M5L25m.txt

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Superradiance would not be as important
as it is if it could not be observed in extended samples.
So now I want to use the last 10 minutes
to show you what is kept and what
has to be dropped when we talk about extended samples.
For pedagogical reasons, let's assume
we have an extended sample.
It's sort of a Cuban cigar, a really thick cigar.
And this is now our extended sample.
And what I need is, I need the cross section of the sample A.
And let's assume the length's L. This
is diameter D. It's a cigar is much, much larger than B.
And yes, we are talking about superradiance.
We are talking about spontaneous emission.
But if you see a long cigar with excited atoms,
you think immediately about lasing action.
A photon is emitted and is amplified along the path.
And, of course, the preferred direction,
where you would expect the maximum effect to happen,
is when the light is emitted along the long axis
of the cigar.
So we want to consider now modes, preferential modes,
along the x-axis.
So, if you now assume that you have many atoms and they emit
light.
If an atom here and here would emit light in this direction,
it may constructively interfere.
But in another direction it will destructively interfere.
But let us now consider, what is the solid angle into which all
the atoms can coherently emit?
Well, you know that from classical optics
the emission into a solid angle of lambda square over A
can be coherent.
Sort of similar to when you have a double slit
and you ask, over what angle do the two slits emit in phase?
You get a bright fringe.
Then you get a dark fringe.
You get the next bright fringe.
The coherence, the angle, over which the path lengths
differences do not to add up to more than lambda
is sort of the diffraction limited angle.
Which for a beam off size D is lambda over D.
And if you take it to the second dimension,
the solid angle is lambda square over D square.
So that's what I'm talking about.
So if you would sort of give all the atoms in your sample
just the right phase that they are
coherent to emit into the x-axis,
they will also coherently emit into a small solid angle.
And the solid angle is given by this number.
So the gist of it is, and I will not completely prove it to you.
I just want to give you a taste.
Is that, therefore, we still have
a superradiant enhancement.
We know the superradiant enhancement previously,
when we had the localized system, was n.
But now we have the n atoms act together.
But they're not acting together for emission into 4 pi.
They are acting together for emission into the solid angle.
And if I write the big N as density N
times L times A square.
The A square cancels out.
And I get N lambda square L. And if you remember
that the cross section of an atom
was lambda square for absorption.
If the atom is excited, the cross section
for amplification of light for stimulated emission
is also lambda square.
So lambda square is the gain cross section.
And what we find now is the superradiant enhancement factor
is nothing else like something which
reminds us of a laser, which reminds us of optical gain.
And actually, the lasing phenomenon
in superradiance in extended samples has a lot of analogies.
In some limits it's even identical.
When we're talking about spontaneous emission.
We're not talking about stimulated emission.
But if you have a system, which is in some excited state,
superradiant Dicke states, and we
are asking what are the spontaneously emitted
photons coming out?
To say different atoms emit into the same mode.
And now you have to add up the fields coherently.
This is the language we have used so far.
Or if you use the language, one atom emits a photon.
And this photon gets amplified on its way out.
Those two languages strongly overlap.
Or in some limits are even identical.
So the amplification of a photon on its way
out, this is behind superradiance.
But when we localize the atoms to lessen the wavelengths,
well the atoms are pretty much emits as a whole.
And there is no path lengths of the size
of the optical wavelengths where you can say the photon
propagates gets amplified.
So we have looked at just what comes out of it.
But in an extended sample, you could even address the question
how do the photons get amplified, magnified, augmented
when they travel from the center to the edge?
So you could actually ask what is
the light intensity as a function of the position
within the cigar?
For localized samples you can't.
So let me just write that down.
So this is analogous to optical amplification
in an elongated inverted medium.