Re: Fresnel Lens

[John Denker <jsd@MONMOUTH.COM>]

At 11:20 PM 4/14/00 -0400, Hugh Haskell wrote:

> Imagine taking an ordinary lens (flat on one side and convex on the
> other), and cutting it into a whole bunch of very thin concentric
> rings. The cross=section of each ring will be a little trapezoid,
> flat on the bottom and tilted on the top, with the degree of tilt
> increasing with the radius of the ring. Now grind each ring's
> cross-section down until the cross-section is just the triangular
> part on top of the trapezoid and reassemble the rings in the order
> they were originally. Voilla! A Fresnel lens!
>
> Since the lens effect occurs at the surface where the refraction
> takes place, all that extra material behind the curved surface is
> just extra weight to haul around. The Fresnel lens is just a way to
> get rid of the extra weight. This is especially valuable for large
> diameter lenses.

That is basically true as far as it goes, but it leaves off an important
part of the physics.

Hint:  Ask yourself why do the rings have the characteristic non-uniform
spacing, with one large bulls-eye in the center, and finer and finer rings
near the edge?

The point is that when constructing a Fresnel according the the foregoing
prescription, you can't just take an arbitrary ring and push it back by an
arbitrary amount.  That would be OK for one ring in isolation, but if you
do it for a bunch of rings the uncontrolled amount of missing glass would
produce an uncontrolled relative phase shift between the various rings, and
the result would (virtually certainly) cause destructive interference and
utterly ruin the lens properties.

So, short version: The trick is to choose ring-boundaries at locations
where it is convenient to push the glass back by an _integer_ times 2pi of
phase shift.

========

Longer version : Here is the construction, starting from scratch:

Start with the usual setup of object point, lens, and image point:

                   L
                   LL
                   LLL
       O           LLL            I
                   LLL
                   LL
                   L

Now the slightly-oversimplified wave-mechanics criterion for focus is that
all rays from O via L to I arrive with the same phase, no matter which part
of the lens they travel through.  The rays that go through the center have
a short path through the air and a long path through the thick part of the
lens.  The rays that go through the edge of the lens have a longer path
through the air but a shorter path through the glass.

While traveling through the glass, the rays pick up more phase per unit
distance.  You figure the lens to put just enough glass in each path so
that the phases line up.

M. Fresnel noticed that physics does *not* require all the phases to be the
same;  the real requirement is that the phases all be the same modulo 2 pi.

If you are going to push back part of the surface, you must push it back by
a distance that is a multiple of
        X0 := lambda / (n2/n1 - 1)

where n2 is the refractive index of the glass, n1 is the refractive index
of the ambient medium (usually air), and lambda is the wavelength in the
ambient medium.

Usually the pushback is a _large_ multiple of X0, but it is always very
precisely an integer multiple.

=========

Remark:  Thinking about this is extremely good exercise for students who
want to grow up to be real physicists.  Reference: Feynman and Hibbs,
_Quantum Mechanics and Path Integrals_.

Remark:  Because of the dependence on lambda in that expression, a Fresnel
will have ugly chromatic aberrations even if the material of which it is
made is perfectly achromatic.

Remark:  Put M. Fresnel on your list of people who were not trained as
physicists but who made extremely significant contributions to physics.