Plane wave packets and beams
This chapter is devoted to an example which is based on plane waves, but the concepts developed have a much broader scope of validity. Completeness problems, related to a partial differential equation, are well defined only when boundary conditions are specified. In general, the weaker the boundary conditions, the wider the set, in order to be complete. A monochromatic plane wave is a simple, exact solution of Maxwell’s equations, but it can not be taken literally, because it is unphysical under several viewpoints. In order to find similar but physically acceptable solutions, one has to introduce approximations. The chapter considers a packet of uniform plane waves that travel along one direction. It focuses on one or more components of the e.m. field, in a cartesian reference frame, in ways that were not significantly affected by the vector nature of these fields.