# Condition for Optical Gain

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1. May 17, 2015

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.

2. May 18, 2015

### 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.

3. May 18, 2015