How to set research goals

This is a question that is often asked by PhD students. Research is typically a highly nonlinear process. It is rare to sit down, identify a problem, the appropriate approach to solving it, and succeed on the first attempt. Therefore, rather than focusing on immediately solving some big outstanding problem, you should start by solving smaller specific problems in order to build up your skill set. Let me give a few examples from my PhD thesis:

1. Our collaborators performed experiments using optically-induced photonic lattices, and we had a new postdoc in the group who had worked on photonic graphene-like lattices during his PhD. So we wanted to explore graphene physics using optically-induced photonic lattices. The effect we wanted to see required a ideal conical band structure in the vicinity of graphene's Dirac points, but the experimental platform had strong anisotropy between the vertical and horizontal axes which broke the symmetry. This motivated me to look at other classes of lattices with square symmetry exhibiting similar conical dispersion relations. I stumbled upon the face-centred square lattice (also known as the Lieb lattice) which has a conical dispersion relation with intersecting flat band, and started studying its properties. This is how I first encountered flat band physics, which has now blossomed into a large and active research field!

2. Parity time symmetry and non-Hermitian photonics were topics starting to attract growing interest. As an introduction to this field, I studied how parity time-symmetric perturbations can affect the properties of vortex solitons in simple ring-shaped photonic lattices, building on the research topic of my honours thesis (vortex solitons in Hermitian ring lattices). While published a short letter on our findings, I did not pursue non-Hermitian photonics further during my PhD. But it did end up being background for work I did a few years later on the combination of topology with non-Hermitian systems, which is now also a booming research direction.

3. In the final year of my PhD I was interested in learning more about quantum photonics and topological systems. Spontaneous parametric down conversion in nonlinear waveguide lattices was a topic being studied theoretically and experimentally by colleagues in our department, which motivated me to study the effect of topological edge states on this process. While the calculations were quite elementary in hindsight, it served as a valuable first introduction to topological effects in lattices and quantum photonics.

The important take home message from all of these examples is that we started out with a rather small and simple problem we wanted to solve, typically involving the combination of two distinct topics or sub-fields. We did not anticipate the eventual important applications at the outset. So don't agonize so much about whether your current research project is solving one of the big open problems in your field. What really matters is whether it allows you to pick up new skills and hopefully identify fresh approaches to solving the big problems!