Optical forces play an important role in revealing the nature of light–matter interactions in scientific fields such as atomic physics, optics, photonics, and nanotechnology. Since the invention of laser tweezers [1,2], optical manipulation has made significant progress in optical trapping, binding, sorting, and transporting of particles by utilizing optical forces [3–6]. Currently, there is intensive research to enhance the ability to control different types of optical forces, for example, to dictate the dynamics and interactions of microparticles and nanoparticles (NPs) [3–6].

The familiar laser tweezers are point-like traps that allow capturing particles thanks to intensity-gradient forces arising from the strong focusing of a laser beam [1–4]. In contrast to the intensity-gradient forces, which are conservative forces that drag particles toward the minima of potential energy, non-conservative optical forces are very versatile being able to pull, rotate, and move particles [7]. The term “non-conservative” refers to the fact that part of the energy provided by the light beam is dissipated (absorbed or scattered) to achieve mechanical action [7].

The phase-gradient optical force is a well-known type of non-conservative force able to move particles [8]. This force arises from phase gradients of a focused light beam. For instance, curve-shaped trapping laser beams have recently been developed that take advantage of both intensity and phase-gradient forces to simultaneously capture and transport particles along any curve [8–13], respectively. This optical transport mechanism relies on the ability to design the phase gradient of the beam to redirect part of the light radiation pressure along the curve. Thus, the intensity-gradient force acts on the perpendicular direction to the curve enabling three-dimensional particle confinement while the phase-gradient force acts in one dimension: in the tangent direction to the curve for propelling them [8–13]. The full potential of the phase-gradient force can be harnessed in higher dimensions. In particular, a customizable phase-gradient force field extended in two dimensions can allow versatile optical manipulation of multiple particles while enabling them to interact in ways that would not be possible using conventional optical tools. However, this is a challenging problem that has not been addressed.

In this paper, we introduce the concept of a surface laser trap (LT) in which a customizable phase-gradient force field acts in two dimensions to both move and confine the particles on demand. We have developed a beam shaping technique that easily allows to conform this transverse phase-gradient force field to any surface, regardless of the shape of its boundaries. For instance, a radial, azimuthal, or arbitrary phase-gradient force field can be tailored onto the surface. It allowed us to achieve autonomous and sophisticated optical transport of individual particles and particle systems across the whole surface, with control over their speed along any trajectory. Surface LTs also have intensity-gradient forces; however, they are too weak to compensate the axial component of the light radiation pressure and therefore the particles have to be confined against a substrate (e.g., cover slip glass). Nevertheless, the intensity-gradient forces at the surface boundary allow the transverse confinement of the NPs to prevent them from escaping from the trap. To accumulate sufficient statistical data regarding the motion of the confined NPs, we have considered surfaces in the form of closed ribbon to set the particles into a continuous cyclic motion across the surface. Surfaces with and without holes are considered as well. Let us underline that the beam’s intensity and phase distributions prescribed on the surface are also customizable.

In contrast to conventional laser traps, the extended optical field of the surface LT facilitates particle–particle and particle–environment interactions. This is particularly interesting for harnessing electromagnetic interactions among the NPs resulting in optical binding forces [14,15]. Optical binding enables light-induced self-assembly between particles without direct contact. It also operates between multiple particles enabling interesting collective behaviors [14]. In the last years, optical binding forces have been applied to create optically bound (OB) arrays of non-contacting NPs working as optical matter systems [16–18]. Different types of OB arrays of metal NPs have been created against a substrate (e.g., cover slip glass) by using a Gaussian or a flat-top trapping beam with a transverse parabolic phase profile. This basic phase distribution gives rise to a radial phase-gradient force necessary for applying a compression force for establishing and maintaining an organized OB array of NPs [16–18]. This radial compression force is sufficiently weak to prevent the collapse of the optical binding configuration within the array of NPs. It does not allow the optical transport of the OB array of NPs, which can only rotate about its mutual center of mass due to the action of optical torque arising from circularly polarized illumination [16–18].

Here we show that it is possible to combine optical binding forces and driving phase-gradient forces in the surface LT, for example, to optically transport OB particles across the surface. In our experiments we have used colloidal gold nanospheres (dispersed in water) of 400 nm in diameter irradiated by an off-resonant trapping laser beam of wavelength λ0=1064nm. These NPs have high electric polarizability so they can exhibit optical binding forces strong enough to create stable OB dimers [19,20], which is the smallest case of optical matter system. We have found that multiple OB dimers (also triangular trimers) can spontaneously be created (without applying compression force) when the NPs traveling across the surface LT are close enough. Under circularly polarized illumination an OB dimer can exhibit optical torque about the mutual center of mass of the bounded NPs [19]. The surface LT also enables the integration of this optical torque with the optical binding and driving optical phase-gradient forces. This allowed us to experimentally demonstrate, for the first time, the optical transport of stable rotating OB dimers with simultaneous control of their rotational and translational motion across the surface LT. Another relevant interaction of the NPs and OB dimers is developed with their surrounding thermal-bath environment (water). This interaction is driven by thermal stochastic motion fluctuations associated with Brownian motion. We have analyzed the effects of these stochastic fluctuations on the motion of individual NPs and rotating OB dimers traveling across the surface.

This article is organized as follows. Section 2 introduces fundamentals for understanding the role played by the optical and thermal stochastic forces acting over the NPs in the surface LT. It also presents the numerical simulation method for predicting the motion of individual NPs. Section 3 presents the beam shaping technique developed for designing and adapting the phase-gradient force field onto the surface. The required experimental system is also described in this section. Section 4 presents the analysis of the experimental results corresponding to single and multiple NPs optically transported in surface LTs for distinct configurations of the phase-gradient force field and surface shapes. The experimental demonstration of rotating OB dimers optically transported in a surface LT is also analyzed in Section 4. Finally, concluding remarks and future perspectives are discussed.

In the electric dipole approximation, the time average optical force exerted over a particle (non-magnetic) at the position r=(x,y,z) can be written as a sum of three components as follows [5–7]: F(r)=14ε0nm2Re(αe)∇|E(r)|2+σextnmcS(r)+σextcnm∇×Ls(r),(1)where σext=kIm(αe) is the extinction cross-section of the dipole particle with electric polarizability αe, ε0 is the permittivity of vacuum, nm is the refractive index of the medium (e.g., water nm=1.33), c is the speed of light, S is the time averaged Poynting vector, and Ls(r)∝Im(E(r)×E*(r)) (where * stands for complex conjugate) is the time-averaged spin density of the light field whose electric field is E(r)exp[−iωt]. In this approximation, the three terms in Eq. (1) can be interpreted as distinct mechanisms by which a beam of light exerts forces on the illuminated particle. The first term in Eq. (1) describes a conservative force, i.e., the familiar intensity-gradient force. The second term represents the flow of energy, i.e., the non-conservative radiation pressure force (often referred to as scattering force) experienced by particles that scatter and/or absorb light. This term is the only contributing one if the incident wave field reduces to a plane wave. The third term is often referred to as curl-spin force (i.e., curl of the spin density) [5,7]. The physical meaning of this last term is less obvious. It is also a non-conservative force that arises in beams with polarization gradients, and it can make illuminated particles circulate in the plane transverse to the light propagation direction [21]. Here, we consider a wave field with both uniform intensity and polarization distributions across the flat surface LT for linear and circular polarization states. Therefore, the curl-spin force is zero in the surface LTs considered in this work.

A laser beam focused in form of flat surface can only provide two-dimensional particle trapping against a surface such as a glass microscope slide. This is because the axial intensity-gradient force is much weaker than the axial component of radiation pressure. Therefore, in a surface LT the axial displacement of the particles (in the light propagation direction) is suppressed by the microscope slide (i.e., the sample’s wall). The motion of the particles can be optically controlled by using the transverse component of the optical force Eq. (1).

Let us first consider the case of linearly polarized light for which the electric field strength is given by E(r)=E0(r)exp[iΦ(r)], where I(r)=nmε0c|E(r)|2/2 is the irradiance (intensity distribution) of the beam. Note that E0(r) is the real-valued amplitude and Φ(r)=kz(r)z+ϕ(r) is the phase distribution, where ϕ(r) is the transverse phase profile on the wavefront of the beam: uz∇ϕ(r)=0 with uz being the unit vector in the z direction (i.e., light propagation direction). The direction of the wave vector k(r)=kz(r)uz+∇ϕ(r) varies with position and the light’s wavenumber k=2πnm/λ0 in the medium fulfills the constraint k2=kz2+|∇ϕ(r)|2, where k≫|∇ϕ(r)| (paraxial limit) [8]. The transverse component of the intensity-gradient force can be written as F∇I(r)=nmRe(αe)∇I(r)/2c. This is an attractive force in the case of a particle with Re(αe(λ))>0 (as in our case) [5,6]. In a surface LT with uniform intensity distribution [i.e., I(r)=const within it, at the focal xy-plane, z=0] the transverse confinement force F∇I(r) only acts at the surface boundary due to the abrupt intensity change at the edge, that in turn can prevent the particles escape from the trap. Therefore, the translational motion (optical transport) of the particles across the surface can only be driven by the transverse component of the radiation pressure force, which can be written as [8] Fdrive(r)=σextnmkcI(r)∇ϕ(r).(2)This expression indicates the crucial role played by the transverse phase gradient ∇ϕ(r) that redirects part of the radiation pressure force in the transverse xy-plane (perpendicular to the light propagation direction) to drive the motion of the particle within the surface LT. The force Fdrive(r), Eq. (2), extends over the flat surface and is further referred to as driving phase-gradient force field (i.e., 2D force landscape). It has been demonstrated that the phase-gradient force acts independently of the state of polarization of the beam [21]. In the current state of the art the one-dimensional version of the driving phase-gradient force has exhaustively been applied to optically transport microparticles/nanoparticles along curve-shaped laser traps for both linearly and circularly polarized beams; see, for example, Refs. [8,10–12].

Since the driving phase-gradient force field acts independently of the polarization state, we can study its performance on the example of a surface LT with linear polarization, without loss of generality. In this case the net transverse optical force exerted over the particle across the surface is given by Fnet(r)=F∇I(r)+Fdrive(r),(3)where the confinement force F∇I(r) only acts at the boundaries of the surface. Thus, the optical transport of the particle within the surface is only controlled by the driving phase-gradient force field. The extinction cross-section given as σext=kIm(αe) is valid for a Rayleigh particle with radius a<0.1λ with λ being the wavelength of the laser beam in the medium. It is often assumed that Eq. (2) still provides a reasonably good estimation of the driving phase-gradient force for slightly larger NPs (a∼100–300nm), where σext is numerically calculated by using the Mie scattering theory [22]. Here, we consider a laser wavelength of λ0=1064nm and a gold NP (Cytodiagnostics, citrate-stabilized spheres dispersed in water) with radius a=200nm. Let us recall that the laser wavelength in the medium is λ=λ0/n=800nm for water (nm=1.33), and thus a=0.25λ and σext(λ)=0.32μm2. The optical transport of this type of NP driven by different 1D phase-gradient forces defined along the curve-shaped laser trap has previously been characterized [23]. Here, we consider an off-resonant laser wavelength to demonstrate the action of the phase-gradient propulsion force field across the surface LTs. For a resonant laser wavelength, the NPs considered in this work can experience photo-thermal heating strong enough to create thermal convective fluid flows enabling interesting propulsive effects, as reported in Ref. [23] for curve-shaped laser traps. The proposed surface LTs can be also applied for resonant NPs, which is out of the scope of this work.

The motion of the particles in the surface LT can be described by the two-dimensional Langevin equation of motion [24–26]: Mr¨P(t)=Fnet(rP(t))−νr˙P(t)+ζ(t),(4)where t is the time, rP(t)=(x(t),y(t))P is the position of the NP of mass M and r˙P(t)=drP/dt is the speed vector v(t) of the NP. The Stokes drag friction coefficient is given by ν=6πaηg (for a nanosphere) with η being the dynamic viscosity of the medium (water in our case) and g being a correction term that accounts for the increased hydrodynamic drag (approximated by the Faxen’s law [27]) near the substrate; see Appendix A. The friction coefficient ν is large enough to neglect the acceleration term, and thus the particle motion is over-damped: no average acceleration takes place, r¨P=0. The term ζ(t) is a fluctuating force associated with the stochastic thermal noise, which is responsible for the Brownian motion of the particle. Both the friction force −νr˙P(t) and random force ζ(t) arise from the interaction of the particle with its environment (i.e., the thermal bath, water). The fluctuating force ζ(t) follows a Gaussian distribution whose effects are summarized by its average over time by giving its first and second moments: ⟨ζ(t)⟩=0 and ⟨ζ(t)ζ(t′)⟩=2νkBTδ(t−t′), where νkBT is a measure of its strength with kB being the Boltzmann’s constant and T the temperature of thermal bath [28]. Note that the delta function of time δ(t−t′) indicates that there is no correlation in the Brownian motion in any time interval. Therefore, the translational motion of the NP described by Eq. (4) results from the action of three different forces: the exerted optical force Eq. (3), the friction force (responsible for dissipation), and the stochastic fluctuating force (adding a thermal noise motion).

When the optical force Fnet is not applied or it is too weak the particle undergoes Brownian fluctuations yielding a random motion. In contrast, if the optical force is stiff [|Fnet|≫|ζ(t)|] the speed of the particle can be approximated by v(t)=Fnet(rP(t))/ν without stochastic fluctuations. In this case the net force streamlines are almost equivalent to the particle trajectory, where the particle speed v(t) is tangent to the streamline at the position rP(t). Note that for a surface with uniform intensity distribution the particle speed is proportional to the applied phase gradient as v(t)∝∇ϕ(rP(t))/ν. Therefore, the phase gradient ∇ϕ(r) dictates the speed and trajectory of the particle at each point of the surface, so the phase ϕ(r) has to be designed accordingly. For instance, the particles can be set into an orbital motion enclosing the surface center by using a proper azimuthal distribution of the phase ϕ(r). The particles can be transported outwards from the surface center toward its outer boundary (or vice versa) by using a proper radial phase gradient. The radial and azimuthal (orbital) phase gradients can be combined to create more sophisticated optical transport configurations within the surface, as, for example, the one depicted in Fig. 1(a). Apart from controlling the particle trajectory through the phase-gradient direction, its strength |∇ϕ(r)| can be varied to speed up or slow down the particles at different regions of the surface. Surprisingly, this versatile optical transport mechanism based on a driving phase-gradient force field Eq. (2) tailored onto a surface has not been previously investigated.

In practice, thermal stochastic fluctuations can alter the particle motion driven by the optical force. To perform a realistic simulation of the particle dynamics within the surface LT, we have numerically computed the over-damped (r¨P=0) Langevin equation Eq. (4) by using the well-known splitting-method time-integration scheme (BAOAB) [26]. This allowed us to predict how the phase gradient ∇ϕ(r) tailored onto the surface LT dictates the optical transport of the particles in the presence of this thermal noise. The rules for designing the required phase distribution ϕ(r) for various configurations of interest are presented in Section 3. The corresponding numerical and experimental results are analyzed in Section 4.

It is well known that two particles excited by a common off-resonant field can form an OB dimer when they arrange themselves to a stable position, where the radial optical forces acting on them are zero [14,19]. The optical binding force arises from the electromagnetic (EM) particle–particle interactions, which can also exert optical torque over the OB dimer for both linearly and circularly polarized illumination [19]. For linearly polarized light this optical torque tends to align the OB dimer parallel to the polarization axis of the beam. In the case of circularly polarized light, the OB dimer can experience stable rotation driven by the optical torque [19], when the inter-particle separation is of the order of a few multiples of the wavelength λ (long-range EM interactions regime) [14] and even of half-wavelength 0.5λ (short-range EM interactions regime) [29]. The expressions for the optical binding forces and the optical torque for the case of dipole-like NPs have been reported elsewhere [19]. Such a dipole-like approximation is suitable for describing the optical binding of NP dimers with a diameter of the order of 0.25λ but it is not accurate for larger particles [19,20]. The numerical simulations and experimental results reported in Refs. [19,20] confirm that stable optical torque can also be achieved for large NPs as the ones considered in this work (gold NP with diameter 0.5λ).

As previously mentioned, a single NP can be transported along a phase-gradient force streamline, as sketched in Fig. 1(a). A rotating OB dimer can also be transported along a phase-gradient force streamline, as in Fig. 1(b), in a circularly polarized surface LT. However, due to the simultaneous rotational and translational motion, the expected deterministic trajectory (without stochastic fluctuations) of each NP comprising the OB dimer resembles to an epicycle orbit, as sketched in Fig. 1(c) for a circular ribbon-like surface. In this context the OB dimer behaves as a particle whose translational motion can easily be analyzed by tracking its mutual center of mass. The experimental results presented in Section 4 confirm the interesting dynamics of the rotating OB dimer traveling across the surface in the presence of stochastic fluctuations.

We consider a laser beam strongly focused in form of flat surface defined as {s(x,y);(x,y)∈R2} at the xy-plane (e.g., at z=0). A flat surface can be bounded by one or multiple curves (including piecewise ones) as required by the considered application. For example, the flat surface shown in Fig. 1(a) is a circular annulus domain D (a circular ribbon-like surface) bounded by inner and outer centered circles (C1 and C2, respectively) both enclosing the origin of coordinates (x=0,y=0). In this context, the shape of the boundaries defines the shape of the flat surface. The size and shape of the inner and outer boundary curves can be arbitrary (including polygons), and they can share a common origin of coordinates or not (see Appendix A).

How does the driving phase-gradient force field [Eq. (2)] adapt to the surface shape to control the particle motion within it? To solve this problem, we have developed a beam shaping technique for tailoring both the intensity and phase distributions onto the surface. As it is well known, for some physical problems the use of a curvilinear coordinate system may be simpler than the common Cartesian coordinate system. The proposed technique is based on the use of an orthogonal curvilinear coordinate system (α,β) and an appropriate conformal mapping transformation that simplifies the complex problem of designing the intensity I(s(α,β)) and phase distribution ϕ(s(α,β)) adapted to any surface shape, and so of its corresponding driving force Fdrive(s(α,β)). This technique, further referred to as curvilinear-conformal beam shaping (CBS), makes easier the design of the phase gradient ∇ϕ(s) required to govern the shape of the trajectory of the particles and their speed for any shape of the surface. It also significantly simplifies the design of sophisticated motion configurations of the particles across any surface, apart from the circular orbital motion and the radial one.

Let us first introduce the CBS technique on the simple yet important case of the circular ribbon-like surface (a circular annulus). In this case the orthogonal curvilinear coordinate system coincides with the common polar coordinates where α∈[0,2π] is the azimuthal angle and β∈[β1,β2] is the radial coordinate, where β1 and β2 are the radius of the inner (C1) and outer (C2) boundaries. Since x=βcosα and y=βsinα the annulus surface is parameterized as s(α,β)=β(cosα,sinα) whose grid curves are sα(β)=s(α=const,β) and sβ(α)=s(α,β=const), which are respectively depicted in Fig. 1(d) as red and blue curves. The phase gradient in this orthogonal curvilinear coordinate system is given by the expression ∇ϕ(α,β)=(∂βϕ)tβ+(∂αϕ)tα/β, where tβ=(cosα,sinα) and tα=(−sinα,cosα) are unit tangent vectors to the grid curve sα(β) (red curve) and sβ(α) (blue curve) at any point s(α,β) of the surface, respectively. To obtain an orbital motion, the required phase function is simply given by ϕo(α)=Qoα,(5)with Qo being an integer number. This yields a helical phase distribution (an azimuthal dependence of the phase) tailored onto the surface, as displayed in Fig. 1(d) for Qo=10. Let us underline that Qo corresponds to the familiar topological charge of an optical vortex beam. As it is well known, an optical vortex beam exhibits a point-like phase singularity [e.g., located at the surface center s(0,0)] yielding a dark hole in the beam intensity distribution (zero intensity around the phase singularity). The presence of this dark hole is characteristic of beams with such helical wave fronts and its size is larger as the value |Qo| increases, regardless of the shape of the surface. Therefore, we have considered an inner curve (of any shape) with a size large enough to enclose such a dark intensity hole. Let us underline that the change in the value of Qo does not alter the size and shape of the surface LT. The orbital phase gradient is ∇ϕo=(Qo/β)tα at any point of the surface and the particle can perform an orbital motion (clockwise for Qo<0) following any of the blue grid curves sβ(α); see Fig. 1(d). The radial motion is obtained by using ϕr(β)=2πQrβ/L,(6)where β∈[β1,β2] and the normalization distance L=β2−β1 is the width of the annulus: the difference among the radius β2 and β1 of the outer (C2) and inner (C1) boundaries. The radial phase gradient is ∇ϕr=2π(Qr/L)tβ so the particle can move radially following any of the red grid curves sα(β), inwards for Qr<0 and outwards for Qr>0. Note that the radial motion configuration depicted in Fig. 1(a) corresponds to ϕr(|β−β0|) that allows inwards and outwards motion simultaneously, with respect to the mean radius β0=(β1+β2)/2. The orbital and radial motion can be combined just by using a phase function written as ϕ(α,β)=ϕo(α)+ϕr(β) because such contributions are separable in this orthogonal curvilinear coordinate system (tβ·tα=0). In this way, a ring attractor-like motion configuration can easily be obtained by using the phase function ϕRing(α,β)=ϕo(α)+ϕr(|β−β0|),(7)where the corresponding driving force streamlines are depicted in Fig. 1(a). This case is interesting because the combined action of the radial and azimuthal components of the driving force drags all the particles in the surface to set them into a ring orbit of radius β0. The analysis of the particle motion driven by these phase functions and more sophisticated ones is presented in Section 4.

The CBS technique allows tailoring phase functions in more complex surface shapes than the circular annulus. This is achieved by using a conformal map transformation of the circular annular surface D to the target surface Γ. Specifically, the conformal map f:D→Γ transforms the family of grid curves (including the inner and outer boundaries) of D into the new geometry Γ, preserving the angles between the crossing grid curves [30]. Note that the conformal map f:D→Γ is a complex valued function and therefore the domains D and Γ are represented in the complex plane: D∈C and Γ∈C [30]. The inverse f−1 is a conformal map of Γ onto the circular annulus D. To illustrate the CBS technique, we consider the example of a domain Γ bounded by an inner rectangle (curve C1′) and an outer triangle (curve C2′), corresponding to the flat triangle-shaped surface displayed in Fig. 1(e). This case corresponds to doubly connected domains where the inner and outer boundaries of both domains are related to each other: C1′=f(C1) and C2′=f(C2). The family of grid curves of the surface Γ are mutually orthogonal and they correspond to an orthogonal curvilinear coordinate system (α′,β′). Indeed, the grid curves in this system (α′,β′) are directly obtained by the conformal mapping of the grid curves of the circular annulus: sα′(β′)=f(sα(β)) (red curves) and sβ′(α′)=f(sβ(α)) (blue curves). Thus, the phase function ϕ(α′,β′) can be written by using the change of coordinates α′=α′(α,β) and β′=β′(α,β) provided by the conformal map: β′exp[iα′]=f(βexp[iα]). In other words, in the triangle-shaped surface a red grid curve sα′(β′) corresponding to α′=const [see Fig. 1(e)] is a conformal image of a red curve sα(β) (with α=const) in the circular annulus D shown in Fig. 1(d). A blue grid curve sβ′(α′) corresponding to β′=const is a conformal image of a blue circle sβ(α) (with β=const) in D. In this way, the CBS technique is used to map any phase distribution ϕ(α,β) prescribed in the circular annulus D to a phase ϕ(α′,β′) prescribed in more complex geometries Γ. For instance, the azimuthal phase ϕo(α) and the radial one ϕr(β) of the circular geometry (D) are mapped onto triangle-shaped geometry (Γ) following its radial (red) and concentric (blue) grid curves, respectively. Note that the phase ϕo(α) is constant in each red grid curve sα(β) of D and therefore ϕo(α′) is constant in each red grid curve sα′(β′) of Γ. Analogously, the phase ϕr(β) is constant in each blue grid curve sβ(α) of D and therefore ϕr(β′) is constant in each blue grid curve sβ′(α′) of Γ; see Figs. 1(d) and 1(e). Thanks to the described conformal mapping, any phase distribution prescribed in the circular annulus (polar coordinates) can be tailored onto any other geometry irrespective of the shape of its boundaries. The CBS technique can also be applied for surfaces defined by one boundary (i.e., a surface without holes), which corresponds to the case of simply connected domains, where the inverse f−1 is a conformal map of Γ onto a disk (an annulus D with β1=0) [30]. This case of a surface LT without holes is studied in Section 4 as well.

Here, we have considered analytic boundary curves (without loss of generality) given by the so-called superformula [31] that allows straightforward generation of a large variety of shapes including polygons and curves with corners; see Appendix A. Alternatively, the boundary curves can be drawn freehand by using Bézier curves if required [32]. Let us underline that the CBS technique can be used to create surface LTs of any shape, phase, and intensity profiles. For instance, to create an open ribbon surface LT its amplitude E0(s) can be set to zero over specific regions as needed; see Appendix A. We have developed a program that allows direct and fast computation of the complex field amplitude of the surface laser beam, ES(s)=E0(s(α,β))exp[iϕ(s(α,β)],(8)typically in less than 2 s. This speed in computation is mainly due to the rapid calculation (a few milliseconds) of the conformal map based on the algorithm reported in Ref. [30].

To experimentally create the surface LT, an input collimated infrared laser beam (Azurlight Systems, ALS-IR-1064-10-I-CP-SF, λ0=1064nm, linear horizontal polarization) is modulated by a spatial light modulator (SLM, Meadowlark Optics, HSP1920-600-1300-HSP8, pixel size of 9.2 μm) in which the complex field amplitude [Eq. (8)] of the surface laser beam is encoded as a computer-generated phase-only hologram [33]. The experimental setup is sketched in Fig. 2 and comprises a high-numerical-aperture (NA) microscope objective lens (MO, Nikon CFI 100×, 1.45 NA, oil immersion nimm=1.512), which allows imaging the gold NPs under darkfield illumination (CoolLED illumination system pE-800, wavelength of 740 nm). The dichroic mirror used to redirect the laser beam to the sample also prevents the camera (sCMOS, Hamamatsu, Orca Flash 4.0, 16-bit gray-level, pixel size of 6.5 μm, operating at 800.7 Hz with an exposure time of 1 ms) from back-reflection laser saturation. The linear polarization of the trapping laser beam can easily be converted to circular polarization by using a tandem of a quarter- and half-wave plates (QWP and HWP) placed before the dichroic mirror, as depicted in Fig. 2. A proper rotation of the QWP and HWP allows compensating for the ellipticity arising from the polarization-dependent response of the dichroic mirror (Thorlabs DMSP950) [34]. Specifically, by using the QWP–HWP tandem the measured phase shift δ between the s and p polarization components of the trapping beam (reflected by the dichroic mirror) is δ=88.7°, which is a value close enough to the ideal one 90° for circularly polarized light.

Note that the MO lens projects a spatially filtered version E(s) of the encoded surface laser beam [Eq. (8)] onto the sample (at the focal plane, z=0). Specifically, the tube lens of the microscope projects the Fourier transform (FT) of the encoded beam at the input aperture of the MO lens, which filters its angular spectrum E˜S(k⊥)=FT[ES(s)] as follows: E˜(k⊥)=E˜S(k⊥)P(k⊥), with k⊥=(kx,ky) being spatial frequencies and P(k⊥) being the pupil function of the MO lens [35]. Therefore, we have used such a filtered version E(s)=FT[E˜(k⊥)] of the surface laser beam in all the simulations presented in this work. The expected intensity distribution |E(s)|2 and the measured intensity distribution are also shown in Fig. 2 for the case: surface in form of ribbon-like circular annulus (area of 94μm2) and triangle-rectangle one (area of 89μm2) as well as a triangular surface without hole (area of 97μm2). The optical power of these trapping beams measured at the input of the MO is 60 mW, 70 mW, and 50 mW correspondingly. The measured intensity distributions deviate from complete uniformity due to residual aberrations in the SLM, relay lenses, and microscope. Nevertheless, these intensities closely match the expected ones |E(s)|2 shown in Fig. 2. The strong focusing of the trapping beam can alter its polarization. The considered NPs are sufficiently small (a=0.25λ) and their optical transport is unlikely to be affected by these smooth intensity variations and eventual polarization gradients via the curl-spin force.

Let us first analyze the NP motion in the circular ribbon-like surface trap with orbital phase ϕo(α)=Qoα and linear polarization. In Fig. 3(a), the orbital phase distribution for Qo=−10 is displayed along with the predicted trajectory of the NP (white track, clockwise orbital motion), which has been obtained from the numerical integration of Eq. (4). The transverse intensity-gradient distribution |∇I(s)| displayed in Fig. 3(a) shows that its maximum value is obtained at the surface boundaries [curves sβ1(α) and sβ2(α)] while it is nearly zero within the surface, as expected from a uniform intensity distribution. This confirms that the confinement force F∇I(s) only acts over the NP at the surface boundaries to avoid its escape from the trap. In this case, the phase gradient is ∇ϕo=(Qo/β)tα and therefore the driving force is Fdrive(s)∝(Qo/β)tα, whose strength over the surface is displayed in the first panel of Fig. 3(b). Note that the force strength |Fdrive(s)| is inversely proportional to the radial coordinate β. Circular driving force streamlines are obtained, as depicted in Fig. 3(b), indicating the direction and orientation of the optical force field [Eq. (3)] exerted over the NP at each point of the surface LT. Let us recall that in absence of the thermal noise force ζ(t) the driving force streamlines are equivalent to the particle trajectory. As previously mentioned, the stochastic fluctuations can affect both the trajectory and speed of the NP. This is noticed in the trajectory and speed values |v(s)| of the NP displayed in the second and third panels of Fig. 3(b) corresponding to the numerical simulation and experiment; see also Visualization 1. Note that the experimental tracking data of the NP (its trajectory and speed) have been obtained from the recorded video of the experiment (Visualization 2) by using the tracking algorithm reported in Ref. [36]. These results show that the applied driving force is sufficiently stiff to set the NP into a well-defined circular trajectory; however, there are slight radial stochastic fluctuations that allow the particle to eventually change its circular orbit. This effect of the thermal noise is observed in the experimental and predicted trajectories, which are in good agreement. The maximum speed value of the NP is obtained at the surface regions where the strength |Fdrive(s)| reaches its maximum value (0.8 pN), in this case at the smallest value of the radius β=β1 as expected. To obtain more detailed information about the NP motion dictated by these forces, the radial vr=vrtβ and azimuthal (orbital) v⊥=v⊥tα components of the speed vector v(t)=vr+v⊥ are analyzed, where v⊥·vr=0. The measured orbital speed v⊥ of the NP is displayed at the fourth panel of Fig. 3(b). The speed histograms (given by its probability density function, PDF) displayed in Figs. 3(c) and 3(d) correspond to the numerical simulation and the experimental data, respectively. As observed in Fig. 3(c) the speed values |v| fit to a distribution with a mean value of 50 μm/s and maximum value of 140 μm/s. The histograms for orbital and radial speeds provide more comprehensive information. Indeed, the histogram of the orbital speed v⊥ reveals a well-defined Gaussian distribution centered at −50μm/s for both the simulation and the experiment, where the negative sign indicates clockwise motion. This result quantitatively confirms a stable directed NP motion dictated by a stiff driving phase-gradient force in the azimuthal direction. For both the simulation and experiment, the histogram of the radial speed vr reveals a well-defined Gaussian distribution centered at 0 μm/s, which is evidence that in the radial direction the Brownian motion fluctuations prevail. This stochastic radial motion eventually places the particle on a new circular orbit (of different radius) dictated by the corresponding driving force streamline. Indeed, the change of orbit due to the random action of the thermal noise in the radial direction is well noticed in the NP trajectories shown in Fig. 3(b), corresponding to the simulation and experiment data. Thus, the NP describes circular orbits with different radii that can fill the entire surface given enough time. The simulation and experiment have been performed for a time of 5 s that allows the NP to visit the entire surface. These results show how the combined action of the optical force and the thermal noise force governs the NP motion in the surface LT.

As previously stated, the combination of orbital and radial phase gradients allows the design of more sophisticated particle motion configurations. To illustrate this fact, we consider three representative examples as depicted in Fig. 4. The orbital (azimuthal) phase distribution ϕo(α)=Qoα (for charge Qo=−10) and its phase gradient are shown in Fig. 4(a). Note that the displayed contour curves of the phase gradient are equivalent to the streamlines of the driving force because Fdrive(s)∝∇ϕ(s). The first row of Fig. 4(b) shows the radial phase distribution ϕr(|β−β0|)=2πQr|β−β0|/L for Qr=−2/3 and the corresponding phase gradient ∇ϕr=2π(Qr/L)sgn(β−β0)tβ, where sgn(·) is the signum function. This radial phase combined with the orbital one results on the phase ϕRing(α,β) given by Eq. (7), whose distribution is displayed at the first row of Fig. 4(c). The contour curves of its phase gradient ∇ϕRing show the ring attractor-like configuration with clockwise motion (Qo=−10). Specifically, the radial phase term ϕr(|β−β0|) in Eq. (7) yields a phase-gradient force component that drags the particles into a circular orbit of radius β0, from which they cannot escape. This example illustrates a combined use of the radial and orbital phase-gradient forces to collect particles distributed across the surface and confine them into a fixed orbit if needed. This type of confinement mechanism governed by the driving force can be deactivated to let the particles distribute over selected regions of the surface. For instance, the phase function ϕ2(α,|β−β0|)=ϕr(|β−β0|)H(α,α1,α2) displayed at the second row of Fig. 4(b) is such that the radial phase is smoothly deactivated in a surface region defined between the angles α1 and α2, where H(α,α1,α2) is a smooth approximation to the Heaviside step function (see Appendix A). The phase function ϕSpray(α,β)=ϕo(α)+ϕ2(α,|β−β0|)(9)and its gradient are shown at the second row of Fig. 4(c), where the ring-attractor region is defined for an arc of 180° (as an example). Thus, in this case there is a region of the surface where the orbital phase gradient operates alone and another part where it acts in combination with the radial one to create a half ring-attractor region. This configuration is interesting because several particles can be captured by this ring-attractor region and subsequently released as a spray of particles into the pure orbital region of the surface.

By employing the latter phase functions, the particle can trace a trajectory that conforms to the surface shape. Nonetheless, it is also feasible to place the particle on orbital paths with shapes entirely distinct from the surface shape. For instance, to easily set the particles into a non-circular (i.e., a polygonal curve) orbital motion, the phase function ϕPoly(α,β)=ϕo(α)+ϕr(β)sin(Nα)(10)can be used, where N is the number of vertices of the polygonal trajectory. Note that this phase function [Eq. (10)] is not separable in the coordinates (α,β). The phase component ϕr(β)sin(Nα) and its gradient are shown at the third row of Fig. 4(b) for N=3. As observed at the third row of Fig. 4(c), the phase distribution ϕPoly(α,β) for N=3 has a phase gradient whose contour curves describe triangular orbits, in spite of the circular shape of the surface.

The numerical simulation of the particle motion predicts the trajectories and clockwise motion corresponding to the previous phase configurations [Eqs. (7)–(10)]; see the first and second rows of Fig. 4 and Visualization 3. The experimental results displayed at the third row of Fig. 5 (see also Visualization 4) are in good agreement with these numerical simulations. Specifically, the results for the ring-attractor motion configuration [phase given by Eq. (7)] are shown in Fig. 5(a) while for the sprayed one [for phase Eq. (9)] and the polygonal one [for phase Eq. (10), N=3] being displayed in Figs. 5(b) and 5(c), respectively. Note that the particle trajectory is not deterministic due to the action of the thermal noise force; however, the dominant role played by the driving phase-gradient force enables its optical transport as expected.

A rotating OB dimer can also be transported along any driving force streamline of the surface LT. Here, we analyze the optical transport of rotating OB dimers on the example of a circular ribbon-like surface trap with orbital phase (with charge Qo=−10) and circularly polarized light. The experiment has been performed for a time of 9 s that is sufficiently large to analyze the optical transport of several OB dimers across the surface; see Fig. 6 and Visualization 5. Specifically, several single NPs travel along different circular orbits and eventually some of them are close enough to form a rotating OB dimer that travels along a circular orbit as well. To further study the rotation of the OB dimer, the polarization of the laser trapping beam has automatically been switched from left circular polarization (CP) to horizontal linear polarization and finally to right CP. The QWP (see Fig. 2) is rotated in a sufficiently fast and smooth manner by using a motorized precision rotation mount (Newport URS100BCC) to switch the laser polarization. In the experiment, left CP is fixed in the interval 0–4.5 s [see Fig. 6(a)], then it is smoothly switched to horizontal linear polarization [time 4.5–6.2 s; see Fig. 6(b)], and finally right CP is fixed for 6.3–9 s [Fig. 6(c)]; see also Visualization 5. The tracking of the NPs and OB dimers confined by the surface LT is shown at the first column of Fig. 6 for each case. The measured orbital trajectory of the OB dimer (of its center of mass) and the driving force streamlines are shown at the second column of Fig. 6 for each case, where the value of the spin frequency Ω(t)/2π of the OB dimer is also displayed. Let us recall that the mutual center of mass of the OB dimer (with identical NPs) is located in polar coordinates at (θc,|rc|), where rc=(r1+r2)/2 with r1 and r2 being the position vectors of each NP as depicted in Fig. 1(b). Here it is important to distinguish the orbital angular frequency ωc=dθc/dt from the spin angular frequency Ω(t)=dθ/dt, which is the rate of change in the rotation angle θ(t) of the OB dimer around its center of mass. In the third column of Fig. 6 the measured angular positions θc(t) are represented as a function of the time along with the corresponding values of spin frequency Ω(t)/2π. The OB dimers display a stable clockwise orbital motion regardless of the light polarization. In Fig. 6(b) the change in the spin frequency of the OB dimer (labeled as D2) is noticed when left CP (time 2–4.5 s, in Visualization 5) is switched to linear polarization (time 4.5–6.2 s, in Visualization 5). The histograms (PDF distribution) of the spin frequency values Ω(t)/2π provide more quantitative information about the rotational motion of the OB dimers in each case; see the fourth column of Fig. 6. Specifically, the mean value of the spin frequency is −5.6Hz and 5.9 Hz for left and right CP, respectively. For linear polarization (only applied in 4.5–6.2 s) there is no optical torque and the histogram of the measured spin frequency Ω(t)/2π fits well to a Gaussian distribution centered at 0 Hz, which is characteristic of the thermal noise. Thus, for linear polarization the OB dimers exhibit random rotation while for circular polarization their rotation is stable, as expected. Let us recall that for gold NPs with a diameter bigger than λ/2 (as in our experiments) the resulting optical torque is negative [19,20]. In particular, Ω(t)<0 (OB dimer in clockwise rotation) for left CP and Ω(t)>0 (OB dimer in counterclockwise rotation) for right CP, as depicted in the fourth column of Fig. 6. The measured tracks of the NPs comprising the rotating OB dimers are also displayed at the fourth column of Figs. 6(a) and 6(c). Such particle tracks reveal epicycle orbits slightly affected by the random action of the thermal noise force. The measured inter-particle separation of each OB dimer is ∼500nm, which is close to 0.5λ. These experimental results demonstrate the ability to govern the translational and rotational motion of the OB dimer across the entire surface for both left CP and right CP. Previous works have only shown rotating OB dimers at a fixed position (without translational motion) [20].

To further explore more sophisticated motion configurations, we have conceived a multidirectional particle transport based on the phase function ϕSplit(α,β)=α(Q1H(βs−β)+Q2H(β−βs))(11)where the term H(β−βs) (Heaviside step function centered at βs=β0−b) is applied to spatially split the phase into two different orbital configurations with charge Q1 and Q2, respectively. Let us first explain its performance with the example of the circular ribbon surface LT displayed in Fig. 7(a). In this type of surface LT, the particles can perform clockwise and counterclockwise orbital motion depending on the surface region where they are, as observed in Fig. 7(b). Specifically, the particles perform an orbital motion in the inner section (clockwise, Q1=−7) and in the outer section (counterclockwise, Q2=7) of the circular ribbon surface. The phases of the inner and outer sections are separated by a thin phase-gap section (of width 2b) defined by β∈[β0−b,β0+b] where ϕSplit(α,β) is constant; see Appendix A. The driving phase-gradient force field created around this transition region that works as a roundabout allows the exchange of particles between the outer and inner sections, as observed in Fig. 7(b). The number of exchange regions indicated in the first panel of Fig. 7(b) coincides with the value of the charge Q2=−Q1. The change of direction of the particle is well noticed in the third panel of Fig. 7(b) where the trajectory and azimuthal (orbital) speed v⊥ of the particle are shown; see also Visualization 6. The corresponding experimental results are displayed in Fig. 7(c) where the particle tracking (Visualization 7) and the time lapse image reveal the multidirectional particle motion as in the numerical simulation.

The CBS technique allows mapping any phase function onto any flat surface shape. Thus, any of the previous phase distributions can be prescribed onto any other surface geometry to obtain an analogous particle motion configuration. For instance, the previous multidirectional particle motion can easily be set into the triangle-shaped surface as shown in Fig. 7(d), which is also shown in Visualization 6 along with the annulus surface trap to help their comparison. Note that the phase ϕSplit(α′,β′) prescribed onto this triangle-shaped surface trap is given by Eq. (11), where the changes of coordinates α′=α′(α,β) and β′=β′(α,β) are provided by conformal map as explained in Section 3. As expected, the particle describes orbits adapted to the geometry of the outer (triangle) and inner (rectangle) boundaries. This is also observed in Fig. 8(a) that corresponds to the case of the same triangle-shaped surface for an orbital phase ϕo(α′)=Qoα′ with charge Qo=−10 (clockwise orbits). Indeed, the particles close to the inner boundary describe a rectangular trajectory while the particles close to the outer boundary describe a triangular trajectory. The particles at intermediate radial positions (between the boundaries) exhibit orbits whose shapes coincide with the driving force streamlines as well. The numerical simulation displayed in Fig. 8(a) (see Visualization 8) is in good agreement with the experiment; see time lapse image and Visualization 9. It is worth noting that the optical transport of the NPs fits both boundaries of the surface, which also evidences a good optical confinement. These results underline the versatility of the developed CBS technique to control the optical force and demonstrate that it is possible to adapt the particle trajectory to any surface, no matter how different its boundaries may be.

Finally, we study the case of surface LTs defined by one boundary and without holes. In this case, a radial phase distribution can be prescribed onto the surface, as, for example, the triangle surface displayed in Fig. 8(b). Specifically, the radial phase ϕr(β′) for Qr=−2 [see Eq. (6)] allows setting an inwards radial motion from the outer boundary (triangular curve) toward the surface center; see the first row of Fig. 8(b). Note that the phase ϕr(β′) has a spatial distribution analogous to a converging (Qr<0) or diverging (Qr>0) phase adapted to the surface geometry. The numerical simulation results shown in Fig. 8(b) confirm that the particle can be set into an inwards radial motion according to the driving force streamlines. The particle trajectory fits the radial force streamline in spite of the thermal noise force because the applied driving force is sufficiently stiff. For the inwards configuration, the particle starts near the surface boundary, and it quickly arrives at the surface center (in 125 ms) where it remains strongly confined by the radial phase-gradient force defined around the surface center. The particles cannot escape from the surface center due to the radial inwards configuration of the driving force at this point of the surface. For an outwards radial phase-gradient force (e.g., for Qr=2) the particles can move from the center of the surface toward its outer boundary, as observed in the second and third rows of Fig. 8(b). In this case, the particles are expelled from the surface trap at the outer boundary because the outward radial phase-gradient force is stiffer than the intensity-gradient force at the surface boundary; see first column (second and third rows) of Fig. 8(b). Therefore, as expected, such a divergent phase distribution yields an outward driving force that expels all the particles leaving the optical trap empty. Is there a method to stop particles from escaping in the case where an outward driving force is applied? Indeed, there exists a method to avoid this issue thanks to the versatility on the design of the phase of the surface LT. Specifically, it consists of placing a contrary (inward) phase-gradient force close to the boundary of the surface in order to stop the particles. This delimits the action of the outwards driving force up to an edge close to the surface boundary. The corresponding phase function is ϕBarrier(β′)=ϕr(|β′−βb′|) that places such an edge barrier at the radius βb′ close to the surface boundary. As shown in Fig. 8(b), second column (second and third rows), the particle travels from the center toward the boundary but it cannot escape because it is strongly confined at the edge curve sβb′(α′), where the outward and inward forces are balanced. Since the radial force is suppressed at the edge curve sβb′(α′), the particle can perform a random motion along this curve due to the action of thermal noise force. The experimental results are displayed at the third column of Fig. 8(b); see also Visualization 10. For the inwards configuration corresponding to ϕr(β′) the driving force drags multiple NPs toward the surface center where they remain strongly confined, as expected. When the delimited outward driving force [corresponding to ϕBarrier(β′)] is applied, all these particles quickly move toward the edge barrier where they remain. As an example, in the experiment we have alternated the inward and the delimited-outward driving force configurations every second (see Visualization 10), with the purpose of visualizing the motion and confinement of NPs with sufficient time. The particle tracks (white tracks for 500 ms) are also displayed at the third column of Fig. 8(b). These results underline the versatility in the design and application of the phase-gradient force tailored onto the surface LT.

The proposed CBS technique allows straightforward design and control of the driving phase-gradient force across the entire surface LT, regardless of the surface shape. It is based on a conformal mapping method that we have developed for tailoring the beam’s complex field distribution onto the surface. Different numerical simulations and experiments have been presented to demonstrate how the phase-gradient force can be exerted over the particles to control their speed and trajectory on the surface. We have found that the phase-gradient force field can easily be structured to enable both the optical transport and confinement of the particles, over specific regions of the surface as needed. For instance, the particles can be guided along a predetermined path and then allowed to disperse like a spray across a specific area of the surface. Moreover, we have shown the ability to set the particles into a multidirectional transport configuration enabling simultaneous clockwise and counterclockwise transport orbits in different regions of the surface. To achieve autonomous distribution of particles in a stable multidirectional transport, we have devised a method for designing a phase-gradient force field that allows particles to change their trajectory at roundabout regions of the surface without user intervention. The design of the phase-gradient force field is so versatile that it can even be configured to create confinement edge barriers to confine the particles within the surface LT.

The versatility of surface LTs with conformable phase-gradient force is beyond the scope of conventional optical manipulation methods. It simplifies the optical manipulation of multiple particles across the surface and allows them to interact in various ways. They can interact with each other mechanically (e.g., collisions) and with their environment, and even electromagnetically among themselves (e.g., OB particles). One of these important interactions is developed with their thermal bath environment (water in our case), which is responsible for slight motion fluctuations associated with Brownian motion. We have experimentally analyzed the effects of these stochastic fluctuations on the motion of the transported particles across the surface. For instance, when a particle is set into a circular transport orbit, the small random fluctuations acting on the perpendicular direction (where there is no driving force component) slightly move the particle at a neighbor circular orbit. This result illustrates an interesting dynamic arising from the combined action of the optical and stochastic thermal force. The thermal force follows the statistics of a white noise that decorrelates in time. The possible hydrodynamic interactions, however, involve non-trivial correlations between the particles and their environment [37,38], whose analysis is out of the scope of this work. The hydrodynamic interaction between multiple particles in a surface LT could enrich the pattern of dynamical behaviors involving translational and rotational motions that arise [39].

The considered gold NPs are large enough (400 nm in diameter) to enhance the forward scattering along the spin angular momentum direction of light creating a recoil negative torque due to momentum conservation [20]. The observation of stable negative torque in an OB dimer under circularly polarized light has been reported elsewhere [20]. Here, for the first time, we have experimentally demonstrated that it is possible to set an OB dimer in simultaneous translational and rotational motion. Specifically, the negative optical torque governs the rotational motion of the OB dimer while the phase-gradient force drives its trajectory and speed across the surface LT. We have found that the trajectory of the mutual center of mass of the two NPs comprising the OB dimer fits the driving force streamline, as expected. Interestingly, these two NPs describe epicycle orbits due to their simultaneous translational and rotational motion; see Fig. 1(c) and Fig. 6. In this context the OB dimer behaves as a quasi-particle where its two NPs travel together at the same speed across the surface LT. As is well known, the stochastic fluctuations influence all degrees of freedom in the particle motion including small random translations and rotations. The analysis of the measured trajectories of the OB dimers indicates that the optical torque exerted over the OB dimer is strong enough to set it into sustained rotation (at ∼6Hz) and the stochastic thermal fluctuations have little effect on it. The rotation direction of the OB dimers can be switched by changing the circular polarization handedness, without interrupting their optical transport across the surface. These experimental results show the independent action of the optical torque and the driving phase-gradient force on the particles. The simultaneous rotational and translational motion of the OB dimer turns it into a type of light-driven miniature rotatory motor of high interest. In the experiments we have also observed rotating OB triangular trimers (Visualization 9) and other small groups of optically bound NPs.

We envision that surface LTs with conformable phase-gradient force field can facilitate the manipulation and study of complex systems of interacting particles including the optical matter ones [16,17,40,41], whose manipulation is challenging by using conventional optical traps. Both the structure and function of the phase-gradient force field can be governed on the surface LT. This can be further applied to create and manipulate light-driven nanomotors and optical matter structures such as ordered arrays of optically bound NPs [17,40]. The optical manipulation of low-refractive-index particles (i.e., lower than the refractive index of their surrounding medium) such as hollow glass microspheres and vesicles can be challenging by using conventional optical traps [42]. It has been recently reported that customized dark traps comprising several laser curves allow manipulating such particles [42]. The proposed surface LTs can be also applied to manipulate low-refractive-index microparticles thanks to the demonstrated conformable phase-gradient force field enabling both confinement and optical transport.