  Electromagnetic waves interacting with three--dimensional periodic structures
occur in many applications of great scientific and engineering interest. These
three dimensional interactions are extremely complicated and subtle, so it is
unsurprising that practitioners find their rapid, robust, and accurate
numerical simulation to be of paramount interest. Among the wide array of
possible numerical approaches, the High--Order Spectral algorithms are often
preferred due to their surpassing fidelity with a moderate number of unknowns,
and here we describe an algorithm that fits into this class. In addition, we
take a perturbative approach to the problem which views the deviation of the
permittivity from a reference value as the deformation and we conduct a regular
perturbation theory. This work concludes a line of research on these methods
which began with two-dimensional problems governed by the Helmholtz equation
and moved to small perturbations in the fully three-dimensional vector Maxwell
equations. We now extend these latter results to large (real) perturbations
constituting a rigorous analytic continuation.
