# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 1 Introduction
The mechanism, underlying the emission of bremsstrahlung photons by relativistic electrons moving inside a magnetic undulator, can be traced back to the same microscopic process (see Figure 3). The undulator is a device characterized by an alternating magnetic field, with a spatial period \(\lambda_{u}\). The use of the Fermi-Weiszacker-Williams [11; 12; 13] approximation was the starting point of the [PERSON] seminal paper [1], which signed the modern point of view to the FEL. In more practical terms, the process of photon emission in the undulator can be considered as the head-on collision of a \"pseudo\" photon with wavelength  
\[\lambda^{*}=2\lambda_{u}, \tag{3}\]  
and an electron with the relativistic factor defined as  
\[\gamma^{*}=\dfrac{\gamma}{\sqrt{1+\dfrac{K_{u}^{2}}{2}}}. \tag{4}\]  
According to Equation (2), we find for the wavelength of the radiated photons  
\[\lambda_{s}=\dfrac{\lambda_{u}}{2\gamma^{2}}\bigg{(}1+\dfrac{K_{u}^{2}}{2} \bigg{)}. \tag{5}\]  
From the point of view of the elementary mechanisms, the role of the undulator-strength parameter is that of including, in the scattering process, the contribution of the scattered wave photon density [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. It is evident that an analogous term (sometimes called Kibble shift) should also appear in Equation (2). The laser field strength parameter is of noticeable importance for our discussion and, in terms of the laser intensity, reads (we use \(K_{l}\) instead of \(a_{0}\) to denote the laser strength parameter either for continuity reasons with the notation of our previous papers or to underline the quasi complete equivalence with the undulator parameter)  
\[K_{l}=\dfrac{eE_{l}\lambda_{l}}{2\pi\pi n_{e}c^{2}}, \tag{6}\]  
Figure 2.0 Emissionin undulator macroscopic picture (**left**) and microscopic picture (**right**); IP stands for interaction point. where \(E_{I}\) is the peak electric field associated with the laser. In practical units, the Equation (6) is usually written as  
Figure 2.1  
with \(\lambda_{I}\) and \(I_{I}\) being the laser wavelength (expressed in nanometer) and the intensity expressed in \(W/m^{2}\), respectively. The laser strength parameter \(K_{I}\) is a key quantity determining the regime of operation of the scattering process itself (for further comments see the second part of the paper). In order to give an idea of the involved orders of magnitude, we note that for an Infrared Laser, (\(\lambda\approx 1\)\(\upmu\)m) \(K_{I}=0.1\) corresponds to a laser intensity slightly larger than \(10^{18}\)\(W/m^{2}\). Even though the process shown in Figure 2 reports a single photon-single electron scattering, in a CBS device, the interaction occurs between the \(N_{e}\) (non interacting) electrons composing the electron bunch and the \(N_{I}\) photons contributing to the laser field. The flux of the scattered X-ray photons (\(\dot{N}_{x}\)) in a head-on scattering depends on the total cross section of the process and on the luminosity (\(L_{0}\)), a pivotal quantity including the numbers of electrons and photons, namely  
Figure 2.2  
where \(\sigma_{Th}\) is the Thomson cross section [26], and \(L_{0}\) writes  
Figure 2.3  
Figure 2.4  
being \(\sigma_{e}\) the r.m.s. electron bunch transverse section, \(\sigma_{I}\) the r.m.s. laser beam transverse section and \(f\) the rate of the collisions. We have considered a round beam, otherwise \(\Sigma_{e,I}\), should be replaced by \(\sqrt{(\sigma_{e,x}^{2}+\sigma_{l,x}^{2})(\sigma_{e,y}^{2}+\sigma_{l,y}^{2})}\). The previous relationships suggest the naive, but importantly, consideration that the output flux is optimized by increasing the number of electrons, (and therefore the electron bunch charge) the number of photons (and thus the laser intensity) and finally by matching and minimizing the electron and laser transverse sections. We can handle Equation (8) in order to obtain a more convenient form, for practical purposes.