Legume: A free implementation of the guided-mode expansion method for photonic crystal slabs

We describe legume, a free electromagnetic solver that implements the guided-mode expansion method for patterned multilayer waveguides, or photonic crystal slabs. legume has a built-in tool for automatic differentiation, which makes it suitable for the inverse design of photonic crystal structures with desired physical properties. Compared to a previous version of the method (M. Minkov et al., 2020 [12]), here we introduce several new features of the code, we discuss additional technical aspects of the method and its numerical implementation. The novel features that are treated in this paper include: (i) the separation of modes according to their mirror symmetry with respect to a vertical symmetry plane of the photonic structure, (ii) the problem of polarization mixing in coupling to far-field radiation modes, and (iii) the description of active two-dimensional layers through a suitably formulated radiation-matter coupling Hamiltonian, allowing to describe the physics of both weakly and strongly coupled exciton-photon modes, the latter leading to photonic crystal polariton eigenmodes. Detailed and direct comparisons with rigorous coupled-wave analysis simulations are used to test the accuracy of the method and the numerical efficiency of the code. These newly added features of the legume code significantly increase the prospective applications of guided-mode expansion, making it a very practical and versatile tool enabling the design of advanced photonic structures and the description of radiation-matter interaction.我们在此介绍legume——一款面向结构化多层波导或光子晶体平板（photonic crystal slabs）的、采用导模展开法（guided-mode expansion）的免费电磁求解器。该软件内置自动微分（automatic differentiation）工具，可用于实现具有目标物理特性的光子晶体结构的反向设计。相较于该方法2020年的先前版本（M. Minkov等，2020 [12]），本文新增了该代码的多项功能，并讨论了该方法及其数值实现的额外技术细节。本文所述的新增功能包括：(i) 根据光子结构垂直对称面的镜像对称性对模式进行分类；(ii) 与远场辐射模式耦合时的偏振混合问题；(iii) 通过合理构建的辐射-物质耦合哈密顿量（Hamiltonian）描述有源二维层，从而可同时刻画弱耦合与强耦合激子-光子模式（exciton-photon modes）的物理特性，其中强耦合情形可得到光子晶体极化激元本征模式（polariton eigenmodes）。本文通过与严格耦合波分析（rigorous coupled-wave analysis）仿真结果的直接细致对比，验证了本方法的精度与该代码的数值效率。legume代码的这些新增功能大幅拓展了导模展开法的潜在应用场景，使其成为一款兼具实用性与通用性的工具，可用于设计先进光子结构并刻画辐射-物质相互作用的物理过程。