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.:

(11.19) | |||

(11.20) |

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:

(11.21) |

for arbitrary smooth and is the most general solution of the 1-dimensional wave equation.

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:

(11.22) |

as the condition for a solution. Then a

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:

0 | |||

0 | |||

0 | |||

0 | (11.23) |

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

(11.24) |

Which basically means for a

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

(11.25) |

(the cancels, ). This means that and have the

*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:

(11.26) |

and

(11.27) |

where and are constants that may be complex. It is worth noting that |E| = v|B| have the same dimensions and that the magnitude of the electric field is greater than that of the magnetic field to which it is coupled via Maxwell's Equations by a factor of the speed of light in the medium, as this will be used a

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^{11.5} reason
that
cannot be complex (while
remains real!)
As an exercise, figure out the complex vector of your choice such that

(11.32) |

Did you get that? What, you didn't actually try? Seriously, you're going to have to at least

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:

(11.33) |

(11.34) |

(11.35) |

So, must be orthogonal to and the difference of their squares must be one. For example:

(11.36) |

works, as do infinitely more More generally (recalling the properties of hyberbolics functions):

(11.37) |

where the unit vectors are orthogonal should work for any .

Thus the **most general**
such that
is

(11.38) |

where (sigh) and are again, arbitrary complex constants. Note that if is complex, the exponential part of the fields becomes:

(11.39) |

This

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.