On the second order characterization of ultrashort pulses

The spectral properties of almost-Gaussian functions are considered and app lied to the characterization of the second-order approximation in the expan sion of the coefficients of almost-perfect optical pulses. Specifically, ad ding small amounts of odd-order Hermite-Gaussians to a Gaussian induces a s econd-order increase in the time-bandwidth product, while the increase in t he time-bandwidth product from adding even-order Hermite-Gaussians is highe r-order and hence smaller. We indicate the class of small perturbations of Gaussian functions which change neither the temporal profile of the intensi ty nor the intensity of the spectral profile.