- #1

Mindscrape

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Say we have a monochromatic plane wave

[tex]\mathbf{E}=\mathbf{E_0}e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}[/tex]

and we want to make a vector

[tex]\mathbf{b}=\mathbf{b_1}+i\mathbf{b_2}[/tex]

where

[tex]\mathbf{b_1}\cdot\mathbf{b_2}=0[/tex]

such that

[tex]\mathbf{E_0}=\mathbf{b}e^{-i\alpha}[/tex]

meaning that

[tex]\mathbf{E}=\mathbf{b}e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t - \alpha)}[/tex]

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.