Abstract

Snell’s Law of geometric optics is a classical result which describes how light propagates when passing from one material to another. Similarly, Fermat’s principle says that light rays trace out paths which minimize a certain functional (optical path length). Both laws are well understood in classical materials (e.g. air, water, glass, etc.), both theoretically and experimentally. For more complex materials, such as liquid crystals, these laws remain important for understanding how light rays propagate through such a material and have recently appeared in studies involving optical interactions with topological solitons as well as Pancharatnam-Berry phase based lenses. I will discuss rigorous formulations of Snell’s Law in the context of liquid crystals (for instance, at the interface of a bi-axial nematic) and other general media such as metasurfaces. I will then discuss various challenges in this context as well as the utility of having such laws in hand mathematically, e.g. for applications to controlling light.