Plane-Wave Propagation in Unbounded Media

Let us look for plane-wave solutions to the Maxwell equations (2.12)- (2.15). What does this statement mean We know that the electromagnetic field (E, H) cannot be arbitrarily specified. Only certain electromagnetic fields, those that satisfy the Maxwell equations, are physically realizable. Therefore, because of their simple form, we should like to know under what conditions plane electromagnetic waves 

E0exp( — k x) and H0exp( —k x) are the amplitudes of the electric and magnetic waves, and / = k x — ut is the phase of the waves. An equation of the form K x = constant, where K is any real vector, defines a plane surface the normal to which is K. Therefore, k is perpendicular to the surfaces of constant phase, and k is perpendicular to the surfaces of constant amplitude. If k and k are parallel, which includes the case k = 0, these surfaces coincide and the waves are said to be homogeneous if k and k are not parallel, the waves are said to be inhomogeneous. For example, waves propagating in a vacuum are homogeneous. 

Let us briefly consider propagation of surfaces of constant phase. Choose an arbitrary origin O and a plane surface over which the phase J is constant (Fig. 2.2). At time t the distance from the origin O to the plane is z, where k x = k z and k z — ut = j . In a time interval At the surface of constant phase will have moved a distance A z, where 

the velocity of propagation of surfaces of constant phase, the phase velocity v, is 

Equations (2.41) and (2.42) are the conditions for transversality k is perpendicular to E0 and H0. It is also evident from (2.43) or (2.44) that E0 and H0 are perpendicular. However, k, E0, and H0 are, in general, complex vectors, and the interpretation of the term perpendicular is not simple unless the waves are homogeneous for such waves, the real fields E and H lie in a plane the normal to which is parallel to the direction of propagation. 

---