$\alpha$ is the chirp parameter, and the FWHM pulse duration is related to the rms duration $\tau$ by $\Delta \tau = \sqrt{4 \ln
2} \cdot \tau$.

By Fourier transforming Eq. (\[pulsec\]), it can be demonstrated (see e.g. [@MILO]) that the spectral intensity is a Gaussian with a FWHM given by $\Delta \omega = (\sqrt{4\ln 2}/\tau) \sqrt{1 +
\alpha^2}$. The time-bandwidth product of the pulse is therefore

$$\begin{aligned}
\Delta \omega\cdot \Delta \tau = 4 \ln 2 \cdot \sqrt{1 + \alpha^2}
\label{TBp}\end{aligned}$$

This is larger than the time-bandwidth product of an unchirped Gaussian pulse, which is just $4 \ln 2$ . In other words, chirping increases the time-bandwidth product by broadening the pulse spectrum while preserving the pulse width. Note that $\Delta \omega
\cdot \Delta \tau = 4 \ln 2$ is the smallest time-bandwidth product for a Gaussian pulse corresponding to the transform-limit (or bandwidth limit, or Fourier limit).

The temporal quality of the pulse can be defined by a quality factor $M_t^2$, defined as the ratio between the time-bandwidth product for real and transform-limited pulse. Hence, one can characterize pulse by specifying its quality through the $M_t^2$ factor and by giving the pulse shape. In our case of interest, $M_t^2 = 