Re: [Phys-l] question about total internal reflection

[John Denker <jsd@av8n.com>]

On 06/10/2010 09:40 AM, Philip Keller wrote:
> 1.  One implication of Snell's law is that for light incident on a
> boundary with a less dense medium, there must be a maximum angle
> where transmission can occur.  This angle can be calculated by
> assuming that the refracted light makes a 90 degree angle with the
> normal in the second medium.  

OK.

> But in fact, there is no refracted
> light -- if you place a sensor along the surface, you fail to detect
> light.  The light has been internally reflected, as a well placed
> sensor would reveal.

Not actually true.

Under conditions of total internal reflection, conservation
of energy tells us that there cannot be any energy flowing
in the transmitted wave ... but if you look closely you see
that the transmitted wave has a distinctly nonzero amplitude
and energy.  The trick is that the energy does not flow.

> 2.  This is not a sudden transition -- in fact, a fraction of the
> light is reflected at any angle of incidence.  The reflected portion
> increases with angle of incidence and the transmitted portion
> decreases to zero..at that same critical angle?  But why is that?  It
> can't just be coincidence that the law that determines the fraction
> of light transmitted at a given angle from one medium to another
> happens to predict zero transmission at precisely the angle that
> Snell's law says it should.

Again:  The wavevector kx goes to zero, but the amplitude 
does not.

> Can anyone point me toward an explanation?

Feynman volume II chapter 33 "Reflections from Surfaces".
See e.g. equation (33.55) and the unnumbered equations
right above that.  Also equation (33.52).  Also all of
section 33-6 "Total internal reflection" ... including
diagrams of how to measure the evanescent wave.