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A beam of light of width $W$ and wavelength $\lambda$ with divergence that is diffraction-limited is refocused with an optical element placed at a distance $D$ from the beam source. Will the refocused beam have worse divergence than the one imposed by diffraction limits? how much worse?

Is there an optical way to improve the beam divergence of the refocused beam? If a 2nd refocusing element, very similar to the 1st one, is placed on front of it, will the divergence get progressively worse? by how much?

Finally, does it matter if the beam is a Gaussian or Fresnel beam?

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If you define the radius of a beam from the standard deviation of its profile (up to a constant factor), then for a given waist radius, gaussian beams are the ones that diverge the less (at infinity).

In paraxial optics, with gauss approximation etc., a perfect spherical lens or mirror will always transform a gaussian beam into another gaussian beam, hence a beam that has minimal divergence for a given waist radius. So, in an ideal case, starting from a diffraction-limited beam (=gaussian), the beam will always be diffraction-limited because it will stay gaussian.

I started writing a detailed answer on how to collimate beams, but I got lost into details and I'm not entirely sure what you want to do. My advice would be to play with a nice and simple gaussian beam software like this to see what optical components you need. Hint: if you want a beam with a very small divergence, you need a beam with a large diameter to start with, so you might have to expand your beam before collimating it.

Fresnel beams may allow you to have a beam that is better collimated around a focal point than a gaussian beam, for a given waist radius. However, their divergences (defined at infinity) are always worse than those of gaussian beams.

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this is the kind of answer i was hoping for! thanks!! –  lurscher Feb 22 '13 at 18:38

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