Let's try to summarise, and sort out some of the confusion:

* As Dale says, a mesh reflector is almost as efficient as a solid
surface of the same shape, even for hole sizes as large as 0.1wl.

* That was a side-track anyway, since the OP doesn't have a mesh wok.

* Any vaguely bowl-shaped reflector - including a wok - will make a big
improvement in cellphone performance, if you locate the cellphone at
the best possible place, line the whole thing up correctly, and manage
to keep your head out of the way.

* A paraboloid is the best shape for a reflector, because only a
paraboloid can focus all the incoming rays to a single point; and all
the incoming energy from a plane wavefront will arrive in-phase at the
focal point. This applies equally to light and radio waves. Optical
ray-tracing theory breaks down if the reflector is only a few
wavelengths in diameter, but a paraboloid is still the best practical
shape to aim for.


As Dale says, Arecibo uses a spherical reflector to allow a few degrees
of beam steering by pointing the feed antenna at different areas of the
dish. However, this is a very special case: the only practical way to
achieve a 1000ft dish was to build it immovably on the ground, so the
designers then had to find some other way to steer the beam, by moving
the feed antenna at the focus. In this one special case, the optimum
shape for the reflector is not a paraboloid but a sphere (because the
geometry of a sphere is the same in any direction, as seem from the
feedpoint at the centre).

The problem of course is that a sphere doesn't *have* a focal point - it
has a smeared-out focal line with phase variations along it. For many
years, long slotted-waveguide feeds were used to collect the available
energy from along this focal line, and to compensate for the phase
variations. By doing this, the Arecibo designers were able to achieve
similar efficiencies to a paraboloid of the same size, and with some
degree of steerability too. The limitation was that a different feed was
required for every frequency of operation, and this obviously restricted
the range of research.

With computer-aided design replacing 1950s slide-rules, Arecibo now has
a sub-reflector system of a very cunning shape that compensates for the
phase variations. Being a reflector it is not frequency-sensitive, and
it brings everything into focus at a conventional single-point feedhorn
which can be changed in relative comfort - that is, if anyone can feel
comfortable suspended out on cables, 500ft in mid-air...

Because Puerto Rico is quite close to the Equator, the Moon passes
overhead from time to time, and this allowed the Arecibo observatory to
do pioneering work on radar mapping of the Moon's surface. It also
turned out that the old 400MHz line feed would work reasonably well on
432MHz, and back in the old days there were occasional empty time-slots
when the dish would appear on EME. Sometime in the mid-1980s I had the
good fortune to work KP4I on what seems to have been the last
opportunity. He was LOUD.


(That's an outsider's view of Arecibo. Chip, please correct any
inaccuracies :-)




--
73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB)
http://www.ifwtech.co.uk/g3sek