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Plasmon-Polaritons on the Surface of a Pseudosphere

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Plasmon-Polaritons on the Surface of a Pseudosphere Plasmon-polaritons on the surface of a pseudosphere. Igor I. Smolyaninov, Quirino Balzano, Christopher C. Davis Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA (Dated: October 27, 2018) Plasmon-polariton behavior on the surfaces of constant negative curvature is considered. Analyti- cal solutions for the eigenfrequencies and eigenfunctions are found, which describe resonant behavior of metal nanotips and nanoholes. These resonances may be used to improve performance of metal tips and nanoapertures in various scanning probe microscopy and surface-enhanced spectroscopy techniques. Surface plasmon-polaritons (SP)1,2 play an important role in the properties of metallic nanostructures. Op- a a2 x2 tical field enhancement due to SP resonances of nanos- z = aln( + ( − 1)1/2) − a(1 − )1/2 (1) tructures (such as metal tips or nanoholes) is the key x x2 a2 factor, which defines performance of numerous scanning The Gaussian curvature of the pseudosphere is K = probe microscopy and surface-enhanced spectroscopy −1/a2 everywhere on its surface (so it may be understood techniques. Unfortunately, it is often very difficult to pre- as a sphere of imaginary radius ia). It is quite apparent dict or anticipate what kind of metal tip would produce 3 from comparison of Figs.1 and 2 that the pseudospherical the best image contrast , or what shape of a nanohole shape may be a good approximation of the shape of a in metal film would produce the best optical through- nanotip or a nanohole. On the other hand, this surface put. Trial and error with many different shapes is of- possesses very nice analytical properties, which makes it ten the only way to produce good results in the exper- very convenient and easy to analyze. iment and/or in the computer simulation. The reason The surface of the pseudosphere may be understood for this situation is that only a few nanostructure ge- as a sector (a horocyclic sector4 in more precise non- ometries have known analytical solutions, which may be Euclidian geometry terms) of the hyperbolic plane, an used in designing a good scanning probe or sensor sur- infinite surface with the metrics face shapes. At the moment, these analytically-solvable geometries are basically limited to the planar, spherical and cylindrical cases. Unfortunately, these geometries ds2 = R2(dΘ2 + sinh2ΘdΦ2), (2) may serve only as very poor approximations of the shapes of many real-life scanning probe tips and nanoholes. It is which (as shown by Hilbert) cannot be entirely fit- easy to notice that in all the known analytically-solvable ted isometrically into the Euclidean space R3. If the cases mentioned above the surface curvature is either edges of the horocyclic sector are glued together along positive or zero. On the other hand, the best results the geodesics CC†(Fig.2(a)), the pseudospherical shape in scanning probe microscopy are often obtained with is obtained. Thus, the small-scale local metrics on the metal tips, which have negative curvature (Fig.1(a)). surface of the pseudosphere is described by equation (2). The opening of real-life nanoholes has negative curva- The physics of quantum fields in hyperbolic spaces of ture too (Fig.1(b)). Thus, understanding of the prop- various dimensions (such as the anti-de Sitter space) is a erties of plasmon-polaritons on the so-called hyperbolic topic of very active current research5,6,7. Since nonlinear surfaces (surfaces of negative curvature) is essential in optics of surface plasmon-polaritons provides an interest- understanding the optics of nanostructures. ing toy model of quantum gravity in a curved space-time background8, the study of plasmon optics on a pseudo- In this paper we consider the properties of surface sphere may have important field-theoretical implications, plasmon-polaritons, which propagate over a surface of a in addition to practical benefits of building good scanning pseudosphere4. Pseudosphere (or anti-sphere, shown in probes and nanosensors. Fig.2(a)) is the simplest surface exhibiting hyperbolic ge- Before we proceed with writing down the Maxwell arXiv:cond-mat/0502532v1 [cond-mat.str-el] 22 Feb 2005 ometry. Its Gaussian curvature is negative and constant equations for surface plasmon-polaritons on a pseudo- everywhere. We are going to show that the SP resonances sphere, let us discuss and emphasize important analyti- of the pseudosphere may play no lesser role in nanopho- cal properties of the two-dimensional optics of SPs on a tonics than the well-established role of spherical plasmon hyperbolic plane. Even though the surface of the pseudo- resonance of metal nanoparticles1. sphere represents only a sector of the plane, this approxi- The pseudosphere is produced as a surface of revolu- mation is useful when the SP wavelength is small. In this tion of the tractrix (Fig.2(b)), which is the curve in which (ray-optics) approximation the SPs behave as free clas- the distance a from the point of contact of a tangent to sical particles on a hyperbolic plane. They move freely the point where it cuts its asymptote is constant. In without scattering along various geodesics (the straight Cartesian coordinates the equation of tractrix is lines of the non-Euclidian geometry on the hyperbolic 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.
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