# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 5 Final Comments  
We have neglected, so far, quantum effects in the study of CBS. We have mentioned that most of our assumption is due to the neglecting of quantum recoil. The interplay between classical and quantum regimes is not as sharp as we have indicated. Some comments in this direction are addressed in this section. In the previous sections, we have developed and summarized different ideas concerning the physics and the phenomenological aspects of the CBS, finalized to the design and construction of quasi-coherent X-ray sources. We have underscored that the relevant theoretical and technological foundations deepen their roots in the research developed since the second half of the last century. In particular the underlying technology is the result of the tremendous effort aimed at providing high-intensity lasers. It is therefore important to emphasize that the breakthrough occurred with the introduction of the Chirped Pulse Amplification technique [51], and accelerators and high-quality beams. Within the last context, a not secondary role has been played by the development of FEL sources in all its declinations (oscillators, SASE, fourth generation of synchrotron radiation sources, etcetera). We have underlined that, if the quantum effects are neglected, most of the formalism resembles that already exploited to understand the details of the radiation emitted in magnetic undulators [52; 53]. This is however a superficial statement, which should be commented on more carefully. To frame the problem in easily understandable physical terms, we list a few guiding parameters [54]  
\[\psi=\frac{eE_{I}\Lambda_{e}}{m_{e}c^{2}}=\frac{E_{I}}{\mathcal{E}_{ss}},\]  
\[\varsigma=\frac{\hbar\omega_{I}}{m_{e}c^{2}}=\frac{\Lambda_{e}}{\Lambda_{I}}, \tag{56}\]  
\[\mathcal{E}_{ss}=\frac{m_{e}c^{2}}{e}\frac{1}{\Lambda_{e}}\cong 1.323\times 10^{18 }V/m,\]  
where \(\mathcal{E}_{ss}\) is the Sauter-Schwinger critical field [54; 55], corresponding to the threshold energy for the creation of an \(e^{\pm}\) pair. The quantities \(\psi\), \(\varsigma\) compare, to the electron mass, the energy gained by an electron under the effect of an external field in a reduced Compton wavelength and the energy of a single laser photon, respectively. Regarding the parameter \(\varsigma\), we note that it is hidden in Equation (1), which, in the limits of negligible angular and intensity dependent shift, yields the following scattered frequency and wavelength shift  
\[\omega_{s}\cong\frac{4\gamma^{2}\omega_{I}}{1+4\gamma\varsigma},\]  
\[\frac{\Delta\lambda_{s}}{\lambda_{s}}=4\gamma\varsigma. \tag{57}\]  
According to Figure 13a, we get that deviations from the classical limit can be observed for \(4\gamma\varsigma>0.1\), which can also be written as  
\[\lambda_{e}>2.5\times 10^{-2}\frac{\lambda_{I}}{\gamma}, \tag{58}\]  
quantum shift effects can, e.g., be observed for energies above \(5~{}GeV\) for a pump laser with \(\lambda_{I}\cong 500~{}nm\) (2.48 \(eV\)) [56]. The same conclusion can be achieved by inspecting the CBS cross section vs. \(4\gamma\varsigma\) reported in Figure 13b, which exhibits deviations larger than 10% with respect to the Thomson cross section for the same amount of \(4\gamma\varsigma\). According to the previous discussion, significant quantum effects can be observed in the process of X-ray photons scattered by energetic electrons. This problem has been addressed in Ref. [55; 57], where the use of FEL X-ray beams to generate narrow band \(GeV\) photons via the backscattering of a \(7~{}GeV\) electron beam has been proposed. In Figure 14, we have reported the differential cross section of the Compton backscattering of a pump laser with different photon energies (from IR to hard X-ray), namely  
Figure 13: (**a**) Comparison between back-scattered photon frequency vs. \(\Delta\lambda_{s}/\lambda_{s}\) with (dot) and without (continuous) recoil contributions. (**b**) CBS cross section normalized to \(\pi r_{0}^{2}\) vs. \(\Delta\lambda_{s}/\lambda_{s}\).