Novel Optical Materials

Over the past decade and a half, there has been a revolution in novel optical materials, with new directions continuously emerging. Some of the areas that we envisage this workshop encompassing include: (i) space-time microstructures (including foldable electromagnetic structures and manipulations of boundaries in time and space, where exciting phenomena have been observed); (ii) topological insulators, where topological constraints in the bulk lead to interesting surface effects and novel devices, arising through breaking space and/or time reversal symmetries and (iii) hyperbolic materials, where the electrical permittivity has both positive and negative values, with a small imaginary part. For quasistatics, in the limit as the imaginary part goes to zero, these are closely related to space-time microstructures, with time replaced by a spatial variable. Radiation can propagate along the characteristics, and this can be used for imaging. 

The goals will be first to expose mathematicians to these developments. Then secondly, to develop a synthesis between mathematicians and other scientists, including physicists, engineers, and material scientists, to better understand these phenomena.  Recently, methods have evolved to achieve homogenization theories in periodic materials when the wavelength is comparable to the unit cell size, as to be contrasted with the classic theory where the wavelength is much larger than the cell size.  The new homogenization gives effective equations for the slowly varying envelope function which modulates the waves. In this context, hyperbolic metamaterials and star-shaped wave forms are of particular interest. A challenge will be to extend these methods to random composites and composites with varying microstructure.