Synthetic Topological Nodal Phase in Bilayer Resonant Gratings

The notion of synthetic dimensions in artificial photonic systems has received considerable attention as it provides novel methods for exploring hypothetical topological phenomena as well as potential device applications. Here, we present nanophotonic manifestation of a two-dimensional topological nodal phase in bilayer resonant grating structures. Using the mathematical analogy between a topological semimetal and vertically asymmetric photonic lattices, we show that the interlayer shift simulates an extra momentum dimension for creating a two-dimensional topological nodal phase. We present a theoretical model and rigorous numerical analyses showing the two nodal points that produce a complex gapless band structure and localized edge states in the topologically nontrivial region. Therefore, our results provide a practical scheme for producing high-dimensional topological effects in simple low-dimensional photonic structures.

Figure 1

Schematic of an asymmetric bilayer grating for synthetic topological nodal phase. The structure consists of two identical thin-film photonic lattices with fill factor $F$, refractive index $n_1$, period $a$, and total thickness $d$. Degree of interlayer shift between two layers is denoted by $S$.

Figure 2

Complex bulk band features of a topological nodal phase Hamiltonian. (a) Real (left) and imaginary (right) energy band structures in 2D synthetic space $(p_x,p_S)$ for ${\mathbf{H}}_{\mathrm{TNP}}$ in Eq. (2). Parameters are $v_g=1.5$, $\kappa =1$, and $\gamma =0.5$. Two Dirac points (DPs) are denoted by white arrows in the real band. White dashed lines in the imaginary band on the right indicate flat bands along the $p_x$ axis at $p_S=0.5\pi$ and $1.5\pi$. Skin color indicates relative even-mode strength for corresponding eigenvectors. (b) Bloch sphere representation $\mathbf{b}$ of the lower-band eigenvector for geometric picture of the quantized complex Berry phase. Line color indicates $\cos (p_S)$ value.

Figure 3

Open boundary complex energy spectra and gapless topological edge states in bilayer gratings. (a) Projected real (left) and imaginary (right) dispersion relations in the normalized frequency and synthetic parameter space. Line color indicates edge localization factor—edge states are in red and bulk Bloch states are in black. (b) Spatial distribution of electric field norm $|E_y(x,z)|$ for the topological edge state at $S=0.5$.

Figure 4

Resonant spectral responses of topological nodal phase under plane wave incidence. (a) Schematic of photonic junction consisting of two bilayer gratings, one on the left with variable $S$ and the other on the right with fixed $S=0$. (b) Resonant reflection spectra depending on $S$. (c) Spatial distribution of electric field norm for the Jakiw-Rebbi-state resonance at $S=0.5$ and $a/\lambda =0.347$ as denoted by red dot in Fig. 4.