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紧聚焦高斯脉冲激光电场的归一化矢势a(η)可以表示为:
$ \boldsymbol{a}(\eta)=a(\cos \varphi \cdot \boldsymbol{x}+\delta \sin \varphi \cdot \boldsymbol{y}) $
(1) $ a=\frac{b_0}{b} a_0 \exp \left(-\frac{\eta^2}{L^2}-\frac{\rho^2}{b^2}\right) $
(2) 式中,a0=0.85×10-9λ0$\sqrt{I}$是通过mc2/e进行归一化的激光振幅,其中m和e分别为电子静止时的质量和电量,c是光速,I和λ0为激光强度峰值和激光波长;η=z-t,z和t表示电子在z轴的坐标和观察点的时间;L是激光的脉宽;δ为偏振参量,椭圆偏振下0 < δ < 1;b=b0(1+z2/zR2)1/2为脉冲传播到z轴时的束腰半径;zR=b02/2为瑞利长度;b0是b的最小值;ρ2=x2+y2代表与传播方向的垂直距离,高斯脉冲形状是分别由横向参数exp(-ρ2/b2)和纵向参数exp(-η2/L2)所决定。定义脉冲相位为:
$ \varphi=c_0 \eta^2+\eta+\varphi_R-\varphi_{\mathrm{G}}+\varphi_0 $
(3) 式中,φ0是激光脉冲的初始相位;c0为激光啁啾参数;φG=z/zR为古伊相位;φR=ρ2/2R(z)与波前曲率有关,R(z)=z(1+1/φG2)为波前曲率半径。在上述定义中,k0-1和ω0-1已将空间坐标和时间坐标归一化,其中k0和ω0为激光的波数和频率。在库仑定律▽·a=0下,矢场纵向分量为:
$ \boldsymbol{a}_z=\frac{2 a[-x \cos (\varphi+\theta)+\delta y \sin (\varphi+\theta)] \boldsymbol{z}}{b_0 b} $
(4) 式中,θ=π-arctan(z/zR)。椭圆偏振激光脉冲与高能电子相互作用的示意图如图 1所示。
设电子沿着-z轴方向移动,激光脉冲沿z轴正方向移动。在真空激光和单电子理论中,激光场中电场强度E=-da/dt,给出真空中单电子与激光发生碰撞运动的基本方程:
$ \nabla^2 \boldsymbol{a}+\mathrm{d} \boldsymbol{a} / \mathrm{d} \boldsymbol{u}=0 $
(5) $ \mathrm{d}(\boldsymbol{p}-\boldsymbol{a}) / \mathrm{d} t=-\nabla \gamma $
(6) $ \mathrm{d} \gamma / \mathrm{d} t=\boldsymbol{u} \cdot(\mathrm{d} \boldsymbol{a} / \mathrm{d} t) $
(7) 式中,u为电子的速度;p=γu为电子动量; γ=(1-u2)-1/2为电子能量。考虑到:
$ \nabla \gamma=(\boldsymbol{u} \cdot \nabla) \boldsymbol{p}+\boldsymbol{u} \times(\nabla \times \boldsymbol{a}) $
(8) 由于矢量势垂直于激光传播方向,电子的速度可以分解为正交分量:
$ \left\{\begin{array}{l} \gamma\left(\mathrm{d} u_x / \mathrm{d} t\right)=\left(1-u_x{ }^2\right)\left(\mathrm{d} a_x / \mathrm{d} t\right)-u_x u_y\left(\mathrm{~d} a_y / \mathrm{d} t\right)+u_y\left(\mathrm{~d} a_x / \mathrm{d} y-\mathrm{d} a_y / \mathrm{d} x\right) \\ \gamma\left(\mathrm{d} u_y / \mathrm{d} t\right)=\left(1-u_y{ }^2\right)\left(\mathrm{d} a_y / \mathrm{d} t\right)-u_x u_y\left(\mathrm{~d} a_y / \mathrm{d} t\right)-u_y\left(\mathrm{~d} a_x / \mathrm{d} y-\mathrm{d} a_y / \mathrm{d} x\right) \\ \mathrm{d} \gamma / \mathrm{d} t=u_x\left(\mathrm{~d} a_x / \mathrm{d} t\right)+u_y\left(\mathrm{~d} a_y / \mathrm{d} t\right) \\ \gamma=\left(1-u_x^2-u_y^2-u_z^2\right)^{-1 / 2} \end{array}\right. $
(9) 式中,ux,uy,uz为高能电子在坐标轴上的速度分量的大小;ax和ay为归一化矢势分量的大小。根据电动力学的知识,电子在加速运动时发射电磁辐射,由下式控制:
$ \frac{\mathrm{d} P(t)}{\mathrm{d} \boldsymbol{{\mathit{\Omega}}}}=\left|\frac{|\boldsymbol{n} \times[(\boldsymbol{n}-\boldsymbol{u}) \times(\mathrm{d} \boldsymbol{u} / \mathrm{d} t)]|^2}{(1-\boldsymbol{n} \cdot \boldsymbol{u})^6}\right|_{t'} $
(10) 式中,$\frac{\mathrm{d} P(t)}{\mathrm{d} {\mathit{\Omega}}} \text { 是被 } \frac{e^2 \omega_0^2}{4 {\rm{ \mathsf{π} }} c}$归一化的单位立体角辐射功率; n为辐射方向; du/dt为电子的加速度。电子辐射方向在笛卡尔坐标系中可以表示为:
$ \boldsymbol{n}=\sin \theta \cos \phi \cdot \boldsymbol{x}+\sin \theta \sin \phi \cdot \boldsymbol{y}+\cos \theta \cdot \boldsymbol{z} $
(11) 式中,θ和ϕ分别表示与激光运动方向的夹角和垂直于碰撞方向的平面上的偏转方位角。此外,Δt为发生碰撞时刻t与观察到碰撞现象时刻t'之间的时间间隔。
$ \Delta t=t-t^{\prime}=S_0-\boldsymbol{n} \cdot \boldsymbol{r} $
(12) 式中,S0表示观测点距离碰撞中心的距离,r为电子位置矢量。
通过上述方程能够得到脉冲激光和高能电子作用的过程中能量和功率随观测角的变化情况。
激光脉冲初始相位对电子辐射的影响
Effect of initial phase of laser pulse on electron radiation
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摘要: 为了研究超短超强椭圆偏振激光初始相位对于高能电子辐射特性的影响, 采用了Lorentz方程与电子能量方程构造高能电子与强激光场的对撞模型的方法, 并使用MATLAB进行数值模拟, 获得了电子的运动轨迹以及激光场空间辐射的功率与能量分布的数据与图像, 对不同的激光初始相位所对应的3维空间辐射特性进行了研究。结果表明, 当激光脉冲撞击电子时, 电子产生辐射, 且辐射功率呈现出双峰形; 高能电子的辐射功率图像在初始相位为0°, 180°和360°时表现为对称型双峰, 而在其它相位下则呈现出非对称型双峰。该结论为超短超强椭圆偏振激光的初始相位3维反探测研究提供了一定的基础。Abstract: In order to study the influence of the initial phase of ultrashort and super elliptically polarized laser on the radiation characteristics of high-energy electrons, the collision model of high-energy electrons and high-energy laser field was constructed by Lorentz equation and electron energy equation, and the numerical simulation was carried out by MATLAB. The data and images of electron trajectory and power and energy distribution of spatial radiation of laser field were obtained. The 3-D radiation characteristics corresponding to different initial laser phases were studied. The results show that when the laser pulse strikes the electron, the electron produces radiation, and the radiation power presents a double peak shape. The radiation power image of high-energy electrons shows symmetrical double peaks when the initial phase is 0°, 180° and 360°. The conclusion provides a certain basis for the study of initial phase 3-D inverse detection of ultrashort and ultra strong elliptically polarized laser.
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