Role of evanescent waves in power calculations for counterpropagating beams

A general expression is obtained for the time-average power passing through a plane transverse to the direction of propagation for two counterpropagating electromagnetic beams. Each beam is represented by its plane-wave spectrum, which contains both propagating and evanescent plane waves. The expression clearly shows that, under certain conditions, the evanescent plane waves contribute to the time-average power passing through the plane. This is in contrast to the case of a single electromagnetic beam, in which only the propagating plane waves contribute to the time-average power passing through the plane. The utility of the expression is demonstrated with a practical example: a line current placed over a dielectric slab. Here the counterpropagating beams are the incident and reflected fields in the region between the current and the slab. The expression is applied to a plane in this region, and it is used to determine the time-average power associated with the evanescent waves passing through this plane. This power is then shown to be equal to the time-average power carried by the guided modes of the slab.