Re: [Phys-L] Each ray travels "as if it knew" all values of t=t1+t2

[John Denker <jsd@av8n.com>]

On 04/13/2014 12:54 PM, Ken Caviness wrote:
> The light acts as if it follows all paths from point A to point B,
> but most of those paths cancel out:  the wave arriving by almost
> every path interferes destructively with the wave arriving via some
> other route.  The minimum time path, however, can't cancel out:  the
> wave arriving that way arrives first, before any other candidate for
> interference. 

That's the correct physics.

I would add that there's nothing particularly quantum-mechanical
about it.  The sum over paths works just fine for classical waves.

> So the light wave traveling along the minimal time
> path is the one that actually occurs, the one that is actually
> observed.

That's the right general idea, and it's true for rays, but there
are special cases where it's not strictly true.

In particular, consider a Fresnel zone plate.  Not a Fresnel lens 
with pieces of refractive material, but a zone plate, with black 
and white rings.  You can focus light with the thing.   A great 
number of paths contribute via constructive interference.  They
do not all have the same path-length, but rather the same length
modulo one wavelength.  Other paths, the ones that that would have 
contributed out-of-phase, are blocked by the black rings.

It is quite remarkable that adding black rings allows you to 
deliver *more* light to the focal point.

The same applies to diffraction gratings in general, and to
holograms.  A Fresnel zone plate is basically the hologram of
a lens.

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

Let's connect this with the question that was originally asked:
In the "ray" description, i.e. in classical geometrical optics, it
does not make sense to say that the wave "knows about" or "cares
about" or is in any way affected by what's going on in places
where the ray doesn't go.

However, the real physics *is* affected by what's going on in places
where the classical ray doesn't go.  In other words, classical rays
are interesting, but we have tons of evidence that says they are
not a completely correct description of the real physics.

=======

Connections are super-important to learning and to thought in
general, so we should note the connection between the "sum over
paths" approach and the Huygens construction.

The Huygens construction, in its original form, is mostly the 
right idea, but not correct as to details.  Wave propagation is 
a second-order differential equation, so it requires /two/ initial 
conditions, but the Huygens construction specifies only one (the 
height of the wavefunction).  This is fixable.
    David A. B. Miller
    "Huygens's wave propagation principle corrected"
    http://www-ee.stanford.edu/~dabm/146.pdf