- #1
JustinLevy
- 895
- 1
Hello, I am having trouble finding solutions for a localized electromagnetic pulse in 3-D. Ideally, I'd like the fields to be zero outside of a region centered on the pulse, but a gaussian wavepacket is fine with me as well.
Let's have the pulse travel in the z direction, the plane wave solutions look like:
[tex] \vec{E}(x,y,z) = \vec{E}_0 \cos(kz - \omega t) [/tex]
where [tex]k = \omega/c[/tex]
The constraint from Maxwell's equations, [tex]0 = \nabla \cdot \vec{E} = -[E_{0}]_z \ k \sin(kz - \omega t)[/tex] shows us that the electric field needs to be transverse to the direction of motion (ie. that E_z = 0).
Using Fourier transforms I can easily build a gaussian pulse (or any other shape) in the z direction using these plane waves. Basically, I can always write a solution of the form:
[tex] \vec{E}(x,y,z) = \vec{E}_0 \ f(kz - \omega t) [/tex]
for any function f. But the result is still an infinite plane in the x-y direction. How do I make this a localized pulse?
I still want the pulse to travel in the z direction, and let's just choose linear polarization in the x direction, and then generalizing this 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) \ f(kz - \omega t) [/tex]
Now seeing the constraint on g(x,y) we find:
[tex]0 = \nabla \cdot \vec{E} = E_{0} \ f(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? What is the correct way to continue to get some localized electromagnetic pulse. Heck, even localized in two dimensions (a continuous laser beam for example) would be useful.
Let's have the pulse travel in the z direction, the plane wave solutions look like:
[tex] \vec{E}(x,y,z) = \vec{E}_0 \cos(kz - \omega t) [/tex]
where [tex]k = \omega/c[/tex]
The constraint from Maxwell's equations, [tex]0 = \nabla \cdot \vec{E} = -[E_{0}]_z \ k \sin(kz - \omega t)[/tex] shows us that the electric field needs to be transverse to the direction of motion (ie. that E_z = 0).
Using Fourier transforms I can easily build a gaussian pulse (or any other shape) in the z direction using these plane waves. Basically, I can always write a solution of the form:
[tex] \vec{E}(x,y,z) = \vec{E}_0 \ f(kz - \omega t) [/tex]
for any function f. But the result is still an infinite plane in the x-y direction. How do I make this a localized pulse?
I still want the pulse to travel in the z direction, and let's just choose linear polarization in the x direction, and then generalizing this 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) \ f(kz - \omega t) [/tex]
Now seeing the constraint on g(x,y) we find:
[tex]0 = \nabla \cdot \vec{E} = E_{0} \ f(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? What is the correct way to continue to get some localized electromagnetic pulse. Heck, even localized in two dimensions (a continuous laser beam for example) would be useful.