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

[Ken Caviness <caviness@southern.edu>]

Thanks for the corrections/supplemental material.  Yes, I shouldn't have called this a quantum-mechanical effect.  I've taught Modern Physics from a book that showed how classical wave relationships lead to an uncertainty principle relating delta_x and delta_k for a wave packet, preparatory to moving to the uncertainty principle in quantum physics.  It was interesting to realize how much of what we tend to think of as quantum is really just a classical wave treatment.  But beyond that are the true quantum, classically forbidden, contributions.

Fresnel zone plates, Miller's correction to Huygen's principle, great stuff -- very nice connections.  Threads like this are why this forum is worth our time.

Thanks,

Ken

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@phys-l.org] On Behalf Of John Denker
Sent: Sunday, 13 April, 2014 5:26 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Each ray travels "as if it knew" all values of t=t1+t2

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
 
_______________________________________________
Forum for Physics Educators
Phys-l@phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l