Far-field radiation of bulk, edge and corner eigenmodes from a finite 2D Su-Schrieffer-Heeger plasmonic lattice

This paper employs a coupled electromagnetic dipole formalism to analyze the far-field radiation of bulk, edge, and corner eigenmodes in a finite 2D Su-Schrieffer-Heeger plasmonic lattice, demonstrating how symmetry breaking and out-of-plane dipolar resonances dictate the darkness, Q-factors, and complex radiation patterns of these topological states.

The Gist

Imagine you have a giant, flat chessboard made not of wood, but of tiny, shiny metal beads (nanoparticles). These beads are so small that they interact with light in a special way, acting like miniature radio antennas that can catch and re-send light waves.

This paper is about figuring out exactly how these beads talk to each other and, more importantly, how they shout their message to the rest of the world (the "far-field").

Here is the breakdown of their discovery, using some everyday analogies:

1. The Setup: The "Su-Schrieffer-Heeger" Chessboard

The researchers built a specific pattern of these beads. Imagine a square grid where the distance between beads isn't always the same. Some neighbors are close, some are far. This creates a "broken" symmetry, like a dance floor where some dancers are holding hands tightly and others are standing apart.

Because of this specific arrangement, the light trapped inside the array behaves in three distinct ways:

• Bulk Modes: The "crowd." These are waves that ripple through the entire middle of the board, like a stadium wave.
• Edge Modes: The "fence." These waves get stuck running along the very edge of the board, unable to go into the middle.
• Corner Modes: The "VIPs." These waves get trapped in the four corners of the board, like a ball bouncing in a corner of a room.

2. The Big Question: Who Can Be Heard?

The main goal of the paper was to answer: If these beads vibrate, can we hear them from far away?

In physics, "hearing from far away" means the light escapes the board and travels into space. If the light stays trapped, the mode is "dark" (silent). If it escapes, it's "bright" (loud).

The researchers used a clever trick: instead of shining a light on the board and seeing what bounces off, they looked at the "eigenmodes." Think of this as listening to the natural hum of a guitar string. They asked, "If this specific pattern of vibration happened, would it make a sound we can hear?"

3. The Discovery: The "Quiet" vs. The "Loud"

The "Dark" Secrets (High Quality, Low Noise)

The researchers found that some patterns are antisymmetric. Imagine a group of people clapping. If half the people clap on the "beat" and the other half clap on the "off-beat," the sound waves cancel each other out. The room goes silent.

• The Result: These "antisymmetric" modes are dark. They don't radiate light into the distance.
• Why is this good? Because they don't waste energy shouting into the void, they hold onto their energy longer. In physics, this is called a High Q-Factor. It's like a bell that rings for a long time without fading. These "dark" modes are perfect for storing energy or making very precise lasers.

The "Bright" Secrets (The Loudmouths)

Other patterns are symmetric. Everyone claps at the same time. The sound waves add up, creating a loud boom.

• The Result: These modes are bright. They radiate light easily.
• The Catch: Because they shout so much, they lose their energy quickly. They have a lower Q-factor (they ring for a shorter time).

4. The Special Case of the Corners

Usually, you'd think a corner is just a tiny spot, so it shouldn't make much noise. But here's the twist:

• The Bulk (Middle) modes at the center of the board are often "dark" because of a rule called the "transversality condition" (light waves can't vibrate in the direction they are traveling).
• The Corner modes, however, are weird. Even though they are tiny, they don't follow the same rules as the big waves in the middle. They manage to find "open channels" to escape.
• The Analogy: Imagine a whisper in a crowded room (the bulk) that gets lost. But a whisper in a corner (the corner mode) might bounce off the walls in a way that lets it escape the room entirely. The paper proves that these corner states are actually loud and can be seen from far away, even if they are tiny.

5. Why Does This Matter?

This research is like learning the secret language of light.

• For Engineers: If you want to build a super-efficient solar cell or a tiny laser, you want to use the "dark" modes (the ones that hold energy).
• For Sensors: If you want to detect a virus or a chemical, you want the "bright" modes (the ones that shout to the detector).
• The Takeaway: By changing the shape of the "chessboard" (breaking the symmetry), scientists can now design materials that either trap light perfectly or shoot it out exactly where they want.

Summary in One Sentence

The paper shows that by arranging tiny metal beads in a specific, broken pattern, we can create "silent" vibrations that hold energy for a long time (great for lasers) and "loud" vibrations that shoot light out from the corners (great for sensors), giving us total control over how light behaves at the nanoscale.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding and controlling the far-field radiation properties of topological plasmonic modes in finite two-dimensional (2D) arrays. While topological photonics has successfully demonstrated the existence of robust edge and corner states (protected by symmetries) in infinite or large periodic systems, there is a gap in understanding how these specific eigenmodes (bulk, edge, and corner) radiate into the far-field when the system is finite and terminated.

Key questions include:

• How do the spatial symmetries and localization of these modes affect their ability to couple to free-space radiation?
• Which modes are "bright" (radiative) and which are "dark" (non-radiative/evanescent)?
• How does the array size influence the radiation patterns and quality factors (Q-factors) of these modes?
• Can symmetry breaking in multipartite unit cells be exploited to tailor these optical properties?

2. Methodology

The authors employ a coupled electromagnetic dipole formalism to model the system. This approach replaces the optical response of each plasmonic nanoparticle with an electric dipole, allowing for a semi-analytical treatment of the collective modes.

• System Model: A finite 2D Su-Schrieffer-Heeger (SSH) lattice composed of plasmonic nanoparticles. The lattice features a unit cell with four sites (A, B, C, D) and alternating inter-particle distances controlled by a parameter $\beta$.
• Eigenmode Analysis: Instead of simulating the response to an external excitation (which convolutes the response of multiple modes), the authors solve the eigenvalue problem of the coupled-dipole matrix. This isolates the contribution of each individual resonant array mode to the far-field.
• Effective Dispersion Bands: For finite arrays where translational symmetry is broken, the authors define "effective dispersion bands" by projecting the real-space eigenmodes onto momentum space ($q$-space) using a Fourier-like transform. This allows them to map finite-size modes to the dispersion bands of the infinite periodic system.
• Far-Field Calculation: The far-field radiation patterns are calculated by summing the fields radiated by all dipoles in the array, accounting for interference effects.
• Symmetry Isolation: To isolate out-of-plane dipolar resonances (which are the focus of the study), the authors utilize elongated nanoparticles (nanospheroids) oriented perpendicular to the lattice plane, acting as projectors for the electric field.

3. Key Contributions

• Eigenmode-Based Far-Field Analysis: The paper establishes a rigorous method to calculate the far-field radiation of specific bulk, edge, and corner eigenmodes in finite topological arrays, separating them from the background continuum.
• Symmetry-Dependent Radiation Rules: The study proves that the radiative nature of a mode is dictated by its symmetry relative to the lattice and the transversality condition ($k \cdot E = 0$).
• Identification of Dark vs. Bright States:
  • Bulk Modes: All out-of-plane bulk $\Gamma$-modes are proven to be dark (Bound States in the Continuum, BICs) due to the transversality condition. However, antisymmetric modes (e.g., $d_{xy}$ symmetry) are significantly darker and possess higher Q-factors than symmetric modes ($s$-symmetry).
  • Edge Modes: The paper distinguishes between bright edge states (symmetric, $E_{3,4}$) and dark edge states (antisymmetric, $E_{1,2}$). Bright edge states remain radiative even in the infinite limit (except at normal incidence), while dark edge states vanish in the far-field due to destructive interference.
  • Corner Modes: Corner states, being quasi-0D and highly localized, do not have a defined crystal momentum. Consequently, they possess open radiation channels and are radiative even at the $\Gamma$ point, unlike bulk modes.
• Size-Dependent Evolution: The authors demonstrate how radiation patterns evolve with array size ($N_c$). As $N_c \to \infty$, bulk modes collapse into single channels (vanishing for dark modes), while edge and corner modes maintain distinct radiation lobes or in-plane concentration.

4. Key Results

• Bulk Modes ($B_1, B_4$):
  • The $B_4$ mode ($s$-symmetry) is dark at $\Gamma$ but becomes radiative at oblique angles. As the array size increases, its radiation intensity decreases, approaching a BIC.
  • The $B_1$ mode ($d_{xy}$-symmetry) is darker than $B_4$. Due to antisymmetry within the unit cell, it experiences destructive interference across the entire $\Gamma-X$ and $\Gamma-Y$ lines, resulting in a much higher Q-factor and near-zero far-field radiation even for finite sizes.
• Edge Modes ($E_{1-4}$):
  • Symmetric Edge Modes ($E_{3,4}$): These are "bright." Even as the array grows, they maintain strong radiation lobes perpendicular to the edges. They are only dark at normal incidence ($\Gamma$).
  • Antisymmetric Edge Modes ($E_{1,2}$): These are "dark." The antisymmetry causes destructive interference that cancels radiation perpendicular to the boundary. Their far-field contribution vanishes as the array size increases.
• Corner Modes ($C_{1-4}$):
  • Corner states are highly localized (0D). Their radiation patterns do not vanish with increasing array size.
  • Symmetric corner modes ($C_4$) radiate with a pattern concentrated around the in-plane direction ($k_z \approx 0$).
  • Antisymmetric corner modes ($C_1$, $d_{xy}$) also radiate, with patterns flattening around $k_z=0$.
  • Crucially, corner states are accessible from the far-field because their lack of translational symmetry prevents the formation of a forbidden radiation channel at $\Gamma$.
• Quality Factors (Q-factors):
  • The highest Q-factors are found in the antisymmetric dark modes (e.g., $B_1$ and $E_{1,2}$).
  • The Q-factor peaks at high-symmetry points ($\Gamma$ and $M$) in the Brillouin zone. The peak at $M$ is attributed to rotational symmetry causing destructive interference, similar to the mechanism at $\Gamma$.

5. Significance

• Tailoring Optical Properties: The work demonstrates that by engineering the symmetry of multipartite unit cells (breaking symmetries), one can selectively enhance or suppress far-field radiation. This allows for the design of plasmonic lattices with specific radiative properties.
• Robustness of Quasi-BICs: The study confirms that antisymmetric modes in plasmonic arrays act as robust Quasi-Bound States in the Continuum (q-BICs). Despite the high optical losses inherent in metals, these symmetry-protected modes achieve significantly higher Q-factors than their symmetric counterparts.
• Applications in Nanophotonics:
  • Lasing and Nonlinear Optics: The high Q-factors of dark modes and the high local density of states (LDOS) of corner states make them ideal candidates for low-threshold lasing and enhanced nonlinear interactions.
  • Sensing and Imaging: Understanding the radiation patterns of topological states is crucial for far-field detection schemes and photoluminescence imaging of topological edge/corner states.
  • Design Guidelines: The paper provides a blueprint for designing plasmonic metasurfaces where specific modes can be isolated for either strong coupling to free space (bright states) or strong near-field confinement (dark states).

In summary, the paper bridges the gap between topological band theory and practical far-field optics, proving that symmetry is the primary lever for controlling whether topological plasmonic modes radiate or remain confined.