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

Let $ \tau$be the lifetime of the level 4 for decay into any of the lower states. As the decay constants are additive, we have

$\displaystyle \frac{1}{\tau_4}= \frac{1}{\tau_{4,1}}+ \frac{1}{\tau_{4,2}}+\frac{1}{\tau_{4,3}}$

A similar convention will be used to denote the lifetime of a state against decay to any of the lower levels. Thus

$\displaystyle \frac{1}{\tau_3}= \frac{1}{\tau_{3,2}}+ \frac{1}{\tau_{3,1}}$

The rate equations may now be written as follows :

$\displaystyle \frac{dN_4}{dt}$

$\displaystyle =$

$\displaystyle W_p(N_1-N_4) - \frac{N_4}{\tau_4}$

(1)

$\displaystyle \frac{dN_3}{dt}$

$\displaystyle =$

$\displaystyle \frac{N_4}{\tau_{4,3}} - \frac{N_3}{\tau_3}$

(2)

$\displaystyle \frac{dN_2}{dt}$

$\displaystyle =$

$\displaystyle \frac{N_4}{\tau_{4,2}}+ \frac{N_3}{\tau_{3,2}}- \frac{N_2}{\tau_{2,1}}$

(3)


These are three independent equations in four variables as the total number $ N= N_1+N_2+N_3+N_4$.
We will now solve these equations in steady state by putting the time derivatives to be zero. From Eqn. (1), we get,

1