Calculating the Cross Section for a Laser's Transition Rate Equations

[Habeebe]

Homework Statement

For a laser, assume single mode operation, only lifetime broadening due to A₂₁ and A₁. Write out the rate equations for the atomic densitions N₁ and N₂ and the flux $\Phi$.

Then there's a diagram where there's 2 stats. Pump rate is P, stimulated emission between 1 and 2 is allowed, and spontaneous emission from 1 to 2 and from 1 to the reservoir.

Homework Equations

The Attempt at a Solution

Below are the rate equations as I have them. My issue is that I don't know how to get the cross section (σ). I can't find it in the book. The example from class was the same as before, but spontaneous decay could happen from both states 2 and 1, and the pump went into both states also. For that case it was $\sigma = \frac{\lambda^2}{\pi}\frac{A_{21}}{A_{21}+A_{2L}+A_{1L}}.$

Rate equations:
$\dot{N_2}=P-\Phi\sigma(N_2-N_1)-A_{21}N_2$
$\dot{N_1}=\Phi\sigma(N_2-N_1)+A_{21}N_2-A_1N_1$
$\dot{\Phi}=\Phi \frac{c}{2d}[ln(R_1R_2+2l\sigma(N_2-N_1)]$EDIT: I'm not sure if it's clear, but I need cross section in terms of wavelength/frequency and the A coefficients, as those are filled in later with actual numbers.

 

[Habeebe]

The cross section was actually given to me, and I just found it, so my problem is solved. I feel like this is something I should be able to figure out though, so if anyone can show how to I'd get to the solution, that'd be cool.