# Analysis of the Effect of Pulse Length and Magnetic Field Strength on Nonlinear Thomson Scattering
## 2 Materials and Methods
\begin{array}{c}\partial_{t}a_{x}=\partial_{t}a_{L}\ cos\varphi- \partial_{t}qa_{L}sin\varphi\\ \partial_{y}a_{x}=\partial_{y}a_{L}cos\varphi-\partial_{y}qa_{L}sin\varphi\\ \partial_{z}a_{x}=\partial_{z}a_{L}\ cos\varphi-\partial_{z}\varphi a_{L}sin \varphi\\ \partial_{t}a_{y}=-\delta[\partial_{t}a_{L}sin\varphi+\partial_{t}qa_{L}cos \varphi]\\ \partial_{x}a_{y}=-\delta(\partial_{x}a_{L}sin\varphi+\partial_{x}qa_{L}cos \varphi)+B_{0}\\ \partial_{z}a_{y}=-\delta[\partial_{z}a_{L}sin\varphi+\partial_{z}\varphi a_{ L}cos\varphi]\\ \partial_{t}a_{x}=-\{\frac{2x}{R_{0}}\partial_{t}a_{L}sin(\varphi+\theta)+ \frac{2x}{R_{0}}a_{L}cos(\varphi+\theta)\partial_{t}\varphi+\delta\left[ \frac{2x}{R_{0}}\partial_{t}a_{L}cos(\varphi+\theta)-\frac{2y}{R_{0}}a_{L} sin(\varphi+\theta)\partial_{t}\varphi\right]\}\\ \partial_{x}a_{z}=-\{\frac{2}{R_{0}}a_{L}sin(\varphi+\theta)+\frac{2x}{R_{0}} a_{L}sin(\varphi+\theta)\partial_{x}\varphi+\delta\left[\frac{2y}{R_{0}} \partial_{x}a_{L}cos(\varphi+\theta)-\frac{2y}{R_{0}}a_{L}sin(\varphi+\theta) \partial_{x}\varphi\right]\}\\ \partial_{y}a_{z}=-\{\frac{2x}{R_{0}}\partial_{y}a_{L}sin(\varphi+\theta)+ \frac{2x}{R_{0}}a_{L}cos(\varphi+\theta)\partial_{y}\varphi+\delta\left[ \frac{2x}{R_{0}}a_{L}cos(\varphi+\theta)+\frac{2y}{R_{0}}\partial_{y}a_{L}cos( \varphi+\theta)-\frac{2y}{R_{0}}a_{L}sin(\varphi+\theta)\partial_{x}\varphi \right]\}.\end{array} \right. \tag{7}\]  
Combining Equations (2)-(7) and the Coulomb norm condition \(\
abla\cdot a=0\), the decomposition is unfolded in a three-dimensional spatial coordinate system, and a set of differential equations is used to realize the spatial and temporal decomposition of the electron motion as follows:\[ \begin{cases}\gamma d_{t}u_{x}=\big{(}1-u_{x}{}^{2}\big{)}\partial_{t}a_{x}+u_{y} \big{(}\partial_{y}a_{x}-\partial_{x}a_{y}\big{)}+u_{z}\big{(}\partial_{z}a_{x}- \partial_{x}a_{z}\big{)}-u_{x}u_{y}\partial_{t}a_{y}-u_{x}u_{z}\partial_{t}a_{ z}\\ \gamma d_{t}u_{y}=\big{(}1-u_{y}{}^{2}\big{)}\partial_{t}a_{y}+u_{x}\big{(} \partial_{x}a_{y}-\partial_{y}a_{x}\big{)}+u_{z}\big{(}\partial_{z}a_{y}- \partial_{y}a_{z}\big{)}-u_{x}u_{y}\partial_{t}a_{x}-u_{y}u_{z}\partial_{t}a_ {z}\\ \gamma d_{t}u_{z}=\big{(}1-u_{z}{}^{2}\big{)}\partial_{t}a_{z}+u_{x}\big{(} \partial_{x}a_{z}-\partial_{z}a_{x}\big{)}+u_{y}\big{(}\partial_{y}a_{z}- \partial_{z}a_{y}\big{)}-u_{x}u_{z}\partial_{t}a_{x}-u_{y}u_{z}\partial_{t}a_ {y}\\ d_{t}\gamma=u_{x}\partial_{t}a_{x}+u_{y}\partial_{t}a_{y}+u_{z}\partial_{t}a_ {z}.\end{cases}  \tag{8}\]  
An electron undergoing relativistic accelerated motion emits radiation, which has a power per-unit solid angle formula:  
\[\frac{dP(t)}{d\Omega}=\left[\frac{\left|\overrightarrow{n}\times\left[\left( \overrightarrow{n}-u\right)\times d_{t}u\right]\right|^{2}}{\left(1- \overrightarrow{n}\cdot u\right)^{6}}\right]_{t^{\prime}} \tag{9}\]  
\[\frac{dP(t)}{d\Omega}\propto\gamma^{8}d_{t}u_{x}{}^{2}, \tag{10}\]  
where the radiated power is normalized by \(\ell^{2}\omega_{0}{}^{2}/4\pi c\); \(\Omega\) is the unit solid angle; and the direction of radiation \(\overrightarrow{n}=\sin(\theta)\cos(\varphi)\overrightarrow{x}+\sin(\theta) \sin(\varphi)\overrightarrow{y}+\cos(\theta)\overrightarrow{z}\). The derivation of Equation (10) is given by [23]. t\({}^{\prime}\) is the time at which the electron interacts with the laser pulse; the following relationship exists between t\({}^{\prime}\) and t:  
\[t=t^{\prime}+R_{0}-\overrightarrow{n}\cdot\overrightarrow{r}, \tag{11}\]  
where \(R_{0}\) is the distance of the observation point from the electron and \(\overrightarrow{r}\) is the electron site vector. It is assumed that the observation point is far away from the point of interaction between the laser and the electron.