field applications including EEHG FELs. In particular, femtosecond laser systems have become the primary method to deal with these applications. The reasons for this are the availability of broadband, efficient lasing media such as titanium-doped sapphire (Ti:Sa), and of techniques like Kerr-lens mode locking and chirp pulse amplification (CPA). In CPA systems, light passes through a number of optical components. Moreover, non-linear effects take place in the amplifying medium. This can degrade the temporal quality of the output pulse, which can be appropriately modeled in a slowly-varying real field envelope and time-dependent carrier frequency approximation. The time-bandwidth product constitutes a proper measure of the departure from the ideal case, in which there are no temporal variations of the carrier frequency. In this Section we quantitatively describe the relation between carrier frequency chirp and corresponding broadening of the spectrum. This leads to a time-bandwidth product exceeding the Fourier limit.

Pulse duration and spectral width
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For our purposes, it is convenient to consider a Gaussian pulse with a linear frequency chirp. This choice is one of analytical convenience only, and may be generalized. The slowly complex field envelope is given by

$$\begin{aligned}
E(t) = A \exp\left[-\frac{t^2}{2 \tau^2}\right]\exp\left[i
\frac{\alpha t^2}{2\tau^2}\right] \label{pulsec}\end{aligned}$$

where 