arXiv:cond-mat/0502532v1 [cond-mat.str-el] 22 Feb 2005 h on hr tct t smtt scntn.In constant. is is asymptote tractrix of its equation cuts the coordinates it Cartesian where point the ino h rcrx(i.(),wihi h uv nwhich in curve the distance is which the (Fig.2(b)), tractrix the of tion eoac fmtlnanoparticles plasmon nanopho- metal spherical in of of role resonance role well-established lesser the no than play tonics may resonances SP pseudosphere the the that show constant of to and going are negative We is everywhere. curvature ge- Gaussian hyperbolic Its exhibiting ometry. surface simplest the is Fig.2(a)) lso-oaios hc rpgt vrasraeo a of surface a pseudosphere over propagate which plasmon-polaritons, in essential nanostructures. is of curvature) optics the negative hyperbolic prop- understanding of so-called the the (surfaces of on surfaces understanding curva- plasmon-polaritons negative Thus, of has erties (Fig.1(a)). (Fig.1(b)). nanoholes curvature too real-life results ture with negative of best obtained have opening the often which The hand, are tips, other microscopy either the metal probe is On scanning curvature in zero. surface or the analytically-solvable positive known above the mentioned is all It in cases that nanoholes. notice and to tips easy probe shapes scanning the real-life of geometries many approximations of these poor very Unfortunately, as only spherical serve may planar, cases. the cylindrical to sur- analytically-solvable and limited sensor these basically or moment, are probe the geometries At scanning ge- good shapes. nanostructure a face be few designing may a which in solutions, only used reason analytical that The known exper- have is the ometries simulation. situation in of- computer results this is the good for shapes in produce different and/or to through- many way iment optical with only best error the and the ten Trial produce would put. film metal in ito niiaewa ido ea i ol produce would contrast tip metal image of best kind the what pre- anticipate to spectroscopy or difficult very dict often surface-enhanced key is it Unfortunately, the and techniques. is scanning microscopy numerous nanoholes) of or probe performance nanos- tips defines of which Op- metal resonances factor, as SP nanostructures. (such to metallic due tructures enhancement of field properties tical the in role h suopeei rdcda ufc frevolu- of surface a as produced is pseudosphere The nti ae ecnie h rpriso surface of properties the consider we paper this In ufc lso-oaios(SP) plasmon-polaritons Surface a ouin o h iefeunisadeigenfunctions and eigenfrequencies the for solutions cal isadnnaetrsi aiu cnigpoemicroscop probe use scanning be various may in techniques. resonances nanoapertures These and nanoholes. tips and nanotips metal of a 4 suopee(rat-pee hw in shown anti-sphere, (or Pseudosphere . lso-oaio eairo h ufcso osatneg constant of surfaces the on behavior Plasmon-polariton rmtepito otc fatnetto tangent a of contact of point the from lso-oaioso h ufc fapseudosphere. a of surface the on Plasmon-polaritons 3 rwa hp fananohole a of shape what or , grI mlaio,QiioBlao hitpe .Davis C. Christopher Balzano, Quirino Smolyaninov, I. Igor nvriyo ayad olg ak D272 USA 20742, MD Park, College Maryland, of University 1 eateto lcrcladCmue Engineering, Computer and Electrical of Department . 1,2 lya important an play Dtd coe 7 2018) 27, October (Dated: de ftehrcci etraegudtgte along together glued are CC sector geodesics horocyclic the the of edges h hsc fqatmfilsi yeblcsae of a research is current spaces space) active very Sitter hyperbolic anti-de of in the topic as fields (such quantum the dimensions (2). various equation on of by metrics physics described local is The pseudosphere small-scale the the of surface Thus, obtained. is ulda emtytrs ftehproi ln,an plane, metrics hyperbolic the with the surface of infinite terms) geometry Euclidian sashr fiaiayradius imaginary of sphere a as − ie ftennEciingoer ntehyperbolic the on geometry non-Euclidian straight freely the (the move of geodesics They various lines plane. along hyperbolic scattering clas- a without free on as this particles behave In sical SPs small. the is wavelength approximation SP approxi- (ray-optics) the this when plane, useful the is of mation pseudo- sector a the a of on only surface represents SPs the sphere of though Even optics analyti- two-dimensional plane. hyperbolic important the emphasize pseudo- of and properties a discuss cal on us let plasmon-polaritons sphere, surface for equations scanning nanosensors. good and building of probes benefits practical to addition implications, in field-theoretical important have may space-time sphere curved a in gravity quantum background of model interest- toy an ing provides plasmon-polaritons surface of optics sasco ahrcci sector horocyclic (a sector a as analyze. surface to it a this easy makes and of hand, which convenient properties, other shape very analytical the the nice On of very possesses nanohole. approximation a good or a nanotip pseudospherical be the may that 2 shape and Figs.1 of comparison from e smtial noteEcienspace Euclidean the into isometrically ted 1 h asincraueo h suopeeis pseudosphere the of curvature Gaussian The eoew rce ihwiigdw h Maxwell the down writing with proceed we Before h ufc ftepedshr a eunderstood be may pseudosphere the of surface The hc a hw yHlet antb nieyfit- entirely be cannot Hilbert) by shown (as which /a r on,wihdsrb eoatbehavior resonant describe which found, are 2 vrweeo t ufc s tmyb understood be may it (so surface its on everywhere z tv uvtr scniee.Analyti- considered. is curvature ative n ufc-nacdspectroscopy surface-enhanced and y = oipoepromneo metal of performance improve to d 8 aln h td fpamnotc napseudo- a on optics plasmon of study the , ds ( 2 x a = † ( + Fg2a) h suopeia shape pseudospherical the (Fig.2(a)), R 2 x a ( 2 d 2 Θ − 2 1) + 1 sinh / 2 ) ia 4 − .I sqieapparent quite is It ). 2 nmr rcs non- precise more in Θ 5,6,7 a (1 d Φ ic nonlinear Since . − 2 ) , x a 2 2 ) R 1 / 3 2 fthe If . K (1) (2) = 2 plane). You do not need to solve anything in this approx- imation! The motion of the surface wave on the curved 2 2 2 2 surface of the hyperbolic plane is a free motion, because 1 ∂ Hφ ω ǫ(ω) 1 ∂ Hφ ∂Hφ 2τ ∂ Hφ 2 2 + 2 Hφ + 2 ( 2 − +e 2 )=0 the curvature is the same everywhere. Since φ = const a ∂α c a ∂τ ∂τ ∂φ curves on the surface of the pseudosphere in Fig.2 are (5) geodesics, a short-wavelength SP wave packet launched We are going to search for surface-wave-like solutions towards the apex of the pseudosphere should freely move with a well-defined angular momentum n of the form straight down the apex without scattering, indefinitely. This is a robust topological property of the pseudosphere, − H = H e καeinφeikτ (6) which is extremely important for the effect of optical field φ 0 enhancement at the apex of the tip, or deep inside the In the most interesting case of n = 0 modes (the modes nanohole. Such robust and predictable shapes are very that correspond to the rays moving straight down the desirable in scanning probe microscopy and spectroscopy. apex), equation (6) gives Unfortunately, this ray-optics approximation breaks down when the tip radius becomes comparable with the a2ω2 SP wavelength. We must solve Maxwell equations in or- k2 − ik − (κ2 + ) = 0 (7) der to find the exact field distribution near the tip apex. 0 c2 Luckily, nice geometrical properties of the pseudosphere in vacuum, and allow us to find exact analytical solution of this prob- lem. Let us introduce rectangular curvilinear coordinates (α, θ, φ) as shown in fig.2, so that the distance from a a2ω2ǫ(ω) k2 − ik − (κ2 + ) = 0 (8) point on the pseudosphere to the z-axis is ρ = asinθ, m c2 and the radius of curvature of the φ = const tractrix is r = a/tanθ (where ~r connects the point on the trac- in metal, where κ0 and κm are the exponents of trix with the center of the osculating circle at this point; the field decay perpendicular to the interface in vac- note that the locations of all the osculating circles form uum and metal, respectively. The boundary condition the evolute of the tractrix, which is called the catenary, κ0 = −κmǫ(ω) leads to the following solution for k and see fig.2(b)). The third coordinate in the orthogonal set, κ0: which is normal to the surface of the pseudosphere is de- fined in the vicinity of the pseudosphere surface as adα, a2ω2ǫ 1 i k = ±( − )1/2 + , (9) since the tractrix is orthogonal to a set of circles of con- c2(ǫ + 1) 4 2 stant radius a whose centers are located on the z-axis, as shown in Fig.2(b). Thus, the spatial metrics in this system of coordinates takes the form a2ω2 κ0 = − (10) c2(ǫ + 1) 2 2 2 2 2 cos θ 2 2 2 2 2 2 ds = a dα + a dθ + a sin θdφ (3) In the Drude model, in which ǫ(ω)=1 − ωp/ω is real, sin2θ the surface-wave-like solution with n = 0 exists if ǫ< −1. In vacuum the SP field is We should emphasize that the so introduced global angular coordinates (θ, φ) on the surface of the pseudo- −κ0α −τ/2 ik∗τ sphere and the local angular coordinates (Θ, Φ) in equa- Hφ = H0e e e , (11) tion (2) are not the same. The local (Θ, Φ) coordinates ∗ provide good representation of the metrics of the pseu- where we have introduced k = k − i/2. The SP dis- dosphere surface only when R<<ρ(θ). persion law of this mode looks like The Maxwell equations in the curvilinear (α, θ, φ) co- ordinates 2 2 2 2 ∗ a ω (1 − ωp/ω ) 1 1/2 k = ±( 2 2 2 − ) (12) c (2 − ωp/ω ) 4 ω2ǫ(ω) ∇×~ ∇×~ H~ = H,~ (4) It is easy to see that in the ω <<ω limit, the gap ap- c2 p pears in the SP spectrum, as usually happens on the sur- 5 (where ǫ(ω) is the dielectric constant of metal), and the faces of negative curvature . In the low-frequency limit continuity of Dα on the metal surface can be used in order to find the dispersion law of surface plasmon-polaritons 2 2 ∗ a ω 1 on the surface of the pseudosphere. Introducing a new k = ±( − )1/2, (13) c2 4 variable τ as τ = −ln(sinθ), which has a clear physical meaning of the normalized arc length of the tractrix φ = where the frequency gap width is related to the curva- const, the Maxwell equations (4) lead to ture as ∆ω = c/2a. 3

In the purely theoretical case of a complete infinite intensity distribution in one of these resonant modes pseudosphere surface the spectrum of surface plasmon- with m = 4 is shown in Fig.3. According to eq.(11), polaritons is continuous. However, any practical STM the field intensity on the surface is proportional to tip is not infinitely sharp (Fig.1(a)). Real tips may be (ρ/a)sin2(k∗ln(a/ρ)), so the SP intensity in SP field approximated by a portion of the pseudosphere surface, maxima attenuates roughly as ρ/a. In effect, a pseudo- which is cut at some ρmin, and hence some τmax. In such spherical tip with a pseudo-radius a>λ/4π (thus, of the case the tip would exhibit a set of resonant eigenfrequen- order of a few hundreds of nanometers) behaves as a good cies, which correspond to the excitation of standing SP optical nanoantenna in the ∼ λ wavelength range. If one waves on the tip surface. The resonant condition is given of the eigenfrequencies of such a nanoantenna is matched ∗ 1,2 by k τmax = πm, where m is integer. While quite simple with the frequency of the spherical plasmon resonance 1/2 and straightforward in terms of the normalized pseudo- at ωp/3 , and a spherical metal nanoparticle is placed ∗ sphere arc length τmax and k , this condition is not as ev- (or formed) at the apex of the pseudosphere, a very ef- ident in terms of the tip length zmax and light frequency ficient way of delivering optical energy to a nanoscale ω (see eqs.(1) and (12)), which explains why the reasons region of space will result. Such plasmon nanolenses are for very good performance of some scanning probe tips being actively developed at the moment9,10. Note that (compared to very bad one of others) were unclear in the limitation a>λ/4π on the pseudo-radius of the res- local scanning probe spectroscopy measurements3. Un- onant pseudospherical nanoantenna originates from the til now the only good experimental strategy was to try frequency gap ∆ω = c/2a in the SP spectrum due to large number of tips at random, until the good one is negative curvature. found. We believe that this situation may be radically In conclusion, plasmon-polariton behavior on the sur- changed now. Instead of somewhat cumbersome equa- faces of constant negative curvature has been considered. tion (1), a simpler looking parametrization may be used Analytical solutions for the eigenfrequencies and eigen- via introduction of a new variable γ as coshγ = a/ρ, so functions have been obtained, which describe resonant that τ = ln(coshγ), and behavior of metal nanotips and nanoholes. These reso- nances may be used to improve performance of metal tips and nanoapertures in various scanning probe microscopy z = a(γ − tanhγ) (14) and surface-enhanced spectroscopy techniques. Thus, analytical calculation of tip resonances becomes This work has been supported in part by the NSF grant a very straightforward task. An example of a field ECS-0304046.

Appl.Opt. 5, S16 (2003). 3 A. Hartschuh, et al., Phys. Rev. Lett. 90, 095503 (2003). FIG. 1: (a) Electron microscope photo of an etched metal 4 R. Bonola, Non-Euclidian Geometry (Dover, New York, tip of a scanning tunneling microscope. (b) 3D view of a 1955). nanohole array produced in a 100 nm thick gold film using 5 E.N. Argyres, et al., J. Phys. A 22, 3577 (1989). focused ion beam milling. The nanohole periodicity is 500 6 T. Hertog, et al., Phys. Rev. Lett. 92, 131101 (2004). nm. Both surfaces exhibit negative curvature around the tip 7 N. Goheer, et al., Phys. Rev. Lett. 92, 191601 (2004). and nanoholes, respectively. 8 I.I. Smolyaninov, Phys.Rev.Lett. 94, 057403 (2005). 9 M. I. Stockman, Phys. Rev. Lett. 93, 137404 (2004). 10 M. I. Stockman, et al., Phys. Rev. Lett. 92, 057402 (2004). 1 H. Raether, Surface Plasmons (Springer, Berlin, 1988). 2 A.V. Zayats and I.I. Smolyaninov, J.Opt. A: Pure 4

FIG. 2: (a) Pseudosphere, which is obtained as a surface of revolution of a tractrix (b), may be used as a good approxi- mation of the shape of a metal tip of a scanning probe micro- scope, or the shape of a nanohole in a metal film. Also shown in (b) is a catenary, which is an evolute of the tractrix.

FIG. 3: SP field intensity distribution of the m = 4 resonant mode on the surface of a pseudosphere. The pseudosphere boundary is shown by dots. ( b )

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