# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 3 Spectral Properties and Inhomogeneous Broadening  
In the previous two sections we have not considered the spectral properties of the Thomson backscattering, which are affected by the structure of the electron beam itself and determine a dilution of the spectral brightness. Here we fix a simple procedure to account for the quoted effects. Before going further, we take a step back and better clarify what we have already mentioned without any thorough discussion. Regarding the definition of Thomson scattering limit of CBS, we note that it occurs if the electron recoil (see Figure 6) does not play any role in the scattering process, or, similarly, when the energy of scattered photons is negligible with respect to e-beam energy. Even though academic, for the range of parameters considered here, we will comment on the effect of quantum corrections to CBS in the final section. As is well known [7], underscored in the introduction and discussed elsewhere in this issue [38], within this approximation, the CBS can be treated using the same procedure adopted to treat radiation emission in the magnetic undulator. The analogy allows a straightforward inclusion of the spectral inhomogeneous broadening and of the non-linear contributions, which occur at larger laser intensities. They are responsible for frequency shift (due to the laser equivalent strength parameter) and for inducing a figure eight-like motion in the electron trajectory, determining the emission at higher harmonics [39]. Figure 5: Log-log plot of the flux function \(F(\chi)\sqrt{\chi}\). Figure 6: Vertex of the Feynman diagram expressing the difference between Compton (**a**) and Thomson Scattering (**b**). In case (**b**), the electron dynamics remain practically unperturbed after the interaction process. The energy of the scattered photons with the inclusion of the effect of the laser wave intensity and of small angular deviation from the head-on collision (see Figure 7) writes  
\[\hbar\omega_{s}=\frac{4\gamma^{2}\hbar\omega_{l}}{1+\frac{K_{l}^{2}}{2}+\left( \gamma\theta\right)^{2}}. \tag{36}\]  
The previous equation refers to the wavelength scattered by a single electron, since the bunch is composed by electrons with different energies and different entrance angles, it will be necessary to evaluate the frequency shift associated with the deviations from the reference values and then make an average on the electron energy and angular distributions. This procedure has been described in articles and textbooks (see [3; 5; 37; 39; 40; 41; 42] and references therein) and will not be repeated here. We just remind the reader that it is sufficient to evaluate the relative frequency shift, composed as  
\[ \begin{split}\frac{\delta\omega}{\omega}=\left(\frac{\delta \omega}{\omega}\right)_{\varepsilon}+\left(\frac{\delta\omega}{\omega} \right)_{K_{l}}+\left(\frac{\delta\omega}{\omega}\right)_{\theta}\\ \varepsilon=\frac{\delta\gamma}{\gamma}\equiv\text{relative energy deviation}\end{split}  \tag{37}\]  
where the different terms are due to deviation in energy (\(\varepsilon\)), laser strength parameter (\(K_{l}\)) and scattering angle (\(\theta\)). The dependence of \(K_{l}\) on the transverse coordinate comes from the laser mode distribution shape, while the angular variables from the relation \(\theta^{2}=x^{\prime 2}+y^{\prime 2}\). Considering that the spectral shape is given by [34]  
\[ \begin{split} S(\
u)=\left[\frac{\sin\left(\frac{\
u}{2}\right)} {\left(\frac{\
u}{2}\right)}\right]^{2}=2 Re\left(\int_{0}^{1}(1-t)e^{-i\
u t} dt\right),\\ \
u=2\pi\,\widetilde{N}_{l}\frac{\omega_{x}-\omega}{\omega_{x}}, \end{split}  \tag{38}\]  
where \(\widetilde{N}_{l}=c\Delta T/\lambda_{l}\) is the number of laser cycles within the pulse duration. The convolution on the electron beam distribution can be evaluated as  
\[ \begin{split}\langle S(\
u)\rangle=\int_{D}S(\
u+\delta\
u)f(x, x^{\prime};y,y^{\prime};\varepsilon,t)d^{5}x,\\ \delta\
u=2\pi\,\widetilde{N}_{l}\frac{\delta\omega}{\omega}, \end{split}