This is to clarify Landau's treatment of polarization in his classical

[Mindscrape]

This is to clarify Landau's treatment of polarization in his classical fields book.

Say we have a monochromatic plane wave
$$\mathbf{E}=\mathbf{E_0}e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$$
and we want to make a vector
$$\mathbf{b}=\mathbf{b_1}+i\mathbf{b_2}$$
where
$$\mathbf{b_1}\cdot\mathbf{b_2}=0$$
such that
$$\mathbf{E_0}=\mathbf{b}e^{-i\alpha}$$
meaning that
$$\mathbf{E}=\mathbf{b}e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t - \alpha)}$$

I'm having trouble seeing how the original plane wave variables relate to the new plane wave representation. Specifically, what are the original components E1, E2, with their phases in terms of b1, b2, and alpha.

I started breaking E_old into unit vectors and projecting it onto b1 and b2 vectors respectively, but didn't really seem to be getting anywhere.

 

[M Quack]

If E0 is written in some basis e1,2 then b1,2 have to be expressed in the same basis. Given that a common phase alpha is factored out, the vectors b1,2 should be real. Write out the real and imaginary parts of E0 from the equation after "such that" and take it from there.

 

[Mindscrape]

Why do they have to be expressed on the same basis, i.e. you are implying that b1 is parallel to e1 and and e2 parallel to b2? That seems to be the approach Landau takes to arrive at the equation for elliptical polarization, but was a constraint imposed afterwards. Can't b1 and b2 really be in any direction though (I suppose they could be expressed by the rotation matrix of some angle in terms of e1 and e2)?

This approach to polarization seems so simple and makes me so feel so stupid for not being able to fill in the gap.