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Chapter 9:
Through most of this book we assume that the vector character of light waves can be ignored. In all practical situations this is not strictly true and we must keep in mind that the results derived are only valid as approximations. Although these approximations are fairly accurate most of the time there are cases when the vector character of the wave plays an important role. The purpose of this chapter is to gain some insight into the vector character of the electromagnetic radiation in order to understand some implications of the scalar approximation and learn how the vector nature can be exploited to our benefits. In homogeneous and isotropic media the solution of the wave equation for electromagnetic radiation usually leads to transverse fields. At a given instant, the electric field E and the magnetic field H are orthogonal to the local propagation direction and to each other [Fig. 9-1 (a)]. It should be noted, however, that there are exceptions to this rule. One such example is the field within the focal region of a tightly focused beam [71]. Not considering these special cases, the view over a plane perpendicular to the propagation direction (z) is shown in Fig. 9-1(b). In general, the field vectors are functions of position and time and the diagrams of Fig. 9-1 refer to a given position in space at a given instant of time. To maintain this picture we have, in principle, to define a special coordinate system for each point. To bypass this difficulty it is customary to treat vector problems only with respect to plane wave propagation. More complicated situations can be solved by a decomposition into a plane wave spectrum [72], and, as long as only linear media are considered, each plane wave component can be treated separately. For most of this chapter we shall also restrict our analysis to plane waves but will briefly discuss in section 9.8 some of the problems encountered with nonplanar wavefronts.
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