Polarization of Plane Waves

We've really done all of the hard work already in setting things up above (and it wasn't too hard). Indeed, the $\mbox{\boldmath$E$}_1$ and $\mbox{\boldmath$E$}_2$ 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 $\mbox{\boldmath$E$}$ in a fairly general way:

$$\mbox{\boldmath$E$}_i = \hat{\mbox{\boldmath$\epsilon$}}_i ... ...dmath$k$} \cdot \mbox{\scriptsize\boldmath$x$} - \omega t)},$$ (9.39)

Then we can return (as we will, over and over) to the curl equations to find:

$$\mbox{\boldmath$B$}_i = \sqrt{\mu \epsilon} \frac{\mbox{\boldmath$k$} \times \mbox{\boldmath$E$}_i}{k}$$ (9.40)

for $i = 1,2$ and $\hat{\mbox{\boldmath$\epsilon$}}_i$ a unit vector perpendicular to the direction of propagation $\mbox{\boldmath$n$}$.

Then generally,

$$\mbox{\boldmath$E$}(\mbox{\boldmath$x$},t) = (\hat{\mbox{\b... ...oldmath$k$} \cdot \mbox{\scriptsize\boldmath$x$} - \omega t)}$$ (9.41)

$$\mbox{\boldmath$B$}(\mbox{\boldmath$x$},t) = \frac{1}{v}(\h... ...boldmath$k$} \cdot \mbox{\scriptsize\boldmath$x$} - \omega t)}$$ (9.42)

where $E_1$ and $E_2$ 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 $E$ corresponds to a simple phase shift in the exponential.)

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:

1. If $E_1$ and $E_2$ have the same phase (but different magnitude) we have Linear Polarization of the $E$ field with the polarization vector making an angle $\theta = \tan^{- 1}(E_2/E_1)$ with ${\bf\epsilon}_1$ and magnitude $E = \sqrt{E_1^2 + E_2^2}$. Frequently we will choose coordinates in this case so that (say) $E_2 = 0$.

2. If $E_1$ and $E_2$ have different phases and different magnitudes, we have Elliptical Polarization. It is fairly easy to show that the electric field strength traces out an ellipse in the $1,2$ plane.

3. A special case of elliptical polarization results when the amplitudes are out of phase by $\pi/2$ and the magnitudes are equal. In this case we have Circular Polarization. Since $e^{i\pi/2} = i$, in this case we have a wave of the form:
   

   $$\mbox{\boldmath$E$} = \frac{E_0}{\sqrt{2}}\left(\hat{\mbox{... ...lon$}}_2\right) = E_0 \hat{\mbox{\boldmath$\bf\epsilon$}}_\pm.$$ (9.43)

   
   
   where we have introduced complex unit helicity vectors such that:
   

   $$\hat{\mbox{\boldmath$\bf\epsilon$}}^\ast_\pm \cdot \hat{\mbox{\boldmath$\bf\epsilon$}}_\mp = 0$$ (9.44)

   $$\hat{\mbox{\boldmath$\bf\epsilon$}}^\ast_\pm \cdot \hat{\mbox{\boldmath$\bf\epsilon$}}_3 = 0$$ (9.45)

   $$\hat{\mbox{\boldmath$\bf\epsilon$}}^\ast_\pm \cdot \hat{\mbox{\boldmath$\bf\epsilon$}}_\pm = 1$$ (9.46)

   
   

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:

$$\mbox{\boldmath$E$}({\bf x},t) = \left( E_+ \hat{\mbox{\bol... ...e\boldmath$k$}\cdot \mbox{\scriptsize\boldmath$x$} - \omega t}$$ (9.47)

It is left as an exercise to prove this. Note that as always, $E_\pm$ are complex amplitudes!

I'm leaving Stokes 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.