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高反射率介质反射镜的设计,通常可以采用两种不同折射率的膜层交替实现,从膜系界面上反射的光束因具有相同位相,会发生相长干涉,可以获得很高的反射率[19]。因此,可以采用H和L两种高、低折射率的材料,周期性地叠加作为膜系结构,比如:(H L)^n,n的取值越高则反射率越高。考虑实际镀膜过程中存在膜层的吸收和散射,设计时必须选择合适的膜系层数来实现高反射率。
这样的初始结构较容易实现相应波段的高反射率,但却无法避免P光和S光在宽角度入射下位相差过大的现象。利用等效多层膜理论:对于以中间一层为中心,两边对称安置的多层膜,却具有单层膜特征矩阵的所有特点,推广到任意多层膜组成对称膜系,最终又形成一个等效单层膜[20−21],改变膜系层数和厚度,以获得不同的等效折射率和等效位相,以实现位相调控的目的。
根据以上原理,设计方法如下:选择两种不同折射率材料,H、L分别代表高、低折射率材料,(H L)^n作为基本膜系结构,设计波长以截止带中心位置处于信号通道为准,在基本膜系结构外层加d1L d2H d3L或d1H d2L d3H等效多层位相调控层,考虑初始膜系结构为 G|(H L)^14 |d1H d2L d3H d4L d5H d6L d7H d8L|Air,其中G为基底(石英),Air为空气,d1~d8代表膜层的厚度系数,为非规整位相调控膜层。根据任务指标设定优化目标,采用Global Modified LM或Global Simplex等优化方法进行优化迭代,优化d1~d8,以获得最佳设计结果。
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反射镜根据系统应用要求,技术指标如表1所示。
表 1 反射镜技术指标
Table 1. Technical index of mirror
Technical index Value Target wavelength/nm 780 Range of incidence angle/(°) 45±7.5 Reflectance ≥99% Phase difference/(°) ≤3 根据上述设计思想,选择薄膜材料时需综合考虑反射镜的工作波段,考察薄膜材料的透明区、折射率、吸收系数及基底和薄膜材料之间的应力匹配、牢固度等因素[22−23]。为此,选择Nb2O5(H)和SiO2(L)作为高、低折射率材料。初始膜系结构为G|(HL)^14 d1H d2L d3H d4L d5H d6L d7H d8L|Air,设计波长890 nm,H的折射率为2.22@890 nm,L的折射率为1.44@890 nm。将初始膜系结构代入光学薄膜设计软件Filmwizard,在优化目标设定时,按照任务要求,分别在37.5°、45°和52.5°三个角度下设置优化目标参数,使得优化结果可以满足入射范围45°±7.5°的任务要求。优化目标设定780 nm处反射率>99%,位相差<3°。通过不断优化每层薄膜的厚度系数,以实现评价函数最小、光谱性能最佳的结果。最终设计结果为:(H L)^14 0.231 H 0.935 L 1 H 1.15 L 1.122 H 1.043 L 1.051 H 0.907 L,设计波长890 nm。设计结果如图2所示。
图 2 位相调控反射镜反射率设计曲线(a)及位相差设计曲线(b)
Figure 2. Reflectance design curve (a) and phase difference design curve (b) of phase control mirror
表 2 目标波长(780 nm)下设计膜系反射率及位相差结果
Table 2. Design result of reflectance and phase difference at target wavelength (780 nm)
Incidence angle/(°) Reflectance Phase difference/(°) 37.5 99.62% 0.22 45 99.57% 0.34 52.5 99.46% 1.09 从表2可以看出,设计能达到预期目标,在目标波长处,45°±7.5°入射范围内实现了99.4%以上的高反射率,且位相差控制在1.1°以内。虽然从图2(b)中可以看出位相差最佳位置在790 nm处,但在大于790 nm时,52.5°入射下位相差急剧变化。考虑研制时不可避免的工艺误差,实际光谱位置会有偏差,结合光谱定位容差考虑,该设计结果是合理可行的。
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设计膜系的最后八层为非规整膜系,对监控精度要求很高。为此,利用Optilayer对设计膜系进行了膜层误差引起光谱变化的灵敏度分析。
如图3所示,膜系最外层的灵敏度较大,在研制过程中需要进行精确控制。对位相调控膜系引入1%的随机误差,做进一步分析,观察其对位相差的影响,引入误差后的位相差曲线如图4所示。从表3看出,在37.5°、45°和52.5°三个角度下,780 nm处的位相差小于2.5°,可以满足反射镜需将位相差控制在3°以内的任务要求。因此,可以依据上述设计膜系开展实际的研制工作。
图 4 37.5°(a)、45°(b)、52.5°(c)入射下位相差误差分析曲线
Figure 4. Phase difference error analysis curves under 37.5° incidence (a), 45° incidence (b), 52.5° incidence (c)
表 3 目标波长(780 nm)下反射镜位相差误差分析结果
Table 3. Phase difference error analysis at target wavelength (780 nm)
Incidence angle/(°) Error +1% Error 0% Error −1% 37.5 0.51 0.22 0.017 45 1.326 0.34 0.654 52.5 2.14 1.09 0.291 -
该反射镜的制备是在德国莱宝公司生产的Lab900-plus型真空镀膜机上完成的,设备配有两把e型电子枪,SiO2采用环形坩埚,Nb2O5采用七穴坩埚。同时,配有四探头石英晶振物理厚度控制系统、OMS5100光学自动控制系统和栅网口径为12 cm的Veeco射频(RF)离子源。样品为Φ50×5 mm的石英基片,研制时参考量子卫星保偏金属反射镜研制工艺参数[14−15],具体如下:当真空到达6×10−2 Pa时,打开工件旋转,设定烘烤温度为200 ℃,保温2 h,真空到达2×10−3 Pa时,打开离子源,充入O2 后进行离子辅助薄膜沉积。工艺参数如表4所示。
表 4 沉积工艺参数
Table 4. Deposition process parameters
Parameter Value Nb2O5 SiO2 Rate/nm·s−1 0.15 0.8 EB oxygenation/sccm 40 30 RF bias voltage/V 600 600 RF discharged current/mA 500 500 Background pressure/Pa 2×10−3 2×10−3 Baking temperature/℃ 200 200 在研制过程中,首先尝试使用光学极值百分比监控策略 (POEM)进行29~36非规整膜层的监控,但效果并不理想。分析其原因,当采用该监控策略时,监控波长选择时需尽可能使监控停止位置距离下一个极值点保持4%×(Tmax−Tmin) 以上距离[24],以减小监控误差。结合设计膜系,因为29层厚度较薄,导致部分膜层未能找到合适的监控波长,使其判停点处于4%×(Tmax−Tmin)区间,因此产生镀制误差。而与此同时,对坩埚材料蒸发消耗量变化、晶振参数修正等方面进行膜厚分析及改进后,提高了晶振控制的精度,通过实验结果发现,29~36层采用晶振监控方式研制出的位相调控反射镜可满足任务要求。因此,最终选择1~28层光学监控、29~36层晶振监控的监控方式。
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反射镜的反射率光谱曲线是在美国 PE 公司生产的 Lamda900型分光光度计上完成的,其配有V-W反射率测试附件,可以进行30°~60°入射下的反射率测试;位相差测试是在美国J. A. WOOLLAM 公司生产的 VASE-32可见近红外椭偏仪上完成的,其可以进行15°~90°入射下的位相差测试,测试原理如图5所示。
$$ \mathrm{\rho }=\frac{{{r}}_{\mathrm{P}}}{{{r}}_{\mathrm{S}}}=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{\psi }\mathrm{e}\mathrm{x}\mathrm{p}\left({i}\mathrm{\Delta }\right)$$ (1) $$ \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{\psi }=\frac{\left|{{r}}_{\mathrm{P}}\right|}{\left|{{r}}_{\mathrm{S}}\right|}\mathrm{\Delta }={\mathrm{\delta }}_{1}-{\mathrm{\delta }}_{2}$$ (2) 式中:$\psi $和Δ为椭偏仪能够测量的样品椭偏参数,tan$\psi $为反射光P光与S光的振幅比,Δ为P光与S光的位相差。测试结果如图6所示。
表 5 目标波长(780 nm)下反射镜反射率及位相差测试结果
Table 5. Test result of reflectance and phase difference at target wavelength (780 nm)
Incidence angle/(°) Reflectance Phase difference/(°) 37.5 99.60% 0.46 45 99.47% 0.36 52.5 99.36% 2.83 研制结果如表5所示,反射镜在入射角度为37.5°、45°和52.5°时,780 nm处反射率大于99.3%,且位相差小于3°,满足任务要求。
最终研制的反射镜其反射率曲线和位相差虽然都满足任务要求,但与设计结果存在一定的差异。主要是由以下两个原因造成的:1) 非规整膜层监控有误差,导致制备结果出现偏差;2)研制时所用监控参比片和试验件厚度为1 mm,与正式产品厚度的5 mm差异较大,导致薄膜沉积后性能产生偏差。
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按照航天产品环境模拟实验要求,对反射镜进行了可靠性及牢固度实验,实验结果见表6。实验项目如下[15]。
1) 附着力实验:采样标准聚酯胶带紧贴样品膜层表面,沿膜层表面垂直拉起,观察膜层是否脱落。
2) 浸泡实验:在45±1 ℃ 的水中浸泡8 h,观察膜层是否脱落。
3) 温度交变实验:在50±1 ℃环境中保持 1 h,然后在−25±1 ℃ 环境中保持1 h,循环三次,观察膜层是否脱落。
4) 湿热实验:在相对湿度 95% 、温度45±2 ℃的大气中保持24 h,观察膜层是否脱落。
表 6 反射镜可靠性及牢固度实验结果
Table 6. Reliability and durability results of mirror
Experiment Result Adhesion experiment Pass Soaking experiment (45±1 ℃) Pass High (50±1 ℃) and low (−25±1 ℃) temperature experiment Pass Temperature (45±2 ℃) and humidity (95%) experiment Pass
Design and fabrication of wide angle phase control mirror
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摘要: 宽角度位相调控反射镜是下一代中高轨量子通信系统中的核心元件,用于宽角度范围内信号光的高效传递与偏振态的精确调控。基于等效多层膜理论,采用介质反射膜堆加非规整位相调控膜的膜系结构,设计了一种宽角度位相调控反射镜。选择Nb2O5和SiO2分别作为高、低折射率薄膜材料,通过误差分析,优化沉积工艺,采用电子束蒸发结合离子辅助沉积的方式,在德国莱宝Lab900-plus设备上制备出该薄膜元件。研制结果表明,反射镜在780 nm处、45°±7.5°入射范围内,其反射率大于99.3%,位相差控制在3°以内,满足量子通信系统的反射率及位相差控制要求,且通过相关环境模拟实验,满足可靠性要求,为该类偏振调控薄膜元件在下一代中高轨量子卫星中的工程应用打下了坚实的基础。Abstract:
Objective In quantum communication, such as quantum key distribution, quantum entanglement, and quantum teleportation, photons are controlled to a specific polarized direction for transmission and decoding. There is a need to establish an effective and stable link of quantum channels between the transmitter and receiver to maintain max channel efficiency and reduce quantum bit error rate. Therefore, it is necessary to do phase control on the thin film optical components in the optical system. In published literatures, most phase control mirrors are designed with a single incidence angle (such as 10°, 22.5°, 35° and 45°), which cannot meet the requirement of optical-mechanics system used in new generation of quantum communication working at medium to high orbit any longer. This research developed a dielectric reflector with high reflectivity and wide angle range for efficient energy transfer and precise phase control, which has wide application prospects for this type of thin film component in next generation of quantum communication systems working at medium to high orbit. Methods The design method is as below. Two materials with different refractive index are selected, the values of which are referred as H and L respectively. (HL) ^ n is used as the basic film system mechanism, and the design value of wavelength is determined based on the signal channel at the center of the cutoff band. Multiple layers equivalent to d1L d2H d3L or d1H d2L d3H for phase control are applied on the surface layer of the film base structure. The initial film structure is: G | (HL) ^ 14 d1H d2L d3H d4L d5H d6L d7H d8L | Air, where G is the substrate , d1-d8 are the thickness coefficients of each film layer respectively. Optimal goals are set based on task indicators, and optimization algorithms are used such as Global Modified LM or Global Simplex for optimal iterations to change film thickness to obtain the best design result. The preparation of this product was completed on Lab900-plus vacuum deposition machine produced by Leybold Company in Germany. The equipment is equipped with two e-type electron guns, with SiO2 using a circular crucible and Nb2O5 using a seven hole crucible. Equipped with a 4-probe quartz crystal oscillator physical thickness control system, OMS5100 optical automatic control system, and a Veeco RF ion source with a grid aperture of 12 cm. The sample is Φ 50 ×5 mm quartz substrate. Results and Discussions During the deposition, precisely controlling the layer of high sensitivity in the film system is needed. So the film thickness fitting analysis is carried out in terms of the evaporation consumption of the crucible material and crystal oscillator parameter correction, which ensured the success of development in the end. The results show that the reflectivity of the reflector reaches over 99.3% at 780 nm at incidence angles of 37.5°, 45°and 52.5°, and the phase difference is less than 3° (Fig.6), meeting the task requirements. It also passed the environmental reliability and firmness tests (Tab.6). Conclusions The wide angle phase control mirror uses Nb2O5 and SiO2 as high and low refractive index materials, quartz as the substrate, and a combination film system of high reflectivity film layer and multi layer phase control film as the initial film system. Optimization algorithms such as Global Modified LM or Global Simplex are used to design high reflectivity and phase control mirror in wide angle range. On Lab900-plus, the equipment of Leybold in Germany, by combining electron beam evaporation with Veeco RF ion source assisted deposition, deposition process was optimized. By combining optical monitoring and crystal oscillator monitoring, the development was successful. The results show that at 780 nm, with an incidence angle range of 45° ± 7.5°, the reflectivity is greater than 99.3%, and the phase difference is controlled within 3°. The product passed environmental reliability tests. The development of this product can enable the design of quantum communication optical systems with a wide angle field of view. How to ensure the product performance over the lifetime is the focus of next stage of work. -
Key words:
- optical thin film /
- phase control /
- mirror /
- polarization /
- wide angle
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表 1 反射镜技术指标
Table 1. Technical index of mirror
Technical index Value Target wavelength/nm 780 Range of incidence angle/(°) 45±7.5 Reflectance ≥99% Phase difference/(°) ≤3 表 2 目标波长(780 nm)下设计膜系反射率及位相差结果
Table 2. Design result of reflectance and phase difference at target wavelength (780 nm)
Incidence angle/(°) Reflectance Phase difference/(°) 37.5 99.62% 0.22 45 99.57% 0.34 52.5 99.46% 1.09 表 3 目标波长(780 nm)下反射镜位相差误差分析结果
Table 3. Phase difference error analysis at target wavelength (780 nm)
Incidence angle/(°) Error +1% Error 0% Error −1% 37.5 0.51 0.22 0.017 45 1.326 0.34 0.654 52.5 2.14 1.09 0.291 表 4 沉积工艺参数
Table 4. Deposition process parameters
Parameter Value Nb2O5 SiO2 Rate/nm·s−1 0.15 0.8 EB oxygenation/sccm 40 30 RF bias voltage/V 600 600 RF discharged current/mA 500 500 Background pressure/Pa 2×10−3 2×10−3 Baking temperature/℃ 200 200 表 5 目标波长(780 nm)下反射镜反射率及位相差测试结果
Table 5. Test result of reflectance and phase difference at target wavelength (780 nm)
Incidence angle/(°) Reflectance Phase difference/(°) 37.5 99.60% 0.46 45 99.47% 0.36 52.5 99.36% 2.83 表 6 反射镜可靠性及牢固度实验结果
Table 6. Reliability and durability results of mirror
Experiment Result Adhesion experiment Pass Soaking experiment (45±1 ℃) Pass High (50±1 ℃) and low (−25±1 ℃) temperature experiment Pass Temperature (45±2 ℃) and humidity (95%) experiment Pass -
[1] 龚威, 史硕, 陈必武. 对地观测高光谱激光雷达发展及展望 [J]. 遥感学报, 2021, 25(1): 501-513. doi: 10.11834/jrs.20210086 Gong Wei, Shi Suo, Chen Biwu. Development and prospect of hyperspectral LiDAR for earth observation [J]. National Remote Sensing Bulletin, 2021, 25(1): 501-513. (in Chinese) doi: 10.11834/jrs.20210086 [2] 周煜东. 基于LD的780nm高光谱分辨率激光雷达系统设计[D]. 西安: 西安理工大学, 2020. Zhou Yudong. Design of 780 nm high spectral resolution lidar based on laser diode[D]. Xi’an: Xi'an University of Technology, 2020. (in Chinese) [3] 韩慧鹏. 国外卫星激光通信进展概况 [J]. 卫星与网络, 2018(8): 44-49. Han Huipeng. Foreign satellite laser communication progress overview [J]. Satellite & Network, 2018(8): 44-49. (in Chinese) [4] 白帅, 王建宇, 张亮, 等. 空间光通信发展历程及趋势 [J]. 激光与光电子学进展, 2015, 52(7): 070001-1. Bai Shuai, Wang Jianyu, Zhang Liang, et al. Development progress and trends of space optical communications [J]. Laser & Optoelectronics Progress, 2015, 52(7): 070001. (in Chinese) [5] 姜会林, 付强, 赵义武, 等. 空间信息网络与激光通信发展现状及趋势 [J]. 物联网学报, 2019, 3(2): 1-8. Jiang Huilin, Fu Qiang, Zhao Yiwu, et al. Development status and trend of space information network and laser communication [J]. Chinese Journal on Internet of Things, 2019, 3(2): 1-8. (in Chinese) [6] 吴金才, 何志平, 舒嵘, 等. 偏振光学系统中相位延迟机理及其应用 [J]. 光子学报, 2013, 42(1): 84-88. doi: 10.3788/gzxb20134201.0084 Wu Jincai, He Zhiping, Shu Rong, et al. Mechanism and application of phase shift in the polarized optical systems [J]. Acta Photonica Sinica, 2013, 42(1): 84-88. (in Chinese) doi: 10.3788/gzxb20134201.0084 [7] Wu Jincai, Zhang Liang, Jia Jianjun, et al. Polarization-maintaining design for satellite-based quantum communication terminal [J]. Optics Express, 2020, 28(8): 10746-10759. doi: 10.1364/OE.387574 [8] Wu Jincai, He Zhiping, Zhang Liang. Polarization study about a telescope-based transmitter for quantum communication [J]. Applied Optics, 2017, 56(30): 8501-8506. doi: 10.1364/AO.56.008501 [9] 庞武斌, 岑兆丰, 李晓彤, 等. 偏振对光学系统成像质量的影响 [J]. 物理学报, 2012, 61(23): 234202-1. doi: 10.7498/aps.61.234202 Pang Wubin, Cen Zhaofeng, Li Xiaotong, et al. The effect of polarization light on optical imaging system [J]. Acta Physica Sinica, 2012, 61(23): 234202. (in Chinese) doi: 10.7498/aps.61.234202 [10] 黄宏, 孙英杰, 孙兵. 大角度激光通信带通滤光片的研究 [J]. 光电技术应用, 2020, 35(1): 35-40. Huang Hong, Sun Yingjie, Sun Bing. Research on band-pass filter with large angle for laser communication [J]. Electro-Optic Technology Application, 2020, 35(1): 35-40. (in Chinese) [11] 陈少杰, 张亮, 吴金才, 等. 空间激光通信中精跟踪系统的实现与优化 [J]. 红外与毫米波学报, 2018, 37(1): 35-46. Chen Shaojie, Zhang Liang, Wu Jincai, et al. Realization and optimization of fine tracking system of free space laser communication [J]. Journal of Infrared and Millimeter Waves, 2018, 37(1): 35-46. (in Chinese) [12] 马小凤, 王丹, 刘定权, 等. 利用等效层的消偏振宽带减反膜设计 [J]. 光学学报, 2007, 27(3): 563-566. Ma Xiaofeng, Wang Dan, Liu Dingquan, et al. Design of non-polarizing broadband antireflection coating using equivalent layer [J]. Acta Optica Sinica, 2007, 27(3): 563-566. (in Chinese) [13] 李大琪, 于天燕, 陈刚, 等. 0.55~0.85μm波段增透膜的相位调控设计与研制 [J]. 光学学报, 2016, 36(7): 0731001G-1. doi: 10.3788/AOS201636.0731001 Li Daqi, Yu Tianyan, Chen Gang, et al. Design and fabrication phase modulated antireflection coatings in 0.55~0.85μm waveband [J]. Acta Optica Sinica, 2016, 36(7): 0731001. (in Chinese) doi: 10.3788/AOS201636.0731001 [14] 段微波, 李大琪, 刘保剑, 等. 空间原子氧对保偏反射镜偏振对比度的影响 [J]. 光学学报, 2018, 38(11): 1131001-1. doi: 10.3788/AOS201838.1131001 Duan Weibo, Li Daqi, Liu Baojian, et al. Effect of spatial atomic oxygen on polarization contrast of polarization maintaining mirror [J]. Acta Optica Sinica, 2018, 38(11): 1131001. (in Chinese) doi: 10.3788/AOS201838.1131001 [15] 余德明, 段微波, 李大琪, 等. 偏振和相位调控反射镜的设计与制备 [J]. 光学学报, 2020, 40(8): 1531001-1. Yu Deming, Duan Weibo, Li Daqi, et al. Design and fabrication of polarization and phase modulated mirror [J]. Acta Optica Sinica, 2020, 40(8): 1531001. (in Chinese) [16] Ma Chong, Chen Gang, Liu Dingquan, et al. Polarization maintaining dichroic beam-splitter and its surface shape control by back side AR coating [J]. Current Optics and Photonics, 2021, 5(5): 1-7. [17] 尹欣, 刘定权, 段微波, 等. 近红外波段偏振编码用分色片的设计与制作 [J]. 红外与毫米波学报, 2012, 31(6): 505-509. Yin Xin, Liu Dingquan, Duan Weibo, et al. Design and fabrication of near-infrared dichroic beam-splitter for polarization state coding [J]. Journal of Infrared and Millimeter Waves, 2012, 31(6): 505-509. (in Chinese) [18] 潘永刚, 张四宝, 刘政, 等. 偏振和位相调控分光膜的设计与制备 [J]. 红外与激光工程, 2022, 51(5): 20210512-1. doi: 10.3788/IRLA20210512 Pan Yonggang, Zhang Sibao, Liu Zheng, et al. Design and fabrication of polarization and phase modulated beam splitter [J]. Infrared and Laser Engineering, 2022, 51(5): 20210512. (in Chinese) doi: 10.3788/IRLA20210512 [19] 唐晋发, 顾培夫, 刘旭. 现代光学薄膜技术 [M]. 杭州: 浙江大学出版社, 2006. [20] 廖延彪. 偏振光学 [M]. 北京: 科学出版社, 2003. [21] 张金豹, 史成浡, 耿浩, 等. 基于等效折射率法的短波通滤光膜制备技术 [J]. 新技术新工艺, 2022(3): 36-40. Zhang Jinbao, Shi Chengbo, Geng Hao, et al. Preparation technology of short wave pass filter film based on method of equivalent refractive index [J]. New Technology & New Process, 2022(3): 36-40. (in Chinese) [22] 刘保剑, 段微波, 李大琪, 等. 退火温度对 Ta2O5/SiO2 多层反射膜结构和应力特性的影响 [J]. 物理学报, 2019, 68(11): 114208-1. doi: 10.7498/aps.68.20182247 Liu Baojian, Duan Weibo, Li Daqi, et al. Effect of annealing temperature on structure and stress properties of Ta2O5/SiO2 multilayer reflective coatings [J]. Acta Physica Sinica, 2019, 68(11): 114208. (in Chinese) doi: 10.7498/aps.68.20182247 [23] 袁文佳, 章岳光, 沈伟东, 等. 离子束溅射制备Nb2 O5光学薄膜的特性研究 [J]. 物理学报, 2011, 60(4): 047803. doi: 10.7498/aps.60.047803 Yuan Wenjia, Zang Yueguang, Shen Weidong, et al. Characteristics of Nb2O5 thin films deposited by ion beam sputtering [J]. Acta Physica Sinica, 2011, 60(4): 047803. (in Chinese) doi: 10.7498/aps.60.047803 [24] 周晟, 刘定权, 王凯旋, 等. 中短波红外双带通低温滤光片的设计与制备 [J]. 红外与激光工程, 2022, 51(9): 20210964-1. doi: 10.3788/IRLA20210964 Zhou Sheng, Liu Dingquan, Wang Kaixuan, et al. Design and fabrication of short and middle wavelength infrared dual band-pass filter at cryogenic temperature [J]. Infrared and Laser Engineering, 2022, 51(9): 20210964. (in Chinese) doi: 10.3788/IRLA20210964