Bulk-selvedge coupling theory for the optical properties of surfaces

We derive expressions for the Fresnel coefficients using a theory which takes into account the coupling between bulk and selvedge; the selvedge is the region near the surface of thickness $l$, $\frac{l}{\lambda }\ll 1$, where $\lambda$ is the wavelength of light, in which the macroscopic Maxwell equations with a bulk dielectric constant cannot be used. The expressions involve selvedge response coefficients, and we show how to evaluate them for a given microscopic model of the surface. Since our expressions describe the coupling of bulk and selvedge to all orders, unlike earlier simple perturbation expressions, the dispersion relations of surface excitations can be determined from their poles. Using jellium metal with random-phase-approximation electron dynamics as an example, we derive a formula for the surface-plasmon dispersion relation, and compare it with earlier approximate formulas.