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

As there is a distribution of velocities, there will be a distribution of resonant frequencies. As $ \nu = \nu_0(1+v_z/c)$, we may substitute $ v_z = c(\nu-\nu_0)\nu_0$in the expression for $ P(v_z)$and get an expression for the lineshape function $ g(\nu)$
$\displaystyle g(\nu) d\nu = \frac{1}{\sqrt\pi}\frac{1}{\Delta\nu^\prime} \exp\left(-\frac{(\nu-\nu_0)^2}{(\Delta\nu^\prime)^2} \right) d\nu$
where
$\displaystyle \Delta\nu^\prime =\nu_0 \sqrt{\frac{2kT}{mc^2}}$
is the distance from the central maximum of the frequency value at which the probability is $ 1/e$times the maximum value. It is often convenient to use the full width at half maximum (FWHM), which is defined as the distance between the points on either side of the central maximum, where the probability (or the intensity) is half its maximum value. Defining this to be $ \Delta\nu$, the expression for the lineshape function becomes
                               $\displaystyle g(\nu) d\nu = \frac{1}{\sqrt\pi}\frac{2\sqrt{\ln 2}}{\Delta\nu} \exp\left(-\frac{(\nu-\nu_0)^24\ln 2}{(\Delta\nu)^2} \right) d\nu$
where the FWHM value $ \Delta\nu$is given by

$\displaystyle \Delta\nu = 2\nu_0\sqrt{\frac{2kT\ln 2}{mc^2}}$
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