On the constraints of electromagnetic multipoles for symmetric scatterers: eigenmode analysis

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 C_4v as an example, we show that each irreducible representation of C_4v 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 D_3h symmetry. Our theoretical analyses are in perfect agreements with the fullwave finite element calculations of the eigenmodes of the symmetric scatters.