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设定一静止于坐标原点的单电子以及一束与该单电子相互作用的、沿+z轴方向传播的紧聚焦高斯激光脉冲,如图 1所示。该脉冲可由归一化矢势[17]表示,归一化系数为mc2/e(其中m为单电子的质量,e表示单电子所带的电荷数,即1.6×10-19,c表示光速,其值为3×108 m/s2),具体形式为:
$ \begin{aligned} \boldsymbol{A}(k) & =A_0 \exp \left(-\frac{h^2}{w^2}-\frac{k^2}{L^2}\right)\left(\frac{w_0}{w}\right) \times \\ & {[\cos \varphi \cdot x+\delta \sin \varphi \cdot y] } \end{aligned} $
(1) 式中,A0表示被mc2/e归一化后的激光脉冲振幅值,该值可由激光波长及峰值激光强度确定;$h=\sqrt{x^2+y^2}$,x, y, z为电子的空间坐标;w表示激光在z处的束腰半径值;w0则对应激光最小的焦点束腰半径,二者关系为w=w0(1+z2/zR2)1/2(zR=w02/2,对应于此光束的瑞利长度);k=z-t,为前面(1)式的A(k)的自变量,t表示时间坐标;L表示激光的脉冲宽度。
高斯激光脉冲的相位φ满足具体关系式:
$ \varphi=k+c_0 k^2+\varphi_R-\varphi_{\mathrm{G}}+\varphi_0 $
(2) 式中,c0是激光脉冲的啁啾参数; φ0表示在激光脉冲与单电子相互作用的初始时刻位相值; φR与波阵面曲率相关,有:
$ \varphi_R=\left(x^2+y^2\right) /[2 R(z)] $
(3) 式中, R(z)值可表示为R(z)=z(1+φG2); φG=z/zR表示古依位相,其值与高斯光束的瑞利长度相关; δ则是激光偏振参数,即本文中着重研究的激光参数。对于偏振参数值,δ=0时对应激光线偏振态,δ=1时对应激光圆偏振态,当δ数值在0到1间变化时,则对应不同状态的椭圆偏振激光。此外,电子的3维空间坐标系及时间坐标系分别被k0-1和ω0-1归一化,k0和ω0分别是激光的波数和频率。
当把光场归一化矢势在3维笛卡尔坐标系的x,y,z坐标轴方向上分解时,矢势的分量可表示为以下形式:
$ \left\{\begin{array}{l} A_x=A_1 \cos \varphi \\ A_y=A_1 \sin \varphi \\ A_z=A_1\left[\frac{-2 x}{w_0 w} \sin (\varphi+\theta)+\frac{\delta 2 y}{w_0 w} \cos (\varphi+\theta)\right] \end{array}\right. $
(4) 式中,系数A1=exp(-k2/L2-h2/r2)(w0/w),角度θ=π-arctan(z/zR)。
模型已设定一初始动量p0=0的单电子被置于笛卡尔坐标系原点处,一高斯激光脉冲沿+z轴方向发射,二者相遇瞬间单电子开始一边振荡一边前进,并向各个方向发出谐波辐射,图 1中n即表示电子辐射方向,与图中标示出的θ和ϕ的取值有关。该单电子受激运动、辐射可分别用下式来描述:
$ \frac{\mathrm{d}(\boldsymbol{p}-\boldsymbol{A})}{\mathrm{d} t}=-\nabla(\boldsymbol{u} \cdot \boldsymbol{A}) $
(5) $ \frac{\mathrm{d} \gamma}{\mathrm{d} t}=\boldsymbol{u} \cdot\left(\frac{\mathrm{d} \boldsymbol{A}}{\mathrm{d} t}\right) $
(6) 式中,p=γu是用mc归一化的电子动量,u是用光速c归一化的电子速度,A则是(1)式中定义的聚焦高斯激光脉冲的归一化矢势,相对论因子γ代表用mc2归一化的电子能量,γ=(1-u2)-1/2。
把矢势在坐标轴上的分解形式即(4)式代入(5)式和(6)式中进行计算,可得到电子在激光场中的坐标、速度、加速度以及能量随时间的变化过程如下:
$ \left\{\begin{array}{l} \gamma \frac{\mathrm{d} u_x}{\mathrm{~d} t}=\left(1-u_x^2\right) \frac{\mathrm{d} a_x}{\mathrm{~d} t}+u_y\left(\frac{\mathrm{d} a_x}{\mathrm{~d} y}-\frac{\mathrm{d} a_y}{\mathrm{~d} x}\right)+u_z\left(\frac{\mathrm{d} a_x}{\mathrm{~d} z}-\frac{\mathrm{d} a_z}{\mathrm{~d} x}\right)-u_x u_y \frac{\mathrm{d} a_y}{\mathrm{~d} t}-u_x u_z \frac{\mathrm{d} a_z}{\mathrm{~d} t} \\ \gamma \frac{\mathrm{d} u_y}{\mathrm{~d} t}=\left(1-u_y^2\right) \frac{\mathrm{d} a_y}{\mathrm{~d} t}+u_x\left(\frac{\mathrm{d} a_x}{\mathrm{~d} y}-\frac{\mathrm{d} a_y}{\mathrm{~d} x}\right)+u_z\left(\frac{\mathrm{d} a_x}{\mathrm{~d} z}-\frac{\mathrm{d} a_z}{\mathrm{~d} x}\right)-u_x u_y \frac{\mathrm{d} a_x}{\mathrm{~d} t}-u_y u_z \frac{\mathrm{d} a_z}{\mathrm{~d} t} \\ \gamma \frac{\mathrm{d} u_z}{\mathrm{~d} t}=\left(1-u_z^2\right) \frac{\mathrm{d} a_z}{\mathrm{~d} t}+u_x\left(\frac{\mathrm{d} a_x}{\mathrm{~d} z}-\frac{\mathrm{d} a_z}{\mathrm{~d} x}\right)+u_y\left(\frac{\mathrm{d} a_z}{\mathrm{~d} y}-\frac{\mathrm{d} a_y}{\mathrm{~d} z}\right)-u_x u_z \frac{\mathrm{d} a_x}{\mathrm{~d} t}-u_y u_z \frac{\mathrm{d} a_y}{\mathrm{~d} t} \\ \frac{\mathrm{d} \gamma}{\mathrm{d} t}=u_x \frac{\mathrm{d} a_x}{\mathrm{~d} t}+u_y \frac{\mathrm{d} a_y}{\mathrm{~d} t}+u_z \frac{\mathrm{d} a_y}{\mathrm{~d} t} \end{array}\right. $
(7) 式中,ux,uy,uz是电子在笛卡尔坐标系中三坐标轴方向上分解的速度分量的大小;ax,ay,az是电子在笛卡尔坐标系中三坐标轴方向上分解的加速度分量的大小。在此式基础上,运用数学工具MATLAB迭代求解即可对电子相对论运动状态进行分析研究。
由电动力学知识可知,做相对论加速运动的电子会在向外放出电荷辐射能量的同时产生X射线,并且其相关特征与电子轨迹直接相关。根据李纳-维谢尔势(Lienard-Wiechert potential),单位立体角内电子的辐射功率可以表示为[20]:
$ \frac{\mathrm{d} P(t)}{\mathrm{d} {\mathit{\Omega}}}=\left[\frac{|\boldsymbol{n} \times[(\boldsymbol{n}-\boldsymbol{u}) \times(\mathrm{d} \boldsymbol{u} / \mathrm{d} t)]|}{(1-\boldsymbol{n} \cdot \boldsymbol{u})^6}\right]_{t^{\prime}} $
(8) 式中,辐射功率$\frac{\mathrm{d} P(t)}{\mathrm{d} {\mathit{\Omega}}} \text { 被 } \frac{e^2 \omega_0^2}{4 {\rm{ \mathsf{π} }} c}$归一化,n为辐射方向的单位向量。
此时认为观测点距离电子与激光脉冲作用点足够远,则由于相对论效应,观测点时间和电子与激光脉冲实际相互作用的时间会产生差异。令t'为电子与激光脉冲相互作用的时间,t为观测的时间,则二者之间满足关系式:t-t'=d0-n·r(t), 其中d0表示观测点与电子间的距离,r(t)是在t时刻的电子位置矢量。
电子与激光脉冲相互作用的过程中,其单位立体角、单位频率间隔的以沿运动轨迹的位置、速度和加速度为自变量的辐射能表达式为:
$ \begin{gathered} \frac{\mathrm{d}^2 I}{\mathrm{~d} \omega \mathrm{d} {\mathit{\Omega}}}= \\ \left|\int_{-\infty}^{+\infty} \exp \{\mathrm{is}[t-\boldsymbol{n} \cdot \boldsymbol{r}(t)]\} \frac{\boldsymbol{n} \times\left[\left(\boldsymbol{n}-\boldsymbol{u}^{\prime}\right) \times \boldsymbol{a}^{\prime}\right]}{\left(1-\boldsymbol{u}^{\prime} \cdot \boldsymbol{n}\right)^2} \mathrm{~d} t\right|^2 \end{gathered} $
(9) 式中,$\frac{\mathrm{d}^2 I}{\mathrm{~d} \omega \mathrm{d} {\mathit{\Omega}}} \text { 被 } \frac{e^2}{4 {\rm{ \mathsf{π} }}^2 c} \text { 归一化, } s=\frac{\omega_{\mathrm{s}}}{\omega_0}, \omega_{\mathrm{s}}$是谐波辐射的频率,ω0是基波辐射的频率,u'是此时被光速c归一化后的电子速度,a'=du'/dt是电子的加速度。通过求解上述(8)式和(9)式两个方程,就可以对进行全时间、全空间和全频谱的高能电子谐波辐射研究。可以看到,当电子的加速度a'=0时,辐射能表达式数值为0,这意味着具有加速度是电子发射电磁波的原因,且电子沿瞬时速度的方向进行辐射。
偏振参数对高能电子运动及辐射特性的影响
Effects of polarization parameters on the motion and radiation characteristics of high-energy electrons
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摘要: 为了探究超强激光偏振参数的梯度变化对场内高能电子运动及辐射特性的影响, 首先以电磁学基本方程为基础, 推导并建立了初始动量为0的相对论性单电子加速模型, 其次编写无近似的数值模拟仿真程序进行迭代计算与理论分析, 取得了不同偏振参数的超强激光作用下单电子的运动以及空间辐射可视化数据。结果表明, 随着偏振参数δ由0到1逐渐增大, 电子的运动轨迹由2维平面振荡逐渐过渡为3维螺旋状前进, 绕旋幅度逐渐增大且轨迹投影逐渐趋向于正圆; 电子的功率辐射空间分布也从平面线性逐渐变为3维涡旋状, 由上下针状分叉逐渐变为平滑连接, 总体变化趋势可按形态划分为δ=0, δ∈(0, 0.6], δ∈(0.6, 0.99]以及δ=1共4个阶段。该结果为高能电子辐射研究提供了多视角的理论及数值依据, 对实际应用中精确探测超强激光各项参数是有帮助的。Abstract: In order to explore the gradient changes of super laser polarization parameters based on the high energy electron motion and the influence of radiation characteristics, based on the basic equation of electromagnetism, a relativistic electron acceleration model was derived and set up with the initial momentum of 0. Then a numerical simulation program of no approximation was developed for iterative calculation and theoretical analysis. Visualized data of single electron motion and space radiation under different polarization parameters were obtained. The results show that with the increase of the polarization parameter δ from 0 to 1, the trajectory of the electron gradually changes from 2-D plane oscillation to 3-D spiral, and the amplitude of rotation increases gradually, and the trajectory projection tends to be positive circle. The spatial distribution of electron power radiation gradually changed from planar linear to 3-D vortex, and gradually changed from up-down needle-like bifurcation to smooth connection. The general change trend can be divided into four stages according to morphology: δ=0, δ∈(0, 0.6], δ∈(0.6, 0.99] and δ=1. The results provide a theoretical and numerical basis for the study of high-energy electron radiation from multiple perspectives, and are helpful for the accurate detection of super-strong laser parameters in practical applications.
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Key words:
- laser optics /
- electronic radiation /
- numerical simulation /
- polarization parameters
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