On Thu, 1 Jul 2004, Alf P. Steinbach wrote:
Yes, thanks, it seems luck is the key word in dechiphering that first
mentioned paper, even the presumably clarifying figures... ;-)
Sketch some pictures as you read this. It's a bit beyond my ascii-art
abilities...

On way to think of this is to consider that we and our telescopes are
much smaller than the source, lens and distances involved. So much so that
on a drawing of that the rays you might trace in geometric optics, we are
tiny even compared to the lines! So you can consider those lines as
pencils of light larger than the earth. Light within them can be
considered parallel rays similar those from a distant object, so if we
point a telescope along that pencil we can image the object like any
normal object. --The gravitaion lens doesn't form images, it just
redirects the light so we can form them, like we do with our eyes and a
(flat) mirror.
A glass or plastic lens that behaves like a gravitational lens does not
have a surface that looks like the surface of a section of a sphere, like
a standard lens, it looks like a 1/x curve**. For a standard lens the
normal to the surface of the lens gets close to the normal to the plane of
the lens as you move towards the center. For a gavitaional lens it gets
further away, so it is impossible for a gravitaional lens to focus in the
usual sense --it can't even form a virtual image like a concave lense
does. But it can cause rays of light that pass by at a given radius to
cross at a particular distance, making the ring associated w/ gravitaional
lenses. Now if you blocked out (or dispersed) most of the light except for
a small bit of the arc, you could image the source, the problem being that
normally there are many ovelapping images in the ring.
Note that even in the case of a radially symetric lens you could see
more than a simple ring: If an source is off center is will produce a
diplaced ring, so say we had a black hole with 3 stars arranged in a
triangle behind it: A red one, a blue one and a yellow one. There would be
3 rings, a red, a blue and a yellow one interlocking or partially
overlapping. Look really cool as a special effect in a movie (hey! Ya
gotta put my name in the credits if ya use it!) You might even be able to
deconvolve an image out of a full ring. (More commonly small/point sources
& distortions in the images are used to try to get info about the shape of
the lensing object.)
Now with aberration, the pencils of light may converge before they get
to the earth, then diverge, or diverge at the outset. It turn out that
there is alway one path where the light remains relatively parallel (that
corresponds to the direct path) allowing light to reach the earth (but
which may be physically blocked by the lensing body). And if things are
radially symmetric (and not to strange, like a toroidal glalaxy
cluster...) there will be some radius that will form a ring -but- there
can be an aberation caused by an over or under mass in part of the lensing
object that causes part of the ring to spread it's pencils out, diluting
their light so we see nothing, on the other side of the ring this
over/under mass may cause a convergance in the pencils before they reach
the earth, after that point they diverge and again we see nothing, again
-but- at some point along what would have been the ring, between converges
too much and diverges is "just right" --since it's along a ring, there are
two points, so we add two images. And thus, images are always added in
pairs.
A more detailed consideration using Huygen's principle show that, in
fact my fingers never left my hands during that hand waving: there is
always a minimum time path for the light to travel. If I add a new local
minima there must be a new local maxima added too and images show up in
set of one, three, five, etc.

And that's about 10% more than I know :-)

3ch

*arguably, in this sense, only the final lens in a telescope makes the
image, all the others do is "Bend the light". In fact, the main purpose of
the large mirror in most telescopes is to gather light from a large area
to a small one --the camera or eyepiece. The same field of view and
magmification could be acheived w/ a much smaller telescope.

**It is not (or at least I don't mean to imply) it -is- a 1/x curve, only
that it looks like one in the sense x^4 "looks" like a parabola. --Some
people have plastic "gravitational lenses". Fun and informative to play
with a bit if you can find one.