 Solution

The temporal intensity is proportional to the squared field modulus:Its half-maximum points satisfy $2a{t}^2=\ln 2$, so the intensity full width at half maximum isWith the chosen $e^{-i\omega t}$ convention, the instantaneous angular frequency is minus the time derivative of the field's argument. HenceA localized Gaussian requires $\mathrm{Re}\Gamma =a>0$. It is positively chirped precisely when $d\omega /dt=2\mathrm{Im}\Gamma >0$, or $b>0$; $b=0$ gives no chirp and $b<0$ gives negative chirp.