@Aladdin Suppose you start with atoms in their ground state i.e. $N_1 = N_{tot}$ and $N_2 = 0$, where $N_{tot}$ is the total density of the atoms. OK so far?
Now suppose we shine in some light with an intensity $\rho$. The rate at which atoms in the ground state are excited to the upper state is:
$$ \frac{dN_1}{dt} = -B_{12}N_1 \rho $$
And let's suppose that natural emission is slow so the decay rate is dominated by stimulated emission, then the rate the atoms decay back to the ground state is:
If you're not sure what I'm saying we need to sort this out before going any further.
What I'm saying is that suppose we have some density $N_1$ of atoms in the ground state and we shine a light on them then the rate that $N_1$ decreases is equal to the rate that the atoms get excited into the upper state.
OK. So we start with all atoms in the 1 state and shine light on them. $N_1$ starts decreasing and $N_2$ starts increasing. But as $N_2$ starts increasing we get stimulated emission at the rate $N_2 B_{21} \rho$ so state 2 atoms start decaying back to state 1.
If we graph $N_1$ and $N_2$ as a function of time it's going to look like this:
We approach a constant value when $R_{12} = R_{21}$ i.e. the rate atoms excite from 1 to 2 is the same as the rate they decay from 2 to 1. Yes?
But if we want to make a laser we need the rate of stimulated emission to be higher than the rate of excitation. That is we want our laser to be a net producer of light i.e. it produces more light than it absorbs.
For a two level system in equilibrium this implies that T has to be negative if N2 has to be greater than N1. Hence it is not possible to obtain population inversion between E2 and E1 in a two level system and it may not be possible to get a LASER beam from absorption and emission between two energy levels
What happens is that $N_3$ starts increasing and can get large because there is no stimulated emission.
But ...
Eventually one of the atoms in state 3 is going to decay by natural emission, and when it does this produces a photon of energy $hf = E_3 - E_1$ i.e. just the right energy to cause stimuated emission from state 3 to state 1.
So that photon causes stimulated emission the first time it hits another atom in state 3, and now we have two photons. Those two photons cause stimulated emission and now we have 4 photons, then 8, then 16 ... you get the idea.
In real lasers there are all sorts of tricks and tweaks used to try and improve the intensity of the laser light, and it can get very complicated. Few lasers actually use the simple three level system I've described.
You'd have to post a link to the 4 level laser that you are talking about for me to see why the four level design has been used and what the advantages of it are.
The 3 level laser I described is what we call a pulse laser. The state 3 atoms build up until a natural decay happens and then the stimulated decay causes all the state 3 atoms to decay in a fraction of the second to produce an intense pulse of light.
But once all the state 3 atoms have decayed the light emission stops, and no more light is emitted until the state 3 population has had time to build up again.
So this type of laser produces very short, intense pulses of light with relative long periods of no emission in between the pulses. OK so far?
But sometimes we want a laser that produces a continuous beam not pulses. For example a laser pointer as used in lecture theatres across the world. So we have to figure out some way to make the light production continuous not pulsed. And that's what four level lasers can do.
So when the pulse starts $N_3$ starts falling rapidly, but because $N_1 \ll N_3$ the rate at which we can replenish the population in state 3 is small and $N_3$ falls essentially to zero. That's why we get a pulse and not a steady beam.
Is this all OK so far? If you're happy with this we can get onto how the four level system avoids this problem.
With the four level system the light we shine in excites atoms from 1 to 4, so we end up with $N_1$ roughly equal to $N_4$.
Then atoms in state 4 decay to state 3, and atoms in state 3 decay to state 2 and emit light with an energy $hf = E_3 - E_2$. Finally the atoms in state 2 decay to state 1.
And we generally try and make $N_3$ and $N_2$ much lower than $N_1$ and $N_4$.
So what happens is that $N_1$ and $N_4$ are roughly equal and roughly constant, because if $N_3$ and $N_2$ are small then the lasing process doesn't change $N_1$ and $N_4$ that much.
So we get a more or less continuous rate for the 1 to 4 process, and therefore a continuous rate for the 4 to 3 process. So the $N_3$ population stays roughly constant and emits light at a roughly constant rate.
