Module 8 : Laser- II
Lecture   : Rate equations for Lasers
 

the equation for the metastable level $ m$is

$\displaystyle \frac{dN_u}{dt} = \frac{N_u}{\tau_{u,m}}-\frac{N_m}{\tau_{m,g}}$

The rate equation for the ground state is obtained from the above two equations by number conservation

$\displaystyle \frac{dN_g}{dt} = -\frac{d}{dt}(N_u+N_m)= - W_p(N_g-N_u)+\frac{N_m}{\tau_{m,g}}$

In the steady state, each of the time derivatives is zero, which gives

$\displaystyle N_u$

$\displaystyle =$

$\displaystyle \frac{W_p\tau_{u,m}}{W_p\tau_{u,m}+1}N_g$

 

$\displaystyle N_m$

$\displaystyle =$

$\displaystyle \frac{W_p\tau_{m,g}}{W_p\tau_{u,m}+1}N_g$

 


The equation for population inversion becomes

$\displaystyle \frac{\Delta N}{N} = \frac{N_m-N_u}{N}= \frac{W_p\tau_{m,g}(1-\beta)-1} {W_p\tau_{m,g}(1+2\beta)+1}\eqno (A)$

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