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紧聚焦高斯脉冲激光电场的归一化矢势为:
$ \boldsymbol{a}(\zeta)=a_{0} a_{1}[\cos \varphi \cdot \boldsymbol{x}+\delta \sin \varphi \cdot \boldsymbol{y}] $
(1) 式中,a1=(r0/r)exp[-ζ2/B2-(x2+y2)/r2],ζ=z-t,t是观测时间,r和B分别为激光的束腰半径和脉宽,r=${r_0}\sqrt {1 + {z^2}/{z_{\rm{R}}}^2} $,r0为脉冲的最小半径,zR=r02/2为脉冲激光的瑞利长度,δ=0时为线偏振。
(1) 式中由mc2/e归一化的激光振幅表示为:
$ a_{0}=e A_{0} /\left(m c^{2}\right)=0.85 \times 10^{-9} \sqrt{I \lambda^{2}} $
(2) 式中,m, e分别为电子静止时的质量和电量,I,λ分别为激光强度的峰值和激光波长,c为光速,A0为激光场矢势的振幅。
(1) 式中激光脉冲相位φ表示为:
$ \varphi=\zeta+c_{0} \zeta^{2}+\varphi_{\mathrm{WC}}-\varphi_{\mathrm{G}}+\varphi_{0} $
(3) 式中,c0为激光脉冲的啁啾参数,φWC=(x2+y2)/[2R(z)]是与波前曲率有关的相位,R(z)=z(1+zR2/z2)为脉冲激光波前曲率半径,φG=z/zR,φ0为脉冲激光的初始相位,线偏振激光下δ=0。空间坐标被k0-1归一化,k0为激光频率。
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沿+z轴方向传播的线偏振紧聚焦脉冲激光与电子作用的示意图如图 1所示,处于静止状态的电子初始位置为坐标原点。
该激光场矢势分量的大小表示为:
$ \left\{\begin{array}{l} a_{x}=a_{1} \cos \varphi \\ a_{y}=a_{1} \sin \varphi \\ a_{z}=\frac{2 a_{1}}{r_{0} r[-x \sin (\theta+\varphi)+\delta y \cos (\theta+\varphi)]} \end{array}\right. $
(4) 式中,θ=π-arctan(z/zR)是观测极角。
在电子的运动整个过程中,电子向各个方向发出谐波辐射,辐射方向n为:
$ \boldsymbol{n}=\sin \theta \cos \phi \cdot \boldsymbol{x}+\sin \theta \sin \phi \cdot \boldsymbol{y}+\cos \theta \cdot \boldsymbol{z} $
(5) 电子在电磁场中的运动状态和能量方程分别为:
$ \mathrm{d}(\boldsymbol{p}-\boldsymbol{a}) / \mathrm{d} t=-\nabla_{\boldsymbol{a}}(\boldsymbol{\gamma} \cdot \boldsymbol{a}) $
(6) $ \frac{\mathrm{d} \sigma}{\mathrm{d} t}=\boldsymbol{\gamma} \cdot \frac{\partial \boldsymbol{a}}{\partial t} $
(7) 式中,▽a仅作用在a上,σ=[1-(v/c)2]-1/2为相对论公式因子,v是电子运动速率,γ=v/c, p=σγ分别为归一化后的电子速度和动量。
通过联立方程组(4)式以及(6)式和(7)式,可以得到:
$ \left\{\begin{array}{l} \sigma \frac{\mathrm{d} \gamma_{x}}{\mathrm{~d} t}=\left(1-\gamma_{x}^{2}\right) \frac{\partial a_{x}}{\partial t}+\gamma_{y}\left(\frac{\partial a_{x}}{\partial y}-\frac{\partial a_{y}}{\partial x}\right)+ \\ \ \ \ \ \gamma_{z}\left(\frac{\partial a_{x}}{\partial z}-\frac{\partial a_{z}}{\partial x}\right)-\gamma_{x} \gamma_{y} \frac{\partial a_{y}}{\partial t}-\gamma_{x} \gamma_{z} \frac{\partial a_{z}}{\partial t} \\ \sigma \frac{\mathrm{d} \gamma_{y}}{\mathrm{~d} t}=\left(1-\gamma_{y}^{2}\right) \frac{\partial a_{y}}{\partial t}+\gamma_{x}\left(\frac{\partial a_{x}}{\partial y}-\frac{\partial a_{y}}{\partial x}\right)+ \\ \ \ \ \ \gamma_{z}\left(\frac{\partial a_{y}}{\partial z}-\frac{\partial a_{z}}{\partial y}\right)-\gamma_{x} \gamma_{y} \frac{\partial a_{x}}{\partial t}-\gamma_{y} \gamma_{z} \frac{\partial a_{z}}{\partial t} \\ \sigma \frac{\mathrm{d} \gamma_{z}}{\mathrm{~d} t}=\left(1-\gamma_{z}^{2}\right) \frac{\partial a_{z}}{\partial t}+\gamma_{x}\left(\frac{\partial a_{x}}{\partial z}-\frac{\partial a_{z}}{\partial x}\right)+ \\ \ \ \ \ \gamma_{y}\left(\frac{\partial a_{z}}{\partial y}-\frac{\partial a_{y}}{\partial z}\right)-\gamma_{x} \gamma_{z} \frac{\partial a_{x}}{\partial t}-\gamma_{y} \gamma_{z} \frac{\partial a_{y}}{\partial t} \\ \frac{\mathrm{d} \sigma}{\mathrm{d} t}=\gamma_{x} \frac{\partial a_{x}}{\partial t}+\gamma_{y} \frac{\partial a_{y}}{\partial t}+\gamma_{z} \frac{\partial a_{z}}{\partial t} \end{array}\right. $
(8) 式中,γx,γy,γz分别为电子沿x,y,z这3个坐标正方向上的速度分量大小。
当电子做相对论加速运动时,辐射功率为P(t),单位立体角Ω内的电磁辐射功率表示为:
$ \frac{\mathrm{d} P(t)}{d \varOmega}=\frac{|\boldsymbol{n} \times[(\boldsymbol{n}-\boldsymbol{\gamma}) \times(\mathrm{d} \boldsymbol{\gamma} / \mathrm{d} t)]|^{2}}{(1-\boldsymbol{n} \cdot \boldsymbol{\gamma})^{6}} $
(9) 式中,dP(t)/dΩ被e2ω02/(4πc)归一化处理。
辐射能量为W(t),则单位立体角内的辐射能量表示为:
$ \begin{gathered} \frac{\mathrm{d} W(t)}{\mathrm{d} \varOmega}= \\ \int_{-\infty}^{\infty} \frac{|\boldsymbol{n} \times[(\boldsymbol{n}-\boldsymbol{\gamma}) \times(\mathrm{d} \boldsymbol{\gamma} / \mathrm{d} t)]|^{2}}{(1-\boldsymbol{n} \cdot \boldsymbol{\gamma})^{6}} \mathrm{~d} t \end{gathered} $
(10) 观测时间t与电子的延迟时间t′二者的关系为:
$ t=t^{\prime}+D_{0}-\boldsymbol{n} \cdot \boldsymbol{d} $
(11) 式中,D0为探测器观测点与电子和激光的作用点之间的距离,同时可认为探测点与作用点之间的距离足够大,d为电子所处位置矢量。
单位立体角与单位频率间隔内的辐射能量为:
$ \begin{gathered} \frac{\mathrm{d}^{2} W}{\mathrm{~d} \omega \mathrm{d} \varOmega}= \\ \left|\int_{-\infty}^{\infty} \frac{\boldsymbol{n} \times[(\boldsymbol{n}-\boldsymbol{\gamma}) \times \boldsymbol{\gamma}]}{(1-\boldsymbol{n} \cdot \boldsymbol{\gamma})^{2}} \exp [\mathrm{i} s(t-\boldsymbol{n} \cdot \boldsymbol{d})] \mathrm{d} t\right|^{2} \end{gathered} $
(12) 式中,d2W/(dωdΩ)被e2/(4πc2)归一化处理;s=ωh/ω0,ωh为谐波辐射频率。
通过求解(9)式~(12)式,最终得到电子和激光作用的整个过程中能量和功率随观测角的变化的情况,从而确定使得电子辐射能量最大的运动方向,进而得到沿该方向运动的电子的辐射能量随频率变化的具体情况。
紧聚焦强激光脉冲中电子的非对称性辐射
Asymmetric radiation of electrons in intensely compact-focused laser pulse
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摘要: 为了获得紧聚焦脉冲激光作用下电子的辐射特性, 采用非线性Thomson散射理论和线偏振紧聚焦激光脉冲与单电子的相互作用模型, 利用MATLAB完成数值模拟, 获得了电子的运动特性、不同观测角度下的功率与能量分布, 以及最大辐射方向上的功率与能量。结果表明, 与平面波激光脉冲作用下的非线性Thomson散射现象相似而又不同, 在紧聚焦激光脉冲下的辐射特性中, 辐射功率脉冲双峰的脉宽分别为0.008fs和0.121fs, 其对称性在时间上不再成立; 辐射能量在低辐射频率0~50ω0内呈现出剧烈振荡的变化特点。该结果对强激光场中电子辐射特性的研究是有帮助的。Abstract: In order to obtain the electronic radiation characteristics under the action of a tightly compact-focused laser, the nonlinear Thomson scattering theory and the model for the interaction of linearly polarized compact-focused laser pulse with a single electron were utilized in this study. And the numerical simulation was completed by using MATLAB. The motion characteristics of the electron, as well as the power and energy distribution at different observation angles were analyzed, especially in the direction of maximum radiation. Compared with the plane-wave laser pulse, it is found that the symmetric-bimodal radiation power pulse no longer holds in time for the pulse width was 0.008fs and 0.121fs, respectively; And the radiation energy shows the dramatic oscillation at low radiation frequency of 0~50ω0. The results can provide great value for the electron emission characteristics in intense laser field.
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Key words:
- laser optics /
- electronic radiation /
- asymmetry /
- linear polarization /
- compact-focused laser
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