Re: [Phys-L] highway mirage

[John Denker <jsd@av8n.com>]

On 04/09/2014 11:08 AM, John Denker wrote:
> As the saying goes, an expert should be able to see things in
> more than one way,

Still true.

> I am not seeing any geometrical optics
> (ray optics) answer.

I think I see how to do it now.

I just got back from a bike ride.  Perhaps being oxygen-
deprived helps me think outside the box.  Or maybe I
just needed a little more time.

> the obvious approach is to divide the graded
> index into a thousand layers, and integrate the equation of
> motion as the ray crosses from layer to layer, using Snell's
> law.

Still true.

There are two main cases, plus a pathological third case.

 1) The usual case is that the ray crosses from layer to 
  layer, and gets refracted by some angle in accordance
  with Snell's law.  The angle is small, because the change
  in index is small, because there are so many layers.

 2) Eventually the ray direction will become so close to
  horizontal that the ray cannot refract.  The angle of
  incidence is so close to 90° that there is no angle of
  refraction that satisfies Snell's equation.  In this case
  the ray undergoes total internal reflection.  The angle
  of reflection is equal to the angle of incidence.  The
  ray now goes back up through the layers.  In the ideal
  case, the upward path is exactly the mirror image of
  downward path.

 3) There will always be a Dedekind cut, some angle right
  at the boundary between refraction and total internal
  reflection.  The ray exhibited a 90 degree angle of
  refraction when it entered this layer.

  Here's the key idea:  We have been hornswoggled by paying
  too much attention to this case.  First of all, it only
  happens on a subset of measure zero among the set of all
  possible initial conditions.  So we would be well justified
  in ignoring it for this reason alone.

  Secondly, the layers are an imaginary construction anyway.
  I am free to shift they layer-boundary half a layer one 
  way or the other, whenever necessary, to make case (3) 
  go away.

So that covers all the bases.  The layered model predicts
internal reflection in a graded-index medium.

==============

In case you're wondering how I figured this out:  I imagined
writing a computer program to actually do the numerical
integration of the equation of motion, layer by layer.  As
far as I can tell, the program would work just fine.  A 
modicum of defensive programming would be necessary, but
there's nothing new or special about that.