[DIYbio] Why does a beam expander allow a smaller focused spot size?

I've read some equations and searched google, but I'm still not
understanding WHY expanding the beam and shortening the focal length
allows light to be focused into a smaller spot. I know Feynman would
probably tell me I shouldn't ask WHY... but I still wonder

The equations here are intelligible, but I'm not finding any
explanations.... to me it seems like you'd have more play with a
longer focal length (i.e. minute variations past the focus would be
less impact as the divergence is slower)
http://www.kellerstudio.de/repairfaq/sam/laserioi.htm#ioicdf

http://specialoptics.com/pdf/wp_laser_beam_expander_theory.pdf

Simon, care to physicize me?

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Nathan McCorkle
Rochester Institute of Technology
College of Science, Biotechnology/Bioinformatics

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