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在非线性汤姆逊散射模型中,对于圆偏振激光考虑了由高斯型包络建模[14-15]。紧聚焦激光与电子相互作用的几何示意图如图 1所示。为了便于分析描述,作者做出如下假设:+z轴延伸方向是图中圆偏振激光脉冲的传播方向,一个电子位于坐标原点上,并且在作用前保持静止状态,当激光脉冲到达坐标原点,与该电子相遇时,受到了激光电场的作用,电子不仅在横向做高频振荡,也会在激光脉冲的有质动力推动下向前运动[16-17]。在电子和激光脉冲二者相互作用时,各个方向都有由电子发出的谐波辐射[18],如果用n来表示电子的辐射方向,那么它的表达式为n =sinθ· cosϕ· x +sinθ sinϕ· y +cosθ· z (极角θ和方位角ϕ如图 1所示)。
由相关理论知识[7]公式推导所得结论,在3维坐标系中,紧聚焦圆偏振激光脉冲电场的矢势可以表示为:
$ {\mathit{\boldsymbol{a}}_1} = {a_0}\exp \left( { - \frac{{{\eta ^2}}}{{{L^2}}} - \frac{{{\rho ^2}}}{{{w^2}}}} \right)\left( {\frac{{{w_0}}}{w}} \right) $
(1) 式中,a0是已经归一化的激光强度振幅, 表达式为a0=eA0/(mc2)=0.85×10-9$\sqrt {I{\lambda _0}^2} $,m, e分别为电子静止时的质量和电量,A0为激光场矢势的振幅,c为光速,I为激光强度峰值,波长λ0=1 μm;L和w分别是圆偏振激光的脉宽和束腰半径, w0是脉冲的最小束腰半径;η=z-t,ρ2=x2+y2是传播方向的垂直距离。
为了便于分析,矢势的正交分量的大小可以表示为:
$ \left\{ {\begin{array}{*{20}{l}} {{a_x} = {a_1}\cos \phi }\\ {{a_y} = {a_1}\sin \phi } \end{array}} \right. $
(2) 圆偏振紧聚焦激光脉冲的相位φ在3维坐标系下可以表示为:
$ \varphi = (z - t) + {C_0}{(z - t)^2} + \varphi - {\varphi _{\rm{G}}} + {\varphi _0} $
(3) 式中,C0是啁啾系数(本文中C0=0,不考虑啁啾), φ0是激光脉冲的初始相位,φG是受波阵面曲率影响的Guoy相位,φR是与曲率半径相关的相位,它可以通过以下公式表示:
$ {\varphi _R} = \frac{{{x^2} + {y^2}}}{{2R(z)}} $
(4) 式中,R(z)是脉冲激光波前的曲率半径,可以表示为:
$ R(z) = z\left( {1 + \frac{1}{{{\varphi _{\rm{G}}}^2}}} \right) $
(5) 根据电子的能量方程和拉格朗日方程[7-10],可以将电子在电磁场中的运动表示为:
$ \frac{{{\rm{d}}\gamma }}{{{\rm{d}}t}} = \mathit{\boldsymbol{v}} \cdot \frac{{\partial \mathit{\boldsymbol{a}}}}{{\partial t}} $
(6) $ \frac{{{\rm{d}}(\mathit{\boldsymbol{p}} - \mathit{\boldsymbol{a}})}}{{{\rm{d}}t}} = - {\nabla _a}(\mathit{\boldsymbol{v}} \cdot \mathit{\boldsymbol{a}}) $
(7) 式中, v为利用光速c进行归一处理的电子速度,a为被mc2/e所归一化的矢势,p =γv为利用mc进行归一化的电子动量,相对论因子γ=(1- v2)-1/2,也就是由mc2进行归一处理的电子能量。而(7)式中的哈密顿算子▽a只作用在a上。
根据(2)式,微分方程(6)式、(7)式可分解得到如下4个微分方程:
$ \begin{array}{c} \gamma \frac{{{\rm{d}}{v_x}}}{{{\rm{d}}t}} = \left( {1 - {v_x}^2} \right)\frac{{\partial {a_x}}}{{\partial t}} + {v_y}\left( {\frac{{\partial {a_x}}}{{\partial y}} - \frac{{\partial {a_y}}}{{\partial x}}} \right) + \\ {v_z}\left( {\frac{{\partial {a_x}}}{{\partial z}} - \frac{{\partial {a_z}}}{{\partial x}}} \right) - {v_x}{v_y}\frac{{\partial {a_y}}}{{\partial t}} - {v_x}{v_z}\frac{{\partial {a_z}}}{{\partial t}} \end{array} $
(8) $ \begin{array}{c} \gamma \frac{{{\rm{d}}{v_y}}}{{{\rm{d}}t}} = \left( {1 - {v_y}^2} \right)\frac{{\partial {a_y}}}{{\partial t}} + {v_x}\left( {\frac{{\partial {a_x}}}{{\partial y}} - \frac{{\partial {a_y}}}{{\partial x}}} \right) + \\ {v_z}\left( {\frac{{\partial {a_y}}}{{\partial z}} - \frac{{\partial {a_z}}}{{\partial y}}} \right) - {v_x}{v_y}\frac{{\partial {a_x}}}{{\partial t}} - {v_y}{v_z}\frac{{\partial {a_z}}}{{\partial t}} \end{array} $
(9) $ \begin{array}{l} \gamma \frac{{{\rm{d}}{v_z}}}{{{\rm{d}}t}} = \left( {1 - {v_z}^2} \right)\frac{{\partial {a_z}}}{{\partial t}} + {v_x}\left( {\frac{{\partial {a_x}}}{{\partial z}} - \frac{{\partial {a_z}}}{{\partial x}}} \right) + \\ {v_y}\left( {\frac{{\partial {a_y}}}{{\partial z}} - \frac{{\partial {a_y}}}{{\partial z}}} \right) - {v_x}{v_z}\frac{{\partial {a_x}}}{{\partial t}} - {v_y}{v_z}\frac{{\partial {a_x}}}{{\partial t}} \end{array} $
(10) $ \frac{{{\rm{d}}\gamma }}{{{\rm{d}}t}} = {v_x}\frac{{\partial {a_x}}}{{\partial t}} + {v_y}\frac{{\partial {a_y}}}{{\partial t}} + {v_z}\frac{{\partial {a_z}}}{{\partial t}} $
(11) 式中,vx,vy和vz分别表示电子在3个坐标轴方向上的速度分量的大小;ax、ay和az分别表示电子在3个坐标轴方向上的加速度分量的大小。利用MATLAB程序求解上述的4个微分方程,可分别求出电子在激光场中的坐标、速度、加速度和能量随时间变化的改变量。
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由电动力学知识,结合参考文献[19]和参考文献[20]中所得相关结论以及第1.1节中所作出的讨论可知,以相对论速度作加速运动的电子会放出电磁辐射,单位立体角内的辐射功率可以表示成:
$ \frac{{{\rm{d}}P(t)}}{{{\rm{d}}\mathit{\Omega }}} = {\left[ {\frac{{{{\left| {\mathit{\boldsymbol{n}} \times \left[ {(\mathit{\boldsymbol{n}} - \mathit{\boldsymbol{v}}) \times {{\rm{d}}_t}\mathit{\boldsymbol{v}}} \right]} \right|}^2}}}{{{{(1 - \mathit{\boldsymbol{n}} \cdot \mathit{\boldsymbol{v}})}^6}}}} \right]_{{t^\prime }}} $
(12) 式中,n为辐射方向的单位向量,P(t)为辐射功率,Ω为立体角,t′是圆偏振激光脉冲与电子相互作用的时间,t是观察点所处的时间,即t=t′+R。R=R0- n · r,R0为观察点距离电子与圆偏振激光脉冲相互作用点的距离,r是电子位置矢量。
因此,单位立体角的总辐射能量可以由所有有效辐射的积分表示,即:
$ \frac{{{\rm{d}}W}}{{{\rm{d}}\mathit{\Omega }}} = \int_{ - \infty }^\infty {\left[ {\frac{{{{\left| {\mathit{\boldsymbol{n}} \times \left[ {(\mathit{\boldsymbol{n}} - \mathit{\boldsymbol{v}}) \times {{\rm{d}}_t}\mathit{\boldsymbol{v}}} \right]} \right|}^2}}}{{{{(1 - \mathit{\boldsymbol{n}} \cdot \mathit{\boldsymbol{v}})}^6}}}} \right]} {\rm{d}}t $
(13) 式中,W为总辐射能量。
初始位置对电子运动轨迹和空间角辐射的影响
Influence of initial position on the trajectory and spatial angular radiation of electrons
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摘要: 为了探究在圆偏振激光脉冲中电子初始位置对其运动轨迹和空间角辐射的影响,根据非线性汤姆逊散射模型、能量方程以及拉格朗日方程推导出了高能电子的空间运动方程,并与MATLAB数值模拟的方法相结合,做出了高能电子空间运动轨迹图和空间角辐射模拟图。结果表明,电子在涡旋横向力的作用下在全空间运动的前部轨迹呈紧密分离螺旋状,而后部轨迹由空间间隔遥远的稀疏圆组成,随着电子初始位置的右移,空间角辐射达到最大值时,极角θ和方位角ϕ的值有不断减小的趋势,在z0=5λ0后趋于稳定,(θ, ϕ)=(23.5°, 175.5°);激光脉冲中电子初始位置的改变对电子的运动轨迹和空间角辐射有较大影响。该结果为后续研究电子初始位置对高能电子辐射特性的影响奠定了基础。Abstract: For the sake of studying the initial position's influence of electrons on the trajectory and spatial angular radiation of high-energy electrons in circularly polarized laser pulses, the spatial motion equation of high-energy electrons was deduced theoretically on the basis of nonlinear Thomson scattering model, energy equation, Lagrange equation, and in combination with the assistance of MATLAB numerical simulation, the spatial motion trajectory diagram and spatial angular radiation simulation diagram of high-energy electrons made. The results show that, under the action of the transverse vortex force, the front trajectory of the electron in the whole space is in a tightly separated spiral shape, and the rear trajectory is composed of sparse circles with distant spatial spacing. With the right shift of the initial position of the electron, the value of the polar angle θ and azimuth ϕ has a decreasing trend when the spatial angular radiation reaches the maximum. Specifically, it becomes stable when z0=5λ0 and (θ, ϕ)=(23.5°, 175.5°). Therefore the change of the initial position of the electron in the laser pulse has a great influence on the trajectory and spatial angular radiation of the electron, which creates the groundwork for the subsequent research of the initial position's influence of the electron on the radiation characteristics of high-energy electrons.
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