This thesis is part of an effort to bring together two very active fields of physics: band topology and photonics. The field of band topology has revealed exotic phenomena such as robust, unidirectional edge states that occur at the interfaces between materials that belong to different topological phases. The Nobel Prize in Physics 2016 was awarded to Thouless, Haldane, and Kosterlitz, for predicting such phases in electronic systems where the topological edge states may revolutionise electronics and quantum computing. There is now great interest in reproducing such topological phases in photonics using photonic crystals: periodic nanostructures with tunable photonic bands. Realising such topological edge states in photonic devices could revolutionise optical data transport and optical quantum computing. In this thesis, we focus on two symmetry-protected topological phases that have been difficult to realise in photonics: the quantum spin-Hall effect (QSHE, protected by the fermionic time-reversal symmetry of electrons) and square-root topological semimetals (protected by chiral symmetry, also known as sublattice symmetry).

We introduce a new topological index for C2T symmetric crystals that emulate the QSHE using the angular momentum of light to mimic the spin of electrons. For example, in 2015 Wu & Hu proposed a photonic analogue of the QSHE where the crystalline symmetries and bosonic time-reversal symmetries of the photons generated a pseudo-fermionic time-reversal symmetry. Subsequent works suggested that this crystal was a trivial phase rather than a non-trivial QSHE phase. However, we believe that our new topological index demonstrates the non-trivial QSHE-like nature of the photonic crystal introduced by Wu & Hu while accounting for all of the valence bands determined from full-wave calculations.

We then study the topology of networks of voids and narrow connecting channels that are formed by the space between closely spaced perfect conductors. In photonics, chiral symmetry is often broken by long-range interactions, but Vanel et al 2017 showed that such void-channel networks can be mapped to analagous mass-spring systems in an asymptotically rigorous manner and therefore have only short-range interactions. We demonstrate that topological tight-binding models, such as square-root semimetals, can be reproduced in these void-channel networks with appropriate boundary conditions.

Finally, we discuss an interesting application of closely spaced nanoscopic metallic particles in the mid-to-far infrared and larger wavelengths. We show that despite being composed of highly dispersive and lossy metals, the effective dielectrics are virtually dispersion-free throughout the infrared spectrum and can be even more transparent than natural dielectrics such as germanium in the far-infrared. The effective index can be tuned locally, allowing us to design gradient-index lenses where light is guided by a continuously varied local refractive index. We propose a novel gradient-index lens that exploits the simultaneous transparency and high metallic filling fraction of the effective dielectrics to create intense ‘doubly-enhanced’ hotspots where light is focused on the microscale and the electric field ‘squeezed’ between the metallic particles on the nanoscale.