Calculating the Polarization of a Plane Wave in Electromagnetic Exercises

• enpires
In summary, the conversation is about calculating the polarization of a plane wave and determining if it is polarized. The direction of propagation is found to be along the \hat{k} vector, and the speaker suggests making a change of basis to \{\hat{u},\hat{v},\hat{w}\} to analyze the polarization direction. However, this leads to strange values for the coefficients and the speaker is unsure of their mistake. Another person in the conversation points out that the polarization direction is clear from the vector preceding the exponent and that a coordinate rotation can be used to align the propagation direction with a principal axis. The conversation ends with a comment about the basis vectors not being perpendicular to each other.
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? :)

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.

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

1. What is polarization of a plane wave?

Polarization of a plane wave refers to the direction of the electric field oscillations of the wave. This direction can be either linear, circular, or elliptical.

2. How is polarization of a plane wave measured?

Polarization of a plane wave can be measured using a polarizer, which is a device that only allows light waves with a specific direction of polarization to pass through. The intensity of the light passing through the polarizer can then be measured to determine the polarization of the wave.

3. What is the difference between linear and circular polarization?

In linear polarization, the electric field oscillations of the wave occur in a straight line. In circular polarization, the electric field oscillations occur in a circular motion. This means that in circular polarization, the direction of the electric field changes as the wave propagates.

4. Can a plane wave have both linear and circular polarization?

No, a plane wave can only have one type of polarization at a time. However, a combination of linearly and circularly polarized waves can create elliptical polarization.

5. What factors can affect the polarization of a plane wave?

The polarization of a plane wave can be affected by the material it passes through, the angle at which it strikes a surface, and the presence of other electromagnetic fields. These factors can cause the wave to change its direction of polarization or become a combination of different polarizations.

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