This thesis explores the connection between symmetry and photonic bandgaps in dielectric structures for practical application in photonic systems. We challenge the common intuition regarding the optimality of certain crystal-symmetries in photonic crystals as well as exploit alternative notions of symmetry to overcome the limitations inherent to them. To do this, we develop upon three relevant problems using numerical tools and experimental verification by collaborators to efficiently probe the properties of photonic structures and develop new intuitions about them.

We begin by developing upon Face-Centred Cubic (FCC) symmetric photonic foams as a self-assembling photonic structure with sizeable complete bandgaps in commercially available materials. We predict the size of the complete photonic bandgaps in FCC and Body-Centred Cubic (BCC) foams as 8.95% and 3.35% respectively at standard index contrast n = 3.4. The continuum of optimal bandgap sizes for all intermediate Body Centred Tetragonal (BCT) symmetric foams between the FCC- and BCC-symmetric foams during uniaxial deformation is charted via parameter sweep, revealing unexpected complexity in the variation of bandgap width during the deformation. We show that there exists BCT-symmetric foam structures with superior bandgap widths to FCC foams, opening a complete photonic bandgap for dielectric contrasts of n > 2.73. We then confirm the robustness of our results by calculating the Density of States (DOS) and subjecting the foam structures to plausible manufacturing defects.

Subsequently, we analyse the formation of photonic band gaps in three-dimensional disordered photonic structures. Here, we explore the photonic bandstructures of tetravalent structures produced under different degrees of order and symmetry: diamond, connected random networks. We demonstrate, for the first time, the presence of a sizable photonic band gap in a stealthy hyperuniform disordered photonic structure. Our calculations of the bandstructure and density of states of an exemplar stealthy hyperuniform structure reveals that these structures can possess a complete bandgap at index contrast as low as n = 2.1. We show a clear correspondence between the photonic bandgaps present in diamond, Connected Random Networks (CRNs), and stealthy hyperuniform structures, and confirm the key role played by spatial and topological uniformity in the formation of the band gaps. Our simulations are in very good agreement with experimental measurements performed in the microwave range on centimetre-size structures.

Finally, we explore the space of designs for hyperuniform and near-hyperuniform patterns as hollow core fiber-optic claddings. Our approach exploits the large number of degrees of freedom offered by hyperuniform designs, which hold the potential for superior performance compared to periodic structuring while being more resilient to fabrication errors. For cylindrically symmetric structures such as hollow core optical fibers, the isotropy of disordered hyperuniform structures implies that the bandgap surrounding the hollow core will be have the same width and center frequency for all azimuthal angles around the hollow core. The new techniques we introduce for constrained generation of near-hyperuniform claddings optimise the isotropy of the structured cladding and provide nearly-optimal guiding of light through hollow core due to the surrounding statistically isotropic photonic band gap. Our calculations of the representative band structure for both crystalline structures and stealthy hyperuniform disordered structures demonstrate the presence of guided modes with reduced losses.