Geens functon ntegal equaton methods fo plasmonc nanostuctues (Ph Couse: Optcal at the Nanoscale) Thomas Søndegaad epatment of Phscs and Nanotechnolog, Aalbog Unvest, Senve 4A, K-9 Aalbog Øst, enma. Intoducton to Geens functons n electomagnetcs Consde the tas of solvng an nhomogeneous opeato equaton of the fom θφ =, () whee s the opeato, s a souce tem, and s the functon we want to calculate. Ths tpe of equaton can be staghtfowadl solved when a Geens functon [,] of the opeato ests and s nown. A Geens functon g of a gven opeato s a soluton to the equaton θg = I, () whee I s the unt opeato, and consequentl a soluton to Eq. () s gven staghtfowadl b φ = g. (3) Othe solutons can be obtaned b addton of a soluton to the homogeneous equaton (Eq. () wth =) o b usng Eq. (3) but wth anothe choce of Geens functon. One appoach to the constucton of a Geen s functon n the case that s a Hemtan opeato wth a complete set of othonomal egenfunctons φ n satsfng θ φ n = n φ n s g = φ n φ n n (4) n A convenent method fo electomagnetcs puposes of addng a homogeneous soluton s to use Eq. (4) whee we add a ve small magna pat *s to the denomnato and obseve the tanston of s ±, and appl lm s = P πδ(). ependng on the sgn of s when tang the lmt we obtan ethe the +s advanced o the etaded Geen s functon. Geens functon elated to the electostatc potental: A well-nown nhomogeneous dffeental equaton n electostatcs fo the electc potental s φ() = ρ()/ε, (5) whee s the chage denst dstbuton, and ε s the vacuum pemttvt. The equall well-nown patcula soluton to ths equaton s gven b
φ() = 4π ρ( )/ε d 3. (6) Although the concept of the Geens functon s usuall not mentoned n elaton to the method of calculatng the potental n Eq. (6) ths equaton s an eample of Eq. (3) wth the Geens functon g, = 4π, whch satsfes g, = δ,.e. n ths case δ s the unt opeato. Geens functon of the opeato of the, and 3 Helmholt equaton: The macoscopc monochomatc electc feld E = E()e ωt geneated b a monochomatc cuent dstbuton J = J()e ωt satsfes the Helmholt equaton + E() = J(), (7) whee = ω/c, and c s the vacuum speed of lght. In addton, f we assume that thee ae no fee chages, the electc feld must satsf E =. A patculal smple stuaton s the one-dmensonal case of popagaton n onl one dmenson, e.g. along the -as, wth feld and cuent vectos pontng pependcula to the -as, and vang onl along the -as, e.g. E() = E() and J() = J(), n whch case the equement E = s automatcall satsfed, and Eq. (7) educes to d d + E() = J(). (8) The elevant Geen s functon n ths case must be a soluton to d d + g, = δ( ), (9) and t s staghtfowad to show that a soluton s gven b g, = e. () Note that ths patcula Geen s functon also satsfes the condton that t behaves as waves popagatng awa fom the souce pont,.e. ths s the etaded Geen s functon. Thus, solutons to Eq. (7) obtaned b E = g, J( )d () satsf the adatng bounda condton,.e. that the waves geneated b the souces J should popagate awa fom the souces. Fo popagaton n two dmensons and s-polasaton,.e. the electc feld (and cuents) s pependcula to the plane of popagaton, the condton E = s also automatcall satsfed. In ths case the Helmholt equaton becomes + + E(, ) = J(, ), () The Geen s functon elated to the opeato actng on E s gven b
g,,, = 4 H (), (3) whee H () s the Hanel functon of the second nd and ode eo. The soluton of nteest s E, = g,,, J(, )d d. (4) Fo completeness we wll also menton the Geen s functon fo the 3 scala Helmholt equaton g, = ep ( ), (5) 4π whch satsfes + g, = δ( ). (6) In the 3 case the wave equaton fo the electc feld can be wtten + E = ωμ J + E, (7) n whch case the esult fo the scala 3 case can be appled to constuct the followng equaton fo E E = g, ωμ J d 3 g, E d 3. (8) Howeve, fom Eq. (7) follows b tang the dvegence on each sde that E = ωμ J, (9) whch when beng combned wth Eq. (8) leads to E = g, ωμ J + g, ωμ J d 3. () Ths can be ewtten as E = ωμ G, J d 3 () n tems of the dadc Geen s tenso G, = I + g,, () whch s a soluton to G, + G, = Iδ( ). (3) Eecse : Constuct the Geen s functon (5) usng Eq. (4). 3
. Scala Geens functon doman ntegal equaton methods fo scatteng calculatons We wll now use the esults fom the pevous secton to constuct ntegal equatons that can be used fo scatteng poblems. In ode to llustate the pncple we wll stat wth the smple case of wave popagaton n one dmenson, and consde eflecton, tansmsson and nea felds of a plane wave beng ncdent on a delectc bae fom the left. = ef = b = ef W Fg. : A plane wave ncdent on a delectc bae fom the left s patall eflected and patall tansmtted. The ncdent plane wave s on the fom n ef E ( ) e. (4) The ncdent feld s a soluton to the wave equaton of a homogeneous mateal wth delectc constant ef : ef E. (5) We ae nteested n solvng the somewhat moe complcated wave equaton on the fom E, (6) whee the delectc constant now depends on the poston and assumes the same value as the bacgound mateal fo values of outsde the bae, and assumes the value of the bae b fo values of nsde the bae. The equatons (5) and (6) can be combned nto E E ef ef E, (7) and b teatng the ght-hand sde as a souce tem smla to the cuents consdeed pevousl we ave at the ntegal equaton fo the electc feld E( ) E ( ) g(, ) ef E( ) d, (8) wth the Geen s functon g, = n ef e n ef, (9) 4
whee n ef = ε ef s the delectc constant of the efeence medum. The ntegal equaton (8) can be solved numecall b dscetng the bae nto N dscete elements n whch the electc feld s assumed constant. The samplng ponts,,... N fo the N elements mght be taen at the cente of the bae, and we ma efe to the coespondng samplng values of the feld as E,E,...E N. Ths esults n the dscete lnea sstem of equatons E, g g, g ef E E,. (3) Note that t s suffcent to dscete onl the egon of the bae snce the ntegand n (7) vanshes fo ponts outsde the bae. Also note that the scatteed feld E-E satsfes the adatng bounda condton, namel that outsde the bae the scatteed feld popagates awa fom the bae. Once the feld nsde the bae has been calculated the equaton (8) can be used to calculate the feld at all othe postons. Eecse : Calculate numecall the electc feld nsde a delectc bae wth delectc constant (a) =4., (b) =4.-., and (c) =4.-4.. In the latte two cases the magna pat of the delectc constant epesents absopton. Use the fee-space wavelength.8m and the bae wdth.5m. (Ths patcula eample was boowed fom [3]). The case of scatteng of a plane s-polaed wave b e.g. a ectangula o clndcal bae, as llustated n Fg., follows the same pncples: (a) = ef (b) = ef = b = b Fg. : A plane wave popagatng n the decton defned b the angle s ncdent on a (a) ectangula o (b) clndcal delectc bae. The ncdent plane wave mght be on the fom cos sn E (, ) e, (3) and the total electc feld can be calculated b solvng the Geen s functon ntegal equaton E(, ) E (, ) g(, ;, ), ef E(, ) d d, (3) 5
whee hee the Geen s functon s gven b () g, ;, H nef. (33) 4 Compaed to the case the numecal tas s slghtl moe dffcult due to the sngulat of the Geen s functon. The ntegal equaton mght be dsceted nto e.g. N squae shaped elements wth aea A og cente n (, ), (, ),... ( N, N ). Agan the coespondng samplng values of the feld s denoted E,E,...E N, whch esults n the lnea sstem of equatons: E E ef E A, g g, ;, d d,,, g A element In the case when t s also possble to ust use g g, ;,. (34). As an appomaton when = we ma consde a ccula element wth the same aea as the squae shaped element, n whch case the adus of the ccle s a A/. Ths esults n g A a Eecse 3: H 4 () ( ) d a 4 a H () ( ) d () ah ( ) a a. (35) Calculate the absolute value of the feld E nsde and outsde a ectangula bae of wdth.5m, heght.m, and delectc constant =4.. The ncdent feld s a plane wave wth the wavelength =633nm, and the angle of ncdence s (a) and (b) 45. ef =. Calculate the feld fo a plane wave ncdent on a clnde of damete.5m. 3. Geen s tenso volume ntegal equaton method In ths secton we wll consde the case of a 3 scatteng poblem and also the possblt of a moe comple efeence stuctue than ust a homogeneous delectc. The equaton that we would le to solve s the vecto wave equaton fo the electc feld E ( ) E -, (36) whee s the elatve delectc constant of the total stuctue consstng of a efeence stuctue and one o moe scatteng obects. Late we wll consde a plana gold suface o gold flm as a efeence stuctue and a numbe of scattees placed on the suface. The bounda condton s that outsde the scatteng obects the soluton must be the sum of a gven ncdent feld and a scatteed feld, whee the latte popagates awa fom the scattees. The ncdent feld E must be a soluton n the case wthout scatteng obects,.e. ( E - ef ), (37) whee ef s the elatve delectc constant fo the efeence stuctue, e.g. the plana gold suface wthout gold scattees on the suface. 6
7 A Geen s tenso G fo the efeence stuctue s defned as a soluton to the equaton ) (, ) ( ef I G -, (38) whee s the ac delta functon, and I s the unt tenso. Compaed to pevous sectons we allow ef to depend on the poston. Smla to pevous sectons we can ewte Eqs. () and () nto E E E ) ( ) ( ) ( ef ef -, (39) whch esults n the vecto ntegal equaton, 3 ef d E G E E. (4) The scatteed feld component wll satsf the adatng bounda condton fo a pope choce of Geen s tenso. It s possble to e.g. use as a efeence stuctue the plana metal suface, a metal flm, o a moe comple stuctue, wthout addtonal numecal costs, f the efeence stuctue Geen s tenso G s nown. Onl n the case of a homogeneous efeence medum can we obtan a smple analtcal epesson fo the Geen s tenso. If the delectc constant n ths case s ef ()=n, whee n s the efactve nde (= n), the Geen s tenso G=G s gven b, 4, I G g e g g. (4) Ths leads to the followng analtc and hghl sngula Geen s tenso 3 3 I G g. (4) Fo a efeence stuctue consstng of plana metal-delectc nteface o metal flm the Geen s tenso can be constucted e.g. b epandng the homogeneous medum Geen s tenso n tems of an n-plane wave numbe, and fo each add a tem fo > whch accounts fo eflecton, and b constuctng a tansmtted tem fo < such that the electomagnetcs bounda condtons ae fulflled at the nteface [4], whch fo souce and obsevaton ponts wth, > esults n,, 4 ; ) ( 3 S e J J J J J J d s p p p ρ ρ G, (43) whee p and s ae the Fesnel eflecton coeffcents fo p- and s-polaed waves, espectvel, =( - ) ½ wth Im( ), and ae the poectons of and on the -plane, =, and, and ẑ ae
coodnate unt vectos n a clndcal coodnate sstem centeed at,.e ρ ρ / ρ ρ. J s the Bessel functon of ode, and means dffeentaton wth espect to the agument. The Fesnel eflecton coeffcents ae functons of. One appoach to solve Eq. (4) s to dscete the scattee nto N volume elements whee the feld and the delectc constant ae assumed constant. Ths leads to a lnea sstem of equatons of the fom E E, G ef, E, (44) whee G V 3, d G. (45) The latte ntegal ove the Geen s tenso s athe dffcult n the case whee = due to the sngulat of the Geen s tenso. In patcula, f an ecluson volume s placed aound the sngulat when cang out the ntegal (45), and the lmt of the ecluson volume gong to eo s obseved, the esult wll depend on the shape of the ecluson volume. Fo ve small volume elements and = the epesson (45) onl depends on the shape of the volume element V and not the se. It has been tabulated fo vaous shapes n Ref. [5]. One method of handlng the sngulat s to convet the volume ntegal nto a suface ntegal [6], whee the sufaces ae placed awa fom the sngulat. Fo the homogeneous medum Geen s tenso ths s done b usng Eqs. (4) and then applng Geen s theoem esultng n V n In g s d s G, (46) I whee V s the suface of volume element, and n s the outwad suface nomal vecto. The appoach of Eqs. (4) and (4) s equvalent to the scete pole Appomaton method (A) ntoduced b Pucell and Pennpace n 973 [7] f we estct samplng ponts and volume elements to be placed on a cubc lattce, and teat each volume element as a pont dpole wth a polaablt equvalent to a small sphee wth the same delectc constant and the same volume as the cubc volume element t has eplaced. In ths sense the A s equvalent to usng, V G G,. (47) Regadng the case = Pucell and Pennpace used the equvalent of G =-I/(3 ef ). Late t was emaed e.g. b ane [8] that n ode fo a (delectc) dpole to not oscllate eactl n phase wth the ncdent feld t s necessa to nclude an magna pat beng equvalent to usng G =-I(/3+ 3 V/6)/ ef. Both epessons ae good appomatons to Eqs. (45). We ma notce that n the case of a homogeneous efeence medum the Geen s tenso depends onl on the dffeence -, n whch case Eq. (4) s a convoluton ntegal. In the case of the A, o when usng volume elements of the same se and shape placed on a lattce, Eq. (4) taes the fom of a dscete convoluton,.e. 8
,, E,, E,,, G,,,, ef E,,. (48) Rathe than solvng Eq. (48) usng Gaussan elmnaton, LU-decomposton etc. and smla schemes wth calculaton tme popotonal to N 3, t mght seem advantageous to solve the equaton usng an teatve appoach whee a tal vecto s optmed untl a convegence ctea s satsfed (see e.g. [8]). Ths pocedue nvolves man mat-vecto multplcatons (convolutons) of the fom n Eq. (48). B calculatng the convoluton b fst applng the Fast Foue Tansfom (FFT), multplng n ecpocal space, and then applng the FFT once moe the calculaton tme (due to the FFTs) can be educed to beng popotonal to NlogN f the numbe of gd ponts along each as s a powe of. Futhemoe, the stoage fo the mat s educed to scale as N. Fom these consdeatons t appeas that both n tems of calculaton tme and compute memo equements t s desable to use e.g. cubc volume elements of the same se and shape placed on a cubc lattce (o the A). In the case of a plana metal-delectc nteface efeence stuctue Eq. (48) wll also contan a summaton ove G S dependng on + nstead of -. Ths case can also be caed out usng the FFT wth esultng calculaton tme ~ NlogN. Actuall, one mao dawbac of usng cubc elements of the same se and shape s that the suface of cuved stuctues wll be epesented wth a sta-cased suface, whch seems to esult n slow convegence, o wose, as we shall see an eample of n the net secton. 4. Geen s tenso aea ntegal equaton method In the case of popagaton n onl and p-polaaton we ae faced wth the same complet as n the pevous secton, namel that the vecto components of the feld ae coupled, and a vecto ntegal equaton s equed. We assume that the stuctue and electomagnetc felds ae nvaant along the -as, and popagaton s n the -plane. The electc feld s gven b E ( ) E ( ) E ( ),. The ntegal equaton s ecept fo a d nstead of d 3 equvalent to the 3 ntegal equaton,.e. E E G, E d ef. (49) The Geen s tenso s equed hee, and fo a homogeneous efeence medum t can be calculated analtcall,.e. G (), G, Ig,, g, H. (5) 4 scetaton and numecal soluton follows the same pncples as consdeed pevousl. We wll now gve an eample of the effect of sta-casng on numecal convegence. The esults fo a plane wave ncdent on a ccula clnde usng two dffeent dscetaton methods ae pesented n Fgs. 3 and 4. Fo a modest clnde delectc constant of 4, clnde damete nm, and wavelength 7nm, convegence s possble usng squae-shaped aea elements wth a sta-cased epesentaton of the clnde suface (Fg. 3, M). The calculated feld along the - and -aes (ogo at the clnde cente) conveges towads the analtcal esult as the se of dscetaton elements deceases. Futhemoe, the calculated feld values ae close to the eact esult. Wth the same dscetaton elements nsde the clnde and specal elements nea the suface followng closel the clnde suface (Fg. 3, M), convegence s dastcall mpoved 9
usng pactcall the same numbe of elements. In ths case the elements of tpe M leads to the coect esult but much slowe compaed to usng the elements of tpe M. (a) (b) Fg. 3: Total feld along the (a) -as and (b) -as though the cente of a ccula clnde wth delectc constant 4 (bacgound delectc constant ) and damete nm. A p-polaed plane wave wth wavelength 7nm s ncdent along the -as. M: calculaton usng onl squae aea elements. M: the same squae aea elements ae used nsde the clnde, and specal elements ae used nea the suface followng closel the actual clnde suface (see nset). s the sde length of the squae aea elements. The sold lne s the analtcal and eact esult. A much moe dffcult case s when the clnde s made of slve wth the delectc constant =-.99-.395 (Fg. 4). The method M conveges ve slowl - f at all. The calculated feld s g-aggng, and t can even have the wong sgn and devate fom the analtcal esult b seveal hunded pecent. Wth method M, howeve, convegence s obtaned. Wth the dscetaton se =nm thee ae clea devatons fom the eact esult. Howeve, wth =.5nm the numecal esult s pactcall dentcal wth the analtcal esult. Compang Fgs. 3 and 4 t appeas that the suface descpton of a scattee becomes nceasngl moe mpotant wth nceasng contast between mateals. Fo metal stuctues wth cuved sufaces convegence s pactcall not possble usng onl squae elements. A dawbac wth espect to method M s that the dsceted veson of Eq. (49) s no longe a dscete convoluton. Theefoe the FFT cannot be appled (to the same etent) to educe the calculaton tme. (a) (b) Fg. 4 Same stuaton as n Fg. 4 ecept that the clnde delectc constant s =-.99-.395.
These eamples llustate that fo a hgh contast between the nvolved delectc constants the numecal teatment of the suface of the stuctue eques seous consdeaton. Smla to the 3 case the Geen s tenso fo a metal suface o metal flm can be constucted b addng wave solutons to the homogeneous medum Geen s tenso such that bounda condtons ae satsfed at the metal sufaces. The scala homogeneous medum Geen s functon can be epanded n the followng wa g, cos e d,, Im( ). (5) An epanson of G n n-plane wave numbes s then obtaned b applng the opeato n Eq. (5) to Eq. (5). Fo each a eflecton tem must then be added, and elsewhee a tansmsson tem must be constucted such that the electc feld bounda condtons ae fulflled fo each. In the case of the souce pont placed above the metal flm ( >) the Geen s tenso fo > becomes G(, )= G (, ) + G S (, ), whee G S, sn ( ) cos ( ) P e d,,(5) 5. Geen s functon suface ntegal equaton method In ths secton we wll consde the Geen s functon suface ntegal equaton method In the pevous secton we found that the teatment of the suface of a stuctue s patculal mpotant when the contast n mateal constants s hgh. As the numecal poblem n the SIEM s educed to fndng felds at the scattee suface the method allows a ve accuate descpton of the stuctue suface (no sta-casng), and futhemoe dscetng the suface compaed to the aea eques fa less samplng ponts. A mno dawbac s that the ntegals n the SIEM ae not convoluton ntegals, and the Fast Foue Tansfom cannot be appled fo fast evaluaton of ntegals as could be the case fo the doman ntegal equaton. The scatteng stuaton we wll consde s llustated n Fg. 5. We wll consde agan p-polaed lght popagatng n the -plane (magnetc feld along the -as, and electc feld n the -plane). The felds and the scatteng obect wll be teated as nvaant along the -as. The magnetc feld s gven b H ( ) H( ),. Outsde the scattee the feld can be dvded nto the ncdent and scatteed feld components,.e. H( ) H ( ) H ( ). scat Fg. 5 Illustaton of scatteng of an ncdent magnetc feld H b a stuctue wth delectc constant suounded b a medum wth delectc constant. The feld at postons nsde and outsde the scattee s gven b the followng suface ntegals: dl H Hs n g (, s) g (, s) n H s, H, (53)
H dl Hs n g (, s) g (, s) n H s, (), (, whee the Geen s functon g, ) H / 4 The subscpt n, (54), and n s the outwad suface nomal vecto. n H ndcates that ths s the nomal devatve of the magnetc feld appoachng s n wth / s the suface fom medum. In Eq. (54) the electomagnetcs bounda condtons have been appled to eplace H n H s. Eq. (54) follows dectl fom applng Gauss theoem to ts ghthand sde and tansfomng the cuve ntegal nto an aea ntegal and then applng ) g (, ) ( ) and ( ( )) H( ). Eq. (53) follows fom consdeng an equaton (,, smla to Eq. (54) fo the total feld ove a doman bounded b the suface of and a ccula suface C placed at nfnt. Fo a poston placed fa fom C we ma advantageousl splt the suface ntegal ove C nto an ntegal ove H and an ntegal ove H scat. The fst ntegal gves H (), and the latte vanshes snce H scat satsfes the adatng bounda condton. We note that H scat = H-H does n fact satsf ths bounda condton at C due to the natue of the Geen s functon chosen (see Eq. (53)). Befoe Eqs. (53,54) can be appled we have to fnd the feld and ts nomal devatve at the suface. Self-consstent equatons can be obtaned fom Eqs. (53,54) b lettng appoach the suface fom ethe sde, n whch case we deal wth the sngulat of n g (, ) n the lmt of appoachng a pont s on the suface b ewtng the ntegals, s as pncpal value ntegals, whee the sngulat fo a smooth suface gves a contbuton of H(s) / dependng on fom whch sde the suface s appoached,.e. H s P Hs n g( s, s) g( s, s) n Hs H s dl, (55) H dl s P Hs n g ( s, s) g ( s, s) n H s, (56) whee P efes to the pncpal value. These equatons ae dsceted such that the suface s dvded nto a fnte numbe of segments on whch H and n H ae assumed constant. Regadng the ntegals n Eqs. (55,56) the vaaton of n g ( s, ) and g s, ) can be taen nto account b subdvdng each segment n, s, ( s e.g. subsegments, whch also allows descbng moe accuatel the actual shape of the suface. The E ) /( ) H( ) and Eqs. (53,54). A few eamples llustatng electc feld s obtaned usng (, convegence of the method appled to metallc nanostuctues ae pesented n Fg. 6.
(a) (b) Fg. 6 (a) Total feld along the - and -aes though the cente of a ccula slve clnde of damete 5nm fo an ncdent p-polaed plane wave of wavelength 7nm popagatng along the -as. The delectc constant of slve used s Ag =-.99-.395. Ponts: SIEM usng 5 samplng ponts. Sold lnes: eact calculaton. (b) Scatteng coss secton vs wavelength calculated wth the SIEM fo a p-polaed plane wave ncdent on a ght-angled slve tangle wth base nm wth shap cones combned wth an even dstbuton of samplng ponts, and wth.5nm oundng of cones combned wth an uneven dstbuton of samplng ponts favoung cones. We obseve fo a plane wave wth wavelength 7nm ncdent on a ccula slve clnde wth damete 5nm that the esult obtaned wth the SIEM usng onl 5 samplng ponts s ecellent, whch s elated to the accuate ncopoaton of the clnde suface n the numecal scheme. On the scale of Fg. 6(a) thee s no notceable dscepanc between the numecal and the eact analtc calculatons. A moe dffcult eample (Fg. 6b) s scatteng of a plane wave b a ght-angled slve tangle of base nm. We consde both oundng the cones b.5nm wth an uneven dstbuton of samplng ponts fo bette esoluton nea the cones, and not oundng the cones and usng an even dstbuton of ponts ove the tangle suface. In the fst case thee s onl a small dffeence between usng 5 and samplng ponts, and the esult s n quanttatve ageement wth Ref. [9]. The 5 ponts beng suffcent hee should be compaed to usng the doman ntegal equaton method and seveal thousand tangula aea elements n Ref. [9]. When the cones ae not ounded we notce that easonable convegence can stll be acheved fo some wavelength egons usng 5- ponts. Howeve, fo wavelengths appoachng the esonance peas at ~36nm and ~45nm convegence s slow. In Ref. [9] t was shown that the feld at these wavelengths s patculal stong at the cones of the tangle. Smla to the othe methods consdeed t s possble to tae account of a plana suface n the efeence medum b usng a Geen s functon that taes nto account eflecton fom the suface []. 3
6. Results obtaned fo plasmonc nanostuctues usng Geen s functon ntegal equaton methods In ths secton we wll consde modelng of plasmonc nanostuctues elated to thee tpes of plasmonc suface waves bound to and popagatng along plana metal-delectc ntefaces (Fg. 7a), namel the suface plasmon polaton (SPP) popagatng along a sngle metal-delectc nteface, the long-ange SPP (LR-SPP) and the shot-ange SPP (SR-SPP) bound to and popagatng along a thn (-nm) metal flm []. The thee tpes of SPPs ae p-polaed waves,.e. when the popagate along the postve -as as n the llustaton n Fg. 7a the electc feld onl has an - and a -component. At least fo SPPs and LR-SPPs, and often fo SR-SPPs at postons outsde the metal, the electc feld s domnated b the -component (E ) llustated n Fg. 7a. The popagaton length s lmted b ohmc losses. It depends on the tpe of SPP and the metal flm thcness. Fo SPPs and wavelength 55nm the ntenst s e.g. educed b a facto e afte popagatng 6m along an nteface between gold and a polme wth efactve nde.54. Fo LR-SPPs and a 5nm gold flm the coespondng length s 6mm, equvalent to thousands of wavelengths, whch s enough to mae these waves nteestng fo ntegated optcs [,3]. LR-SPPs ae, howeve, loosel bound wth most of the feld located outsde the metal flm. The SR-SPP on the othe hand s stongl bound but the popagaton length s lmted to a few fee-space wavelengths. B placng obects at the metal ntefaces, o va flm temnatons, the SPP waves can be manpulated va scatteng pocesses, esultng n plasmonc devces e.g. fo communcaton on an optcal chp, sgnal pocessng and sensng [4] (Fg. 7b). We wll consde clndcal gold scattees aanged on a heagonal lattce on a gold suface (SPP bandgap stuctue) [,5-7], peodcall epeated gold dges on ethe sde of a thn gold flm (LR-SPP gatng) [8-], and a flm of shot length,.e. a metal stp (SR-SPP esonato), whee SR-SPPs popagatng bac and foth fom standng wave esonances [-4]. SR-SPPs ae conta to LR-SPPs effcentl eflected at stp temnatons as the ae stongl bound. (a) (b) Fg. 7 (a) Illustaton of suface plasmon polatons (SPPs) bound to and popagatng along a plana metal suface, and long- and shot-ange SPPs bound to and popagatng along a thn metal flm. (b) Eamples of plasmonc nanostuctues: SPP bandgap stuctue wth clndcal metal scattees placed on a plana metal suface on a heagonal lattce, LR-SPP gatng wth metal dges placed smmetcall on each sde of a thn metal flm, and a SR- SPP esonato wth thn metal stps suppotng standng waves of SR-SPPs popagatng bac and foth along the - as. Eample fo a SPP bandgap stuctue: It was pevousl shown [7] that gold scattees of heght 5nm and adus 5nm aanged on a heagonal lattce wth lattce constant 45nm on a plana gold suface, whee the suoundng delectc s vacuum, ehbt a bandgap fo SPP waves aound the wavelength 8nm. In Fg. 8a an eample s gven of the feld magntude showng that the SPP beam can be edected b 3 usng a bent channel of mssng scattees n the SPPBG stuctue. The whte dots epesent the scattees. The tansmsson calculaton though the bend fo thee dffeent bend confguatons (Fg. 8b) shows that movng thee scattees n the bend egon easl leads to an mpovement n tansmsson b a few db. Tansmsson was evaluated fom the feld ntenst n 4
a small bo placed nea the et of the bent wavegude. The athe poo oveall tansmsson s due to out-ofplane scatteng and absopton. (a) (b) Fg. 8 A plana gold suface s consdeed wth an aa of clndcal gold suface scattees of heght 5nm and adus 5nm placed on a heagonal lattce (lattce constant 45nm). A channel wth a 3 bend has been ntoduced b omttng scattees. The gold suface and gold scattees ae suounded b vacuum. A SPP beam of Gaussan nplane pofle s ncdent on the left-sde of the channel. (a) Electc feld magntude E 3nm above the gold suface. (b) Tansmsson spectum fo the bend fo thee dffeent aangements of scattees n the bend egon. Ths eample was calculated wth the volume ntegal equaton method [7]. Eample fo a LR-SPP gatng: A calculaton fo a LR-SPP wave popagatng along a 5nm gold flm and beng ncdent on an aa of gold dges s pesented n Fg. 9. The 6 gold dges on each sde of the flm (LR-SPP gatng n Fg. b) ae nm hgh, 3nm wde, and placed on a lattce wth lattce constant 5nm. The gold flm and dges ae suounded b a delectc wth efactve nde n=.543. Tansmsson (T) and eflecton (R) nto LR-SPPs, and out-of-plane scatteng (OUPS), ae pesented n Fg. 9a. Notce the clea Bagg eflecton pea fo wavelengths aound 543nm=n and the coespondng tansmsson dp. Whle the bandgap of the gatng s easl dentfable fom the eflecton pea t s moe dffcult to dentf the bandgap fom the tansmsson dp, whch s elated to a elatvel hgh out-of-plane scatteng loss fo wavelengths shote than the bandgap ange of wavelengths. The gatng ma suppot leaage fee Bloch waves bound to the peodc metal stuctue fo wavelengths longe than the bandgap but ths s not the case fo the shote wavelengths, whee the gatng effcentl couples lght nto out-of-plane popagatng waves [8]. A calculaton of the feld magntude above the metal flm and dges (Fg. 9b) llustates fo the wavelength 55nm that the out-of-plane scatteng ognates to a lage etent fom the couplng of the LR-SPP nto the gatng fo = whee the gatng stats (t ends at =8m). 5
(a) (b) Fg. 9 LR-SPP gatng wth a 5nm thc gold flm wth 6 gold dges of heght nm and wdth 3nm placed on a lattce wth lattce constant 5nm smmetcall on each sde of the gold flm. The efactve nde of the suoundng medum s.543. A LR-SPP popagatng along the flm s ncdent on the aa of dges. (a) Reflecton (R) and tansmsson (T) nto LR-SPP waves, and out-of-plane scatteng (OUPS). The powe s nomaled to the powe of the ncdent LR-SPP. (b) Nea-feld mage of the magntude of the electc feld n the -plane above the gold flm and dges fo the wavelength 55nm. Whte dashed lnes ndcate begnnng and end of the gatng. Ths eample was calculated wth the aea ntegal equaton method [8]. Eample fo a metal nano-stp esonato: An eample s gven n Fg. fo a etadaton-based plasmon esonance nvolvng shot-ange (and slow) SPPs. Smla stuctues wee ecentl nvestgated n Refs. [-4]. Scatteng and nea felds ae shown fo a plane wave ncdent on a slve metal stp of heght nm and wdth 84nm (=nm) and two slve stps of the same total wdth sepaated b a small 5nm gap (=5nm). The plane wave s popagatng along the - as (nset Fg. a). In the case of the sngle slve stp (=nm) a scatteng esonance ests at the wavelength =54nm. The esonance s elated to standng waves of fowad and bacwad popagatng (along the -as) SR-SPP waves [-4]. As the sngle stp s splt n the mddle nto two stps sepaated b =5nm the esonance pea s blue-shfted to =45nm. The elatvel lage shft compaed to the small modfcaton of the stuctue (=5nm) can be eplaned fom the fact that the modfcaton s made at the poston whee the magntude of the electc feld nsde the stp s mamum n the case of =nm at esonance (Fg. (b) - uppe half). A sgnfcant feld magntude enhancement compaed to the magntude E of the ncdent feld s found n the gap between the two stps (Fg. b - lowe half, Fg. c). Ths s a consequence of the electc feld of SR-SPP waves beng domnated b the -component nsde the slve stps, and the fact that ths feld component umps acoss the slve-a nteface n the gap egon gvng a feld magntude ncease of Ag / A, whee Ag and A ae the delectc constants of slve and a, espectvel. The feld magntude enhancement n the cente of the gap s app. a facto 4 (Fg. c). Ths tpe of calculaton can qualtatvel eplan feld enhancements obseved fo esonant optcal antennas [5]. Compaed to pevous wo [-4] the wdth of the stps (4nm) s shote and the scatteng peas (Fg. a) ae shape. The feld magntude enhancement s, howeve, not lage because of a smalle Ag at the shote esonance wavelength. Related SPP esonato stuctues based on gap SPPs [6], stpe SPPs [7], and SPPs n annula apetues [8] have ecentl been demonstated epementall. Nanostp esonatos smla to Fg. wee demonstated epementall n Ref. [4]. 6
(a) (b) (c) Fg. : (a) Scatteng coss secton fo a pa of slve nano-stps suounded b a of heght nm, wdth 4nm, and sepaaton (see nset). The ncdent p-polaed plane wave s popagatng along the -as. (b) Magntude of the electc feld elatve to the ncdent wave fo sepaatons = and 5nm at the esonance wavelength. (c) Feld magntude along the -as though the cente of the slve stps coespondng to the felds n (b). Acnowledgements The autho gatefull acnowledges fnancal suppot fom the ansh Reseach Councl fo Technolog and Poducton. S.I. Bohevoln s acnowledged fo man useful dscussons on SPP mcostuctues. Lage pats of ths tet wee based on ef. [9]. Refeences [] E.N. Economou, Geen s Functons n Quantum Phscs,(Spnge-Velag Beln Hedelbeg, New Yo, 979). [] P.M. Mose and H. Feshbach, Methods of Theoetcal Phscs, Chapte. 7 (McGaw-Hll, New Yo, 953). [3] O.J.F. Matn, A. eeu, and C. Gad, J. Opt. Soc. Am. A, 73 (994) [4] L. Novotn, B. Hecht, and.w. Pohl, J. Appl. Phs. 8, 798-86 (997). [5] A.. Yaghan, Poc. IEEE 68, 48 (98). [6] J. Nachamn, IEEE Tans. Antennas Popagat. 38, 99 (99). [7] E.M. Pucell and C.R. Pennpace, The Astophscal Jounal 86, 75-4 (973). [8] B.T. ane, The Astophscal Jounal 333, 848-7 (988). [9] J.P. Kottmann, O.J.F. Matn,.R. Smth, and S. Schult, Opt. Epess 6, 3 (). [] J. Jung and T. Søndegaad, Phs. Rev. B 77, 453 (8). [] H. Räthe, Suface Plasmons (Spnge, Beln, 988). [] S.I. Bohevoln, J. Eland, K. Leosson, P.M.W. Sovgaad, and J.M. Hvam, Phs. Rev. Lett. 86, 38 (). [3] W.L. Banes, A. eeu, and T.W. Ebbesen, Natue (London) 44, 84 (3). [4] K. Knepp, H. Knepp, I. Itan, R.R. asa, and M.S. Feld, J. Phs.: Condens. Matte 4, R597 (). [5] S.C. Ktson, W.L. Banes, and J.R. Sambles, Phs. Rev. Lett. 77, 67 (996). [6] M. Ketschmann, Phs. Rev. B 68, 549 (3). [7] T. Søndegaad and S.I. Bohevoln, Phs. Rev. B, 7, 549 (5). [8] T. Søndegaad and S.I. Bohevoln, Phs. stat. sol. (b) 4 (5), 364-9 (5); Phs. Rev. B 73, 453 (6). [9] M.U. Gonále et al., Phs. Rev. B 73, 5546 (6). [] F. Lópe-Teea, F.J. Gacía-Vdal, and L. Matín-Moeno, Phs. Rev. B 7, 645 (5). [] S.I. Bohevoln et al., Opt. Comm. 5, 38-33 (5). [] T. Søndegaad and S.I. Bohevoln, Phs. Rev. B, 75, 734 (7). [3] T. Søndegaad and S.I. Bohevoln, Opt. Epess, 5, 498 (7). [4] T. Søndegaad, J. Beemann, A. Boltasseva, and S.I. Bohevoln, Phs. Rev. B 77, 54 (8). [5] P. Mühlschlegel, H.-J. Esle, O.J.F. Matn, B. Hecht, and.w. Pohl, Scence 38, 67 (5). [6] H.T. Maa and Y. Kuoawa, Phs. Rev. Lett. 96, 974 (6). [7] H. tlbache et al., Phs. Rev. Lett. 95, 5743 (5). [8] M.J. Locea, A.P. Hbbns, J.R. Sambles, and C.R. Lawence, Phs. Rev. Lett. 94, 939 (5). [9] T. Søndegaad, Phs. Stat. Sol. (b) 44, 3448-6 (7). 7
Soluton to eecses: Eecse : Eecse 3: Result fo = (4 ponts nsde the bae): Result fo =45 (4 ponts nsde the bae): Result fo a clnde of damete.5m: 8
9