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Transform limited pulse

A transform limited pulse of an optical wave is the shortest pulse theoretically possible given the spectral bandwidth. This is achieved by removing the chirp across the full bandwidth. A pulse with no chirp is knows as transform limited, Fourier-transform limited, or bandwidth limited.

The transform limited pulse duration Δτ is linked to the spectral bandwidth Δν via the Fourier Transform. A minimum product of the two can be defined as the Time Bandwidth Product TBP. $$TBP = \Delta\tau \Delta\nu.$$ $$TBP \text{ is unitless}$$ $$\Delta\tau \text{ in seconds}$$ $$\Delta\nu \text{ in Hz}$$

Based on the shape of the spectrum, the TBP has a unique value: $$TBP_{Gauss} = \frac{2 Log[2]}{\pi} \approx 0.441$$ $$TBP_{Sech^2} = \frac{(2 Log[1+\sqrt{2}])^2}{\pi} \approx 0.315$$ $$TBP_{Lorentz} \approx 0.142 $$

The widths Δτ and Δν are measured at FWHM.
The TBP for Gaussian shape is 1.4 larger compared to the TBP for Sech². This means that for pulses with identical spectral bandwidth Δν the pulse duration of the Gaussian shape is 1.4 times longer compared to the Sech² shape.
If the transform limited pulse duration Δτ or the spectral bandwidth Δν is know for a given shape, the other quantity can be calculated:
$$\Delta\tau = \frac{TBP}{\Delta\nu}$$ $$\Delta\nu = \frac{TBP}{\Delta\tau}$$ If the spectral shape is not equal to one of the shapes above, the TBP is different, and the transform limited pulse duration will differ. A Fourier Transform calculation of the spectrum is required to determine the transform limited pulse duration and the appropriate TBP.
A Fourier Transform calculator for arbritary shapes can be found here: Fourier Transform.