# Homework Help: Electromagnetic wavepacket

1. Nov 19, 2007

### JustinLevy

Electromagnetic waves

1. The problem statement, all variables and given/known data
Find the solution of Maxwell's equations in vaccuum for a continuous beam of light of frequency $\omega$ travelling in the z direction with a gaussian profile in the x and y directions.

2. Relevant equations
Maxwell's equations in vaccuum.
$$\nabla \cdot \vec{E} = 0$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla \times \vec{E} = - \frac{\partial}{\partial t} \vec{B}$$
$$\nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial}{\partial t} \vec{E}$$

These of course can be combined to give the wave equation:
$$\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} \vec{E} = \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \vec{E}$$

3. The attempt at a solution

I already know what the plane wave solutions look like in the z direction:
$$\vec{E}(x,y,z) = \vec{E}_0 \cos(kz - \omega t)$$
where $$k = \omega/c$$

No polarization was specified, so let's choose linear polarization in the x direction. Our solution should then be similar for polarization in the y direction, and we can get the general solution of any polarization by adding these with arbitrary amplitudes and phases.

So, for linear polarization in the x direction and generalizing the above with dependence in the x-y direction it would be something like:
$$\vec{E}(x,y,z) = \hat{x} \ E_0 \ g(x,y) \ \cos(kz - \omega t)$$

Now seeing the constraint on g(x,y) we find:
$$0 = \nabla \cdot \vec{E} = E_{0} \ \cos(kz - \omega t) \frac{\partial}{\partial x}g(x,y)$$

Which seems to say there can't be any dependence on x! What!?

What am I doing wrong?
I've tried searching around the net and haven't found any good hints on this problem either. Please help.

Last edited: Nov 19, 2007