Surface plasmon polariton propagation near an index step

In this work we study theoretically the scattering of p-polarized light of frequency w from a system consisting of a dielectric medium (prism) characterized by a dielectric constant Eo in the region x(3) > D; a metal film characterized by a complex, frequency-dependent dielectric function epsilon (1)(omega) in the region 0 < x(3) < D; a dielectric film characterized by a dielectric constant epsilon (2) in the region zeta (x(1)) < x(3) < 0; and vacuum (epsilon (3) = 1) in the region x(3) < <zeta>(x(1)) The light, whose plane of incidence is the x(1)x(3)-plane, is incident through the prism. For the surface profile function zeta (x(1)) we take the form zeta (x(1)) = -d0(x(1)) or zeta (x(1)) = -d0(x(1))0(L -x(1)), where 0(x(1)) is the Heaviside unit step function. Thus we have a dielectric film of thickness d and dielectric constant epsilon (2) covering the half of the lower surface (x(3) = 0) of the metal film defined by x(1) > 0, or a dielectric film of thickness d and dielectric constant Ea covering the part of the lower surface (x(3) = 0) Of the metal film defined by 0 < x(1) < L. The reduced Rayleigh equation for the amplitude of the light scattered back into the prism, R(q/k), is obtained, and solved by the Wiener-Hopf method, and the result is used to calculate the intensity of the scattered field in the far field region as a function of x(1) for a fixed value of x(3) for several values 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 an index step on that interface. A brief discussion of the transmission of light through this system is also given.