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enpires

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I'm doing some electromagnetic exercises, but I got stuck in calculating the polarization of a plane wave.

The complex field associated to the wave is the following

[itex]\vec{E} = (\sqrt{2}\hat{x}+\hat{y}-\hat{z})e^{-2\pi 10^6(y+z)} = \hat{p}e^{-2\pi 10^6(y+z)}[/itex]

It is easy to calculate that the direction of propagation is

[itex]\hat{k}= \frac{1}{\sqrt{2}}(\hat{y}+\hat{z})[/itex]

but now I want to calculate if the wave is polarized and how.

If the propagation direction was along one of the axis was pretty easy, I just need to look at the components in front of the exponential. But now that I can't how can I do it?

My idea was to make a change of the basis, from [itex]\{\hat{x},\hat{y},\hat{z}\}[/itex] to [itex]\{\hat{u},\hat{v},\hat{w}\}[/itex] where

[itex]\hat{u} = \frac{1}{\sqrt{2}}(\hat{x}+\hat{z}) ,\quad \hat{v} = \frac{1}{\sqrt{2}}(\hat{x}+\hat{y}) ,\quad \hat{w} = \frac{1}{\sqrt{2}}(\hat{y}+\hat{z})[/itex]

Then rewrite [itex]\hat{p}[/itex] in this base, which became (I don't consider the square root factor, it doesn't change the result since I'm just looking how the components are related to each other)

[itex]\sqrt{2}\hat{x}+\hat{y}-\hat{z} = a\hat{u} + b\hat{v} + c\hat{w}= a (\hat{x}+\hat{z}) + b(\hat{x}+\hat{y}) + c(\hat{y}+\hat{z})[/itex]

Which gives me values for all a,b and c. And that's strange (c should be zero since is a plane wave so there shouldn't be anything along this direction).

So... What's my mistake? :)