Geometric potential for plasmon polaritons on curved surfaces

G Della Valle (1) , S Longhi (1)

1 Dipartimento di Fisica

Surface plasmon polaritons (SPPs) on a curved metal-dielectric interface are shown to experience a geometric potential V top like quantum particles confined on a curved surface. As opposed to electrons or photons constrained by a squeezing potential, no ambiguities arise in the definition of the geometric potential for SPPs. A general form of V top for SPPs on a generalized cylindrical surface is derived by a multiple-scale asymptotic analysis of vectorial Maxwell's equations for the case of a curvature radius larger than the SPP wavelength. G Della Valle et al., Geometric potential for plasmon polaritons on curved surfaces...2 Geometry plays a major role in defining the dynamical properties of quantum particles and classical waves confined on a curved space [ 1, 2 ]. For instance, it is well known that surface geometry influences the motion of a quantum particle constrained on a curved surface [ 1, 3, 4, 5, 6, 7, 8, 9 ], like electrons in low-dimensional nanostructures [ 10, 11, 12 ]. A similar behavior occurs for optical waves squeezed on a curved thin dielectric guiding layer [ 13 ]. As in the 'classical ' (e.g. newtonian or ray-optics) limits the motion is ruled by the surface metric and the force-free trajectories are geodesics, in the full wave regime an additional frictional potential V top, referred to as the geometric potential, is found [ 3, 4, 5, 6 ]. Though V top vanishes in the classical limit (as it should), its dependence on the surface topology has been a matter of great controversies which have been especially discussed in the quantum mechanical context [ 4, 7, 8, 9 ]. An accepted approach, originally proposed by Jensen, Koppe and da Costa [ 5, 6 ], is to squeeze the quantum particle on the surface by a physical confining potential. This approach has been recently adopted for confined optical waves as well [ 13 ]. Even in this case some ambiguities remain due to the freedom in selecting the squeezing potential [ 9 ]. Such difficulties cast a shadow over the physical relevance of any geometric potential for three-dimensional particles or waves constrained on a surface, because the way they sense the surrounding space strongly depends on the squeezing potential. Some authors also argued that ideal squeezing procedures generally correspond to unrealistic restrictions [ 7 ]. In this Letter we show that a geometric potential, which does not suffer from any ambiguity, does exist at a curved metal-dielectric interface for surface plasmon polaritons (SPPs), i.e. for coupled modes of plasmons and photons. As compared to electrons or photons squeezed on a surface, SPPs are intrinsically two-dimensional quasi particles and do not require any squeezing potential. SPPs have attracted special attention in recent years for their relevance in subwavelength optics and nanophotonics [ 14, 15 ]. In spite of the considerable efforts devoted to study SPPs in different curved geometries [ 16, 17, 18, 19, 20, 21 ], including SPPs scattering and radiation at bends or interfaces [ 22, 23, 24, 25 ], the concept and properties of a geometric potential for SPPs has been overlooked. Here we consider SPPs on a generalized cylindrical surface [ Fig. 1(a) ] and derive, by a multiple scale asymptotic analysis of full vectorial Maxwell's equations, a general form of V top. The predictions of the asymptotic analysis, such as the appearance of bound states sustained by the geometric potential, are confirmed by full-vectorial numerical simulations in curved Ag-glass interfaces. Let us consider SPPs at optical frequency ω = 2πc 0 /λ on a curved metal-dielectric interface defined by a generalized cylindrical surface and let us introduce a set of local curvilinear coordinates (σ, η, z) around the surface, as shown in Fig. 1(a). If we normalize the spatial variables to the reduced wavelength λ = λ/(2π) = 1/k and indicate by u = (c 0 B σ, E η, E z) and v = (E σ, c 0 B η, c 0 B z) the mixed components of electric E and magnetic B fields, Maxwell's equations in the curvilinear coordinate system can be cast in the following

Mots clés: 73.20.Mf; 03.65.Ge; 42.25.-p

G Della Valle, S Longhi. Geometric potential for plasmon polaritons on curved surfaces. Journal of Physics B: Atomic, Molecular and Optical Physics, 2010, 43 (5), pp.51002. ⟨10.1088/0953-4075/43/5/051002⟩. ⟨hal-00569868⟩