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
and
, let's
pick
so that e.g.:

(9.22) | |||

(9.23) |

which are plane waves travelling to the right or left along the -axis for any complex , . In one dimension, at least, if there is no dispersion we can construct a fourier series of these solutions for various that converges to any well-behaved function of a single variable.

[Note in passing that:

(9.24) |

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.

Note that
and
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

where , , and are constant vectors (which may be complex, at least for the moment).

Note that applying
to these solutions in the HHE
leads us to:

(9.25) |

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:

(9.26) |

or (dividing out nonzero terms and then repeating the reasoning for ):

(9.27) |

Repeating this sort of thing using one of the the curl eqns (say,
Faraday's law) one gets:

(9.28) |

*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:

(9.29) |

(9.30) |

(9.31) |

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,
:

Note well the combination , as it will occur rather frequently in our algebra below, so much so that we will give it a name of its own later. So much for the ``simple'' monochromatic plane wave propagating coherently in a dispersionless medium.

Now, kinky as it may seem, there is no real^{9.5} reason
that
cannot be complex (while remains real!)
As an exercise, figure out the complex vector of your choice such that

(9.36) |

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:

(9.37) |

(9.38) |

(9.39) |

(9.40) |

(9.41) |

Thus the **most general**
such that
is

(9.42) |

(9.43) |

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.