I have seen stated that ultrashort pulses have a broad bandwidth.
In the link above, it is stated that a "Gaussian pulse with a 1 ps pulse duration(...) has an optical bandwidth of $\approx 0.44$ THz."
But was is the mathematical formula relating these terms?
Fourier analysis: $$ \Delta \omega \Delta \tau \ge \frac{1}{2} $$ where $\Delta \tau$ is the pulse duration and $\Delta \omega$ the bandwidth. It is because to have a wave form whose amplitude goes up and back down again in a time $\Delta \tau$ (and then stays down) you have to add together sinusoidal waveforms with a range of frequencies, so that they reinforce each other during the time $\Delta \tau$ and cancel each other out at other times. For this to be possible they must be getting out of step with one another in a time $\Delta \tau$ so their frequencies cannot all be very similar: they have to be spread apart by separations of order $1/\Delta \tau$ or more.
