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Electromagnetic wavepacket

  1. Nov 19, 2007 #1
    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 [itex]\omega[/itex] travelling in the z direction with a gaussian profile in the x and y directions.

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

    These of course can be combined to give the wave equation:
    [tex]\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}[/tex]

    3. The attempt at a solution

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

    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:
    [tex] \vec{E}(x,y,z) = \hat{x} \ E_0 \ g(x,y) \ \cos(kz - \omega t) [/tex]

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

    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
  2. jcsd
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