Plane waves can propagate in any direction. Any superposition of these waves, for all possible , is also a solution to the wave equation. However, recall that and are not independent, which restricts the solution in electrodynamics somewhat.
To get a feel for the interdependence of
so that e.g.:
[Note in passing that:
If there is dispersion (where the velocity of the waves is a function of the frequency) then the fourier superposition is no longer stable and the last equation no longer holds. Each fourier component is still an exponential, but all the velocities of the fourier components are different. As a consequence, any initially prepared wave packet spreads out as it propagates. We'll look at this shortly (in the homework) in some detail to see how this works for a very simple (gaussian) wave packet but for now we'll move on.
are connected by having to satisfy
Maxwell's equations even if the wave is travelling in just one direction
(say, in the direction of a unit vector
); we cannot choose
the wave amplitudes separately. Suppose
Note that applying to these solutions in the HHE leads us to:
This has mostly been ``mathematics'', following more or less directly from the wave equation. The same reasoning might have been applied to sound waves, water waves, waves on a string, or ``waves'' of nothing in particular. Now let's use some physics and see what it tells us about the particular electromagnetic waves that follow from Maxwell's equations turned into the wave equation. These waves all satisfy each of Maxwell's equations separately.
For example, from Gauss' Laws we see e.g. that:
Repeating this sort of thing using one of the the curl eqns (say, Faraday's law) one gets:
If is a real unit vector in 3-space, then we can introduce three real, mutually orthogonal unit vectors such that and use them to express the field strengths:
We have carefully chosen the polarization directions so that the
(time-averaged) Poynting vector for any particular component pair
points in the direction of propagation,
Now, kinky as it may seem, there is no real11.5 reason that cannot be complex (while remains real!) As an exercise, figure out the complex vector of your choice such that
Of course, I didn't really expect for you to work it out on such a sparse hint, and besides, you gotta save your strength for the real problems later because you'll need it then. So this time, I'll work it out for you. The hint was, pretend that is complex. Then it can be written as:
Thus the most general such that is
Fortunately, nature provides us with few sources and associated media that produce this kind of behavior (imaginary ? Just imagine!) in electrodynamics. So let's forget it for the moment, but remember that it is there for when you run into it in field theory, or mathematics, or catastrophe theory.
We therefore return to a more mundane and natural discussion of the possible polarizations of a plane wave when is a real unit vector, continuing the reasoning above before our little imaginary interlude.