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分束器是一个重要的衍射光学元件,在干涉仪、光子晶体等领域都有着广泛的应用。分束器有着高衍射效率和高均匀性的要求。近年来,人们对衍射光分束器进行了广泛研究,如Lingwei Guo等人提出了一种大型高效的1×9达曼光栅分束器[1];Dong Cheon Kim等人使用伴随态方法[2]优化光学元件的拓扑结构,得出高性能、多功能的广角衍射分束器[3];Jinzhe Li等人提出了新的矢量迭代傅里叶变换算法(IFTA)来实现准连续超表面分束器的设计[4]。
为了满足分束器在高衍射效率和高均匀性方面的要求,文中基于全局拓扑优化的深度学习模型[5]对超构光栅分束器展开了一系列的研究。基于全局拓扑优化的深度学习模型的结构如图1(a)所示,由两个密集连接层和四个转置卷积层依次连接而成,其输出结果为由100个超构光栅分束器的设计参数构成的张量
${{n}}$ ,用以表示超构光栅每个周期的折射率分布;输入的参数为入射高斯光束的波长$ \lambda $ 和分束光的偏折角$ \theta $ ,以及和张量${{n}}$ 大小维度完全相同的随机生成的张量$ {{{\textit{z}}}} $ 。超构光栅分束器由一系列厚度为325 nm的${\rm{Si}}$ 柱子和${{\rm{SiO}}}_{2}$ 衬底组成,如图2(b)所示。超构光栅分束器的衍射效率$ \eta $ 和均匀性$ UE $ 的定义为:图 2 偏折角为150°的超构光栅分束器对应的衍射效率和目标函数值。(a)、(d)超构光栅分束器平均衍射效率和平均目标函数值随训练次数的变化;(b)、(c)训练100次和1000次时超构光栅分束器的衍射效率分布;(e)、(f)训练100次和1000次时超构光栅分束器的目标函数值分布
Figure 2. Diffraction efficiency and target function of metagrating beam splitter with deflection angle 150°. (a), (d) Average diffraction efficiency and the average target function values of the metagrating beam splitter vs training times; (b), (c) Distribution of diffraction efficiency of the metagrating beam splitter after 100 and 1000 times of training; (e), (f) Distribution of target function values of the metagrating beam splitter after 100 and 1000 times of training
$$ \eta ={\eta }_{1}+{\eta }_{-1} $$ (1) $$ UE=\frac{{|\eta }_{1}-{\eta }_{-1}|}{{\eta }_{1}+{\eta }_{-1}} $$ (2) 式中:
$ {\eta }_{1} $ 、$ {\eta }_{-1} $ 为+1级和−1级的衍射效率。根据分束器+1衍射级和−1衍射级的衍射效率,定义了描述超构光栅分束器衍射效率和均匀性的目标函数(即深度学习模型中的损失函数):$$ FoM={\left({\eta }_{1}-{\eta }_{{\rm{tgt}}1}\right)}^{2}+{\left({\eta }_{-1}-{\eta }_{{\rm{tgt}}-1}\right)}^{2} $$ (3) 式中:
${\eta }_{{\rm{tgt}}}$ 为目标衍射效率。在深度学习模型的每一次训练中,$ FoM $ 关于神经网络每一个权重值的偏导数将会调整该权重值,使深度学习模型朝着最小化损失函数的方向优化,从而使超构光栅分束器的两个衍射级的衍射效率趋向于目标衍射效率。设定工作波长为900 nm,分束光的偏折角设置为150°,+1级和−1级的目标衍射效率均设置为50%,训练次数为1000次。深度学习模型的运行结果如图2所示。随着训练次数的增加,由图2(d)可以看出深度学习模型的目标函数减小至趋近于0,而超构光栅分束器的两个衍射级的平均衍射效率均稳定在40%左右,如图2(a)所示。而图2(b)和图2(c)则分别表示当训练次数为100和1000时的100个超构光栅分束器的衍射效率,蓝点表示+1级的衍射效率,红点表示−1级的衍射效率。当训练100次时,超构光栅分束器的衍射效率分布较为散,而当训练1000次时,大部分超构光栅分束器两个衍射级的衍射效率都集中在40%附近,因此具有良好的均匀性。图2(e)和图2(f)则表示当训练次数为100和1000时超构光栅分束器对应的目标函数值分布,当训练完1000次时,超构光栅分束器对应的目标函数值绝大部分都很接近0。
为了进一步验证超构光栅分束器的实际光学性能,将分束光的偏折角设置为150°和120°,各训练1000次后从结果中选取了(150°时
$ {\eta }_{1} $ 和$ {\eta }_{-1} $ 为41.25%和43.68%,$ \eta $ 为84.93%,$ UE $ 为1.5%;120°时$ {\eta }_{1} $ 和$ {\eta }_{-1} $ 为48.28%和46.85%,$ \eta $ 为95.13%,$ UE $ 为2.86%)最优超构光栅分束器,使用FDTD Solutions仿真软件进行了全波模拟,得出超构光栅分束器的场分布,如图3所示。图3(a)、图3(b)分别表示偏折角为120°和150°时超构光栅每个周期的结构,图3(c)、图3(d)分别表示偏折角为120°和150°时超构光栅分束器的$ {E}_{x} $ 分布,图3(e)、图3(f)分别表示偏折角为120°和150°时超构光栅分束器的场分布。这些结果表明基于全局拓扑优化深度学习模型设计出的超构光栅分束器实现了良好的均匀性和较高的衍射效率。图 3 偏折角为120°和150°时对应的超构光栅分束器结构和全波模拟结果。(a)、(b)偏折角为120°和150°时对应的超构光栅分束器结构;(c)、(d) 偏折角为120°和150°时对应的超构光栅分束器的
$ {E}_{x} $ 分布;(e)、(f) 偏折角为120°和150°时对应的超构光栅分束器的场分布Figure 3. Structure and the full wave simulation results of the metagrating beam splitter with deflection angle of 120° and 150°; (a), (b) Structure of the metagrating beam splitter with the deflection angle of 120° and 150°; (c), (d)
$ {E}_{x} $ distribution of the metagrating beam splitter with the deflection angle of 120° and 150°; (e), (f) Electric field distribution of the metagrating beam splitter with the deflection angle of 120° and 150°
Global topology optimized metagrating beam splitter based on deep learning
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摘要: 将深度学习模型应用于超构光栅分束器的逆向设计,可以在全局范围内获得具有良好均匀性和高衍射效率的结构。利用基于全局拓扑优化的深度学习模型,围绕超构光栅分束器的结构设计和衍射效率及均匀性等光学性能展开了一系列的研究。在波长为900 nm的入射光下,基于全局拓扑优化深度学习模型设计出大角度高衍射效率超构光栅分束器,设计的分束角为120°与150°时衍射效率分别达到95%与85%。Abstract: With the help of the deep learning model applied in the inverse design of the metagrating beam splitter, good uniformity and high diffraction efficiency can be obtained. The structure design, diffraction efficiency and uniformity of the metagrating beam splitter was studied by using the global topology optimization neural networks. Under the working wavelength of 900 nm, the beam splitter with splitting angle of 120° and 150° designed based on the global topology optimization networks had high diffraction efficiencies of 95% for 120° and 85% for 150°.
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Key words:
- metagrating /
- beam splitter /
- deep learning
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图 2 偏折角为150°的超构光栅分束器对应的衍射效率和目标函数值。(a)、(d)超构光栅分束器平均衍射效率和平均目标函数值随训练次数的变化;(b)、(c)训练100次和1000次时超构光栅分束器的衍射效率分布;(e)、(f)训练100次和1000次时超构光栅分束器的目标函数值分布
Figure 2. Diffraction efficiency and target function of metagrating beam splitter with deflection angle 150°. (a), (d) Average diffraction efficiency and the average target function values of the metagrating beam splitter vs training times; (b), (c) Distribution of diffraction efficiency of the metagrating beam splitter after 100 and 1000 times of training; (e), (f) Distribution of target function values of the metagrating beam splitter after 100 and 1000 times of training
图 3 偏折角为120°和150°时对应的超构光栅分束器结构和全波模拟结果。(a)、(b)偏折角为120°和150°时对应的超构光栅分束器结构;(c)、(d) 偏折角为120°和150°时对应的超构光栅分束器的
$ {E}_{x} $ 分布;(e)、(f) 偏折角为120°和150°时对应的超构光栅分束器的场分布Figure 3. Structure and the full wave simulation results of the metagrating beam splitter with deflection angle of 120° and 150°; (a), (b) Structure of the metagrating beam splitter with the deflection angle of 120° and 150°; (c), (d)
$ {E}_{x} $ distribution of the metagrating beam splitter with the deflection angle of 120° and 150°; (e), (f) Electric field distribution of the metagrating beam splitter with the deflection angle of 120° and 150° -
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