I am studying polarization of EM waves. Where does this general form come from? 
This is a snapshot from The Physics of Waves by Georgi. I am wondering if anyone could explain where this general form of polarization vector comes from? Thank you very much. 
 A: A simple linear(ly) polarized transversal wave propagating in the $\hat z$ direction if polarized along the $\hat x$ axis can be written as $\textbf{A} = \hat x A cos(\omega t -\kappa z)$ where $A$ is the oscillation amplitude and $\kappa = 2\pi / \lambda$. Now if you superimpose on the wave $\textbf{A}$ another linearly polarized wave $\textbf{B} = \hat y B cos(\omega t -\kappa z +\phi)$  that also propagates in the $\hat z$ direction but oscillates along the $\hat y$ direction with amplitude $B$ and phase $\phi$ relative to the first one then the sum of the two waves will be $$ \textbf{C} = \hat x A cos(\omega t -\kappa z) + \hat y B cos(\omega t -\kappa z +\phi) $$ This wave can also exist just like the other two if the medium of propagation is linear and thus linear superposition holds. 
Now assume that $\phi = -\pi/2$ in which case $ \textbf{C} = \hat x A cos(\omega t -\kappa z) + \hat y B sin(\omega t -\kappa z) $. For any $\alpha= \omega t -\kappa z$ as $\alpha$ runs over all values in $[0, 2\pi]$ the end point of the vector $ \textbf{C} = \hat x A cos(\alpha) + \hat y B sin(\alpha) $ describes an ellipse of semi-major and semi-minor axes $A$ and $B$, and is called elliptic polarization. In the special case of $A=B$ you get circular polarization for obvious reasons and is used exclusively in earth-satellite radio links, for example.
The complex notation in your textbook comes from the Euler formula replacing the trigonometric functions.
$$ \textbf{C} = \hat x A cos(\omega t -\kappa z) + \hat y B sin(\omega t -\kappa z) \\ = \Re [\hat x A e^{\mathfrak{j}( \omega t -\kappa z))}] +\Re [\hat y B e^{\mathfrak{j}( \omega t -\kappa z -\pi/2))}] $$ or
$$ \tilde {\textbf{C}} = \hat x \tilde{A}  -\mathfrak{j} \hat y \tilde{B}  $$
where the tilde denotes the complex amplitude.
Now if the two linear polarized components are not aligned with the coordinate axes you still get the endpoint describing an ellipse as is in your textbook just with tilted major/minor axes relative to the coordinate axes.
