Using group theoretic and topological concepts, together with tunneling phenomena, we geometrically design interfacial wave networks that contain splitters which partition energy in 2, 3, 4 or 5 directions. This enriches the valleytronics literature that has, so far, been limited to 2-directional splitters. Additionally, we describe a design paradigm that gives greater detail, about the relative transmission along outgoing leads, away from a junction; previously only the negligible transmission leads were predictable. We utilise semi-analytic numerical simulations, as opposed to finite element methods, to clearly illustrate all of these features with highly resolved edge states. As a consequence of this theory, novel networks, with directionality tunable by geometry, ideal for applications such as beam-splitters, switches and filters are created. Coupling these novel networks, that contain multi-directional energy-splitters, culminates in the first realization of a topological supernetwork.

Strategically combining four structured domains creates the first ever three-way topological energy-splitter; remarkably, this is only possible using a square, or rectangular, lattice, and not the graphene-like structures more commonly used in valleytronics. To achieve this effect, the two mirror symmetries, present within all fully-symmetric square structures, are broken; this leads to two nondistinct interfaces upon which valley-Hall states reside. These interfaces are related to each other via the time-reversal operator and it is this subtlety that allows us to ignite the third outgoing lead. The geometrical construction of our structured medium allows for the three-way splitter to be adiabatically converted into a wave steerer around sharp bends. Due to the tunability of the energies directionality by geometry, our results have far-reaching implications for applications such as beam-splitters, switches and filters across wave physics.

We create a passive wave splitter, created purely by geometry, to engineer three-way beam splitting in electromagnetism in transverse electric and magnetic polarisation. We do so by considering arrangements of Indium Phosphide dielectric pillars in air, in particular we place several inclusions within a cell that is then extended periodically upon a square lattice. Hexagonal lattice structures are more commonly used in topological valleytronics but, as we discuss, three-way splitting is only possible using a square, or rectangular, lattice. To achieve splitting and transport around a sharp bend we use accidental, and not symmetry-induced, Dirac cones. Within each cell pillars are either arranged around a triangle or square; we demonstrate the mechanism of splitting and why it does not occur for one of the cases. The theory is developed and full scattering simulations demonstrate the effectiveness of the proposed designs.

Predictive theory to geometrically engineer devices and materials in continuum systems to have desired topological-like effects is developed here by bridging the gap between quantum and continuum mechanical descriptions. A structured elastic plate, a bosonic-like system in the language of quantum mechanics, is shown to exhibit topological valley modes despite the system having no direct physical connection to quantum effects. We emphasise a predictive, first-principle, approach, the strength of which is demonstrated by the ability to design well-defined broadband edge states, resistant to backscatter, using geometric differences; the mechanism underlying energy transfer around gentle and sharp corners is described. Using perturbation methods and group theory, several distinct cases of symmetry-induced Dirac cones which when gapped yield non-trivial band-gaps are identified and classified. The propagative behavior of the edge states around gentle or sharp bends depends strongly upon the symmetry class of the bulk media and we illustrate this via numerical simulations.

In solid-state physics, including photonics and wherever periodic lattice structures occur, it is essential to establish the fundamental features associated with wave propagation through the lattice: This is achieved using Bloch waves, the reciprocal lattice, and the reduction, using periodicity, to consider the irreducible Brillouin zone. A general approach, although widely accepted as not being perfectly legitimate, is to plot the dispersion relations around the edges of the Brillouin zone. We show definitively that this can be dangerous and that an important mode of practical significance is missed if this is done in too cavalier a fashion: This missing mode is illustrated for the design of endoscopes based on spring-mass (discrete) periodic structures and photonic crystals.

Galvanised by the emergent fields of metamaterials and topological wave physics, there is currently much interest in controlling wave propagation along structured arrays, and interfacial waves between geometrically different crystal arrangements. We model array and interface waves for structured thin elastic plates, so-called platonic crystals, that share many analogies with their electromagnetic and acoustic counterparts, photonic and phononic crystals, and much of what we present carries across to those systems. These crystals support several forms of edge or array-guided modes, that decay perpendicular to their direction of propagation. To rapidly, and accurately, characterise these modes and their decay we develop a spectral Galerkin method, using a Fourier-Hermite basis, to provide highly accurate dispersion diagrams and mode-shapes, that are confirmed with full scattering simulations. We illustrate this approach using Rayleigh Bloch modes, and generalise high frequency homogenisation, along a line array, to extract the envelope wavelength along the array. Rayleigh-Bloch modes along graded arrays of rings of point masses are investigated and novel forms of the rainbow trapping effect and wave hybridisation are demonstrated. Finally, the method is used to investigate the dispersion curves and mode-shapes of interfacial waves created by geometrical differences in adjoining media. arXiv:1812.07531v1 [physics.class-ph]

The coupled light-matter modes supported by plasmonic metasurfaces can be combined with topological principles to yield subwavelength topological valley states of light. We give a systematic presentation of the topological valley states available for lattices of metallic nanoparticles: All possible lattices with hexagonal symmetry are considered, as well as valley states emerging on a square lattice. Several unique effects which have yet to be explored in plasmonics are identified, such as robust guiding, filtering and splitting of modes, as well as dual-band effects. We demonstrate these by means of scattering computations based on the coupled dipole method that encompass the full electromagnetic interactions between nanoparticles.

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies.The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice near the edges of the Brillouin zone. The leading order solution is expressed as a product between the standing wave solution and long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones.