Photonic band structure design using persistent homology

 

A short summary of our paper recently published in APL Photonics and presented at the APS March Meeting (talk slides available here).

Optimising the design of photonic crystals is challenging due to the large number of available degrees of freedom. We have shown how a machine learning technique known as persistent homology can be applied to classify the shape of photonic band structures and speed up the design process.

Persistent homology studies the shape of datasets over a range of different scales. Shapes are quantified by computing the numbers of topological features, such as holes or loops. Features persisting over a wide range of scales typically represent its true shape, while features with low persistence may be identified as noise and reliably discarded.

In the context of photonic crystals there are different important notions of shape. The shape of constant frequency lines or surfaces in the photonic band structure determines the radiation profile of emitters at that frequency embedded in the photonic crystal. The modes of photonic crystals also have abstract shapes characterised by topological invariants, which currently attract a lot of interest as a means of designing robust cavities and waveguides for light. 

We applied persistent homology to characterise low energy modes in a honeycomb photonic lattice and identify ranges of lattice parameters supporting interesting looped “moat band” and multi-valley dispersion relations. In the future, it will be interesting to apply persistent homology to study the properties of more complex many-body quantum systems.