Plasmon‑Enhanced Spin–Orbit Dynamics of Gold Nanodimers under Circularly Polarized Illumination

Background

Optical binding—wherein two microparticles or nanoparticles become stably trapped by the interference of scattered light—has been extensively studied under linearly polarized (LP) illumination [1–6]. The equilibrium separations of the bound pair typically correspond to integer multiples of the wavelength in the surrounding medium, and the dimer aligns perpendicular to the polarization direction. Recent experiments with circularly polarized (CP) plane waves have shown that the helicity of the longitudinal spin and orbital rotation of silica nanodimers matches the incident light’s handedness [11], while focused CP Gaussian beams can induce a *negative* orbital torque, reversing the rotation direction [12,13,14]. Single‑particle spin rates of up to 3.5 kHz have also been measured for gold nanoparticles under CP Gaussian illumination [15–18]. These observations underscore the importance of longitudinal and transverse SOC in structured light fields [19–34], particularly in the near field of plasmonic Au or Ag particles where collective electron motion amplifies the effect [28–31].

In this work we extend the theoretical framework to a gold nanodimer supported on a substrate, illuminated by a left‑handed CP Gaussian beam. The substrate confines the particles to the focal plane and eliminates reflection at the interface (water‐like refractive index). We employ the MMP method to compute the electromagnetic field, from which we derive optical forces and torques, thereby revealing the SOC‑driven dynamics of the dimer and identifying conditions for negative orbital torque.

Methods

Figure 1 depicts the geometry: two identical Au nanospheres (radius = 100 nm) sit on a substrate and are illuminated from above by a normally incident left‑handed CP beam. The interparticle center‑to‑center distance is d, the beam waist is w₀, and the focal plane coincides with the nanoparticle centers. The electric fields of the plane wave and Gaussian beam are provided in the Appendix. Because the substrate’s refractive index matches that of the surrounding water, no reflection occurs; the substrate simply constrains the particles’ motion to the focal plane.

The optical force on particle j (j = 1, 2) follows from the time‑averaged Maxwell stress tensor:

\mathbf{F}^j=\int_{S_j}\mathbf{T}\cdot\mathbf{n}\,dS\quad (1)

where n is the outward normal to the surface S_j and

\mathbf{T}=\tfrac12\operatorname{Re}\{\varepsilon\mathbf{E}\bar{\mathbf{E}}+\mu\mathbf{H}\bar{\mathbf{H}}-\tfrac12(\varepsilon\mathbf{E}\cdot\bar{\mathbf{E}}+\mu\mathbf{H}\cdot\bar{\mathbf{H}})\mathbf{I}\} \quad (2)

All fields are the exterior total fields. In cylindrical coordinates the radial component F_r indicates attraction or repulsion, while the azimuthal component F_θ determines the direction of orbital rotation. The longitudinal orbital torque about the optical axis is F_θ d/2.

The spin torque on particle j is obtained from the angular‑momentum flux:

\mathbf{M}^j=\int_{S_j}\mathbf{x}^j\times\mathbf{T}\cdot\mathbf{n}\,dS\quad (3)

with x^j = x – x_c^j the position relative to the particle center. The longitudinal component M_z drives spinning about the optical axis; the transverse components M_r and M_θ arise from the twisted field at the focal plane. The permittivity of Au at 800 nm is (–24.062, 1.507) [38].

Configuration of the Au nanodimer on a substrate illuminated by a normally incident left‑handed CP Gaussian beam of waist w₀. The beam axis lies in the focal plane of the nanospheres, and d is the center‑to‑center distance.

Results and Discussion

We evaluate the forces and torques on two 100‑nm Au nanospheres in water, illuminated by a 25 MW cm^–2 left‑handed CP plane wave or a focused Gaussian beam (λ = 800 nm, w₀ = 500 nm). The particles are free to move in the focal plane. Figure 2a (plane wave) and 2b (Gaussian beam) plot F_r and F_θ versus interparticle distance d. For the plane wave, F_r crosses zero at stable‑equilibrium separations d₁ = 603 nm and d₂ = 1204 nm, each with a negative slope, indicating an optical spring that keeps the particles at integer multiples of the wavelength in the medium (d_m = mλ/n). The Gaussian beam yields slightly smaller equilibrium distances (585 nm and 1131 nm) due to the additional gradient force.

At these orbits, the azimuthal force F_θ generates an orbital torque F_θ d/2. For the plane wave, F_θ > 0, so the dimer rotates counter‑clockwise in the focal plane, matching the light’s handedness. Conversely, the Gaussian beam produces F_θ < 0, resulting in a negative orbital torque and a clockwise rotation—opposite to the incident helicity—consistent with experimental observations [12,13,14]. This reversal originates from the twisted electromagnetic field of the Gaussian beam [23]. Using Stokes’ law, the terminal orbital speed is v_T = F/(6πrη) with η = 0.001 kg m^–1 s^–1. For F_θ = –4 pN at d = 585 nm, the angular velocity is approximately –7 kHz, matching the experimentally measured –4 kHz for silver dimer rotations [12]. In the short‑range regime (d < 291 nm), the radial force becomes negative, pulling the particles together; the resulting spiral trajectory leads to eventual collision, as illustrated by the 2D streamline maps in Figure 2c,d. Figure 3 displays the spin torques M_r, M_θ, and M_z versus d for both illumination types. The longitudinal spin torque M_z retains the handedness of the incident CP light in both cases. Transverse spin torques are significantly larger for the Gaussian beam, peaking near the first stable‑equilibrium orbit (585 nm). Applying Stokes’ law for a spinning sphere, the terminal angular velocity is ω_T = M/(8πr³η), yielding spin rates on the order of 10 kHz, in line with the measured 3.5 kHz for single Au nanoparticles [15]. These results demonstrate that the orbital and spin motions of the dimer are tightly coupled through the SOC of the twisted near‑field. Figure 4 explores how the beam waist influences the orbital and spin torques. As w₀ decreases, the magnitude of the negative orbital torque increases, reaching a turning point at w₀ ≈ 1150 nm where F_θ = 0. Simultaneously, the transverse spin torques grow while the longitudinal spin torque diminishes. This behavior underscores the role of beam focusing in tailoring the SOC: tighter beams enhance field distortion, amplifying the curl of the spin angular momentum and thus the negative orbital torque and transverse spin. In summary, the observed negative orbital rotation and pronounced transverse spinning stem from the curl of the light field’s spin angular momentum, independent of the beam’s intrinsic orbital angular momentum. The helicity of both orbital and spin motions follows the incident light’s handedness, providing a controllable handle on nanodimer dynamics.

Conclusions

Our theoretical study of a gold nanodimer under circularly polarized illumination reveals that long‑range optical binding establishes stable‑equilibrium orbits at integer‑multiple wavelengths, with each particle acquiring both spin and orbital torques. A focused CP Gaussian beam can reverse the orbital rotation direction by generating a negative orbital torque, while simultaneously enhancing transverse spin torques as the beam waist shrinks. In the short‑range regime, mutual attraction drives a spiral trajectory leading to collision. The transition between long‑ and short‑range interactions occurs near half a wavelength in the medium. These findings elucidate the mechanisms of plasmon‑enhanced SOC in nanoscale systems and suggest practical routes to manipulate nanodimer motion in optical tweezers, photonic circuits, and related nanotechnologies.

Future work will examine the correlation between optical spin/orbital torques and the local angular‑momentum densities of the electromagnetic field, as well as extend the analysis to twisted near fields of metamaterials [40–43].

Abbreviations

CP:

Circularly polarized

EM:

Electromagnetic

LH:

Left‑handed

LP:

Linearly polarized

MMP:

Multiple multipole

MP:

Microparticle

NP:

Nanoparticle

SOC:

Spin–orbit coupling