Strong spatial dispersion in wire media in the very large wavelength limit

S ong spa ial dispe sion in wi e media in he e y la ge wa eleng h limi P. A. Belo ,1,2 R. Ma que ´s,3S. I. Maslo ski,1I. S. Ne edo ,1,4 M. Sil ei inha,5C. R. Simo ski,1,2 and S. A. T e yako 1 1Radio Labo a o y, Helsinki Uni e si y o Technology, P.O. Box 3000, FIN-02015 HUT, Finland 2Physics Depa men , S . Pe e sbu g Ins i u e o Fine Mechanics and Op ics, Sablinskaya 14, 197101 S . Pe e sbu g, Russia 3Depa men o Elec onics and Elec omagne ism, Facul ad de Fı ´sica, Uni e si y o Se illa, A enida Reina Me cedes s/n, 41012 Se illa, Spain 4Ins i u e o Radio Enginee ing and Elec onics, Russian Academy o Science, Zelyonaya 38, 410019 Sa a o , Russia 5Ins i u o Supe io Te ´cnico, Ins i u o de Telecomunicac¸o ˜ es, A enida Ro isco Pais, 1049-001 Lisboa, Po ugal 共Recei ed 27 No embe 2002; published 25 Ma ch 2003兲 I is ound ha he e exis composi e media ha exhibi s ong spa ial dispe sion e en in he e y la ge wa eleng h limi . This ollows om he s udy o la ices o ideally conduc ing pa allel hin wi es 共wi e media兲. In ac , ou analysis e eals ha he desc ip ion o his medium by means o a local dispe si e uniaxial dielec ic enso is no comple e, leading o unphysical esul s o he p opaga ion o elec omagne ic wa es a any equencies. Since nonlocal cons i u i e ela ions ha e been usually conside ed in he pas as a second- o de app oxima ion, meaning ul in he sho -wa eleng h limi , he a o emen ioned esul p esen s a ele an heo e ical in e es . In addi ion, since such wi e media ha e been ecen ly used as a cons i uen o some disc e e a i icial media 共o me ama e ials兲, he epo ed esul s open he ques ion o he ele ance o he spa ial dispe sion in he cha ac e iza ion o hese a i icial media. DOI: 10.1103/PhysRe B.67.113103 PACS numbe 共s兲: 78.20.Ci, 41.20.Jb, 42.70.Qs, 78.70.Gq Causali y imposes ha all ma e ial media mus be dispe - si e. In mos cases his beha io esul s in local dispe si e cons i u i e ela ions, i.e., in equency-dependen cons i u- i e pe mi i i y and pe meabili y enso s. Nonlocal dispe - si e beha io 共i.e., spa ial dispe sion兲, which esul s in con- s i u i e ope a o s depending also on he spa ial de i a i es o he mean ields 共o , o plane elec omagne ic wa es, on he wa e- ec o componen s兲, is usually conside ed as a small e ec , meaning ul in he sho -wa eleng h limi . Spe- ci ically, spa ial dispe sion will always appea when he highe -o de e ms in he se ies expansion o he cons i u i e pa ame e s in powe se ies o he dimensionless pa ame e a/␭(ais he la ice cons an o he c ys al and ␭ he wa e- leng h inside he medium兲a e no neglec ed.1Thus, i is usually assumed ha nonlocal dispe si e ea ions a e only meaning ul when ␭app oaches a. The usually weak na u al op ical ac i i y o some ma e ials is a well-known example o he applica ion o his p inciple.1When such p inciple is ansla ed o he analysis o disc e e a i icial media, also called me ama e ials, i would imply ha nonlocal dispe si e cons i u i e ela ions a e only expec ed o be a small e inemen o he local cons i u i e ela ions usually con- side ed. Howe e , he e is a leas a coun e example o his assump ion. The pa allel wi e medium is a medium o med by a egu- la la ice o ideally conduc ing wi es wi h small adii com- pa ed o he la ice pe iods and he wa eleng h, see Fig. 1. I has been known in mic owa e applica ions o a long ime2–4 as an a i icial dielec ic, also called odded medium and qua- sis a ic models o he e ec i e pe mi i i y a e a ailable.5 Recen ly, some a en ion o wi e media has been paid also in op ics 共e.g., Re s. 6,7兲and in he ealiza ion o le -handed media8,9 as composi e media made om la ices o long con- duc ing wi es and spli ing esona o s10–12 共see discussions co esponding o ha in13兲. The elec omagne ic eponse o he speci ic wi e medium shown in Fig. 1 is analyzed in Re s. 3,7 ollowing di e en app oaches. Bo h analysis a e ca ied ou o wa e p opaga ion pe pendicula o he wi es and show ha , o elec ic- ield pola iza ion along he wi es, he medium is cha ac e ized 共i a/␭,b/␭Ⰶ1) by a equency- dependen e ec i e dielec ic cons an gi en by ␧⫽␧0 冉 1⫺k0 2 k2 冊 ⫽␧0 冉 1⫺ ␻ 0 2 ␻ 2 冊 .共1兲 The cons an ␻ 0共 he co esponding wa e numbe k0 ⫽ ␻ 0 冑 ␧0 ␮ 0) in Eq. 共1兲plays he ole o an equi alen ‘‘plasma equency.’’ Thus, his medium is o en called ‘‘a - i icial plasma’’ since he ideal 共collisionless兲elec on plasma is desc ibed by he same equi alen pa ame e . I he wi es FIG. 1. The geome y o wi e media: A la ice o pa allel ide- ally conduc ing hin wi es. PHYSICAL REVIEW B 67, 113103 共2003兲 0163-1829/2003/67共11兲/113103共4兲/$20.00 ©2003 The Ame ican Physical Socie y67 113103-1 a e assumed o be e y hin, so ha hei pola iza ion in he di ec ion o hogonal o he wi es can be neglec ed, he e ec- i e pe mi i i y o elec ic- ield pola iza ion o hogonal o he wi es is ␧0. The a o emen ioned analysis sugges s ha his wi e me- dium could be modeled as a uniaxial dielec ic wi h he ol- lowing local pe mi i i y dyadic: ␧ ¯ ¯ ⫽␧z0z0⫹␧0共x0x0⫹y0y0兲,共2兲 whose pe mi i i y in he axial di ec ion, ␧, would be gi en by Eq. 共1兲. Howe e , i will be shown in he ollowing ha his nai e hypo hesis leads o unphysical esul s and mus be subs i u ed by a nonlocal dispe si e ela ion. In ac , assum- ing ha he medium can be desc ibed by he uniaxial dyadic 共2兲, he dispe sion equa ion o ex ao dina y plane wa es (Ez⫽0) wi h he wa e ec o (qx,qy,qz)Tin his uniaxial dielec ic eads14,15 ␧0共qx 2⫹qy 2兲⫽␧共k2⫺qz 2兲,共3兲 whe e k⫽ ␻ 冑 ␧0 ␮ 0is he phase cons an o he hos ma ix. On he o he hand, hese ex ao dina y wa es co espond o he well-known TM 共 o z) se o modes, allowed by he in a iance o he bounda y condi ions along z. Thus, o any ex ao dina y wa e a eling wi h a phase cons an qzalong he zaxis, he Ez ield mus sa is y he Helmhol z equa ion 再 ⳵ ⳵ x2⫹ ⳵ ⳵ y2⫹共k2⫺qz 2兲 冎 Ez⫽0, 共4兲 wi h he bounda y condi ion Ez⫽0 on he wi es. I is clea om his equa ion ha any ‘‘plane’’ex ao dina y wa e mus sa is y k共qx,qy,qz兲⫽ 冑 k2共qx,qy,0兲⫹qz 2.共5兲 This las esul is incompa ible wi h Eq. 共1兲–共3兲, as can be easily seen by subs i u ion o Eq. 共1兲in o Eq. 共3兲. Howe e , i we choose ␧共k,qz兲⫽␧0 冉 1⫺k0 2 k2⫺qz 2 冊 共6兲 ins ead o Eq. 共1兲, hen Eqs. 共2兲and 共3兲become compa ible wi h Eq. 共5兲, gi ing he ollowing dispe sion equa ion o he plane wa e: q2⬅qx 2⫹qy 2⫹qz 2⫽k2⫺k0 2,共7兲 whe e we ha e assumed ha qz⫽k共 he case wi h qz⫽kwill be analyzed a he end o his pape 兲. The abo e a ionale sugges s ha he conside ed wi e media s ill can be desc ibed by he pe mi i i y dyadic 共2兲, bu he axial pe mi i i y ␧ mus be a nonlocal pa ame e o he o m 共6兲. The con en- ional exp ession 共1兲would be only a pa icula case o Eq. 共6兲, alid o wa e p opaga ion in he x-yplane. The main di e ence be ween he local uniaxial model, Eq. 共1兲, and he p oposed nonlocal model, Eq. 共6兲, o he pa allel wi e medium is ha he nonlocal model p edic s a s op band 共a equencies below ␻ 0⫽k0/ 冑 ␧0 ␮ 0) o ex ao - dina y wa es p opaga ing along any di ec ion in he media. On he con a y, Eqs. 共1兲–共3兲p edic p opaga ion o ex ao - dina y wa es a any equency p o ided qz⬎k⫽ ␻ 冑 ␮ 0␧0. Thus, bo h models p edic quali a i ely e y di e en beha - io s, e en nea he cu o plasma equency ␻ 0whe e q2 →0共i.e., a/␭→0). Tha is, he nonlocali y o he p oposed cons i u i e ela ions a ec s he elec omagne ic esponse o he medium e en in he e y la ge wa eleng h limi , hus being impo an o any alues o he a/␭ a io inside he medium. O he ele an di e ences be ween he p edic ions o bo h models will be de eloped along his pape . The igo ous p oo o Eq. 共6兲is based on he local- ield app oach which is desc ibed in de ail in Re . 16. In Re . 16 he low- equency s op band o he wi e medium has been analyzed, as well as i s high- equency band-gap s uc u e. This analysis e eals ha , in he hin wi e medium 共Fig. 1兲 and o qz⫽k, wo se s o modes can p opaga e: o dina y 共wi h Ez⫽0) and ex ao dina y 共wi h Ez⫽0) wa es. The o dina y wa es do no in e ac wi h he wi es and p opaga e in he hos media. Fo ex ao dina y wa es, an explici dis- pe sion equa ion connec ing he wa e ec o q ⫽(qx,qy,qz)Twi h he wa e numbe o he hos iso opic ma ix k⫽ ␻ 冑 ␧0 ␮ 0has been de i ed in Re . 16. I can be w i en as 1 ␲ ln b 2 ␲ 0 ⫹1 bkx (0) sinkx (0)a coskx (0)a⫺cosqxa ⫹兺 n⫽0 冉 1 bkx (n) sinkx (n)a coskx (n)a⫺cosqxa⫺1 2 ␲ 兩 n 兩 冊 ⫽0. 共8兲 He e kx (n)deno es he xcomponen o n h Floque mode wa e ec o : kx (n)⫽⫺j 冑 冉 qy⫹2 ␲ n b 冊 2 ⫹qz 2⫺k2,Re 兵 冑 共兲 其 ⬎0. 共9兲 The o he no a ions a e clea om Fig. 1. Nume ical solu ion o his dispe sion equa ion shows ha he e exis s a low- equency s op band o all p opaga ion di ec ions 共excep o he pa icula case o qz⫽k ha will be analyzed la e 兲. Le us simpli y he dispe sion equa ion 共8兲 o he quasis a ic case a,bⰆ ␲ /k. Using he Taylo expansion o sin(x) and cos(x) unc ions o small a gumen s we ob ain Eq. 共7兲wi h k0 2⫽2 ␲ /s2 ln s 2 ␲ 0 ⫹F共 兲 ,共10兲 whe e s⫽ 冑 ab, ⫽a/b, and F共 兲⫽⫺ 1 2ln ⫹兺 n⫽1 ⫹⬁ 冉 co h共 ␲ n 兲⫺1 n 冊 ⫹ ␲ 6.共11兲 The e o e, we ha e shown he sui abili y o he sugges ed app oach o he desc ip ion o he wi e medium, wi h k0 gi en by Eq. 共10兲. Pa ame e k0he e plays he ole o he BRIEF REPORTS PHYSICAL REVIEW B 67, 113103 共2003兲 113103-2 wa e numbe co esponding o he plasma equency. Mo e exac ly, i indica es he uppe edge o he low- equency s op band. Na u ally, k0as a unc ion o wo la ice pe iods aand bis a symme ic unc ion: k0(a,b)⫽k0(b,a). I means ha unc ion F( ) has he ollowing p ope y: F(1/ )⫽F( ). Fo he commonly used case o he squa e g id (a⫽b), F(1) ⫽0.5275. Exp ession 共10兲looks simila o he app oxima e exp essions o he plasma equency de eloped ea lie in Re s. 2–4,6, bu o hin wi es i is mo e accu a e and akes in o accoun he geome y o he la ice. No ice ha he dis- pe sion equa ion 共7兲 o ex ao dina y wa es is indi e en o he di ec ion o he wi es’ axis: he wa e- ec o componen s qx,qy,qzen e in o his equa ion comple ely symme ically. I means ha , wi hin he low- equency s op band, he ex- ao dina y wa e decays wi h he same decay ac o along all di ec ions in space. The same can be said o p opaga ion in he i s equency passband. This iso opy o he dispe sion equa ion is a he su p ising since he medium is s ongly aniso opic. Howe e , i can be shown om he e y unda- men al ac s summa ized in Eqs. 共5兲and 共3兲by assuming, as usual, ha Eq. 共1兲is alid o ex ao dina y wa es p opaga - ing in he x-yplane. I is possible o ansi Eq. 共6兲 om he spec al domain (q, ␻ ) o he physical domain ( , ). The ollowing nonlocal ma e ial equa ion can be de i ed om Eq. 共6兲using he double Fou ie ans o m: D共x,y,z兲⫽␧0E共x,y,z兲⫹ ␧0k0 2c 2z0 冕 ⫺⬁ 冕 z⫺c( ⫺ ⬘) z⫹c( ⫺ ⬘) ⫻Ez共x,y,z⬘, ⬘兲dz⬘d ⬘,共12兲 whe e c⫽1/ 冑 ␧0 ␮ 0is he speed o ligh in he hos ma ix. He e, he a ea o in eg a ion in he z- plane is he ligh cone 兩 z⫺z⬘ 兩 ⬍c( ⫺ ⬘). In o he wo ds, he ke nel in he Fou ie con olu ion is u关c( ⫺ ⬘)⫺ 兩 z⫺z⬘ 兩 兴, whe e u(x) is he Hea iside s ep unc ion. I means ha he poin (x,y,z) in- side he wi e medium 共desc ibed as a dispe si e con inuum兲 a momen is a ec ed by he zcomponen s o elec ic ields coming om he domain „x,y,z⫾c( ⫺ ⬘)…su ounding 共along he wi e axis兲 his poin du ing all he pas ime ( ⬘ ⬍ ). This esul illus a es he consis ency o he epo ed model om he ela i is ic s andpoin . In he ollowing we will desc ibe some ele an e ec s in he analyzed pa allel wi e medium, associa ed wi h he spa- ial dispe sion. Re ac ion and e lec ion o plane wa es a a plane in e ace show s ong di e ences be ween he local and nonlocal models. Le us conside an in e ace be ween an iso opic dielec ic wi h he pe mi i i y ␧1and a uniaxial dielec ic wi h ␧desc ibed by Eq. 共1兲. The in e ace is in he y-zplane and i is illumina ed by a plane wa e coming om he iso opic dielec ic. The wa e ec o and elec ic- ield ec o lie in x-zplane (qy⫽0,Ey⫽0). The incidence angle o he plane wa e is ␪ .I ␧1⬎␧0,␧⬍0, and sin2( ␪ ) ⬍␧0/␧1 he wa e will be comple ely e lec ed, bu o sin2( ␪ )⬎␧0/␧1some pa o he wa e will be ansmi ed h ough he in e ace. This ansmi ed wa e will be an ex- ao dina y wa e, as i ollows om i s pola iza ion s a e, and can be exci ed a any equency. This amazing beha io disappea s when he nonlocal model summa ized in Eq. 共6兲 is used. Indeed, i he nonlocal axial pe mi i i y 共6兲is used o he wi e medium, we obse e ha no ansmission inside he wi e medium is possible o k⬍k0.A k⫽k0 ansmis- sion is possible only in he case o he no mal incidence. Only i k⬎k0, a e ac ed wa e appea s. Le us nex conside he guidance o elec omagne ic wa es in a pa allel-pla e wa eguide in ini e in he xand y di ec ions and bounded by pa allel pe ec ly conduc ing planes o hogonal o he zaxis. Sepa a ion be ween he con- duc ing walls is d. We assume ha his wa eguide is illed wi h a wi e medium wi h he wi es along he zdi ec ion. We will conside eigenwa e p opaga ion along he xaxis o he TM01 mode (Hy,Ex,Ez⫽0). Fo wa eguides illed by a lo- cal uniaxial dielec ic wi h aniso opy axis along he zdi ec- ion we ha e om Eq. 共3兲 ␧0qx 2⫽␧共k2⫺qz 2兲,qx⫽ 冑 ␧ ␧0 冋 k2⫺ 冉 ␲ d 冊 2 册 .共13兲 I ␧⬎0, Eq. 共13兲gi es a cu o o k⬍ ␲ /dand p opaga ion o k⬎ ␲ /d. In con as , i ␧⬍0, p opaga ion is allowed when k⬍ ␲ /d共and o bidden o k⬎ ␲ /d). Wi hin his pass- band a backwa d wa e (dq/d ␻ ⬍0) p opaga es, as one can see in Fig. 2 共 hin solid lines兲. This amazing beha io disappea s i one ills he wa e- guide wi h he analyzed nonlocal wi e medium. Using Eq. 共7兲, we ha e in his case qx 2⫹qz 2⫽k2⫺k0 2,qx⫽ 冑 k2⫺k0 2⫺ 冉 ␲ d 冊 2 ,共14兲 and we ob ain he usual equency beha io : cu o o k ⬍ 冑 ( ␲ /d)2⫹k0 2and p opaga ion o k⬎ 冑 ( ␲ /d)2⫹k0 2.An inc ease o he cu o equency is obse ed compa ed o he FIG. 2. No malized p opaga ion ac o s in a pa allel-pla e wa e- guide s no malized equency o di e en ypes o wa eguide illing: Thin solid lines, uniaxial dielec ic wi h a nega i e pe mi - i i y; hick solid line, wi e medium; dashed line, emp y wa eguide. The wi e medium and he uniaxial dielec ic ha e he same k0 ⫽ ␲ /(2d). BRIEF REPORTS PHYSICAL REVIEW B 67, 113103 共2003兲 113103-3 case when he e is no illing medium, as one can see in Fig. 2共 hick solid line and dashed line兲. Le us inally analyze he p opaga ion o plane wa es along he wi e medium o he pa icula case o qz⫽k.In his case, he nonlocal pe mi i i y along he zaxis 共6兲be- comes in ini e. To a oid he singula i y p oblem we use ma- e ial equa ion o he o m E⫽␧ ¯ ¯ ⫺1D. In his case, he Max- well equa ions ha e plane-wa e solu ions o all equencies. Fo hese wa es he ans e se wa e ec o q⬜⫽(qx,qy)Tis a bi a y. The wa es a e ans e se wi h espec o he wi e axis: Hz⫽0 and Ez⫽0. The elec ic ield is pa allel o he ans e se wa e ec o , q⬜⫻E⫽0. Such wa es can be in e p e ed as ansmission-line modes p opaga ing along he pa allel wi es. In ac , a se o Nin i- ni e pa allel wi es can be iewed as a sys em o coupled ansmission lines. This sys em can suppo Ndegene a e ansmission-line wa es wi h Hz⫽0, Ez⫽0, and phase con- s an qz⫽k. The elec ic ield o hese wa es can be ob ained om E⫽⫺共ux ⳵ x⫹uy ⳵ y兲 ␾ 共x,y兲exp共⫺jkz兲,共15兲 whe e ␾ (x,y) is a quasielec os a ic po en ial aking cons an bu a bi a y alues ␾ n(n⫽1,2,...,N) a each wi e. In ac , he plane wa e wi h ans e se wa e numbe q⬜and qz⫽k co esponds o he ansmission-line wa e wi h ␾ n ⫽ ␾ 0exp(⫺jq⬜• n), whe e n⫽(xn,yn)Tis he loca ion o he n h wi e in he ans e se x-yplane. In summa y, i has been shown ha pa allel wi e media possess e y s ong spa ial dispe sion e ec s a any equen- cies, including he e y la ge wa eleng h limi . An analy ical model o he nonlocal pe mi i i y dyadic o hese media has been p esen ed and discussed. Inconsis ency o he local model o pa allel wi e media wi h non anishing wa e- ec o componen along he wi es has been shown. D ama ic di e ences in he p edic ed beha io o ha media, a ising om he use o he con en ional local and/o he nonlocal model o he pe mi i i y a e shown. Finally, he p oposed nonlocal model o he pe mi i i y has been ound o be also sui able o he desc ip ion o he ansmission-line modes o he s uc u e. We eel ha he epo ed esul s open he ques- ion o he ole o spa ial dispe sion in he adequa e cha ac- e iza ion o disc e e me ama e ials as e ec i e media, a leas i a bi a y di ec ions o p opaga ion and/o pola iza ion o he elec omagne ic ield should be conside ed in he analysis. In addi ion, an example has been p esen ed o an e ec i e medium in which spa ial dispe sion is impo an a any equency, in con as wi h some commonly assumed ideas abou he physical ele ance o his e ec . 1L. Landau, E. Li schi z, and L. Pi ae ski, Elec odynamics o Con inuous Media 共Pe gamon P ess, Ox o d, 1984兲. 2J. B own, P og. Dielec . 2, 195 共1960兲. 3W. Ro man, IRE T ans. An ennas P opag. 10,82共1962兲. 4R. King, D. Thiel, and K. Pa k, IEEE T ans. An ennas P opag. 31, 471 共1983兲. 5S. Maslo ski, S. T e yako , and P. Belo , Mic owa e Op . Tech- nol. Le . 35,47共2002兲. 6J. 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