In this paper an asymptotic solution to the wave equation for a thin curved layer with periodically inhomogeneous permittivity is derived. The wave equation and field components are expanded in the thickness h of the layer. A propagator, i.e., an operator that maps the fields from one side of the surface to the other, is derived. This propagator is used to derive a higher order impedance boundary condition. The solution is expressed as an asymptotic expansion in the thickness h for a layer coated on a perfect electric conductor (PEC). The problem treated is two-dimensional and the TE case is considered. A couple of numerical examples are given with comparisons to solutions obtained by integral equation based techniques.