Dynamically Controllable Two-Dimensional Microvehicle by Coordinated Optical Pulling-Lateral Force

Harnessing exotic optical forces promises a plethora of biophysical applications and novel light-matter interactions. The exotic optical pulling force (OPF) and optical lateral force (OLF) have been studied separately, yet synthesizing both candidates simultaneously remains an unsolved challenge and could offer a more powerful manoeuvre of particles. Here, we report a coordinated scheme to harness these two forces together and present a dynamically controlled two-dimensional (2D) microvehicle. The strategy is to leverage unexplored helicity-dependent features of both forces, while the particle size and incident angle of light can also reverse optical forces. The underlying physics of the pulling-lateral force is beyond the dipole approximation, and can be the combined effect from the linear momentum transfer, spin-orbit interactions, etc. Notably, the ratio of both forces can be dynamically and arbitrarily controlled by the ellipticity of incident light solely. The configured 2D microvehicle provides a nontrivial recipe other than using metastructures which require exquisite designs and subtle fabrication processes.

The OLF was recently found on a chiral particle placed above or at an interface by the coupling of chirality and the linearly polarized light [29], also for achiral particles due to spin-orbit interaction inside circularly polarized light beams or interference patterns [27], [40], [41].The sign of OLF is normally determined only by the helicity of light, while it could be reversed with the size and chirality of a particle [42], and incident angle of light when the particle size is within the Mie regime [29], [43].However, the sign reversal effect on an achiral nanoparticle for a fixed helicity was not reported.We will show that, due to the momentum transfer, the OLF can also be reversely subject to the light helicity.
Here, we propose an exquisite scheme to harness these two forces synergically to configure an elegant two-dimensional (2D) microvehicle.Notably, the ratio of two forces is dynamically controlled, covering from negative infinity to positive infinity via only the light ellipticity, namely only by orientating the half-wave plate.Thus, the particle experiencing these two coordinated forces can move in an arbitrary direction.Meanwhile, precisely delivering the particle to a target position facilitates the particle-assembly [44], drug delivering [45], cell targeting [46], etc, which is normally realized by the translation of laser beam in optical trapping [47], or by a purposeful phase profile [48], [49], [50].Light-powered microvesicles are attracting growing attention because of versatile controls of force and torque, enabling complex manoeuvre of particle.By virtue of recent advances in nanofabrication technology, metastructures can be peeled off from the substrate and move at the interface of air and water illuminated with differently polarized light waves.For instance, translational nano-meta-vehicles are realized by the directional side scattering or spin-orbit interactions [26], [51], [52], [53].The metastructure can be designed to be selfstabilized when levitated inside a collimated light beam, acting as a promising light sail [54], [55].Optical torques have also been widely deployed in engineered metastructures to assist more sophisticated trajectories [56], [57], [58], [59].So far, there lacks a cooperative way to configure a 2D microvehicle that moves in an arbitrary trajectory using optical forces without optical torques.Our subtle 2D microvehicle with coordinated OPF and OLF extends the degree of freedom in the optical manipulation, and inspires new advances in handy and accessible light-powered robots for multifunctional applications in physical and biomedical sciences.

II. DESIGN
We consider a simple scenario that a dielectric spherical particle with a permittivity ε p is floating at the interface of air (ε air = 1) and water (ε m = 1.33), as shown in Fig. 1(a).An obliquely incident linearly polarized plane wave passing through the half-wave plate and the quarter-wave plate becomes an elliptically polarized light beam (wavelength λ = 532 nm), which is then impinged on the particle.By orientating the half-wave plate with different angles ϕ, the polarization direction of the linearly polarized light will be altered.After that, the light beam further passing through the quarter-wave plate will become an s, left-hand circular, p, right-hand circular or s polarization, for ϕ = 0, 45°, 90°, 135°or 180°, respectively.Different polarized light beams will exert optical forces in distinct directions, which can be composed of forces in the x-(F x , consisting of the optical radiation pressure force and OPF) and y-(F y , OLFs) directions.Consequently, the particle can move towards four directions in four quadrants.This capability relies on the dynamic and continuous control of the optical forces in the x-and y-directions, as shown in Fig. 1(b).F x and F y can be zero at different values of ϕ, causing the ratio of F y and F x (F y /F x ) to transit from negative infinity to positive infinity [red curve in Fig. 1(b)].By continuously controlling the angle from ϕ 1 to ϕ 4 , the particle can move along a complex trajectory.For instance, the particle can move along the +y-direction when ϕ ≈ 132°(F x ≈ 0), as shown in Fig. 1(c).When it encounters an obstacle, it can turn to top-right or top-left by setting ϕ to 90°−132°and 132°−180°, respectively.Remarkably, the moving direction covers angles in 360°by orientating ϕ from 0−180°.
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III. RESULTS
To find an optimal incident angle for F x and F y varying between negative and positive values at different ϕ, so that F y /F x can cover a range from negative infinity to positive infinity, we plot the map of F x with θ and ϕ, as shown in Fig. 2(a).For incident angles at which F x does not reverse its sign, such as large incident angles (θ > 65°), they are not suitable for configuring the 2D microvehicle.Thus, θ = 24°is chosen in Figs.1(b), 2(b) and (c).The mechanism of negative F x (OPF) is based on the momentum transfer from air to particle and further to water.Since permittivities of air (ε air ), particle (ε p ) and medium (ε m ) satisfy the relationship of ε air < ε m < ε p , the photon energy scattered into the medium can be larger than that incident into the particle as predicted by the Minkowski's theory, which describes that the momentum of a photon in a dielectric medium can be given as p = n hω/c, where n is the refractive index of the medium.Consequently, F x can be negative under a large forward scattering momentum.For instance, the force vectors on a particle when θ = 52°and ϕ = 0 show a dominant net force in the −x-direction (OPF), as shown in Fig. 2(b).This linear momentum increase has been demonstrated by floating an oil droplet at the interface of air and water [23], [40].However, the interpretation of this OPF effect is based on the geometrical optics where the particle radius a λ.This effect has a strong correlation with the polarizations of light [Fig.2(a)], as well as the particle size, as shown in Fig. 2(c).Normally, a large radius (a ≥ 600 nm) is required to induce this OPF, while it has some exceptions when ϕ = 90°(1/2π, s polarization), showing the emergence of OPF when a = 450 nm.The particle can be regarded as a dipole when the radius a λ, thus only the radiation pressure occurs.Simulations of optical forces in this article are performed in COMSOL using the rigorous Minkowski stress tensor, which is given as [31], [60], [61]: Here, n is the unit outward normal to the integral surface, and δ ij is Kronecker delta.Simulations are conducted by integrating over the homocentric sphere with a radius 30 nm larger than the particle at the interface.The optical force can be balanced by a fluid drag force expressed as F drag = 6f D πηav, where η is the viscosity of liquid, v is the particle velocity, and f D is the drag coefficient depending on the viscosity ratio of upper and lower media [62], [63].By approximately setting f D = 0.6 [62], the particle velocity is calculated [Fig.2(d)], showing maxium velocities approximately a few tens of microns per second.
To coordinate with F x and realize a 2D microvehicle, we plot the map of F y with θ and ϕ, as shown in Fig. 3(a).F y = 0 when the light beam is linearly polarized (ϕ = 0 and 180°, p polarization; ϕ = 90°, s polarization).For most incident angles except 11°< θ < 16°, F y < 0 and F y > 0 when 0 < ϕ < 90°and 90°< ϕ < 180°, respectively, indicating the correlation of OLF with the helicity of light.The sign of OLF is widely interpreted by the theory of spin-orbit interaction using the dipole approximation, where the particle can be regarded as two dipoles above and beneath water [40].
The equation of a dipole near a surface can be given as , where p = αeff E 0 is the induced dipole moment near a surface, αeff is the polarizability of particle, E 0 and E D are the incident field and re-scattered field by the dipole, respectively.Predicted by the dipole approximation, most OLFs coincides well with the helicity of light.The unexpected sign reversal could be attributed to the momentum transfer between air, particle and water as also indicated in Fig. 2(a) for the sign reversal of F x in an overlapped range of incident angle (11°−16°).The emergence of OLF can be visualized in Fig. 3(b), where the far-field energies are scattered dominantly to the +yand −y-directions for ϕ = 45°and 135°, indicating the negative and positive OLFs, respectively.Compared with F x , the sign of F y is more sensitive to particle radius as shown in Fig. 3(c).F y remains negative for small particles, e.g., a < 200 nm, when 0 < ϕ < 90°, as implied by the dipole theory.It then oscillates around the axis of F y = 0 for some larger particles and eventually has the sign remaining unchanged for even larger particles.The sign of F y can be easily tuned between negative and positive by orientating the angle ϕ.
Fig. 4(a)−(d) show F x and F y with the relationship of permittivities of particle and medium.For a fixed medium, e.g., water with ε m = 1.33 2 , smaller ε p facilitates the generation of negative  F x for some specific θ and ϕ.In contrast, larger ε m is beneficial to the negative F x [Fig.4(c)], which can be comprehended by the Minknowski's theory: p m = n m hω/c [64], implying a larger momentum scattered from the particle into the medium than that incident into the particle for a larger n m .Larger ε p also induces a strong oscillation on the OLF, which will cause the reversal of sign when the overall magnitude of OLF is small, as shown in Fig. 4(b).The OLF increases and then decreases with the increment of ε m [Fig.4(d)], thus water (ε m = 1.33 2 ) is well suitable for the generation of OLFs.The portion of particle immersed in water can be dynamically tuned with different densities of the medium.For example, by dissolving a certain amount of KCl into water, one can get a relatively large range of density from 1−1.17 g/cm 3 at a temperature of 20°, while the polystyrene particle has a density ∼1.05 g/cm 3 .Whereas the refractive index of the solution only has a narrow range of 1.33−1.38.We then model the F x and F y with different positions of particle, as shown in Fig. 4(e) and (f), respectively.F x has a maximum OPF when z ≈ −50 nm, while F y increases then decreases with the increment of z.F x changes sign from negative to positive when z is greater than 400 nm as the interface system is gradually diminished.Since F x is typically larger than F y , setting z from 0 to 300 nm should be feasible for configuring the microvehicle.It is noted that metallic particles are not feasible for microvehicles because of high densities and large radiation pressure forces.
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IV. CONCLUSION
In summary, we unveil unexpected phenomena in optical forces by illuminating a single particle with an elliptically polarized light beam.The effect of momentum transfer correlates strongly with the light polarizations and gives rise to the extraordinary OPF.The spin-orbit interactions generate the reversible OLF with different spins, which could also induce unexpected OLFs together with the momentum transfer between different materials.The polarization-dependent optical forces in two orthogonal directions (x-and y-directions) can be coordinated to facilitate configuring a simple 2D microvehicle capable to move towards any direction as the ratio of F y and F x ranges from negative infinity to positive infinity.
The sign of OPF shows strong correlations with light polarizations, which is one of the findings and key strategies for the 2D microvehicle.Another strategy is the reversible OLF, which broadens conventional understanding that the OLF on a single achiral particle depends intuitively on the light spin.Though new characteristics of exotic forces depend closely on the incident angle of light and properties of the medium and particle, the tuning of polarization solely is capable of implementing a 2D microvehicle by synchronizing the OPF and OLF.Compared with current design of microvehicle using metastructures, this scheme eases the design, fabrication/synthesis, and potential experimental implementation for transporting microscopic objects.It can also be extended to a larger scale system (e.g., millimetres or centimetres in dimension) using electromagnetic waves with longer wavelengths (e.g., microwave).We believe that our study not only showcases anomalous and coordinated optical pulling-lateral forces on a single sphere with a simple illumination of the plane wave, but also opens up a new paradigm for extending the degree of freedom in optical manipulation, having great potential in exploiting light-matter interactions, and biological applications, such as tumour targeting, optical binding and drug delivery [64], [65].

Fig. 1 .
Fig. 1.Synchronization of optical pulling-lateral forces for configuring a simple 2D microvehicle.(a) Schematics of the movement of a particle placed at the interface of air (n air = 1) and water (n m = 1.33) towards an arbitrary direction, which can be realized simply by controlling the polarization of the plane wave, namely by orientating the half-wave plate with an angle ϕ.The obliquely incident light beam is elliptically polarized with a wavelength λ = 532 nm and an incident angle θ.The directions of force can be determined by the coordination of the OLF (optical lateral force, F y ), OPF (optical pulling force, negative F x ) and ORPF (optical radiation pressure force, positive F x ).(b) Simulations of F x , F y and F y /F x with ϕ.F x and F y equal zero at different ϕ, so that F y /F x can be dynamically tuned in a range from negative infinity to positive infinity.Particle radius a = 750 nm, θ = 24°, n p = 1.45 (oil).(c) Illustration of the detour of particle by controlling of the orientation angle of the half-wave plate.The particle can move towards the first, second, third and fourth quadrants when ϕ 1 ≈ 90°−132°, ϕ 2 ≈ 132°−180°, ϕ 3 ≈ 0−48°and ϕ 4 ≈ 48°−90°, respectively.Thus, by tuning of ϕ from 0−180°, the particle can move towards an arbitrary direction.

Fig. 3 .
Fig. 3. Anomalous optical lateral forces.(a) Reversible OLFs, F y with θ and ϕ.Most F y remain negative and positive on the left and right sides of ϕ = 90°, respectively, coinciding with the consequences of spin-orbit interaction.The exceptions happen when 11°< θ < 16°, which may arise from the momentum transfer between the air, particle and medium similar to the mechanism to induce the OPF in Fig. 2(a).a = 750 nm, n p = 1.45.(b) Polar plot for the far-field scattering.The far-field light scatters dominantly to the +yand −y-directions for ϕ = 45°and 135°, indicating the negative and positive OLFs, respectively.a = 750 nm, n p = 1.45, θ = 80°.(c) OLFs with particle radius for different polarized light.The OLFs oscillate with the increment of particle radius.The oscillation could induce the reversal of sign for smaller particles, e.g., a < 300 nm.n p = 1.45, θ = 24°.

Fig. 4 .
Fig. 4. Optical pulling forces and lateral forces under different parameters.(a) F x with permittivity of particle ε p for various θ and ϕ.The OPF can only occur when ε p is small, e.g., ε p < 2.2.(b) F y with permittivity of particle ε p for various θ and ϕ.The oscillation of F y with ε p will reverse the sign of F y when the magnitude of F y is small for θ = 45°and ϕ = 24°.In (c) and (f), ε m = 1.33 2 .(c) F x with permittivity of medium ε m for various θ and ϕ.Larger ε m can induce the OPF more easily.(d) F y with permittivity of medium ε m for various θ and ϕ.The large ε m (e.g., ε m > 1.5) tends to decrease F y .In (c) and (f), ε p = 1.45 2 .(e) F x with the position of particle z.The large z will diminish the OPF by weakening the momentum transfer effect at the interface.×5 means the data is multiplied by 5. (f) F y with the position of particle z.F y increases and then decreases with the increment of z.In (a)−(f), a = 750 nm.