Mechanical separation of chiral dipoles by chiral light

We calculate optical forces and torques exerted on a chiral dipole by chiral light fields and reveal genuinely chiral forces in combining the chiral contents of both light field and dipolar matter. Here, the optical chirality is characterized in a general way through the definition of optical chirality density and chirality flow. We show in particular that both terms have mechanical effects associated respectively with reactive and dissipative components of the chiral forces. Remarkably, these chiral force components are directly related to standard observables: optical rotation for the reactive component and circular dichroism for the dissipative one. As a consequence, the resulting forces and torques are dependent on the enantiomeric form of the chiral dipole. This suggests promising strategies for using chiral light forces to mechanically separate chiral objects according to their enantiomeric form.

Introduction -A new area in optics has emerged through the discovery of exotic light modes with complex field topologies [1][2][3]. Light fields carrying orbital angular momenta, optical vortices and singular non-diffractive beams have been studied extensively, providing a unique framework to identify and demonstrate new light-matter interactions, both at the classical and quantum levels [4][5][6]. A clear illustration is how chirality has come again to the forefront in optics [7][8][9][10]. For instance, so-called 'superchiral' fields provide fascinating perspectives in the context of ultra-high resolution spectroscopy of chiral molecules [11][12][13].
Numerous experiments have demonstrated intriguing opto-mechanical effects based on spin and orbital angular momentum transfers, related to optical negative forces and tractor beams [14,15]. Yet, the processes through which complex light fields can transfer energy to material systems are still giving rise to fundamental questions, currently fueling much effort and debate [16][17][18][19].
In this Letter, we combine both chiral fields and chiral systems to reveal, in the dipolar regime, motional effects that are specific to the coupling between chirality of light and chirality of matter. We show that the forces induced by the chiral fields on chiral dipoles can be decomposed into reactive and dissipative components [20]. From the field point of view, this decomposition involves the concepts of chirality density and chirality flow which are treated separately. From the matter point of view, the two components are associated with optical rotation and circular dichroism respectively. In analogy with the achiral case, we perform a spin-orbit decomposition of the chirality flow in connection with the chiral components of the optical force and torque exerted on the system. Finally, we emphasize that the signs of the chiral force and torque are directly related to the enantiomeric form of the dipolar distribution. This leads to the possibility of separating mechanically chiral objects through chiral optical forces.
Chiral light fields -We first consider arbitrary electromagnetic (E, H) fields in vacuum with an energy flow (i.e. Poyting vector) S(r, t) = E × H and an energy den- * Corresponding author: genet@unistra.fr and magnetic W (H) parts-connected through a continuity equation ∂ t W + ∇ · S = 0 for the conservation of energy. It is also possible to characterize an associated optical chirality through a chirality density and a chirality flow Both quantities are related too by a continuity equation ∂ t K + ∇ · Φ = 0 (see [9]). Light fields exert instantaneous force F and torque Γ on both electric and magnetic dipoles with [18]: Chiral dipoles -A small chiral object interacting with (E, H) can be described as a coupled system of induced electric P and magnetic M dipole moments [21]. In the linear and harmonic regimes, the system writes as . For simplicity here, we assume isotropic responses so that the complex polarizability tensors (α, β, χ) become complex scalars.
Specific to chirality is the complex mixed electricmagnetic dipole polarizability χ [22]. Its sign determines the enantiomeric form associated with the dipolar system (4). Note that the electric and magnetic polarizabilities are invariant through enantiomeric changes since they are respectively quadratic forms of the electric and magnetic dipolar moments [23]. One aim of this Letter is to identify the effect of such a ±χ sign change on optical forces exerted by light on a chiral dipole. The core of the discussion therefore consists in evaluating the force and the torque induced on such a chiral dipole (4) immersed in a chiral field characterized by Eqs. (1,2).
The time-averaged expressions of both quantities in arXiv:1306.3708v1 [physics.optics] 16 Jun 2013 Eq. (3) immediately display additive achiral and chiral contributions according to: where Φ (E) = E ×Ė/ω and Φ (H) = H×Ḣ/ω are the electric and magnetic ellipticities, is the time-averaged Poynting vector, the vector fields are the known electric and magnetic expressions [18,19], and is an additional term characterizing the mechanical coupling between the chiral dipole and the chiral field.
Reactive and dissipative components -For the achiral contribution to the force F α,β , we can extend the decomposition for the electric dipole to the magnetic case in a straightforward way from the duality of Maxwell's equations [19]. The reactive component of the averaged achiral force is determined, as far as the fields are involved, by the real parts of f 0 and g 0 . Precisely, one derives a conservative force given that Re[f 0 ] = ∇( E 0 2 )/2 and Re[g 0 ] = ∇( H 0 2 )/2. These terms are associated with electric and magnetic gradient forces. Because α/ε 0 and β/µ 0 have the dimension of a volume, ∇ W (E) and ∇ W (H) correspond to the reactive optical force density carried respectively by the electric and magnetic field.
In contrast, the dissipative component of the achiral force is not conservative and, as expected from the electric dipolar case [17][18][19], it turns out to be related to the electric and magnetic orbital parts of the Poynting vector with using the relations In this case, the electric and magnetic orbital parts give rise to a dissipative optical force density carried by the electric and magnetic fields, respectively. These orbital parts can also be interpreted as a generalization of the electric and magnetic phase gradient [20]. Remarkably from Eq. (5), the chiral contribution can also be separated into reactive and dissipative components as: The reactive component is proportional to the in-phase component of χ which is associated with optical rotation for chiral systems. This component is conservative since ]. Noteworthy, this term is directly related to the chirality density of an harmonic field given that in this case, the density given in Eq. (1) is time-independent with K(r) = Im [(E 0 · H * 0 )] ω/(2c 2 ). One can thus write the conservative force where cχ has the dimension of a volume, enabling to interpret cK/ω as the chiral equivalent of the energy densities W appearing in Eq. (7).
Here in the harmonic regime, the chirality flow Φ(r) = ω(ε 0 Φ (E) + µ 0 Φ (H) )/2 is time-independent and 2cΦ/ω contributes to the chiral dissipative force just as the energy flow Π does for the achiral dissipative forces in Eq. (8). In fact, Eq. (11) can be understood as an analogous spin-orbit separation Φ = Φ O + Φ S for the chirality flow with This decomposition is substantiated by looking at the chiral part of the torque: showing that, similarly to the case of an achiral electric dipole [19], the spin part Φ S is related to the torque applied to the chiral dipole by the field, therefore not involved in the transfer of chiral radiation pressure F d. χ . It is remarkable that Γ χ is solely determined from the time-averaged Poynting vector of the field. This implies, as physically expected, that a torque will be exerted as soon as the dipole is chiral, independently of the chiral nature of the interacting electromagnetic field.
In fact, the expressions obtained above highlight an interesting symmetry: the energy flow Π determines the chiral contribution of the torque when the electric and magnetic ellipticities (summed into the chirality flow Φ -Eq. (2)) give the achiral contribution to the torque, as seen in Eq. (5). As a consequence, the curl of Π defines the spin part Φ S of the chirality flow, when the curls of the ellipticities define the spin parts Π S of the energy flow.
These simple results directly show that the chiral content of an electromagnetic field can have a mechanical action on matter through the real and imaginary parts of the linear chiral susceptibility (i.e. χ for a chiral dipole). This discussion leads us naturally to investigate below two specific situations where the forces are stemming either from the mechanical action of the chirality density or the chirality flow of the light field. A selection can also be performed by a wavelength resonant excitation of either Re [χ] or Im [χ] which are Kramers-Kronig related [21].
Examples -While considering an arbitrary chiral system, there is usually an important hierarchy |β| < |χ| < |α| in the different susceptibilities involved in Eq. (4). Indeed, for a chiral object of size a and an harmonic field of wave vector k = ω/c, the quantities α, χ and β scale as successive orders in ka. Therefore, while χ/α ∼ ka, the much smaller ratio β/α ∼ (ka) 2 1 in the dipolar limit allows us to merely drop the magnetic parts of the force and torque from the discussion.
We first consider a 1D-standing wave along the z-axis, made of two counter-propagating circularly polarized planar waves of same frequency ω. We focus on the case where the two waves have opposite handedness (same handedness leads to no net force nor torque): We start by deriving the reactive part of the achiral force that stems from the inhomogeneous repartition of the electromagnetic energy between the electric and magnetic fields: with the field intensity I 0 = |E 0 | 2 ε 0 /µ 0 . For such a field, the time-averaged Poynting vector Π and its orbital and spin parts are zero, so that the radiation pressure < F d. α,β > vanishes. The chirality density is zero, and as a consequence the reactive part of the chiral force F r.
χ is also zero. Nevertheless the uniform chirality flow Φ = ωε 0 |E 0 | 2 /2(0, 0, 1) t yields an homogeneous dissipative chiral force in the propagation direction: From the hierarchy in the different susceptibilities discussed above, the total net force is dominated by its achiral reactive component -Eq. 2D-standing wave of complex amplitude E 1 in the (x, z)plane, with a wave of complex amplitude E 2 propagating in the y-direction, all waves are planar and linearly polarized: E 0 = E 1 e ıϕ cos kz + E 2 e ıky , 0, −E 1 cos kx t H 0 = 0, ıH 1 (sin kx + e ıϕ sin kz) , −H 2 e ıky t with a phase difference ϕ between the two standing waves. Even though the interfering waves are linearly polarized, the resulting field gives an inhomogeneous chirality density K(r) = −ωI 1,2 cos kx sin(ky + θ)/(2c 2 ), with a coupling term E * 1 H 2 = I 1,2 e ıθ . This density generates a periodic potential in the x and y-directions within determines the motion of the chiral dipole. It results in a reactive component of the chiral force which is presented in Fig. for in-phase amplitudes (θ = 0), for two chiral dipoles with opposite chirality χ. This force induces an alternative succession of attrac-tive (resp. repulsive) positions where the chirality density is maximum (resp. minimum) if Re[χ] > 0 (resp. Re[χ] < 0). In contrast with the configuration presented above, this one exhibits a situation where it is the chiral reactive force, related to the real part of χ, that yields an opposite mechanical effect on two dipolar enantiomers.
Conclusion -These results unveil chiral forces that stem directly from the chiral properties of the lightmatter interaction. In addition to the usual optical gradient forces and radiation pressures, the motion of a chiral dipole is defined within a new force field which directly depends on the enantiomeric form of the chiral dipole. We emphasize that such a chiral dipole is actually a model for the linear response of a chiral molecule to a monochromatic field, that is therefore widely employed in the field of molecular chirality [21]. In this context, we note that the reactive and dissipative decomposition of the chiral force actually links motional effects to two standard observables used for characterizing optically active systems: optical rotation (Re [χ] associated with chirality density) and circular dichroism (Im [χ] associated with chirality flow). This connection could open promising perspectives for separating objects according to their chirality. Our proposal appears realistic for either specifically manufactured or macromolecules, aggregates, biological objects, such as proteins. Importantly thus, it can be relevant for molecular deracemization. However, one should be cautious with respect to thermal fluctuations and Brownian motions that might overwhelm any dynamical signature related to optical chiral forces exerted on such small systems as molecules [24,25]. Nevertheless, we have assessed statistical methods allowing to recover weak external force signatures super-imposed on thermal fluctuations that could be effective in this context [26]. We therefore hope that our results will generate further discussions and experiments.