Topological Edge Plasmon FWM in Graphene

• Topological edge plasmon FWM is a nonlinear optical process that exploits chiral edge states in engineered graphene metasurfaces to achieve frequency conversion.
• The system leverages a magnetic-field-induced bandgap and over 100x field enhancement to enable robust, defect-immune propagation at sub-10 nW pump powers.
• Coupled-mode theory and FEM simulations validate linear dispersion, perfect phase matching, and high nonlinear gain, underscoring its potential for integrated photonic applications.

Topological edge plasmon four-wave mixing (FWM) refers to nonlinear optical interactions—specifically FWM—between topologically protected edge states of plasmons in engineered graphene metasurfaces. In such systems, a periodic array of nanoholes induces a wide topological bandgap in the terahertz regime when time-reversal symmetry is broken by a perpendicular static magnetic field. The resulting chiral edge plasmons, confined to subwavelength scales and immune to backscattering by defects, exhibit extreme nonlinearity due to the intrinsic third-order susceptibility of graphene and electric field enhancement at the edge. This allows net FWM gain at remarkably low pump powers and underpins prospects for ultra-low-power and robust photonic quantum devices (You et al., 2019).

1. Plasmonic Metasurface Geometry and Topological Bandgap

Graphene is patterned with a hexagonal lattice of air holes—lattice constant $a = 400$ nm, hole radius $r = 120$ nm—forming a plasmonic metasurface immersed in a perpendicular magnetic field $B_\perp$. The surface conductivity tensor acquires gyrotropy at terahertz frequencies due to $B_\perp$, parametrized by

$$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$

where $\sigma_L, \sigma_H$ arise from the Kubo formula. At $B_\perp = 0$ the system restores parity-time symmetry and supports massless Dirac-cone bulk plasmons at the $K, K'$ points. For $B_\perp \ne 0$, the breaking of time-reversal symmetry opens a nontrivial bandgap $\Delta\nu$ of several THz. The computed bandgap Chern number $r = 120$0 entails a single unidirectional chiral edge mode per edge, which is robust to defects and immune to backscattering (You et al., 2019).

2. Edge-Mode Dispersion and Electromagnetic Profiles

A finite-width, infinitely long ribbon geometry demonstrates a single unidirectional edge band $r = 120$1 within the bulk bandgap. Near $r = 120$2 THz, the dispersion is approximately linear: $r = 120$3 with group velocity $r = 120$4. Finite-element method (FEM) simulations show that the edge-plasmon electric fields are strictly confined to the metasurface boundary, decaying evanescently into the bulk within tens of nm ($r = 120$5m). In contrast, bulk modes at the same frequencies are delocalized over the graphene sheet. The edge field enhancement factor $r = 120$6 exceeds 100, enabling large local fields for a given input power.

3. Nonlinear Maxwell Formalism and Coupled-Mode Theory

The dominant nonlinearity in magnetized graphene is characterized by the third-order susceptibility $r = 120$7 m$r = 120$8V$r = 120$9. The nonlinear surface current at frequency $B_\perp$0 ($B_\perp$1) relevant for FWM processes is: $B_\perp$2 with $B_\perp$3 nm being the effective graphene thickness. Maxwell’s equations are projected onto the edge modes and recast through perturbative coupled-mode theory (CMT), yielding: $B_\perp$4 where $B_\perp$5 is the phase-mismatch, $B_\perp$6 are linear loss coefficients, and $B_\perp$7 is the effective four-wave mixing parameter.

4. FWM Gain, Effective Nonlinearity, and Thresholds

The effective waveguide nonlinearity is quantified by

$B_\perp$8

with full-wave simulations yielding $B_\perp$9 W$B_\perp$0m$B_\perp$1. Degenerate FWM gain can alternatively be characterized by $B_\perp$2 W$B_\perp$3m$B_\perp$4. These values exceed typical silicon nanowire values by more than ten orders of magnitude.

In the undepleted-pump, small-signal regime, signal power amplification follows: $B_\perp$5 where the net gain coefficient is $B_\perp$6 and $B_\perp$7, with $B_\perp$8 the edge-plasmon lifetime and $B_\perp$9 the group velocity. The threshold pump power for net gain is $$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$0. For $$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$1 ps and $$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$2, $$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$3 nW enables full net gain at sub-10 nW pump powers.

5. Figures of Merit and Scaling Behavior

The following table summarizes key physical parameters and scaling trends for the system (You et al., 2019):

Parameter	Typical Value	Significance
Group velocity $$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$4	$$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$5 ($$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$6 m/s)	Slow light, enhances nonlinearity
Plasmon lifetime $$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$7	Up to 50 ps (with hBN-supported graphene)	Boosts net gain and propagation length
Field enhancement $$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$8	$$\sigma_s(\omega)=\begin{bmatrix}\sigma_L(\omega) & \sigma_H(\omega)\ -\sigma_H(\omega) & \sigma_L(\omega)\end{bmatrix}$$9	Increases local intensity
Mode confinement area $\sigma_L, \sigma_H$0	$\sigma_L, \sigma_H$1	Deep-subwavelength confinement

All these factors synergistically increase the intensity per input power, thereby dramatically enhancing the effective nonlinear interaction strength $\sigma_L, \sigma_H$2.

6. Full-Wave Simulations and Topological Robustness

Finite-element simulations detail degenerate FWM with a pump at $\sigma_L, \sigma_H$3 THz, a seeded signal at $\sigma_L, \sigma_H$4 THz, and idler generation at $\sigma_L, \sigma_H$5 THz. These simulations confirm perfect phase matching ($\sigma_L, \sigma_H$6) and unidirectional propagation of all participating modes, even along sharp bends. The topologically protected edge channels exhibit defect-immune transport, and the exponential growth of signal and idler power with distance persists as long as $\sigma_L, \sigma_H$7 ps, overcoming intrinsic graphene loss. Energy conservation is maintained throughout evolution.

7. Prospects for Applications in Photonics and Quantum Technology

Topological edge plasmon FWM processes support the development of ultra-low-power-consumption photonic components, including switches, frequency converters, and on-chip sources of nonclassical light for quantum communications. The deep-subwavelength confinement, immunity to disorder-induced backscattering, and the uniquely high nonlinear figure of merit of graphene edge plasmons distinguish this platform as highly promising for robust, scalable, and integrated active photonic systems at the nanoscale (You et al., 2019).