Compacted dimensions and singular plasmonic surfaces

Abstract

Abstract:In advanced field theories there can be more than four dimensions to space, the excess dimensions described as compacted and unobservable on everyday length scales. We report a simple model, unconnected to field theory, for a compacted dimension realised in a metallic metasurface periodically structured in the form of a grating comprising a series of singularities. An extra dimension of the grating is hidden, and the surface plasmon excitations, though localised at the surface, are characterised by three wave vectors rather than the two of typical two-dimensional metal grating. We propose an experimental realisation in a doped graphene layer. One Sentence Summary: Plasmonic excitations of a singular metallic grating serve as a model for compacted dimensions.Main Text: A conventional two dimensional object is characterised by two quantum numbers. For example the frequencies of surface plasmons on a periodic surface are labelled by the components of their momentum projected onto the surface axes. We describe theoretically systems that instead require three quantum numbers to label them: the two conventional in-plane momenta plus a third momentum corresponding to a compacted dimension hidden from view inside a singularity. Compacted dimensions are ingredients of advanced string theories (1,2) where the extra dimensions in a 4+N dimensional theory are said to be compacted and so not directly observed on everyday length scales. As far as we know our singular surfaces are the only physically realisable model of this curious effect. We give two instances of how this might be done.We make use of the technique of transformation optics (3-5) which exploits the invariance of Maxwell's equations under a coordinate transformation: only the values of ε, µ are affected by the transformation. We use this theory to compact a dimension through a singular transformation that compresses one of the dimensions of a 3D system into one or more singular points. An example of the process is given (Fig. 1) for a 3D system (Fig. 1A), periodic in one of the dimensions and translationally invariant in the two other directions. The blue shaded areas are metallic and support surface plasmons (6) whose spectrum is characterised by three wave vectors: k x , k y , k u where k u is the wave vector heading out of the plane of the paper.Our intent is to show that the x dimension can be hidden using 2D conformal transformations where the x, y coordinates are represented by a complex number z = x + iy . Conformal transformations in 2D have the property of conserving the permittivity and permeability, ε, µ , in the plane of the transformation so that in this plane we are working with the same materials in all coordinate frames. Under each successive transformation the /page 2 spectral properties are preserved, and the modes once calculated in the initial frame can be found in the other frames through the properties of the transformation.In the first step we compress x = −∞ to a point at the origin,(1) which gives rise to Fig. 1B. This transfor...

Surface second-harmonic generation from metallic-nanoparticle configurations: A transformation-optics approach

“…The analytic predictions of TO have been found to be in excellent agreement with numerical results (for subwavelength particles), and verified experimentally by several groups [9][10][11][12]. In addition, CT allowed addressing various other physical effects such as the moulding of surface plasmon polariton propagation [13], the emergence of non-local effects in metallic nanostructures [14,15], van der Waals interactions at the nano-scale [16], graphene [17] and singular [18] metasurfaces, and plasmon-exciton interactions [19] and strong coupling [20].…”

Plasmon Localization Assisted by Conformal Symmetry

“…Notably, our symmetry argument empowers us with deep insight into the scattering properties of these metasurfaces. Furthermore, by coupling the plasmon to large momenta, the eigenmodes of this system are extremely localized near the valley points of the structure, a feature previously studied in the context of singular surfaces [38][39][40]. Hence, the advantage of conformal metasurfaces for the enhancement of light-matter interactions is twofold: on the one hand the long dwelling of the plasmon near the excitation region opens interesting opportunities for, e.g.…”

Nonlocal effects in plasmonic metasurfaces with almost touching surfaces

“…A conventional metasurface [12][13][14][15] is characterized by two wave vectors, such that the selection of k-vectors is discretized. On the other hand, singular metasurfaces feature three wave vectors, owing to the additional length scale introduced by the singularity, which allows surface modes to exist over a continuum of quantum numbers [16][17][18] .…”