Closed-form low frequency solutions for electromagnetic waves through a frequency selective surface

A simple closed-form approximate solution is given to the problem of transmission of a low-frequency electromagnetic wave through a frequency selective surface (FSS). FSS are periodic metal plates (or their complimentary apertures) sandwiched between dielectric slabs. At low frequencies, the induced currents on the metal plates may be approximated by a known function with a constant coefficient to be determined by the boundary conditions. Based on such a "one-mode" approximation, we derive a closed-form solution for the scattered field for FSS with multiple narrow rectangular slots, with a single wide rectangular aperture, and a circular aperture. When compared with the available exact solutions, we find that the one-mode approximation is valid when the period
a
of the FSS is such that
a/\lambda
is small enough that no grating lobe appears, e.g.,
a < \lambda
for normal incidence.