What is the role of degeneracies in the condition for optical gain?

[James Brady]

I'm a novice studying laser physics and I came a across the condition for optical gain:

$$\frac{N_2}{g_2} > \frac{N_1}{g_1}$$

This is a basic set up where N_1 is the number of atoms in the lower energy state and N_2 is the number of atoms in the higher energy state. g_1, and g_2 are the degeneracies for the two states respectively. I understand that for optical gain you need N_2 to be higher than N_1, but why are the degeneracies factored in? It seems like it the number of possible ways to be in a certain energy state shouldn't matter, only the number of states themselves.

Can someone explain why the degeneracies are in the denominators? I do not see why having many possible ways to obtain an energy state would effect that states relative population. Also, I guess the key term which we are trying to achieve for a laser is "population inversion" or having more atoms in the high state than the low state.

 

[blue_leaf77]

For now imagine there is only one electron sitting in one the degenerate levels in the upper energy state, where the degeneracy is $g_2$-fold. Then one photon comes in, this photon will have $1/g_2$ chance of interacting with the single electron in the upper energy state. (This problem is analogous to that where one is throwing a dice, when the dice lands he will have 1/6 chance of getting, e.g. 2-dot surface facing up, 2/6 chance of getting 2-dot OR 5-dot surface facing up, and so on). Hence if there are $N_2$ electrons in the upper energy state, our previous photon will encounter an electron with $N_2/g_2$ chance. And remember interacting with upper state electron will trigger stimulated emission.
But there is also the lower energy state with $g_1$-fold degeneracy. Using the same argument as above, if there are $N_1$ electrons, the incoming photon will encounter an electron from the lower state with $N_1/g_1$ chance, which means $N_1/g_1$ chance of triggering absorption.
The generalized definition for a population inversion is actually the number of emission events being larger than the absorption events, therefore for degenerate levels one must have $N_2/g_2 > N_1/g_1$. If you want a more mathematical derivation, go check a book titled Principles of Laser by O. Svelto in the beginning of the first chapter.

 

[James Brady]

Oh I actually kind of get it now, so pretty much adding degeneracy levels increases the area where the electron could be, thus making it harder to hit with the photon...?

Thank you for the book recommendation. I will look into that one.