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研究CPP的匀滑特性,实际上就是分析畸变波前与CPP面形之间的相位叠加。如图1所示,在光路中,在聚焦透镜前放置CPP,从而实现改善焦斑质量的目的。
在标量衍射理论下,相位叠加后的远场满足傅里叶变换关系。设畸变波前相位为
${\phi _1}$ ,CPP面形相位为${\phi _2}$ ,叠加后的波前位相是$\phi $ ,则有:$$\phi = {\phi _1} + {\phi _2}$$ (1) $${E_\phi } = F\{ \exp [ - j\phi ]\} = {E_{\phi 1}}*{E_{\phi 2}}$$ (2) $${I_\phi } = {\left| {{E_\phi }} \right|^2}$$ (3) 式中:E为远场光振幅;
${I_\phi }$ 为远场光强;F{}为傅里叶变换;*为卷积。从公式上看,复振幅的卷积看上去像是匀滑,实际上,干涉散斑也包含在公式(2)中,而散斑既不均匀也不平滑,所以不能根据卷积运算来解释CPP的匀滑机理。这里笔者利用梯度直方图来对这一现象进行解释,梯度直方图可以认为是几何光学描述下的角谱,CPP面型梯度方向对应角谱方向,直方图则对应角谱的相对强度[13]:
$${I_\phi } = hist(\nabla \phi )$$ (4) 式中:hist( )为直方图计算;
$\nabla $ 为梯度计算,在一维情况下实际上计算的是斜率;${I_\phi }$ 为CPP的角谱,也可以认为是焦斑光强。注意到公式(4)中的焦斑光强${I_\phi }$ 与公式(3)中的不完全一致,实际上描述的是公式(3)的包络,或者说低通滤波结果。这是由于几何光学方法忽略了光学位相,因此,不含干涉效应。 -
根据大数定律可知,在足够多独立实验的情况下,概率密度函数等于实验直方图。由于CPP的孔径尺寸远大于随机面型的自相关长度,在物理上是满足大数定律的。可以将CPP面形函数的概率密度与其远场直方图之间的关系表示为:
$$hist(\xi ) = PDF(\xi )$$ (5) 式中:
$\xi $ 为一个随机函数,表示了CPP的面形。另外两个独立随机变量${\xi _1}$ 与${\xi _2}$ 之和的概率密度等于各自概率密度的卷积,可以写为:$$PDF({\xi _1} + {\xi _2}) = PDF({\xi _1})*PDF({\xi _2})$$ (6) 虽然CPP的面型具有随机分布的特性,但一个已经加工好的元件其表面数据是确定的,因此,可以将任意一个CPP看做随机函数的一个实例。
如果笔者把公式(4)中的
$\phi $ 及$\nabla \phi $ 看作随机变量,可以令$\nabla \phi = \xi $ 。此外,${\phi _1}$ 和${\phi _2}$ 分别代表CPP和其他输入(比如加工误差)面型,注意到${\phi _1}$ 和${\phi _2}$ 是完全独立无关的随机量,满足公式(5)的条件,于是根据公式(1)、(4)~(6)可以推得:$$hist(\nabla \phi ) = hist(\nabla {\phi _1})*hist(\nabla {\phi _2})$$ (7) 公式的物理意义是叠加后的焦斑包络等于叠加前焦斑包络的卷积。注意到卷积就是匀滑,就是低通滤波。这意味着通过CPP的畸变光束,其远场光强将被去掉很多中高频成分。这样,最终得到了CPP匀滑的数学解释。
与公式(2)的形式相比较,公式(7)描述的是光强包络的卷积而非振幅卷积。在大小相当的情况下,具有平顶焦斑的CPP和具有高斯焦斑的CPP比较,后者的匀滑能力要更强,因为前者对应到频谱空间是一个sinc函数的滤波器,有一些残留高频部分,而后者是高斯函数滤波器,基本没有残留的高频部分。
应用统计几何光学方法分析CPP,与常规的标量衍射方法相比,简化了焦斑计算,降低了波前叠加分析的困难,得到了关于CPP匀滑的数学公式。这对匀滑特性的物理图像有一个更清晰的描述和解释。
Influence of phase additive effect on beam smoothing character of continuous phase plate
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摘要: 在大型激光系统的运行过程中,为了对相位畸变导致的焦斑分布不均匀进行改善,在光路通常会使用连续位相板(CPP) 来进行远场束匀滑。根据CPP 面型的随机特性,利用统计的方法对位相板与畸变波前相位的叠加特性进行了计算,系统研究了连续位相板对光束波前分布实现控制的机理。从CPP面形的概率密度与远场直方图之间的关系出发,推导了畸变波前通过CPP 后远场光强分布的表达式,从理论上解释了这种束匀滑器件的工作原理及特性。通过数值模拟计算了不同畸变光束经过CPP后的远场直方图,对结果进行比较并分析了不同面型特性对最终束匀滑效果的影响。结果证明:位相板能在焦斑光强上起到卷积滤波的作用,从而实现光束匀滑效果。从原理上解释了CPP 在具有小相关长度时具有更高匀滑效果这一特性,为实际面型设计和优化提供理论基础。另外,应用统计几何光学方法进行分析,可有效降低波前叠加分析的难度。Abstract: During the operational process of high power laser system, the uniformity of focal spot will impact the experiment and application seriously. To improve the focal spot quality, a continuous phase plate (CPP) should be used in the light path for far field beam smoothing. As a phase element CPP has different functions, such as decoherece and beam shaping. In this paper the smoothing performance of CPP was concerned. Good performance of beam smoothing depended on a reasonable surface figure distribution. According to the random characteristic of phase plate surface, the statistical method was employed to study the beam smoothing mechanism. According to the relationship between the probability density and far field histogram of the surface of CPP, the expression of the superposition intensity envelope of distorted beam and CPP surface figure was deduced. The formula proved that the function of a CPP for focal spot was just a convolution filtering. So the mathematical explanation on the beam smoothing mechanism was achieved. Furthermore, using this analysis model the reason that the CPP with short correlation length will have the capability of better beam smoothing was explained theoretically. Numerical simulations were done to show the beam smoothing performance of CPP. The far field histograms with different distorted beams are calculated and compared. The results show that after the phase additive of the distorted beam and CPP a new wavefront was generated. If the law of large numbers is satisfied, when the correlation length and the gradient of the wavefront is small, the light focus spot distribution is smooth and uniform. The statistical geometrical optical method used in this paper can reduce the analysis difficulty on the phase additive effectively.
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Key words:
- continuous phase plate /
- statistics character /
- phase additive /
- beam smoothing
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图 5 畸变A通过CPP的焦斑。(a)
$hist(\nabla \phi + \nabla {\phi _A})$ 计算结果;(b)$hist(\nabla \phi )*hist(\nabla {\phi _A})$ 计算结果Figure 5. Focal spot of distortion A passing through CPP. (a) Calculation result of
$hist(\nabla \phi + \nabla {\phi _A})$ ; (b) Calculation result of$hist(\nabla \phi )*hist(\nabla {\phi _A})$ 图 6 畸变B通过CPP的焦斑。(a)
$hist(\nabla \phi + \nabla {\phi _B})$ 计算结果;(b)$hist(\nabla \phi )*hist(\nabla {\phi _B})$ 计算结果Figure 6. Focal spot of distortion B passing through CPP. (a) Calculation result of
$hist(\nabla \phi + \nabla {\phi _B})$ ; (b) Calculation result of$hist(\nabla \phi )*hist(\nabla {\phi _B})$ 图 9 二维畸变通过CPP的焦斑。(a)
$hist(\nabla {\phi _{CPP}} + \nabla {\phi _{abb}})$ 计算结果;(b)$hist(\nabla {\phi _{CPP}})*hist(\nabla {\phi _{abb}})$ 计算结果Figure 9. Focal spot of 2D distortion passing through CPP. (a) Calculation result of
$hist(\nabla {\phi _{CPP}} + \nabla {\phi _{abb}})$ ; (b) Calculation result of$hist(\nabla {\phi _{CPP}})*hist$ ($\nabla {\phi _{abb}}$ ) -
[1] 黄金勇, 赵恒, 胡庆, 等. 大口径平面光学元件波前梯度数控抛光[J]. 光学 精密工程, 2019, 27(7): 1473-1480. doi: 10.3788/OPE.20192707.1473 Huang Jinyong , Zhao Heng, Hu Qing, et al. Large aperture optical element wavefront gradient controlled by computer numerical controlled polishing [J]. Optics and Precision Engineering, 2019, 27(7): 1473-1480. (in Chinese) doi: 10.3788/OPE.20192707.1473 [2] 杨勋, 徐抒岩, 马宏财, 等. 径向温度梯度对轻量化反射镜面形精度的影响[J]. 光学 精密工程, 2019, 27(7): 1553-1560. Yang Xun, Xu Shuyan, Ma Hongcai, et al. Influence of radial temperature gradient on surface figure of lightweight reflective mirror [J]. Optics and Precision Engineering, 2019, 27(7): 1553-1560. (in Chinese) [3] 张博文, 王小勇, 赵野, 等. 天基大口径反射镜支撑技术的发展[J]. 红外与激光工程, 2018, 47(11): 1113001. doi: 10.3788/IRLA201847.1113001 Zhang Bowen, Wang Xiaoyong, Zhao Ye, et al. Progress of support technique of space-based large aperture mirror [J]. Infrared and Laser Engineering, 2018, 47(11): 1113001. (in Chinese) doi: 10.3788/IRLA201847.1113001 [4] Kato Y, Mima K, Miyanaga N, et al. Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression [J]. Phys Revi Letts, 1984, 53(11): 1057-1060. doi: 10.1103/PhysRevLett.53.1057 [5] 王震, 付文静, 张蓉竹. 飞秒激光多脉冲烧蚀金属铁的数值模拟[J]. 红外与激光工程, 2019, 48(7): 0706002. doi: 10.3788/IRLA201948.0706002 Wang Zhen, Fu Wenjing, Zhang Rongzhu. Numerical simulation of femtosecond laser multi-pulse ablation of metal iron [J]. Infrared and Laser Engineering, 2019, 48(7): 0706002. (in Chinese) doi: 10.3788/IRLA201948.0706002 [6] 刘克俭, 苗锡奎, 徐晨阳, 等. 半主动激光制导能量传输与模拟技术[J]. 中国光学, 2019, 12(2): 256-264. doi: 10.3788/co.20191202.0256 Liu Kejian, Miao Xikui, Xu Chenyang, et al. Semi-active laser-guided energy transmission and simulation technology [J]. Chinese Optics, 2019, 12(2): 256-264. (in Chinese) doi: 10.3788/co.20191202.0256 [7] Néauport J, Ribeyre X, Daurios J, et al. Design and optical characterization of a large continuous phase plate for laser integration line and laser Megajoule facilities [J]. Appl Opt, 2003, 32(14): 2377-2382. [8] Yang Chunlin, Yan Hao, Wang Jian, et al. A novel design method for continuous-phase plate [J]. Opt Express, 2013, 21(9): 11171-11180. doi: 10.1364/OE.21.011171 [9] Arieli Y. Continuous phase plate for non-uniform illumination beam shaping using the inverse phase contrast method [J]. Opt Comm, 2000, 180(4): 239-245. [10] Zhang Dazhi, Wan Yongjian, Zhang Rongzhu, et al. Surface statistical characteristics and smoothing analysis of continuous phase plate [J]. Optik, 2012, 123(22): 2062-2067. doi: 10.1016/j.ijleo.2011.09.032 [11] 陈波, 王菡子, 韦辉, 等. 用于惯性约束聚变束匀滑的完全连续相位板设计方法[J]. 光学学报, 2001, 21(4): 480-484. doi: 10.3321/j.issn:0253-2239.2001.04.024 Chen Bo, Wang Hanzi, Wei Hui, et al. , Desigen of fully continuous phase plates for beam smoothing in ICF [J]. Acta Optica Sinica, 2001, 21(4): 480-484. (in Chinese) doi: 10.3321/j.issn:0253-2239.2001.04.024 [12] Marozas John A. Fourier transform–based continuous phase-plate design technique: a high-pass phase-plate design as an application for OMEGA and the National Ignition Facility [J]. J Opt Soc Am A, 2007, 24(1): 74-83. doi: 10.1364/JOSAA.24.000074 [13] Yang Chunlin. Using the gradient histogram to analyze the continuous phase plate [J]. Opt Laser Tech, 2015, 65: 137-141. doi: 10.1016/j.optlastec.2014.07.018 [14] Yang Chunlin. Analysis on the focal spot characteristics of random Gauss phase plate [J]. Optik, 2017, 130: 601-607. doi: 10.1016/j.ijleo.2016.10.084