Homework 4. 1 Electromagnetic surface waves (55 pts.) Nano Optics, Fall Semester 2015 Photonics Laboratory, ETH Zürich

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1 Homework 4 Contact: frmmerm@ethz.ch Due date: December 04, 015 Nano Optcs, Fall Semester 015 Photoncs Laboratory, ETH Zürch The goal of ths problem set s to understand how surface plasmon polartons, electromagnetc surface waves at metal-delectrc nterfaces can be excted and exploted for sensng applcatons. In the second part, we determne the optcal forces actng on a small polarzable partcle n a tghtly focused laser beam, leadng to the concept of optcal tweezers, an ndspensable tool of modern scence. Regardng your solutons, please stck to the format suggested n Homework 3. 1 Electromagnetc surface waves (55 pts.) In ths problem, we consder electromagnetc felds at the nterface between two dfferent materals. In partcular, we focus on surface waves whose felds propagate along the nterface, but decay exponentally perpendcularly to the nterface, as sketched n Fg. 1(a). The nterface les n the plane z = 0. Such evanescent surface waves allow the manpulaton of electromagnetc felds on length scales smaller than the wavelength and consttute a foundaton of nano-optcs. We start by examnng some propertes of the solutons of Maxwell s equatons n the separate meda. To ths end, we consder a p-polarzed electromagnetc wave n a source-free medum wth materal parameters µ and ε at frequency ω, whose complex electrc feld ampltude s of the form E x, E (r) = 0 e (k x,x+k z, z). (1) E z, (a) z medum 1 medum x (b) vacuum metal delectrc α Fgure 1: (a) Illustraton of a surface wave propagatng along the nterface between two meda, whose feld decays exponentally n the drecton perpendcular to the nterface. (b) Kretschmann confguraton for the exctaton of a surface plasmon on a vacuum-metal nterface va total nternal reflecton at a thn metal flm on a delectrc. (a) ( pts.) Use a Maxwell equaton to show the followng relaton k x, E x, + k z, E z, = 0. () (b) (3 pts.) Calculate the magnetc feld H (r) resultng from E (r). (c) ( pts.) Show that the followng condton needs to be satsfed n order to fulfll Maxwell s equatons k z, H y, = ωε ε 0 E x,. (3) 1

2 (d) (4 pts.) Convnce yourself usng another Maxwell equaton as well as the magnetc feld calculated n (b) together wth Eq. () that the followng dsperson relaton s satsfed k x, + k z, = ω c n. (4) (e) (4 pts.) We nvestgate the nterface between medum 1 n the regon z > 0 (wth permttvty ε 1 and permeablty µ 1 = 1) and medum n the regon z < 0 (ε and µ = 1). Use the boundary condtons for electromagnetc felds to derve the followng system of equatons for the magnetc feld ( ) ( ) kz1 /ε 1 k z /ε H1y = 0. (5) 1 1 (f) (4 pts.) A homogeneous soluton of Maxwell s equatons,.e., a soluton of the system of equatons n Eq. (5), exsts, when the determnant of the characterstc matrx vanshes. Show that the parallel wavevector of the surface wave s gven by H y k x = ω c ε 1 ε ε 1 + ε. (6) We assume from now on that we are dealng wth an nterface between a delectrc (ε 1 1) and a metal. Let the delectrc functon of the metal be descrbed by the Drude model ε (ω) = 1 ω p/ω, (7) where ω p s the materal specfc plasma frequency and we assumed the metal to have no Ohmc losses. (g) (6 pts.) Sketch the Drude permttvty and determne n whch frequency range the metal shows metallc optcal propertes (ε < 0) and where t dsplays delectrc propertes (ε > 0). Label your axes approprately. Examne the refractve ndex of a metallc materal and examne the resultng wave vector to argue why an electromagnetc wave cannot enter the metal but s reflected. (h) (6 pts.) Calculate the parallel wavevector for the homogeneous soluton gven by Eq. (6) under the assumpton that the delectrc functon of the metal s gven by the Drude permttvty. The delectrc medum s stll characterzed by ε 1 1. For whch frequences do you obtan solutons whch propagate along the nterface? Hnt: You fnd two branches n the dsperson relaton. The branch at lower frequences s called surface plasmon polarton (SPP). () (8 pts.) Plot the dsperson relaton ω(k x ) for the parallel wavevector at the nterface. To ths end, plot ω vertcally as a functon of k x. Mark the plasma frequency ω p as well as the surface plasmon frequency ω SPP = ω p / ε Assume that the delectrc above the metal s vacuum wth ε 1 = 1. Add the dsperson relaton of lght n vacuum (called the lghtlne) to your plot and determne ts slope. Hnt: The lghtlne s gven by ω(k x ) for a plane wave n a homogeneous medum (here, we consder vacuum) propagatng along the x-drecton. Accordngly, the lghtlne marks the maxmally achevable k x n the homogeneous medum at a gven frequency ω. For completeness, we remark that upon allowng also for propagaton n the y-drecton, the metasurface of maxmally achevable n-plane wavevector ω(k x, k y ) s termed the lghtcone.

3 (j) (6 pts.) To understand the nature of the two branches of the dsperson relaton ω(k x ) better, we now consder the z-components of the solutons n the two meda. Show that n medum the followng relatons holds (wth j) kz, = kx ε. (8) ε j For whch range of frequences are the solutons for the metal-vacuum nterface evanescent n the drecton perpendcular to the nterface and propagatng along the nterface? For whch frequency range are the solutons propagatng both along the nterface and perpendcular to t? We have convnced ourselves, that surface plasmons are solutons of Maxwell s equatons whch can propagate along the nterface between a metal and a delectrc and whose felds fall off exponentally nto the adjacent meda. We now turn towards the queston how these suface plasmons can be excted. For smplcty, we stck wth our assumpton of medum 1 beng vacuum wth ε 1 = 1. (k) (3 pts.) Consder the metal-vacuum nterface and determne under whch angle α relatve to the nterface normal a plane wave would have to mpnge n order to excte a surface plasmon. Hnt: Keep n mnd that the parallel component of the wavevector has to be conserved at the nterface. (l) (3 pts.) We now consder the so-called Kretschmann confguraton for the exctaton of surface plasmon polartons. Here, a metal flm s deposted onto a delectrc medum wth ε d > 1, wth the metal flm thckness of the order of the metal s skn depth [see Fg. 1(b)]. Determne the angle α as a functon of the refractve ndex of the delectrc n d and the plasma frequency of the metal ω p under whch a plane wave at frequency ω has to mpnge from the delectrc medum onto the metal flm n order to excte a surface plasmon on the vacuum-metal nterface. (m) (4 pts.) Determne the maxmum frequency ω max (as a functon of refractve ndex n d of the delectrc) at whch a surface plasmon can be excted. Illustrate ths frequency by addng the lght lne n the delectrc medum to your dsperson relaton from problem (). There s a sensor famly whose operatng prncple reles on the generaton of surface plasmon polartons. In the Kretschmann confguraton, surface plasmons are excted wth a monochromatc laser at the approprate angle, whch shows n a relatvely low reflectvty of the nterface. The top sde of the metal flm, where the surface plasmons are propagatng, s exposed to the analyte, whch s typcally n an aqueous soluton. The metal surface s functonalzed such that certan analytes preferentally stck to t. The presence of these analytes changes the effectve refractve ndex at the surface of the metal flm and accordngly, the dsperson curve of the surface plasmon shfts. The reflected ntensty of the laser amed at the lower surface s now ncreased, snce the angle of ncdence does not correspond to the exctaton angle for surface plasmons any more. The outstandng strength of such SPP sensors les n ther senstvty to changes n refractve ndex n the very proxmty of the surface thanks to the evanescent felds decayng exponentally nto the delectrc medum. Optcal Tweezers [45 pts.] In 1971, Arthur Ashkn nvestgated delectrc partcles exposed to a tghtly focused laser beam. He dscovered that partcles could be trapped n the laser focus. By now, ths technque termed optcal tweezers has found wdespread applcatons n bology and physcs for manpulaton of tny objects and measurement of mnute forces. 3

4 x F θ z Fgure : Optcal force actng on a sub-wavelength partcle n the focus of a tghtly focused Gaussan beam. In ths exercse we consder a delectrc partcle wth no materal loss n the focus of a tme-harmonc Gaussan beam wth angular frequency ω. The partcle s smaller than the wavelength λ and we model t as a dpolar scatterer wth purely real electrostatc polarzablty α = 4πε 0 a 3 ε 1 ε +. (9) (a) (3 pts.) Usng the electrodynamc correcton to the electrostatc polarzablty derved n Homework 3, show that the electrodynamc polarzablty s, to frst order n the electrodynamc correcton, gven by α = α + α. (10) Express the magnary part α usng the wavenumber k, the electrostatc polarzablty and numercal constants. (b) (3 pts.) The force actng on a dpole p n an electromagnetc feld s gven by F(r) = 1 { } Re p E (r) (11) where [x, y, z]. Show that ths force can be wrtten as F(r) = α { } Re E (r) E (r) + α { } Im E (r) E (r). (1) The frst term n Eq. (1), whch can be expressed as F grad = (α /4) (E E), s referred to as the gradent force. From ths expresson we see that F grad s the gradent of the potental (E E). Consequently, F grad s conservatve, whch means that F grad = 0. The second term F scatt s referred to as the scatterng force. It cannot be expressed as the gradent of a potental. Consequently, F scatt s nonconservatve and accordngly F scatt 0. Consder a Gaussan beam n vacuum polarzed along the x-axs and focused under an angle θ = 53. Ths hgh numercal aperture stretches the paraxal approxmaton, but t turns out that our results are stll an excellent representaton of the physcal realty. The wavelength s λ = 1064 nm and the feld strength at the focus s E 0 = 10 7 V/m. The partcle s radus s a = 150 nm and ts delectrc constant s ε =.1. The orgn of our coordnate system s centered at the focus of the Gaussan beam as sketched n Fg.. (c) (4 pts.) Calculate the power carred by the Gaussan beam. (d) (8 pts.) Gve an expresson for the gradent force F grad along the x-axs and plot t n the range x = [ λ... λ]. 4

5 (e) (3 pts.) Expand F (x) grad (x, 0, 0) (x-component of the gradent force along the x-axs) to lnear order n x. For small dsplacements from the focus, the gradent force can now be wrtten as F xgrad κ x x,.e., the partcle behaves as f t were attached to the focus by a sprng wth sprng constant κ x. Determne the force constant κ x. (f) (3 pts.) In the lnear approxmaton, the equaton of moton for a partcle movng along the x-axs s mẍ + κ x x = 0. Assume that the specfc densty of the partcle s ρ =.5 g/cm 3 and calculate the oscllaton frequency Ω of the partcle along the x-axs. (g) (8 pts.) Gve an expresson for the gradent force F grad along the z-axs and plot t n the range z = [ λ... λ]. Calculate the sprng constant κ z descrbng the restorng force n the lnear approxmaton. Along whch drecton, x or z, s the restorng force actng on the partcle stronger? (h) (4 pts.) The scatterng force F scatt pushes the partcle away from the focus along the optcal axs (z). For the values gven above, plot F scatt along the z-axs n the range z = [ λ... λ]. Optonal: Fnd an analytcal expresson for F (z) scatt. () (4 pts.) Plot the total force F along the z-axs n the range z = [ λ... λ] and determne from your plot the dsplacement z of the equlbrum poston due to the scatterng force. (j) (5 pts.) One fnds expermentally that t s mpossble to stably trap too large partcles. Gve a quanttatve argument why ths s the case. 5