Re: [Phys-l] About light interference and energy preservation

[John Denker <jsd@av8n.com>]

On 03/19/2010 05:35 AM, CARABAJAL PEREZ, MARCIAL ROBERTO wrote:

> We were reading in class the book "Please Explain" by Isaac  Asimov,
> and answer 78 explains about light interference. It says that when
> two white light waves interfere destructively on a screen, we can
> obtain darkness in the screen, but energy should be converted to
> another form: heat (screen temperature raises up) . A student asked
> me about the case that white ligth interference were in perfect
> vacuum, ¿ in wich way can the energy conservation principle be
> preserved ? .¿ Energy will be manifested as heat ?. 

That's easy.  Asimov's answer is fiction.  Not science
fiction, just fiction.

Here's an easy proof.  Proof by contradiction.  Consider the 
following scenario:
  If we (hypothetically!) say that destructive interference 
  produces heating, then what about constructive interference?  
  Where does the energy come from?  In this scenario, it must
  come from negative heating!

  It seems Asimov has unwittingly invented the long-sought 
  cold ray (counterpart to the familiar heat ray).  It can be 
  used as the basis for the fabled microwave refrigerator 
  (counterpart to the familiar microwave oven).

This scenario is completely preposterous.  In the real world,
interference has got nothing to do with dissipation.  And 
dissipation has got nothing to do with interference.

Also the student's disproof is perfectly valid.  If you take
away the screen, the absurdity of the "heating" story is
made clear.  Sometimes it helps to draw the picture;  some
pictures of interference patterns without screens can be
found at
  http://www.av8n.com/physics/wave-add.htm

Even better than "artists's conception" drawings are the real
interference patterns.  You can easily set up such patterns
in a ripple tank.

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

The actual physics much more interesting than the fake 
"heating" story.  The actual energy budget for interference 
works like this:

You may have heard that "one and one makes two".  Indeed
it is proverbially true that "one and one makes two".  But 
it's not actually true.

For waves, the energy goes like the square of the amplitude.
Here is the "addition table" for two waves:
                                           Superposition 
    Wave A          Wave B      relative    of A and B
  ampl. energy   ampl. energy    phase     ampl.  energy
    1     1       1      1       180        0       0
    1     1       1      1        90        1.4     2
    1     1       1      1       -90        1.4     2
    1     1       1      1         0        2       4

That is, in terms of energy:
 -- Sometimes one and one makes zero (destructive interference).
 -- Sometimes one and one makes four (constructive interference).
 -- Sometimes one and one makes two (waves that add in quadrature).

The classical result, i.e. one and one makes two, does not
come from having all waves in quadrature phase, but rather
from taking the *average* over all phases.  If you average
the for numbers in the rightmost column in the table (above),
it averages out to 2.

So, the real rule is:  one and one makes two /on average/.

This is discussed in more detail, with pictures, at
  http://www.av8n.com/physics/wave-add.htm

Energy is conserved because locally less energy in one place
is always accompanied by locally more energy someplace else.
Interference just relocates the energy.  There's nothing
mysterious about relocating the energy of a wave;  at the
macroscopic level we do that all the time using lenses and
mirrors.  (In fact, at the microscopic level, the operation
of a lens can be understood in terms of interference.  See
Feynman volume I chapter 31, "The Origin of the Refractive
Index".)

In scattering theory, the proposition that "locally more 
energy in one place is always accompanied by locally more
energy someplace else" is called the optical theorem.  If
this weren't true, it would not only violate conservation
of energy i.e. the first law of thermodynamics, it would
also violate the second law of thermodynamics, violate
Liouville's theorem, violate the Heisenberg uncertainty
principle, violate unitarity, et cetera.  It would be a
Bad Thing.

When we say that interferences causes less energy in one
place and more energy someplace else, the two places are
not necessarily nearby.  For example, consider the first-
order, second-order, third-order etc. beams coming off a
diffraction grating;  the bright beams can be very far
apart, with dark in between.  (This is the same physics 
as at two-slit diffraction pattern, except that there are
thousands of slits, so the bright beams are farther apart.)

This is quite fundamental physics.  It applies to light
waves, sound waves, and the quantum-mechanical wave
functions that describe electrons, protons, and everything
else.  It also has innumerable practical applications.  
For starters, interference -- including the phase-dependence 
and nonlinearity of the energy -- is key to understanding 
how lasers work.  And how it is possible for clouds to
disappear, reappear, and change shape so quickly.  And why
the AR coating on a lens is desirable, and why it works.
And zillions of other things.