Circuit model for extraordinary transmission through periodic array of subwavelength stepped slits

Transcription

1 1 Crcut model for extraordnary tranmon through perodc array of ubwavelength tepped lt Amn Khava, Maoud Edalatpour and Khahayar Mehrany Abtract Two crcut model are propoed for analytcal nvetgaton of extraordnary tranmon through perodc array of ubwavelength tepped lt n a thck metallc plate. A cacade model frt developed and then a tub model uggeted for cae that the cacade model not accurate. The parameter of the propoed crcut are gven by cloed form expreon. Both ymmetrc and aymmetrc tepped lt are condered. Index Term tepped lt, extraordnary tranmon, crcut model E I. INTRODUCTION XTRAORDINARY tranmon (EOT) through ubwavelength perodc array of hole or lt, reported by Ebbeen et. al. [1], ha been a hot topc nce t dcovery []-[5]. Although th phenomenon ha been qualtatvely decrbed by urface wave [7]-[9], the reonance of TEM mode [10] and artfcal urface conductvty [11], ome crcut model are recently propoed to provde both qualtatve and quanttatve nght to the ubject [1]-[15]. Crcut model have been developed for two-dmenonal array of hole [1] and for one-dmenonal array of metallc lt [13]-[15]. In the abovementoned work on one-dmenonal array of metallc lt, the lt are aumed to have no cut. Introducng a cut lead to a tepped lt whch ha been recently nvetgated n the context of EOT [16]-[19] and t ha been hown that the ntroducton of the tep change the reonant wavelength ubtantally [17]. It ha alo been demontrated that adjutable phae reonance can be realzed n compound metallc gratng [18]. The am of th paper to generalze the crcut model preented n [13]-[15], to the metallc gratng wth tepped lt. Frt, a cacade model developed. Although, th model accurate n many cae, t wll fal f the juncton are too cloe to each other. For thee cae a tub model propoed. Smlar to our prevou work [15], arbtrary angle of ncdence and arbtrary urroundng medum are tuded. Parameter of the approprate crcut model are gven by cloed form expreon to avod reortng to addtonal numercal effort for ther extracton. In th paper, we alo how that by adjutng the length and place of a tep, a flat-top band pa flter can be degned. An analytcal crcut model can be very ueful for talorng the pectral repone of thee tructure. Th paper organzed a follow: Secton II and III are devoted to the detaled decrpton of the propoed crcut model: the cacade model and the tub model, repectvely. Numercal reult are then gven n ecton IV, where the accuracy of the propoed model compared agant numercal reult obtaned by ung rgorou approache. Fnally, the concluon are made n ecton V. II. THE CASCADE MODEL The geometry of the tructure to be tuded hown n Fg. 1. The perodc arrangement of tepped lt form a metallc gratng wth perod d. The lt are flled wth a delectrc wth refractve ndex of n. The gratng aumed to have three layer n the z drecton. Each layer a mple lamellar gratng enumerated by = 1,, 3. The wdth of lt and ther heght n the th layer are denoted by w and h, repectvely. The gratng regon andwched between two homogenou regon wth refractve ndce of n 1 and n 3. The magnetc permeablty everywhere equal to that of the free pace. The tructure llumnated by a polarzed (magnetc feld along the y ax) unform plane wave whoe wave vector ncdent at angle wth repect to the Oz ax. The free pace wavelength of the wave. The propoed crcut model made of tranmon lne and reactve lumped crcut element. The former repreent propagatng wave wthn layer of the gratng (between the lt), and the latter repreent the evanecent wave torng electromagnetc energy at ether the juncton formed between adjacent layer of the gratng, or the two flat nterface formed between the gratng regon and the urroundng homogenou meda. Manucrpt receved??? Th work wa upported n part by Natonal Elte Foundaton (Bonyad Mell Nokhbegan) under Shahd Dr. Chamran Award. The author are wth Electrcal Engneerng Department, Sharf Unverty of Technology, Tehran , Iran (e-mal: mehrany@harf.r).

2 Z co / n, 1,3 (3) 0 Fg. 1. The geometry of the tructure under tudy: perodc array of tepped lt under the ncdence of a unform plane wave. If h large enough, the two juncton nde the gratng can be aumed to be decoupled. Wth th aumpton a cacade model a hown n Fg., propoed for the tructure. The length of the tranmon lne correpondng to the gratng layer naturally et to be h. Smlarly, t propagaton contant;, and charactertc mpedance; Z, are et to be equal to the propagaton contant and charactertc mpedance of the domnant TEM mode upported by a parallel plate wavegude whoe plate are eparated by a dtance w [15]: Z k n 1,,3 (1), 0 w 0, 1,,3 () k n d 0 where k 0 = / and are the free pace wave number and mpedance, repectvely. Equaton (1) ha been gven by aumng perfect conductvty; however, t can be traghtforwardly modfed to take nto account the effect of loe [13]. where 1 and 3 are the zeroth dffracted order angle n regon 1 and 3, repectvely. It worth notng that 1 =. The wavelength-dependent capactor C 1 and C 3 account for the evanecent feld at the upper and lower nterface between the gratng regon and the urroundng homogenou meda. Ther approxmate value are preented n [15]: t C 1 C, 1,3 (4) 1 / C where n ( 1 n d and C ) C t w n Ln cc d d, 1,3 c 0 0 Here, c 0 denote the peed of lght n free pace. A uggeted n [15] a good emprcal formula for 4 f 1/ 1 (5) ; where f = 1 w /d tand for the fll factor of the 'th layer of the gratng. It hould be mentoned that (5) vald n the range. C The only parameter reman to be determned are C 1 and C 3, repreentng the effect of juncton between the adjacent gratng layer. Fortunately, very accurate cloed-form expreon ha been provded n [0] to take the effect of hgher order mode excted at the juncton of two wavegude. Fgure 3(a) and 3(b) how ymmetrc and aymmetrc juncton of two wavegude wth arbtrary wdth w and w j, repectvely. The uceptance of the capactor modelng the ymmetrc juncton (Fg. 3(a)) [0] a 1 dn 1 a 1 a a B ln (6) 0 4a 1 a A where a w j / w and 1 a A 1 a a n w / n w / 1 3a 1 a (7) Fg.. The propoed crcut to model the tructure hown n Fg. 1 for polarzed ncdent wave. The crcut termnated by Z 1, C 1 and Z 3, C 3 at t nput and output termnal; repectvely. The Z 1 and Z 3 mpedance repreent electromagnetc wave mpedance n the ncdent and tranmon homogenou regon havng refractve ndce of n 1 and n 3 ; repectvely. It therefore eay to how that [15] w n Symmetrc (a) w j Fg. 3. (a) Symmetrc and (b) aymmetrc juncton of two gude. In accordance wth Fg. 3, the larger and the maller wdth are denoted by w and w j, repectvely. w n Aymmetrc (b) w j

3 3 To obtan the uceptance of the capactor modelng the aymmetrc juncton, n (6) and (7) hould be replaced wth / [0]. It hould be noted that the abovementoned equaton are nvald n the range nw and nw for ymmetrc and aymmetrc cae, repectvely. III. THE STUB MODEL For a cut wth mall heght the tructure wll be mlar to Fg. 4. In th cae the evanecent wave excted at the two juncton nde the gratng can couple to each other through the econd layer. Th make the cacade model naccurate, that wll be demontrated n the next ecton. Here, we develop another crcut model for uch cae. for a parallel plate wavegude were already gven n the equaton (1) and (). It alo obvou that the length of the tub and t correpondng tranmon lne h = w w 1. It here aumed that w 1 = w 3. The mpedance Z m the mpedance of the metal that the tub termnated to t. The value of Z m can be calculated by the followng equaton [1]: Z m 1 (8) Z tub where m the permttvty of the metal. m Fg. 5. (a) The propoed tub model for the tructure hown n Fg. 4(b). (b) The crcut whoe nput mpedance Z. The reactve element whoe value are denoted by B a, B b, B c and B d are for takng nto account the effect of T-juncton. Approxmate cloed-form expreon for thee element have been gven n [0]: dn w 1 1 ln 1 B tan (9) a 0w B b 1.1dn w w 0 w w B c w w (10) d (11) w 0 w Fg. 4- The (a) ymmetrc and (b) aymmetrc tructure of Fg. 1 wth mall h, for whch the tub model hould be ued. Conder a unt cell of the tructure hown n Fg. 4 (b), t reemble a wavegude that a tub n ere wth t. From th pont of vew the crcut model hown n Fg. 5 (a) can be attrbuted to the aymmetrc tructure. In th crcut the tub modeled wth the ere mpedance Z whch the nput mpedance of the crcut hown n Fg. 5 (b). In th crcut Z and are the charactertc mpedance and the tub tub propagaton contant of a parallel plate wavegude whoe plate are eparated by a dtance w = h. Thee parameter B d dn ew w ln w w w w w w 3 (1) where w = w 1 = w 3, = w /(w) and e the Neper number. For the ymmetrc cae (ee Fg. 4 (a)) the juncton' model ha not been gven n [0], however we extend the propoed model to the ymmetrc cae by ung an ntutve reaonng. In th cae ntead of one tub of length h, we have two tub of length h /. Therefore, h n Fg. 5 (b) mut be replaced wth h /. Furthermore, n th cae two ymmetrc tub are on two de of the wavegude. Thank to the ymmetry of the tructure each tub eparated from the other by a perfect electrc conductor wall dvdng the man wavegude nto two

4 4 equal halve. Hence, n (11) and (1) w hould be changed to w/. Fnally, n Fg. 5 (a) we put ntead of the ere mpedance, Z. Z IV. NUMERICAL EXAMPLES Let u examne the accuracy of the propoed model n a numercal example. Conder a tructure wth ymmetrc juncton whoe parameter are n accordance wth Fg. 1: = 0, n 1 = 1.4, n = 1, n 3 = 1.4, h 1 = h = h 3 = d = cm and w 1 = w 3 = 0.1 cm. The alumnum ued a the conductor whoe dc conductvty et to dc = S/m. The tranmon pectrum of th tructure for w = 0. cm, w = 0.4 cm and w = 0.6 cm are plotted n Fg. 6(a), (b) and (c), repectvely. The reult have been obtaned by ung the propoed cacade model (old lne), HFSS mulaton (crcle) and the rgorou approach of [] (pont), n whch the regularzaton preented n [3] ha been appled. Excellent agreement oberved between all method. It can be een from thee fgure that the preence of tepped lt reult n the dplacement of reonance frequence. It fortunate uch a dplacement geometrcally adjutable. For example, two reonance frequence are merged together n Fg. 6(c) and a relatvely flat-top pectral repone ha been created around C / = A notable beneft of the analytcal crcut model, propoed n th letter, that the pectral repone of EOT can be effcently talored. A another example conder a tructure wth the ame parameter a n the prevou example, but t juncton are aymmetrc. The accuracy of the propoed model for th cae demontrated n Fg. 7(a), (b) and (c) where the tranmon depcted n term of C / for w = 0. cm, w = 0.4 cm and w = 0.6 cm, repectvely. The reult are obtaned by ung the propoed cacade model (old lne), HFSS mulaton (crcle) and the rgorou approach of [] (pont). Once agan, an excellent agreement oberved between the propoed model and the rgorou approache. Fg. 6- Tranmtted power through an array of tepped lt veru C/ calculated by ung the propoed cacade model (old lne), HFSS mulaton (crcle) and the rgorou approach of [] (pont). The tructure parameter are: = 0, n 1 = 1.4, n = 1, n 3 = 1.4, h 1 = h = h 3 = d = cm and w 1 = w 3 = 0.1 cm. The reult are llutrated for three dfferent wdth of w : (a) w = 0. cm, (b) w = 0.4 cm and (c) w = 0.6 cm. The juncton are aumed to be ymmetrc. Fg. 7- The ame a Fg. 6, but the juncton are aumed to be aymmetrc. Now, let u compare the accuracy of the cacade model agant the tub model. The tructure ha many parameter and each of them can contrbute to the accuracy of reult. However, from Fg. 5 and equaton (9)-(1), t obvou that

5 5 h (= w w 1 ), w (= h ) and w (= w 1 = w 3 ) are the three parameter that drectly affect the accuracy of the tub model. For the ake of mplcty, we only examne the effect of thee three crucal parameter on both model. Although t not a complete comparon, t can gve u a rule of thumb for electng the approprate model. Conder a tructure under normal ncdence of a plane wave wth = 5d. The tructure parameter are: n 1 = n = n 3 = 1, and h 1 = h 3 = d = cm. The relatve error n calculatng power tranmon computed a a functon of w / and h / for both model. Regon at whch the relatve error of the cacade and tub model more than 10% are llutrated by black and gray color n Fg. 8, repectvely. The error map are plotted n Fg 8 (a), (b), (c) and (d), for four cae, w = 0.1d, 0.d, 0.3d and 0.4d, repectvely. (a) (b) (c) (d) Fg. 9- Tranmtted power through an array of tepped lt veru C/ calculated by ung the propoed tub model (old lne), HFSS mulaton (crcle) and the rgorou approach of [] (pont). The tructure parameter are: = 0, n 1 = 1.4, n = 1, n 3 = 1.4, h 1 = h 3 = d = cm, h = w 1 = w 3 = 0.1 cm and w = 0.6 cm. The reult are llutrated for the both (a) aymmetrc and (b) ymmetrc juncton. Fg. 8. Regon wth more than 10% relatve error n calculaton of tranmtted power: the black and gray regon for the cacade and tub model, repectvely. Four cae are depcted: (a) w = 0.1d, (b) w = 0.d, (c) w = 0.3d and (d) w = 0.4d. The juncton are aumed to be aymmetrc. A a rule of thumb, t can be concluded that the tub model hould be ued for h > w. Although t cannot be taken a a general and accurate rule but t ueful n many cae. To further demontrate the accuracy of the propoed tub model the tranmon of th tructure plotted a a functon of C / n Fg. 9 (a) and (b) for aymmetrc and ymmetrc juncton, repectvely. The heght of the econd layer choen mall enough (h = 0.1 cm) o that the tub model reman accurate. V. CONCLUSION In concluon, extraordnary tranmon through ubwavelength array of tepped lt ha been tuded by ung analytcal crcut model. Frt, a cacade model wa preented, whch compoed of tranmon lne correpondng to dfferent part of lt, and capactance for modelng juncton. Another model ha been alo propoed n whch the econd layer of the gratng ha been regarded a a tub. It demontrated that the cacade model utable when the two juncton nde the gratng are far enough apart, otherwe the tub model hould be ued. The crcut parameter have been gven n cloed form expreon, thu the model can be employed for effcent nvetgaton of extraordnary tranmon n uch tructure. The reult of the propoed crcut model have been compared wth rgorou method n numercal example and excellent agreement have been oberved. REFERENCES [1] T. W. Ebbeen, H. J. Lezec, H. F. Ghaem, T. Tho, and P. A. Wolff, " Extraordnary optcal tranmon through ub-wavelength hole array, " Nature, vol. 391, pp , Feb [] C. Genet and T.W. Ebbeen, "Lght n tny hole" Nature, vol. 445, pp , Jan [3] F. J. Garca-de-Abajo, " Colloquum: Lght catterng by partcle and hole array," Rev. Mod. Phy., vol. 79, no. 4, pp , Oct [4] F. J. Garca-Vdal, L. Martn-Moreno, T. W. Ebbeen, and L. Kuper, " Lght pang through ubwavelength aperture," Rev. Mod. Phy., vol. 8, no. 1, pp , Mar [5] R. Gordon, A. G. Brolo, D. Snton, and K. L. Kavanagh, " Reonant optcal tranmon through hole array n metal flm: phyc and applcaton" Laer Photonc Rev., vol. 4, no., , Feb. 010.

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