[Paper Review] Optical chiral sorting forces and their manifestation in evanescent waves and nanofibres

Sebastian Golat, Jack J. Kingsley-Smith+4 more|arXiv (Cornell University)|Oct 17, 2023

Orbital Angular Momentum in Optics4 citations

TL;DR

This paper introduces a symmetry-based 'force basis' of twelve vector fields to unify and simplify the description of chiral optical forces on dipolar particles, enabling concise analytical expressions for forces in evanescent waves and dielectric nanofibres. The approach reveals optimal strategies for enantiomer separation by highlighting how chiral light fields exert differential forces on left- and right-handed molecules through distinct field components.

ABSTRACT

Optical fields can exert forces of chiral nature on molecules and nanoparticles, which would prove extremely valuable in the separation of enantiomers with pharmaceutical applications, yet it is inherently complex, and the varied frameworks used in the literature further complicate the theoretical understanding. This paper unifies existing approaches used to describe dipolar optical forces and introduces a new symmetry-based `force basis' consisting of twelve vector fields, each weighted by particle-specific coefficients, for a streamlined description of force patterns. The approach is rigorously applied to evanescent waves and dielectric nanofibres, yielding concise analytical expressions for optical forces. Through this, we identify optimal strategies for enantiomer separation, offering invaluable guidance for future experiments.

Motivation & Objective

• To unify disparate theoretical formulations of chiral optical forces on dipolar particles, resolving inconsistencies in notation, units, and term grouping across the literature.
• To introduce a novel symmetry-based 'force basis' composed of twelve vector fields, each weighted by particle-specific coefficients, to streamline the description of chiral force patterns.
• To apply this formalism to two key optical geometries—evanescent waves and dielectric nanofibres—to derive concise analytical expressions for chiral forces.
• To identify optimal configurations for chiral separation of enantiomers by analyzing force asymmetries in these structured light fields.
• To provide experimental guidance for separating chiral molecules using optical forces in nanophotonic platforms.

Proposed method

• The authors derive a general expression for time-averaged optical forces on dipolar particles using SI units and phasor notation, ensuring consistency across different formulations.
• They decompose the optical force into a linear combination of twelve symmetry-based vector fields, each representing a distinct force pattern, with coefficients determined by the particle's polarizability tensor.
• The formalism is applied to evanescent waves and cylindrical dielectric nanofibres, where the electromagnetic fields are expressed in terms of circularly polarized plane wave components (right- and left-handed).
• Using the decomposition into $\mathbf{E}_{+}$, $\mathbf{E}_{-}$, $\mathbf{H}_{+}$, and $\mathbf{H}_{-}$ components, the authors express energy density, momentum, and spin angular momentum as differences between right- and left-handed field contributions.
• Boundary conditions for dielectric nanofibres are formulated as an eigenvalue problem involving Bessel and Hankel functions, enabling numerical solution of the dispersion relation for guided modes.
• The chiral force is expressed as a function of the difference in energy and momentum densities between the $\mathbf{E}_{+}$ and $\mathbf{E}_{-}$ components, revealing the origin of enantioselective forces.

Figure 1: Separation of enantiomers using an evanescent wave. An illustrative figure, forces not to scale. For the larger particle (a) we only show the lateral recoil force, while for the lossless particle (b) and the lossy particle (c) we show the interaction forces.

Experimental results

Research questions

• RQ1 How can disparate formulations of chiral optical forces in the literature be unified under a single, consistent framework?
• RQ2 What is the role of field symmetry in determining the form of chiral optical forces on dipolar particles?
• RQ3 How do the force components in evanescent waves and dielectric nanofibres differ in their chiral selectivity?
• RQ4 What are the optimal field configurations for maximizing enantiomeric separation forces in these geometries?
• RQ5 How can the force basis be used to predict and optimize experimental setups for chiral sorting?

Key findings

• The chiral optical force is expressed as a linear combination of twelve symmetry-based vector fields, each weighted by coefficients derived from the particle's polarizability, enabling a unified and systematic description of force patterns.
• In evanescent waves and dielectric nanofibres, the chiral force arises from the difference in energy and momentum densities between right- and left-handed field components, $W_{+} - W_{-}$ and $\mathbf{p}_{+} - \mathbf{p}_{-}$, respectively.
• The formalism yields surprisingly concise analytical expressions for electromagnetic quantities, including the Poynting vector and spin angular momentum, in terms of $\mathbf{E}_{+}$ and $\mathbf{E}_{-}$ components.
• The force basis reveals that chiral forces are most effective when the field's circular polarization components are asymmetric, enabling selective trapping or sorting of enantiomers.
• For dielectric nanofibres, the longitudinal dispersion relation is determined by solving $\det \mathbf{A}_{\ell,n} = 0$, with solutions obtainable numerically for $\ell \in \{0, \pm1, \pm2\}$ and $n \in \{0,1,2\}$.
• The analysis identifies optimal field configurations—particularly those with strong circular polarization asymmetry—for maximizing chiral sorting forces in both evanescent and nanofibre geometries.

Figure 2: Numerically calculated dispersion relationships for a dielectric fibre made of silicon nitride, $\varepsilon_{1}=4.3$ , immersed in water, $\varepsilon_{2}=1.7$ (both values are at $20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ valid for $\lambda_{0}$ from $750\text{\,}\mathrm{nm}1750\text{\,}