Re: [Phys-L] highway mirage

[John Denker <jsd@av8n.com>]

On 04/10/2014 08:10 AM, Carl Mungan wrote:
> Well... I'm still bothered by it. If you go back to my original
> message, I did consider what you're mentioning here. Mathematically,
> if the index truly varies continuously, I don't see that total
> internal reflection (TIR) can occur because there are no layers. The
> angle smoothly and asymptotically bends toward 90 degrees.

The claim that it goes "smoothly and asymptotically" towards
90° involves some kind of "zero times infinity" argument.
Such arguments are well known to be invalid.  If you assume
that the zero dominates the infinity, you get one answer.
If you assume that the infinity dominates the zero, you get
a very different answer.  I say again that if you do it with
any large but finite number of layers, you get the right 
answer (except on a subset of measure zero) ... and then
if you take the limit as the number of layers goes to zero,
you get the right answer.

You could get into exactly the same kind of trouble with
elementary mechanics.  Throw an object upward into the air.
The vertical component of velocity will go "smoothly and 
asymptotically" to zero.  You could argue that the object
will spend an infinite amount of time at this height.  No
matter where the object might go next, it will take an
infinite amount of time to get there, because the velocity
is zero.

> So now I have two explanations in mind:
>
> 1. The index variation in the real world isn't mathematically smooth.
> There will be "layers" at some sufficiently fine level and hence TIR
> will occur.
>
> 2. The rays in the real world aren't mathematical lines of zero cross
> section. The tops of the wavefronts will "refract" differently than
> the bottoms, with the effect of bending the light waves upward.
>
> Is one explanation better than the other? Are the two explanations
> related to each other? 

The second argument is incomparably better than the first.

The whole idea of "rays" is weird in multiple ways.  The 
idea that the ray has zero size *and* propagates in a straight
line is self-contradictory.  A wide /plane wave/ propagates 
in a straight line.  A wave only one wavelength wide would 
spread out like crazy, producing a diffraction pattern.  You 
can split the difference with some kind of Gaussian beam, but 
then the behavior depends on as-yet unspecified details, such 
as how wide you want the beam to be.

As for the existence of layers, you can have it either way.
It is straightforward to engineer a step in the index, where
the step is abrupt relative to any reasonable beam width.  It
is also straightforward to engineer a gradient in the index,
where the gradient is gradual relative to any reasonable
beam width.  Both of these depend on atoms being small compared
to light waves.  (If we were dealing with gamma radiation the
picture would be more complicated.)

> Which would you use to explain highway mirages in an intro course? 

I would go with the wave mechanics every time.  This is an 
idea that you get to use over and over again.
  -- physical optics
  -- quantum mechanics
  -- the classical principle of least action (including
   Fermat's principle of least time) is easily understood
   as the classical limit of the underlying wave mechanics.
   Mathematically speaking, it is obtained by the method of
   stationary phase.

You can model the mirage physics (i.e. graded index) in a
ripple tank.  Tilt the whole tank slightly.  Reference:
   http://www2.ups.edu/physics/faculty/physics122labs/lab2.doc