Re: Fresnel Lenses

[John Denker <jsd@MONMOUTH.COM>]

I'm beginning to think there are two types of critter that go by the name
Fresnel.

*) In the lighthouses I've seen, the goal is evidently collimation not
imaging, and the Fresnel has only about N=10 segments.  The difference
between something that scales like N and something that scales like sqrt(N)
won't be completely overwhelming.  Unphased rings must be OK in this case.

*) OTOH there are Fresnels that have _thousands_ of rings and produce
images, not just collimation.  I don't see how unphased rings could
possibly work in this case.

At 09:26 AM 4/15/00 -0700, Leigh Palmer wrote:

> Lenses for application to collimation, as is
> the case with most Fresnel lenses, are frequently cast instead of
> being ground.

1) I realize that collimation is not the same as imaging.  For applications
such as stage lighting, the task is to throw a lot of energy in the right
general direction.  Sharp focusing would be not only be an unnecessary
complication, it would be counterproductive.

OTOH I'm still confused about several other applications.
  1a) I've seen Fresnels 10 inches on a side, sold for affixing to windows
or glass doors, to create fisheye views.  That's image quality, not
collimation quality.
  1b) I've seen quite a few lighthouses, green/white airport beacons, and
suchlike that would IMHO benefit from sharper focusing.  One would think
that a shorter-and-brighter pulse would be much easier to notice.
  1c) For burning-glass applications, one often wants the highest possible
peak temperature, which calls for sharp focusing.

2) The cast-versus-ground argument is very weak.  For years it has been
possible to cast optically-perfect surfaces.  Many eyeglass lenses are
cast, and there are other important technological applications for
precision casting.

> Any old piece of lumpy glass with a smooth surface (a cast drinking
> glass, for example) ... After all, one can still see an image through a
> cast drinking glass, though a distorted one.

An excellent point.  But I wonder how much of that has to do with the
adaptive focusing powers of the eye?  When I try to use a lumpy drinking
glass to form a recognizable image on a passive image-plane (like an index
card), it doesn't work very well.

> >Suppose we have one of these alleged lenses with randomly-phased
> >rings.  What happens to the energy that (because of destructive
> >interference) does not go into the main image?
> >   a) Does it go into "nearby" locations in the image plane, which might be
> >somewhat useful, or
> >   b) Does it go into random far-away side lobes, which are useless for
> >lighthouses and for every other purpose I can imagine?
> >
> >I suspect (b).
>
> As I have already explained, a) is a pretty good answer.

Yes, (a) is the right answer.  I missed that.  That answers the physics
puzzle that was bothering me.

To see this in more detail, let's build up a Fresnel, ring by ring.  Start
with the light that comes from the central "bullseye" section of the
lens.  It focuses to form an Airy disk, which we will call disk #1.  The
contributions of the next ring, if added in ideal phase, will contribute
strongly constructively to the middle of disk #1, contribute strongly
destructively to the outer parts of disk #1, and contribute _weakly_
everywhere else.  Ideally this produces a smaller Airy disk, disk #2.  If
we add in the second ring with non-ideal phase, all the strong
contributions are still within disk #1, but they might well produce a donut
shape rather than any small disk.  Additional rings will randomly dump
energy into this general area.  They won't sharpen the image nearly as much
as they would in an ideal lens, but they won't broaden it beyond disk #1,
and the energy will go up as we add more rings.

> >>For a plastic lens this means it can be formed from sheet stock.
> >
> >If the plastic is good enough to implement micron-scale figuring _within_ a
> >given ring, is it not good enough to implement micron-scale control over
> >the step height?
>
> Well, that is a question you can think about if you know something
> about creep or thermal expansion, or if you have ever seen a
> Fresnel lens made of a flexible sheet material that, because it is
> nonrigid, can't possibly be held to micron tolerances across its
> lateral extent.

I have thought about it, and the bending argument seems very weak.  To
first order, simply bending a lens doesn't change the amount of material
(hence the amount of phase shift) that a given ray sees.  As a well known
example, putting a thin plano-convex lens in backwards makes very little
difference.

Consider this:
               L
               LL
               LLL
               LLL
               LLL
               LL
               L

which, if bent quite a bit, becomes this:

                 L
                LL
               LLL
               LLL
               LLL
                LL
                 L