Module 7 : Laser- I
Lecture   : Line Broadening in Lasers
 

$\displaystyle \frac{d^2x}{dt^2} + 2\gamma \frac{dx}{dt}+ \omega_0^2 x =0$

where $ 2\gamma = \alpha/m >0$. (It may be noted that in classical electromagnetic theory the radiation damping term is proportional to $ d^2v/dt^2$, whereas our model assumes it to be proportional to $ v$itself. This is actually of no consequence as the solution being sine and cosine functions, any even number of derivative would do.)
The solution of the above equation for small damping ( $ \gamma^2 << \omega_0^2$) gives damped oscillations

$\displaystyle x = x_0e^{-\gamma t}\sin(\omega t + \phi)$
  where $ \omega = \sqrt{\omega_0^2-\gamma^2}\simeq \omega_0$.
 

Since the dipole moment and hence the electric field of the radiation emitted is proportional to $ x$, we have

$\displaystyle E(t)$

$\displaystyle =$

$\displaystyle E_0e^{-\gamma t}\sin\omega_0t\ \ {\rm for }\ t\ge 0$

 

 

$\displaystyle =$

$\displaystyle 0 \ \ \ {\rm for} \ t<0$

 

   
   
   
   
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