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.