-
设测试光T到达BS 2时的琼斯矩阵[19]为,其中,a是测试光的振幅;ω0是角频率;φ是参考光与测试光的相位差。同时参考光R的琼斯矩阵为,其中, b为参考光的振幅,λ/4波片的琼斯矩阵,微偏振阵列相机中检偏器的琼斯矩阵,其中,α为检偏器的偏振角度。因此到达检偏器上的光可以表示为:
$ \boldsymbol{E}=\boldsymbol{P} \cdot \boldsymbol{Q} \cdot(\boldsymbol{T}+\boldsymbol{R}) $
(1) 因此到达每个检偏器上的光强为两个分量的强度之和:
$ I=\boldsymbol{E}_x{ }^2+\boldsymbol{E}_y{ }^2 $
(2) 式中: Ex和Ey为x轴和y轴方向的光矢量。
分别代入α为0°、45°、90°、135°得到I1、I2、I3、I4。因此参考光与测试光的相位差φ=arctan[(I3-I1)/(I2-I4)],由此公式计算的相位分布在-π和+π之间,发生了2π的相位跃变,本文中使用同步相位展开和去噪算法[20]进行相位解缠以恢复真实相位。
等离子体的折射率基于Abel逆变换[21]得出,其计算公式如下:
$ n-n_0=-\frac{\lambda}{\pi} \int_r^R \frac{\mathrm{d} \varphi / \mathrm{d} x}{\sqrt{x^2-r^2}} \mathrm{~d} x $
(3) 式中: λ为探测光波长;n为等离子体折射率;n0为空气折射率;r和R表示到原点的距离。
在高真空环境下,因为电子对等离子体折射率起主导作用,一般可表示为[22]:
$ n=1-\frac{e^2}{8 \pi^2 \varepsilon_0 m c^2} \lambda^2 N $
(4) 式中:e为元电荷;m为电子质量;ε0为真空介电常数;c为真空中的光速;N为电子密度。
因此, 采用单波长法进行测量时,可对电子密度进行近似计算:
$ N=-\frac{8 \pi^2 \varepsilon_0 m c^2}{e^2 \lambda^2}(n-1) $
(5) -
2维轴对称流体动力学模型可以仿真等离子体羽流在背景气体中的膨胀过程。高真空环境下,由于背景气体含量很少,等离子体的动力学演化几乎不受影响,因此对模型进行了一定的简化,将等离子体羽流视为非粘性流。真空中等离子体的膨胀运动可以用以下的方程组表示:
$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \boldsymbol{v})=0 $
(6) $ \frac{\partial \rho \boldsymbol{v}}{\partial t}+\nabla \cdot\left(\rho \boldsymbol{v}^2\right)=-\nabla p $
(7) $ \begin{gathered} \frac{\partial \rho\left(E+\boldsymbol{v}^2 / 2\right)}{\partial t}+\nabla \cdot\left[\rho \boldsymbol{v}+\left(E+\boldsymbol{v}^2 / 2\right)\right]= \\ -\nabla \cdot(p \boldsymbol{v}) \end{gathered} $
(8) 式中: ρ是质量密度; v表示等离子体羽流的速度; p是压强; E表示比内能。
这里将羽流视为理想气体,遵循理想气体定律,则压强和内能密度可以表示为:
$ p=\left(1+x_{\mathrm{e}}\right) \frac{\rho k T}{m} $
(9) $ \rho E=\frac{\rho}{m}\left[\frac{3}{2}\left(1+x_{\mathrm{e}}\right) k T+I_0 x_{\mathrm{i}}\right] $
(10) 式中: xe和xi为电子和离子的电离率;k为玻尔兹曼常量;T为气体温度;m为气体质量;I0为铝的1级电离能。
假设等离子体处于局部热平衡状态,且模型只考虑1阶电离,结合萨哈方程与电荷守恒方程[23],即可以计算等离子体中离子的数密度。
对本实验中1.333×10-4 Pa真空度下所产生的等离子体羽流进行了数值仿真。由实验可知,激光结束后等离子体的径向和轴向尺寸分别约为1 mm和0.1 mm,初始等离子体中心处的蒸汽密度约为1.3×1021 cm-3、温度约为2 eV。由于等离子体初始电子密度及温度服从高斯分布,所以等离子体电子密度n(x, y)及温度T(x, y)为:
$ \left\{\begin{array}{l} n(x, y)=n_0 \exp \left(-\frac{x^2}{2 X_0^2}-\frac{y^2}{2 Y_0^2}\right) \\ T(x, y)=T_0 \exp \left(-\frac{x^2}{2 X_0^2}-\frac{y^2}{2 Y_0^2}\right) \end{array}\right. $
(11) 式中: X0和Y0分别为激光结束时等离子体的径向和轴向尺寸;n0和T0为激光结束时等离子体中心处蒸汽密度与温度。
-
由数值仿真计算结果可以看出,2维流体动力学模型可以仿真等离子体羽流的动态演化过程,如图 6所示。等离子体的中心电子密度在5 ns时约为1.3×1021 cm-3,到50 ns时衰减为1.4×1020 cm-3。
为了较为直观地分析,对前50 ns等离子体中心电子密度的数值仿真结果与实验结果进行对比,如图 7所示。可以看出,数值仿真的计算结果与实验结果较为吻合,存在偏差的原因有以下几点:(a)数值仿真运用2维轴对称流体动力学模型,忽略了等离子体的黏性应力等参数,忽略背景气体的作用力; (b)在考虑电离方程时只考虑了铝离子的1阶电离而未考虑2阶、3阶电离,这是模型以后需要改进的地方; (c)在实验过程中,真空泵的气压值不能稳定地停在固定值,导致气压与实际产生偏差。
高真空激光等离子体的同步移相干涉诊断及仿真
Simultaneous phase-shifting interferometer diagnosis and simulation of high vacuum laser plasma
-
摘要: 为了解决在高真空环境下, 等离子体膨胀迅速、外围羽流引起的条纹偏移小、单幅干涉条纹图难以检出的问题, 采用同步移相干涉测试技术得到了1. 333×10-4 Pa和1. 333×10-3 Pa真空度下激光诱导铝等离子体电子密度分布; 同时采用2维轴对称流体动力学模型, 对高真空环境下激光诱导等离子体的膨胀过程进行了数值仿真, 得到了电子密度的2维分布, 并分析了数值仿真结果存在偏差的原因及改进方法。结果表明, 等离子体的中心电子密度在50 ns时下降至1. 4×1020 cm-3; 数值仿真结果与实验结果吻合较好, 验证了模型的正确性。该研究为高真空下激光等离子体的研究提供了一定的参考。Abstract: In order to solve the problem of rapid plasma expansion and small fringe offset caused by peripheral plumes in the high vacuum environment, it was difficult to detect a single interference fringe pattern. The electron density distributions of the laser-induced aluminum plasma under 1.333×10-4 Pa and 1.333×10-3 Pa pressure were obtained by using simultaneous phase-shifting interference. At the same time, the expansion process of laser-induced plasma under a high vacuum environment was simulated numerically by using a 2-D axis ymmetric fluid dynamics model, and the two-dimensional distribution of electron density was obtained and analyzed for the reasons for the deviation of numerical simulation results and the improvement methods. The results show that the central electron density of plasma decreases to 1.4×1020 cm-3 at 50 ns. The numerical simulation results are in good agreement with the experimental results, which verifies the reliability of the model. This research provides some reference for the study of laser plasma under high vacuum.
-
[1] 陈亮, 游利兵, 王庆胜, 等. 紫外激光诱导击穿光谱的应用与发展[J]. 激光技术, 2017, 41(5): 619-625. CHEN L, YOU L B, WANG Q Sh, et al. Application and development of UV laser induced breakdown spectroscopy[J]. Laser Technology, 2017, 41(5): 619-625(in Chinese). [2] 丁宇, 杨淋玉, 陈靖, 等. 基于激光诱导击穿光谱法的铝合金中Mg元素定量分析[J]. 激光与光电子学进展, 2022, 59(13): 1314006. DING Y, YANG L Y, CHEN J, et al. Quantitative analysis of mg e-lement in aluminium alloy based on laser-induced breakdown spectro-scopy[J]. Laser & Optoelectronics Progress, 2022, 59(13): 1314006(in Chinese). [3] 林华中, 王英, 何正浩, 等. 激光触发真空开关光谱和导通特性实验研究[J]. 激光技术, 2017, 41(1): 24-28. LIN H Zh, WANG Y, HE Zh H, et al. Experimental study on spectrum and conduction properties of laser triggered vacuum switch[J]. Laser Technology, 2017, 41(1): 24-28(in Chinese). [4] 柯伟, 陈敏源, 袁欢, 等. 激光诱导等离子体中聚焦程度对真空开关真空度检测的影响[J]. 高电压技术, 2022, 48(10): 4224-4232. KE W, CHEN M Y, YUAN H, et al. Influence of focusing degree in laser-induced plasma on vacuum detection of vacuum switch[J]. High Voltage Engineering, 2022, 48(10): 4224-4232(in Chinese). [5] OJEDA-G-P A, DÖBELI M, LIPPERT T. Influence of plume properties on thin film composition in pulsed laser deposition[J]. Advanced Materials Interfaces, 2018, 5(18): 1701062. doi: 10.1002/admi.201701062 [6] 邓钟炀, 贾强, 冯斌, 等. 脉冲激光沉积高性能薄膜制备及其应用研究进展[J]. 中国激光, 2021, 48(8): 0802010. DENG Zh Y, JIA Q, FENG B, et al. Research progress on fabrication and applications of high-performance films by pulsed laser deposition[J]. Chinese Journal of Lasers, 2021, 48(8): 0802010(in Chinese). [7] 赵万芹, 梅雪松, 王文君. 超短脉冲激光微孔加工(上)——理论研究[J]. 红外与激光工程, 2019, 48(1): 140-148. ZHAO W Q, MEI X S, WANG W J. Ultrashort pulse laser drilling of micro-holes (part 1)-Theoretical study[J]. Infrared and Laser Engineering, 2019, 48(1): 140-148(in Chinese). [8] 吴常顺, 冯国英, 刘彩飞. 脉冲激光对单晶硅打孔的研究[J]. 红外与激光工程, 2016, 45(2): 140-145. WU Ch Sh, FENG G Y, LIU C F. Research on drilling hole of single crystal by pulse laser[J]. Infrared and Laser Engineering, 2016, 45(2): 140-145(in Chinese). [9] 罗锦锋, 宋世军, 王平秋, 等. 激光等离子体对硅表面微纳粒子除去机理研究[J]. 激光技术, 2018, 42(4): 567-571. LUO J F, SONG Sh J, WANG P Q, et al. Study on removal mechanism of micro-/nano-particles on silicon surface by laser plasma[J]. Laser Technology, 2018, 42(4): 567-571(in Chinese). [10] VERHOFF B, HARILAL S S, FREEMAN J R, et al. Dynamics of femto-and nanosecond laser ablation plumes investigated using optical emission spectroscopy[J]. Journal of Applied Physics, 2012, 112(9): 93303. doi: 10.1063/1.4764060 [11] FARID N, HARILAL S S, DING H, et al. Emission features and expansion dynamics of nanosecond laser ablation plumes at different ambient pressures[J]. Journal of Applied Physics, 2014, 115(3): 33107. doi: 10.1063/1.4862167 [12] CAO Sh Q, SU M G, LIU J, et al. Expansion dynamics and compression layer in collinear double-pulse laser produced plasmas in a vacuum[J]. Physics of Plasmas, 2020, 27(5): 052101. doi: 10.1063/5.0004184 [13] 王俊霄, 王树青, 张雷, 等. 激光诱导等离子体在背景气体中膨胀的二维轴对称模型[J]. 山西大学学报(自然科学版), 2022, 45(5): 1254-1261. WANG J X, WANG Sh Q, ZHANG L, et al. A two-dimensional axisymmetric model of laser-induced plasma expansion in background gas[J]. Journal of Shanxi University(Natural Science Edition), 2022, 45(5): 1254-1261(in Chinese). [14] GAO Ch L, MIN Q, LIU J Zh, et al. Time evolution of copper-aluminum alloy laser-produced plasmas in vacuum[J]. Journal of Quantitative Spectroscopy and Radiative Transfer, 2021, 274: 107855. doi: 10.1016/j.jqsrt.2021.107855 [15] 丁煜. 动态马赫-曾德尔干涉仪及其关键技术研究[D]. 南京: 南京理工大学, 2019: 1-12. DING Y. Research on key technology of dynamic Mach-Zehnder interferometer[D]. Nanjing: Nanjing University of Science & Technology, 2019: 1-12(in Chinese). [16] 郑东晖. 空间相移干涉测量方法及其关键技术研究[D]. 南京: 南京理工大学, 2019: 17-32. ZHENG D H. Research on spatial phase-shifting interferometry and its key technologies[D]. Nanjing: Nanjing University of Science & Technology, 2019: 17-32(in Chinese). [17] KEMAO Q. Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations[J]. Optics and Lasers in Engineering, 2007, 45(2): 304-317. doi: 10.1016/j.optlaseng.2005.10.012 [18] 左芬, 陈磊. 同步移相抗振光干涉测量技术研究进展[J]. 激光与光电子学进展, 2006, 43(11): 43-48. ZUO F, CHEN L. Development of anti-vibration technology in si-multaneous phase-shifting interferometry[J]. Laser & Optoelectronics Progress, 2006, 43(11): 43-48(in Chinese). [19] 钱克矛, 缪泓, 伍小平. 一种用于动态过程测量的实时偏振相移方法[J]. 光学学报, 2001, 21(1): 64-67. QIAN K M, MOU H, WU X P. A real-time polarization phase shifting technique for dynamic measurement[J]. Acta Optica Sinica, 2001, 21(1): 64-67(in Chinese). [20] PINEDA J, BACCA J, MEZA J, et al. SPUD: Simultaneous phase unwrapping and denoising algorithm for phase imaging[J]. Applied Optics, 2020, 59(13): D81-D88. [21] 郑峰, 张宏超, 陆健, 等. 激光等离子体干涉条纹的图像处理方法研究[J]. 南京理工大学学报(自然科学版), 2009, 33(5): 668-671. ZHENG F, ZHANG H Ch, LU J, et al. Image processing of interference fringes of laser-induced-plasma[J]. Journal of Nanjing Unibersity of Science & Technology (Natural Science Edition), 2009, 33(5): 668-671(in Chinese). [22] 张楚蕙, 陆健, 张宏超, 等. 双脉冲激光诱导铝等离子体的双波长干涉诊断[J]. 红外与激光工程, 2022, 51(2): 441-447. ZHANG Ch H, LU J, ZHANG H Ch, et al. Dual-wavelength interferometric diagnosis of double-pulse laser induced aluminium plasma[J]. Infrared and Laser Engineering, 2022, 51(2): 441-447(in Chinese). [23] BOGAERTS A, CHEN Zh Y, GIJBELS R, et al. Laser ablation for analytical sampling: What can we learn from modeling?[J]. Spectrochimica Acta, 2003, B58(11): 1867-1893.