Laser:population rate equation

[Clau]

Homework Statement

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.

Homework 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
$$N_{2}A_{21}W(\omega_{12}$$

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?

 

[Jhero]

it is important to carefully consider all factors and variables in a system before writing population rate equations. In this case, the pumping power (P3 and P2) should indeed be included in the equations for levels 3 and 2, respectively. This power will affect the rate at which particles are excited to these levels, and thus it should be considered in the overall population dynamics. Additionally, it may be helpful to also include terms for spontaneous emission and stimulated absorption processes, as these can also affect the population rates. Overall, it is important to carefully consider all relevant factors and to include them in the population rate equations to accurately describe the system.