# What is the mathematical relation between the bandwidth of a light pulse and its duration?

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.
• @oliver of course it is a common misconception to think of this as HUR because $t$ is not an operator in QM: see physics.stackexchange.com/q/53802/36194 – ZeroTheHero Apr 2 at 19:48