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

$\displaystyle F(t)$

$\displaystyle =$

$\displaystyle f_0e^{-2\pi i \nu_0t}\ \ {\rm for } \ \mid t\mid \le \frac{\Delta}{2}$

 

 

$\displaystyle =$

$\displaystyle 0, \ \ {\rm otherwise}$

 

The wave is not strictly monochromatics, as would have been the case if the wave train was for indefinite duration. The spread in frequency in the wave is found by taking a Fourier transform of the wave train,
 

$\displaystyle F(\nu)$

$\displaystyle =$

$\displaystyle f_0\int_{-\Delta/2}^{\Delta/2}e^{-2\pi i(\nu-\nu_0)t}dt$

 

 

$\displaystyle =$

$\displaystyle f_0\Delta t \left[\frac{\sin\pi(\nu-\nu_0)\Delta t}{\pi(\nu-\nu_0)\Delta t} \right]$

 

The intensity pattern which is given by the square of $ F(\nu)$is shown. Though there is no unique way of defining the frequency spread, we note that the first zero of the intensity pattern occurs at $ \nu-\nu_0 = \pm 1/\Delta t$. Thus one can define the spread of frequency to be given by $ \Delta\nu = 1/\Delta t$. This is simply a statement of the classical uncertainty principle connecting the band width to coherence time $ \Delta t$,

$\displaystyle \Delta t\Delta\nu \sim 1$
1