<PERSON> EM Conundrum: Power Spectrum Polarization dependence for Thomson Scattering
I'm having some issues wrapping my head around what <PERSON> did for his derivation of nonrelativistic Thomson scattering. I'm using Gaussian units although I don't think there's a conflict here
Nonrelativistic electric field for an accelerated particle is given by (implicit retarded time)
$\vec{E} = \frac{q}{rc} (\hat{n} \times (\hat{n} \times \dot{\vec{\beta}})$
The power pattern (angular distribution of power) just follows from
$\frac{dP}{d\Omega} = \frac{c}{4\pi}r^{2}|\vec{E}|^{2} = \frac{e^{2}}{4\pi c}|\hat{n} \times (\hat{n} \times \dot{\vec{\beta}})|^{2}$
In the instance of <PERSON> scattering, we have an incident plane wave driving oscillations of an electron up and down along a fixed axis. Now I can solve this just fine and get the correct $\sin^{2}\theta$ dependence. 394
My question rests more on what the hell <PERSON> is doing.
In equation 14.120 he'll just jump and say that
$\frac{dP}{d\Omega} = \frac{e^{2}}{4\pi c^{3}}|\epsilon^{*} \cdot \dot{\vec{v}}|^{2}$
Where he says that $\epsilon$ is the polarization state that the power is radiated into.
OK so I tried to figure this out which I imagined would be sort of straightforward. I presume that the vector $\hat{n} \times (\hat{n} \times \hat{\beta})$ is the polarization of radiated light, so simply working out the geometry should do it right? Using a triple product vector identity, I get
$\hat{n} \times (\hat{n} \times \hat{\beta}) = \hat{\beta} - (\hat{\beta}\cdot \hat{n})\hat{n}$
If I calculate the square magnitude as the original power pattern formula says, I get
$1 - (\hat{\beta}\cdot \hat{n})^{2}$
If assume I have an electric field incident along the $\hat{z}$ which drives oscillations in the $z$ direction, then following spherical geometry I get the $\sin^{2} \theta$ shape of the power pattern.
Trying <PERSON>'s method, using $\hat{n} \times (\hat{n} \times \hat{\beta})$ as the polarization doesn't work! In fact, it gives the square of the previous answer. What the heck is going on?
EDIT: I'm starting to suspect that this is just awful notation for what <PERSON> is referring to. Does he mean that we are just projecting the vector amplitude down into a particular polarization basis and calculating the relative amplitudes that way? 346