Classifying topology in photonic crystal slabs with radiative environments

Abstract

In the recent years, photonic Chern materials have attracted substantial interest as they feature topological edge states that are robust against disorder, promising to realize defect-agnostic integrated photonic crystal slab devices. However, the out-of-plane radiative losses in those photonic Chern slabs has been previously neglected, yielding limited accuracy for predictions of these systems' topological protection. Here, we develop a general framework for measuring the topological protection in photonic systems, such as in photonic crystal slabs, while accounting for in-plane and out-of-plane radiative losses. Our approach relies on the spectral localizer that combines the position and Hamiltonian matrices of the system to draw a real-picture of the system's topology. This operator-based approach to topology allows us to use an effective Hamiltonian directly derived from the full-wave Maxwell equations after discretization via finite-elements method (FEM), resulting in the full account of all the system's physical processes. As the spectral FEM-localizer is constructed solely from FEM discretization of the system's master equation, the proposed framework is applicable to any physical system and is compatible with commonly used FEM software. Moving forward, we anticipate the generality of the method to aid in the topological classification of a broad range of complex physical systems.

Figures (4)

• Figure 1: Gapless environment for photonic crystal slabs. (a) Schematic of a free-standing photonic crystal slab in a three-dimensional (3D) geometry. Photonic structures are inherently 3D and are usually surrounded by an homogeneous material that features a light cone. As such, out-of-plane radiative losses, depicted by the red arrows, are inherent to such structures. The photonic crystal is a triangular lattice with lattice constant $a=1µm$ composed of dielectric rods, $\bar{\epsilon}_{jj} = 14$ for $j=x,y,z$, of radius $r=0.37a$ and height $t = 0.5a$ embedded in a gyro-electric material slab, $\bar{\epsilon}_{jj} = 1$ and $\bar{\epsilon}_{xy} = -0.4i$, of thickness $t = 0.5a$. (b) Band structure of the photonic crystal slab in (a) over the first Brillouin zone, for a transverse magnetic-like polarization. The photonic band gap at around $\omega = 0.42 [ 2\pi c/a]$ is the "topological band gap" known from extrapolation from the two-dimensional photonic crystal approximation. The shaded region depicts those frequencies and wavevectors that are at, or above, the light line of the surrounding air. The red line depicts the light line, $\omega = c|k|$. Above the light line, photonic crystal slabs generally exhibit resonances, not bound states.
• Figure 2: Position coordinate of the degrees of freedom used for constructing the position matrices within the finite-element method discretization. (a) Schematic of the discretization into finite elements of the two-dimensional (2D) simulation domain composed of a single dielectric rod (red shaded region) at the center of a triangular unit cell. (b) Zoom-in of (a) where the mesh nodes and extended mesh nodes are depicted using blue and green markers, respectively. In this context, the 2D structure is solved for transverse magnetic polarization ($H_z \neq 0$) and the shape function used are the curl elements. The position of the extended mesh nodes corresponding to the unknown weighting coefficient $E_n$ are located at $(x_n, y_n)$.
• Figure 3: Probing of the local topology in two-dimensional photonic Chern system examples. (a) Design of the 2D photonic Chern heterostructure. The inner parallelogram is a non-trivial topological lattice with lattice constant $a=1µm$ made of dielectric rods, $\bar{\epsilon}_{jj}=14$ for $j=x,y,z$, with radius $r=0.37a$ embedded in a gyro-electric background, $\bar{\epsilon}_{jj} = 1$, $\bar{\epsilon}_{xy} = -0.4i$, that breaks time-reversal symmetry. The outer lattice is a topologically trivial lattice with lattice constant $a=1µm$ composed of air rods, $\bar{\epsilon}_{jj}=1$, with radius $r=0.35a$ in a dielectric background with $\bar{\epsilon}_{jj}=5.5$. (b) Spectrum of the FEM-localizer $\sigma(\hat{L}_{(x,y_0,\omega_0)})$ normalized by $10^{-4} \lVert H_{\text{eff},c}(\omega_0)\rVert$ and the local Chern number $C_{(x,y_0,\omega_0)}^{\textrm{L}}$ along the green line in (a) at $y_0 = 0$ and frequency $\omega_0 = 0.37 [2 \pi c/a]$, with $\kappa = 1.5 [ 10^{-4} \lVert H_{\text{eff},c}(\omega_0)\rVert/\lVert X\rVert ]$. (c) Design of the photonic Chern quasicrystal based on a Penrose tiling. The dielectric rods are located at the vertices of the Penrose tiling in a gyro-electric background, $\bar{\epsilon}_{jj} = 1$, $\bar{\epsilon}_{xy} = -0.4i$. (d) Spectrum of the FEM-localizer $\sigma(\hat{L}_{(x,y_0,\omega_0)})$ normalized by $10^{-4} \lVert H_{\text{eff},c}(\omega_0)\rVert$ and the local Chern number $C_{(x,y_0,\omega_0)}^{\textrm{L}}$ along the green line in (a) at $y_0 = 0$ and frequency $\omega_0 = 0.37 [2 \pi c/a]$, with $\kappa = [ 10^{-4} \lVert H_{\text{eff},c}(\omega_0)\rVert/\lVert X\rVert ]$.
• Figure 4: Probing of the local topology in a photonic Chern slab system. (a) Design of the photonic Chern slab. The inner parallelogram is a non-trivial topological lattice with lattice constant $a=1µm$ made of dielectric rods, $\bar{\epsilon}_{jj}=14$ for $j=x,y,z$, with radius $r=0.37a$ and height $t = 0.5a$ embedded in a gyro-electric slab, $\bar{\epsilon}_{jj} = 1$, $\bar{\epsilon}_{xy} = -0.4i$, with thickness $t = 0.5a$. The outer lattice is a topological trivial lattice with lattice constant $a=1µm$ composed of air rods, $\bar{\epsilon}_{jj}=1$, with radius $r=0.35a$ and height $t = 0.5a$ in a dielectric slab, $\bar{\epsilon}_{jj}=5.5$, with thickness $t$. (b) Spectrum of the FEM-localizer $\sigma(\hat{L}^{(NH)}_{(x,y_0,\omega_0)})$ normalized by $10^{-4} \lVert H_{\text{eff},c}(\omega_0)\rVert$ along the green line in (a) at $y_0 = 0$ and frequency $\omega_0 = 0.37 [2 \pi c/a]$, with $\kappa = 1.5 [ 10^{-4} \lVert H_{\text{eff},c}(\omega_0)\rVert/\lVert X\rVert ]$.