## 留言板

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

 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.
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.

## 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

• 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)
###### 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).

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##### 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
###### 1. 同济大学 物理科学与工程学院 精密光学工程研究所，先进微结构材料教育部重点实验室，上海 2000922. 中国工程物理研究院 激光聚变研究中心，等离子体物理重点实验室，四川 绵阳 621900
基金项目:  National Key Research and Development Program of China (2017YFA0206001); National Natural Science Foundation of China (11805179)
###### 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

### English Abstract

 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.
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.
• 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.

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