On Fri, 25 Mar 2016 07:48:23 -0400, Pat wrote:

I want to create an RF "window" in the shed wall. How big does it
need to be to let 2400 MHz signals pass through it?

The minimum would be a half wave slot cut through the aluminum foil.
The width will vary depending on the type of antenna and its proximity
to the slot. Unless you have a half wave dipole near the slot, it
won't work. If you're using a patch antenna or something similar,
probably a half wave by half wave square hole would be a good minimum.
Of course, bigger is better as it reduces any edge diffraction
effects.

At 2.4GHz, 1 wavelength = 12.5cm so a 6.25 cm long slot would be the
minimum.

A full
wavelength? Half wavelength? Should it be rectangular or would a
vertical slot work?

See first paragraph.

If you want to get fancy, you can cut a 1/2 wave slot in the foil, and
turn it into an antenna. It's called a "slot antenna". Watch out for
the non-obvious change in polarization. A horizontal slot antenna
produces a vertically polarized signal.
http://www.antenna-theory.com/antennas/aperture/slot.php

Opening the shed door (a full size metal door)
temporarily restores a strong signal, but I was hoping a much smaller
opening would work equally well. For aesthetic reasons, I can't
experiment by cutting various size holes.

I suggest you look into what "rickman" suggested. If there's no foil
backed insulation in the attic area, and the roof tiles are not RF
attenuators or reflectors, moving the radio to the attic should be
easy enough. You could also install a fake plastic vent pipe on the
roof, and shove the antenna or radio into the pipe. Use your
imagination.

What's the smallest hole I
can cut in the foil coated insulation and be reasonably assured it
will work?

6.25 x 6.25cm However, bigger is better and I think twice as large
would be more usable. Be sure that the radio is fairly close to the
hole.

If your unspecified radio has an external antenna connector, find a
cheap patch or panel wi-fi antenna on eBay, punch a small hole in the
wall, mount the antenna on the outside of the building, and you're
done. Something like this:
http://www.ebay.com/itm/381490123431
Note that all the gain specifications are lies.

--
Jeff Liebermann
150 Felker St #D http://www.LearnByDestroying.com
Santa Cruz CA 95060 http://802.11junk.com
Skype: JeffLiebermann AE6KS 831-336-2558