I reviewed some basic laser physics from a class. Laser requires a gain medium with state > 2. Here's a quick analysis behind why.

Following Einstein's conventions, 3 things can happen in the system: spontaneous emission, stimulated emission, and photon absorption.

Let's look into each component.

1.Spontaneous emission

$$ \Bigg(\frac{dN_2}{dt}\Bigg)_{spontaneous} = -A_{21} N_{2} $$

$$ \Bigg(\frac{dN_1}{dt}\Bigg)_{spontaneous} = A_{21} N_{2} $$

2.Stimulated emission

$$ \Bigg(\frac{dN_1}{dt}\Bigg)_{- absorption} = B_{21} N_{2} \rho(v) $$

3.Photon absorption

$$ \Bigg(\frac{dN_1}{dt}\Bigg)_{+ absorption} = -B_{12} N_{1} \rho(v) $$

Combining the above and since Einstein showed that \( B_{12} = B_{21} \) because of thermodynamics,

$$ \frac{dN_1}{dt} = A_{21} N_2 + B N_2 \rho(v) - B N_1 \rho(v) $$

$$ \frac{dN_2}{dt} = - A_{21} N_1 + B N_1 \rho(v) - B N_2 \rho(v) $$

Simplifying,

$$ \frac{dN_1}{dt} = A_{21} N_2 + \rho(v) B (N_2-N_1) $$

$$ \frac{dN_2}{dt} = - A_{21} N_2 + \rho(v) B (N_1-N_2) $$

At steady state \( \frac{dN_1}{dt} = \frac{dN_2}{dt} = 0 \),

$$ 0 = 2 \rho(v) B (N_1 - N_2) - 2 A_{21} N_2 $$

$$ \frac{N_1 - N_2}{N_2} = \frac{A_{21}}{\rho(v)B} $$

$$ \frac{N_1}{N_2} = \frac{A_{21}}{\rho(v) B} + 1 $$

$$ \frac{N_2}{N_1} = \frac{\rho(v) B}{A_{21} +\rho(v) B} $$

$$ \frac{N_2 + N_2}{N_1 + N_2} = \frac{\rho(v) B}{A_{21} +\rho(v) B} $$

$$ \frac{N_2}{N_1 + N_2} = \frac{\rho(v) B}{2(A_{21} +\rho(v) B)} $$

Now \( \frac{N_2}{N_1 + N_2} \) is the fraction of state 2 out of total (state 1 + state 2). In order for the population inversion to happen, this ratio needs to be greater than 1/2.

Now let's take a limit where \( \frac{B}{A_{21}}\to\infty \) i.e. pumping the gain medium:

$$ \lim_{\frac{B}{A_{21}}\to\infty} \frac{\rho(v) B}{2(A_{21} +\rho(v) B)} = 1/2 $$

The result of the limit shows that the maximum achievable ratio is 1/2, which means that population inversion is not theoretically possible in a 2-state gain medium.