Spin-momentum locked modes on anti-phase boundaries in photonic crystals

TL;DR

This work investigates spin-momentum locked edge modes along anti-phase boundaries in photonic crystals formed by half-period shifts. It shows that circularly polarized magnetic-dipole sources can selectively excite edge modes that propagate unidirectionally depending on spin, and that tuning geometric parameters such as $R$ and the boundary offset $t$ can induce band inversion, reversing edge-mode propagation. It also demonstrates that gradual shift boundaries support similar edge modes and that a two-orthogonal dipole source can enhance directionality. Together these results establish a route to design chiral photonic waveguides based on anti-phase boundaries, with spin-controlled emission and tunable edge-state dispersion.

Abstract

An anti-phase boundary is formed by shifting a portion of photonic crystal lattice along the direction of periodicity. A spinning magnetic dipole is applied to excite edge modes on the anti-phase boundary. We show the unidirectional propagation of the edge modes which is also known as spin-momentum locking. Band inversion of the edge modes is discovered when we sweep the geometrical parameters, which leads to a change in the propagation direction. Also, an optimized source is applied to excite the unidirectional edge mode with high directivity.

Figures (7)

• Figure 1: (a) Unit cell of photonic crystal with $d$ the diameter of cylinders, $a_0$ the length of diamond edge, and $R$ the distance between the center of the diamond and the center of the cylinders. $\varepsilon_d$ and $\varepsilon_A$ are the relative permittivities of the cylinders and surrounding environment respectively. (b) Anti-phase boundary (red dashed line) formed by shifting the photonic crystal along $\overrightarrow{a}_2$ by one-half period $t=-a_0/2$ where $\overrightarrow{a}_1$ and $\overrightarrow{a}_2$ are lattice vectors of the crystal. The angle between $\overrightarrow{a}_1$ and $\overrightarrow{a}_2$ is $\pi/3$.
• Figure 2: The red hexagons are unit cells of the crystal for $R=a_0/3$ while $\overrightarrow{a}_1^{'}$ and $\overrightarrow{a}_2^{'}$ are lattice vectors.
• Figure 3: (a) Dispersion relation of the super-cell which is periodic in $\overrightarrow{a}_2$ direction and of 8 unit cells on each side of anti-phase boundary in $\overrightarrow{a}_1$ direction. Label $k_2$ means the projection of k vector onto $\overrightarrow{a}_2/|\overrightarrow{a}_2|$. The green-shaded region is the projected band diagram of the bulk modes. Red and blue lines represent the odd modes and even modes respectively. The diameter of cylinder and distance between cylinder center and diamond center are $d=0.24a_0$ and $R=0.345a_0$. The relative permittivities are $\varepsilon_d=11.7$ and $\varepsilon_A=1$. (b) Real part of $E_z$ distributions at points $P_1$, $P_2$ and $P_3$ as shown in (a). The black arrows indicate the time-averaged Poynting vectors over a period. (c) Real part distributions of $E_z$ of magnetic dipoles $(\hat{x}-i\hat{y})/\sqrt{2}$ (left) and $(\hat{x}+i\hat{y})/\sqrt{2}$ (right) are plotted. The red arrows represent the time-averaged Poynting vectors. (d) $|E_z|$ are plotted for the driven modes excited by magnetic dipoles $(\hat{x}-i\hat{y})/\sqrt{2}$ (left) and $(\hat{x}+i\hat{y})/\sqrt{2}$ (right) respectively. The yellow arrow indicates the location of the source, which is at the center of the unit cell. The normalized frequency of the source is chosen to be $f_0a_0/c=0.46$.
• Figure 4:
• Figure 5: Dispersion relations of the super-cells with (a) $R=0.345a_0$ and (b) $R=0.3a_0$ when tuning the offset $t$ in units of $a_0$. $|E_z|$ distributions are plotted for the edge modes with $R=0.345a_0$ when (c) $k_2a_0/2\pi=0$ and (d) $k_2a_0/2\pi=0.1$.