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Rigorous modal analysis of plasmonic nanoresonators

Wei Yan, Rémi Faggiani, Philippe Lalanne

arXiv:1711.05011v3physics.optics

TL;DR

Open, dispersive nanoresonators are difficult to analyze transparently because conventional solvers do not explicitly compute their resonance states. The paper uses auxiliary fields, a finite-element QNM solver, and closed-form excitation coefficients to reconstruct scattered fields. Increasing the retained states yields accurate predictions, including for complex backgrounds where PML-modes are needed.

  • Problem

    Open non-Hermitian nanoresonators lack mature modal-expansion tools that transparently represent resonance states across general geometries and dispersive materials.

  • Method

    The authors combine auxiliary fields, a finite-element solver for QNMs and PML-modes, and closed-form modal excitation coefficients for scattered-field reconstruction.

  • Results

    Retaining 200 PML-modes is necessary for quantitative prediction of the scattering cross-section in the tested complex-background case.

  • Takeaways & Limitations

    The formalism supports physically interpretable analysis of Fano interference, quenching, and coupling with continuum modes in complicated plasmonic resonators.

Abstract

from arXiv · show

The specificity of modal-expansion formalisms is their capabilities to model the physical properties in the natural resonance-state basis of the system in question, leading to a transparent interpretation of the numerical results. In electromagnetism, modal-expansion formalisms are routinely used for optical waveguides. In contrast, they are much less mature for analyzing open non-Hermitian systems, such as micro and nanoresonators. Here, by accounting for material dispersion with auxiliary fields, we considerably extend the capabilities of these formalisms, in terms of computational effectiveness, number of states handled and range of validity. We implement an efficient finite element solver to compute the resonance states, and derive new closed-form expressions of the modal excitation coefficients for reconstructing the scattered fields. Together, these two achievements allow us to perform rigorous modal analysis of complicated plasmonic resonators, being not limited to a few resonance states, with straightforward physical interpretations and remarkable computation speeds. We particularly show that, when the number of states retained in the expansion increases, convergence towards accurate predictions is achieved, offering a solid theoretical foundation for analyzing important issues, e.g. Fano interference, quenching, coupling with the continuum, which are critical in nanophotonic research.

I. Introduction

The paper develops a comprehensive modal framework for open, dispersive nanoresonators, addressing interpretability and computational limitations of conventional scattering solvers. Auxiliary fields linearize dispersive materials and support QNM analysis for general three-dimensional resonators.

  • Motivation: Conventional time- and frequency-domain Maxwell solvers do not explicitly compute resonance states, limiting interpretability and requiring repeated calculations across driving conditions.The modal formalism instead operates at complex frequencies set by the resonator’s natural states.
  • Gap: Existing QNM theory contains important ingredients but is often scarcely presented or restricted to peculiar geometries.The paper targets arbitrary three-dimensional shapes, materials, and potentially non-uniform backgrounds.
  • Framework: The framework generalizes modal analysis to open resonators by combining finite-element QNM computation with PML-based treatment of outgoing-wave conditions.The discretized mapped operator contains both physical QNMs and numerical PML-modes forming a complete basis.
  • Significance: The joint numerical and theoretical development aims to improve physical insight and computational speed in nanophotonic resonance analysis.The authors position it as a step toward a comprehensive modal theory analogous to optical waveguide theory.
  • Augmented-field approach: Auxiliary fields convert the nonlinear eigenproblem of Lorentz-Drude dispersive materials into a linear eigenproblem.The hidden material variables are reintroduced explicitly, enabling a standard linear formulation in the augmented state space.

B. QNM eigensolver implementation

The authors implement an auxiliary-field QNM eigensolver in COMSOL using finite-element formulations for dispersive plasmonic resonators. The solver computes QNMs and PML-modes for complex three-dimensional geometries while accurately resolving curved and multiscale features.

  • Implementation: The QNM solver is built from the augmented formulation and transformed into a standard quadratic eigenvalue problem.Stable algorithms can then be applied after the coupled equations are discretized.
  • Implementation: The coupled partial differential equations are converted into weak formulations and solved with COMSOL’s built-in iterative eigensolver.The implementation uses user-defined COMSOL interfaces for the discretized equations.
  • Capabilities: Finite-element meshes accurately represent arbitrary three-dimensional plasmonic resonators combining curved shapes, planar interfaces, and sharp corners.The authors report superior accuracy and performance compared with prior finite-difference solvers.
  • Bowtie example: For the silver bowtie antenna, the solver computes transverse and longitudinal QNMs together with PML-modes.The eigenstate spectrum includes transverse modes at nonzero frequencies and longitudinal modes such as bulk plasmons and static states.
  • Bowtie example: Figure 1 displays eigenstate energies and decay rates alongside field-intensity maps for QNMs Q1–Q3 and PML-mode P1.PML-mode intensity is concentrated predominantly inside the PML region.

III. Reconstruction: Orthogonality relation, modal excitation coefficients

The reconstruction theory supplies orthogonality relations and closed-form modal excitation coefficients for dispersive nanoresonators. These results enable scattered-field reconstruction from QNMs and PML-modes while preserving the physical interpretation of the modal basis.

  • Orthogonality relation: The augmented formulation yields orthogonality relations for the complete set of QNMs and PML-modes through unconjugated Lorentz reciprocity.The resulting normalization is formulated over the entire mapped space, including the PMLs.
  • Orthogonality relation: The normalization relation differs from conjugate-energy formulations because the QNM eigenproblem is non-Hermitian.It is consistent with earlier QNM normalization work while accounting for the non-Hermitian augmented problem.
  • Modal excitation: A closed-form expression is derived for the modal excitation coefficients used to reconstruct the scattered field at real frequencies.The coefficients are applicable to resonators composed of dispersive materials in non-uniform backgrounds.
  • Modal excitation: The excitation coefficients use overlap integrals between each resonance mode and the incident field over the resonator domain.The weighting functions and background permittivity enter the analytic reconstruction expression.
  • Interpretation: The modal expansion explicitly exposes Fano-like interference between dominant eigenvectors and can provide highly accurate predictions when the relevant basis is complete.The mapped-operator eigenvectors form a complete set absent accidental degeneracies such as exceptional points.

IV. Numerical results and validation

The numerical tests assess the QNM-expansion formalism across resonator geometries, illumination conditions, physical interpretations, computational speed, exactness, and convergence. The supplied passage frames these tests as validation of the modeling capabilities.

  • Validation: The authors test the formalism on selected examples to clarify resonant physics and analyze complex situations involving many modes.Additional tests examine computational speed, exactness, and convergence as the number of retained eigenstates changes.

A. Spectral and temporal analysis of antenna in uniform backgrounds

In a uniform-background bowtie antenna, QNM expansions accurately reproduce spectral and temporal responses while exposing how individual modes shape the dynamics. Increasing the retained modes supports convergence, fast computation, and interpretation of interference features.

  • Spectral analysis: No more than 10 QNMs significantly affect the bowtie’s visible extinction and absorption spectra.Longitudinal QNMs are not excited under the stated plane-wave illumination.
  • Temporal analysis: The temporal expansion reconstructs the bowtie response to a 10-fs Gaussian pulse accurately compared with COMSOL data.The pulse has central frequency 0.5ω_p and bandwidth 0.058ω_p.
  • Computational performance: The temporal response is computed in about 10 minutes on an ordinary desktop, with modal-coefficient evaluation much faster than QNM computation.The method is presented as a potential fast alternative to conventional FDTD for temporal responses.
  • Temporal interpretation: Off-resonant QNMs follow the incident pulse, whereas two dominant modes produce the long-time oscillatory tail through beating.One highlighted dark mode has Q≈220 and is energy-matched to the driving pulse.
  • Interference analysis: QNM superposition interprets steep asymmetric Fano resonances as spectral mode interference shaped by geometry and driving-field parameters.The modal representation separates the roles of individual modes in the resulting lineshapes.

B. Quenching in nanoresonators

The QNM expansion captures quenching in metallic nanoantennas only when many modes are retained. It matches Green-tensor and COMSOL results and links near-surface loss to high-order localized plasmons.

  • Decay-rate enhancement: With 500 QNMs, the dipole decay rate rises from 25 to 200 as separation from the silver nanorod decreases from 20 nm to 2 nm.The prediction agrees closely with fully vectorial Green-tensor data from COMSOL.
  • Decay-rate enhancement: A single dominant QNM fails to predict the decay-rate increase at small emitter–surface separations.Earlier QNM treatments using one dominant mode are contrasted with the 500-QNM result.
  • Modal origin: Quenching is attributed to excitation of high-order plasmon modes that accumulate near the surface-plasma frequency of flat interfaces.As the separation vanishes, these modes resemble those of a flat interface and become less dependent on antenna shape.
  • Ohmic quenching: Accurate absorbed-power-density distributions at 2 nm require a high truncation rank, with 500 QNMs agreeing quantitatively with COMSOL.The loss builds up directly beneath the emitter as the truncation rank increases.

C. Nanoresonators in complex backgrounds: importance of PML-modes

For resonators in non-uniform backgrounds, QNMs alone are incomplete because branch-cut physics enters the Green tensor. Including PML-modes restores accurate reconstruction of scattering and radiation patterns.

  • Completeness: QNM completeness breaks down for 2D systems and 3D resonators in non-uniform backgrounds such as thin-film substrates.The breakdown is associated with branch cuts in the complex-frequency Green tensor.
  • Complex-background example: The nanobullet-on-semiconductor-slab example requires PML-modes for accurate reconstruction.The slab has infinite transverse extent, creating a complex background environment.
  • Mode spectrum: The complex eigenfrequency spectrum contains vertical PML-mode branches associated with PML-truncated slab-waveguide modes.These branches are regularly spaced along the horizontal frequency axis.
  • Mode identification: QNMs and PML-modes are distinguished by their different sensitivity to PML-parameter variations.QNMs remain insensitive, while PML-modes shift when the PML parameters change.
  • Reconstruction accuracy: Retaining 20 QNMs alone cannot accurately predict scattering, whereas adding 200 PML-modes yields quantitative agreement with COMSOL.The same improvement applies to far-field radiation diagrams.

V. Conclusion

The formalism combines a FEM-based solver for dissipative, dispersive resonators with general modal-excitation coefficients to reconstruct fields. It supports faithful analysis of complex geometries and non-uniform backgrounds while remaining open to numerical extension.

  • Conclusion: The method’s two main building blocks are a robust FEM-based QNM solver and a general modal-excitation coefficient expression.Together they reconstruct responses for dissipative and dispersive nanoresonators.
  • Conclusion: The formalism provides faithful predictions for plasmonic nanoresonators with complex geometries in non-uniform backgrounds.The conclusion presents this as progress toward deploying modal theories for nanoresonator analysis.
  • Future developments: Future numerical development may incorporate dispersive PMLs to compute QNMs over an extended spectral range.The paper also identifies optomechanical cooling and charge-carrier coupling as possible application areas.

SUPPLEMENTARY INFORMATION

The supplementary information documents the authorship and affiliation of the work and provides derivations, implementation details, and numerical evaluations of the modal approach.

  • Wei Yan, Rémi Faggiani, and Philippe Lalanne authored the work.
  • The authors are affiliated with Laboratoire Photonique, Numérique et Nanosciences at IOGS-Univ. Bordeaux-CNRS in Talence, France.
  • The supplementary information derives the modal formulae, details the COMSOL QNM eigensolver, and evaluates solver accuracy and reconstruction convergence.

1. DEFINITIONS AND NOTATION

The formalism models dispersive resonators by linearizing Maxwell’s equations with auxiliary fields and representing the electromagnetic state as an augmented vector.

  • A single-pole Lorentz permittivity is generalized to N poles with spatially dependent material parameters.
  • Two auxiliary fields are introduced for each Lorentz pole to linearize the source-free Maxwell equations with respect to frequency.
  • For a single-pole Lorentz material, the augmented electromagnetic vector contains H, E, P, and J components, with an external source term.
  • Quasi-normal modes satisfy source-free Maxwell equations, and their eigenfrequency and eigenvector are represented within the augmented formulation.
  • The auxiliary-field matrix supports derivation of unconjugated Lorentz reciprocity and QNM/PML-mode orthogonality, differing from a Hermitian dispersive-system matrix.

2. ELECTROMAGNETIC THEOREMS

The electromagnetic theorems establish reciprocity and energy-balance relations for dispersive resonators with auxiliary fields, including expressions for stored energy, absorption, radiation, and quality factor.

  • 2.1. Unconjugated form of the Lorentz reciprocity theorem: The auxiliary-field Lorentz reciprocity theorem is derived from two Maxwell solutions and supports QNM and PML-mode orthogonality.
  • 2.1. Unconjugated form of the Lorentz reciprocity theorem: The theorem applies to both the continuous unbounded operator and its finite discretization bounded by perfectly matched layers.
  • 2.1. Unconjugated form of the Lorentz reciprocity theorem: The derivation uses the auxiliary-field operator, divergence theorem, and algebraic rearrangements to obtain the reciprocity relation.
  • 2.2. Poynting theorem: Stored energy combines electromagnetic energy with the electrons’ mechanical kinetic and potential energy in dispersive materials.
  • 2.2. Poynting theorem: The Poynting theorem equates electromagnetic-energy decay with absorption and radiation losses minus input power.
  • 2.2. Poynting theorem: For a source-free QNM, input power vanishes, giving −2Im(ω̃)We = Pabs + Prad and a quality-factor relation.

3. MODAL FORMALISM: THEORETICAL RESULTS

The modal formalism establishes completeness, orthogonality, and reconstruction properties for QNMs and PML-modes in finite PML-mapped spaces. PML-modes are essential when QNMs alone are incomplete, while their numerical identification and physical interpretation depend on the system and PML behavior.

  • 3. MODAL FORMALISM: THEORETICAL RESULTS: The theoretical results apply to QNMs and PML-modes of a discretized operator defined on finite space bounded by PMLs.The open problem is replaced by a closed mapped-space eigenproblem.
  • 3.1. Completeness of QNMs and PML-modes: Completeness is assumed away from exceptional points, enabling a biorthogonal eigenstate basis constructed with adjoint eigenstates.Exceptional points can coalesce eigenstates and prevent completeness.
  • 3.2. Orthogonality of QNMs and PML-modes: Unconjugated Lorentz reciprocity yields the orthogonality relation for QNMs and PML-modes in PML-mapped space.Perfect electric/magnetic conductor boundaries make the outer PML surface integral vanish.
  • 3.3. Importance of PML-modes: In open systems, QNM completeness depends on geometry and background: branch cuts in 2D or non-uniform backgrounds make pole expansions incomplete.For simple 1D and 3D resonators in uniform backgrounds, completeness inside the resonator is established.
  • 3.3. Importance of PML-modes: Finite PML-mapped spaces guarantee completeness and can recover relevant open-system QNMs when PMLs accurately enforce outgoing-wave conditions near operating frequencies.The recovered QNM-like spectrum is reliable only over the PMLs’ effective spectral interval.
  • 3.3. Importance of PML-modes: When QNMs are complete, accurate reconstructions often use only QNMs, whereas PML-modes extend completeness outside the resonator in the closed mapped system.The numerical examples report accurate predictions with only QNMs retained in such cases.
  • 3.3. Importance of PML-modes: When QNMs are incomplete, PML-modes must be retained for completeness over the mapped space, and a large number may be required for accuracy.This situation occurs for 2D systems or complex backgrounds, including a nanobullet on a semiconductor slab.
  • 3.3. Importance of PML-modes: PMLs distort modes outside their finite spectral range, while numerical modes with large damping can be difficult to distinguish from PML-modes and usually have negligible reconstruction impact.Such modes generally have very high spatial frequencies and are weakly excited by incident fields.

4. MODAL FORMALISM: IMPLEMENTATION AND NUMERICAL TOOLBOX

The implementation converts dispersive QNM equations into finite-element weak formulations and packages the resulting solver with reconstruction tools. Validation against an independent pole-search approach supports its accuracy and faster parallel computation for complex plasmonic resonators.

  • 4.1. Eigenequations and weak formulations: Auxiliary fields linearize the dispersive QNM problem, allowing the PML-mapped operator to be discretized and solved as a finite-dimensional linear eigenproblem.The implementation uses finite elements to model curved boundaries and complex geometries accurately.
  • 4.1. Eigenequations and weak formulations: The formulation can be written as a quadratic eigenproblem with stiffness, damping, and mass matrices, suitable for efficient companion linearization.The matrices are denoted K̂, D̂, and M̂.
  • 4.1. Eigenequations and weak formulations: Weak forms use arbitrary smooth test functions and are directly implemented in COMSOL’s Weak Form PDE environment and built-in eigensolver.The auxiliary-field equation is omitted for non-dispersive dielectric resonators and extended for multiple Lorentz poles.
  • 4.2. Eigensolver implementations and toolbox package: The toolbox combines a COMSOL QNM eigensolver with Matlab codes for reconstructing fields, spectra, radiation diagrams, and temporal responses.Users can define new geometries and materials to compute QNMs and PML-modes.
  • 4.2. Eigensolver implementations and toolbox package: The solver uses finite PML-bounded domains with Cartesian, cylindrical, and spherical stretched PMLs whose coordinate transformations are frequency independent.Dispersive PMLs may offer better performance, but the implementation chooses frequency-independent PMLs for simplicity.
  • 4.2. Eigensolver implementations and toolbox package: Auxiliary fields are restricted to dispersive domains, including PML regions only when those PMLs absorb dispersive background media.Curl-type basis functions handle discontinuities of normal electromagnetic-field components across material interfaces.
  • 4.2. Eigensolver implementations and toolbox package: The implementation uses COMSOL’s built-in eigensolver with direct matrix preconditioning in the numerical examples.This is an implementation choice rather than a separate modal-formalism result.
  • 4.3. Accuracy of the QNM eigensolver: The bow-tie accuracy test compares computed eigenfrequencies with an independent freeware pole-search method based on iterative Padé fitting of electromagnetic quantities.The comparison uses the same electromagnetic software, mesh, and PMLs but different algorithms.

5. NUMERICAL TESTS

The numerical tests assess exactness, convergence, and computational speed across increasingly complex plasmonic resonators. Increasing the retained modes improves accuracy, while computational gains depend on the number of modes required.

  • 5.1. Exactness: The modal method approaches the exact Mie-theory extinction cross section for a metallic sphere, with accuracy limited by finite-element discretization.
  • 5.1. Exactness: Increasing the truncation rank M progressively reduces numerical error until it reaches a plateau comparable to the frequency-domain FEM solver.
  • 5.1. Exactness: 10^-3 to 10^-5: refining the mesh substantially improves absolute accuracy, showing discretization limits the modal method’s precision.
  • 5.3. Convergence performance: For the nanobullet, retaining only QNMs recovers the overall spectral shape weakly, whereas M > 100 modes achieves below 1% relative error.
  • 5.5. Computational speed: Around 10 QNMs provide good bowtie accuracy in 7 mins and 16s, compared with 2 hours and 54 mins for 200 frequency-domain FEM samples.
  • 5.5. Computational speed: Speed advantages increase when frequencies, polarizations, or incidence angles are swept, but diminish when accuracy requires more eigenstates.
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