Optical forces, helicity, angular momentum and how they are all intertwined

The theoretical description of optical forces and torques on micron_sized particles is a crucial area of research and has formed the foundation for advancements in optical trapping and manipulation technologies. In this study, we derive analytical expressions for optical forces and torques on micron_sized spherical particles illuminated by focused Laguerre_Gaussian (LG) beams, employing the well_defined helicity multipolar decomposition of electromagnetic fields and Mie theory. We developed a multifunctional program, Multipolar Optical Forces Toolbox, based on this theoretical framework. The program, available on GitHub, was used to generate optical trapping stability maps. These maps predict trap stability across a wide range of system parameters and serve as a practical tool for designing advanced optical trapping experiments. Our analysis reveals the important role of helicity p and orbital angular momentum l on the dynamics of particles trapped off_axis in LG beams and demonstrates the unique nature of the tangential torque. Our findings also highlight notable differences in longitudinal optical forces resulting from pure helicity modifications in Gaussian beams. Furthermore, we showcase the ability of LG beams to isolate Mie resonances, offering a novel approach to locate the spectral positions of the resonances of high multipolar modes. These insights deepen the understanding of helicity in LG optical traps and pave the way for the development of more advanced optical manipulation techniques.

💡 Research Summary

This paper presents a comprehensive theoretical framework for calculating optical forces and torques on micron‑sized spherical particles illuminated by tightly focused Laguerre‑Gaussian (LG) beams. The authors combine a helicity‑based multipolar decomposition of the electromagnetic field with exact Mie scattering theory, and they implement the resulting formalism in an open‑source software package called the Multipolar Optical Forces Toolbox (MOFT).

Key conceptual advances:

1. Helicity eigen‑multipoles – By forming linear combinations of electric and magnetic multipoles, the authors define modes Aₚ^(n)(j,m_z) that are eigenstates of the helicity operator (p = ±1). A paraxial circularly polarized beam carries a single helicity; after passing through an aplanatic high‑NA objective the beam still contains only that helicity, which greatly simplifies the description of focused structured light.
2. Beam Shape Coefficients (BSCs) – Closed‑form expressions for the BSCs of on‑axis and off‑axis LG beams are derived. The total angular momentum along the propagation axis is m_z = ℓ + p, where ℓ is the topological charge (orbital angular momentum) and p the helicity (spin). These coefficients enter directly into the scattered field expansion.
3. Mie‑based scattering with helicity – The scattered field is expressed as a superposition of the same helicity mode and the opposite helicity mode, weighted by α_j = –a_j + b_j and β_j = a_j – b_j, where a_j and b_j are the usual electric and magnetic Mie coefficients. This formulation makes explicit how helicity is (or is not) conserved: only for a dual particle (a_j = b_j for all j) does the scattered field retain the incident helicity. For ordinary dielectric spheres duality is not exact, but near‑dual conditions can be engineered.
4. Exact force and torque from the Maxwell Stress Tensor – Using the multipole expansion, the authors integrate the Maxwell Stress Tensor over a spherical surface to obtain analytical expressions for the three components of the optical force (F_x, F_y, F_z) and the torque (τ_x, τ_y, τ_z). The tangential torque τ_φ is shown to depend linearly on the total angular momentum m_z, while the longitudinal force F_z exhibits a striking helicity dependence even for a pure Gaussian beam (ℓ = 0). Flipping the helicity changes the sign of F_z, a phenomenon absent in conventional scalar‑beam treatments.

Practical implementation:
The MOFT toolbox (available on GitHub) automates the calculation of BSCs, Mie coefficients, and the resulting force/torque fields for arbitrary particle size, refractive index, beam parameters (ℓ, p, numerical aperture, power), and particle displacement. By scanning the parameter space, the authors generate “stability maps” that indicate regions where three‑dimensional trapping is stable. These maps reveal how increasing ℓ widens the off‑axis stable region, while the sign of p can either enhance or suppress axial confinement.

Novel physical insights:

• Tangential torque and helicity–orbital coupling – The torque that tends to rotate a particle around the beam axis (optical wrench) is proportional to ℓ p, confirming that both spin (helicity) and orbital angular momentum contribute multiplicatively.
• Helicity‑controlled axial force – For a Gaussian beam (ℓ = 0) the axial force reverses when the helicity is switched, providing a new knob for axial positioning without altering beam intensity.
• Force‑based Mie resonance spectroscopy – When ℓ matches the order of a high‑order Mie mode, the longitudinal force exhibits sharp resonant peaks. By measuring the force spectrum as a function of wavelength, one can locate the spectral positions of high‑multipole resonances without direct scattering measurements. This offers a non‑invasive method to probe particle material properties.

Overall, the paper demonstrates that a helicity‑centric multipolar approach not only yields exact analytical results comparable to the most sophisticated GLMT implementations, but also provides clear physical intuition about how spin and orbital angular momenta intertwine in optical trapping. The open‑source toolbox translates these theoretical advances into a practical design aid for researchers developing advanced optical tweezers, optical spanners, and contact‑free spectroscopic techniques.