Efficient method for computing resonant modes in periodic photonic structures via transverse impedance operators

Periodic photonic structures, such as photonic crystal slabs and crossed gratings, are periodic in two independent directions and are sandwiched between two homogeneous media. They support rich resonance phenomena and are highly desirable for practical applications. Accurate and efficient computation of resonant modes is therefore critical for both device design and analysis. A common approach employs perfectly matched layers (PMLs) to truncate the variable perpendicular to the periodic directions, yielding a generalized eigenvalue problem on an extended domain. However, PML-based methods enlarge the computational domain, may introduce spurious modes, and require case-dependent parameter tuning. Here we propose a PML-free method for computing resonant modes. We select a bounded region only slightly larger than the original structure, introduce exterior and interior transverse-impedance (TI) operators on the upper and lower interfaces, and thereby obtain a nonlinear eigenvalue problem (NEP). These operators are accurately approximated by small matrices, with the exterior TI obtained analytically and the interior TI computed via a fast, memory-efficient scheme. The resulting low-dimensional matrix NEP is solved using a contour integral method. Numerical results show that our method is highly efficient, free from spurious modes, and can accurately capture degenerate resonances. © 2026 The Author(s).