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聚焦高斯脉冲激光场可由经$\frac{m c^{2}}{e}$归一化的激光振幅a0(其中,m与e分别为电子的静止质量和电荷量)、激光脉宽L、光束半径w、相位φ等参量描述。其中$a_{0}=0.85 \times 10^{-9} \lambda_{0} \sqrt{I}$,I为峰值光强,λ0为激光波长,λ0=1μm。光束半径w是坐标z的函数,其间的关系为$w(z)=w_{0}\left(1+\frac{z^{2}}{z_{\mathrm{R}}^{2}}\right)^{1 / 2}$,其中w0为该激光的束腰半径,$z_{\mathrm{R}}=w_{0}^{2} / 2$为瑞利长度。
笛卡尔坐标系下激光脉冲的相位φ具有如下关系式[22]:
$ \varphi=\eta+\varphi_{R}-\varphi_{\mathrm{G}}+\varphi_{0} $
(1) 式中,η=z-t,t为时间;$\varphi_{R}=\frac{x^{2}+y^{2}}{2 R(z)}$为波阵面曲率相关相位,$R(z)=z\left(1+\frac{z_{\mathrm{R}}^{2}}{z^{2}}\right) ; \varphi_{\mathrm{G}}=\frac{z}{z_{\mathrm{R}}}$为Guoy相移,其与初始位相φ0均由激光本身所决定。
于是高斯激光脉冲的矢势可用下式表示[22]:
$ \boldsymbol{a}(\eta)=a_{0} a_{1} \frac{w_{0}}{w}(\cos \varphi \cdot \boldsymbol{x}+\delta \sin \varphi \cdot \boldsymbol{y}) $
(2) 式中, al=exp(-η2/L2-ρ2/w2),ρ2=x2+y2, δ为偏振参量,对于线偏振光,δ=0;x和y分别表示x, y方向单位矢量。应注意的是,上述模型中的时空坐标已分别被ω0-1和k0-1归一化,其中ω0和k0分别是激光的频率和波数。
直角坐标系中光场的矢势分量可以表示为[22]:
$ \left\{\begin{array}{l} a_{x}=a_{1} \frac{w_{0}}{w} \cos \varphi \\ a_{y}=a_{1} \frac{w_{0}}{w} \sin \varphi \end{array}\right. $
(3) 根据库伦规范$\nabla \cdot{ \boldsymbol{a}}=0$($\nabla \cdot{ \boldsymbol{a}}$表示对矢势a求散度),矢势的纵向分量为[22]:
$ a_{z}=\frac{2 a_{1}}{w^{2}}[-x \sin (\varphi+\theta)+\delta y \cos (\varphi+\theta)] $
(4) 式中, θ=π-arctan φG。
线偏振高斯激光脉冲与单电子相互作用的示意图如图 1所示。图中ϕ与θ分别代表电子与激光相互作用发出辐射的方位角与极角。这里,假设激光脉冲沿+z轴传播,初始时刻电子静止于z轴。
电子在电磁场中的运动可以用拉格朗日方程和电子的能量方程描述[22]:
$ \frac{\mathrm{d}(\boldsymbol{p}-\boldsymbol{a})}{\mathrm{d} t}=-\nabla_\boldsymbol{a}(\boldsymbol{u} \cdot \boldsymbol{a}) $
(5) $ \frac{\mathrm{d} \gamma}{\mathrm{d} t}=\boldsymbol{u} \cdot \frac{\partial \boldsymbol{a}}{\partial t} $
(6) 式中,$\nabla_\boldsymbol{a}$表示求a的梯度,u是用光速c归一化的电子速度,p=γu是经mc归一处理的电子动量,γ=(1-u2)-1/2是相对论因子,也即归一化的电子能量。
将(3)式和(4)式代入(5)式和(6)式,得到如下方程组[22]:
$ \left\{\begin{array}{l} \gamma \frac{\mathrm{d} u_{x}}{\mathrm{~d} t}=\left(1-u_{x}^{2}\right) \frac{\partial a_{x}}{\partial t}+u_{y}\left(\frac{\partial a_{x}}{\partial y}-\frac{\partial a_{y}}{\partial x}\right)+u_{z}\left(\frac{\partial a_{x}}{\partial z}-\frac{\partial a_{z}}{\partial x}\right)-u_{x} u_{y} \frac{\partial a_{y}}{\partial t}-u_{x} u_{z} \frac{\partial a_{z}}{\partial t} \\ \gamma \frac{\mathrm{d} u_{y}}{\mathrm{~d} t}=\left(1-u_{y}^{2}\right) \frac{\partial a_{y}}{\partial t}+u_{x}\left(\frac{\partial a_{x}}{\partial y}-\frac{\partial a_{y}}{\partial x}\right)+u_{z}\left(\frac{\partial a_{y}}{\partial z}-\frac{\partial a_{z}}{\partial y}\right)-u_{x} u_{y} \frac{\partial a_{x}}{\partial t}-u_{y} u_{z} \frac{\partial a_{z}}{\partial t} \\ \gamma \frac{\mathrm{d} u_{z}}{\mathrm{~d} t}=\left(1-u_{z}^{2}\right) \frac{\partial a_{z}}{\partial t}+u_{x}\left(\frac{\partial a_{x}}{\partial z}-\frac{\partial a_{z}}{\partial x}\right)+u_{y}\left(\frac{\partial a_{z}}{\partial y}-\frac{\partial a_{y}}{\partial z}\right)-u_{x} u_{z} \frac{\partial a_{x}}{\partial t}-u_{y} u_{z} \frac{\partial a_{y}}{\partial t} \\ \frac{\mathrm{d} \gamma}{\mathrm{d} t}=u_{x} \frac{\partial a_{x}}{\partial t}+u_{y} \frac{\partial a_{y}}{\partial t}+u_{z} \frac{\partial a_{z}}{\partial t} \end{array}\right. $
(7) 式中,ux, uy, uz分别为电子在相应坐标方向上的速度分量,ax,ay,az分别表示矢势a在x, y, z这3个坐标轴上的分量。
通过求解以上偏微分方程组,可以确定电子在与激光相互作用过程中运动和能量的变化过程。
由电动力学相关知识,以相对论速度运动的电子会向空间发出电磁辐射。单位立体角内的辐射总能量可以表示为[21]:
$ \frac{\mathrm{d} W(t)}{\mathrm{d} \mathit{\Omega}}=\int_{-\infty}^{t_{0}}\left[\frac{\left|\boldsymbol{n} \times\left[(\boldsymbol{n}-\boldsymbol{u}) \times \frac{\mathrm{d} \boldsymbol{u}}{\mathrm{d} t}\right]^{2}\right|^{2}}{(1-\boldsymbol{n} \cdot \boldsymbol{u})^{6}}\right]_{t^{\prime}} \mathrm{d}t $
(8) 式中,$\frac{\mathrm{d} W(t)}{\mathrm{d} \mathit{\Omega}}$已被$\frac{e^{2} \omega_{0}^{2}}{4 \pi c}$归一化; n为能量辐射方向;t是观察点相对于电子与激光相互作用时间t′的延迟时间,其间的关系为:
$ t=t^{\prime}+R_{0}-n \cdot \boldsymbol{r} $
(9) 式中, R0是观察点与电子和激光相互作用点之间的距离,r表示电子的位矢。
电子初始位置对高能电子空间辐射的影响
Influence of electron's initial position on spatial radiation of high-energy electrons
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摘要: 为了探究高能电子辐射与其初始位置间的关系, 依据拉格朗日方程构建了单个高能电子与高斯激光脉冲相互作用发生散射的模型, 并采用数值模拟的方法通过MATLAB获得了电子运动轨迹及散射光的空间辐射特性, 具体分析讨论了电子初始位置对空间能量辐射的影响。结果表明, 初始状态静止的高能电子经与线偏振紧聚焦强激光相互作用, 其在平面内沿+z方向先做振荡运动, 然后沿直线行进; 最大辐射能量及其辐射方向均受到电子初始位置的较大影响, 前者随电子初始位置朝z轴正向移动出现极大值, 而后者在方位角恒定的同时极角逐渐减小并最终稳定; 全空间最大辐射能量在电子初始位置位于(0, 0, -7λ0)(λ0为激光波长)、极角和方位角分别为23.5°和180°时取得。此结果说明通过合理设置电子的初始位置可以获得强度尽可能大的辐射。Abstract: In order to study the relationship between the radiation of high-energy electrons and the electron's initial position, a scattering model of a single high-energy electron interacting with a Gaussian laser pulse was constructed according to the Lagrange's equation. And the method of numerical simulation was adopted to obtain the trajectory of the electron and the spatial radiation characteristics of the scattered light by MATLAB. The influence of the initial position of the electron on the space energy radiation was discussed in detail. The results show that the initially static high-energy electron first oscillates in the +z direction in a plane, and then travels along a straight line after interacting with the linearly polarized tightly focused intense laser. Both the maximum radiated energy and its corresponding radiation direction are greatly affected by the electron's initial position, while a peak value of the former exists as the initial position of the electron moves to the positive direction of z axis, and the azimuth angle of the latter stays unchanged while the polar angle gradually decreasing but finally stabilizing. The maximum radiation energy in the whole space is obtained when the electron is initially set at (0, 0, -7λ0) (λ0 is the wavelength of the laser) with the polar angle and the azimuth angle being 23.5° and 180°, respectively. The research indicates that the highest possible intensity of radiation can be obtained by setting the electron's initial location reasonably.
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