Optics - Electric Field (E0) of a Pulse

[mooeypoo]

Hey guys,

 

I missed class the other day and we have hw that are going to be in the exam we have this week. I'm confused and stumped and I can't find info online. There's a strong possibility I'm overcomplicating stuff because I'm stressing myself out, but.. well.. that's why I'm trying to get help here.

 

Naturally, I'm not looking for the solution alone, but since I missed the class, I''m not entirely sure I know if I'm doing it right. Also, the professor doesn't go by the book, so it makes things a lot harder.

 

So, here's the first question I'm in trouble with:

Find the electric ([math]E_0[/math]) (in V/cm) and magnetic (B) (in gauss) fields for 10 mjoule pulse of 100 fs duration focused to 1 cm spot.

So, here's what I think (absolutely not sure, tho, again, I missed class):

 

[math]t=100fs[/math]

[math]Area=1 cm^2[/math]

[math]P=10 \text{mjoule} = 10^7 \text{joule} = 6.2*10^{25} eV[/math]

 

And I start with the wave equation:

 

[math]E=E_0 \exp{(100\omega - kz)}[/math]

 

And

 

[math]\frac{P}{\text{Area}}= \frac{E_0^2}{2 \mu v}[/math]

[math]\frac{6.2*10^{25} eV}{1}= \frac{E_0^2}{2 \mu c}[/math]

[math]6.2*10^{25}\mu c=E_0^2[/math]

[math]E_0=\sqrt{6.2*10^{25}\mu c}[/math]

 

Is that right? I'm not sure also about my units, and also, where is the time info (the fact it's a 100fs pulse) fit into this?

 

And that would mean that to find B, I'm still using

 

[math]B=\frac{1}{c}E_0 = \frac{\sqrt{6.2*10^{25}\mu c}}{c}[/math]

 

Is that right?

 

Thanks in advance, guys, and... well.. I probably will post a few more of these.

 

~moo

[swansont]

Not knowing the exact context, I'm not sure what the point of the problem is. Is this a Gaussian beam? The spot size is usually the 1/e size, and you can get the amplitude from that.

 

The time comes into it because a shorter pulse has a higher intensity. 1 Joule in 1 sec is 1 Watt, but 1 Joule in 100 fs is 10 PetaWatts. That most definitely affects the field strength. You've used the pulse energy rather than the power, which is why your units aren't working.

 

 

Also, mJoule is millijoule, but that only affects the magnitude of the answer.

[mooeypoo]

Ah, milli, I thought Mega for some reason.. you're right, Mega should've been Mj and not mj. Bah.

 

Okay, other than that, I assume it's a Gaussian beam just because that's the usual pulse we're dealing with.

 

I don't know if I understand where the time effect comes in, though, swansont? where do I get it mathematically into the equations? How would it affect what I did..?

 

In other words, what I did until now (forget the milli/mega for a second) is right if this is a continous wave but wrong for a pulse? I'm trying to understand the methods so I can understand which to pick and when.

[swansont]

You used the energy instead of the power in your calculation:

 

[math]P=10 \text{mjoule} = 10^7 \text{joule} = 6.2*10^{25} eV[/math]

 

You should be using [math]P = \frac{E}{\Delta t}[/math]

[mooeypoo]

Alright, trying again, then:

 

[math]E=10mJoule = 10*10^{-9}J=6.2*10^{9}eV[/math]

[math]t=100fs = 10^{-13}s[/math]

[math]A=1cm=0.01m[/math]

 

So:

 

[math]P=\frac{E}{\triangle t} = \frac{6.3*10^{9}eV}{10^{-13}s}=6.2*10^{22}eV/s[/math]

 

[math]\frac{P}{A}=\frac{E_0^2}{2\mu c}[/math]

 

So:

 

[math]\frac{6.2*10^{22}}{\pi *10^{-4}}=\frac{E_0^2}{2\mu c}[/math]

 

[math]E_0^2=1.24*10^{27}\frac{\mu c}{\pi}[/math]

[math]E_0=3.52*10^{13}\sqrt{\frac{\mu c}{\pi}}[/math]

 

And so, to find the magnetic field:

 

[math]B=\frac{1}{c}E_0=\frac{3.52*10^{13}}{c}\sqrt{\frac{\mu c}{\pi}}=\frac{3.52*10^{13}}{3*10^8}\sqrt{\frac{\mu c}{\pi}}=1.2*10^5\sqrt{\frac{\mu c}{\pi}}[/math]

 

Is that right? seems right... I hope...

[swansont]

milliJoule is 10^-3 (you've got the number for a nanoJoule)

 

You didn't carry through the factor of c in your calculation of B

 

——

 

I don't think the conversion to eV is necessary or even desired, unless you have your other terms (permeability and c) in appropriate units.

 

So check the units; I would have expected the permittivity in the equation for E (but of course that's related to permeability, depending on where you put factors of c)