# Laser:population rate equation

by Clau
Tags: equation, laserpopulation, rate
P: 11
1. The problem statement, all variables and given/known data
Simultaneous Laser Emission on Two Lines
Consider the multi-level system show in the figure where amplification can occur at frequencies $$\omega_{12}$$ and $$\omega_{21}$$. The arrow "triplets" are meant to indicate the 3 Einstein absorption and emission processes that take place between levels 2 and 3. Write down the population rate equations for levels 1, 2 and 3.

The figure is attached.

My problem is: I don't know if I have to introduce the pumping power inside the population rate equations.

2. Relevant equations
Ni=number of particles in level i
Between levels 1 and 2 we have:
$$\downarrow \uparrow \downarrow$$
first down arrow: spontaneous trasition rate
$$N_{2}A_{21}$$
up arrow: absorption rate
$$N_{1}B_{12}W(\omega_{12})$$
second down arrow: stimulated emission rate
$$N_{2}A_{21}W(\omega_{12}$$

3. The attempt at a solution
For level 3 I have:
$$dN_{3}/dt = - N_{3}A_{31} + N_{1}B_{13}W(\omega_{13}) - N_{3}B_{31}W(\omega_{13})$$

Level 2:
$$dN_{2}/dt= - N_{2}A_{21} + N_{1}B_{12}W(\omega_{12}) - N_{2}B_{21}W(\omega_{12})$$

Level 1:
$$dN_{1}/dt = - dN_{3}/dt - dN_{2}/dt - N_{1}R{1}$$

Should I add P3 to level 3 and P2 to level 2?
Attached Files
 figure.doc (20.5 KB, 67 views)

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