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Re: combining laser beams



Carl E. Mungan wrote:

Okay John, spill the beans.

Here you go:

Bean #1:
A few miles before we get to Liouville we
pass through Phase Space.

Phase space (by definition) is a space with enough
dimensions so that a point in the space fully
specifies the state of "the" object.

(This use of the word "phase" is not the
same nor even closely analogous to the
"phase" angle of a wave.)

In geometric optics, the usual object of interest is
a ray, that is, a point in phase space fully specifies
what the ray is doing. A beam is a bundle of rays,
represented by a cloud of points in phase space.

The points in phase space move as a function of
time in accordance with the laws of motion, tracing
out paths in phase space. Given any one point in
time along the path, you can reconstruct the full
path (i.e. all times past and future).

An immediate corollary of these definitions is
that paths in phase space never cross, split, or
merge. (Otherwise you would have some point that
has two different destinies, contrary to the
defining property that the location of the point
fully determines what's going to happen.)

In D=2 geometric optics, suppose there is a beam
travelling more-or-less in the X direction. A
nice traditional choice of axes in phase space
is Y (how far off-axis the ray is at the moment)
and dY/dX (the slope). Since the equation of
motion is a 2nd-order differential equation,
specifying these two numbers is a sufficient
set of boundary conditions.

(The extension to D=3 is straightforward but
I won't complicate things by discussing it.)

The idea of merging beams so that they come
together and stay together violates the corollary
given above: paths in phase space cannot merge.
-- Two beams can cross (same X but different dY/dX).
This can be done by shining two beams on the
same target. A condenser lens also falls into
this category.
-- You can put two beams side-by-side (same dY/dX
but different X).
-- There is no hope of combining beams so that
they come together (same X) and stay together
(same dY/dX). Two rays that are the same now
must have been the same at all past times.

This applies equally to laser beams, flashlight
beams, particle beams, and everything else.

I believe this is a 100% complete answer to the
original question, unless I misunderstand the
question.

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

Bean #2:
There is a slightly more restricted notion of
phase space, wherein there are N pairs of axes,
and the elements of each pair are dynamically
conjugate. (The example given above, X and
dX/dY, complies with this restriction.)

This leads to Liouville's theorem: area in
phase space is conserved. That is, if you have
a bundle of rays that cover a certain area at
time T1, then the same rays will cover an equal
amount of area at any other time.

Reference:
http://mitpress.mit.edu/SICM/book-Z-H-44.html

If you define brightness as the amount of
energy per unit area per unit solid angle,
then Liouville tells us that brightness is
conserved by any apparatus where you can
keep track of the whole phase space (i.e.
excluding absorbers and emitters, which
do of course uphold Liouville's theorem
but for which a detailed analysis is beyond
the scope of this note).

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

Bean #3:
It turns out that
-- Liouville's theorem,
-- the 2nd law of thermodynamics, and
-- the uncertainty principle
are AFAICT equivalent. The same physical principles
that enforce one enforce the other two. Any exception
to one of them could be used, with trivial modifications,
to violate the other two.