If you want to follow a "ray" then a useful model is given by (s is measured
along the ray; v is a UNIT vector in the present direction of ray travel; n
is the spatially varying index of refraction) ==>
d/ds(nv) = GRAD(n) .
This says that the "ray" will always bend toward the direction of GRAD(n).
Thus, if n everywhere increases in the vertically upward direction, then a
ray will continually bend upward - even if it is presently travelling
horizontally.
-----Original Message-----
From: Bob Sciamanda
Sent: Friday, April 11, 2014 5:47 PM
To: phys-l@phys-l.org
Subject: Re: [Phys-L] highway mirage
Carl, I think your problem lies in trying to apply Snell's law outside of
its valid scope. Snell's law is a special case of Huygen's principle,
applied to a plane wave travelling from one homogeneous medium into a second
homogeneous medium, with a clearly defined discontinuity at the interface.
For a continuously (spatially) varying index of refraction, the wavefront
(not a "ray") must be followed using Huygen's principle.
-----Original Message-----
From: Carl Mungan
Sent: Tuesday, April 08, 2014 1:44 PM
To: phys-l@phys-l.org
Subject: [Phys-L] highway mirage
There seems to be a problem with typical textbook discussions of the
highway mirage of a pool of water appearing on a hot road. If the
index n monotonically decreases as we drop in altitude toward a hot
road, shouldn't a ray of light from the sky bend toward 90 degrees
(so that it is traveling parallel to the road)? What bends it upward?
One fix I have heard of is to say that once the ray gets close to
being horizontal, it reaches the critical angle and totally
internally reflects off the warmer layer of air below it.
Sounds good at first, but that explanation seems as flawed as the
textbook discussions on closer thought. The equation for critical
angle is ArcSin[n2/n1]. If n2 = n1-dn, then the critical angle
approaches 90 degrees as dn -> 0 for a continuous function. There is
no single well-defined critical angle unless you have layers of air
with steps in the index between them.
So who has a better fix? I suppose we could get small discontinuous
steps due to fluctuations.
Or perhaps one could say that even before it reaches the critical
angle, more and more of the light begins to reflect as the angle
approaches 90 degrees. Again this sounds reasonable, but I'd like to
see it quantitatively checked before I accept it. It should be
experimentally testable, because it suggests that a single incident
ray will gradually reflect into a fan of rays.
Finally I suppose one could abandon ray optics and consider
distortions of the actual wavefronts.