ABSTRACT

This thesis presents studies of band engineering in structured photonic systems, in which geometry, symmetry, and periodic patterning are used as design principles to access topological phases, flat bands, and nonlinear optical responses that are unavailable in unstructured settings. Both the dispersion and the geometric structure of Bloch states provides the unifying theoretical language connecting the works presented here. We begin by revisiting the Wu-Hu model of topological photonic crystals, establishing a careful algebraic classification of its topological properties using topological quantum chemistry and clarifying the topological nature of its edge states. We then construct a photonic flatband with fragile topology in a coupled resonator system, demonstrating the Wannier obstruction associated with fragile topology which serves as a versatile platform to explore the interplay between quantum geometry and many-body physics in flatband systems. In twisted bilayer photonic crystals, we develop a theoretical framework that incorporates both near-field interlayer coupling and far-field radiation on equal footing, identifying unique features that extend photonic moiré physics beyond merely an analogue of electronic moiré systems. Turning to engineered perturbations of photonic lattices, we demonstrate that a density wave-like displacement of the air holes in a photonic graphene slab simultaneously controls the band dispersion, bulk topology, and far-field radiation. The perturbation folds the intrinsic flat band of photonic graphene above the light line, drives a tunable band inversion, and produces a Friedrich-Wintgen bound state in the continuum alongside Jackiw-Rebbi interface states that inherit the flatness of the bulk band. Finally, we show that the bulk photovoltaic effect, which is forbidden in bulk transition metal dichalcogenides by inversion symmetry, can be activated and enhanced for chiral photodetection by patterning the material surface with a chiral metasurface. The metasurface engineers a symmetry-broken near-field environment that unlocks the geometric photocurrent intrinsic to the bulk material without modifying its crystal structure. Taken together, these works establish band engineering in structured photonic systems as a coherent and versatile  research platform, demonstrating that the interplay between geometry, symmetry, and quantum geometry in photonic crystals provides a rich design space for controlling light and its interactions with matter.