Surface plasmon polariton propagation near an index step

In this work, we study theoretically the scattering of p-polarized light of frequency ω from, and its transmission through, a system consisting of a dielectric medium (prism) characterized by a dielectric constant ϵ0 in the region x3 > D; a metal film characterized by a complex, frequency-dependent dielectric function ϵ1(ω) in the region 0 < x3 < D; a dielectric film characterized by a dielectric constant ϵ2 in the region ζ(x1) < x3 < 0; and vacuum (ϵ3 = 1) in the region x3 < ζ(x1). The light, whose plane of incidence is the x1x3-plane, is incident through the prism. For the surface profile function ζ(x1) we assume the form ζ(x1) = −dθ(x1)θ(L − x1), where θ(x1) is the Heaviside unit step function. Thus, we have a dielectric film of thickness d and dielectric constant ϵ2 covering the part of the lower surface (x3 = 0) of the metal film defined by 0 < x1 < L. The reduced Rayleigh equation for the amplitude of the light scattered back into the prism, R(q|k), is derived, as is the reduced Rayleigh equation for the amplitude of the light transmitted into the vacuum, T(q|k). These integral equations are solved numerically and the results are used to calculate the intensity of the scattered field in the near- and far-field regions, and the intensity of the transmitted light in the near-field region, for several values of L and of the wavelength of the incident light. The results provide information about the scattering of the surface plasmon polariton at the metal-vacuum interface, excited by the incident light, by index steps on that interface, which can be used to determine the thickness of the dielectric film d, its extent L, and its dielectric constant ϵ2.