van der Waals and retardation (Casimir) interactions of an electron or an atom with multilayered walls

We use the quantized surface mode technique to evaluate the interaction V of a ‘‘particle’’ (an electron or atom) with two sets of plane parallel walls of arbitrary thicknesses and arbitrary permittivities; one set is on one side of the particle and the other set is on the other side. It is shown that a set of walls on either side of the particle can be treated as a single wall with an effective reflection coefficient scrR, a function of the frequency and angle of incidence of an incoming plane wave. The two sets of walls have thereby been effectively reduced to one wall on each side. Using a modified version of the standard image technique—the locations of the images are the usual ones, but the strengths are modified—one finds that the closed form for V can be reexpressed as a sum of interactions of the set of images on a given side with the wall on the other side. Taking various limits (thin walls, thick walls, metallic walls, dilute media) reproduces many known interactions and also provides a number of results not previously obtained. The latter results enable us to give quantitative estimates of the error incurred in approximating a wall of finite thickness by a wall of semi-infinite thickness, or approximating a wall with large permittivity by a wall of infinite permittivity. In an attempt to make the paper user friendly, we provide a tabulation of many of the short-range (van der Waals-like) and long-range (retardation) interactions now known, with equation references.