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Re: [Phys-L] A waves question



On 06/22/2012 10:33 AM, Peter Schoch forwarded the question:
If a wave can transit energy that is proportional to its amplitude
squared, and energy can't be created or destroyed, what happens to
the energy when two waves destructively interfere?

There is a deep -- and I mean reeeeally deep -- principle of physics
that says you can never have a situation where two waves go in and
one wave comes out ... there have to be *two* waves coming out.

In the situation in question, one of the output waves will exhibit
constructive interference and the other will exhibit destructive
interference in equal measure. The overall energy is conserved.

In more detail, consider the case of a half-silvered mirror. It
necessarily has _four_ ports and all four must be taken into account.


D |
|
|
|/
A _ _ _ _ _ _ /_ _ _ _ _ _ C
/|
/ |
|
|
B |


If we treat A and B as inputs then C and D are outputs. You can choose
the relative phase of A and B such that there is zero energy at C ... in
which case all the energy comes out at D.

In other words, you cannot simply write
C = transmitted + reflected
D = reflected + transmitted
without regard for phases.

Instead, in the simplest case, you get
C = +transmitted + reflected
D = -reflected + transmitted
where that minus sign really matters. Conservation of energy requires it.

========================
Connections:

1) The same physics shows up at RF and microwave frequencies in 3dB splitters.

It is easy to buy a splitter that "looks" like a three-port device, but as we
have just seen, it *must* be a four-port device. The seemingly-missing input
port actually exists, but is not accessible. It is terminated in a 50-Ω
resistor internally.

2) The same physics shows up in 20dB directional couplers.

In general, you can write things in terms of the S-matrix:

[ C ] [ cos θ sin θ ] [ A ]
[ ] = [ ] [ ]
[ D ] [ -sin θ cos θ ] [ B ]

for some mixing angle θ. So this is connected to a rotation in some abstract space.

3) There is more at stake here than conservation of energy ... namely conservation
of phase space. There are two different principles here, neither of which is a
corollary of the other.

Some famous people have gotten this wrong. Indeed whole groups of exceedingly smart
people have gotten this wrong, and fooled themselves for years at a time.

In particular, consider the case where one of the input ports (say B) is terminated
in a 50-Ω resistor internally, as in ye olde off-the-shelf 3dB mixer. Then we seemingly
have one input port and two output ports. If you treat it as a 3-port device, energy
is conserved ... but if you look closely you find that
-- the S-matrix is not unitary
-- the 2nd law of thermodynamics is being violated,
-- the Heisenberg uncertainty principle is being violated,
-- Liouville's theorem is being violated,
-- the optical "brightness theorem" is being violated,
-- etc.
which is not a N-way coincidence but rather N ways of saying the same thing.

To get out of this mess, you have to treat it as a four-port device. In particular,
you have to consider the thermal fluctuations and quantum fluctuations that are
emitted by the resistor on port B ... which (after splitting) appear in the output
ports. That is, the output ports contain contributions from input A *and* input B,
even if input B is coming from a cold load.

To say the same thing another way, "cold load" is not equivalent to "zero voltage".
There will be nonzero fluctuations in the voltage.

This famously showed up in connection with quantum-nondemolition back-action-evading
measurement devices, which in turn showed up in connection with gravity-wave antennas.
There is more to this story ... but a crucial starting point is realizing that
a) you need to account for all the ports, and
b) you need to make sure the equation of motion is unitary.

To say the same thing another way: The /correct/ equation of motion is unitary. If
you write down something that is not unitary, it's not the correct equation of motion.

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

For more about the addition of waves, see
http://www.av8n.com/physics/wave-add.htm

For more about conservation of phase space, see
http://www.av8n.com/physics/liouville-intro.htm

For an application to the phase space of a thin lens, see
http://www.av8n.com/physics/phase-space-thin-lens.htm