Surfac plasmon rsonanc is snsitiv mchanism for obsrving slight changs nar th surfac of a dilctric-mtal intrfac. It is commonl usd toda for discovring th was in which protins intract with thir nvironmnt, and ar valuabl in th production of nw pharmacuticals. Th phnomnon of surfac plasmon rsonanc has bn known for th last cntur, but has maturd to bcom a stat-of-th-art tchnolog within th last 30 ars. W will discuss thm simplst configuration supporting surfac plasmons, th intrfac of a smiinfinit mtal with a smi-infinit dilctric. B taking th curl of th curl of th lctric fild, (or similarl, of th H fild) on can obtain th wav quation dscribing th propagation of light: E E ε c t = 0 (1) B assuming a harmonic tim dpndnc, w can rwrit th E fild as E=E(r) -iwt. Substituting this back into (1), w obtain th Hlmholtz quation: E( r) + ko εe( r) = 0 () W now want to invstigat how this oscillating lctric fild can bhav at a mtaldilctric intrfac. In ordr to do this, w first dfin our propagation gomtr, as shown in Figur 1. Figur 1
It is thn apparnt that ε is a function of z, with z = 0 dfining th surfac at th intrfac. W st th dirction of propagation on th surfac in th x dirction and assum thr is no variation along th dirction. On th surfac, th wav can now b writtn as: E ( r) = E( z) iβx (3) β is thn th wav vctor of th surfac wav. W will latr show that β is rlatd to th dilctric constants of th surrounding mdia and thus to th proprtis of th fr lctrons of th mtal. Substituting (3) back into th Hlmholtz quation ilds th following xprssion, which givs us th gnral form of our surfac wav: E ( z) + ( ko ε β ) E( z) iβx = 0 (4) In ordr to discovr th mannr in which th fild dpnds on z, w nd to find xprssions for ach componnt of th lctric (and H) fild. This is don b taking th curl of E and H as givn b Maxwll s quations, laving a total of six quations which can b rducd and sparatd into two sts of solutions corrsponding to TE and TM polarization. For TM polarization, onl th H, E x, and E z componnts of th filds rmain, rsulting in th following dfining quations: 1 H Ex = i ωεoε Ez H β = H ωεoε + ( k o ε β ) H = 0 (5) (6) (7) For TE polarization, onl th E, H x, and H z componnts of th filds rmain and th following quations ar obtaind:
E 1 Hx = i ωμ H z + ( k o E β = E ωμo o ε β ) E = 0 (8) (9) (10) W can now assum an xponntiall dcaing solution, so that th fild is confind to th surfac. For TM mods, for xampl, w hav that: H( z) = A iβx k z (11) for z > 0 and: H(z) = A1 iβx k1z (1) for z < 0. Th corrsponding lctric filds can b found b substituting (11) and (1) into (5) and (6). In ordr to solv for A i, β, and k i, w appl boundar conditions. From Maxwll s quations, w know that th tangntial componnts of th H and E fild must b continuous. B quating th H in (11) and (1), w find that A 1 =A. Now quating E x in both rgions lads to th following rlation: k k = 1 ε ε 1 (13) It is important to not hr th ngativ sign on th right sid of th xprssion. Sinc th wav vctors in th two mdia hav th sam sign, it is clar that th dilctric constant of mdia 1 and mdia must b opposit. Thus, surfac plasmons can onl xist at th intrfac of a conductor and insulator. Now, w would lik to find an quation rlating β to ε 1 and ε. Plugging quations (11) and (1) into (7) ilds th following xprssions:
k = β k o ε k = β ko ε 1 1 (14) (15) Putting (14) and (15) into (13) lads to our xprssion for β: β = ε 1ε k o ε 1 + ε (16) Having discovrd th possibilit of a surfac wav xcitd b TM polarizd light, w now considr th possibilit of surfac wavs xcitd b TE polarization. If w assum that E = A 1 iβx -k1z for z > 0 and E = A 1 iβx -kz for z < 0, as bfor, and quat E in both rgions and H x in both rgions, w find that w must hav: k = k 1 (17) As notd bfor, k 1 and k must hav th sam sign, and thrfor w s that it is impossibl to satisf th boundar conditions and thus surfac plasmons cannot b xcitd using TE polarizd light. W can now tak a closr look at th proprtis of surfac plasmons b xamining (16). From th Drud modl, w know that w can writ th dilctric constant of our mtal (assuming ngligibl damping) as: p ω ε1 = 1 ω (18) Whr ω p is th plasma frqunc of th fr lctrons and ω is th frqunc of th incidnt light (and thus of th surfac wav). Plotting ω as a function of β allows us to s xplicitl how th surfac plasmon frqunc dpnds on th wavvctor, as shown in Figur.
Figur W can s that as β bcoms largr, th frqunc asmptoticall approachs a maximum valu. In ordr to find this maximum frqunc, w st β =, and thus ε 1 + ε = 0. Using (18), w find: ωsp = ωp 1+ ε (19) Whr ω sp is th cutoff frqunc. It is thus impossibl to xcit surfac plasmon rsonanc with a frqunc highr than ω sp will b futil. W can discovr anothr intrsting proprt of surfac plasmons from Figur. Examination of th light lin of air and th disprsion of a surfac plasmon xcitd at a mtal-air intrfac, w s that th two nvr cross. Thus it is impossibl to xcit a plasmon b simpl shining light incidnt from air on a smooth intrfac. On altrnativ is to us prism coupling. Sinc glass has a highr rfractiv indx than air, its light lin will hav a smallr slop, and thus will intrsct with th plot of th surfac Plasmon in air at som β. Howvr, prism coupling rquirs th us of a thin mtallic film rathr than a smi-infinit mtal, lading to a mor complx situation than discussd hr. Grating coupling is anothr possibilit, in which th x componnt of k o can b mad to match β b pattrning th surfac of th mtal with shallow gratings.
In summar, w hav shown that a surfac wav of oscillating fr lctrons drivn b an incidnt lctromagntic wav can xist at th intrfac btwn a mtal and a dilctric. Practicall, ithr prism coupling to a thin mtallic film or th us of a grating is rquird to achiv rsonanc; howvr, th idas introducd hr provid a basis for furthr calculations.
Bibliograph Wangsnss, Roald K. Elctromagntic Filds. John Wil and Sons Inc, 1979, 1986. Print Mair, Stfan A. Plasmonics Fundamntals and Applications. Springr Scinc+Buisnss Mdia LLC, 007. Print
Surfac Plasmon Polaritons at Mtal-Dilctric Intrfacs Courtn Bard Phsics 545