Generalized Non-Hermitian Hamiltonian for Guided Resonances in Photonic Crystal Slabs

We develop a generalized non-Hermitian Hamiltonian formalism for guided resonances in photonic crystal slabs, derived directly from Maxwell's equations through a systematic guided-mode expansion. By expanding the electromagnetic fields over the complete mode basis of an unpatterned slab and systematically integrating out radiative Fabry--P\'erot channels, we obtain the analytical operator structure of the Hamiltonian, which treats guided-mode coupling and radiation losses on equal footing. The resulting Hamiltonian provides explicit expressions for both dispersive and radiative coupling terms in terms of modal overlap integrals and Fourier components of the permittivity modulation. For specific geometries, the Hamiltonian coefficients can be extracted from full-wave simulations enabling accurate modeling without phenomenological assumptions. As a case study, we investigate hexagonal lattices with both preserved and broken $C_6$ symmetry, demonstrating predictive agreement for complex band structures, near-field distributions, and far-field polarization patterns. In particular, the formalism reproduces symmetry-protected bound states in the continuum (BICs) at the $\Gamma$ point, accidental off-$\Gamma$ BICs near the $\Gamma$ point, and the emergence of chiral exceptional points (EPs). It also captures the tunable behavior of eigenmodes near the $K$ point, including Dirac-point shifts and the emergence of quasi-BICs or bandgap openings, depending on the nature of $C_6$ symmetry breaking. We further demonstrate in the Appendix that the same formalism extends naturally to other symmetry classes, including $C_2$ (1D grating) and $C_4$ (square lattice) photonic crystal slabs. This approach enables predictive and efficient modeling of complex photonic resonances, revealing their topological and symmetry-protected characteristics in non-Hermitian systems.