Microwave Surface Impedance of a Periodic Medium

The surface impedance of a medium is calculated, the complex but scalar conductivity of which is a periodic function in a direction parallel to the surface. Radiation is assumed to impinge vertically onto the surface. It is shown that there are two independent field configurations according to whether the induced microwave current flows in a direction parallel to the conductivity pattern or perpendicular to it. These two configurations lead to different surface impedances, the one obtaining in the perpendicular configuration being generally larger. The calculation is carried through when the conductivity is of the form σ(y) = σ0 + 2σ1 cos (ky), and a local relation between current and fields is assumed. Equations are also derived for the more general problem when the periodic conductivity is an arbitrary Fourier series and a nonlocal relation between current and fields is assumed. These problems are of interest in measurements of the microwave surface impedance of hard super-conductors.