## 4 Inhomogeneous Broadening Partial Amplitudes and High Gain \(Fel\) Equation
In the following we make the assumption that the electron bunch is sampled into a series of slices with a longitudinal size of the order of a coherence length [48; 49; 50]:  
\[\lambda_{c}=\frac{\lambda}{4\pi\sqrt{3}\rho}. \tag{50}\]  
During the interaction the radiation is mismatched with respect to the electron bunch reference by a quantity depending on the slippage length. The computation of this aspect of the problem requires a certain amount of computation which, for the homogeneous case (no energy spread) has been accomplished in [51] and for the present case is summarized below. According to the aforementioned reference and to the analytical steps outlined in [46; 47], the problem is solved, in relatively easy terms, by replacing the detuning parameter with the operator  
\[\bar{v}_{0}\rightarrow\hat{v}_{0}=\bar{v}_{0}+i\frac{\lambda_{c}}{L_{g}} \partial_{\xi} \tag{51}\]  
where \(\zeta\) is the bunch coordinate. The replacement (51) in the partial amplitudes accounts for the action of the \(FEL\) dynamics on the optical packet, expressed by the simple Gaussian (to avoid any misunderstanding, we emphasize that \(\zeta\) now indicates the optical bunch coordinate and should not be confused with the pendulum coordinate in Equation (1))  
\[a_{0}(\zeta)=\frac{1}{\sqrt[4]{2\pi\sigma_{\zeta}^{2}}}e^{-\left(\frac{\zeta} {2\pi\zeta}\right)^{2}}. \tag{52}\]  
The action of the partial amplitudes on the initial packet is specified by  
\[ \begin{split}& g_{k}(\hat{z},\hat{v}_{0},\hat{\mu}_{c})a_{0}( \zeta)=g_{k}(\zeta,\hat{z},\bar{v}_{0},\hat{\mu}_{c})e^{\left(\hat{S}_{k} \partial_{\zeta}+D_{k}\partial_{\zeta}^{2}\right)}a_{0}(\zeta),\\ & S_{k}=\frac{\lambda_{c}}{2\Sigma_{k}^{2}L_{g}}\bigg{\{}\bigg{[} \frac{(\pi\mu_{\epsilon}\hat{z})^{2}\xi_{k}}{12}+\lambda_{k}\xi\bigg{]}-i\bar{ v}_{0}\bigg{\}},\qquad\qquad D_{k}=\frac{1}{2\Sigma_{k}^{2}}\bigg{(}\frac{ \bar{z}\lambda_{c}}{L_{g}}\bigg{)}^{2}.\end{split}  \tag{53}\]  
The action of the exponential operator containing the shift (first-order derivative) and diffusive operators (second-order derivative) on the initial packet is provided by  
\[e^{\left(\hat{S}_{k}\partial_{\zeta}+D_{k}\partial_{\zeta}^{2}\right)}a_{0}( \zeta)=\frac{1}{\sqrt[4]{2\pi}}\sqrt{\frac{\sigma_{\zeta}}{\sigma_{\zeta}^{2 }+D_{k}}}\exp{\left[-\frac{1}{4}\frac{\left(\zeta+\zeta_{k}-i\frac{\bar{v}_{0 }\lambda_{c}\xi}{2\Sigma_{k}L_{g}}\right)^{2}}{\sigma_{\zeta}^{2}+D_{k}} \right]}. \tag{54}\]  
A result which allows a few speculations:  
1. The wave packet is shifted back by a quantity \[\zeta_{k}=\frac{1}{24\Sigma_{k}}\Big{[}(\pi\mu_{\epsilon})^{2}\xi_{n}+12 \lambda_{k}\Big{]}\frac{\lambda_{c}}{L_{g}}\.\] (55) This means that the optical packet is moving back with respect to the bunch frame propagating at velocity \(c\), the associated group velocity is \[v_{g\,k}=\left(1-\frac{1}{c}\frac{d}{dt}\zeta_{k}\right)c\.\] (56)  
The radiation moves slower than \(c\), by the effect of the interaction itself. The gain dilution due to the energy spread counteracts the velocity slow down with respect to the case of negligible energy spread. The complex refraction index derived from Equation (56) can be written as  
\[n_{g,k}\simeq 1+\frac{1}{c}\frac{d}{dt}\zeta_{k}. \tag{57}\]
2. The optical packed undergoes a longitudinal diffusion specified by \[\sigma_{k}=\sqrt{D_{k}}=\frac{1}{\sqrt{2\Sigma_{k}}}\bigg{(}\frac{\varepsilon \lambda_{c}}{L_{g}}\bigg{)}\] (58) which ensures that in one gain length the additional packet width is essentially a coherence length. This effect has been studied in the theory of mode-locked \(FEL\) operation within the context of the oscillator theory; the relevant discussion can be found in [2]. In the forthcoming section we apply the result obtained so far to the slice evolution.