An efficient method for calculating resonant modes in biperiodic photonic structures

TL;DR

Addresses efficient computation of resonant modes in biperiodic photonic structures where outgoing radiation conditions complicate truncation and often rely on PMLs. Introduces transverse impedance operators to split the unit cell into exterior and interior subdomains, reducing Maxwell's equations to a small matrix nonlinear eigenvalue problem $\mathbf{T}(\omega)\mathbf{h}=0$ that is solved by the contour integral method. The interior TI matrix is obtained as a Schur complement and computed by a fast FEM-based solver, while the exterior TI is explicit from Rayleigh expansions, yielding a compact $(P \times P)$ system. Numerical results on a PhC slab and a sphere lattice demonstrate memory- and compute-efficiency, absence of spurious solutions, and accurate handling of degenerate resonances and bound states in the continuum. The method is poised to enable rapid, reliable design and analysis of a wide range of biperiodic photonic devices.

Abstract

Many photonic devices, such as photonic crystal slabs, cross gratings, and periodic metasurfaces, are biperiodic structures with two independent periodic directions, and are sandwiched between two homogeneous media. Many applications of these devices are closely related to resonance phenomena. Therefore, efficient computation of resonant modes is crucial in device design and structure analysis. Since resonant modes satisfy outgoing radiation conditions, perfectly matched layers (PMLs) are usually used to truncate the unbounded spatial variable perpendicular to the periodic directions. In this paper, we develop an efficient method without using PMLs to calculate resonant modes in biperiodic structures. We reduce the original eigenvalue problem to a small matrix nonlinear eigenvalue problem which is solved by the contour integral method. Numerical examples show that our method is efficient with respect to memory usage and CPU time, free of spurious solutions, and determines degenerate resonant modes without any difficulty.

Figures (4)

• Figure 1: A PhC slab with a square lattice of elliptic air holes: (a) 3D view and (b) top view.
• Figure 2: A square lattice of dielectric spheres on top of a uniform dielectric slab.
• Figure 3: Complex $\omega$ plane with integration contours $\mathcal{C}_1$ (blue) and $\mathcal{C}_2$ (pink). The frequencies $\omega_k$, $k=1,2,3$ (asterisks) and $\omega_4$ (circle) calculated using contour $\mathcal{C}_2$ are shown in the inset. The results obtained by contours $\mathcal{C}_1$ and $\mathcal{C}_2$ are nearly identical.
• Figure 4: $\hbox{Im}(E_z)$ of four resonant modes at $z=D$.