So, I've looked at several dozen helibowls and talked to makers of said
items..
There is no published design, per se.
The instructions, as I was told, are "go and buy a cheap mixing bowl"
The ones I saw used things like plastic cups as a form, on which a wire
or piece of copper tape was spiraled. I suppose you'd use the usual
non-critical guideline of circumference comparable to wavelength, so
20cm circumference is 6 cm diameter, etc. But I would say that the
notorious red beer cup is too big. What you want is the smaller cups,
but not the short fat ones.
The "stove burner liners" have also been used (they're basically a bowl
with a hole in the middle).
The late (nov 2009) Don Spitzmesser, who is generally given credit for
inventing it, apparently said, "the bowl is a ground plane, who cares
what shape it really is"
Cheng's MS thesis
http://etd.ohiolink.edu/view.cgi/Cheng%20ChinYuan.pdf?ohiou1176838193
actually talks about modeling it and has, of all things, somewhat
unreadable design drawings from a commercial manufacturer on page 53 of
the pdf
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