Fig. 1. (a) Sketch of different types of particle motions that can be dictated by phase-gradient forces tailored onto a surface. In a surface LT simultaneous translation and rotation motion of an OB dimer is possible, and thus its NPs can describe a cycloid (b) or epicycle orbit (c). A set of several grid curves (blue and red curves) is depicted for circular (d) and triangular (e) ribbon-like surfaces. The proposed technique allows tailoring the phase of the trapping beam onto any surface. Orbital (azimuthal) and radial phase distributions are shown in (d) and (e) as an example.

Fig. 2. Experimental setup: optical trapping system (an inverted darkfield microscope and an SLM) and imaging system (sCMOS camera). A collimated input laser beam (NIR, λ0=1064  nm) illuminates the SLM, where the surface laser beam [Eq. (8)] has been encoded as a hologram. This beam is optically projected onto the sample to create the surface LT, by using relay lenses RL1, the microscope’s tube lens, and objective (MO, Nikon CFI 100×, 1.45 NA). The relay lenses RL1 and RL2 have a focal length of 200 mm and 150 mm, respectively. The expected and measured intensities are shown.

Fig. 3. (a) Orbital (azimuthal) phase ϕo(α)=Qoα with charge Qo=−10 and intensity gradient of the circular ribbon-like surface trap (linearly polarized), displayed along with the expected trajectory (white track) of the NP (radius a=200  nm) provided by the simulation. (b) The strength of the driving phase-gradient force along with its streamlines is shown. The expected and measured trajectories of the particle are also displayed; see Visualization 1. (c) Histograms of the particle speed values provided by the motion simulation. (d) Darkfield time-lapse image of the NP that reveals the particle trajectory in the surface LT; see Visualization 2. The histograms of the speed values obtained from the experiment are also shown in (d).

Fig. 4. (a) Orbital phase ϕo(α)=Qoα with charge Qo=−10 and its normalized phase gradient |∇ϕo|/k0 are shown. The displayed contour curves indicate the direction and orientation of the phase gradient. (b) The phase and its normalized gradient are shown for radial phase ϕr(|β−β0|) and ϕ2(α,|β−β0|) as well as for a non-separable phase function ϕr(β)sin(Nα) (at the third row). The orbital phase ϕo(α) can be combined with the latter ones to obtain the three representative phase configurations shown in (c): ring-attractor [Eq. (7)], spray [Eq. (9)], and polygonal [Eq. (10)].

Fig. 5. Driving force Fdrive(s) tailored onto a surface trap (linearly polarized) and the expected NP motion (simulation, trajectory, and speed; Visualization 3) are shown at the first and second rows, corresponding with the phase functions: (a) ϕRing(α,β) [Eq. (7) with Qo=−10], (b) ϕSpray(α,β) [Eq. (9) with Qo=−10 and Qr=−2/3], and (c) ϕPoly(α,β) [Eq. (10) with N=3, Qo=−10, and Qr=−2/3]. The third row shows the experimental results: (a) and (b) are time-lapse images of the NP motion whereas (c) displays the measured trajectory and speed of an NP; see Visualization 4.

Fig. 6. Experimental results. Orbital (translational) and rotational motion of an OB dimer obtained from the measured tracking of its NPs; Visualization 5. (a) The OB dimer D1 exhibits stable clockwise rotational motion under left circular polarization (CP) of the surface laser beam. The trajectory of D1 and the histogram of its rotation frequency Ω(t)/2π values are displayed. The tracks of the NPs (NP1 and NP2) comprising D1 reveal the expected epicycle orbits. (b) The rotation of the OB dimer D2 stops when the linear polarization is applied (time 4.5–6.2 s), as clearly noticed in the displayed histograms. (d) Under right CP the OB dimer exhibits stable counterclockwise rotation. In all the cases the orbital motion is clockwise, which is governed by the driving phase-gradient force (orbital phase with Qo=−10).

Fig. 7. (a) Simulation. Phase ϕSplit(α,β) for charges Q2=−Q1=7, intensity gradient of the corresponding surface beam (linearly polarized), and trajectory (white track) of a single NP. (b) The strength and streamlines of the driving force are shown along with the NP trajectory and its speed |v|. The exchange of NPs is achieved in the regions indicated as Ei=1–4. The trajectory and orbital speed v⊥ of the NP reveal the change of counter/clockwise orbits; Visualization 6. (c) Experiment. The displayed darkfield images show the trajectories of several NPs revealing their multidirectional transport; Visualization 7. This type of autonomous transport can be achieved regardless of the surface shape, as the triangular trap shown in (d), also shown in Visualization 6 for comparison (simulation).

Fig. 8. (a) Triangle surface trap with rectangular hole (circularly polarized), for orbital phase ϕo(α′)=Qoα′ with charge Qo=−10. The clockwise orbits of the NPs adapt to the surface shape as observed in the simulation (Visualization 8) and experiment (time-lapse image, and Visualization 9). (b) Triangular surface LT without hole (circularly polarized). Inwards and outwards radial motions are obtained by using the radial phase ϕr(β′), Eq. (6), with charge Qr=−2 and 2, respectively. The delimited-outward motion is achieved by using the phase ϕBarrier(β′)=ϕr(|β′−βb′|), which locally yields an inwards force acting as an edge barrier to prevent leaking NPs from the trap as observed in the simulation and experiment [bottom panel in (b)]; see Visualization 10.

Fig. 9. (a) Set of grid curves for different surfaces (domains) whose boundaries curves parameterized by C(α)=R(α)(cos α,sin α) are given by the radius R(α) according to Eq. (12): circle q=(1,1,0,1,1,1), smooth triangle q=(1,1,3,6,6,6), rectangle q=(1,2/3,4,15,15,15), sharp triangle q=(1,1,3,30,55,55), and smooth pentagon q=(1,1,5,4,4,4). (b) Phase distribution (orbital phase, charge Qo=−10) of the surface laser beam [Eq. (8)] for closed ribbons. (c) Phase distribution for open ribbons. (d) The phase distribution ϕSplit(α,β) (with Q2=−Q1=7) is shown for different values of the phase-gap width 2b, for both the circular ribbon and the triangular one. The corresponding intensities are shown in (e).