The scattering and resonant properties of optical scatterers/resonators are determined by the relative ratios among the associated multipole components, the calculation of which usually is analytically tedious and numerically complicated for complex structures. Here we identify the constraints as well as the relative relations among electromagnetic multipoles for the eigenmodes of symmetric scatterers/resonators. By reducing the symmetry properties of the vector spherical harmonic waves to those of the modified generating functions, we systematically study the required conditions for electromagnetic multipoles under several fundamental symmetry operations, i.e., 2D rotation and reflection operations and 3D proper and improper rotations. Taking a 2D scatterer with C4v as an example, we show that each irreducible representation of C4v can be assigned to corresponding electromagnetic multipoles, and consequently the constraints of the electromagnetic multipoles can be easily extracted. Such group approach can easily be extended to more complex 3D scatterers with higher symmetry group. Subsequently, we use the same procedure to map out the complete relation and constraint on the electromagnetic multipoles of a 3D scatterer imposed by D3h symmetry. Our theoretical analyses are in perfect agreements with the fullwave finite element calculations of the eigenmodes of the symmetric scatters.

中文翻译：

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关于对称散射体的电磁多极约束：本征模分析。

光学散射体/谐振器的散射和谐振特性由相关的多极子组件之间的相对比率确定，对于复杂结构，其计算通常在分析上很繁琐并且在数值上很复杂。在这里，我们确定了对称散射体/谐振器本征模的约束以及电磁多极之间的相对关系。通过将矢量球形谐波的对称性降低到修改后的生成函数的对称性，我们系统地研究了电磁多极在几种基本对称操作（即2D旋转和反射操作以及3D正确和不正确旋转）下的条件。以C4v的2D散射体为例，我们表明，C4v的每个不可约表示都可以分配给相应的电磁多极，因此可以轻松地提取电磁多极的约束。这样的分组方法可以容易地扩展到具有较高对称性的分组的更复杂的3D散射体。随后，我们使用相同的过程绘制出D3h对称性对3D散射体的电磁多极的完整关系和约束。我们的理论分析与对称散射本征模的全波有限元计算完全吻合。随后，我们使用相同的过程绘制出D3h对称性对3D散射体的电磁多极的完整关系和约束。我们的理论分析与对称散射本征模的全波有限元计算完全吻合。随后，我们使用相同的过程绘制出D3h对称性对3D散射体的电磁多极的完整关系和约束。我们的理论分析与对称散射本征模的全波有限元计算完全吻合。