# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 1 Introduction  
Free Electron Laser (FEL) [1] and Compton Back Scattering (CBS) [2] devices are complex systems, which embed different physical environments, characterized by a large number of specific parameters. The relevant theoretical description, necessarily non trivial, requires massive numerical simulations to explore the associated phenomenology and the design details. The use of scaling formulae [3; 4; 5; 6] has played an important role in the design of FEL devices operating in the SASE and oscillator configurations and has been recently employed to discuss the feasibility of FELs driven by plasma-accelerated beams. The logical steps underlying the reported strategy are summarized in Figure 1. Namely, scaling/empirical formulae are used to determine the space of parameters fixing the working point, and home-made/commercially available codes are used to refine the first evaluation and eventually to benchmark the first step. CBS X-ray devices aim at providing sources of photons with energy above tens of \(keV\) with adequate luminosity. In this review, we collect different ideas, provided in the past, by several Authors, regarding Compact Light Sources and present the basic tools for fixing the working point of an actual CBS-based device. The elementary physical mechanism underlying devices of this type is the head-on collision of a (sufficiently) energetic beam with a (sufficiently) intense and coherent optical source [7]. The latter may also be provided by the field of an undulator, which is seen by the electrons as a coherent ensemble of photons [8; 9]. The geometry of the CBS process is reported in Figure 2. The energy of the backscattered photon (\(\mathcal{E}_{s}\)) in terms of the laser/electron energies(\(\mathcal{E}_{l}\), \(E_{e}\)), of the electron kinematic variables (\(\beta\), \(\gamma\)) and of the scattering angles (\(\vartheta_{i,f}\) see Figure 2) is specified by (the polarization of the scattered photons is essentially that of the input field)  
\[\mathcal{E}_{s}=\frac{(1-\beta\,\cos(\vartheta_{i}))\mathcal{E}_{l}}{(1-\beta \cos(\vartheta_{f}))+\frac{\mathcal{E}_{l}}{E_{e}}(1-\cos(\vartheta_{f}- \vartheta_{i}))}. \tag{1}\]  
For the details of the symbols, notations and abbreviations see Nomenclature, if not explicitly mentioned in the text. Figure 1: FEL design strategy flow chart: (1) Input data (beam and undulator parameters and the desired wavelength emission) (2) \(S/E\longrightarrow WP\) = use of Scaling/Empirical formula to find the working point (3) Numerical Simulations (NS) = use of numerical codes to refine the details of the preliminary definition of the space parameters (4) Loop the process, changing the input data and/or the working point, until the benchmark with semi-analytical model is true. Figure 2: Kinematic and geometric variables of Compton Backscattering: input (\(\vec{p}\)) and output (\(\vec{p}^{\prime}\)) electron momenta, \(\vec{k}\), \(\vec{k}^{\prime}\) wave vectors of incident and scattered photon, respectively; orientation and amplitude of the incident \(\theta_{i}\), scattered \(\theta_{f}\) photon angles and \(\phi=\theta_{i}-\theta_{f}\) relative difference. The wavelength of the scattered radiation for a head-on diffusion (\(\theta_{i}\cong\pi\), \(\theta_{f}\cong 0\)), for (ultra-)relativistic electrons (\(\beta\cong 1-\dfrac{1}{2\gamma^{2}}\)) and in the Thomson limit (assuming \(E_{e}\ll\mathcal{E}_{l}\)) is  
\[\lambda_{s}=\dfrac{\lambda_{l}}{4\gamma^{2}}. \tag{2}\]  
For further comments on the scattering geometry and the relevant kinematics, we address the reader to ref. [10]. It should be mentioned that, albeit we refer to the process under study as \"Compton\" scattering, most of our forthcoming discussion applies to the Thomson scattering, namely the physical conditions in which the quantum recoil and hence the ratio \(\mathcal{E}_{l}/E_{e}\) does not play any role. We find indeed that the \"quantum\" term correction \(\mathcal{E}_{l}/E_{e}=\Lambda_{c}/(\gamma\lambda_{l})<<1\) is negligible for the experimental conditions discussed here.