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

Abstract

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--Pérot 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 $Γ$ point, accidental off-$Γ$ BICs near the $Γ$ 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.

Figures (10)

• Figure 1: Generalized guided-mode expansion. Guided resonances in a PhC slab result from periodic permittivity perturbations coupling guided modes and Fabry--Pérot modes of an unpatterned slab waveguide.
• Figure 2: Geometry of hexagonal lattices. a) Triangular lattice. b) Honeycomb lattice. c) First Brillouin zone in momentum space.
• Figure 3: Guided mode basis. a) Eigenmodes operating at the $\Gamma$ point and above the light cone are described by six guided modes $\lvert\Gamma_n\rangle$ of wave vector $\mathbf{\Gamma_n}$, with $n=1\rightarrow 6$. b) Eigenmodes operating at the $K$ point and above the light cone are described by three guided modes $\lvert K_n\rangle$ of wave vector $\mathbf{K_n}$, with $n=1\rightarrow 3$. The green region indicates the first Brillouin zone, the blue regions indicate the second Brillouin zone, and the yellow region indicates the third Brillouin zone.
• Figure 4: General near-field patterns. Calculated magnetic near-field profiles $\mathbf{H}_{\lvert\Omega_n\rangle}^{(\Gamma)}$ for the eigenmodes at the $\Gamma$ point. Arrows represent the electric field vectors $\mathbf{E}_{\lvert\Omega_n\rangle}^{\Gamma}$.
• Figure 5: Eigenmodes near $\Gamma$ for triangular and honeycomb lattice design. a,d) Real part of the photonic band energies as a function of the in-plane wavevector. The zoomed-in insets highlight the crossing along $\Gamma K$(blue box) and anticrossing $\Gamma M$ (green box) between the third and fourth bands of the triangular lattice. b,e) Quality factors of the photonic bands. Green and blue dashed lines indicate reference curves proportional to $1/k^4$ and $1/k^2$, respectively. Red scatters represent numerically simulated photonic bands, while black lines show their corresponding analytical fitting using the effective theory. c,f) Far-field polarization textures (i.e., the orientation of radiated polarization) of the photonic bands hosting monopolar, quadrupolar, and hexapolar modes.