  In the recent years, photonic Chern materials have attracted substantial
interest as they feature topological edge states that are robust against
disorder, promising to realize defect-agnostic integrated photonic crystal slab
devices. However, the out-of-plane radiative losses in those photonic Chern
slabs has been previously neglected, yielding limited accuracy for predictions
of these systems' topological protection. Here, we develop a general framework
for measuring the topological protection in photonic systems, such as in
photonic crystal slabs, while accounting for in-plane and out-of-plane
radiative losses. Our approach relies on the spectral localizer that combines
the position and Hamiltonian matrices of the system to draw a real-picture of
the system's topology. This operator-based approach to topology allows us to
use an effective Hamiltonian directly derived from the full-wave Maxwell
equations after discretization via finite-elements method (FEM), resulting in
the full account of all the system's physical processes. As the spectral
FEM-localizer is constructed solely from FEM discretization of the system's
master equation, the proposed framework is applicable to any physical system
and is compatible with commonly used FEM software. Moving forward, we
anticipate the generality of the method to aid in the topological
classification of a broad range of complex physical systems.
