  We consider a class of potential problems on a periodic half-space for the
modelling of electrified oil films, which are used in the development of novel
switchable liquid optical devices (diffraction gratings). A boundary integral
formulation which reduces the problem to the study of the oil-air interface
alone is derived and solved in a highly efficient manner using the Nystr\"{o}m
method. The oil films encountered experimentally are typically very thin and
thus an interface-only integral representation is important for avoiding the
near-singularity problems associated with boundary integral methods for long
slender domains. The super-algebraic convergence of the proposed methods is
discussed and demonstrated via appropriate numerical experiments.
