Plasmonic systems exhibit unique interactions with light due to the collective oscillations of conduction electrons at metal-dielectric interfaces. These interactions give rise to optical forces that can manipulate nanoparticles, enhance light-matter coupling, and enable precise control over nanoscale phenomena. Theoretical models for these forces rely on rigorous electromagnetic frameworks, including Maxwell stress tensor formulations and multipole expansions, which provide insights into gradient forces, scattering forces, and optomechanical coupling.

The optical forces in plasmonic systems arise from momentum transfer between incident light and the induced charge density in metallic nanostructures. The Maxwell stress tensor offers a fundamental approach to calculating these forces. It describes the electromagnetic momentum flux and is defined as a second-rank tensor combining electric and magnetic field components. For a time-harmonic field, the time-averaged force density acting on a particle is derived from the divergence of the stress tensor. In plasmonic systems, the near-field enhancements significantly amplify these forces, particularly at resonant frequencies where localized surface plasmons are excited.

Gradient forces originate from spatial variations in the electric field intensity. In plasmonic nanostructures, such as dimers or bowtie antennas, the field gradients are exceptionally strong due to subwavelength confinement. The gradient force is proportional to the real part of the polarizability of the nanoparticle and the gradient of the electric field intensity squared. For a spherical nanoparticle with radius much smaller than the wavelength, the dipole approximation holds, and the gradient force can be expressed in terms of the Clausius-Mossotti factor. Plasmonic resonances enhance the polarizability, leading to forces that can trap nanoparticles with high precision.

Scattering forces result from momentum transfer due to light absorption and re-radiation. These forces are proportional to the imaginary part of the polarizability and the Poynting vector of the incident field. In plasmonic systems, scattering forces are influenced by the radiative and non-radiative decay channels of surface plasmons. For example, a gold nanoparticle under resonant illumination experiences significant scattering forces due to enhanced absorption and subsequent photon re-emission. The balance between gradient and scattering forces determines the stability of optical trapping in plasmonic fields.

Optomechanical coupling in plasmonic systems bridges optical forces and mechanical motion. The coupling arises from the dependence of plasmon resonances on the geometry and relative positions of nanostructures. For instance, in a system of coupled plasmonic nanoparticles, the optical force modulates the interparticle gap, which in turn shifts the resonant frequency. This feedback mechanism can be modeled using coupled mode theory or numerical simulations solving Maxwell’s equations alongside Newton’s equations of motion. The optomechanical coupling strength depends on the overlap between the optical mode and mechanical displacement, quantified by the optomechanical coupling rate.

Multipole expansions provide a systematic way to analyze optical forces beyond the dipole approximation. For larger nanoparticles or higher-order plasmon modes, quadrupole and octupole contributions become significant. The force can be decomposed into terms involving electric dipole, magnetic dipole, and higher-order moments, each interacting with the incident field and its spatial derivatives. The multipole expansion reveals how higher-order modes contribute to trapping stiffness and rotational dynamics in plasmonic systems. For example, a silver nanorod exhibits not only dipole resonances but also quadrupole modes that influence the force profile under oblique illumination.

The Maxwell stress tensor formalism can be applied to complex geometries using numerical methods such as finite-difference time-domain or boundary element techniques. These simulations compute the near-field distributions around plasmonic structures, from which the stress tensor and resulting forces are derived. The accuracy of these models depends on the discretization of the computational domain and the incorporation of material dispersion through appropriate dielectric functions. For gold and silver, Drude-Lorentz models accurately capture the frequency-dependent permittivity in visible and near-infrared regimes.

Theoretical studies also explore the role of thermal effects in plasmonic force generation. Local heating due to plasmon dissipation alters the surrounding medium’s refractive index and viscosity, indirectly modifying optical forces. Coupled thermo-electromagnetic models solve the heat equation alongside Maxwell’s equations to quantify these effects. The temperature rise depends on the absorption cross-section and thermal conductivity of the nanoparticle and its environment.

Quantum corrections become relevant at sub-nanometer gaps where nonlocal effects and electron tunneling occur. Hydrodynamic models and quantum mechanical treatments extend classical electrodynamics to account for electron density variations at atomic scales. These corrections adjust the plasmon resonance condition and the resulting optical forces, particularly in systems like nanoparticle-on-mirror configurations.

In summary, theoretical models of optical forces in plasmonic systems integrate electromagnetic, mechanical, and thermal phenomena. The Maxwell stress tensor provides a comprehensive framework for force calculation, while multipole expansions capture the contributions of higher-order plasmon modes. Gradient and scattering forces dominate particle dynamics, with optomechanical coupling introducing feedback between light and motion. Numerical simulations and quantum corrections further refine these models, enabling precise predictions for applications in nanomanipulation and photonic device design. The continued development of these theoretical tools will advance the understanding and utilization of plasmonic forces in emerging technologies.