In the presence of absorption Maxwell's equations for a macroscopic dielectric show a time convolution. In particular, the time evolution is unidirectoral. We show how Maxwell's equations can be extended to a larger set of equations with a unitary time evolution exp[iKt] in a suitable Hilbert space. We then consider the situation where spatial periodicity is present. Such systems are called "photonic crystals" and are at present intensely studied in view of technological applications. After making a Bloch-Floquet decomposition and a complex dilatation we find, using analyticity and compactness argument, that the complex dilated version of K has spectrum consisting of areas in the lower complex half plane. We shall present a few numerical results for a two-dimensional case. Some physical implications are also discussed.
Heinz Siedentop