## 4 Inhomogeneous Broadening Partial Amplitudes and High Gain \(Fel\) Equation  
The high gain integral equation becomes slightly more complicated if the effect of non ideal e-beam qualities is included. The averages in Equation (2) need to be extended to the beam distribution (energy, spatial, angular, etc.). With these premises and limiting ourselves to the effect of the energy spread only, we find [45]  
\[\frac{d}{d\tau}a=i\pi g_{0}\int_{0}^{\tau}a(\tau-\tau^{\prime})\tau^{\prime}e ^{-iv_{0}\tau^{\prime}-\frac{1}{2}(\pi\mu_{\tau}\tau^{\prime})^{2}}d\tau^{ \prime},\qquad\mu_{\epsilon}=4N\sigma_{\epsilon} \tag{46}\]  
obtained after taking the average of a Gaussian relative energy distribution with r.m.s. \(\sigma_{\epsilon}\). The solution of Equation (46) cannot be obtained in analytical terms; however, the method of partial amplitude expansion pioneered by [PERSON] in references [46; 47] yields a fairly useful analytical mean allowing the solution of, e.g., Equation (46) in the form (we use the \(\tilde{v}_{0},\tilde{z}\) variables, more appropriate for the forthcoming discussion)\[\tilde{a}(\tilde{z},\tilde{v}_{0})=\tilde{a}_{0}\Bigg{(}1+\sum_{k=1}^ {k}\tilde{\tau}\bigg{(}\frac{\tilde{z}}{\sqrt{3}}\bigg{)}^{3k}g_{k}(\tilde{z}, \tilde{v}_{0},\tilde{\mu}_{\iota})\Bigg{)}, \tag{47}\] \[g_{k}(\tilde{z},\tilde{v}_{0},\tilde{\mu}_{\iota})=\frac{M_{k} \gamma_{k}}{\Sigma_{k}}\exp\Biggl{[}-\frac{(\tilde{\mu}_{\iota}\tilde{z})^{2} (\beta_{k}+i\tilde{\varsigma}_{k}(\tilde{v}_{0}\tilde{z}))+12(i\lambda_{k}( \tilde{v}_{0}\tilde{z})+(\tilde{v}_{0}\tilde{z})^{2})}{24\Sigma_{k}^{2}} \Biggr{]},\] \[M_{k}=\frac{1}{(3k)!},\qquad\gamma_{k}=(3k+1)\sqrt{\frac{3k+2}{ 2k(k+1)}},\qquad\Sigma_{k}=\sqrt{\gamma_{k}^{2}+\frac{1}{12}(\tilde{\mu}_{ \iota}\tilde{z})^{2}},\] \[\beta_{k}=2\frac{4k+1}{k+1},\qquad\lambda_{k}=2(3k+1)\frac{3k+2 }{k+1},\qquad\xi_{k}=2\frac{(2k+1)(k-1)}{(k+1)(3k+1)}.\]  
The previous equation can be profitably used to get information of a practical nature. The low gain expansion (\(k=1\)) yields for the intensity growth  
\[\mid\tilde{a}(\tilde{z},\tilde{v}_{0})\mid^{2}\simeq\mid\tilde{a}_{0}\mid^{2} \left[1+2\frac{M_{1}}{3\sqrt{3}\Sigma_{1}}\tilde{z}^{3}\cdot\exp\left(-\frac{ \beta_{1}\tilde{\mu}_{\iota}^{2}+\tilde{v}_{0}^{2}}{24\Sigma_{1}^{2}}\tilde{z }^{2}\right)\sin\left(\frac{\tilde{v}_{0}\left(12\lambda_{1}+\xi_{1}(\tilde{ \mu}_{\iota}\tilde{z})^{2}\right)}{24\Sigma_{1}^{2}}\tilde{z}\right)\right] \tag{48}\]  
which provides an idea of the interplay between inhomogeneous broadening and the other mechanisms contributing to the laser evolution. In Figure 6 we have reported the analogues of Figure 2b for different values of the inhomogeneous broadening parameter. The plot yields quite a good description of how the intensity growth is diluted by the presence of a non zero energy spread. Other effects due to angular and spatial distribution of the electron beam can be introduced too. From the conceptual point of view, they do not imply any problem, but for the introduction of a further convolution term in Equation (46). The induced detrimental effects are expressed through appropriate \(\mu\)-parameters, analogous to those associated with the energy spread. They are listed in [18]. It is important to stress that, from the practical point of view, the effect of non perfect electron beam qualities produces an increase of the gain length, which can be parameterized as [17; 18]  
\[L_{g}(\tilde{\mu}_{\epsilon})=\left(1+0.185\frac{\sqrt{3}}{2}\tilde{\mu}_{ \epsilon}^{2}\right)L_{g} \tag{49}\]  
The interest in the method of partial amplitudes stems also for the fact that it can be generalized to more complicated situations, involving, e.g., the pulse propagation contributions.