Re: combining laser beams

["John S. Denker" <jsd@MONMOUTH.COM>]

Carl E. Mungan wrote:

> Perfect! This is exactly the right spirit of discussion on the theory
> side of the issue. Now, can you me some pointers on what restrictions
> Liouville puts on absorbers and emitters?

Let's lay some groundwork first.

Rather than beams, let's consider electrical signals
(EM waves) travelling down a coax cable.  The phase
space for this at each point is two-dimensional.
Recommended variables are q and phi, the local
charge-density and flux-density.  You can convert
those to the more-conventional voltage and current
by dividing by the capacitance-per-unit-length
and inductance-per-unit-length, but I prefer q
and phi because they are dynamically conjugate
as they stand:  delta(q)delta(phi) has units of
action (same units as hbar, the natural units of
measure in phase space).

Now suppose you have two coax cables, with a
20dB directional coupler:


       __________    _________
                 \__/
                  __
       __________/  \_________


You do not have a pair of D=2 phase spaces;  you have
one D=4 phase space.  The S-matrix for the directional
coupler has exactly the same structure as a rotation
matrix.  It is a unitary transformation that rotates
some of the signal from coax A into coax B.  There
are obviously a couple of D=2 subspaces, but area in
these subspaces is !!not!! conserved.  It is only the
volume in the big D=4 phase space that is conserved
in accordance with Liouville's theorem.

Next consider an attenuator.  It could be built as
follows:

       _______    ___________    ___________    ______
              \__/           \__/           \__/
               __             __             __
          zz__/  \__zz   zz__/  \__zz   zz__/  \__zz


That is, you rotate a little bit of signa out of your
main coax and absorb it in a resistor ("zz").  Then
you do it again, rotating out a little bit more.

BUT every time you do that, you rotate a little bit of
signal !!from!! some resistor !!into!! your main coax.
This is required since the coupler is unitary.  This
is the essence of the fluctuation-dissipation theorem.
You can add this to the list of fundamentally-equivalent
ideas:
 -- Liouville theorem: conservation of phase space
 -- 2nd law of thermodynamics
 -- Heisenberg uncertainty principle
 -- fluctuation-dissipation theorem.

Your coax has one mode (actually two, one for each
direction of propagation) while each of the N resistors
contains something like 10^23 internal modes.   So the
phase space for the system has something like 2N*10^23
dimensions.  A decrease in the brightness of the mode
you care about is accompanied by a verrrry small increase
in many of the other modes.  The increase is
unobservable because we are treating each resistor
as a two-terminal black box;  we agree that we won't
try to Fourier-analyze its internal modes.


> It certainly seems that a laser amplifier increases
> brightness of the optical beam at the expense of the entropy dumped
> out as waste heat

That's a nice guess, but misses an important point.
The actual physics is more interesting than that.

Yes, the laser amplifier increases brightness by
taking energy and phase-space from a bunch of
other modes of the system.

But the physics does not require the laser to
produce much in the way of "waste heat".  What
is actually required is that the laser or any
other "ordinary" amplifier must add noise to
the output beam.  Here's a simple argument to
convince you this must be true: take a small
signal, run it through an amplifier, then
measure the amplified signal as accurately
as you can subject to the uncertainty principle.
If the amplification were noise-free, you would
be able to infer the properties of the input
signal to !!better!! than what the uncertainty
principle allows.  The amplifier necessarily
adds enough noise to enforce the uncertainty
principle.

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

While we're on the subject, note that the
original request to combine beams runs afoul
of the 2nd law of thermodynamics.

Let one beam be the black-body radiation
coming from object A.  Let the second beam
be the black-body radiation coming from
body B.  Let the combined beam fall on
object C.  If all three start out at the
same temperature, C will heat up while
the others cool off........