# Gaussean beam

1. Sep 27, 2006

### Quasi Particle

hello.

I've got a gaussean beam, which is collimated with a diameter of 2mm, and a wavelength of 1112nm. I need to focus it to a beam waist of 25µm, but my lens with the smallest focal length I have is still f=300mm, so I need to build a telescope:

---collimated beam (w0)-----|lens f)-----beam with w'0-----|lens f')-----beam with w"0-----

I found the following formulae:

-focussing a collimated beam with a lens:
$$w'_{0}=\frac{\lambda}{\pi w_{0}} f$$
where $$w'_0$$ is the new and $$w_0$$ is the old beam waist.

-focussing a beam with the lens in the waist of the original beam:
w"0 = w'0 / [1+(z'0/f')^2]^-1/2
$$w"_0=\frac{w'_0}{\sqrt{1+{\left(\frac{\pi {w'_0}^2}{\lambda f'}\right)}^2}}$$
where w"0 is the second beam waist

now, from the second formula I wanted to calculate $$w'_0$$ so I can determine f (when f'=300) but ended up with
$$w'_0=\sqrt{\sqrt{\frac{{w'_0}^2}{a^2}+\frac{1}{4a^4}}-\frac{1}{2a^2}}$$
with $$a=\frac{\pi}{\lambda f}$$.

Is that right? It looks quite strange and I get something on the order of 10e-10 which seems very small for a beam waist.
Is it right the right ansatz, in the first place? The aim is to calculate how to make the telescope, i.e. what focal length and distance the first lens must have. This is a real problem btw., but it's so much like textbook problems that I posted it here.

Any help is appreciated, I'd be happy to be pointed into the right direction or given some material on this. I searched the forum and the web without much success, but maybe I've been looking in the wrong places

Thanks in advance for taking the time =D

____

EDIT: I don't seem to be able to make latex display some of the formulae. Sorry for the inconvenience.

Last edited: Sep 28, 2006
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