Effect of Surface Roughness on Propagation of Surface Plasmon Polaritons Along Thin Lossy Metal Films

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1 Effect of Surfce Roughne on Propgtion of Surfce Plon Polriton Along Thin Loy Metl Fil Hod Fdr*, Abolghe Zeidbdi Nezhd**, nd Air Borji*** Deprtent of Electricl nd Coputer Engineering, Ifhn Univerity of Technology, Ifhn, IRAN ** Abtrct: In thi pper urfce plon polriton (SPP tht propgte long thin loy etl fil with urfce irregulritie re tudied. Two ce of etl lb (with infinite width nd etl trip (with finite width re conidered. Surfce roughne i odelled grting with deterinitic profile. To nlyze the lb tructure the ethod of reduced Ryleigh eqution i ued. The reult how tht ntiyetric ode i trongly ffected with roughne. Accurcy of the reult i deontrted through coprion with HFSS iultion. The effect of urfce roughne on the etl trip tructure i only iulted by HFSS oftwre. It how tht the ttenution of SPP ode i increed due to the urfce roughne. Keyword: Surfce plon polriton (SPP, reduced Ryleigh eqution, urfce roughne 1. Introduction Surfce Plon Polriton (SPP re urfce electrognetic wve tht re bound to the interfce between two edi where the rel prt of their perittivity hve oppoite ign. The field coponent of SPP decy eponentilly wy fro the interfce in trnvere direction uch tht the lterl dienion of thee wve i ller thn the free pce wvelength. Therefore, plonic wveguide hve ttrcted lot of ttention for ppliction in high denity integrtion of opticl coponent. Nobel etl uch gold nd ilver ehibit negtive dielectric contnt t viible nd infrred wvelength, thu propgtion of bound wve t etl dielectric interfce h been widely tudied in recent yer [1]-[]. A etl fil ndwiched between dielectric lyer with infinite width (lb nd finite width (trip hown in Fig.1 i often ued plonic wveguide. Thi tructure, with ooth urfce, w nlyzed by Berini with ei-nlyticl ethod of line [1]. Berini chrcterized the effect of vrying width, thicne nd bcground perittivity on diperion of the purely bound ode tht upported by thee wveguide. One of the fundentl ode of thee tructure i long rnge ode tht h long propgtion length. The penetrtion depth of long rnge ode i le thn penetrtion depth of hort rnge one o the ttenution due to borption in the etl i reduced. Tight confineent of the field to the etl urfce e SPP very enitive to urfce roughne. There re everl tudie on the effect of urfce irregulritie on the diperion curve of SPP ode tht propgte on the interfce of lole etl nd dielectric [4]-[7]. In thi pper we invetigte the effect of urfce roughne on the diperion reltion of SPP ode tht propgte long loy etllic lb or trip. Surfce roughne i odelled periodic grting with inuoidl profile. In the ce of lb wveguide the nlyticl ethod of reduced Ryleigh eqution i ued to obtin the diperion curve of yetric nd ntiyetric ode. Thi tructure i lo iulted with HFSS eigenolver nd the reult re in cloe greeent with the nlyticl ethod. For the etl trip wveguide there i no nlyticl olution, thu only the HFSS eigenolver i ued. Figure 1: etl lb wveguide etl trip wveguide with one grting boundry. Slb wveguide.1 Theory The tructure i hown in Fig.1. It conit of lb wveguide (dielectric /etl/ dielectric tht h two different interfce: one of the i ooth (flt nd the contin periodic grting whoe profile i ζ (. The thicne of the lb i t nd it reltive perittivity i ε ( r ω, urrounded by dielectric of reltive perittivity 468

2 ε r1 nd ε r. Only the SPP ode tht propgte perpendiculr to the groove of grting (long i re conidered, o TM nd TE polriztion re decoupled. A it i well nown, the infinitely wide yetric tructure upport only two bound TM urfce ode with field coponent H y, E nd E z. The coponent of the field in the plne perpendiculr to the direction of propgtion hve either yetric or ntiyetric ptil ditribution with repect to the Y i. The yetric ode h ll ttenution contnt but ntiyetric ode ehibit lrge ttenution becue it field penetrte ore into the etl [1]. According to [4] the gnetic field cn be preented : for z>t ( = ( ω ( α ( ω H y 1, z A ep j z (1 for z< ζ ( in = ( = ( ω ( + β ( ω H y, z D ep j z ( for = ζ ( <z<t ( ( [ ( ω ( γ ( ω H, z = ep j B ep z + y = ( ω ep ( γ ( ω ] ( C z where = + π / L, A ( ω, B ( ω, C ( ω nd D ( ω re Fourier coefficient nd α ( ω, β ( ω, γ ( ω re defined : ( = ( 1/ α ω ε r 1 ( = ( 1/ β ω ε r ( 1/ ( = ( γ ω ε ω (4 r with Re{ φ ( ω, I{ φ ( ω φ α, β, γ > < =. After pplying the boundry condition nd oe nipultion the following eqution re obtined [4]: M M 11 1 n M n A 1 D n M n = -N, -N-1,...N-1, N =, n= -N, -N-1,...N-1, N ( α t ε ( ω α r ep ( γ 1 ε γ r1 ε r ( ω α K ep ( γ t 1 n ε γ r1 ep M = J t + + M 11 n n 1 = L n n (5 (6 (7 ( α t ( ε ω α r ep ( γ 1 ( 1 ε ( ω α r K ( γ t ( ε ( ω n n r ε γ r 1 ep M = J t + n n ε ω γ ε γ r r + ep 1 (8 Ln = n n r r ( ε M ε β Eqution (5 i the o clled reduced Ryleigh eqution nd Fourier coefficient in thee eqution re defined by: L ( 1 ( π r ω = ep ep( γ ( ω ζ ( r L L J d j L ( 1 ( π r ω = ep ep ( γ ( ω ζ ( r L L K d j L ( 1 ( π r ω = ( β ( ω ζ ( r L L L d ep j ep (1 After truncting the Floquet ode to N, dienion of the tri becoe (N + 1 (N + 1. The diperion reltion of SPP i obtined by equting the deterinnt of the coefficient A nd D in (5 to zero. In thi pper inuoidl profile function i conidered, nely ζ ( = ζ co( π / L. Thu, the Fourier coefficient becoe odified Beel' function: ( ( ω = r r ( γ ζ ( ( ω = ( β ζ K I L I r r ( r ( ω = ( 1 ( β ζ J I r r 1 (9 (11. Nuericl reult To verify the forultion, we lo reproduce the reult of ooth lb ( ε r1 = ε r nd ζ = nd copre our reult with thoe of Berini which were obtined with different ethod. The free pce wvelength i et to λ =.6 µ, the reltive perittivity of the ilver fil t thi wvelength i r ( 19 j.5 ε ω = +, nd the perittivity of the urrounding edi i ued to be ε r 1 = ε r = 4. The diperion curve for ilver lb with L = 5 n nd different vlue of ζ =, ζ =.1 L, ζ =.L were obtined. Figure to 4 how the rel nd iginry prt of / function of lb thicne t for yetriclie nd ntiyetric-lie ode. In order to chieve convergence, it i enough to chooe N = 4. For ζ = the urfce becoe flt nd the reult re in good greeent with [1]. In [1] the proble i olved by ethod of line (MOL which i n ccurte ei nlyticl ethod. The ttenution of the ode i increed with decreing the thicne of the fil becue the field penetrte ore into the loy ilver 469

3 fil. In the ode with decreing the thicne the ttenution i decreed nd the ode evolve towrd the TEM wve upported by the bcground. Thi ode i long rnge SPP..5 L=5n, ζ =.L R e { / Thicne ( Re { / Thicne ( L=5n, ζ =.L I { / I { / Thicne ( 1-7 Figure : propgtion contnt rel prt iginry prt for ζ = I { / Re { / / in ter of ilver lb thicne L=5n, ζ =.1L Thicne ( L=5n, ζ =.1L Thicne ( 1-7 Figure : propgtion contnt rel prt iginry prt for / in ter of ilver lb thicne ζ =.1L Thicne ( 1-7 Figure 4: propgtion contnt rel prt iginry prt for ζ =.L / in ter of ilver lb thicne In figure nd 4 it i oberved tht with increing the depth of the grting (roughne the rel nd iginry prt of propgtion contnt of ode re increed. However, the overll behviour of nd ode do not chnge, i.e. with decreing the thicne of the fil, the ttenution of ode i increed nd tht of the ode i decreed. A the eprtion between the top nd botto interfce incree, the nd ode plit into pir of uncoupled SPP ode loclized t the ilver /dielectric interfce. For the flt interfce, ζ =, the nd ode becoe degenerte but in the preence of urfce roughne the degenercy i broen uch tht the ode converge to the SPP wve tht propgte on flt interfce nd the ode converge to the SPP tht propgte on grting interfce. When there i grting in one ide of the lb, the ptil ditribution of the field i not truly yetric or ntiyetric bout Y i lthough they re loclized ner one of the interfce. The ode field ditribution h iu t the flt interfce while the ode h iu t the grting interfce. In the ce ζ the ode h 47

4 cut-off thicne, becue the ode cnnot evolve into TEM wve upported by the bcground. For coprion, the etl lb with ζ =.L i lo iulted with eigenvlue olver of HFSS oftwre. Fig. 5 how the gnitude of the electric field of nd ode for t =.6 µ. It i oberved tht the ode field re loclized t the grting interfce but the field of ode re loclized t the flt interfce therefore the propgtion contnt of ode i enitive to the height of grting or urfce roughne. Moreover, with increing the lb thicne the propgtion contnt of nd ode converge towrd two different vlue. Mode Mode Figure 5: Mgnitude of electric field of yetric nd ntiyetric ode The diperion curve tht i obtined fro HFSS iultion i hown in Fig.6. The reult re in good greeent with reult of reduced Ryleigh eqution ethod. I { / R e { / Thicne ( µ Thicne (µ Figure 6: propgtion contnt / in ter of ilver lb thicne rel prt iginry prt for ζ =.L uing HFSS Figure 7: unit cell for iulting etl trip wveguide with one grting boundry. Strip wveguide.1 Nuericl ethod ipleenttion There i no nlyticl ethod for the nlyi of the effect of grting on the etl trip with finite width ebedded in lole dielectric teril, o only HFSS i eployed. With the ue of eigenolver odule in HFSS the rel nd iginry prt of the propgtion contnt cn be clculted [8].The ethod tht i ued in thi pper differ lightly with [8]. In [8] the perittivity of loy ilver i function of frequency deterined by Johnon nd Chrity [9]. But in thi iultion in order o rein conitent with Berini reult, only ingle λ =.6 µ, thu Silver i frequency i conidered ( defined with perittivity ( 19 j.5 ε ω = + nd dielectric hot with contnt perittivity ε r 1 = 4. The tructure i illutrted in Fig.1, the yetry cn be ued to reduce the coputtionl doin, therefore, only hlf of the tructure i nlyzed hown in Fig.7. In the direction of propgtion the periodic boundry condition re ued. Periodic boundry condition enure tht the field in the lve plne differ fro the field in the ter plne within phe dely ϕ o tht the field coponent tify g ( z d ep ( iϕ g ( z r + = where d i the ditnce between the two plne. For ech vlue of ϕ eigenolver provide cople eigen frequency for ech ode. The rel prt of the coputed eigen frequency hould be equl to frequency f tht correpond to λ =.6 µ. Therefore, we ut weep the vlue of ϕ nd deterine ϕ tht tifie the bove condition. Hving deterined ϕ, the rel prt of propgtion contnt cn be found fro zr = ϕ / d. The iginry prt of propgtion contnt i clculted ccording to zi = I { f / vg where v g i the group velocity given by ( Re{ / =. A entioned in [] d hould be vg d f dzr choen ufficiently ll for non-periodic proble o tht the folded diperion curve in the Brillouin zone boundry occur t frequency beyond the pectrl rnge tht i conidered. Creful review of the field ditribution i generlly required to identify the ode of interet tht hve their field loclized cloe to the etllic trip. 471

5 .6 Figure 8: unit cell for iulting flt trip wveguide of Berini. Nuericl reult To copre our reult with [1] the flt ilver trip i lo iulted nd oe of the reult reported by Berini in [1] re reproduced. The tructure i hown in Fig.8. Utilizing the yetry of the tructure, only one qurter of the trip i iulted. The o clled E or M yetric boundrie in HFSS re ued to obtin different ode of trip wveguide. The ode re divided into four ctegorie depending on the yetry of their field nd re lbelled ccording to the noenclture propoed by Berini. Therefore, pir of letter i ued to indicte tht E y i yetric or ntiyetric with repect to verticl or horizontl plne, repectively. The upercript indicte the nuber of i of E y long the lrger dienion nd when there i no iu no upercript i ued. Here only the bound ode i conidered o the ubcript b which en tht the ode re bound to urfce i eliinted. Siultion reult for W = 1 µ nd d =. µ re depicted in Fig. 9. In the region tht thicne of the fil i ll t <.4 µ the ode denity becoe very lrge, with ode pced cloely in frequency thu identifying the olution of interet becoe difficult o the reult i given for t.4 µ. The,,, nd ode re the fundentl ode upported by the tructure. The reult hve good greeent with Berini reult. The field relted to the ode tht re ntiyetric with repect to horizontl plne penetrte ore into the loy ilver thn yetric ode, o thee ode re highly ttenuted. Siultion reult of the tructure tht i depicted in Fig.7 re hown in Fig.1. Preter of the inuoidl grting re L = 5 n nd ζ =.L. Coprion of Fig.9 nd Fig.1 how tht propgtion contnt of ll i ode re increed due to the urfce roughne. Iginry prt of the propgtion contnt of thee ode re lo increed becue of the grting on the etl trip. Thi en tht with increing roughne of the urfce the bound SPP wve becoe ley nd ttenution i increed. I { z \ Re { z \ ,, Thicne ( ,, Thicne ( Figure 9: Siultion reult of flt trip wveguide of Berini rel prt iginry prt of / z 1 The field ditribution of, ode t t =.14 µ i hown in Fig.11 nd Fig.1. A entioned in [1] the ode upported by etl fil of finite width re in fct uper ode which re creted fro coupling of edge nd corner ode upported by ech etl/dielectric interfce defining the tructure. In thi tructure, Becue of the eitence of grting on one interfce, the upper ode y be creted fro the coupling of diiilr interfce ode. The coupled ode hould hve iilr propgtion contnt nd hre the field yetry with repect to the center verticl i. In Fig. 11 for intnce, it i een tht the grting edge ode h two eter nd i of higher order thn the flt edge ode which h no eteru. In thi tructure, the grting interfce h higher phe contnt thn the flt interfce. Becue uperode i creted fro coupling of edge ode hving iilr propgtion contnt, it hould be epected tht in thi tructure different edge ode y couple to crete uperode. Higher-order ode hve, in generl, ller vlue of phe contnt copred to lower-order ode, o in thi tructure ll uperode re copried of grting edge ode of the e order or higher thn the flt edge ode, hown in Fig. 11 nd Fig

6 Re{ z \ I{ z \ ,, Thicne ( ,, Thicne ( Figure 1: Siultion reult of etl trip wveguide with one grting boundry rel prt iginry prt of / z 4. Concluion In thi pper the effect of urfce roughne on SPP wve tht propgte on two tructure, loy etllic lb nd loy etl trip re invetigted. The reult how tht the roughne tht i odelled by grting of inuoidl profile ffect the ode tht re ntiyetric with repect to horizontl plne. The rel nd iginry prt of the propgtion contnt of thee ode incree which en increed ttenution due to the roughne. In the ce of lb wveguide the reult of reduced Ryleigh eqution ethod re coptible with HFSS reult. Reference [1] P. Berini, Plon-polriton wve guided by thin loy etl fil of finite width bounded ode of yetric tructure, Phy. Rev. B, vol. 61, No. 15, pp , Apr.. [] P. Berini, Plon-polriton ode guided by etl fil of finite width bounded by different dielectric, Opt. Epre, vol. 7, No. 1, 6 Nov.. [] P. Berini, Plon-polriton wve guided by thin loy etl fil of finite width bounded ode of yetric tructure, Phy. Rev. B, vol. 6, 1 Mr.1. [4] M. M. Auto, G. A. Fri, nd A. A. Mrdudin, Surfce polriton on etl fil with grting urfce, Surfce Science, vol.167, No.1, pp , Mrch [5] Bernrdo L, D. L. Mill, nd A. A. Mrdudin, Surfce polriton on lrge-plitude grting, Phy. Rev. B, Vol.,No. 1, pp , My [6] C. Kuo, nd M. Moghdd, A Theoreticl Anlyi of bc cttering enhnceent due to urfce plon fro ultilyer tructure with rough interfce, IEEE Trn. Antenn Propgt, vol. 56, NO. 4, Apr. 8. [7] F. Toigo, A. Mrvin, V. Celli, nd N.R. Hill, Opticl propertie of rough urfce: Generl theory nd ll roughne liit, Phy. Rev. B, vol. 15, NO.1, Jun [8] A. Degiron, nd D. R. Sith, Nuericl iultion of longrnge plon, Opt. Epre, vol. 14, No. 4,pp , 6 [9] P. B. Johnon, nd R. W. Chrity, Opticl Contnt of the Noble Metl, Phy. Rev. B, vol. 6, No.1, pp , De Figure 11: field ditribution of of grting trip with t =.14 µ 1 Figure 1: field ditribution of of grting trip with t =.14 µ 47