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:

(9.44) |

(9.45) |

Then generally,

(9.46) |

(9.47) |

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:

(9.48) *unit helicity vectors*such that:

(9.49) (9.50) (9.51)

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:

(9.52) |

I'm leaving Stokes parameters^{9.6} 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.