# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 2 Fixing a More Realistic Model  
In this section we put the \"theoretical frame\" of the introductory section on a more firm basis, including the details associated with electron and laser beam distribution details. The assumption underlying the definition of brightness in Equation (20) is that the laser spot envelopes the electron beam (see Figure 4). Furthermore, it implicitly contains the hypothesis that the transverse phase space angular divergence of the scattered X-rays matches that of the electron beam. In the numerical example presented in the previous section, we have assumed a large repetition rate, a small emittance and a very low \(\beta_{T}\) value at the IP. We have accordingly followed a naive optimization procedure, which simply maximizes the repetition rate and minimizes the laser and electron transverse dimension at IP. The electron bunch length should be furthermore sufficiently short to avoid the hour-glass effect, as understood by inspecting Figure 4.  
The weakness of the discussion developed in the introductory section is displayed in Figure 4, where we have reported the CBS collision region, indicating the hour-glass effect associated with the incoming laser and output X-ray beam diffraction behavior [33]. Any reasonable model should therefore include the spatial and temporal dependence of the laser and electron distribution. Assuming a cylindrical symmetry we can write the number of photons scattered per unit volume and unit time as  
\[\frac{d^{3}N_{x}}{c\,dVdt}=\sigma_{Th}(1+\beta_{0})\bar{N}_{l}(r,z,t)\bar{N}_ {e}(r,z,t), \tag{21}\]  
with \(dV=2\pi rdrdz\) as the volume element in cylindrical coordinates, \(\bar{N}_{l,e}\) as the laser and electron number distribution and \(\beta_{0}\) as the average electron on axis reduced velocity. It is evident that Equation (21) is substantially equivalent to our starting Equation (8), with the only difference that, in order to evaluate the number of scattered X-photons, it is necessary to perform an integration on the transverse volume and on the longitudinal coordinate. We will follow the indications in Ref. [33] to model the electron and laser photon distributions by the use of the same formalism. We exploit indeed the fact that the relevant longitudinal and transverse envelopes can be viewed as equivalent, if the transport is provided by linear optical elements (lens, quadrupoles, etcetera) where aberrations do not play any significant role [34; 35]. We write therefore  
\[ \begin{split}&\bar{N}_{p}(r,z,t)=\frac{N_{p}}{A_{p}}\frac{1}{ \sigma_{p}(z)^{2}}e^{-\frac{r^{2}}{\sigma_{p}(z)^{2}}}e^{-\frac{t_{p}(z)^{2}}{ \Delta\tau_{p}^{2}}},\\ & p\equiv l,e.\end{split}  \tag{22}\]  
The previous function is nothing but a Gaussian in time and transverse coordinate, the quantities \(\sigma_{p}(z)\) and \(\Delta\tau_{p}\) are the r.m.s. of the longitudinal and temporal distributions  
Figure 4: Laser and scattered X-ray beam longitudinal profiles. The screen in green represents the detector and \(z_{d}\) is the relevant longitudinal coordinate. (see Ref. [33] for further details). and \(A_{p}\) is the normalization constant. The specific forms for the \(e/l\) dependence on z is given below  
\[ \begin{split}\sigma_{e}(z)&=\sigma_{e}\sqrt{1+\left( \frac{z}{\beta_{T}}\right)^{2}},\\ \sigma_{l}(z)&=\sigma_{l}\sqrt{1+\left(\frac{z}{Z_{R }}\right)^{2}},\end{split}  \tag{23}\]  
where \(\sigma_{e},~{}~{}\beta_{T}\) are the already introduced electron-beam section and Twiss parameter, respectively. The laser section \(\sigma_{l}\) and \(Z_{R}\) are associated with the beam waist and Rayleigh length, namely  
\[ \begin{split}\sigma_{l}^{2}&=\frac{W_{0}^{2}}{2},~{}~ {}~{}W_{0}\equiv\text{laser beam waist},\\ Z_{R}&=\frac{\pi\,W_{0}^{2}}{\lambda_{l}}\\ A_{e}&=\pi^{3/2}c\Delta\tau,\\ A_{l}&=2\Big{(}\frac{\pi}{2}\Big{)}^{3/2}c\Delta t. \end{split}  \tag{24}\]  
In passing, we remind the reader of close analogy between the Rayleigh length and the \(\beta_{T}\) parameter [34; 35; 36].