M3L17d.txt

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If I wanted to be present you saturation, power broadening,
and all that, in the purest form,
I would just present you with the optical Bloch equations.
We can solve them, and them we have everything we want.
A result which explains saturation, a result which
explains power broadening.
However-- and you do some of it in your homework--
however, what I want to show here
is that saturation is actually a general feature which
will have a system, if you have sort of three rates, which
I will explain to you in a moment-- very similar
to Einstein's A and B coefficients--
that all such systems have saturation.
And then, you may find, you may immediately
solve the optical Bloch equation for monochromatic radiation,
but for broadband radiation, we usually
don't use the optical Bloch equations
because there is-- for infinitely broad light,
there is no coherence for which we need optical Bloch
equations.
If you only have the optical Bloch equations,
you have solved for saturation in one limiting case,
and you don't see that the concept of saturation
is much broader.
So, therefore, let us assume that we
have a two-level system, and we couple two levels
with a rate which you can think of the rate of absorption,
the rate of stimulated emission, and I call
the rate the unsaturated rate.
In addition, there is some dissipation,
some spontaneous decay described by gamma.

So R u is the unsaturated rate for absorption
and for stimulated emission.
Of course, you know, even before you solve those equations,
that there must be some saturation built in.
If you would look at the fraction of atoms
in the excited state, and you change the laser power--
which means changing the situated rate--
things cannot go on-- things cannot shoot up forever,
because you cannot put more than 100% of the population
into the excited state.
However, the fact that when we increase the laser power,
we do upwards absorption and downward stimulation,
means you won't even get 100%, the maximum you can get is 50%.

And what I'm just drawing for you is,
this is a phenomenon of saturation,
and now we want to understand the details.
So, using this rate equation, we have
defined-- this is now a definition-- the saturated rate
is the next transfer from A to B. Well,
the next transfer is-- because we
have an absorption stimulated emission--
the next transfer is the unsaturate times the population
difference.
And this is our saturated rate, but of course, we
normalize everything per atom, so therefore,
our saturated rate has a rate coefficient S
times the total number of atoms, or the total population
in both states.
Eventually we are interested in steady state.
You can immediately solve the rate equation
in terms of steady state, which is done there.
And we find that for those, we can now eliminate one
the states from the equations because we
have the steady state ratio, and then we
find that the saturated rate is gamma over 2,
times an expression which involves the saturation
parameter.
So in other words, it's just almost
previous solutions of very simple equations
which describe the saturation parameter, the saturation
phenomenon I outlined for you at the beginning.

This solution has the two limiting cases
which we want to see.
That at very low power, at the very low unsaturated rate,
the saturated rate is the unsaturated rate
because there is no saturation.
And secondly, if you would go to infinity power,
the saturated rate becomes gamma over 2,
because we have equalized the population between ground
and excited state, one-half of the atoms
are in the excited state, and they dissipate or scatter light
with the rate gamma.