Classical “Dressing” of a Free Electron in a Plane Electromagnetic Wave

The behavior of a free electron in a electromagnetic wave is one of the most commonly discussed topics in classical electromagnetism. Yet, several basic issues remain to be clarified. These relate to the question: to what extent can net energy be transferred from an electromagnetic pulse (such as that of a laser) in vacuum to a free electron? These issues are made more complex by quantum considerations, including the role of the “quasimomentum” of an electron that is “dressed” by an electromagnetic wave [1]. As a small step towards understanding of the larger issues, we consider a simpler question here. The response of a free electron to a plane electromagnetic wave is oscillatory motion in the plane perpendicular to the direction of the wave, in the first approximation. Thus, the electron has momentum transverse to the direction of the wave. However, the wave contains momentum only in its direction, and the radiated wave contains no net momentum (in the nonrelativistic limit). How is momentum conserved in this process? The general sense of the answer has been given by Poynting [2], who noted that an electromagnetic field can be said to contain a flux of energy (energy per unit area per unit time) given by