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

Fraction of atoms in level 4 which arrive at level 3 is given by the ratio $ \tau_4/\tau_{4,3}$and the fraction of atoms in level 3 which radiatively make a transition to level 2 is $ \tau_3/\tau^{rad}_{3,2}$. We have used a superscript rad to indicate that only the radiative component of the transition from 3 to 2 is considered. Thus

$\displaystyle \eta = \frac{\tau_4}{\tau_{4,3}}\frac{\tau_3}{\tau^{rad}_{3,2}}$

Substituting this in the previous equation, we get, for $ \Delta N/N$
$\displaystyle \boxed{\frac{\Delta N}{N} = \frac{(1-\beta)W_p\eta\tau_{3,2}^{rad... ...+ \left[ (1+\beta)+ 2\frac{\tau_{4,3}}{\tau_3}\right]W_p\eta\tau_{3,2}^{rad} }}$
To ensure that most of the atoms excited by pumping participate in laser transition, the life time in the level 3 must be the longest. Using $ \tau_{4,3}\ll \tau_3$, we may ignore this term in the denominator of the above. Further, $ \tau_{3,2}\gg \tau_{2,1}$. Using these, one can see that $ \beta$is small and approaches zero.

 

Thus $ \Delta N/N$ramains positive and population inversion occurs even for very small pumping power. For $ \beta\rightarrow 0$, the expression for $ \Delta N/N$is given by
$\displaystyle \boxed{\frac{\Delta N}{N}= \frac{ W_p\eta\tau_{3,2}^{rad}}{1+ W_p\eta\tau_{3,2}^{rad}} }$
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