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We propose a framework that connects the spatial symmetries of a metasurfaceto its material parameter tensors and its scattering matrix. This provides asimple yet effective way to effortlessly determine properties of a metasurfacescattering response, such as chirality or asymmetric transmission, and which ofits effective material parameters should be taken into account in the prospectof a homogenization procedure.  In contrast to existing techniques, this approach does not require any apriori knowledge of group theory or complicated numerical simulation schemes,hence making it fast, easy to use and accessible. Its working principleconsists in recursively solving symmetry-invariance conditions that apply todipolar and quadrupolar material parameters, which include nonlocalinteractions, as well as the metasurface scattering matrix. The overall processthus only requires listing the spatial symmetries of the metasurface.  Using the proposed framework, we demonstrate the existence of multipolarextrinsic chirality, which is a form of chiral response that is achieved ingeometrically achiral structures sensitive to field gradients even at normalincidence.

Spatial Symmetries in Multipolar Metasurfaces: From Asymmetric Angular Transmittance to Multipolar Extrinsic Chirality