From: Mark Holm 
To: atm{at}shore.net
Reply-To: Mark Holm 


>
> I thought at this level it was about phase interference, not angles.
> (I.e wave nature of light, not particle.)
>
>


Since I have been reading Feynman on quantum electrodynamics, I'll chip in a bit
here.  According to QED: what matters is travel time; light is particles
not waves; and what you get out of the travel time calculation is a
probability function.  The only thing you can calculate is the probability
that a photon will get from point A to point B.  (I really did mean the
only thing.  QED doesn't calculate anything else about photons.  OK, it
does calculate the probability that a photon has a certain polarization.) 
The really awkward thing
is you have to calculate the path time for all possible paths and do a
vector sum of the results.  Then you square the magnitude of the resulting
vector to get the probability.  Although more accurate than any other
theory, it is rather
impractical for calculations involving complex systems, so we still use both ray
and wave theory for certain types of work, despite the fact that both are wrong.

Now when the light is traveling through a uniform medium, the travel time
turns out to be strictly proportional to path length (phase), so the
calculation becomes very similar to the path length calculation used for
wave theory.  (If we throw in refractive index, the calculations become
similar even for non-uniform media.)  In a lot of situations, the
probability distribution of a QED calculation will end up looking just like
the intensity distribution calculated by wave theory.  (Which is why wave
theory is useful in a lot of situations.)

Thinking of angles is ray theory.  Although ray theory still has its uses,
one has to keep firmly in mind that it is Wrong, Wrong, Wrong.

Ray theory's strengths are: 1. It is relatively easy to understand.  2. It
is well suited to the techniques of analytic geometry.  3. Its results can
often be
understood in terms of geometric ideas such as astigmatism, spherical
aberration, etc.  4. Because of the power of analytic geometry, it is often
possible to solve problems by substituting mind boggling brain power for
masses of computation.  The mathematical successes of the greats of ray
theory are truly inspiring.

Ray theory's big weaknesses: 1. It doesn't work at all for any diffraction
effects!  Diffraction effects are not just window dressing on optical
theory. They are critically important in a number of areas, including when
you are talking about the formation of high quality images.  That is why we
call them "diffraction limited" images.  The big name wave theory
methods treat all of optics as a form of diffraction theory.  Thus you have
Fraunhoffer diffraction theory, Fresnel diffraction theory, etc.  These
aren't just theories of diffraction, they are theories of all optics, and
were the best around before quantum mechanics.  (Still they are useful in a
lot of situations.)  2. It pretty much falls on it's face for predicting
anything involving path length differences of less than about 1/4 wave.  (I
am mixing my theoretical metaphors here.)  Properly applied ray theory can
get you up to about the level of Raleigh's criterion, but it won't get you
much farther.  (Except that it can help you understand the geometric shapes
of the surfaces that are contributing to residual aberrations.)  What it
can't do past this level is predict the brightness distribution in the
diffraction pattern and that is what we use to decide whether one image is
better than another.  (Actually, it can't predict that at any level, but we
know from experience that if the ray theory says aberrations are greater
than a certain amount, the diffraction pattern will be seriously messed up.
 If the aberration is bad enough, the diffraction pattern will even start
to look like the pattern calculated by ray tracing. (If you don't look too
closely.)

Wave theory is a better approximation, and will be fine for most uses an
ATM cares to make of it.  The main thing against wave theory is that you
have to do lots of calculations with lots of significant figures: a real
drawback in the days before computers.  The geometry of the calculations
stay the same as for ray theory, in the most simple minded form of wave
theory, but the emphasis changes.  Instead of measuring angles and
calculating ray positions at the image, the emphasis is on path lengths,
and calculating phase differences at the
image.  The great mathematical minds found ways to extract simplifying
results out of wave theory, just as for ray theory, but they have been less
well written
up in typical ATM sources.  Actually, with computers, it is sometimes
easier just to brute force through the calculations rather than try to
understand the nifty math.  One nifty math trick that helps keep the
calculation load down is the application of Fourier transforms.  Roughly 50
years ago, Fourier transforms
were the hot thing in optics as well as a lot of other areas, and are now
standard fare in a very wide range of physics applications.  There are
several other types of related math "transform" tricks that help
with aspects of wave theory.

If you want to understand something of how that ultra thin layer of
aluminum atoms on your mirror manages to reverse the direction of incident
light, you will have to break down and go with QED, because even wave
theory won't cut that.

Oh, did I mention that ray theory is wrong?

BTW, I should say, to keep from making myself out to be more qualified than
I really am, that I am not mathematically terribly talented.  I have
trouble with the math of any of these theories, so I can't claim a lot of
expertise with any of them.  I do often have the ability to see and
understand the basic physics underpinning these theories. (When someone
much more gifted (such as Richard Feynman or my college physics professors)
explains it to me.)  That is why I can
hold forth on this particular topic with reasonable certainty that I am not
far from the truth.

Mark Holm
mdholm{at}telerama.com

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