we are excluding metallic vias (wires perpendicular to the plane of the FSS) which could potentially connect currents from different strata of the FSS structure. With this type of a stratified structure in mind, we can then use a plane wave expansion for the fields in and around the FSS, since plane waves are the eigenfunction solution to the vector wave equations in source-free media. To solve equations () and () for a free-standing, doubly periodic surface, we consider an infinite 2D periodic surface occupying the entire x-y plane, and assume a discrete plane wave expansion for all currents, fields and potentials (Tsao [1982], Scott [1989], Fourier optics): where for mathematical simplicity, we assume a rectangular lattice in which α only depends on m and β only depends on n. In the equations above, and, where lx, ly are the dimensions of the unit cell in the x,y directions respectively, λ is the free space wavelength and θ0, φ0 are the directions of an assumed incident plane wave, with the FSS regarded as lying in the x-y plane. In (), the root is taken which has a positive real part and non-positive (i.e., either negative or zero) imaginary part). Integral equation for free-standing PEC FSS Substituting equations () into () and () yields the spectral domain Greens function relating the radiated electric field to its source currents (Scott [1989]), where we now consider only those components of the field vectors lying in the plane of the FSS, the x-y plane: where, One notices the branch point singularity in the equation above (the inverse square root singularity), which is no problem thanks to the discrete spectrum, as long as the wavelength never equals the cell spacing. With this, the electric field boundary condition on the surface of PEC material within a unit cell becomes (Scott [1989]): where again, we are restricting our attention to the x,y components of currents and fields, which lie in the plane of the scatterer. Equation () is not strictly correct, since only the tangential components of electric field are actually zero on the surface of the PEC scatterers. This inexactness will be resolved presently when () is tested with the current basis functions, defined as residing on the surface of the scatterer. In this type of problem, the incident field is considered a plane wave expressed as in the x-y plane. Method of moments (MoM) solution