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).


Basically correct. Although the tracking is better than just a few degrees:-)

Arecibo was initially designed to be a survellance instrument, passively
listening to Soviet communications through inadvertant moonbounce. It also was
designed, initially, as an ionospheric heating facility.

Through the huge luck of its overengineering, it was found to be able to track
quite accurately, and the feeds and carriage houses were designed to
accommodate a greater tracking range.

Today the site no longer has chicken wire (and hasn't for 30 years); and the
feeds are Gregorians that accommodate very large bandwidths and spectral
ranges. Most of the recent upgrades were funded by the NSF, and also the Seti
Institute.

73,
Chip N1IR