Dipole-quadrupole model and multipole analysis of resonant membrane metasurfaces

Membrane-metasurfaces, formed by periodic arrangements of holes in a dielectric layer, are gaining attention for their easier manufacturing via subtractive techniques, unnecessity of substrates, and access to resonant near-fields. Despite their practical relevance, their theoretical description remains elusive. Here, we present a semi-analytical dipole-quadrupole model for the multipole analysis of numerically-obtained reflection and transmission spectra in metasurfaces excited at arbitrary angles. Dipole models are generally sufficient to study traditional metasurfaces made of solid nanostructures. However, the inclusion of electric and magnetic quadrupoles is necessary to study membrane-metasurfaces, which offer an ideal platform to showcase our method. We demonstrate the importance of choosing the optimal position of a symmetric membrane-metasurface's unit cell to ensure the sufficiency of the dipole-quadrupole approximation. We show that our formalism can explain complex phenomena arising from inter-multipole interference, including lattice anapole and generalized Kerker effects, Fano resonances, and quasi-bound states in the continuum. We also present the applicability of the method to membrane-metasurfaces with non-centrosymmetric unit cells (e.g., conical holes and surface voids). By enabling a deeper insight into the coupling mechanisms leading to the formation of local and collective resonances, our method expands the electromagnetics toolbox to study, understand, and design conventional and membrane-metasurfaces.