Resonant subspace approximation for photonic crystal slabs
Resonant approximations are indispensable for the interpretation and efficient modeling of photonic crystal slabs, yet most of them describe an eigenmode as a pole in the complex energy plane at a fixed set of the remaining parameters. Tracking such poles and their hybridization across the Brillouin zone or upon variation of structural parameters is labor-intensive and severely limits the use of resonant approximations in band-structure calculations and structural optimization. Here we introduce a resonant subspace approximation that treats the photon energy and all other parameters on an equal footing. Considering a wide class of photonic crystal slabs that can be split into two non-resonant parts, we show that their resonances arise solely from the round-trip propagation of coupled Fourier harmonics between these parts, in direct analogy with Fabry--Pérot and waveguide modes. We generalize the scalar round-trip phase to round-trip and phase matrices, whose smooth dependence on all parameters allows us to project the problem onto a small resonant subspace defined at a single anchor point and to extrapolate it throughout a local region of parameter space of arbitrary dimensionality. As a result, rigorous computations at only a few points suffice to reconstruct the band structure, modal linewidths, complex hybridization, and optical spectra as functions of energy, wavevector, geometric dimensions, or permittivity within seconds. We demonstrate the accuracy and versatility of the approach on a strong hexagonal silicon grating, resolving intricate mode hybridization, symmetry-protected features, and subtle geometry-controlled effects that are hardly accessible to straightforward computations. The method is fast, accurate, and readily extensible, offering a practical route to the exploration, design, and optimization of resonant photonic crystal slabs.