Calculating the Polarization of a Plane Wave in Electromagnetic Exercises

[enpires]

Hello everybody :)
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
$\vec{E} = (\sqrt{2}\hat{x}+\hat{y}-\hat{z})e^{-2\pi 10^6(y+z)} = \hat{p}e^{-2\pi 10^6(y+z)}$
It is easy to calculate that the direction of propagation is
$\hat{k}= \frac{1}{\sqrt{2}}(\hat{y}+\hat{z})$
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 $\{\hat{x},\hat{y},\hat{z}\}$ to $\{\hat{u},\hat{v},\hat{w}\}$ where
$\hat{u} = \frac{1}{\sqrt{2}}(\hat{x}+\hat{z}) ,\quad \hat{v} = \frac{1}{\sqrt{2}}(\hat{x}+\hat{y}) ,\quad \hat{w} = \frac{1}{\sqrt{2}}(\hat{y}+\hat{z})$
Then rewrite $\hat{p}$ 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)
$\sqrt{2}\hat{x}+\hat{y}-\hat{z} = a\hat{u} + b\hat{v} + c\hat{w}= a (\hat{x}+\hat{z}) + b(\hat{x}+\hat{y}) + c(\hat{y}+\hat{z})$

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? :)

 

[Claude Bile]

The wave is polarized - I don't understand what you mean by "how" it is polarized: the polarization direction is quite clear from the vector that precedes the exponent.

The polarization is indeed orthogonal to the propagation direction as required (easily checked by taking the dot-product). If you want to rotate p so that the propagation direction lies along a principal axis, you can always employ a coordinate rotation (which is more intuitive than employing a transformation to a non-orthogonal basis set IMO).

Claude.

 

[dauto]

$\{\hat{u},\hat{v},\hat{w}\}$ are not perpendicular to each other.