Maxwell’s Equations and Plane Wave Propagation

Abstract

In this chapter, many of the equations and concepts fundamental to later developments are presented. The discussion beg ins with a statement of Maxwell’s equations and a short discussion of what additional information is necessary to find solutions to them. This discussion leads to the concept of constitutive relations. The presentation turns to the discussion of wave propagation in the medium with the simplest constitutive relations, that of free space. Monochromatic plane waves are taken up as the simplest of solutions to Maxwell’s free space equations. Energy propagation is introduced through the construct of Poynting’s vector. This vector then is used in a discussion of the physical reality of plane wave solutions, which by nature must be truncated. The discussion then turns to that of polarization and its representation in terms of complex numbers. The following section takes up the more realistic case of the propagation of polychromatic plane waves. The section’s presentation is motivated by a discussion of the nature of thermal electromagnetic sources and the concepts of stimulated and spontaneous emission. The Poynting vector is then generalized to the polychromatic case. The introduction of the Wiener-Khintchine theorem allows discussion to turn to spectra of nonthermallight sourees. A brief discussion of quasi-monochromatic Jones vectors is then used to introduce the concept of polychromatic polarization and its description in terms of Stokes parameters.