\sqrt{1 +
\alpha^2}> 1$ for Gaussian pulses with linear frequency chirp.

Finally, it should be noted that considerations analogous to those just discussed above, can be proposed for the electron beam microbunching. For example the current envelope of a Gaussian, chirped electron beam can be described similarly as in Eq. (\[pulsec\]), with a chirp parameter $\alpha_m$. A time-bandwidth product can be defined, and a quality factor $M_{t,m}$ can be defined as well.

\[sub:con\] Constraint on temporal phase variation for the output Ti:Sa laser pulse
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There are several simplifying assumptions that will be used in our analysis. As has been the case for the analysis presented in the previous paragraph, we restrict our attention to a microbunched electron beam with Gaussian shape. This is not a significant restriction, and extensions are not difficult to consider.

We introduce the following criterion: we consider the electron beam microbunching nearly transform-limited when the performance ratio $M_{t,m}^{-2}$ is down not more than $1/\sqrt{2}$. For a microbunching with Gaussian shape and linear frequency chirp, this criterion will be satisfied under the restriction that the microbunch chirp parameter $\alpha_m < 1$.

A specific example of a microbunched beam with Gaussian profile could be realized in the 