Field localisation and spin-momentum locking in zero-dimensional dissipative topological photonic interface state

TL;DR

The paper addresses how to realize and tailor zero-dimensional Jackiw-Rebbi-like interface states in dissipative one-dimensional topological photonic crystals. It develops a spatiotemporal CMT approach that maps the system to a non-Hermitian $1+1$D Dirac framework with a complex mass, revealing that the real part localizes the field while the imaginary part drives energy flow toward the interface and induces spin-momentum locking via transverse SAM. The work derives a closed-form mode-volume expression showing confinement strengthens with larger bulk band gaps, and validates the theory with FDTD simulations and experimental measurements in plasmonic gratings, achieving consistent extraction of coupling constants from both band gaps and field patterns. This provides a predictive framework for designing dissipative topological photonic interfaces with enhanced light-matter interactions, enabling robust, tunable field localization and spin-controlled photonic functionalities.

Abstract

Topological photonic systems support edge states that are robust against disorder and perturbation. Depending on the symmetry and dimensionality of the bulk systems, different edge states emulating soliton, quantum integer and quantum spin Hall effects have been realized. A major concern in photonics is how one can shape the strength and polarisation of electromagnetic fields to suit different applications. Here, we show zero-dimensional (0D) interface state arising from one-dimensional (1D) dissipative topological photonic crystals exhibit strong field localisation and spin-momentum locking thanks to its complex classical analogue Dirac mass parameter. By using spatiotemporal coupled mode theory to formulate 1D photonic crystals and their corresponding Jackiw Rebbi-like (JR) interface state, we find the interaction between two energy bands at high symmetry points plays a major role in defining not only the topological triviality of the crystals but also its complex Dirac mass parameter. More importantly, when two topological trivial and nontrivial bulk systems are brought together to form a JR state, while the real part of the Dirac mass parameter governs the spectral and spatial field localisations of the interface state, the imaginary part gives rise to a net flow of energy towards the interface and a transverse spin angular momentum, resulting in a strong spin-momentum locking. We verify our theory by 1D plasmonic crystals using finite-difference time-domain simulations as well as far-field angle-resolved spectroscopy and imaging.

Figures (7)

• Figure 1: Figure showing the eigenenergy $E(k)$ of the SSH model plotted as a function of momentum $k$ in the first Brillouin zone for 3 different cases. As the value of $\Delta$ changes from $<-1$ to $>-1$, eigenvector $\vec{v}(\pi)$ remains unchanged. The inversion eigenvalue at $\pi$ stays at $\lambda(\pi)=1$. However, the eigenvector $\vec{v}(0)$ changes from $\vec{v}(0)=^T$ to $\vec{v}(0)=[-1,1]^T$ after $\Delta$ passes the Dirac point $\Delta=-1$.
• Figure 2: (a): Schematic diagram showing the different parameters of an interface state. $W_{1,2}$ are the slit widths, $H_{1,2}$ are the slit heights, and $P_{a,b}$ are the grating periods. We choose gold as the material of the gratings for surface plasmon propagation with moderate absorption. In all simulations and experiments, we fixed $H_{1,2}=30$nm and $P_{a,b}=800$nm. By varying $W_{1,2}$ we effectively change $\tilde{\omega}_{1,2}$ in our simulations and experiments. The diagram also shows the spatial decay of the electric field predicted by our theory and verified using simulations and experiments. As our interface state exhibits spin-momentum locking, the x-component of the Poynting vectors points toward the center of the interface state, while the directions of the SAM arise from the propagating surface plasmon. (b) SEM image of FIB-fabricated PmCs forming an interface state. Both gratings have a period of 800nm. The grating on the left has a slit width of 200nm, while the grating on the right has a slit width of 600nm.
• Figure 3: (a)-(d): Simulated band structures. P-polarised light is incident on the grating at angles ranging from $-3^{\circ}$ to $3^{\circ}$ along the $\Gamma$-X direction. The reflectivity profiles were collected to form a band structure. The $(0,\pm1)$ SPP modes exhibit band inversion symmetry for $W<400$nm and $W>400$nm. The band gap obtained at normal incidence is used to calculate the coupling constant $\omega_c'$ via Eqt.\ref{['eigenvalues and eigenvectors']}. (e)-(h): Measured band structures of the fabricated PmCs. The experimental band structures are similar to the simulation results in (a)-(d). As with the simulations, $\omega_c'$ is determined from the measured band gaps. (d) and (h) show the band structure when an interface state is formed using $W=200$nm and $W=600$nm gratings. The JR state appears at the center of the band gap.
• Figure 4: Poynting vector and transverse SAM simulated for the 200-600nm interface state. (a): Poynting vector. (b): Transverse SAM. The y-axes are in arbitrary unit. The envelopes follow the form predicted by theory, and the two sides carry opposite signs, demonstrating spin-momentum locking in the topological plasmonic JR state. (c): Comparison between the slopes of $P_x$, SAM, and the field pattern ($\omega_c'$). In section \ref{['section: Theory']} we predicted their spatial decay should be dominated by the $e^{-m'x}$ term. The similarity of the trends confirms the theoretical prediction.
• Figure 5: Simulated near-field patterns plotted on a logarithmic scale. P-polarised light at frequency $\omega_0$ is normally incident on the interface. A spatial monitor records the near-field pattern. As shown in Eqt.\ref{['log_E^2']}, the envelope of the log-scale intensity should be a straight line. The slope of the linear fit is used to calculate $\omega_c'$. (b): Real-space image obtained using the orthogonal imaging scheme. The interface state emits light at the resonance frequency after excitation with p-polarised light. The signal is averaged along the direction of the slits and used to determine $\omega_c'$. (c): Intensity profiles captured by field-pattern imaging, plotted on a logarithmic scale. The decaying profile was fitted using linear regression; the slope was used to calculate $\omega_c'$ as shown in Fig.\ref{['fig: omega_c_and_dirac_mass']}(b).