Re: [Phys-L] highway mirage

[John Denker <jsd@av8n.com>]

On 04/10/2014 08:10 AM, Carl Mungan wrote:

> So now I have two explanations in mind:
>
> 1. The index variation in the real world isn't mathematically smooth.
...

> 2. The rays in the real world aren't mathematical lines of zero cross section. 

Here's a third way of looking at it.

I don't recommend the "rays and layers" approach, but if
you insist on using that approach, it can be made to work.
I recommend the wave mechanics approach, but as the saying 
goes, an expert should be able to see things both ways.

One key ingredient is to use the /correct/ physics for the
rays and layers.  In particular:  There is always some 
reflection off the interface between layers.

To say the same thing another way:  As we approach the 
critical angle, there is *not* a sudden switchover from 
100% refraction to 100% total internal reflection.

Look at the right-hand panel here:
  http://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Fresnel_reflection.svg/800px-Fresnel_reflection.svg.png
which is discussed about 40% of the way down the page at
  http://en.wikipedia.org/wiki/Fresnel_equations

If you have a huge number of layers, there will be near-
total internal reflection from several of the layers 
before the critical angle is reached.  So, even if you
think the angle of the refracted ray goes "smoothly and
asymptotically" to 90°, it doesn't matter, because the
energy in the refracted ray went to zero long before it
got anywhere near the asymptote.

As for the textbooks that claim you can understand what's
going on using Snell's law alone, that's clearly wrong,
because Snell only describes refraction, not reflection.
In the rays+layers approach, at some point you have to 
come to a layer where Snell's law doesn't apply.

OTOH if you take the wave-mechanics approach, you get a
unified view of the whole situation, including the 
coefficient of reflection along with the coefficient of 
refraction, as well as the angles and everything else, 
consistent from beginning to end.  Also you get an easy 
connection between a graded index and a simple lens (lower
index <--> less glass), consistent from beginning to end.

If you insist on the rays+layers approach, it can be made
to work, but I don't recommend it.