Creating stable trapping force and switchable optical torque with tunable phase of light

Abstract

Light-induced rotation of microscopic objects is of general interest as the objects may serve as micromotors. Such rotation can be driven by the angular momentum of light or recoil forces arising from special light-matter interactions. However, in the absence of intensity gradient, simultaneously controlling the position and switching the rotation direction is challenging. Here, we report stable optical trapping and switchable optical rotation of nanoparticle (NP)-assembled micromotors with programmed phase of light. We imprint customized phase gradients into a circularly polarized flat-top (i.e., no intensity gradient) laser beam to trap and assemble metal NPs into reconfigurable clusters. Modulating the phase gradients allows direction-switchable and magnitude-tunable optical torque in the same cluster under fixed laser wavelength and handedness. This work provides a valuable method to achieve reversible optical torque in micro/nanomotors, and new insights for optical trapping and manipulation using the phase gradient of a spatially extended light field.

Fig. 1.. Discovery of an intriguing optomechanical phenomenon by a new type of holographic optical trap (HOT).

(A) Schematic of different types of in-plane optical trapping and manipulation by HOTs. Left: Optical trapping by conventional intensity gradient force of a focused laser spot (0D, no phase gradient). Middle: Optical transportation by phase gradient force in a 1D optical curve with intensity gradient along the short axis and constant phase gradient along the long axis. Right: The proposed 2D-PGT for NP trapping and assembly using phase gradient force. (B) Schematic of optical torque reversal in a LANC by increasing the optical phase gradient with fixed laser power, wavelength, and handedness of circular polarization. In conventional optical driving, although the rotational direction of an object can be changed by tuning some optical parameters (such as the wavelength, handedness of circular polarization, and direction of orbital angular momentum), the sign of optical torque is seldom reversible.

Fig. 2.. Phase gradient–controlled trapping and self-assembly of Au NPs in a 2D-PGT.

(A) Illustration of the x-z view of the reconstructed 2D-PGT from the designed hologram. The dashed color line indicates the manipulation plane (x-y plane at z = 0), where the in-plane phase gradient force is driving the NPs toward the trap center. The optical field propagated along the +z direction. (B) The measured phase distribution of the 2D-PGT. Scale bar, 1 μm. (C) The measured intensity map of the 2D-PGT (the white line plots the intensity profile along the dashed x axis), where the measured trajectories of four Au NPs being driven to the center are overlaid on the map. These trajectories were tracked in ImageJ (62) according to four experimental videos. The laser intensity is 1.03 mW/μm². Except where otherwise stated, the NPs are always 250 nm in diameter. (D) Trajectories of a Au NP confined by a 2D-PGT with a phase parameter of −0.69 rad/μm². The background arrows represent the phase gradient force vectors. NPs can be trapped when phase gradients vectors are all pointing to the center of the optical field (i.e., the creation of a phase gradient potential well in the x-y plane). (E) Histograms for the particle displacement along the x and y directions corresponding to (D). (F) Measured phase gradient–dependent trapping stiffness of a Au NP. (G) Dark-field images of NPs driven by the phase gradient in a 2D-PGT to form a self-assembled hexagonal cluster. The images were recorded by a CMOS detector with a frame rate of 250 frames per second.

Fig. 3.. Tunable motion of LANC micromotors driven by optical phase gradients.

(A) Dark-field image sequences of a 19-Au-NP LANC in three different 2D-PGTs, all with the same left-handed circular polarization. Scale bar, 1 μm. The laser intensity is fixed at 4 mW/μm². The color spots mark the same NPs throughout the sequences, revealing the motion of the cluster. With different optical phase gradients, LANCs in the left (ξ = −0.41 rad/μm²), middle (ξ = −0.58 rad/μm²), and right (ξ = −0.63 rad/μm²) columns exhibit clockwise, stationary, and counterclockwise rotation, respectively. (B) Measured NP trajectories of the LANC with different ξ. (C) The angular trajectories of the NP components relative to the center of the LANC. Note that a negative slope corresponds to negative optical torque. The dotted lines indicate the moments when the phase profile is changed. (D) Similar angular trajectories of the same NPs that show bidirectional rotation of the LANC in real time, where a positive slope indicates a positive optical torque. (E) Calculated optical torques (Γ) for an ideal LANC micromotor (i.e., an exact hexagonal 19-Au-NP lattice) with increasing lattice constant and optical phase gradient. (F) Dynamics simulations of the nearest neighbor distance in the LANC versus ξ. The measured distances from experiments are plotted with red stars. (G) Calculated torques corresponding to (F). Compared with an exact hexagonal NP lattice (E), here, the different crossover value ξ (−0.58 rad/μm²) is due to the inhomogeneous lattice spacing in a real LANC. The rotational direction of the LANC micromotors in the experiments are marked by red arrows, while the “×” means stationary.

Fig. 4.. Phase gradient–controlled light scattering in LANCs.

(A) Illustrations of light scattering in the y direction by a single NP in an ideal LANC when the LANC is illuminated by 2D-PGTs with different phase gradients. The color maps represent the phase profiles, which resemble a paraboloid of ϕ(x, y) = ξ(x² + y²). The red and black arrows represent directions of light scattering from the positive (y2) and negative (y1) sides of the NP, respectively. The lattice spacing of the LANC is fixed at 620 nm. (B) Calculated scattering intensities of y1 and y2 in a series of 2D-PGTs with a negative sign of ξ. As a reference, the gray line shows the calculated optical torque of the LANC. The crossover from negative to positive torque occurs where y1 equals y2. (C) Calculated scattering intensities of y1 and y2 in a series of 2D-PGTs with a positive sign of ξ. The dashed red and black lines represent scattering intensities of y2 and y1 in a plane wave, respectively. a.u., arbitrary units.

Fig. 5.. Tailoring optical torque via tuning the in-plane phase gradient.

(A) Phase gradient of another type of 2D-PGT whose phase profile is defined by a paraboloid function with ξ > 0. The measured trajectories of three Au NPs show that opposite phase gradients can repel NPs. (B) The difference between the optical torque acting on an ideal 19-Au-NP LANC calculated from 2D-PGTs with tunable ξ and a plane wave. The lattice spacing of the LANC is fixed at 620 nm. (C) Calculated optical torques of the same LANC in different types of 2D-PGTs versus lattice constants. The 2D phase profiles of types I and I* resemble a paraboloid of ϕ(x, y) = ξ(x² + y²) + ϕ₀ (where ξ < 0, ϕ₀ > 0 for type I and ξ > 0, ϕ₀ < 0 for type I*); types II and II* resemble a tapered surface of $\$\phi (x,y)=2\sqrt{2}\xi \sqrt{x^2+y^2}+{\phi }_{}\$$ (where ξ < 0, ϕ₀ > 0 for type II and ξ > 0, ϕ₀ < 0 for type II*); types III and III* resemble a hyperbolic paraboloid of ϕ(x, y) = ξ(x² – y²) (where ξ < 0 for type III and ξ > 0 for type III*). Here, −2 μm ≤ x ≤ 2 μm, −2 μm ≤ y ≤ 2 μm, |ξ| = 0.7, and |ϕ₀| = 5.6. (D) The corresponding phase profiles and phase gradients of these flat-top 2D-PGTs. The arrows represent the phase gradient force vectors, whose magnitude varies spatially for types I(*) and III(*) but remains constant for type II(*) due to the linear phase profiles.

Fig. 6.. 2D-PGT as a general tool for bidirectional rotation of LANC micromotors.

(A) Calculated optical torques exerted on ideal LANCs (620-nm lattice spacing) with different numbers of particles (top, the diameter of each NP is 250 nm) and size (bottom, the NP number is 19) in three 2D-PGTs with different ξ. (B) Calculated optical torques exerted on ideal LANCs consisting of 19 NPs (250-nm diameter, 620-nm lattice spacing) with different materials. Calculations were done in a weak (ξ = −0.39 rad/μm²) and a strong (ξ = −1.32 rad/μm²) 2D-PGT for materials with different real and imaginary parts of the refractive index.