advantage over other – strictly numerical - solutions to Maxwell's equations for FSS. And that is that it yields a matrix equation of very small dimensionality, so it is amenable to solution on virtually any type of computer. The dimension of the matrix is determined by the number of current basis functions on each individual scatterer and can be as small as 1×1 for a dipole at or below resonance. The matrix elements however take longer to compute than with volumetric approaches such as FEM. Volumetric approaches require that a volume surrounding the unit cell be gridded accurately, and can require many thousands of elements for an accurate solution, though the matrices are usually sparse. Floquet's principle The spectral domain method is based on Floquet's principle, which implies that when an infinite, planar, periodic structure is illuminated by an infinite plane wave, then every unit cell in the periodic plane must contain exactly the same currents and fields, except for a phase shift, corresponding to the incident field phase. This principle allows all currents, fields and potentials to be written in terms of a modified Fourier series, which consists of an ordinary Fourier series multiplied by the incident field phase. If the periodic plane occupies the x-y plane, then the Fourier series is a 2-dimensional Fourier series in x, y. Plane wave spectrum As in Fourier optics, the Floquet–Fourier series expansion of fields and currents in the plane of the FSS leads immediately to the discrete plane wave spectrum representation of the fields on either side of the FSS. Field equations for 2D PEC frequency selective surfaces Perfectly electrically conducting (PEC) periodic surfaces are not only the most common but also the easiest to understand mathematically, as they admit only electric current sources J. This section presents the spectral domain method for analyzing a free-standing (no substrate) PEC FSS. The electric field E is related to the vector magnetic potential A via the well-known relation (Harrington [2001], Scott [1989], Scott [1997]): and the vector magnetic potential is in turn related to the source currents via (Harrington [2001], Scott [1997]): where Plane wave expansion of the fields in source-free media Frequency-selective surfaces are frequently stratified in the direction normal to the plane of the surface. That is, all dielectrics are stratified and all metallic conductors are considered stratified as well, and they will be regarded as perfectly planar. As a result,