency inevals, he powes of he -efleced ode can even each %. The maeials of he measuface s uni cell offeing hese emakable chaaceisics can be odinay dielecics. The poposed opimized alldielecic sucues can be used in seveal componens and devices ha use he mechanism of anomalous eflecion o boos hei efficiency. Thei applicaion ange spans fom wavefon manipulaion, imaging, and beam seeing o anieflecion coaings, absobes, and couples. The consideaion of inclusions of diffeen shapes pe uni cell as well as of diffeen duy cycles may assis fuhe in achieving enhanced anomalous eflecion effecs fo a wide

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