Module 7 : Laser- I
Lecture   : Line Broadening in Lasers
  are depleted. If $ u(\nu_s)$is the energy density of the signal, the rate of depletion of number of photons $ N_{ph}$is

$\displaystyle -\frac{dN_{ph}}{dt}=-\frac{d}{dt}\left[\frac{u(\nu_s)d\nu}{h\nu_s}\right]=\frac{-d}{dt}\left[n_1\right]$

where we have used the fact that the incident photons are nearly monochormatic.

Thus equation (A) becomes

$\displaystyle \frac{\partial u(\nu_s)}{\partial t}= B_{21} u(\nu_s) g(\nu_s, \nu_0)h\nu_s\left[n_2-\frac{g_2}{g_1}n_1\right]\eqno(B)$

Consider laser photons travelling through a slab of active material of thickness dx. If $ c$is the speed of light in the material, as the signal advances from $ x$to $ x+dx$, we may use $ \frac{\partial} {\partial t} = c\frac{\partial}{\partial x}$to rewrite the above as

$\displaystyle \frac{\partial u (\nu_s)}{\partial x}=\frac{1}{c} B_{12} u(\nu_s) g(\nu_s,\nu_0) h\nu_s\left[n_2-\frac{g_2}{g_1}n_1\right] $

Integrating

$\displaystyle u(\nu_s)=u_0(\nu_s)\left[-h\nu_s g(\nu_s,\nu_0) B_{21}(n_2-\frac {g_2}{g_1}n_1)\frac{x}{c}\right]$
1