Fourier Transform of short pulse I'm trying to take the fourier transform of a short laser pulse, represented by $E(t) = E_oe^{-(t/\Delta T)^2}\times e^{-i \omega t}$
E is the electric field of the laser pulse. $E_o$ and $\Delta T$ are both constants. Specifically I want to know if there are beats in the fourier transform, and what their frequency is. Any help would be great.

The $e^{-i\omega t}$ factor in your pulse means only a phase delay by $\omega t$, otherwise the pulse is a Gaussian pulse which means that the Fourier transform will also be Gaussian and it will be $$\large \mathcal{E}(f)=E_0\Delta T\sqrt{\pi}e^{-\Delta T^2 \pi^2\left(f+\frac{\omega}{2\pi}\right)^2}$$  
So, the power will be mostly concentrated in the frequency range $\displaystyle \left[0,\frac{1}{\sqrt{2}\pi \Delta T}\right]$ 
Also, the frequency domain pulse has its peak at the $\frac{-\omega}{2\pi}$ frequency and then it monotonically decreases as $f$ increases. 
