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Re: Spherical Mirrors



At 23:43 4/29/00 -0400, Jennifer wrote:
...Suppose you have a sphere with a small hole in it.
The inside surface is reflective. If light passes into the sphere, would
it be possible to "trap" any and all light inside the sphere? Of course,
this is assuming that ALL wavelengths of light are reflected to the SAME
focal point....

Jennifer

It seems to me you are justified in proposing the sphere as a light trap.
For a device with a perfectly reflecting surface, the only perfect escape
would be a return along the path of the incident ray.
If we accept that rays traversing a small pinhole can be considered
as a point source of light, then after one reflection, the entry point
would need to be a focus in order to serve as the exit, and you have
established that the focus is elsewhere.

Newton relates in his 'Optics' that he considered spherical concave
mirrors too. He pointed to glass as a superior material for such mirrors
in comparison with metal because he found (as others did before and since)
that the metal would tarnish, which being rubbed away with a very soft
leather, still disturbed the figure of the mirror.

We like to think that he 'invented' the reflecting telescope, and it is
certain that he personally used pitch coated laps and abrasive (in his case
made from a refined 'putty') to grind them to a good figure.

You may be surprised to see that in his 'Optics', he wrote:
" But because metal is more difficult to polish than glass, and is afterwards
very apt to be spoilt by tarnishing, and reflects not so much light as
glass quick-silvered over does, I would propound to use instead of the metal
a glass ground concave on the foreside and as much convex on the backside,
and quicksilvered over on the convex side."

I expect he silvered the rear surface to overcome the varying thickness of
an early reflecting film in the now preferred front surface.

At any rate Newton reminds us that reflection is associated with some loss
of light (then and now) so that if we suppose as little as 0.1% of the light
intensity is lost at each bounce (and it is surely more) it would not require
many bounces before the light would have been essentially lost, distributed
by heating the mirror surface.

It is interesting to calculate how many bounces would be needed to
dissipate 999/1000 of the light energy, given that only 1/1000 of it is lost
at each bounce.

Because we model the loss as happening at discrete times, we prefer the
compound interest equation familiar at your Savings & Loan, rather than
the (continuous) exponential decay expression favored by Rutherford to
model the activity of materials like those extracted at such great personal
cost by Marie Curie and her husband.

Your loan officer uses S = P(1 + I%/100)^Y
where S = Sum accumulated
P = initial Principal
I = fractional interest as a percent
Y = term, that is the time interval count.

Here, we have
P = 1 let us say,
I = -0.1%
Y = number of bounces - unknown
and S the remaining value we set to 0.001, one thousandth intensity.

0.001 = 1(1 - 0.1/100)^Y
Simplify to 0.001 = 0.999^Y
and take logs
so log (0.001) = Y log(0.999)
and Y = log(0.001)/log(0.999) = -3/-0.0004345 = 6904 bounces.

This certainly seems like quite an impressive 'trap' sequence,
but light travels very quickly, of course. Could we hope to monitor
the gradual decay? How long does this take?

Taking a sphere of internal diameter 1 meter (no bounce could travel
further...) then we see there is a distance of 6906 meters during
which this dimming process happens - at a speed of 300 thousand
kilometers/sec that takes 6906 / 3 X 10^8 = 23 microseconds

There are certainly many detectors capable of registering this decay time.
A phototransistor, a photodiode, a photomultiplier tube.

But you may find that the reflective coatings you can easily buy or make
are not as good as the reflectivity figure I pulled out of the air of 99.9%

I would not be surprised to see coatings specified at 2 or 3% loss per
reflection. This would make the experimental task much harder.

Brian



brian whatcott <inet@intellisys.net>
Altus OK