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

\'Alvaro Buend\'ia; Jos\'e Antonio S\'anchez Gil; Jos\'e Luis Pura; Vincenzo Giannini

arxiv: 2510.08063 · v2 · pith:426C33J4new · submitted 2025-10-09 · ❄️ cond-mat.mes-hall · physics.optics

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

\'Alvaro Buend\'ia , Jos\'e Luis Pura , Vincenzo Giannini , Jos\'e Antonio S\'anchez Gil This is my paper

Pith reviewed 2026-05-21 21:34 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.optics

keywords Su-Schrieffer-Heeger latticeplasmonic nanoparticlesfar-field radiationeigenmodesout-of-plane dipolessymmetry breakingfinite arraystopological photonics

0 comments

The pith

Antisymmetric modes in finite 2D SSH plasmonic lattices radiate less and show higher Q-factors than symmetric ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an eigenmode analysis using a coupled electromagnetic dipole formalism to isolate far-field radiation contributions from bulk, edge, and corner modes in a finite 2D Su-Schrieffer-Heeger array of plasmonic nanoparticles. It examines out-of-plane dipolar resonances and exploits symmetry breaking in multipartite unit cells to control radiation. Antisymmetric modes turn out darker with higher quality factors than symmetric counterparts. Bulk modes at the Gamma point are completely dark owing to their out-of-plane character. Edge and corner states require additional in-plane symmetries to suppress radiation, and their patterns grow more complex and in-plane concentrated as array size increases.

Core claim

By employing a coupled electromagnetic dipole formalism, the contribution of each array mode to the far-field radiation is isolated for bulk, edge and corner out-of-plane eigenmodes in a finite 2D SSH plasmonic nanoparticle array. The breaking of symmetries in multipartite unit cells is exploited to tailor the optical properties and far-field radiation of the resonant modes. Antisymmetric modes are darker and have higher Q-factors than their symmetric counterparts. The out-of-plane nature of the dipolar resonances imposes that all bulk Γ-modes are dark, while corner and edge states need extra in-plane symmetries to cancel the far-field radiation; radiation patterns in turn become more more

What carries the argument

Coupled electromagnetic dipole formalism that isolates the far-field radiation contribution of each array mode.

Load-bearing premise

The coupled electromagnetic dipole formalism accurately captures the far-field radiation and eigenmode isolation for out-of-plane dipolar resonances in the finite array without significant contributions from higher-order multipoles or full retardation effects.

What would settle it

Direct experimental comparison of measured far-field radiation intensity and quality factors between symmetric and antisymmetric modes in fabricated finite 2D SSH plasmonic nanoparticle arrays of different sizes.

read the original abstract

Subwavelength arrays of plasmonic nanoparticles allow us to control the behaviour of light at the nanoscale. Here, we develop an eigenmode analysis, employing a coupled electromagnetic dipole formalism, which permits us to isolate the contribution to the far-field radiation of each array mode. Specifically, we calculate the far-field radiation patterns by bulk, edge and corner out-of-plane eigenmodes in a finite 2D Su-Schrieffer-Heeger (SSH) array of plasmonic nanoparticles with out-of-plane dipolar resonances. The breaking of symmetries in multipartite unit cells is exploited to tailor the optical properties and far-field radiation of the resonant modes. We prove that the antisymmetric modes are darker and have higher Q-factors than their symmetric counterparts. Also, the out-of-plane nature of the dipolar resonances imposes that all bulk $\Gamma$-modes are dark, while corner and edge states need extra in-plane symmetries to cancel the far-field radiation; radiation patterns in turn become more complex and concentrated along the array plane with increasing array size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper isolates far-field radiation from bulk, edge, and corner modes in a finite SSH plasmonic array via coupled dipoles and uses symmetry to explain which ones stay dark.

read the letter

The main point is that this work takes the SSH lattice into a finite 2D plasmonic nanoparticle array, runs a coupled-dipole eigenmode decomposition, and separates the radiation patterns coming from bulk, edge, and corner states. They focus on out-of-plane dipoles and show how breaking symmetries in the unit cell controls darkness and Q-factors. Antisymmetric modes come out darker with higher Q than symmetric ones, bulk Gamma modes are dark from the out-of-plane orientation alone, and edge or corner modes need extra in-plane symmetry to cancel radiation. Patterns get more complex and stay in the plane as the array grows. That combination of mode isolation plus explicit symmetry arguments for radiation looks new relative to the cited literature. The calculations follow standard coupled-dipole equations applied to the SSH geometry, which is a reasonable way to get concrete trends for device design in this subfield. The soft spot is the dipole approximation itself. It assumes point dipoles dominate and that higher multipoles or retardation do not spoil the symmetry cancellations, especially at larger sizes. The abstract gives no error bars, convergence checks, or full-wave comparisons, so the claimed darkness ordering and Q-factor differences could shift if those effects matter for real plasmonic particles. This is a minor-to-moderate concern rather than a load-bearing flaw, but it needs addressing. The paper is for people working on topological plasmonics or subwavelength arrays who care about radiation control and dark modes. A reader who wants practical symmetry rules for suppressing or directing emission in finite lattices will get usable guidance from the mode-by-mode patterns. The thinking is clear and the methods are standard, so it deserves a serious referee who can ask for validation of the approximation against more complete simulations. I would send it to review.

Referee Report

1 major / 2 minor

Summary. The manuscript develops an eigenmode analysis employing a coupled electromagnetic dipole formalism to isolate the far-field radiation contributions of bulk, edge, and corner out-of-plane eigenmodes in a finite 2D Su-Schrieffer-Heeger plasmonic nanoparticle array. It exploits symmetry breaking in multipartite unit cells to show that antisymmetric modes are darker and possess higher Q-factors than symmetric counterparts, that all bulk Γ-modes are dark due to the out-of-plane dipolar character, and that corner and edge states require additional in-plane symmetries to suppress radiation; radiation patterns are reported to grow more complex and concentrate in the array plane with increasing size.

Significance. If the symmetry-based conclusions hold under the employed model, the work supplies a transparent, analytically grounded route to predict and engineer darkness and Q-factors in topological plasmonic lattices without parameter fitting. The emphasis on finite-size effects and the explicit separation of mode contributions via the dipole formalism are useful for guiding experimental design of subwavelength directional sources or high-Q resonators.

major comments (1)

[Coupled electromagnetic dipole formalism (methods section)] The central claims on mode darkness, Q-factor ordering, and radiation cancellation rest on the coupled electromagnetic dipole formalism accurately capturing far-field radiation for out-of-plane resonances. No quantitative estimate of higher-multipole or retardation corrections, nor comparison to full-wave simulations for representative array sizes, is provided; this assumption is load-bearing for the assertion that bulk Γ-modes are dark and that antisymmetric modes are systematically darker.

minor comments (2)

[Abstract] The abstract asserts that radiation patterns become more complex and in-plane concentrated with array size, but does not report any quantitative metric (e.g., directivity or integrated side-lobe level) to support the qualitative statement.
[Introduction or model section] Notation for the multipartite unit-cell symmetries and the precise definition of the antisymmetric versus symmetric eigenmodes could be clarified with an explicit table or diagram early in the text to aid readers unfamiliar with the SSH geometry.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the single major comment below and are prepared to revise the work to strengthen the presentation of the coupled-dipole results.

read point-by-point responses

Referee: [Coupled electromagnetic dipole formalism (methods section)] The central claims on mode darkness, Q-factor ordering, and radiation cancellation rest on the coupled electromagnetic dipole formalism accurately capturing far-field radiation for out-of-plane resonances. No quantitative estimate of higher-multipole or retardation corrections, nor comparison to full-wave simulations for representative array sizes, is provided; this assumption is load-bearing for the assertion that bulk Γ-modes are dark and that antisymmetric modes are systematically darker.

Authors: We agree that the dipole approximation is central to the quantitative claims. The formalism is employed precisely because it permits an exact decomposition of the far-field into individual eigenmode contributions while retaining the symmetries of the finite SSH lattice. For the subwavelength particles and out-of-plane dipolar resonances considered, the leading radiation term is the electric dipole; symmetry-enforced cancellations (antisymmetric pairs, bulk Γ-point out-of-plane cancellation, and additional in-plane ordering for edge/corner states) are therefore expected to survive small higher-multipole or retardation corrections. Nevertheless, we acknowledge that an explicit validation would increase confidence. In the revised manuscript we will add (i) an order-of-magnitude estimate of the relative strength of quadrupole and magnetic-dipole terms based on the particle-radius-to-wavelength ratio and (ii) a direct comparison of far-field patterns and Q-factors for a representative small array (e.g., 4×4) obtained from both the coupled-dipole model and full-wave FDTD simulations, confirming that the darkness ordering and radiation patterns remain consistent within the parameter regime of the study. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard coupled-dipole analysis applied to SSH geometry

full rationale

The paper develops an eigenmode analysis using the coupled electromagnetic dipole formalism to isolate far-field contributions from bulk, edge, and corner modes in a finite 2D SSH plasmonic array. Claims that antisymmetric modes are darker with higher Q-factors, that all bulk Γ-modes are dark due to out-of-plane dipoles, and that corner/edge states require additional in-plane symmetries follow directly from symmetry arguments within this standard framework. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the derivation remains self-contained against external benchmarks of the dipole approximation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard coupled-dipole approximation and the topological properties of the SSH lattice; no new particles or forces are introduced.

free parameters (1)

inter-particle spacing and radius
Geometry parameters that define the finite array and coupling strengths in the dipole model.

axioms (1)

domain assumption Coupled electromagnetic dipole formalism is sufficient to isolate far-field radiation of each eigenmode
Invoked in the abstract as the method that permits isolation of mode contributions.

pith-pipeline@v0.9.0 · 5738 in / 1389 out tokens · 110400 ms · 2026-05-21T21:34:09.223758+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We develop an eigenmode analysis, employing a coupled electromagnetic dipole formalism... out-of-plane dipolar resonances... antisymmetric modes are darker and have higher Q-factors... all bulk Γ-modes are dark
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the out-of-plane nature of the dipolar resonances imposes that all bulk Γ-modes are dark

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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