Anisotropy and oblique total transmission at a planar negative-index interface

Transcription

1 Ansoropy and oblque oal ransmsson a a planar negave-ndex nerface Le Zhou, C.T. Chan and P. Sheng Deparmen of Physcs, The Hong Kong Unversy of Scence and Technology Clear Waer Bay, Kowloon, Hong Kong, Chna Absrac We show ha a class of negave ndex (n) maerals has neresng ansoropc opcal properes, manfes n he effecve refracon ndex ha can be posve, negave, or purely magnary under dfferen ncdence condons. Wh dsperson aen no accoun, reflecon a a planar negave-ndex nerface exhbs frequency selecve oal oblque ransmsson ha s dsnc from he Brewser effec. Fne-dfference-me-doman smulaon of realsc negave-n srucures confrms he analyc resuls based on effecve ndces. PACS numbers: e, Jb, n 1

2 Maerals whose permvy ε and permeably µ are smulaneously negave are sad o possess a negave refracve ndex n, wh many unusual properes [1]. Negave-n meallc resonang composes and wo dmensonal (2D) soropc negave-n maeral have been consruced [2,3], and negave lgh refracon was observed [4]. The unconvenonal properes of such maerals have drawn an ncreasng amoun of aenon n boh scence and engneerng [5-10]. Recenly, he ansoropc wave characerscs of 2D soropc negave-n maerals [3-4] have been suded [8,9]. A blayer srucure composed of ansoropc negave-n maerals has been proposed o be a perfec lens [10]. In hs wor, we are neresed n he opcal ansoropy and reflecon/refracon properes a a planar nerface wh a negave-n maeral, such as hose descrbed n Ref. [2], whose wave characerscs have ye o be examned. We fnd he maeral o possess an angle-dependen effecve refracve ndex n( θ ), whch can be eher negave, posve, or purely magnary a dfferen wave-vecor drecons. Whle was poned ou ha mos negave-n nerfaces had reflecance near uny for ncden propagang wave [10], we fnd he presen one o exhb frequency selecve oblque oal ransmsson, whose governng physcs s dfferen from ha of he Brewser angle phenomenon [11]. Fne-dfference-me-doman (FDTD) smulaons [12] on a realsc meallc resonance srucure confrmed he predced effec. Consder a homogeneous meda characerzed by a dagonal permvy marx whε < 0, ε = ε = 1, and a dagonal permeably marx wh µ < 0, µ = µ = 1 zz zz [13]. For EM waves ravelng along he z drecon wh E feld polarzed along he y-axs, he maeral exhbs smulaneous negave ε and µ, leadng o a negave n. Ths corresponds exacly o he meallc mea-maeral recenly fabrcaed [2]. The wave equaon for B feld can be wren as: [ ε! (! ( µ "! B))] = ω "! B/ c, (1) 1 1 where c s he speed of lgh, ε, µ are respecvely he nverses of permvy and 2

3 permeably marces, ω and! are angular frequency and wave vecors. Le! be "! confned n he xz plane and E polarzed along he y axs. Ths wll be he case suded!! n hs paper. We noe ha E # y mples B y = 0, and B = 0 means B x x = B z z. Pung he above wo expressons no Eq. (1), we!! ge [( ε µ ) + ( ε µ ) ] B = ω B/ c, whch leads o a dsperson relaon z zz x ω = ( c) / n( θ) wh an angle-dependen refracon ndex n( θ) = [( ε µ ) cos θ + ( ε µ ) sn θ]. (2) zz Here,θ s he angle beween! and he z axs. Fgure 1 shows a comparson of hs angle dependen effecve refracon ndex wh hose for ar and for an ordnary non-absorbng ansoropc maeral, where he angle represens he wave-vecor drecon and radus gves he modulus of he ndex. For he ordnary non-absorbng ansoropc maeral, n( θ ) s real n he enre angle regme, and resembles an ellpsod [11]. Ansoropc negave-n maeral, however, has a real n( θ ) (s sgn wll be denfed laer) n some angular range, mplyng he suppor for ravelng waves, and has a purely magnary n( θ ) n ohers, forbddng any ravelng waves. In addon, f ε µ < 1, we fnd ha he condon n( θ ) = 1can be me a some parcular angles. The mplcaon of hs condon s addressed laer. 3

4 Forbdden Permed Permed Forbdden y X z Re(n) Im(n) n of ar ordnary x Fg. 1 Refracve ndex as a funcon of he wave-vecor drecon for a negave-n maeral whε = µ = 0.5, µ = 1 (open squares for he real par, sold crcles zz for he magnary par). For comparson, he refracve ndex as a funcon of angle s also shown for ar (sold lne), and an ordnary ansoropc maeral wh ε = µ = 0.5, µ = 1 (dashed lne). zz We now consder he reflecon/refracon properes a a planar nerface beween ar and such a negave-n maeral. Due o he specal angular varaon n n, here are wo dsnc cases correspondng o wo dfferen orenaons of he nerface. Consder frs a planar xy nerface beween ar and a negave-n maeral. Translaonal nvarance along surface drecons a he nerface means he conservaon of he parallel! componen: θ = θ, sn( θ ) = n( θ )sn( θ ), (3) r where θ, θ and θ are respecvely he ncden, reflecon and refracon angles. The r ray dagram s schemacally depced n Fg. 2, where he subscrps,r, sand for he ncden, refleced and ransmed waves, respecvely. Wh fxed by x, here are x 4

5 wo choces for he sgn of n he second meda. Causaly requres ha he Poynng z "! "! ""!! vecor S E H (dfferen from he vecor!) nsde he second meda should pon away from he nerface so ha he ransmed wave carres energy away from he z source. Based on hs creron, we choose > 0 for hs case. I s raher easy o chec ha he oher choce ( z < 0 ) gves an unphyscal drecon for S! (subscrp denoes "! "! ransmsson). We noe ha a negave ε means D s anparallel o E for he "! ""! ransmed wave; and µ µ zz ndcaes ha B s non-collnear wh H. The Poynng vecor s specfed by an angle, θ s, whch s dsnc from θ for he s wavevecor and should be carefully deermned [14]. I should be noed ha f θ were regarded as he refracon angle, hen he refracon here corresponds o a posve ndex.. E, D, S X D. E H B θ r, S r. E r, D r H, B θ θ r θ s H r,b r Ar our meda S y X z x Fg. 2 Schemac ray dagram for lgh reflecon/refracon a he xy nerface beween ar and he presen suded negave-n maeral. The nerface s mared by he sold lne and he surface normal by he doed lne. 5

6 θ (degree) θ s θ (a) Reflecance 0.5 (b) θ (degree) Fg. 3 (a) Refracon angleθ (doed lne), s θ (dashed lne) and (b) he reflecance as funcons of he ncdence angle θ a he xy nerface beween ar and a negave-n maeral whε = µ = 0.5, µ = 1. zz Applyng he usual boundary condons o he parallel componens of E and H a he nerfaces, he Fresnel formula for he reflecance R s derved o be 6

7 n( θ ) cosθ cos θ µ R = ( ) n( θ ) cosθ + cos θ µ 2. (4) For he reflecance shown n Fg. 3(b), we noe ha R ncreases as θ ncreases, smlar o ha of an ordnary maeral, and he refraced beams are confned nsde he permed regme [see Fg. 3(a)]. Snce n( θ ) s an ncreasng funcon of θ and dverges when θ approaches he boundary of he permed regme, s clear ha one can always fnd a θ for any θ whch sasfes Eq. (3). r, S r E r, D r. θ r H r, B r E. D X θ H B, S. θ H, B E, D Ar S θ s y X Our meda z x Fg. 4 Schemac ray dagram for lgh reflecon/refracon a he yz nerface beween ar and he negave-n maeral. Le us now consder he second case n whch he nerface s a he yz plane. The ray dagram s shown schemacally n Fg. 4, where θ s relaed o θ va sn θ = n '( θ )sn θ, (5) n whch n '( θ) = [( ε µ ) cos θ + ( ε µ ) sn θ ], n whch he angles are zz conssenly defned wh respec o he normal o he nerface. Agan, he choce > 0 x adoped here s deermned by he condon ha S! should pon away from he 7

8 nerface n he second meda. The mos srng feaure n hs case s he negave "! refracon (see he drecon of S n Fg. 4) [15]. A smple calculaon gves he correspondng Fresnel formula o be ' n ( θ ) cosθ cos θ µ zz 2 ' n ( θ ) cosθ + cos θ µ zz R = ( ). (6) Reflecon a hs nerface (Fg. 5(b)) shows an unusual ncdence angle dependence n he case of ε µ < 1. Tha s, whle he nerface oally reflecs he EM waves ncden from he normal, can be shown ha here exss a crcal value for he c 1 ncdence angle, θ = sn [ ε µ ], above whch Eq. (5) can be sasfed, mplyng ha a refraced waves can be coupled no he negave-n medum (see Fg. 5(a)). Dsnc from he convenonal case, here θ s a decreasng funcon of θ, caused by he unusual angular dependence of ' n ( θ ). The fac ha here exss a parcular 0 ncden angle θ such ha ' 0 0 n ( θ ) = 1 ndcaes ha a soluon θ = θ = θ can be found for Eq. (5). Accordng o Eq. (6), he maeral becomes non-reflecng a hs ncdence angle (rememberng ha µ zz = 1). Away from hs parcular ncdence angle, θ devaes qucly fromθ, leadng o srong reflecon. 8

9 θ (degree) θ θ s θ = θ (a) Reflecance 0.5 θ c (b) θ (degree) Fg. 5 (a) Refracon angleθ (doed lne), s θ (dashed lne) and (b) he reflecance as funcons of he ncdence angle θ a he yz nerface beween ar and a negave-n maeral wh ε = µ = 0.75, µ = 1. Perfec ransmsson occurs a θ = θ. zz We emphasze ha he physcs underlyng he absence of reflecon here s nrnscally an ansoropc effec and s dfferen from ha for he zero reflecvy a he Brewser angle n a convenonal soropc maeral [11]. A he nerface beween 9

10 ar and a convenonal soropc maeral characerzed by ε and µ, he reflecance for "! ""! he S-polarzed wave ( E # y) and he P-polarzed wave ( H # y) are respecvely: R S cosθ cos θ / Z cosθ cosθ Z = ( ), R = ( ), (7) cosθ cos θ / cosθ cosθ 2 2 P + Z + Z where snθ = n snθ and n = ε µ, Z = µ / ε are respecvely he refracon ndex and he mpedance. Zero-reflecon aes place a an ncdence angle sasfyng θ + θ = π /2 for he P wave ncden on a convenonal delecrc maeral wh µ = 1. A smple generalzaon shows ha he same phenomenon exss for he S wave ncden on a magnec maeral (wh ε = 1 ). Such an angle, deermned 1 byθ = an ( n ), s called he Brewser angle [11]. The physcs accounng for such a B phenomenon s ha he vbraon of elecrons n he second meda can no generae he refleced beam whch ravels perpendcular o he ransmed beam (because of θ + θ = π /2) [11]. The zero-reflecon dscovered here, however, requresθ = θ. The physcs here s governed by he ansoropy whch maes he maeral ransparen a a parcular oblque ncdence angle (dcaed by he condon n'( θ ) = 1) and dar a ohers (see Fg. 1). The presen effec can no exs n an soropc maeral. If one sees a zero-reflecon soluon sasfyng θ = θ from Eq. (7) whch descrbes an soropc maeral nerface, he only possbly s ε = µ = 1[16]. In fac, f one removes he ansoropy from our suded sysem descrbed n Fg. 5, he oal ransmsson phenomenon dsappears (ndependen of wheher one aes µ = µ = 1 or µ = µ = 0.5). Also, should be noed ha he presen effec can only exs n a zz dspersve meda, snce he condon ε µ < 1 s usually a characersc of a dspersve meda. Ansoropy n a dspersve meda s he ey elemen o ge he oal ransmsson effec. zz All negave-n maerals are hghly dspersve [3,4]. The fac ha ε and µ can vary from very negave values o very posve ones near he resonance suggess ha any such maeral could always have a frequency wndow n whch he condons ε, µ < 0 and ε µ < 1 are sasfed. In wha follows, we consder a 32mm-hc slab composed by a dspersve ansoropc maeral wh effecve ε, µ gven by 10

11 ε ( f ) = f 12 f 7 µ ( f ) = f , where f denoes he frequency measured n GHz. A smple calculaon shows ha bohε and µ are negave andε µ < 1 n frequency range of GHz. We employed he ransfer marx mehod o numercally calculae he ransmance hrough such a 32mm-hc slab as he funcon of he ncdence angle. Whn he range of GHz, we fnd ha here s always a specfc ncdence angle for whch he slab becomes oally non-reflecng. Dashed lne n Fg. 6(a) shows he ransmance hrough he slab as he funcon of he ncdence angle for f = 4.4 GHz. Toal ransmsson occurs a abouθ = 75 %. Srong reflecon s seen a oher ncdence angles [17]. Due o he dsperson of ε and µ, he ransmsson s frequency selecve under a fxed ncdence angle. Dashed lne n Fg. 6(b) shows he ransmsson specra hrough he 32mm-hc slab for θ = 30 %. We fnd oal ransmsson a abou f = 4.55 GHz, and almos zero ransmsson ousde of he frequency range GHz. We noce ha he auxlary ransmsson peas n he specra are nduced by he Fabry-Pero nerferences [11] beween he mulply scaered felds on slab s wo nerfaces. These auxlary peas are dependen on he slab hcness, whereas he man pea s ndependen on he slab hcness. The sold lne n Fg. 6 s an average over he ransmances of 50 slabs, each wh a random hcness devaon ( ± 20% ). We see ha he peas due o Fabry-Pero resonances dsappear upon averagng, leavng behnd only he pea due o he aforemenoned non-reflecng condon. 11

12 f = 4.4 GHz Averaged One slab (a) Transmance θ = 30 o Averaged One slab θ (degree) (b) Frequency (GHz) Fg. 6 Transmance (a) as a funcon of θ a f = 4.4 GHz, and (b) as a funcon of frequency for θ = 30 %. Dashed lne s for a sngle 32mm-hc slab, sold lne denoes he resul averaged over 50 slabs wh ± 20% hcness varaons. 12

13 We have performed FDTD smulaons [12] o demonsrae he oal ransmsson effec on a realsc resonance srucure. The buldng bloc of our desgned maeral s shown n he nse o Fg. 7. Here we used a meallc for n he mddle o creae a negaveε, and spl rngs [18] on he lef and rgh o creae a negave µ. The un cell s hen repeaed wh lace consans d = 16, d = 7.5 mm o le he yz plane, wh wo y 1.6mm-hc delecrc plaes (wh ε = 4 ) employed as subsraes o separae he for and spl-rngs. Fnally, he resulng 3.8mm slab s repeaed n he x drecon wh lace consan d x = 6 mm o consruc a layered srucure. We frs employed he FDTD smulaon [19] o calculae he normal ransmsson specra of a slab of he desgned maeral wh hree un cells along he z drecon (22.5mm-hc) and nfne n he xy plane. From hese calculaed ransmsson specra (usng boh ampludes and phases) we hen derved ε and µ as funcons of frequency. These are shown n Fg. 7. We fnd a frequency range where boh ε and µ are negave andε µ < 1. We hen consder anoher slab of he same maeral wh hree un cells along he x drecon (18mm-hc) and nfne n he yz plane. The ransmance hrough such a slab a frequency 4.51 GHz s calculaed by he FDTD smulaons as a funcon of he ncdence angle. The resul s shown n Fg. 8 by he sold squares. Reasonably good agreemen s seen wh he resul obaned for a homogeneous slab (shown by he sold lne) wh effecve properes shown n Fg. 7. Boh show he oal ransmsson a some oblque ncdence angle. I should be emphaszed ha he FDTD resuls are he numercal soluons of Maxwell s equaons wh he mcrosrucures fully aen no accoun. The only approxmaon s he perfec-meal boundary condon mposed on he meal surfaces, whch holds well n he mcrowave regme. Ths demonsraes ha he oal oblque ransmsson can ndeed occur a a negave-n nerface. z 13

14 ε eff & µ eff x y z ε µ 3.5mm 15mm 0.5mm ar gap 3.8mm 1.6mm 5mm 12mm Frequency (GHz) Fg. 7 Effecve ε and µ as funcons of frequency for our desgned maeral from he FDTD smulaon resuls. The buldng bloc srucure of he desgned maeral s shown n he nse. 14

15 1.0 Transmance 0.5 Model FDTD f = 4.51 GHz θ (degree) Fg. 8 Transmance a 4.51 GHz as a funcon of he ncdence angle hrough a slab of our desgned maeral wh hree un cells n he x drecon, calculaed by he FDTD smulaons (sold squares) and wh effecve ε and µ shown n Fg. 7 (sold lne). In shor, we have shown ha a class of ansoropc negave-n maeral has neresng angle dependen opcal properes. In parcular, allows frequency selecve oal oblque ransmsson, whch s dfferen from he Brewser effec. Such a unque propery has been demonsraed n a realsc meallc sysem wh he help of FDTD smulaons. Acnowledgemen: Ths wor was suppored by Hong Kong RGC hrough HKUST6145/99P and CA02/03.SC01. 15

16 References: [1] V. C. Veselago, Sov. Phys. Usp. 10, 509 (1968). [2] D. R. Smh, Wlle J. Padlla, D. C. Ver, S. C. Nema-Nasser, and S. Schulz, Phys. Rev. Le., 84, 4184 (2000). [3] R. A. Shelby, D. R. Smh, S. C. Nema-Nasser, and S. Schulz, Appl. Phys. Le., 78, 489 (2001). [4] R. A. Shelby, D. R. Smh, S. Schulz, Scence 292, 77 (2001). [5] P. Maros and C. M. Sououls, Phys. Rev. B 65, (2001); Phys. Rev. E 65, (2002). [6] D. R. Smh and S. Schulz, P. Maros and C. M. Sououls, Phys. Rev. B 65, (2002). [7] M. Bayndr, K. Aydn, and E. Ozbay, P. Maros and C. M. Sououls, Appl. Phys. Le., 81, 120 (2002). [8] I. V. Lndell, S. A. Treyaov, K. I. Nosnen, and S. Ilvonen, Mcro. Op. Technol. Le., 31, 129 (2001) [9] L. B. Hu and S. T. Chu, Phys. Rev. B 66, (2002). [10] D. R. Smh and D. Schurg, Phys. Rev. Le., 90, (2003). [11] See, for example, M. Born and E. Wolf, Prncples of Opcs, (2nd e., Pergamon Press LTD., Oxford, London, 1964) [12] K. S. Yee, IEEE Trans. Anennas and Propagaon 14, 302 (1966). [13] We noe none of he cases dscussed n Fg. 2 of Ref. [10] corresponds o our presen case. "! "! s [14] Here we adoped he convenon o ae θ ( θ ) posve f ( S ) s n he oppose sde o he normal wh respec o he ncden one. [15] Some auhors also poned ou ha negave refracon could be realzed n oher nds of ansoropc negave-n maerals [8-10]. [16] We noe ha he oblque angle non-reflecon phenomenon can also occur a a negave-ndex nerface whε = µ = 1, see Ref. [1] and J. B. Pendry, Phys. Rev. Le., 85, 3966 (2000). [17] For presen sysem whou absorpon, we have ransmance + reflecance = 1. [18] J. B. Pendry e. al., IEEE Trans. Mcrowave Theory Tech. 47, 2075 (1999). 16

17 [19] Smulaons were performed usng CONCERTO 2.0, Vecor Felds Lmed, England,