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

$\displaystyle E(\omega)$

$\displaystyle =$

$\displaystyle \frac{1}{\sqrt{2\pi}}\int_0^\infty E_0e^{-\gamma t} e^{i(\omega_0-\omega t}dt$

 

 

$\displaystyle =$

$\displaystyle \frac{1}{\sqrt{1\pi}}\frac{1}{i(\omega-\omega_0)-\gamma}$

 


The magnitude of the electric field is
$\displaystyle E(\omega) = \frac{1}{\sqrt{2\pi}}\left[\frac{1}{(\omega-\omega_0)^2+\gamma^2} \right]^{1/2}$
 

since the intensity $ I(\omega)$is proportional to $ \mid E(\omega)\mid^2$, the normalized intensity is

$\displaystyle I(\omega) = I_0(\gamma/\pi)\left[\frac{1}{(\omega-\omega_0)^2+\gamma^2} \right]$
where $ \int_0^\infty I(\omega)d\omega = 1$. The distribution of intensity is called Lorentzian distribution. The maximum intensity $ I_{\rm max}= I_0/\pi\gamma$at $ \omega=\omega_0$. The full width at half the maximum intensity is $ 2\gamma$. The lineshape function can be easily inferred from the intensity distribution function, and is given by

$\displaystyle g(\nu) = \left(\frac{\Delta\nu}{2\pi}\right)\frac{1}{\left[(\nu-\nu_0)^2+ \left(\frac{\Delta\nu}{2}\right)^2\right]}$
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