How Does Polarization Arise in the Solution of the Wave Equation?

[mahblah]

Homework Statement

The wave equation is
$\nabla^2 \mathbf{A}(\mathbf{r},t) = \frac{1}{c^2} \frac{\partial^2 \mathbf{A}(\mathbf{r},t)}{\partial t^2}$

I want to get a solution for the vector potential A.

Homework Equations

we can use the Fourier transformation

$\mathbf{A}(\mathbf{k},\omega) = \int{d\mathbf{r}}\int{dt \mathbf{A}(\mathbf{r},t) \exp{[-i(\mathbf{k \cdot r} - \omega t)]}}$

$\mathbf{A}(\mathbf{r},t) = \int{\frac{d\mathbf{k}}{(2\pi)^3}} \int{\frac{d\omega}{2 \pi} \mathbf{A}(\mathbf{k},\omega) \exp{[-i(\mathbf{k \cdot r} - \omega t)]}}$

to get

$\left( \mathbf{k}^2 - \frac{\omega^2}{c^2}\right) \mathbf{A}(\mathbf{k},\omega) =0$

(so $\omega = c k$ )

The Attempt at a Solution

Now the the solution is (formally)

$\mathbf{A}(\mathbf{r},t) = \sum_{\lambda =1,2} \int{\frac{d\mathbf{k}}{(2\pi)^3}} \int{\frac{d\omega}{2 \pi} A_\lambda(\mathbf{k},\omega) \hat{\epsilon}_\lambda(k) \cos{(\mathbf{k \cdot r} - \omega t + \varphi_\omega)} \delta(\omega -ck)}$

but i don't understand well why we have 2 polarization vector and form where they come...

thanks all,
MahBlah

 

[mark726]

Dear MahBlah,

The two polarization vectors, denoted as \hat{\epsilon}_\lambda(k), represent the two possible orientations of the electric field in a transverse electromagnetic wave. These vectors are orthogonal to the direction of propagation, which is given by the wave vector \mathbf{k}. The two possible orientations of the electric field are perpendicular to each other and to the direction of propagation, and are commonly referred to as the "x-polarization" and "y-polarization".

These polarization vectors come from the fact that the wave equation is a second-order partial differential equation, which allows for two independent solutions. These solutions correspond to the two polarizations of the electric field.

To better understand the origin of these polarization vectors, you can think of them as representing the two possible directions in which the electric field can oscillate in a transverse wave. For example, in a plane wave traveling in the z-direction, the electric field can oscillate in the x-direction or the y-direction. These two possible orientations are represented by the two polarization vectors.

I hope this helps clarify the concept of polarization in the solution to the wave equation. Keep up the good work in your studies of electromagnetism!