Re: Cat's eyes

[John Denker <jsd@MONMOUTH.COM>]

At 12:25 AM 9/4/00 -0400, Ludwik Kowalski wrote:

> Here is an illustration, a flat mirror and an ideal
> (no aberrations) thin lens.

OK, you understand the recommended model system.

> Case 1
>
> Mirror is horizontal, incident beam at 45 is reflected
> by 45 degrees (on the other side). A lens of f=10 cm is
> intercepting the beam and the focal point is on the
> mirror's surface. The beam intercepted by the lens is
> reflected and does not pass through the lens. That
> what I had in mind by saying that the mirror must
> be perpendicular to the beam in order to produce
> the retroreflection.

That tells me you don't have enough numerical aperture.  The reflected beam
must go _somewhere_;  just extend the lens enough to accommodate it.

> Case 2
>
> Same as above but the angle of incidence is much
> smaller than 45, say 5 degrees. In this case the
> central (co-axial) ray will hit the mirror at 5 degrees
> and will be reflected at 5 degrees, on the other side.
> Will it be refracted to be "retroreflected"? Yes, by
> the principle of reversibility (the principle would be
> violated if the answer were not). The same is true
> for any other ray which passes the lens twice.

Fine.  Exactly so.  You have mentioned all the key ideas.

This explains why retroreflection works whenever the cat is looking in your
general direction -- QED.

> But what is true for an idealized system may not be
> true for real eyes.

Why not?

> I do not know the anatomy of a cat's eye

It's not important.

> Thick short focus lenses are likely to be much
> different from thin lenses, as far as the
> retroreflection is concerned.

No, they're not different.  All you need is
 -- The principle of reversibility you enunciated above, and
 -- the condition of being "in focus".

> and I do not have a good ray tracing software ...
> It is not easy to answer such question, for me.

All you need is a (few cm)^2 of paper, to calculate the efficiency of
retroreflection as a function of angle (and of numerical aperture).
-- The D=1 model system that focuses to a line requires only high-school
trigonometry techniques.  The D=2 system that focuses to a plane is a
trivial corollary if you assume a rectangular aperture.
-- Now consider a D=1 model of a spherical eye.  Let there be no lens other
than the aqueous humor itself, spherical (i.e. circular, since we're in
D=1), with an index cleverly arranged so that its focal length equals its
diameter.  The calculation is in some ways even easier than the previous one.

> Why don't we observe the same "glowing eyes" effect
> in people?

We do.  It shows up in flash photography, when the flash is too nearly
aligned with the camera lens.  This is the main reason you see
photographers exhibit the stereotypical "statue of liberty" pose -- holding
the flash in one hand, arm extended, as far as possible from the camera.