Octupole Plasmon Resonance in Nanostructures

• Octupole plasmon resonance is the excitation of collective l=3 electron oscillations in metallic nanostructures, characterized by distinctive spectral features and spatial field patterns.
• It is modeled using classical electrodynamics and generalized Mie theory to predict resonance conditions and optimize near-field enhancements in nanoscale systems.
• The phenomenon enables advanced applications like surface-enhanced Raman spectroscopy and nonlinear optics through engineered nanoparticle geometries and periodic metasurfaces.

Octupole plasmon resonance refers to the excitation of collective electron oscillations with angular momentum quantum number $l=3$ in a plasmonic nanostructure. These resonances arise naturally as higher-order multipolar modes in metallic nanoparticles and nanostructured arrays, manifesting distinct spectral features, spatial field patterns, and near-field enhancements. Octupole modes play a nontrivial role in optical, electron, and near-field spectroscopies, especially in subwavelength, high-density plasmonic systems, and are increasingly leveraged for advanced photonic functionalities.

1. Fundamental Principles and Theoretical Description

In classical electrodynamics, the localized surface plasmon (SP) resonances in a spherical metal nanoparticle are indexed by angular momentum $l$. The dipole ($l=1$), quadrupole ($l=2$), and octupole ($l=3$) are the lowest, most prominent multipole orders. The SP resonance frequency for mode $l$ is approximately given by

$$\omega_l = \omega_p / \sqrt{\epsilon_\text{core}(\omega_l) + \frac{l+1}{l} \epsilon_b}$$

where $\omega_p$ is the bulk plasmon frequency, $\epsilon_\text{core}(\omega_l)$ is the dielectric response of bound electrons, and $\epsilon_b$ is the surrounding medium's dielectric constant (Raza et al., 2015).

The resonance condition, valid in the nonretarded limit, becomes

$l\,\epsilon(\omega) + (l+1)\,\epsilon_b = 0$

For octupole ($l=3$), this fixes the resonance near $\epsilon(\omega) = -\frac{4}{3}\epsilon_b$. In generalized Mie theory, these plasmon resonances are expressed in the electric multipole Mie coefficient $t_{2l}$, where for $l=3$,

$$t_{6} = \frac{m\,\psi_3(ka)\,\psi'_3(k_ba) - \psi_3(k_ba)\,\psi'_3(ka)}{\xi_3(k_ba)\,\psi'_3(ka) - m\,\psi_3(ka)\,\xi'_3(k_ba)}$$

with $m=\sqrt{\epsilon}/\sqrt{\epsilon_b}$, and $\psi_l,\,\xi_l$ the Riccati–Bessel and Hankel functions, $k$ and $k_b$ are wavenumbers in particle and matrix respectively (Nordebo et al., 2018).

The optimal material for maximizing octupole resonance absorption or scattering is given by the complex conjugate or real part of the corresponding plasmonic pole position,

$$\epsilon_{\text{opt},3}(k_0a) \approx -\frac{4}{3}\epsilon_b + (k_0a)^2 F_3(k_0a,\epsilon,\epsilon_b) - i (k_0a)^7 D_3 + \ldots$$

where $F_3$ and $D_3$ are analytic functions whose forms are detailed in Mie theory asymptotics (Nordebo et al., 2018).

2. Octupole Resonance in Nanostructures and Arrays

In extended and complex geometries, higher-order multipole modes are strongly influenced by particle shape, interparticle coupling, and the photonic environment.

• Silver Octopods: Multi-armed nanostars show multipolar resonances (primarily dipole and quadrupole) whose splitting and hybridization, due to cubic symmetry and arm coupling, closely resemble the emergence of higher-order (including octupole) modes. The overall plasmonic response is tuned via geometric ratios $L/R$ (arm length/core radius), $r/R$ (arm thickness/core radius), and effective radius $a=(3V/4\pi)^{1/3}$. The spectral position, field strength, and mode structure are sensitive to these ratios, and the discrete dipole approximation can resolve these effects (0907.5451).
• Periodic Arrays and Metasurfaces: In a metasurface of gold-disc dimers, octupole resonances arise naturally and become pronounced as array periods (lattice constant $\Lambda$) are reduced below the wavelength. The effective dipole polarizability $\tilde\alpha_x$ vanishes due to collective interactions, while the octupole polarizability $\tilde\gamma_{xxx}$ increases, rendering octupole-induced near-field enhancement dominant. Analytical expressions,

$$\tilde{\alpha}_x = \frac{\alpha_x}{1 - \alpha_x\,\zeta_{d}}, \quad \tilde{\gamma}_{xxx} \approx \frac{\gamma_{xxx} + \gamma^{(\text{iii})}_{xxx}\alpha_x\zeta''_{d}}{1-\alpha_x\,\zeta_d}$$

capture these effects, where $\gamma_{xxx}$ is intrinsic, $\gamma^{(\text{iii})}_{xxx}$ relates to the second spatial derivative of $E_x$, and $\zeta_d,\,\zeta''_d$ encode interparticle interactions (Sehrawat et al., 21 Oct 2025).

3. Experimental Detection and Measurement Techniques

Octupole plasmon resonances have been unambiguously observed in electron energy-loss spectroscopy (EELS) of individual metal nanoparticles, with spectral features traceable to higher angular momenta (Raza et al., 2015).

• EELS of Silver Nanoparticles: Both dipole and higher-order (quadrupole, octupole, and beyond) modes are visible in EELS spectra for radii $R \gtrsim 4$ nm. When the electron beam is positioned at the nanoparticle surface ($b \approx R$), HO modes can even dominate the spectral response. For $R < 4$ nm, HO modes (including octupole) disappear due to strong nonlocal damping explained by the generalized nonlocal optical response (GNOR) theory. GNOR incorporates spatial dispersion and electron diffusion, predicting size-dependent damping that increases with multipolar order.
• Spectral Decomposition: High-resolution EELS can resolve separate features for dipolar and HO resonances, assigned via Gaussian fitting. For discs and dimers, near-field mapping, current density analysis, and scattering-current multipole expansion techniques are used (for instance, extracting $O_{xxx}$ via current expansion) (Sehrawat et al., 21 Oct 2025).

4. Field Enhancement and Hotspots

Octupole modes yield substantial near-field enhancement, crucial for applications such as SERS and nonlinear optics.

• Hotspot Formation: Octupolar excitation creates intense local fields, with enhancement factors up to $50\times$ the incident field in networked nanoparticle structures. In gold-disc dimer metasurfaces, an octupole resonance (near $692\,\textrm{nm}$) coupled to a surface lattice resonance leads to peak field amplification in dimer gaps, surpassing the dipole mode at longer wavelengths (e.g., $1184\,\textrm{nm}$) (Sehrawat et al., 21 Oct 2025).

Structure	Resonant Order	Enhancement Location
Silver Octopod	Quadrupole/split + hybrid	Arm junctions
Gold-disc dimer array	Octupole	Dimer gaps

Electric hotspots are dominant for SERS; magnetic hotspots arise due to induced current loops and can be harnessed for trapping magnetic nanoparticles and engineering negative permeability metamaterials (0907.5451).

5. Symmetry, Geometry, and Resonance Tunability

The manifestation and efficiency of octupole (and other multipole) plasmon resonances are strongly governed by particle symmetry, orientation, and geometric tunability.

• Symmetry Considerations: Nanoparticles with cubic ($O_h$-like) symmetry, such as silver octopods, possess resonant modes—dipole, quadrupole, octupole—that are only moderately sensitive to orientation relative to incident fields. Extinction spectra are robust under changes in incident polarization, enabling efficient excitation and practical device integration.
• Geometric Control: In periodic arrays, decreasing the lattice period diminishes dipole response due to collective quenching but amplifies the octupole channel. By adjusting arm length, thickness, or array period, one may red-shift or boost octupole resonance selectively. This ability facilitates "shape spectroscopy," where the geometry is engineered for targeted optical response (0907.5451, Sehrawat et al., 21 Oct 2025).

6. Applications and Research Implications

The properties of octupole plasmon resonances are being actively leveraged in advanced photonic and sensing applications.

• Surface-Enhanced Raman Spectroscopy (SERS): The large and spatially overlapping electric field enhancements generated at octupole resonance sites dramatically increase Raman signal, supporting the detection of extremely low analyte concentrations (0907.5451).
• High-Density Resonator Arrays: In metasurfaces and photonic circuits, the persistence and enhancement of octupole response in closely packed arrays enable dense hotspot formation, critical for sensing and nonlinear effects (Sehrawat et al., 21 Oct 2025).
• Nonlinear and Quantum Photonics: Octupole-enhanced near-fields contribute to increased nonlinear interactions and may be further studied for quantum-size and nonlocal screening effects, especially as fundamental theory transitions from classical to GNOR and more advanced models (Raza et al., 2015, Nordebo et al., 2018).

A plausible implication is that systematic tuning of material permittivity and structure enables near-optimal absorption or scattering properties for specific multipole orders, with optimization strategies built on Mie theory and constrained quadratic forms (Nordebo et al., 2018).

7. Future Directions and Open Issues

Further investigations are warranted into:

• Exploiting other multipoles, including magnetic octupoles and higher orders, particularly in designed meta-atoms and multi-component arrays.
• The role of near-field interaction terms (such as higher derivatives of array factors) in enhancing multipole excitations.
• Incorporating substrate engineering and waveguiding architectures to further concentrate and manipulate octupole-induced optical fields.
• Quantifying quantum and nonlocal effects in ultra-small (sub-4 nm) plasmonic nanoparticles via advanced optical and electron spectroscopies, using models beyond GNOR (Raza et al., 2015).

The advancement of analytical and numerical multipole extraction, verified by simulation platforms and experimental measurement, will continue to provide essential tools for dissecting and optimizing octupole plasmon resonances in nano-optics.