Spatial dispersion is the effect where media respond not only to a signal at one particular point, but to signals in an area around that point. While temporal dispersion is a well studied topic, spatial dispersion is relatively unexplored. This thesis investigates the behaviour of electromagnetic waves in spatially dispersive, inhomogeneous media. In particular, two types of inhomogeneity are considered: media formed from two homogeneous regions with a common interface, and those with a periodic structure. For a material made of two homogeneous regions joined together we establish a set of boundary conditions to describe the behaviour of waves at this interface. These boundary conditions are additional to the standard ones provided by Maxwell’s equations. The conditions found are shown to reduce to those established previously by Pekar in the case of a boundary between a spatially dispersive region and a purely temporally dispersive region. The polarisation is also found for a spatially dispersive medium with periodic structure. Numeric solutions are found and non-divergent modes are identified. Analytic solutions are also found for small magnitudes of the inhomogeneity. Most interestingly these results show that, for certain conditions, there exist coupled mode solutions. This is an unusual phenomena which arises as a result of the spatial dispersion in the system.
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