Microscopic calculation of electromagnetic fields in refraction at a jellium-vacuum interface

For a long-wavelength electromagnetic wave of frequency $\omega$, incident on a jellium-vacuum interface, the spatial dependence of the vector potential, $\overrightarrow{\mathrm{A}}(\overrightarrow{\mathrm{r}};\omega )$, is evaluated in the surface region. A simple integral equation is derived which relates $\overrightarrow{\mathrm{A}}(\overrightarrow{\mathrm{r}};\omega )$ to the jellium nonlocal conductivity tensor $\overleftrightarrow{\sigma }(\overrightarrow{\mathrm{r}};\overrightarrow{\mathrm{r}},\omega )$; numerical calculations based on this equation are reported, in which the random-phase approximation to $\overleftrightarrow{\sigma }(\overrightarrow{\mathrm{r}};\overrightarrow{\mathrm{r}},\omega )$ was used—the required single-electron wave functions were evaluated via the self-consistent surface-barrier potentials of Lang and Kohn. Families of graphs of $\overrightarrow{\mathrm{A}}(\overrightarrow{\mathrm{r}};\omega )$ are presented, for fixed bulk electron concentration as a function of frequency and for fixed $\frac{{\omega }_p}{\omega }$ (where ${\omega }_p$ is the plasma frequency) as a function of bulk electron concentration. The sensitivity of $\overrightarrow{\mathrm{A}}(\overrightarrow{\mathrm{r}};\omega )$ to the shape of the surface potential barrier, at fixed $\omega$ and bulk electron concentration, is also explored. The use of the results obtained for $\overrightarrow{\mathrm{A}}(\overrightarrow{\mathrm{r}};\omega )$ is proposed for the calculation of refraction effects in surface photoemission and in reflection spectroscopy.