Electromagnetic interaction in dipole grids and prospective high-impedance surfaces

Transcription

1 ADIO SCIENCE, VOL.,, doi:.29/2s36, 25 Elecomagneic ineacion in dipole gids and pospecive high-impedance sufaces C.. Simovsi Depamen of Physics, S. Peesbug Insiue of Fine Mechanics and Opics, S. Peesbug, ussia S. Zouhdi Laboaoie de Génie Elécique de Pais, Supélec, Gif-Su-Yvee, Fance V. V. Yaseno Depamen of Physics, S. Peesbug Insiue of Fine Mechanics and Opics, S. Peesbug, ussia eceived 2 Mach 2; evised 8 Januay 25; acceped 2 Mach 25; published Ocobe 25. [] A wo-dimensional gid of isoopic dipole scaees excied by an obliquely inciden plane wave is consideed. Analyical fomulas ae obained fo he so-called ineacion consan of he gid. To validae ou heoy, a compaison is done beween he obained appoximae elaion fo he ineacion consan, he esul of he numeical summaion of dipole fields, and a nown accuae esul fo a special case. The heoy esuls in he gid impedance of aays which does no depend on he wave incidence angle. We show ha he gid of nealy esonan elecic dipoles uns ou o be pospecive as a consiuive elemen of novel high-impedance sufaces whose impedance is unique fo he whole angula specum of plane waves. Ciaion: Simovsi, C.., S. Zouhdi, and V. V. Yaseno (25), Elecomagneic ineacion in dipole gids and pospecive high-impedance sufaces, adio Sci.,,, doi:.29/2s36.. Inoducion [2] High-impedance sufaces (HIS) ae esonan sucues wih vey small hicness behaving as a magneic wall a esonance. Vaious applicaions wee poposed fo HIS [see, e.g., Sievenpipe, 999; Yang e al., 999]. The main dawbac of hese HIS is he dependence of he suface impedance on he angle of incidence q. The impedance bounday condiions ae only saisfied fo a unique incidence of a plane wave. I is impossible o inoduce a unique suface impedance fo hese HIS in he siuaions whee he angula specum of adiaion is lage. The goal of he pesen pape is o develop a heoy of HIS whose suface impedance is independen of q. This is impoan fo many pacical applicaions, as fo example, in anenna subsaes, whee one needs o obain he impedance bounday condiions fo he whole angula specum of he anenna adiaion. [3] In pacical cases, a HIS is a sucue consising of a plana fequency selecive suface (FSS) locaed on he suface of a meal-baced shield. The main goal is o Copyigh 25 by he Ameican Geophysical Union. 8-66/5/2S36 obain a esonance a low fequencies whee he subsae hicness h is vey small compaed o he wavelengh (e.g., h.2l) and he peiod of he FSS a is also small (e.g., a.l). Belov e al. [998] poposed he ceaion of a novel HIS wih a gid of esonan elecic dipoles locaed a a small heigh h ove he gound plane. Dipole scaees esonaing a vey low fequencies (so ha he size of he scaee is of he ode of one enh he esonan wavelengh in fee space) ae nown. These scaees can be esonan magneic dipoles [e.g., Pendy e al., 999], pais of esonan magneic and elecic dipoles [e.g., Khaina e al., 998] o esonan elecic dipoles [e.g., Simovsi e al., 23a]. Le us conside he case of an FSS fomed by nealy esonan elecic dipoles. We show below ha he esonan fequency w of he whole sucue (FSS plus he shield) is lowe han he esonan fequency w of a fee sanding FSS. As a esul, one can obain a magneic wall a he wavelengh l of he ode l a, whee a is he peiod of he FSS and h < a. [] Belov and Teyaov [22] sudied a HIS fomed by a dipole gid and a meal gound plane sepaaed by fee space a nomal incidence. In he pesen pape we give a moe deailed and geneal analysis of he dipoles of

2 muual coupling in plana aays. We also sugges a HIS based on such gids whose suface impedance does no depend on he plane wave incidence angle. [5] Le a plana squae gid of dipole paicles (ou FSS) be locaed ove he gound plane (x y). The gid is assumed o be pacically isoopic in he (x y) plane. We define he suface impedance efeed o he gid plane as he faco beween he angenial componens of he elecic field (aveaged ove he gid peiod a) and he magneic field (also aveaged) a he uppe side of FSS as E ¼ Z s n H þ : The gid impedance Z g elaes E o he aveaged suface cuen induced in he gid (which is equal o he disconinuiy of he magneic field acoss he FSS), E ¼ Z g J ¼ Z g n ðh þ H Þ: ðþ Teyaov and Simovsi [23] and Simovsi e al. [23b] showed ha he suface impedance Z s of an abiay plana aay locaed on he shield ineface is pacically elaed o he gid impedance of he same aay by a simple equaion Z s ¼ Z gz in Z g þ Z in ; ð2þ whee Z in (defined by he elaion E = Z in n H )ishe suface impedance of he shield. Fo meal-baced shields Z in is he inpu impedance of he ansmission line fomed by he spacing beween he FSS and he gound plane. In fac, equaion (2) is an exac elaion which follows diecly fom he definiions of Z g and Z in. The appoximaion is he ansmission line model fo Z in which neglecs he influence of he FSS on Z in and vice vesa. This appoximaion loos ough, bu acually i is no so. I leads o significan eos only in he case when he gid peiod a is vey lage compaed o h (his was shown fo he special case when he FSS is an aay of paches by Teyaov and Simovsi [23]). In he case whee h.2a he eo of he ansmission line model is sill vey small [Teyaov and Simovsi, 23]. In ou pevious sudies [Simovsi e al., 23b, 2] his fomula was successfully validaed fo hee diffeen inds of diffeen FSS. [6] The fomulas fo Z in TE,TM can be easily obained fo a simple dielecic shield [see, e.g., Teyaov and Simovsi, 23]. This impedance is inducive, and i means ha he esonan fequency w of HIS is always lowe han he esonan fequency of Z g which we denoe as w. Acually, he FSS has a seies esonance, so Z g is capaciive a he fequencies which ae lowe han w. Theefoe we mus have w < w. I means ha he gid peiod a can acually be of he ode of l / a he esonance of he HIS o even smalle. Thus on a suface whose dimension is l l one can place many ( ) uni cells of ou HIS. [7] Boh fequency and angula dependencies fo Z in of a simple shield in he ansvese elecic (TE) case of he plane wave incidence (when E is paallel o he ineface) ae diffeen fom hose in he ansvese magneic (TM) case. Howeve, hee is a vaian of he shield when Z in does no depend on q, and is fequency dependence is he same fo boh cases of he wave incidence. Simovsi e al. [23b] have consideed Z in fo shields in which he dielecic is pefoaed by meal vias and have shown ha if he subsae hicness is vey small (h ) boh Z in TE and Z in TM ae pacically independen of q and equal jm wh. Theefoe all we need is o find a FSS whose gid impedance would be independen of q fo boh TE and TM polaizaions of he inciden wave. Then he suface impedance Z s (w, Q) should be independen of q, oo. We show below ha he gid impedance of a dense dipole aay is wealy dependen on q since he ineacion of dipoles is pacically quasisaic. [8] Since he subsae hicness h is vey small, o obain a paallel esonance fo he whole sucue one needs o have a vey small capaciive impedance fo he FSS. Theefoe he FSS of small elecic dipoles mus be nealy esonan [see, e.g., Simovsi e al., 23a]. Since he dipoles ae elecically sho he idea of hei inducive loading is eviden. In his pape we do no discuss he poblem of he opimal design of small esonan dipoles. Pacical ealizaions of such FSS will be suggesed and sudied in he nex fuue. 2. Gid Impedance of he Plana Aay of Dipoles [9] The soluion fo Z g can be obained as a funcion of he so-called ineacion consan of he gid denoed below as b. This faco expesses he ineacion field (he field poduced by all dipoles p i of he gid excep he efeence dipole p a he poin whee he efeence dipole is locaed), E in ¼ bp : ð3þ Fo a given case of he inciden wave polaizaion (TE o TM) he coefficien b depends on he gid geomey, fequency and incidence angle. The dipole momen of he ih dipole paicle diffes fom p by nown phase shif p i ¼ p e j i ; ðþ whee is he angenial componen of he inciden wave veco and i is he adius veco of he ih dipole (see Figue ). 2of

3 [] To obain he gid impedance we ae ino accoun ha he aveaged elecic field on he gid suface is a sum of inciden and efleced wave fields and ha he aveaged cuen a he oigin is popoional o he dipole momen p, E ¼ E inc þ E ; J ¼ jw p S : Then we obain fom (5), (6), and (8) he elaion ð8þ J ¼ jw S a E : ð9þ e ðþ j w b 3 6p eff c 3 Figue. [] Le us assume ha we now boh he polaizabiliy of an individual dipole a and he ineacion consan b of he gid. Then, fo he efeence dipole momen and fo he local field we have he following elaions: p ¼ ae loc ; Eloc ¼ E inc þ bp : ð5þ Hee he inciden field E inc is aen a he oigin. In he pesen wo we do no conside he influence of he dielecic ineface on which he gid is locaed in pacical sucues. Noice ha in he quasi-saic limi he influence of he dielecic ineface o he polaizaion of dipoles can be descibed in ems of he unifom effecive pemiiviy of a hos medium eff =( s + )/2, whee s is he subsae pemiiviy [Teyaov and Simovsi, 23]. Thus we assume he gid o be locaed in he unifom hos medium of pemiiviy eff. Fo he nomal incidence, he imaginay pa of b is aleady nown [Teyaov and Viianen, 2]: w 3 wh ImðÞ¼ b 6p eff c3 2S ; ffiffiffiffiffiffi h ¼ m ; ð6þ whee S = ab is he gid uni cell suface (in ou isoopic case a = b). I allows us o wie (3) in he fom E in Plane wave incidence on he FSS. w 3 ¼ eðþp b þ j 6p eff c 3 p þ E : ð7þ In his elaion E = jwhp /2S is he ampliude of he angenial componen of he zeo-ode Floque mode poduced by he gid (in ohe wods, E is he angenial componen of he efleced field a he oigin). We asse ha fo ahe dense gids, fomula (7) sill holds even fo he non-nomal incidence. Below we show ha i is valid fo boh TE and TM cases. Fo he TE case one has E = jwhp /2S cos q and fo he TM case E = jwh cos qp /2S. Le he dipola paicles be lossless. Then he elaion [Sipe and Van Kanendon, 97] w 3 ImðaÞ ¼ 6p eff c 3 : ðþ holds. Wih (9), (), and () we come o a fomula fo Z g which is valid fo boh TE and TM polaizaions, Z g ¼ E J ¼ S jw e a eðþ b : ðþ [2] Expessing he dipole momen in ems of he induced cuen and he effecive elecic lengh of he paicle we obain e!! ¼ e Eloc ¼ e jweloc ; ð2þ a p I p l eff whee he effecive lengh l eff is he inegal of he nomalized cuen disibuion along he paicle and I p is he induced cuen a he cene of he paicle (o which he cuen disibuion is nomalized). The mehod of induced elecomoive foce allows us o expess I p hough he same effecive lengh and he paicle inpu impedance Z p (also efeed o is cene). The paicle impedance can be wien in ems of paicle inducance and capaciance Z p = p + j(wl / wc). Hee p is he adiaion esisance of he dipole paicle which does no influence Z g. Thus we obain E locl eff I p ¼ E ¼ ; Z p p þ j wl wc e wwl ¼ ð wc Þ : a Subsiuing his fomula ino (2) and (2) ino () we obain Z g ðw; qþ ¼ j S leff 2 wl þ j S e½bw; q wc w ð Þ Š: ð3þ l 2 eff 3of

4 I follows fom (3) ha he gid esonan fequency w coesponding o Z g = is p ffiffiffiffiffiffi diffeen fom ha of he individual paicle w ind =/ LC. This is he esul of he dipole ineacion. [3] Thus we yield he poblem of he gid impedance fo he gid of nealy esonan dipoles o he poblem of he gid ineacion consan b which is exploed below. Fo he case of nomal incidence (q = ) analyical expessions fo e(b) can be found in wo by Maslovsi and Teyaov [999]. In he case when a = b p his expession uns ou o be vey simple [see also Collin, 99; Teyaov and Viianen, 2]: ; 3595wh eðþ b a 3 : ðþ One of he goals of he pesen sudy is o show ha his quasi-saic elaion pacically wos well above he nomal incidence. A ahe low fequencies a i wos fo boh TE and TM polaizaions unil he angles q 85 (he em wos means ha he elaive eo fo b does no exceed 5%). A lowe fequencies a.5 i wos unil he gazing incidence and he eo (always smalle han %) does no gow vesus q. In fac, a vey low fequencies his simples fomula emains valid fo >, ha is, in he domain of complex incidence angle. This fac maes he esul () applicable fo an analyical sudy of eigenwaves in he gids of esonan dipoles. Such a sudy has become vey applicable ecenly in view of gowing pacical inees in he opical popeies of silve nanopaicles [Webe and Fod, 2]. In his case he em low fequencies means he condiion a 2p, whee > coesponds o he wave popagaion along he gid suface. Noice ha he imaginay pa of he ineacion consan Im(b) does no ene he dispesion equaion fo such a wave [Webe and Fod, 2]. Howeve, in he pesen wo we concenae on he micowave applicaion of his esul and do no sudy moe is opical aspec. [] Fomula () shows ha he dipole ineacion inceases he effecive capaciance C eff of he FSS compaed o C wihou changing he inducance L. We define his effecive capaciance hough he elaions w 2 ¼ LC eff ; Z g ðw ; qþ ¼ : Then, fom (3) and () we obain C eff ¼ CC in C þ C in ; ð5þ whee C in is he negaive ineacion capaciance defined as eðþ¼ b leff 2 C ; in which is equal o C in ¼ eff a 3 :3595l 2 eff and whose absolue value is of couse lage han he paicle pope capaciance C. In fac, he pope capaciance of a plana meal paicle occupying he suface S p < a 2 is smalle han he capaciance of he pach occupying he whole uni cell of he gid (suface S = a 2 ). Theefoe C < eff a < jc in j), and he esul fo C eff given by (5) uns ou o be posiive. [5] We spli he inciden field ino he TE and TM componens (see Figue ) and find b sepaaely fo he TE and TM cases. Thee is no coss polaizaion fo boh hese cases and he dipole momens ae dieced along E inc. Fo he special case of TE incidence (namely, when he plane of incidence is he OXZ plane), he explici and accuae soluion was obained by Collin [99]. We will conside he geneal case of oblique incidence pesened in Figue and validae he heoy by compaison wih he esul fom Collin [99]. elaion () uns ou o be a limi case fo dense gids. 3. Mehod of Sudy [6] Conside a gid of dipole paicles (shown as cosses in Figue ) illuminaed by a plane wave. The wave veco of he inciden field conains he veco angenial componens which can be epesened hough he uni veco as =, whee = sin q and we have also x = cos F, y = sin F. Fo genealiy le he gid be doubly peiodic wih peiods a and b (see Figue ) and le he veical dipole momens be induced in paicles. Then he dipole momen of he efeence paicle can be pesened as p = z p z + p, whee z is he uni nomal veco o he gid plane and p is he pojecion of p on he gid plane. This pojecion conains wo componens: p, which is paallel o and p z which is ohogonal o i (i.e., ohogonal o he plane of incidence). The TM-polaized wave induces he componens p z and p. The angenial dipoles p do no conibue o he veical componen of he ineacion field. Theefoe one can wie wo scala elaions fo he TM case: E in z ¼ b z p z ; E in ¼ b p : ð6þ The TE-polaized wave induces only he componen p z : E in z ¼ b zp z : ð7þ [7] To expess any componen of E in hough he coesponding componen of p we should summaize he paial dipole fields E i poduced by all paicles wih of

5 Following (8) he elaions fo can be wien as ¼ D Z Z 2p 2 sin 2 a þ 3 cos 2 a 2 þ j e jþ ð cos a Þ da d ð2þ Figue 2. Illusaion of he mehod of calculaion of he ineacion field. i 6¼. Taing ino accoun () his expession can be wien as [see Jacson, 999] E i ¼ 2 ðn p p nþ e j i j i i þð3np p Þ þ j e ji j i i 2 : ð8þ 3 i Hee n = i / i is he pola uni veco on he plane (x y). The infinie double seies of em (8) diveges fo eal. To obain a convegen seies one can inoduce he infiniesimal imaginay pa of and hen apply he wellnown Kumme mehod fo condiionally convegen seies. This was done by Belov e al. [998]. Howeve, he elaions deived by Belov e al. [998] ae so complicaed ha i is hadly possible o deduce () explicily fom hem. Maslovsi and Teyaov [999] suggesed a simple appoximae mehod o obain b fo he case of he nomal wave incidence. In he pesen wo we genealize his mehod o he oblique incidence. [8] In he gid plane we conside a cicle of adius ceneed a he efeence dipole (see Figue 2). Ouside he cicle he discee disibuion of he dipole momens is eplaced by coninuous polaizaion wih a suface densiy P = p e j( xx+ y) y /S. We pesen he ineacion field as a finie sum of paial fields poduced by dipoles locaed inside he cicula hole (denoed as E hole )plushe field poduced by a shee of polaizaion P wih a hole of adius (denoed as E shee ): E in ¼ E hole þ E shee ¼ X E i þ E shee : ð9þ i <;i6¼ [9] We inoduce he noaions E shee ¼ p ; Ez shee ¼ z p z ; Ez shee ¼ z p z : z z ¼ D Z ¼ D Z Z 2p Z 2p 2 2 j e jþ ð cos a Þ da d ð2þ 2 cos 2 a þ 3sin 2 a 2 þ j e jþ ð cos a Þ da d: ð22þ Hee D =/p S. [2] Using he well-nown epesenaion of he Bessel funcions Z 2p we obain whee e jg cos a cos na da ¼ ð jþ n 2pJ n ðgþ; ¼ DðI þ I 2 þ I 3 þ I Þ ð23þ I 3 ¼ I ¼ 3 z ¼ DðI þ I Þ ð2þ z ¼ DðI 2 þ 2I 3 þ I Þ; ð25þ I ¼ I 2 ¼ Z Z Z Z J ð Þ J 2 ð Þ J ð Þe j d ð26þ J 2 ð Þe j d ð27þ 2 þ j 2 þ j e j d e j d: ð28þ ð29þ 5of

6 [2] Calculaions ae given in Appendix A. The esul is as follows: ¼ jwh S Xl j n¼ þ Xl n n! ðn þ Þ! X2n ðj 2n þ 3 j m¼ n¼ nþ ðn þ Þ! ðn þ 2Þ! # 2ðnþÞ ðjþ 2nþ e j ð3þ z z ¼ jwh S þ Xl j n¼ X 2n ðj n n! 2 m¼! # 2n þ 3 þ j þ nþ ðn þ Þ! 2 ðjþ 2nþ 2ðnþÞ ¼ jwh S Xl j n¼ n n! ðn þ Þ! e j ð3þ X2n ðj þ n2 þ 2n þ 7 þ ð2n þ 3Þj m¼ nþ ðn þ Þ! ðn þ 2Þ! # 2ðnþÞ ðjþ 2nþ e j : ð32þ [22] In equaions (3) (32), l + is he numbe of ems which mus be aen ino accoun in he seies fo obaining he needed accuacy. The lage he adius, he lage is he needed numbe l. To mae l smalle one should minimize, which also educes he numbe of paial dipole fields which one should ae ino accoun in (9). Does i lead o significan eos if we ae as small as he adius of he gid uni cell (in his case we avoid he explici summaion of he dipole fields and have E hole = )? The answe o his impoan quesion is given below.. Opimizaion of he Hole adius and Validaion of he Theoy [23] In Figue 3 we pesen he esul fo he ineacion field in he pacically ineesing special case a = b. The calculaions coespond o he TE incidence in he plane OXZ. The chosen paamees ae as follows: a =. and q = p/. The eal pa of he ineacion field is shown in he op plo of Figue 3, and he imaginay pa is shown in he boom plo. The ineacion field in he plos is Figue 3. Nomalized ineacion field vesus nomalized cicle adius fo = p/6, TE incidence: (op) eal pa and (boom) absolue value of imaginay pa. E hole (dashed line) and E hole + E shee (dos) ae compaed wih he exac soluion E in ex (solid line). nomalized and can be named as dimensionless ineacion consan b n = E in a 3 /p = a 3 b. The saigh hoizonal line denoes he nomalized exac soluion E in ex. I was menioned above ha Collin [99] summaized he seies of he dipole fields fo he special case when he incidence plane is OXZ and he wave is TE polaized. The coesponding ineacion field is expessed hough he combinaion of single and double seies [Collin, 99, p. 78], E in z ¼ 2 p 2S þ X m¼ a p 2p 2pb X p C log a þ j þ a G m G m mp X n¼ m¼ g 2 n K ðg n maþ a 2 jg j þ p pb 3 :2 2 b 2 2 log b þ 2 b 2 þ 3 b 288 p 2 b 2 j 3 b 3 : ð33þ 8 Hee C e.577 is he Eule consan and is denoed G 2 m ¼ 2pm 2 2 þ 2 a ; g2 n ¼ 2pn 2 2 þ 2 b : Boh single and double seies ae convegen. Fomula (33) has been used o calculae he exac value of he nomalized ineacion consan denoed as E in ex which is pacically equal o.3595 j.3 fo a =. and q = 6of

7 5. The dashed line in Figue 3 coesponds o he diec sum of he dipole fields denoed as E hole. When we incease he numbe of dipoles conibuing ino he ineacion field (i.e., when we incease /a) he esul oscillaes aound he exac soluion. The dos line in Figue 3 epesen he igh-hand side of (9), ha is, he sum of E hole and E shee which pacically coincide wih he exac soluion saing fom values of /a which ae less han uniy. If a he em E hole vanishes and E shee becomes he ineacion field. [2] Maslovsi and Teyaov [999] found ha he opimal adius of he hole fo nomal wave incidence on he isoopic gid of dipoles is op = a/.38. Figue 3 illusaes he fac ha he same value of emains opimal fo oblique incidence. Fo boh TE and TM cases we obained he same esul: E hole + E shee is pacically independen of saing fom = a/.38. [25] Thus we can undesand he paamees given by expessions (3) (32) as he hee ineacion consans b, b z and b z, in which we subsiue = op = a/.38. If we ae ineesed in he imaginay pa of he ineacion consan, he ode of uncaion l depends on q and a. When boh a and q ae small enough one can pu l =. Then (3) (32) can be simplified as follows: b ¼ jwh sin 2 q S j op!# þ 3j op 8 þ 2 2 op 8 b z ¼ jh þ þ sin 2 q S j op!# þ 3j op 2 2 op e j op e j op b z ¼ jwh þ sin 2 q S j op!# þ 7j op op 8 e j op : ð3þ ð35þ ð36þ [26] When q appoaches p/2 he numbe l in (3) (32) needs o be chosen popely. Im(b) apidly gows and elaions (3) (36) become invalid fo he imaginay pa. Howeve, we will see below ha fo e(b), fomulas (3) and (36) wo pefecly even fo lage angles. The eal pas of he igh-hand sides of equaions (3) and (36) ansi o he esul () no only fo q = bu also fo 7of Figue. Nomalized ineacion field vesus gid peiod: (op) eal pa and (boom) imaginay pa. Ou appoximae heoy (solid line fo many ems and dos fo one em) is compaed wih he exac soluion (cosses). q 6¼ wih a elaive eo of he ode (a) 2. In fac, his eo is much smalle han (a) 2 as i follows fom he moe accuae fomulas (3) and (32), whose eal pas give equaion () wih an eo of he ode (a) 2 /. [27] We compaed (36) wih he nown esul (33) fo diffeen cell sizes a and diffeen q. In Figue he esul of compaison fo diffeen a and q = p/6 is pesened. The goal of his compaison is o validae ou heoy. Cosses denoe he exac soluion (33). The fequency bound of he validiy of ou heoy fo his incidence angle can be esimaed as a =.5. Fo a he appoximaion l = (when fomula (32) is subsiued by (36)) leads o an eo smalle han 5%. [28] In Figue 5 he esul of he compaison fo diffeen q is given fo a =.5 (TE polaizaion, plane of incidence is OXZ, same as above). The goal of his compaison was o validae he heoy and o pove ha () wos fo all angles. When q is appoaching 8 we need o ae l = 5 fo Im(b). Fo q >85 even l =5is no enough. When q =89 he needed accuacy fo Im(b) was eached wih l =. The imaginay pa of b n uns ou o be pacically equal o Im(b n )=(a) 3 /6p a/2 cos q. This is equivalen o he esul Im(b) = 3 /6p E /p. Theefoe we confimed fomula (7) fo he TE case. Fo e(b) he appoximaion () gives an excellen ageemen wih he exac heoy, as we can see on op of Figues and 5. The same esul e(b) independen of q and equal o he igh-hand side of ()) was obained fo he TM case (fomula (3)). The esul of (3) fo Im(b) does no diffe (visibly) fom (a) 3 /6p a cos q/2. Thus fomula (7) holds also fo he TM case. I means ha all he long fomulas we deived fo he ineacion consan

8 obained a vey good ageemen. Howeve, in he pesen wo we do no emphasize on his esul since he gid of veically polaized dipoles is no pacically ineesing fo he design of HIS. Figue 5. Nomalized ineacion field vesus incidence angle fo a =.5: (op) eal pa and (boom) imaginay pa. Appoximae heoy (solid and dashed lines) is compaed wih he exac soluion (dos). ae no necessay fo elecically dense gids (a.5). Fo all cases of incidence, we have a simple esul fo imaginay pa of b expessed by same fomula (7) as fo nomal incidence. Anohe impoan esul is ha he saic fomula () is also valid fo all cases of incidence when a.5. Then fom (5) we obain he esul fo he effecive capaciance of a dipole gid C eff ðw; qþ ¼ > C: C :3595l2 eff eff a 3 Fo sho linea dipoles (meal sics) of lengh l (which is pacically wice as lage as hei effecive lengh l eff ) pepaed fom he wie wih adius he pope capaciance is well nown: C ¼ p eff l 2log l ; l < a: The ineacion of dipoles allows us o eep he angula sabiliy of he gid impedance and o incease he effecive capaciance of he gid, ha is, o decease he esonan fequency of he whole sucue w. [29] Finally, noice ha esul () was poved as an accuae low-fequency asympoic one fo b and b z. Fo he ineacion consan b z of he aay wih veical polaizaion he low-fequency appoximaion (a 2p) follows fom (35) and coesponds o he same fomula () wih opposie sign. This esul is nown [see, e.g., Collin, 99]. In he lieaue hee is also a geneal and ahe involved fomula fo b z obained by Simovsi e al. [999, equaions () and (2)]. We compaed esul (3) wih he esul fo b z fom Simovsi e al. [999] and 5. Conclusion [3] In his pape we pesened and validaed an appoximae analyical model of he full-wave ineacion in a fequency selecive suface ealized as a gid of dipoles and illuminaed by obliquely inciden plane waves. The main esul is he vey wea influence of he angle of incidence on he gid impedance of such a FSS fo pacically ineesing cases of dense gids (when a.5). Locaing ou FSS on he ineface of a mealbaced dielecic shield conaining veical meal vias allows us o obain a high-impedance suface whose suface impedance heoeically does no depend on he angle of incidence fo boh possible polaizaions of he inciden wave. Such a HIS, which can be used as an anenna subsae, will behave wihin he whole esonance band as a suface wih a unique and a vey high suface impedance fo a whole angula specum of he anenna adiaion and will no suppo suface waves. Moeove, a all fequencies fo which he heoy is valid he impedance bounday condiions will be saisfied on he ineface of ou sucue (excep, pehaps, evanescen hamonics). In ohe wods, he suface impedance of ou sucue is unique fo all plane waves, and he em high-impedance suface is in ou case moe jusified han fo HIS descibed in he lieaue. Appendix A: Evaluaion of Inegals [3] To calculae he inegals (26) (29) we use he seies epesenaion fo Bessel funcions J n ðzþ ¼ X q¼ ð q! ðq þ nþ! Þ q Z 2qþn : 2 ðaþ Subsiuing (A) ino (26) and exchanging he ode of summaion and of inegaion (i is possible since he inegand is a coninuous funcion) we obain I ¼ X n¼ ð n! 2 Þ n Z 2n e j d: ða2þ 2 Le us ae ino accoun he abulaed inegal Z q e n Xq n d ¼ e m¼ q! m n q mþ ; ð > ; eðnþ > Þ: ða3þ 8of

9 The condiion e(n) > in ou case implies he pesence of (pehaps infiniesimal) losses in he space since coesponds o Im() <. Subsiuing (A3) ino (A2), afe a simple algeba we come o X j I ¼ je n¼ 2n X 2n n n! 2 m¼ ðj : ðaþ [32] The esul fo I 2 can be obained by subsiuing (A) and (A3) ino (27), X j ð2n þ 2Þ! 2nþ2 2nþ2 X ðj I 2 ¼ je n¼ nþ : n! ðn þ 2Þ! m¼ ða5þ [33] To calculae I 3 and I we use he e j s ¼ j þ s sþ e j : ða6þ This allows us o inegae he igh-hand side of (28) by pas I 3 ¼ J ð Þ e j Z J ð Þ e j d: ða7þ The subsiuion of (A) ino he fis em of he ighhand side of (A7) and he subsiuions of (A) and (A3) ino he second em give I 3 ¼ je j X ðjþ 2n 2n n¼ n n! 2 þ X 2 n¼ n n! ðn þ Þ! # 2nþ2 X 2n ðj : ða8þ m¼ [3] elaion (A6) allows o ansfom (29) o he fom whee I ¼ 3J 2ð Þ e j þ I 5 þ I 6 ; I 5 ¼ 6 I 6 ¼ 3 Z Z J 2 ð Þ e j d 2 J 3 ð Þ e j d: ða9þ Using (A) and (A3), we obain and X j I 5 ¼ 6je n¼ X j I 6 ¼ 6je n¼ n¼ nþ n! ðn þ 2Þ! ð2n þ 2Þ! nþ2 n! ðn þ 3Þ! 2nþ2 X 2n m¼ 2nþ 2nþ2 X m¼ ðj ðaþ ðj : ðaþ [35] Subsiuing (A) ino he fis em of he igh hand-side of (A9) and subsiuing (A) and (A) ino he second and hid ems, we come o he following elaion: I ¼ 3je j X ðjþ 2nþ 2nþ2 n¼ n n! ðn þ 2Þ! þ X # ðn þ Þ 2nþ2X ðj 2 n : ða2þ n! ðn þ 2Þ! m¼ To obain (A2) we used he exchange of summaion indices n new = n old + while subsiuing (A) ino (A9) and applied he elaion X n¼ F n ¼ X n¼ F n F : [36] To obain he final expession fo we should unify (A), (A5), (A8) and (A2) as i follows fom (23). Thus we come o he elaion X 2nX ðj n¼ n n! 2 m¼ X ðjþ 2n 2n n¼ n n! 2 X ð2n þ 2Þ! 2nþ2 2nþ2 X ðj n¼ nþ n! ðn þ 2Þ! m¼ X 2nþ2X ðj 2 n¼ n n! ðn þ Þ! m¼ 3 X ðjþ 2nþ 2nþ2 n¼ nþ n! ðn þ 2Þ! 3 X # ðn þ Þ 2nþ2X ðj 2 n¼ n : n! ðn þ 2Þ! m¼ ða3þ ¼ jh S e j [37] To mae his cumbesome expession moe compac we change he indices of summaion so ha he 9of

10 geneal em of evey seies in (A3) conains he idenical faco ( /) 2n+2. This ansfoms (A3) ino (3). The same way we obain (3) and (32) (following o (2) and (25)) fom (A), (A5), (A8) and (A2). efeences Belov, P. A., and S. A. Teyaov (22), esonan eflecion fom dipole aays locaed vey nea o conducing planes, J. Elecomagn. Waves Appl., 6, Belov, P. A., C.. Simovsi, and M. S. Kondajev (998), Analyical sudy of elecomagneic ineacions in wodimensional bianisoopic aays, pape pesened a Bianisoopics 98, 7h Inenaional Confeence on Complex Media, NATO, Baunschweig, Gemany, 8 June. Collin,.E.(99),Field Theoy of Guided Waves, IEEE Pess, Piscaaway, N. J. Jacson, J. D. (999), Classical Elecodynamics, 3d ed., John Wiley, Hoboen, N. J. Khaina, T. G., S. A. Teyaov, A. A. Sochava, C.. Simovsi, and S. Bolioli (998), Expeimenal sudies of aificial omega media, Elecomagneics, 8, Maslovsi, S. I., and S. A. Teyaov (999), Full-wave ineaion field in 2D aays of dipole scaees, In. J. Elecon. Commun., 53, Pendy, J. B., A. J. Holden, D. J. obbins, and W. J. Sewa (999), Magneism fom conducos and enhanced nonlinea phenomena, IEEE Tans. Micowave Theoy Tech., 7, Sievenpipe, D. F. (999), High-impedance elecomagneic sufaces, Ph.D. disseaion, Univ. of Calif., Los Angeles. (Available a hp:// ThesisDan.pdf) Simovsi, C.., M. S. Kondaiev, and P. A. Belov (999), Elecomagneic ineacion of chial paicles in 3D aays, J. Elecomagn. Waves Appl., 3, Simovsi,C..,B.Sauviac,andS.L.Posvinin(23a), Homogenizaion of an aay of S-shaped paicles, locaed on a dielecic ineface, Pog. Elecomagn. es.,, Simovsi, C.., S. A. Teyaov, and P. de Maag (23b), Aificial HIS heoeical analysis fo oblique incidence, Poc. IEEE Anennas Popag. Annu. Symp.,, Simovsi, C.., P. de Maag, and I. V. Melchaova (2), High-impedance sufaces wih angula and polaizaion sabiliy, pape pesened a 27h Euopean Space Agency Anenna Woshop on Innovaive Peiodic Anennas, Saniago de Composela, Spain, 9 Mach. Sipe, J. E., and J. Van Kanendon (97), Macoscopic elecomagneic heoy of esonan dielecics, Phys. ev. A, 9, Teyaov, S. A., and C.. Simovsi (23), Dynamic model of aificial eacive impedance sufaces, J. Elecomagn., Waves Appl., 7, 3 5. Teyaov, S. A., and A. J. Viianen (2), Plane waves in egula aays of dipole scaees and effecive medium modelling, J. Op. Soc. Am. A Op. Image Sci., 7, Webe, W. H., and G. M. Fod (2), Popagaion of opical exciaions by dipola ineacions in meal nanopaicle chains, Phys. ev. B, 7, 2529-(-8). Yang, F.-., e al. (999), A novel TEM waveguide using uniplana compac phoonic band-gap (UC-PBG) sucue, IEEE Tans. Micowave Theoy Tech., 7, C.. Simovsi and V. V. Yaseno, Depamen of Physics, Sae Insiue of Fine Mechanics and Opics, Sablinsaya, S. Peesbug 97, ussia. (simovsy@phd.ifmo.u) S. Zouhdi, Laboaoie de Génie Elécique de Pais, Supélec, Plaeau de Moulon, F-992, Gif-Su-Yvee Cedex, Fance. of