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椭圆偏振紧聚焦激光脉冲在3维坐标系中的相位可表示为[18]:
$ \varphi=\eta+c_{0} \eta^{2}+\varphi_{R}-\varphi_{\mathrm{G}}+\varphi_{0} $
(1) 式中,c0是激光脉冲的啁啾参数;φ0为初始相位; η=z-t,z为电子在z轴的坐标,t为观察点的时间; φR=(x2+y2)/ 是与激光脉冲在轴处的波阵面曲率半径R(z)=z(1+zR2/z2)有关的相位,而φG=z/zR则是与高斯光束从-∞传到+∞时将经历π位相变化相关的古依位相,其中zR=w02/2为激光脉冲的瑞利长度,而w0为激光脉冲的最小半径。
下式常被用于表示紧聚焦高斯脉冲激光电场的归一化矢势[18]:
$ \boldsymbol{a}(\eta)=a_{0} \exp \left(-\eta^{2} / L^{2}-\rho^{2} / w^{2}\right)\left(w_{0} / w\right) \cdot\\ \;\;\;\;\;[\cos \varphi \cdot {\boldsymbol{x}}+\delta \sin \varphi \cdot {\boldsymbol{y}}] $
(2) 式中, a0是被mc2/e归一化的激光振幅,电子的静止质量和电量则分别由m和e表示,其数值分别为9.1×10-31kg和1.6×10-19C; ρ2=x2+y2; L是激光的脉宽; w0是激光脉冲的最小半径,w是激光脉冲的束腰半径,且满足w0=w/(1+z2/zR2)1/2; 文中取偏振参量δ=0.9表示激光为椭圆偏振; x,y分别表示x,y方向单位向量。
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如图 1所示,椭圆偏振激光沿着+z传播,撞击处于原点的静止电子,由于激光脉冲具有质动力,电子会随着撞击方向移动[19],同时散发出谐波辐射,n为辐射方向,且有:
$ \boldsymbol{n}=\sin \theta \cos \phi \cdot \boldsymbol{x}+\sin \theta \sin \varphi \cdot \boldsymbol{y}+\cos \theta \cdot \boldsymbol{z} $
(3) 式中,ϕ和θ分别是图 1中所示的观测方位角和极角。
用洛伦兹方程来描述电子在电磁场中的运动[18]:
$ \mathrm{d}(\boldsymbol{p}-\boldsymbol{a}) / \mathrm{d} t=-\nabla_\boldsymbol{a}(\boldsymbol{u} \cdot \boldsymbol{a}) $
(4) 同时有电子的能量方程为[18]:
$ \frac{\mathrm{d} \gamma}{\mathrm{d} t}=\boldsymbol{u} \cdot \frac{\partial \boldsymbol{a}}{\partial t} $
(5) 式中, u, a和p分别是归一化后的电子速度、矢势以及电子动量。$\nabla_\boldsymbol{a}$表示求a的梯度,归一化系数分别为mc2/e, mc2和mc2,且p=γu,γ=(1-u2)-1/2也是mc2归一化的电子能量。
做相对论加速运动的电子会释放电磁辐射[20],辐射的单位立体角的功率计算公式为:
$ \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]|^{2}}{(1-\boldsymbol{n} \cdot \boldsymbol{u})^{6}}\right]_{t^{\prime}} $
(6) 式中, 辐射功率P(t)已被e2ω02/(4πc)归一化, ω0为激光频率,Ω为单位立体角,t′是电子的延迟时间。t′和t之间存在如下关系[20]:
$ t=t^{\prime}+R_{0}-\boldsymbol{n} \cdot \boldsymbol{r} $
(7) 式中, R0为观测点与电子的距离, r为电子位矢。此处假定观测点远离激光与电子的作用点。
单位立体角单位频率间隔内辐射强度为[20]:
$ \begin{gathered} \frac{\mathrm{d}^{2} I}{\mathrm{~d} \mathit{\omega} \mathrm{d} \mathit{\Omega}}= \\ \left|\int_{-\infty}^{\infty} \frac{\boldsymbol{n} \times[(\boldsymbol{n}-\boldsymbol{u}) \times \boldsymbol{u}]}{(1-\boldsymbol{n} \cdot \boldsymbol{u})^{2}} \exp [\mathrm{is}(t-\boldsymbol{n} \cdot \boldsymbol{r})] \mathrm{d} t\right|^{2} \end{gathered} $
(8) 式中, I为辐射强度,ω为辐射频率,d2I/dωdΩ被e2/(4π2c)归一化, s=ωs,b/ω0, ωs,b为散射所产生辐射的频率,电子谐波辐射的时间以及频谱特性可以通过求解(8)式得到。
脉宽对激光撞击电子辐射峰值影响的模拟计算
Simulation calculation of the influence of pulse width on the peak radiation of laser impact electron
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摘要: 为了研究激光脉宽对撞击电子后产生的辐射能量分布, 采用模拟计算的方法, 以Lorentz方程以及电子辐射方程为基础, 建立了紧聚焦激光作用于静止单电子模型, 并通过MATLAB软件模拟了不同脉宽下的激光脉冲与电子作用后产生的电子辐射能量分布, 对飞秒紧聚焦椭圆偏振激光脉冲的脉宽与电子间的辐射功率峰值进行了深入研究。结果表明, 当紧聚焦激光脉冲遇到静止单电子, 在激光脉冲撞击电子时, 电子会发出辐射; 散射辐射在散射方向的中心呈尖锥状积累; 随着激光脉宽的增加, 辐射功率分布逐渐呈现出双峰形; 脉冲宽度越宽, 电子辐射功率峰值越小, 脉宽为10λ0时的峰值功率仅为脉宽为0.1λ0时峰值功率的1%(初始脉宽λ0=3.33fs), 同时辐射功率达到峰值所需时间越长, 最高峰的持续时间越长, 频谱函数的截止频率越低, 高频分量变少, 谐波次数增加。该结果对激光空气等离子体诊断方面具有重要意义。Abstract: In order to study the radiation energy distribution of laser pulse width after impact electron, the method of simulation calculation was adopted. Based on Lorentz equation and electron radiation equation, a single electron model of compact focusing laser acting on static single electron was established. The distribution of electron radiation energy produced by laser pulse and electron action under different pulse width was simulated by MATLAB software. The pulse width of femtosecond compact focusing elliptic polarization laser pulse and the peak radiation power between electrons were studied. The simulation results show that when the compact focused laser pulse encounters a stationary single electron, the electron radiates while the laser pulse hits the electron. The scattering radiation accumulates in a sharp cone at the center of the scattering direction. With the increase of laser pulse width, the distribution of radiation power gradually presents a double peak. The wider the pulse width, the smaller the peak value of electron radiation power. When the pulse width is 10λ0, the peak power is only 1% of that when the pulse width is 0.1λ0, and the initial pulse width is λ0=3.33fs. At the same time, the wider the pulse width, the smaller the peak value of the electronic radiation power, the longer the required realization of the peak radiation power, the longer the duration of the peak, the lower the cut-off frequency of the spectrum function, the less high frequency components, and the increase of the harmonic frequency. The results are of great significance to the diagnosis of laser air plasma.
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Key words:
- laser optics /
- electron radiation /
- compact focused laser /
- ellipsometry /
- numerical simulation
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Figure 2. Influence of different laser pulse width on power distribution of electron Thomson scattering radiation
a—spatial distribution of radiation power under pulse width λ0
b—spatial distribution of radiation power under pulse width 2λ0
c—spatial distribution of radiation power under pulse width 3λ0Figure 3. Influence of elliptical laser on the power of electron Thomson scattering under different pulse width
a—relationship between observation and radiation under pulse width λ0 b—relationship between observation and radiation under pulse width 2λ0 c—relationship between observation and radiation under pulse width 3λ0
图 5 Time spectrum of electron radiation power peak at different laser pulse width
a—temporal profile of power peak point under laser pulse λ0 b—temporal profile of power peak point under laser pulse 2λ0 c—temporal profile of power peak point under laser pulse 3λ0 d—temporal profile of power peak point under laser pulse 4λ0 e~h—amplification of peak Fig. 5a~Fig. 5d respectively
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