Realizing topologically protected ghost surface polaritons by lattice transformation optics

TL;DR

The paper addresses how to control ghost surface polaritons (GSPs) at interfaces by introducing lattice transformation optics (LTO) to gyromagnetic photonic crystals. By applying a simple shear transformation to a square lattice, the authors reshape the Brillouin zone and steer oblique, unidirectional GSPs while controlling topological invariants such as the Chern number $C$. They show that GSPs can be made topologically protected and even reversed by a negative shear, with the Chern number behavior determined by the Brillouin-zone orientation via $C' = \mathrm{sign}\left(\det\left( \dfrac{\partial(k_x',k_y')}{\partial(k_x,k_y)}\right)\right) C$. The work also distinguishes lattice-transformation-based routing from simple geometric deformation, showing tunable wavefronts and robust edge modes, and positions LTO as a general framework for engineering band structures and topological properties in periodic systems. This approach opens new avenues for tunable polariton control, sensing, and energy transport in photonic crystals.

Abstract

While conventional surface waves propagate along the surface and decay perpendicularly from the interface, the ghost surface polaritons show oblique propagation direction with respect to the interface. Here, we have discovered topologically protected ghost surface polaritons by applying the lattice transformation optics method to gyromagnetic photonic crystals. By introducing the transformation optics method to periodic systems, we develop the lattice transformation optics method to engineer the band structures and propagation directions of the surface polaritons. We show that a simple shear transformation on the square lattice can tailor the propagation directions with ease. The reversed ghost surface polariton is discovered by setting a negative shear factor. Interestingly, we find the topological invariant Chern number will change sign when the orientation of the Brillouin zone flipped during the transformation. Our findings open up new avenues for studying ghost surface polaritons and provide a general engineering method for periodic systems.

Figures (6)

• Figure 1: Schematic of topologically protected GSPs and reversed GSPs designed by lattice transformation optics. (a) The square lattice is transformed into a periodic structure with curvy boundaries. (b) $|E_z|$ field distributions of the original square lattice (middle) and transformed lattice (top: transformation $\bar{\bar{J_1}}$, bottom: transformation $\bar{\bar{J_2}}$) excited by the source. The white star-shaped marks are line currents along the z-axis with normalized frequency $fa/c=0.543$. The unit cells of the original structure and transformed structures are marked by yellow.
• Figure 2: Band analysis of the original unit cell and the transformed unit cells. (a) Band diagrams of the second band of the corresponding unit cells shown in blue. Middle: A square lattice with a YIG rod in the center. Left and right: The diamond lattice is transformed from the square lattice in the middle under the matrix $\bar{\bar{J_1}}$ and $\bar{\bar{J_2}}$ respectively. (b) Projected band diagram of the second and third band of the three unit cells shown in (a). The red curve is the dispersion relation of the super-cells formed by the three unit cells. The blue dot is located at $k_x=0.75\pi/a, fa/c=0.543$. (c) $|E_z|$ field distributions of surface modes at the blue dot in (b)
• Figure 3: Berry curvatures of the second band of the original structure and the transformed structures. Middle: Original square lattice. Left and right: The transformed lattices under shear transformations $\bar{\bar{J_1}}$, $\bar{\bar{J_2}}$ respectively.
• Figure 4: A comparison between the GSPs and conventional surface polaritons when tuning the shear coefficient. (a) The GSPs created by the lattice transformation optics, where both the shape of the YIG rod and the material parameters are transformed according to the TO. (b) Only the shape of lattice is transformed, while the shape of the rod and the material parameters remain the same as the original structure. The white star-shaped marks are line currents along the z-axis with normalized frequency $fa/c=0.543$. The unit cells of the transformed structures are marked by yellow.
• Figure 5: Transformation from a square lattice to a lattice with curved periodic boundaries. The period of the square lattice is $a=1\mathrm{m}$ and the side length is $0.4\mathrm{m}$ for the smaller square. The material is set as $\epsilon=2\epsilon_0$, $\mu=2\mu_0$ for the smaller square and it is surrounded by the vacuum. $|E_z|$ field distributions at $k_x=\pi/3$, $k_y=\pi/2$, and $f=0.74c/a$ are plotted for the original structure and transformed structure.