留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Suppression of higher diffraction orders using quasiperiodic array of rectangular holes with large size tolerance

Wei Lai Chen Yong Wang Shaoyi Fan Quanping Zhang Qiangqiang Zhang Zhong Wang Zhanshan Cao Leifeng

魏来, 陈勇, 王少义, 等. 大公差宽容度矩形孔准周期阵列实现对高阶衍射的有效抑制[J]. 强激光与粒子束, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117
引用本文: 魏来, 陈勇, 王少义, 等. 大公差宽容度矩形孔准周期阵列实现对高阶衍射的有效抑制[J]. 强激光与粒子束, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117
Wei Lai, Chen Yong, Wang Shaoyi, et al. Suppression of higher diffraction orders using quasiperiodic array of rectangular holes with large size tolerance[J]. High Power Laser and Particle Beams, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117
Citation: Wei Lai, Chen Yong, Wang Shaoyi, et al. Suppression of higher diffraction orders using quasiperiodic array of rectangular holes with large size tolerance[J]. High Power Laser and Particle Beams, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117

大公差宽容度矩形孔准周期阵列实现对高阶衍射的有效抑制

doi: 10.11884/HPLPB202032.200117
详细信息
  • 中图分类号: O436.1, O434.13

Suppression of higher diffraction orders using quasiperiodic array of rectangular holes with large size tolerance

Funds: National Key Research and Development Program of China (2017YFA0206001); National Natural Science Foundation of China (11805179)
More Information
    Author Bio:

    Wei Lai (1983—), male, PhD candidate, engaged in X-ray optics and plasma diagnostics; future718@yeah.net

    Corresponding author: Cao Leifeng (1967—), male, PhD, engaged in X-ray optics and plasma diagnostics; leifeng.cao@caep.cn
  • 摘要: 传统光栅的基础研究和应用研究进展一直备受关注。然而,高阶衍射污染使传统光栅获得的光谱纯度受到严重影响。为了抑制高阶衍射贡献,人们提出了许多单级或准单级光栅的设计方案,但它们对高阶衍射的抑制效果不可避免地受到加工精度的限制。提出了一种准周期矩形孔阵列光栅,通过优化矩形孔的概率密度分布函数,获得了比以往设计更大的加工误差宽容度。对这种光栅的衍射特性进行了分析研究。理论计算表明,即使孔径相对误差超过20%,光栅也可以完全抑制二阶、三阶和四阶衍射,五阶衍射效率与一阶衍射效率之比小于0.01%,大大降低了对加工精度的要求。
  • Figure  1.  Schematic of the distribution of holes in the array: from δ-function to arbitrary positive function ${\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right)$

    Figure  2.  (a) Design for a quasi-periodical hole array: in each lattice (d×d square), a rectangular hole with side lengths of d/6 and d is positioned randomly on the ξn axis and has a trapezoidal distribution profile shown in (b)

    Figure  3.  Comparison between far-field diffraction patterns of a quasi-periodical array shown in Fig. 2 when HL=100 and a periodic grating with a duty cycle of 1∶5 and 201 periods

    Figure  4.  (a) Intensity profiles of Fig. 3(a)(c) across the line qλ/2d (indicated by yellow dashed lines in Fig. 3); (b) numerical results of Eq. (1) across the p-axis for the rectangular holes with side lengths of d/5 and d (red dot), and 2d/15 and d (black line).

  • [1] Kallman T, Evans D A, Marshall H, et al. A census of X-ray gas in NGC 1068: Results from 450 ks of Chandra high energy transmission grating observations[J]. Astrophys J, 2014, 780: 121.
    [2] Wang Q D, Nowak M A, Markoff S B, et al. Dissecting X-ray emitting gas around the center of our galaxy[J]. Science, 2013, 341(30): 981-983.
    [3] Tzanavaris P, Yaqoob T. New constraints on the geometry and kinematics of matter surrounding the accretion flow in X-ray binaries from Chandra high-energy transmission grating X-ray spectroscopy[J]. Astrophys J, 2018, 855: 25. doi:  10.3847/1538-4357/aaaab6
    [4] Shpilman Z, Hurvitz G, Danon L, et al. A combined sinusoidal transmission grating spectrometer and X-ray diode array diagnostics for time-resolved spectral measurements in laser plasma experiments[J]. Rev Sci Instrum, 2019, 90: 013501. doi:  10.1063/1.5051486
    [5] Kühne M, Müller P. Higher order contributions in the synchrotron radiation spectrum of a toroidal grating monochromator determined by the use of a transmission grating[J]. Rev Sci Instrum, 1989, 60: 2101. doi:  10.1063/1.1140836
    [6] Sokolov A A, Eggenstein F, Erko A, et al. An XUV optics beamline at BESSY II [C]//Proc of SPIE. 2014: 92060J.
    [7] Suits A G, Heimann P, Yang Xueming, et al. A differentially pumped harmonic filter on the chemical dynamics beamline at Advanced Light Source[J]. Rev Sci Instrum, 1995, 66: 4841. doi:  10.1063/1.1146161
    [8] Heimann P A, Koike M, Hsu C W, et al. Performance of the vacuum ultraviolet high-resolution and high-flux beamline for chemical dynamics studies at the Advanced Light Source[J]. Rev Sci Instrum, 1997, 68: 1945. doi:  10.1063/1.1148082
    [9] Zhou Hongjun, Wang Guanjun, Zheng Jinjin, et al. Higher order harmonics contribution and suppression in metrology beamline [C]//Proc of SPIE. 2010: 75445
    [10] Cao Leifeng, Förster E, Fuhrmann A, et al. Single order X-ray diffraction with binary sinusoidal transmission grating[J]. Appl Phys Lett, 2007, 90: 053501. doi:  10.1063/1.2435618
    [11] Zang Huaping, Wang Chuanke, Cao Leifeng, et al. Elimination of higher-order diffraction using zigzag transmission grating in soft X-ray region[J]. Appl Phys Lett, 2012, 100: 111904. doi:  10.1063/1.3693395
    [12] Liu Yuwei, Zhu Xiaoli, Gao Yulin, et al. Quasi suppression of higher-order diffractions with inclined rectangular apertures gratings[J]. Sci Rep, 2015, 5: 16502. doi:  10.1038/srep16502
    [13] Fan Quanping, Liu Yuwei, Wang Chuanke, et al. Single-order diffraction grating designed by trapezoidal transmission function[J]. Opt Letts, 2015, 40: 2657. doi:  10.1364/OL.40.002657
    [14] Liu Ziwei, Shi Lina, Pu Tanchao, et al. Two-dimensional gratings of hexagonal holes for high order diffraction suppression[J]. Opt Express, 2017, 25(2): 1339. doi:  10.1364/OE.25.001339
    [15] Kuang Longyu, Wang Chuanke, Wang Zhebin, et al. Quantum-dot-array diffraction grating with single order diffraction property for soft X-ray region[J]. Rev Sci Instrum, 2010, 81: 073508. doi:  10.1063/1.3464197
    [16] Li Hailiang, Shi Lina, Wei Lai, et al. Higher-order diffraction suppression of free-standing quasiperiodic nanohole arrays in the X-ray region[J]. Appl Phys Lett, 2017, 110: 041104. doi:  10.1063/1.4974940
    [17] Gao Yulin, Zhou Weimin, Wei Lai, et al. Diagnosis of the soft X-ray spectrum emitted by laser-plasmas using a spectroscopic photon sieve[J]. Laser & Particle Beams, 2012, 30(2): 313-317.
    [18] Chen Yong, Wei Lai, Qian Feng, et al. Higher order harmonics suppression in extreme ultraviolet and soft X-ray[J]. Chin Phys B, 2018, 27: 024101. doi:  10.1088/1674-1056/27/2/024101
    [19] Wei Lai, Qian Feng, Yang Zuhua, et al. Diffraction properties of quasi-random pinhole arrays: suppression of higher orders and background fluctuations[J]. J Mod Opt, 2017, 64(21): 2420. doi:  10.1080/09500340.2017.1367853
    [20] Wei Lai, Chen Yong, Wang Shaoyi, et al. Suppression of higher diffraction orders in the extreme ultraviolet range by a reflective quasi-random square nano-pillar array[J]. Rev Sci Instrum, 2018, 89: 093110. doi:  10.1063/1.5034764
    [21] Hua Y L, Gao N, Xie C Q. Fabrication of ultralarge single order diffraction grating for soft X-ray monochromator [C]//IEEE Symposium on Design, Test, Integration and Packaging of MEMS / MOEMS. 2016: 1–4.
  • [1] 李井元, 方黎勇, 胡栋材, 齐晓世.  基于变换矩阵的BGA X-ray图像倾斜识别及校正方法 . 强激光与粒子束, 2018, 30(10): 109001-. doi: 10.11884/HPLPB201830.180092
    [2] 张耀锋, 黄建微, 胡涛, 张明昕, 江艳.  电子直线加速器X射线转换靶设计 . 强激光与粒子束, 2013, 25(09): 2393-2396. doi: 10.3788/HPLPB20132509.2393
    [3] 阳庆国, 李泽仁, 杨礼兵, 陈光华, 黄显宾, 蔡红春, 李晶, 肖沙里.  Z箍缩等离子体K壳层X射线自辐射单色成像 . 强激光与粒子束, 2012, 24(05): 1081-1084. doi: 10.3788/HPLPB20122405.1081
    [4] 刘冰, 王旭平.  钽铌酸钾晶体二次电光效应的应用研究进展 . 强激光与粒子束, 2012, 24(02): 261-266.
    [5] 秋实, 刘国治, 张治强, 侯青, 张庆元.  高功率微波介质击穿中X射线和紫外线的初步诊断 . 强激光与粒子束, 2011, 23(07): 0- .
    [6] 但加坤, 李剑峰, 杨礼兵, 黄显宾, 李军, 任晓东, 张思群, 欧阳凯, 段书超.  Z箍缩X射线在金属表面产生电荷分离现象 . 强激光与粒子束, 2010, 22(03): 0- .
    [7] 但加坤, 李剑峰, 黄显宾, 杨礼兵, 张思群, 欧阳凯, 李军, 段书超.  斜入射脉冲X射线产生电流实验 . 强激光与粒子束, 2010, 22(12): 0- .
    [8] 陈实, 蒋吉昊, 李剑峰.  X射线在金属表面产生Compton电流引起的电磁辐射时空分布 . 强激光与粒子束, 2009, 21(09): 0- .
    [9] 顾牡, 李达, 倪晨, 刘小林, 刘波, 黄世明.  双层塑料靶丸的X射线相衬成像 . 强激光与粒子束, 2009, 21(10): 0- .
    [10] 黎航, 曹柱荣, 赵宗清, 巫顺超, 董建军, 易荣清, 陈凯.  微通道板2.0~5.5 keV X射线透过率标定 . 强激光与粒子束, 2008, 20(06): 0- .
    [11] 阳庆国, 李泽仁, 彭其先, 陈光华, 叶雁, 刘寿先.  激光驱动X射线单色背光照相系统优化设计 . 强激光与粒子束, 2008, 20(12): 0- .
    [12] 田友伟, 陆云清.  高能电子在强激光场中的同步辐射获得超短X射线脉冲 . 强激光与粒子束, 2008, 20(10): 0- .
    [13] 段黎明, 廖平, 张平, 李胜颚.  高能X射线工业CT数据传输系统的设计 . 强激光与粒子束, 2008, 20(09): 0- .
    [14] 李勤, 石金水, 禹海军, 王莉萍, 何晖.  狭缝法测量X射线斑点大小 . 强激光与粒子束, 2006, 18(10): 0- .
    [15] 田友伟, 余玮, 陆培祥, 何峰, 马法君, 徐涵, 钱列加.  激光同步辐射作为阿秒X射线辐射源的特性研究 . 强激光与粒子束, 2005, 17(11): 0- .
    [16] 仇云利, 郭弘, 刘明伟, 唐华, 邓冬梅.  X射线在折射率连续变化等离子体介质中的传播 . 强激光与粒子束, 2004, 16(05): 0- .
    [17] 牛胜利, 彭玉, 王建国, 乔登江.  X光导管传输特性的蒙特卡罗模拟 . 强激光与粒子束, 2004, 16(12): 0- .
    [18] 郑志坚, 曹磊峰, 张保汉, 丁永坤, 江少恩, 李朝光.  X光Gabor波带片编码成像技术实验研究 . 强激光与粒子束, 2003, 15(08): 0- .
    [19] 彭永伦, 杨莉, 王民盛, 李家明, .  等离子体X射线吸收谱及发射谱研究 . 强激光与粒子束, 2002, 14(03): 0- .
    [20] 张钧.  X光辐射通过稀疏波区的反照率 . 强激光与粒子束, 2001, 13(03): 0- .
  • 加载中
图(4)
计量
  • 文章访问数:  48
  • HTML全文浏览量:  35
  • PDF下载量:  4
  • 被引次数: 0
出版历程
  • 收稿日期:  2020-05-11
  • 修回日期:  2020-06-23
  • 网络出版日期:  2020-06-25
  • 刊出日期:  2020-06-24

Suppression of higher diffraction orders using quasiperiodic array of rectangular holes with large size tolerance

doi: 10.11884/HPLPB202032.200117
    基金项目:  National Key Research and Development Program of China (2017YFA0206001); National Natural Science Foundation of China (11805179)
    作者简介:

    Wei Lai (1983—), male, PhD candidate, engaged in X-ray optics and plasma diagnostics; future718@yeah.net

    通讯作者: Cao Leifeng (1967—), male, PhD, engaged in X-ray optics and plasma diagnostics; leifeng.cao@caep.cn
  • 中图分类号: O436.1, O434.13

摘要: 传统光栅的基础研究和应用研究进展一直备受关注。然而,高阶衍射污染使传统光栅获得的光谱纯度受到严重影响。为了抑制高阶衍射贡献,人们提出了许多单级或准单级光栅的设计方案,但它们对高阶衍射的抑制效果不可避免地受到加工精度的限制。提出了一种准周期矩形孔阵列光栅,通过优化矩形孔的概率密度分布函数,获得了比以往设计更大的加工误差宽容度。对这种光栅的衍射特性进行了分析研究。理论计算表明,即使孔径相对误差超过20%,光栅也可以完全抑制二阶、三阶和四阶衍射,五阶衍射效率与一阶衍射效率之比小于0.01%,大大降低了对加工精度的要求。

English Abstract

魏来, 陈勇, 王少义, 等. 大公差宽容度矩形孔准周期阵列实现对高阶衍射的有效抑制[J]. 强激光与粒子束, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117
引用本文: 魏来, 陈勇, 王少义, 等. 大公差宽容度矩形孔准周期阵列实现对高阶衍射的有效抑制[J]. 强激光与粒子束, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117
Wei Lai, Chen Yong, Wang Shaoyi, et al. Suppression of higher diffraction orders using quasiperiodic array of rectangular holes with large size tolerance[J]. High Power Laser and Particle Beams, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117
Citation: Wei Lai, Chen Yong, Wang Shaoyi, et al. Suppression of higher diffraction orders using quasiperiodic array of rectangular holes with large size tolerance[J]. High Power Laser and Particle Beams, 2020, 32: 072002. doi: 10.11884/HPLPB202032.200117
  • Conventional diffraction gratings comprising regular grooves are used to disperse incident light and provide spectral information for many applications over wavelengths ranging from the infrared to SXRs[14]. Generally, spectral unscrambling and monochromatization only need one diffraction order of a grating. However, from the grating equation, at the same incident angle, the nth order (n=2, 3, 4, …) diffraction of the nth order harmonics (λ/n) of the fundamental wavelength λ is diffracted at the same angle as the first diffraction order of λ. This coincidence introduces harmonics overlapping or contamination[56], which is particularly strong for extreme ultraviolet (EUV) and soft X-ray (SXR) range [79] light.

    The ideal sinusoidal grating may be a radical solution, which gives only 0th and ±1st orders of diffraction but has not been realized for EUV and SXR gratings because of limitations in the fabrication technology[10]. Nevertheless, several single-order or quasi-single-order binary diffraction structures have been developed in recent years by a few novel methods. One method is to use periodic arrangements of nanoscale holes array with complex profiles, making structures as the binary sinusoidal grating[10], the zigzag grating[11], the inclined rectangular aperture grating[12], the trapezoidal transmission grating[13] and the hexagonal holes gratings[14]. All of these can suppress the higher order diffractions in some directions, but the size bias of the holes will reintroduce higher order diffractions, especially the 2nd order[12]. Another method introduces quasi-periodical hole arrays rather than grooves to form a grating. It can achieve an approximate integral sinusoidal transmissivity by setting an appropriate probability density distribution function that describes the probability of occurrence of a hole at some coordinate. For example, the quantum-dot-array grating[15] employs a number of holes, which obeys the sinusoidal distribution function in each period and can theoretically suppress all the higher order diffractions. However, the size of the processed holes is close to 1/50 of the period to avoid overlapping for this distribution function is continuous, which greatly increase data volume and processing time with electron beam lithography tools. To decrease the data volume, the spectrum photon sieve[16-18] uses holes with the size close to half of the grating period and a non-continuous rectangular distribution function to obtain an approximate integral sinusoidal transmissivity. The size bias of the holes restrains the inhibitive effects on the ±3rd order diffractions[1920].

    From the above analysis, the key to a single-order or quasi-single-order grating is selecting appropriate holes and their probability density distribution function. The machined holes are mainly affected by the data volume[2021] or size deviation[12], making it difficult to meet the design requirements. It is shown that any fine optimization of the hole design is futile. Therefore, the probability density distribution function will be the only optimization parameter. In this paper, we propose a non-continuous trapezoidal probability density distribution function, which is a higher order approximation of a sinusoidal function and theoretically suppresses the ±2nd, ±3rd, and ±4th orders by the statistical effect of the distribution, no matter what size of the holes. Besides, its non-continuity makes the maximum size of the holes at the period direction reach 1/6 of the period. Therefore, it yields both larger process tolerance in the size and lower data volume that greatly benefit the processing and application in spectrum measuring and unscrambling in the waveband distributed across the infrared to the X-ray regions.

    • For a metal membrane containing an array with a number of identical holes, the Fraunhofer diffraction intensity distribution is

      $$I(p,q){\rm{ = }}{\left| {{\rm{FT}}\left[ {g(\xi ',\eta ')} \right]} \right|^2} \otimes {\left| {{\rm{FT}}\left( {{L_n}} \right)} \right|^2}$$ (1)

      where $ \otimes $ denotes convolution, px/z, qy/z, z denotes the distance between the plane of the array and the plane of the diffraction pattern, which are denoted by (x, y), $g(\xi ',\eta ')$ represents the binary amplitude transmission of a single hole, ${L_n}$ the distribution function of the coordinates (ξn, ηn) of the centers of the holes, and FT(·) the Fourier transform. Normally, ${L_n}$ is an exact two-dimensional periodic array of δ-functions corresponding to a rectangular lattice:$\left( {{\xi _n},{\eta _n}} \right){\rm{ = }}\left( {h{d_1},l{d_2}} \right)$, where h denotes the ordinal value of the rows of the lattices ($h = 0, \pm 1, \pm 2, \cdot \cdot \cdot , \pm H$), l the ordinal value of the columns ($l = 0, \pm 1, \pm 2, \cdot \cdot \cdot , \pm L$); d1 and d2 denote the periods of the array in the ξ and η directions, respectively. The total number of holes is N=(2H+1)(2L1). A given high order of diffraction cannot be suppressed completely by optimizing only $g(\xi ',\eta ')$ because fabrication errors are unavoidable.

      However, ${L_n}$ may be generalized to a two-dimensional periodic array of arbitrary identical positive functions.

      $$ {L_n} = \frac{{\rm{1}}}{{N\displaystyle\iint\limits_{\left( {{\xi }_{n}},{{\eta }_{n}} \right)}{{\mathit{\Gamma}} \left( {{\xi }_{n}},{{\eta }_{n}} \right){\rm{d}}{{\xi }_{n}}{\rm{d}}{{\eta }_{n}}}}} {\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right) \otimes \sum\limits_{(h,l)} \delta \left( {{\xi _n}{\rm{ - }}h{d_1},{\eta _n} - l{d_2}} \right) $$ (2)

      where ${\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right)$ is the distribution function over one period and $\delta \left( {{\xi _n},{\eta _n}} \right)$ is a δ-function (Fig. 1). The probability density distribution ${L_n}$ for one hole satisfies

      Figure 1.  Schematic of the distribution of holes in the array: from δ-function to arbitrary positive function ${\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right)$

      $$\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{L_n}{\rm{d}}{\xi _n}{\rm{d}}{\eta _n}} } = 1$$ (3)

      The square of its Fourier transform is

      $$\!\!{\left| {{\rm{FT}}\left( {{L_n}} \right)} \right|^2}{\rm{ = }}N{\rm{ + }}N{\rm{(}}N{\rm{ - 1)}}\!\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } \!{{L_{nm}}} {C_n}C_m^*{\rm{d}}{\xi _n}{\rm{d}}{\eta _n}{\rm{d}}{\xi _m}{\rm{d}}{\eta _m}} } } \!\!\!\!\!\!\!\!\!\!\!\!$$ (4)

      where ${C_n} = \exp \left[ {{\rm{i}}k\left( {{\xi _n}p + {\eta _n}q} \right)} \right]$, k denotes the wave number, and nm. Cn and $C_m^*$ are not independent variables, hence we must use the joint distribution function Lnm of (ξn, ηn) and (ξm, ηm), which reads

      $$ \;\!{L_{nm}} \!=\!\frac{{\rm{1}}}{N(N-1){{\left( \displaystyle\iint\limits_{\left( {{\xi }_{n}},{{\eta }_{n}} \right)}{\!{\mathit{\Gamma}} \left( {{\xi }_{n}},{{\eta }_{n}} \right){\rm{d}}{{\xi }_{n}}{\rm{d}}{{\eta }_{n}}} \right)}^{2}}} \left( {{\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right) \otimes \!\!\sum\limits_{(h,l)} \delta \left( {{\xi _n} - h{d_1},{\eta _n} - l{d_2}} \right)} \right)\! \left( {{\mathit{\Gamma}} \left( {{\xi _m},{\eta _m}} \right) \otimes\! \!\sum\limits_{(h',l') \ne (h,l)} \delta \left( {{\xi _m} - h'{d_1},{\eta _m} - l'{d_2}} \right)} \right)\!\!\!\!\!\!$$ (5)

      where the coordinates of (h, l) and (h′, l′) are different in each term. The mean diffracted intensity distribution yields[19]

      $$\overline {I\left( {p,q} \right)} = {I_0}\left( {p,q} \right)\left[ {N + {B^2}\left( {{D^2} - N} \right)} \right]$$ (6)
      $$ \left\{ \begin{array}{l} {I_0}\left( {p,q} \right) = {\left| {{\rm{FT}}\left[ {g(\xi ',\eta ')} \right]} \right|^2} \; \\ B = {\rm{FT}}\left[ {{\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right)} \right]\Bigg/ {\displaystyle\iint\limits_{\left( {{\xi _n},{\eta _n}} \right)} {{\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right){\rm{d}}{\xi _n}{\rm{d}}{\eta _n}}} \; \\ D = \dfrac{{\sin \left[ {(2H + 1)k{d_{\rm{1}}}p/2} \right]}}{{\sin (k{d_{\rm{1}}}p/2)}} \dfrac{{\sin \left[ {(2L + 1)k{d_{\rm{2}}}q/2} \right]}}{{\sin (k{d_{\rm{2}}}q/2)}} \end{array} \right. $$ (7)

      The first term in the bracket of Eq. (6) represents a linear superposition of the diffraction intensity from all holes; the second involves an interference factor between all pairs of holes. Note that, unlike a traditional grating, there is a modulation factor B that gives another degree of freedom to suppress the higher orders of diffraction rather than ${I_0}(p,q)$. Clearly, if ${\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right)$ is a sinusoidal function, the diffraction pattern retains only the 0th and ±1st orders. Because the sinusoidal function is a continuous distribution function, $g(\xi ',\eta ')$ must be a δ-function to avoid holes overlapping. However, a large number of small holes (corresponding to δ-functions) which are reduced considerably in dimension from that of one period, are very difficult to fabricate, especially for the EUV and SXR gratings. The next best solution is to suppress the higher orders adjacent to the ±1st orders, which have a real impact on the performance of grating spectrometers and monochromators.

    • It is noted that an amplitude grating with a duty cycle of 1/2 produces only 0th and odd orders of diffraction, while an amplitude grating with a 1/3 duty cycle eliminates ±3tth orders of diffraction (t=1, 2, 3,…). Setting the normalized distribution function for one period as

      $${\mathit{\Gamma}} \left( {{\xi _n},{\eta _n}} \right){\rm{ = }}{\mathit{\Gamma}} \left( {{\eta _n}} \right) {\mathit{\Gamma}} \left( {{\xi _n}} \right){\rm{ = }}\delta \left( {{\eta _n}} \right) \left[ {{\rm{rect}}\left( {{\rm{2}}{\xi _n}/{d_1}} \right) \otimes {\rm{rect}}\left( {3{\xi _n}/{d_1}} \right)} \right]$$ (8)

      and substituting it into Eq. (7), we obtain the modulation factor

      $$B = {\rm{sinc}} \left( {{d_1}p/3\lambda } \right){\rm{sinc}} \left( {{d_1}p/2\lambda } \right)$$ (9)

      From Eq. (6), this design clearly suppresses the ±2nd, ±3rd, and ±4th orders of diffraction along the p-axis of the diffraction plane, regardless of the hole shape and size. In this instance, reducing the requirements of processing precision is very beneficial.

      Indeed, the distribution function of Eq. (8) is a trapezoidal function. Fig. 2 gives one example, the design consists of a number of rectangular holes with side lengths d/6 and d (d1d2d). Each hole is randomly positioned in a square lattice (d×d square), and the central location (ξn, ηn) of the holes obey the probability density distribution function ${\mathit{\Gamma}} \left( {{\xi _n}} \right)$ in Eq. (8). In every lattice, there is only one hole.

      Figure 2.  (a) Design for a quasi-periodical hole array: in each lattice (d×d square), a rectangular hole with side lengths of d/6 and d is positioned randomly on the ξn axis and has a trapezoidal distribution profile shown in (b)

      From Eqs. (6), (7), and (9), the mean diffracted intensity peak for each order obtains

      $$\overline {I(m\lambda /d,0)} = \left\{ {\begin{array}{*{20}{l}} {{N^2}{I_0}(0,0),}&{m = 0}\\ {\left[ {N + N\left( {N - 1} \right)\dfrac{{{\rm{3}}\sqrt {\rm{3}} }}{{{m^{\rm{4}}}{{\rm{{\text{π}} }}^2}}}} \right]{I_0}(m\lambda /d,0),}&{m = \pm (6t \pm 1),t = 1,2,3, \cdots }\\ {N{I_0}(m\lambda /d,0),}&{m = \pm 2t, \pm 3t,t = 1,2,3, \cdots } \end{array}} \right.$$ (10)

      where m denotes the diffraction order. The 0th order has the same intensity as that of a periodic grating with a duty cycle of 1∶5; the diffraction intensity of the 1st order is about 52.7% of that of a periodic grating with a duty cycle of 1∶5 when $ N \gg 1 $, and there are no diffraction peaks for the ±2tth and ±3tth orders. The ratio of the diffraction efficiency of the nearest higher order (5th order) to that of the 1st order is found to be

      $$\overline {I(5\lambda /d,0)} {\rm{/}}\overline {I(\lambda /d,0)} {\rm{ = \frac{1}{156\;25}}}$$ (11)

      which represents a decrease by 625 times than that of a periodic grating with a duty cycle of 1∶5 when N$\gg $1.

    • The intensity distributions based on the calculated results of Eq. (6) and the numerical results from Eq. (1) (Fig. 3(a) and (b), respectively) indicate that our model, Eqs. (6)–(10), agrees well with the numerical results. Compared with a periodic grating with a duty cycle of 1∶5 under identical conditions (Fig. 3(c) and (d)), the ±2nd to ±4th orders of diffraction seem to vanish and the ±1st orders attenuate significantly, meaning that part of the energy of incident light is diffracted elsewhere. The distinctive background intensity distributions (the two “wings” seen in Fig. 3(a) and (b)) confirm this well. Its hollow structure is a typical characteristic of the diffraction pattern from a quasi-periodical array.

      Figure 3.  Comparison between far-field diffraction patterns of a quasi-periodical array shown in Fig. 2 when HL=100 and a periodic grating with a duty cycle of 1∶5 and 201 periods

      For a quasi-periodical structure, intensity fluctuations in its diffraction pattern are inevitable. The numerical results (Fig. 3(b)) obtained using Eq. (1) show the fluctuations appearing as speckles in the two “wings”. In addition, there are also many background fluctuations between different diffraction peaks in the profile (Fig. 3(d)) across the p-axis in the numerical results obtained using Eq. (1). Their mean value accords with the theoretical value of Eq. (6), which is comparable to the background level of a periodic grating.

      Fig. 4(a) shows the intensity profiles of Fig. 3(a)(c) across the line $q=\lambda /\left( {2}d \right)$(indicated by a yellow dashed line in Fig. 3). The background intensity of a quasi-periodical array is much higher than that of a periodic grating, verifying that the diffracted energy from the higher orders is transferred to the background and increases the mean background level. However, the p-axis is the direction that is applied in the spectroscopic measurements, where there is no significant increase in the background intensity.

      Figure 4.  (a) Intensity profiles of Fig. 3(a)(c) across the line qλ/2d (indicated by yellow dashed lines in Fig. 3); (b) numerical results of Eq. (1) across the p-axis for the rectangular holes with side lengths of d/5 and d (red dot), and 2d/15 and d (black line).

      The size bias of a microstructure has a large impact on the suppression of higher orders. The process bias of an inclined rectangular aperture grating needs to be controlled within ±9% of the target value to ensure the suppression ratio is less than 1/20. This precision is very difficult for electron beam lithography technology[12]. To verify the process tolerance of our design on the suppression of higher order diffractions, Fig. 4(b) shows the effects of process errors on this design, where the side lengths in the direction of the periods of the rectangular pinholes are 120% and 80% of the design values in Fig. 2. This quasi-periodical array still restrains the 2nd to 4th orders, and the ratio of the diffraction efficiency of the 5th order diffraction to that of the 1st order is always below 0.01%. The elimination of the ±2tth and ±3tth (t=1, 2, 3 …) order diffractions produced by the design is a statistical effect of distribution of a number of holes, being independent of the process accuracy of the individual hole size. Accordingly, the design gains a higher tolerance to process errors.

    • For higher-order diffraction suppression, a quasi-periodical array of rectangular holes has been proposed displaying a larger process tolerance than that previously reported. A general analytical description has been established to describe the diffraction properties of the quasi-periodical holes array using an arbitrary periodic probability density distribution function, as well as to reveal the physical mechanism behind the effective suppression of the higher orders and large process tolerance of the design. Its binary structure easily accommodates infrared to X-ray light fields. The quasi-periodical array grating has potential applications in devices performing high-accuracy spectral measurements and monochromators as well as a standard for other devices that require high process accuracy.

参考文献 (21)

目录

    /

    返回文章
    返回