The innovation engine for new materials

Danielle Woods

Danielle Woods


University of California, Santa Barbara



Site Abroad: 

Trinity College Dublin, Ireland


Robert McGuinness

Faculty Sponsor(s): 

Paul Eastham

Faculty Sponsor's Department: 


Project Title: 

Calculation of Hamiltonian and Band Structure for a Photonic Topological Insulator

Project Description: 

Topological insulators are materials with an insulating interior but conductive surface. In two dimensions, surface states are prevented from back-scattering and scattering into the bulk, leading to unique properties for propagation of states. These materials and corresponding phenomena have been widely studied in the context of electrons, but they can also be analyzed for photons. In this project, we considered the case of a two-dimensional photonic crystal. The properties of electromagnetic wave states in this type of topological structure can be characterized by determining the material’s refractive index surfaces and band structure. To make these calculations, we modelled the problem with a mathematical analogue to the Schrödinger equation. This task can be simplified by taking a paraxial approximation of the index surface to the optical axis, but this approximation is not valid in every case. Our goal was to derive the corresponding topologically non-trivial Hamiltonian without making such assumptions. Using the relations between the wavevector components and refractive indices, we developed a program to numerically calculate Hamiltonian matrices from initial wavevector magnitudes. We then constructed plane-wave band structures with a simple periodic potential and no optical activity vector. From our results, we found that the Hamiltonian is produced from real eigenvalues but is non-Hermitian, unlike the effective paraxial Hamiltonian. Additionally, the refraction index surfaces impose a maximum condition for the possible wavevector components, therefore restricting the allowed bands. Future work will involve further developing the band structures and producing Chern number calculations for our model.