  The analysis of fields in periodic dielectric structures arise in numerous
applications of recent interest, ranging from photonic bandgap (PBG) structures
and plasmonically active nanostructures to metamaterials. To achieve an
accurate representation of the fields in these structures using numerical
methods, dense spatial discretization is required. This, in turn, affects the
cost of analysis, particularly for integral equation based methods, for which
traditional iterative methods require O(N^2) operations, N being the number of
spatial degrees of freedom. In this paper, we introduce a method for the rapid
solution of volumetric electric field integral equations used in the analysis
of doubly periodic dielectric structures. The crux of our method is the ACE
algorithm, which is used to evaluate the requisite potentials in O(N) cost.
Results are provided that corroborate our claims of acceleration without
compromising accuracy, as well as the application of our method to a number of
compelling photonics applications.
