We discussed as, in linear regime, the presence of a beta2 (GVD) affects a pulse initially chirped in time domain (i.e. adding a phase in time domain). In such case, the pulse can indefinitely broaden upon traveling along z into the material OR, it can get narrower down to a minimum and then broaden again. Hence, providing some chirp it can be useful, for instance, if one aims to get a minimum beam waist at a given point in space (the sample) and NOT in the z position where the pulse is generated (the laser output). Note that starting with a pulse chirped in time domain, the initial bandwidth is also broadened with respect to 1/T0, i.e. the pulse is initially non fourier Transform limited. In principle, one could start with a pulse chirped in frequency domain (only a phase factor in frequency domain) hence modifying the initial time duration. This situation is similar to what is done pre-compensating pulse broadening. See for instance Chirped Pulse Amplification.
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