We've really done all of the hard work already in setting things up
above (and it wasn't too hard). Indeed, the
and
defined a few equations back are just two independent polarizations of a
transverse plane wave. However, we need to explore the rest of the *physics*, and understand just what is going on in the whole
electrodynamic field and not just the electric field component of same.

Let's start by writing in a fairly general way:

(11.40) |

where you will note that we have converted over to the notation with real, since there is no real reason to treat separately for a while. Then we can turn (as we will, over and over in the pages ahead) to the either of the curl MEs to find (using Faraday's Law in this case):

(11.41) |

with for such that for the two independent directions of polarization perpendicular to .

Then generally,

(11.42) |

(11.43) |

where and are (as usual) complex amplitudes since there is no reason (even in nature) to assume that the fields polarized in different directions have the same phase. (Note that a complex corresponds to a simple phase shift in the exponential, see preliminary section on complex numbers if this is not clear.)

The *polarization* of the plane wave describes the relative *direction, magnitude,* and *phase* of the *electric* part of the
wave. We have several well-known cases:

- If
and
have the same phase (but arbitrarily different
magnitudes) we have
**Linear Polarization**of the field with the polarization vector making an angle with and magnitude . Frequently we will choose coordinates in this case so that (say) . - If
and
have different phases
*and*different magnitudes, we have**Elliptical Polarization**. It is fairly easy to show that the electric field strength in this case traces out an*ellipse*in the plane. - A special case of elliptical polarization results when the
amplitudes are out of phase by
and the magnitudes are equal. In
this case we have
**Circular Polarization**. Since , in this case we have a wave of the form:(11.44)

where we have introduced complex*unit helicity vectors*such that:

0 (11.45) 0 (11.46) (11.47)

As we can see from the above, elliptical polarization can have positive
or negative **helicity** depending on whether the polarization vector
swings around the direction of propagation counterclockwise or clockwise
when looking into the oncoming wave.

Another completely general way to represent a polarized wave is via the unit helicity vectors:

(11.48) |

It is left as an exercise to prove this. Note that as always, are

I'm leaving Stokes parametersStokes Parameters out, but you
should read about them on your own in case you ever need them (or at
least need to know what they are). They are relevant to the issue of
*measuring* mixed polarization states, but are no more general a
description of polarization itself than either of those above.