Asymptotic electromagnetic transition conditions for thin layers: the arithmetic-mean field and susceptibility hybridization
Effective transition conditions that replace a thin polarizable layer by a zero-thickness sheet are commonly applied in electromagnetic models of two-dimensional (2D) materials and metasurfaces. These conditions usually take the arithmetic mean of the fields on either side as the polarizing field at the boundary, and characterize the layer by the $0^\mathrm{th}$ moment of its susceptibility, but both choices are typically imposed rather than derived. We derive transition conditions for a finite-thickness, in-plane homogeneous, in-plane isotropic layer with a general nonlocal susceptibility, using a single-scale asymptotic expansion in $η= l\max(k_0,q)$, where $l$ is the layer thickness, $k_0$ the free-space wavenumber and $q$ the in-plane wavenumber. The sheet limit must be taken after resumming the out-of-plane depolarizing field, as this ensures that the resulting sheet susceptibility is independent of the material thickness. The arithmetic mean then emerges as the leading-order polarizing field for general susceptibility profiles. Asymmetries in the layer profile modify the effective susceptibility appearing in the transition conditions, rather than altering the arithmetic-mean field prescription. If the spatial profile is not even in both out-of-plane coordinates, in-plane and out-of-plane response hybridize, turning the diagonal volumetric susceptibility into a $q$-dependent effective tensor surface susceptibility with off-diagonal components. Mirror symmetry alone does not remove this hybridization. Comparison with exact solutions for local and separable layers shows that the derived conditions remain accurate to $O(η)$ across propagating and evanescent excitations, whereas the standard sheet conditions fail once hybridization becomes appreciable.