Analytical formulation of chiral optomechanics

The scattering problem for optically active spheres has been originally studied by Bohren4,36. Recently, the approach was extended to deal with aggregates of optically active particles37. We consider an optically active particle with an average refractive index \({n}_{{\rm{p}}}=\sqrt{\varepsilon /\mu }\) immersed in a non-dispersive, homogeneous medium of refractive index n m . The staring point for a macroscopic description of optical activity are the Drude-Born-Fedorov (DBF) constitutive relations4,5:

$$\overrightarrow{D}=\varepsilon \overrightarrow{E}+\frac{\alpha \varepsilon }{k}\overrightarrow{

abla }\times \overrightarrow{E},$$ (2)

$$\overrightarrow{B}=\mu \overrightarrow{H}+\frac{\beta \mu }{k}\overrightarrow{

abla }\times \overrightarrow{H}\mathrm{.}$$ (3)

If time symmetry is imposed, α = β. The adimensional chirality parameter β is related to the chiral refractive indices, \({n}_{L}=\bar{n}\mathrm{/(1}-\beta \bar{n})\) and \({n}_{R}=\bar{n}\mathrm{/(1}+\beta \bar{n})\), for left and right circularly polarized waves, respectively, where \(\bar{n}={n}_{{\rm{p}}}/{n}_{{\rm{m}}}\) is the average refractive index relative to the medium (see Supplementary Information):

$$\beta =\frac{1}{2}(\frac{1}{{n}_{R}}-\frac{1}{{n}_{L}})\mathrm{.}$$ (4)

Thus, the real part of β is related with the different speed of propagation and its imaginary part with the different absorption of RCP and LCP electromagnetic waves. In addition, if the real part of β is positive the material is a right-handed medium and the LCP radiation propagates with a slower phase velocity, and viceversa.

Note that for an isotropic chiral medium the BDF constitutive equations satisfy the time-reversal symmetry, the duality transformations, and their validity is supported by experiments1,2. Moreover, it has been recently shown that conservation of energy is preserved both globally and locally38. All these facts consolidate the physical significance of the BDF relations and give a strong argument for their use in practical calculations. Noteworthy, other formulations of the constitutive relations are also suitable to describe optical activity39, for isotropic reciprocal chiral media40, and for composite chiral media41. However, a distinction between different approaches is possible only for arbitrarily time-dependent (not simply harmonic) processes or for inhomogeneous media. The various constitutive relations are equivalent to each other for time-harmonic fields. Here we consistently use the BDF relations in the framework of electromagnetic scattering theory from isotropic optically active particles from which we find a Helmholtz-type equation:

$${

abla }^{2}[\begin{array}{c}\overrightarrow{E}\\ \overrightarrow{H}\end{array}]+{{\mathscr{K}}}^{2}[\begin{array}{c}\overrightarrow{E}\\ \overrightarrow{H}\end{array}]=\mathrm{0,}$$ (5)

where \({\mathscr{K}}\) is the non-diagonal chiral matrix:

$${\mathscr{K}}=\frac{k}{1-{\beta }^{2}\mu \varepsilon }(\begin{array}{cc}\beta \mu \varepsilon & i\mu \\ -\,i\varepsilon & \beta \mu \varepsilon \end{array})$$ (6)

and k is the wavevector in vacuum. Thus, since \({\mathscr{K}}\) is non-diagonal, in a chiral medium the fields are coupled with each other during their propagation. Bohren has shown that any field in the chiral medium can be described as an overlap of the circularly polarized fields, \({\overrightarrow{Q}}_{L}\) and \({\overrightarrow{Q}}_{R}\) (Bohren decomposition)4,36,37:

$$\{\begin{array}{rcl}{\overrightarrow{E}}_{P} & = & {\overrightarrow{Q}}_{L}-\frac{i}{{n}_{{\rm{p}}}}{\overrightarrow{Q}}_{R}\\ {\overrightarrow{H}}_{P} & = & -\,i{n}_{{\rm{p}}}{\overrightarrow{Q}}_{L}+{\overrightarrow{Q}}_{R}\mathrm{.}\end{array}$$ (7)

We now expand these internal fields in magnetic and electric multipoles, \({\overrightarrow{J}}_{Llm}^{\mathrm{(1)}},\,{\overrightarrow{J}}_{Rlm}^{\mathrm{(1)}},\,{\overrightarrow{J}}_{Llm}^{\mathrm{(2)}},\,{\overrightarrow{J}}_{Rlm}^{\mathrm{(2)}}\), with parity indices p = 1, 2 respectively8. The numerical values of the superscripts are related to the parity operator, i.e., by reflecting a magnetic or electric vector, its sign changes or does not change, so that the eigenvalue is either −1 (for p = 1) or 1 (for p = 2). The natural optical activity can be considered as generated by an appropriate superposition of these vector fields, involving interference between electric and magnetic multipoles. In particular, we have that:

$$(\begin{array}{rcl}{\overrightarrow{Q}}_{L} & = & \sum _{lm}\,{C}_{Llm}[{\overrightarrow{J}}_{Llm}^{\mathrm{(1)}}+{\overrightarrow{J}}_{Llm}^{\mathrm{(2)}}]\\ {\overrightarrow{Q}}_{R} & = & \sum _{lm}\,{C}_{Rlm}[-\,{\overrightarrow{J}}_{Rlm}^{\mathrm{(1)}}+{\overrightarrow{J}}_{Rlm}^{\mathrm{(2)}}],\end{array}$$ (8)

with indices l = 0, 1, ... and m = −l, ..., 0, ..., l related to the angular momentum of the multipoles, and:

$${\overrightarrow{J}}_{Llm}^{\mathrm{(1)}}={j}_{l}({k}_{L}r){\overrightarrow{X}}_{lm}^{\mathrm{(1)}}(r),\,{\overrightarrow{J}}_{Rlm}^{\mathrm{(1)}}={j}_{l}({k}_{R}r){\overrightarrow{X}}_{lm}^{\mathrm{(1)}}(r),$$ (9)

$${\overrightarrow{J}}_{Llm}^{\mathrm{(2)}}=\frac{1}{{k}_{L}}\overrightarrow{

abla }\times {\overrightarrow{J}}_{Llm}^{\mathrm{(1)}},\,{\overrightarrow{J}}_{Rlm}^{\mathrm{(2)}}=\frac{1}{{k}_{R}}\overrightarrow{

abla }\times {\overrightarrow{J}}_{Rlm}^{\mathrm{(1)}},$$ (10)

where the chiral wavevectors are k L = n L k and k R = n R k. The expansion coefficients, C Llm and C Rlm , represent the so-called optical activity tensor that describes completely all phenomena related with chirality. By exploiting this approach, imposing the boundary conditions, the full scattering problem by chiral particles can be solved in the T-matrix formalism42 (see Supplementary Information), where the incident and scattered fields are expanded8,43 in terms of spherical Bessel J-multipoles with expansion coefficients \({W}_{lm}^{p^{\prime} }\) for the incident fields and spherical Hankel H-multipoles with coefficients \({A}_{lm}^{(p)}\) for the scattered fields:

$${A}_{lm}^{(p)}={T}_{lml^{\prime} m^{\prime} }^{(pp^{\prime} )}{W}_{l^{\prime} m^{\prime} }^{p^{\prime} }\mathrm{.}$$ (11)

Here \({T}_{lml^{\prime} m^{\prime} }^{(pp^{\prime} )}\) represent the elements of the T-matrix for the chiral particle. For the specific case of a chiral sphere, the T-matrix has a simple expression37 written in terms of coefficients, \({R}_{l}^{(pp^{\prime} )}\), in close analogy with the Mie coefficients, which are combinations of Riccati-Bessel and Riccati-Hankel functions dependent on the sphere size parameter and on the chiral refractive indices (see Supplementary Information):

$${T}_{lml^{\prime} m^{\prime} }^{(pp^{\prime} )}=-\,{R}_{l}^{(pp^{\prime} )}{\delta }_{ll^{\prime} }{\delta }_{mm^{\prime} }\mathrm{.}$$ (12)

In contrast to the Mie case, as expected, the T-matrix here is non-diagonal with respect to the parity index, p, since parity in not conserved for a chiral sphere, but the matrix is still symmetric, i.e., \({R}_{l}^{(pp^{\prime} )}={R}_{l}^{(p^{\prime} p)}\).

We can now exploit this approach to calculate the chiral optical forces generated in the chiral material-chiral radiation optomechanical interaction. First, we focus on a plane wave illumination, studying the simple example of a chiral sphere. Then we extend the approach to the case of optical tweezers.

Plane wave illumination

For a chiral plane wave illumination we can write a simple expression for the radiation pressure force9,44,45 along the propagation direction, \({\hat{k}}_{0}\), in terms of the incident intensity, I 0 , the extinction and scattering cross sections, \({\tilde{\sigma }}_{ext}\) and \({\tilde{\sigma }}_{scat}\), and the anisotropy parameter, \({\tilde{g}}_{i}\):

$${\overrightarrow{F}}_{rad}=\frac{{n}_{{\rm{m}}}}{c}{I}_{0}[{\tilde{\sigma }}_{ext}-{\tilde{\sigma }}_{scat}{\tilde{g}}_{i}]\,{\hat{k}}_{0}=\frac{n}{c}{I}_{0}{\tilde{\sigma }}_{rad}{\hat{k}}_{0}\mathrm{.}$$ (13)

The general expressions for the cross sections and anisotropy parameter are then expressed in terms of the chiral matrix elements, \({R}_{l}^{(pp^{\prime} )}\):

$${\tilde{\sigma }}_{ext}^{\eta }=\frac{2\pi }{{k}_{{\rm{m}}}^{2}}\sum _{lpp^{\prime} }\,\mathrm{(2}l+\mathrm{1)}{R}_{l}^{(pp^{\prime} )}\,[{\delta }_{pp^{\prime} }-{(-1)}^{\eta }\,\mathrm{(1}-{\delta }_{pp^{\prime} })]$$ (14)

$${\mathop{\sigma }\limits^{ \sim }}_{scat}^{\eta }=\frac{2\pi }{{k}_{{\rm{m}}}^{2}}\,\sum _{lp}\,\sum _{p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}}\,(2l+1){R}_{l}^{(pp{\rm{^{\prime} }})\ast }{R}_{l}^{(pp{\rm{^{\prime} }}{\rm{^{\prime} }})}[{\delta }_{p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}}-{(-1)}^{\eta }\,(1-{\delta }_{p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}})]$$ (15)

$$\begin{array}{lll}{\mathop{\sigma }\limits^{ \sim }}_{scat}^{\eta }{{\mathop{g}\limits^{ \sim }}_{i}}^{\eta } & = & \frac{2\pi }{{k}_{{\rm{m}}}^{2}}\sum _{lp}\,{\rm{R}}{\rm{e}}\{\frac{l(l+2)}{l+1}\sum _{p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}}\,{R}_{l}^{(pp{\rm{^{\prime} }})}{R}_{l+1}^{(pp{\rm{^{\prime} }}{\rm{^{\prime} }})\ast }[{\delta }_{p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}}-{(-1)}^{\eta }\,(1-{\delta }_{p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}})]\\ & & +\,\frac{2l+1}{l(l+1)}\sum _{p{\rm{^{\prime} }}{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}{\rm{^{\prime} }}}\,{R}_{l}^{(pp{\rm{^{\prime} }}{\rm{^{\prime} }}{\rm{^{\prime} }})}{R}_{l+1}^{(p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}{\rm{^{\prime} }})\ast }[1-{\delta }_{pp{\rm{^{\prime} }}}][{\delta }_{p{\rm{^{\prime} }}{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}{\rm{^{\prime} }}}-{(-1)}^{\eta }(1-{\delta }_{p{\rm{^{\prime} }}p{\rm{^{\prime} }}{\rm{^{\prime} }}})]\},\end{array}$$ (16)

where δ is the Kronecker delta, k m = n m k is the wavevector in the surrounding medium, and η represents a polarization index, with η = 1 for LCP, i.e., a polarization unit vector \({\hat{c}}_{1}=(\hat{x}+i\hat{y})/\sqrt{2}\), or η = 2 for RCP, i.e., a polarization unit vector \({\hat{c}}_{2}=(\hat{x}-i\hat{y})/\sqrt{2}\). Thus, it is possible to show that all observables (cross sections, optical forces) are separated into a helicity independent part, corresponding to the value for linear polarization, and a second part that is summed or subtracted depending on the light helicity (see Supplementary Information). As an example, the expression for the extinction cross section for an optically active sphere is \({\tilde{\sigma }}_{ext}^{\mathrm{1,2}}={\tilde{\sigma }}_{ext}^{0}\pm {\tilde{\sigma }}_{ext}^{h}\), with:

$${\tilde{\sigma }}_{ext}^{0}=\frac{2\pi }{{k}_{{\rm{m}}}^{2}}\sum _{l}\,\mathrm{(2}l+\mathrm{1)}{Re}\{{R}_{l}^{\mathrm{(11)}}+{R}_{l}^{\mathrm{(22)}}\},\,{\tilde{\sigma }}_{ext}^{h}=\frac{2\pi }{{k}_{{\rm{m}}}^{2}}\sum _{l}\,\mathrm{2(2}l+\mathrm{1)}{Re}\{{R}_{l}^{\mathrm{(12)}}\mathrm{\}.}$$ (17)

The same helicity dependent splitting holds for the optical force, \({\overrightarrow{F}}_{rad}^{\mathrm{1,2}}={\overrightarrow{F}}_{rad}^{0}\pm {\overrightarrow{F}}_{rad}^{h}\) and for the optical torque, \({\overrightarrow{{\rm{\Gamma }}}}_{rad}^{\mathrm{1,2}}={\overrightarrow{{\rm{\Gamma }}}}_{rad}^{0}\pm {\overrightarrow{{\rm{\Gamma }}}}_{rad}^{h}\) (see Supplementary Information). In fact, we can generalize the Marston and Crichton result46 for a chiral sphere by considering the torque, \({\overrightarrow{{\rm{\Gamma }}}}_{rad}^{\mathrm{1,2}}=\pm \,{\tilde{\sigma }}_{abs}^{\mathrm{1,2}}{I}_{0}/\omega \), in terms of the chiral absorbtion cross section, \({\tilde{\sigma }}_{abs}^{\mathrm{1,2}}\), light intensity, I 0 , and frequency, ω. Thus, while for a non-chiral spherical particle the optical torque by a linearly polarized plane wave is zero, for a chiral sphere it is non-zero and it is related to the different absorption cross sections for LCP and RCP light.

We apply this theoretical approach to the case of a plane wave incident on a chiral spherical particle with a radius of 2 mm consisting of randomly oriented copper-coated stainless steel helices embedded in an epoxy resin47,48 with a volume fraction of 0.3% (see the sketch in Fig. 1a). This is a composite material that exhibits a frequency dependent chiral behaviour47,49. In Fig. 2a we show the effective dielectric constant of the composite material as obtained from combining the measured dielectric constant of the epoxy resin and the calculated dielectric constant of the chiral inclusions obtained from the scattering amplitude47 (see Supplementary Information for details). Instead in Fig. 2b we show the chirality parameter β in the microwave range as obtained from the work by Luebbers et al.47. Starting from these parameters, we show in Fig. 2c the calculated \({\tilde{\sigma }}_{rad}\) in a spectral range around 6 GHz where the imaginary part of β has a resonance. Due to the spherical symmetry of the particle, the only component of \({\overrightarrow{F}}_{rad}\) different from zero is in the direction of the incident field, i.e, a longitudinal radiation pressure. A chiral gap opens in the radiation cross section, and hence in the optical force, around the resonance at 6 GHz. Thus, we can directly observe that in a chiral material the amplitude of the optical forces is directly linked to its dichroism.

Figure 1 We investigate chiral optical forces in two different configurations. In (a) we investigate the optical force of a chiral plane wave incident on a composite chiral sphere made of an epoxy resin with embedded copper-coated stainless steel helices47,48,49. In (b) a chiral Gaussian beam is tightly focused through a high numerical aperture lens (chiral optical tweezers) and the optical trapping behaviour of an optically active particle is investigated. Full size image

Figure 2 Optical properties and mechanical effects of light for chiral composites made of copper-coated stainless steel helices embedded in an epoxy resin47. (a) Real (blue) and imaginary (red) part of the electric permittivity. (b) Real (blue) and imaginary (red) part of the associated chirality parameter. (c) Radiation pressure cross section for a spherical particle with a radius of 2 mm and for different incident polarization, RCP (blue lines) and LCP (red lines). Note how a chiral gap opens in the radiation force at about 6 GHz related to the resonance of the chiral helices. (d) Frequency dependence of the optical torque, ωΓ rad /I 0 , for linear incident polarization related to the different absorption of LCP and RCP light by the chiral spherical particle. Full size image

In Fig. 2d, we also show results related to the optical torque, Γ rad ω/I 0 , along the propagation, z, axis and for an incident linearly polarized light, that is a superposition of LCP and RCP light. Since LCP and RCP have different absorption and the optical torque is a consequence of the absorption of circularly polarized photons, we observe that a dichroic absorption spectrum directly reflects the performance of the spectrum of optical forces and torques in that range of frequencies.

Chiral optical tweezers

We now consider the case of chiral optical tweezers, i.e., the optical trapping behaviour of an optically active particle in a tightly focused laser beam (see sketch in Fig. 1b). The starting point of our calculations is the angular spectrum representation50 of the focal fields generalized for a chiral Gaussian beam incident on a high numerical aperture objective lens9,33,51. In particular, we consider an incident laser beam propagating along the z-axis (see Fig. 1b), with 10 mW of power at λ = 632 nm and a Gaussian profile. The beam is focused through an aplanatic lens, with numerical aperture NA = 1.2 and filling factor of 2, that is the waist, w 0 , of the Gaussian beam is two times larger than the size of the back aperture of the (realistic) aplanatic lens33 (see section S3 of the Supplementary Information). The radiation force and torque acting on the particle are obtained by integrating the Maxwell stress tensor within the T-matrix formalism33,52,53,54:

$${\overrightarrow{F}}_{rad}={\oint }_{S}\langle {{\rm{T}}}_{{\rm{M}}}\rangle \cdot \hat{n}\,{\rm{d}}S,\,{\overrightarrow{{\rm{\Gamma }}}}_{rad}=-\,{\oint }_{S}(\langle {{\rm{T}}}_{{\rm{M}}}\rangle \times \overrightarrow{r})\cdot \hat{n}\,{\rm{d}}S,$$ (18)

where \(\overrightarrow{r}\) is the vector position, \(\hat{n}\) is the outward normal unit vector, 〈T M 〉 is the averaged Maxwell stress tensor in the Minkowski form55 that describes the mechanical interaction of light and matter9, and the integration is carried out over a closed surface S surrounding the scattering particle. The Minkowski form of the Maxwell stress tensor is consistent with a choice of the Minkowski-type definition of light momenta in a medium as opposite to the Abraham-type one. While the Abraham-Minkowski dilemma has been going on for quite some time55, a possible solution has been considered56 where the Abraham and Minkowski momenta are two different ones, namely the kinetic momentum, where Abraham’s theory is concerned, and the canonical momentum, where Minkowski’s theory is concerned. Recently, the dilemma has been also discussed in the context of complex media57,58, where it was shown that while the Abraham-type quantities describe the energy flux and the group velocity of the wave, the Minkowski-type quantities describe the actual momentum and angular momentum carried by the wave57,58. However, no experiment has given final evidence in favour of one form or the other. Since most results in optical trapping and manipulation do not depend qualitatively on the momenta definition, we consistently employ the Minkowski momenta definition, which in fact are the ones most often employed for optical tweezers9.

Since we are interested in studying the chirality-dependent dynamics in the trap, we consider the typical case of a particle made with a material that exhibits optical activity in the visible, for this reason we chose a nanosphere with radius a = 200 nm immersed in water (n m = 1.33) with an average refractive index n p = 1.5 and the chiral parameter, β = 0.05. These values are operational values that are not strictly connected to real experimental parameters, but they are consistent with values for composite particles similar to the ones studied in Fig. 2. The dynamic quantities, \({\overrightarrow{F}}_{rad}\) and \({\overrightarrow{{\rm{\Gamma }}}}_{rad}\), are calculated starting from their definitions based on the Maxwell stress tensor33,59. In particular, we need to consider the multipole amplitudes \({\tilde{W}}_{lm}^{(p)}\) of a tightly focused beam. The expansion of a focused beam around the focal point is obtained by exploiting the angular spectrum representation9,33,50,51 (see section S3 in the Supplementary Information).

The expression for the radiation force along the direction of a unit vector \(\hat{u}\), i.e., \({F}_{rad,\hat{u}}={\overrightarrow{F}}_{rad}\cdot \hat{u}\) can be obtained through the knowledge of the scattered amplitudes \({\tilde{A}}_{lm}^{(p)}\) related to the incident focal fields through the particle T-matrix. We can then plot the radiation force as a function of particle displacement in each spatial direction, x, y, and z. Since the trapping position of the particle in the axial (z) direction is typically offset from the centre of the coordinate system (taken at the nominal focus of the lens) due to the offset of the scattering force, we calculate the transverse forces at the equilibrium position, that is the z coordinate at which the axial force vanishes. To better compare optical forces on different systems, it is generally convenient to plot the dimensionless force efficiencies along the three cartesian directions, Q i = cF i /n m P with i = x, y, z, c the light velocity in vacuum, and P the incident laser power.

In Fig. 3, we report the computed components of the optical trapping efficiencies, Q i , around the focal region, for different state of polarization of the incident field. As expected, the plots have a relative maximum and minimum approximately at the particle radius. In fact, at these points the greatest optical force is exerted because the sphere is located in the maximum gradient of intensity. This is consistent with the picture of the optical gradient force in approximated optical trapping regimes for particles much smaller or much larger than the light wavelength9,60. In proximity of the trapping point the radiation force can be linearized as a harmonic restoring force:

$$(\begin{array}{lll}{F}_{rad,x}(x,\,0,\,{z}_{0}) & = & -{\kappa }_{x}x\\ {F}_{rad,y}(0,\,y,\,{z}_{0}) & = & -{\kappa }_{y}y\\ {F}_{rad,z}(0,\,0,\,z) & = & -{\kappa }_{z}(z-{z}_{0})\end{array}$$ (19)

Figure 3 Optical trapping efficiencies for a chiral nanoparticle (a = 200 nm) along the focal region axes for LCP (a), RCP(b) and linear polarization (c). In the case of chiral field we have that Q x = Q y , while for a linearly polarized incident beam a polarization anisotropy is evident that is related to the diffraction limited focusing33. Optical trapping appears to be much stronger for the LCP beam, as we expect for a left-chiral (positive β) nanoparticle. Full size image

Thus, optical tweezers are approximated with an effective harmonic potential with spring constants or trap stiffnesses κ x , κ y κ z . These quantities are important because are the measured quantities when performing optical tweezers calibration61. To calculate the optical trap stiffnesses, we calculate the slope of the force-position plots at the equilibrium position, where the force vanishes.

At this stage, it is very interesting to investigate the dependence of the trap stiffnesses on the dimension of the chiral particle, i.e., the size-scaling of chiral optical trapping. The results of this study are reported in Fig. 4, where it is possible to compare the trap stiffnesses for different polarization states of the incident field. We note that in non-chiral (β = 0) cases the optical forces, and hence the trap stiffnesses, are the same for LCP and RCP radiation (see Fig. S2 and Sect. S2 of the Supplementary Information) and all equations then reduce to the standard Mie scattering. In contrast, for right-chiral particles with a positive chiral parameter (β = 0.05), the stiffness values are systematically higher for the LCP polarization. Thus, this indicates that the optical tweezers are an excellent experimental tool to discriminate the chiral nature of a particle of any size. Moreover, at a fixed wavelength, there is the possibility to optimize the trapping efficiency choosing the appropriate dimension for the investigated particle. For the case under study, we found the κ-values at about a ≈ 300 nm are up to four times higher than those obtained for a > 500 nm. This optimum value is related to the tight focusing used in optical tweezers that maximizes the optical force density when the particle size is of the order of the diffraction limited laser spot. We finally note that the modulation observed in the stiffnesses for larger particle size is explained in terms of interference between the different multipoles occurring in the scattering process62,63.

Figure 4 Transverse (a) and axial (b) optical trapping stiffnesses as a function of the particle radius for 10 mW incident power. Different polarization states result in different optical trapping forces. The siffnesses for LCP light (blue) are about three times larger than the ones RCP illumination (red). For linear polarization (yellow) optical trapping stiffnesses lie in between the circular polarization results. Full size image

In order to show the flexibility of the approach, we study two examples of chiral optical tweezers applied to anisotropic particles: a chiral nanowire and a particle cluster with properties that correspond to the amino acids discovered in the Murchison meteorite35,37. We extend the theory of chiral optical forces to non-spherical particles exploiting the cluster model as developed in the T-matrix approach64,65,66,67. We first use the same material properties as the spherical case and we consider a nanowire with a length of 2 μm and a transverse size of 400 nm, modeled as a chain of 5 spheres with a = 200 nm radius (see the sketch in the inset of Fig. 5a). Figure 5a–c show the calculated trapping efficiencies for LCP (a), RCP (b), and linear (c) polarization. The decrease in the slope, i.e. the trap stiffness, of the nanowire axial trapping efficiency is apparent and it is related to the extended shape of the particle. This is expected as it has been observed68,69 and modeled in non-chiral nanowires52,54. The chiral nature of the nanowire yields a very different optical trapping force for the different polarization of the incident light in both the transverse and the axial directions.

Figure 5 (a–c) Transverse (red, blue lines) and axial (yellow) optical trapping efficiencies for a model chiral nanowire for LCP (a), RCP (b), and linear (c) polarization. The optical activity of the nanowire yields the different optical forces for the different polarization states. For circularly polarized light the components in x,y are superposed because of symmetry. Instead for linear polarization the polarization splitting is evident due to the diffraction limited focusing33,52,68. (d–f) Transverse (red, blue lines) and axial (yellow) optical trapping efficiencies for a cluster with optical properties corresponding to the amino acids in the Murchison meteorite35,37 and a geometry sketched in the inset of (d). Results are shown for LCP (d), RCP (e), and linear (f) polarization. Note how the axial equilibrium point (when the force is zero), z 0 , is shifted from the nominal focus of the objective lens (origin of the z axis) because of the increased scattering force related to photon absorption. Full size image

Finally, we consider a model cluster with optical properties corresponding to the amino acids discovered in the Murchison meteorite35,70,71. These carbonaceous chondrites contain organic matter and have a huge significance in connection with the hypotheses of the extraterrestrial formation of prebiotic molecules and the origin of homochirality in organic matter72,73,74,75. In an astrophysical context it has also been pointed out that the near-field generated by a solar (blackbody) spectrum around interplanetary dust might generate optical trapping forces for atoms or small particles76. Thus, local near-field depolarization73,74,75 might produce chiral optical forces that could influence the sticking of adatoms or small particles to yield chiral molecules or structures. Furthermore, optical tweezers might have interesting applications as non-destructive contactless tools for the capture of interplanetary, cometary, or atmospheric dust in situ77. Indeed, sophisticated versions of optical tweezers (that can be modeled with the T-matrix approach) based on complex beams78,79 could be employed in the near future for interstellar dust or planetary rover missions. Thus, having a better knowledge on the optomechanical properties of exemplar dust particles might give a more precise idea on what to expect in those contexts and help engineering the optical fields required to distinguish optically active particles through their optomechanical interactions.