Shaping of light beams along curves in three dimensions

Generation for high-dimensional caustics and artificially tailored structured caustic beams

We theoretically propose and demonstrate topological parabolic umbilic beams (PUBs) with high-dimensional caustic by mapping catastrophe theory into optics. The PUBs are first experimentally observed via dimensionality reduction. Due to the high-dimensionality, such light beams exhibit rich caustic structures characterized by optical singularities where the high-intensity gradient appears. Further, we propose an improved caustic approach to artificially tailored structured beams which exhibit significant intensity gradient and phase gradient. The properties can trap and drive particles to move along the predesigned trajectory, respectively. The advantages for structured caustic beams likely enable new applications in flexible particle manipulation, light-sheet microscopy, and micromachining.

Zero-order free holographic optical tweezers

Holographic optical tweezers (HOTs) use spatial light modulators (SLM) to modulate light beams, thereby enabling the dynamic control of optical trap arrays with complex intensity and phase distributions. This has provided exciting new opportunities for cell sorting, microstructure machining, and studying single molecules. However, the pixelated structure of the SLM will inevitably bring up the unmodulated zero-order diffraction possessing an unacceptably large fraction of the incident light beam power. This is harmful to optical trapping because of the bright, highly localized nature of the errant beam. In this paper and to address this issue, we construct a cost-effective, zero-order free HOTs apparatus, thanks to a homemade asymmetric triangle reflector and a digital lens. As there is no zero-order diffraction, the instrument performs excellently in generating complex light fields and manipulating particles.

Spatio-temporal structuring control of a vectorial focal field

Focal field modulation has attracted a lot of interest due to its potential in many applications such as optical tweezers or laser processing, and it has recently been facilitated by spatial light modulators (SLMs) owing to their dynamic modulation abilities. However, capabilities for manipulating focal fields are limited by the space-bandwidth product of SLMs. This difficulty can be alleviated by taking advantage of the high-speed modulation ability of digital micromirror devices (DMDs), i.e., trading time for space to achieve fine focus shaping. In this paper, we propose a new, to the best of our knowledge, technique for achieving four-dimensional focal field modulation, which allows for independent manipulation of the focal field's parameters (including amplitude, phase, and polarization) in both the space and time domains. This technique combines a DMD and a vector field synthesis system based on a 4-f system. The high-speed modulation ability of DMDs enables versatile focus patterns to be fast switchable during the exposure time of the detector, forming multiple patterns in a single recording frame. By generating different kinds of focal spots and lines at different moments during the exposure time of the detector, we can finally get complete multifocal spots and lines. Our proposed method is effective at improving the flexibility and speed of the focal field modulation, which is beneficial to applications.

Composite Diffraction-Free Beam Formation Based on Iteratively Calculated Primitives

To form a diffraction-free beam with a complex structure, we propose to use a set of primitives calculated iteratively for the ring spatial spectrum. We also optimized the complex transmission function of the diffractive optical elements (DOEs), which form some primitive diffraction-free distributions (for example, a square or/and a triangle). The superposition of such DOEs supplemented with deflecting phases (a multi-order optical element) provides to generate a diffraction-free beam with a more complex transverse intensity distribution corresponding to the composition of these primitives. The proposed approach has two advantages. The first is the rapid (for the first few iterations) achievements of an acceptable error in the calculation of an optical element that forms a primitive distribution compared to a complex one. The second advantage is the convenience of reconfiguration. Since a complex distribution is assembled from primitive parts, it can be reconfigured quickly or dynamically by using a spatial light modulator (SLM) by moving and rotating these components. Numerical results were confirmed experimentally.

Cycloid-structured optical tweezers

We designed novel cycloid-structured optical tweezers based on a modified cycloid and holographic shaping techniques. The optical tweezers realize all the dynamic characteristics of the trapped particles, including start, stop, and variable-velocity motions along versatile trajectories. The superiority of the tweezers is experimentally verified using polystyrene micro-sphere manipulation. This work provides a novel platform for more complex manipulations of particles.

Collecting, transporting and sorting micro-particles via the optical slings generated by a liquid crystal q(φ)-plate

We disclose a transporting/collecting optical sling generated by a liquid crystal geometric phase optical element with spatial variant topological charge, which shows the intriguing repelling/indrawing effect on the micro-particle along the spiral orbit. Two proof-of-concept prototypes, i.e., an optical conveyor and a particle collector, are demonstrated. Based on the distinct dynamic characteristics of the micro-particles in different sizes, we conceptually propose a design for particle sorting. Thus, our proposed method to generate a spiral optical sling with spatial variant orbital angular momentum for on-demand collecting, transporting and sorting micro-particles is substantiated, which can find extensive applications in bio-medicine, micro-biology, etc.

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

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.

Self-healing property of the self-rotating beam

In this study, we demonstrate the self-healing of self-rotating beams with asymmetric intensity profiles. The proposed self-rotating beam exhibits an asymmetric intensity profile and self-healing properties in free-space propagation. In addition, the rotation direction and beam intensity profile of the self-rotating beam can be adjusted using the parameters a and b in the phase function. The effects of the position and size of the obstruction on the self-healing property of a self-rotating beam were studied both experimentally and numerically. The simulation and experimental results demonstrate that a self-rotating beam can overcome a block of obstacles and regenerate itself after a characteristic distance. Transverse energy flows were used to explain the self-healing properties. Moreover, the beam rotates during propagation, which can be used to capture and manipulate microscopic particles in a three-dimensional space. It is expected that these rotating beams with self-healing properties will be useful in penetrating obstacles for optical trapping, transportation, and optical therapy.

Energy flow inversion in an intensity-invariant focusing field

Dependence of light intensity on energy flow is the most intuitive presentation of an optical field. This dependence, however, also limits the applications to the interaction of the light field with matter. For further insight into this, we demonstrate a novel case of the optical field, named as the counterintuitive chiral intensity field (CCIF), in the highly focusing situation: the energy flow reverses during the propagation but the intensity distribution pattern is kept approximately invariant. Our results show that, in this process, the mode correlation decreases rapidly while the intensity correlation remains invariant in the focus area. Furthermore, this property is still valid even if the pattern helicity and number of spiral arms are changed. This work deepens the understanding of the relationship between energy flow and field intensity, and it will offer diversified operations in many applications, such as optical micromanipulation, optical fabrication, etc.

Chiro-optical fields with asymmetric orbital angular momentum and polarization

In this paper, we proposed a flexible method for generating asymmetric chiro-optical fields. Different from most of the chiro-optical fields superimposed by vortex beams which are rotationally symmetric, the asymmetric chiro-optical field has a locally controllable orbital angular momentum (OAM) and polarization state. By using a helix phase plate (HPP) calculated based on coordinates transformation of the perfect vortex, the OAM controllability of a single chiro-optical field could be achieved. Then, by using the transformation matrix method, several discrete chiro-optical fields with different rotation angles and topological charges were stitched together as a multi-lobed chiro-optical field with asymmetric OAM on each side-lobe. Furthermore, we designed two HPPs that can be loaded into two spatial light modulators to modulate the polarization state of each side-lobe of the asymmetric chiro-optical field independently. The proposed asymmetric chiro-optical field breaks the characteristics of uniform OAM and polarization distribution of conventional chiro-optical fields, which may have potential applications in optical tweezers, communications, and enantiomer-selective sensing.

Self-rotating beam in the free space propagation

We introduce a class of self-rotating beams whose intensity profile tends to self-rotate and self-bend in the free space propagation. The feature of the self-rotating beams is acceleration in the three-dimensional (3D) space. The acceleration dynamics of the self-rotating beams is controllable. Furthermore, multiple self-rotating beams can be generated by a combined diffractive optical element (DOE) simultaneously. Such a beam can be viewed as evolution of a vortex beam by changing the exponential constant of phase. We have generated this beam successfully in the experiment and observed the expected phenomenon, which is basically consistent with the result of the numerical simulation. Our results may provide new insight into the self-rotating beam and extend potential applications in optical imaging.

Spirally rotating particles with structured beams generated by phase-shifted zone plates

The emerging field of structured beams has led to optical manipulation with tremendous progress. Beyond various methods for structured beams, we use phase-shifted zone plates known as beam-shaping diffractive optical elements to generate beams whose phase exclusively or both phase and intensity are twisted along a curve. These beams can trap and guide particles on open curved trajectories for continuous motion, not necessarily requiring a closed symmetric intensity distribution. We show the feasibility and versatility of the proposed method as a promising technique in optical manipulation in which the trajectory of the spiral rotation and the rate of rotation of trapped particles can be controlled.

Experimental visualization of various cross sections through a butterfly caustic

Optical caustics and wavefronts of butterfly beams (BBs) derived by using a catastrophe theory determined by potential functions depending on the state and control variables are reported. Due to the high dimensionality for the control variables, BBs can be manipulated into various optical light structures. It is also demonstrated that these curious beams have relatively simple Fourier spectra that can be described as polynomials, and another way to generate BBs from the Fourier spectrum's perspective is provided. The dynamics for BBs are investigated by potential functions. Our experimental results agree well with the theoretical predictions. In addition to micro-manipulation and machining, these novel, to the best of our knowledge, caustic beams will pave the way for creating waveguide structures since they display high-intensity formations that evolve along curved trajectories.

Observation of optical vortex knots and links associated with topological charge

Knots and links, as three-dimensional topologies, have played a fundamental role in many physical fields. Despite knotted vortex loops having been shown to exist in the light field, the three-dimensional configuration of vortex loop is fixed due to their topological robustness, making the fields with different topologies independent of each other. In this work, we established the mapping between the torus knots/links and the integer topological charge of the optical vortex, and demonstrated the change of the intermediate state with fractional charges. Furthermore, we experimentally observed the transformation process of the three-dimensional topological structure by only changing the topological charge. Remarkably, we revealed two different reconnection mechanisms associated with the odd or even index of the torus topology. We hope these results may provide new insight for the study of singular optics and evolution in other physical fields.

Designing optical fields in inhomogeneous media

Designing optical fields with predetermined properties in source-free inhomogeneous media has been a long-sought goal due to its potential utilization in many applications, such as optical trapping, micromachining, imaging, and data communications. Using ideas from the calculus of variations, we provide a general framework based on the Helmholtz equation to design optical fields with prechosen amplitude and phase inside an inhomogeneous medium. The generated field is guaranteed to be the closest physically possible rendition of the desired field. The developed analytical approach is then verified via different techniques, where the approach's validity is demonstrated by generating the desired optical fields in different inhomogeneous media.

Formation of Inverse Energy Flux in the Case of Diffraction of Linearly Polarized Radiation by Conventional and Generalized Spiral Phase Plates

Recently, there has been increased interest in the shaping of light fields with an inverse energy flux to guide optically trapped nano- and microparticles towards a radiation source. To generate inverse energy flux, non-uniformly polarized laser beams, especially higher-order cylindrical vector beams, are widely used. Here, we demonstrate the use of conventional and so-called generalized spiral phase plates for the formation of light fields with an inverse energy flux when they are illuminated with linearly polarized radiation. We present an analytical and numerical study of the longitudinal and transverse components of the Poynting vector. The conditions for maximizing the negative value of the real part of the longitudinal component of the Poynting vector are obtained.

Engineering phase and polarization singularity sheets

Optical phase singularities are zeros of a scalar light field. The most systematically studied class of singular fields is vortices: beams with helical wavefronts and a linear (1D) singularity along the optical axis. Beyond these common and stable 1D topologies, we show that a broader family of zero-dimensional (point) and two-dimensional (sheet) singularities can be engineered. We realize sheet singularities by maximizing the field phase gradient at the desired positions. These sheets, owning to their precise alignment requirements, would otherwise only be observed in rare scenarios with high symmetry. Furthermore, by applying an analogous procedure to the full vectorial electric field, we can engineer paraxial transverse polarization singularity sheets. As validation, we experimentally realize phase and polarization singularity sheets with heart-shaped cross-sections using metasurfaces. Singularity engineering of the dark enables new degrees of freedom for light-matter interaction and can inspire similar field topologies beyond optics, from electron beams to acoustics.

Caustics of Non-Paraxial Perfect Optical Vortices Generated by Toroidal Vortex Lenses

In this paper, we consider the comparative formation of perfect optical vortices in the non-paraxial mode using various optical elements: non-paraxial and parabolic toroidal vortex lenses, as well as a vortex axicon in combination with a parabolic lens. The theoretical analysis of the action of these optical elements, as well as the calculation of caustic surfaces, is carried out using a hybrid geometrical-optical and wave approach. Numerical analysis performed on the basis of the expansion in conical waves qualitatively confirms the results obtained and makes it possible to reveal more details associated with diffraction effects. Equations of 3D-caustic surfaces are obtained and the conditions of the ring radius dependence on the order of the vortex phase singularity are analyzed. In the non-paraxial mode, when small light rings (several tens of wavelengths) are formed, a linear dependence of the ring radius on the vortex order is shown. The revealed features should be taken into account when using the considered optical elements forming the POV in various applications.

Optical manipulation of a dielectric particle along polygonal closed-loop geometries within a single water droplet

We report a new method to optically manipulate a single dielectric particle along closed-loop polygonal trajectories by crossing a suite of all-fiber Bessel-like beams within a single water droplet. Exploiting optical radiation pressure, this method demonstrates the circulation of a single polystyrene bead in both a triangular and a rectangle geometry enabling the trapped particle to undergo multiple circulations successfully. The crossing of the Bessel-like beams creates polygonal corners where the trapped particles successfully make abrupt turns with acute angles, which is a novel capability in microfluidics. This offers an optofluidic paradigm for particle transport overcoming turbulences in conventional microfluidic chips.

Twin curvilinear vortex beams

We report on a novel curvilinear optical vortex beam named twin curvilinear vortex beams (TCVBs) with intensity and phase distribution along a pair of two- or three-dimensional curves, both of which share the same shape and the same topological charge. The TCVBs also possess the character of perfect optical vortex, namely having a size independent of topological charge. We theoretically demonstrate that a TCVB rather than a single-curve vortex beam can be created by the Fourier transform of a cylindrically polarized beam. The behavior of TCVBs generated through our method is investigated by simulation and experiment, including interference experiments for identifying the vortex property of the TCVBs. The TCVBs may find applications in optical tweezers, such as trapping low refractive index particles in the dark region between two curves and driving them moving along the curvilinear trajectory.

Swallowtail-type diffraction catastrophe beams

We demonstrate a universal approach for generating high-order diffraction catastrophe beams, specifically for Swallowtail-type beams (abbreviated as Swallowtail beams), using diffraction catastrophe theory that was defined by potential functions depending on the control and state parameters. The three-dimensional curved caustic surfaces of these Swallowtail catastrophe beams are derived by the potential functions. Such beams are generated by mapping the cross sections of the high-order control parameter space to the corresponding transverse plane. Owing to the flexibility of the high-order diffraction catastrophe, these Swallowtail beams can be tuned to a diverse range of optical light structures. Owing to the similarity in their frequency spectra, we found that the Swallowtail beams change into low-order Pearcey beams under given conditions during propagation. Our experimental results are in close agreement with our simulated results. Such fantastic catastrophe beams that can propagate along curved trajectories are likely to give rise to new applications in micromachining and optical manipulation, furthermore, these diverse caustic beams will pave the way for the tailoring of arbitrarily accelerating caustic beams.

Spiral Caustics of Vortex Beams

We discuss the nonparaxial focusing of laser light into a three-dimensional (3D) spiral distribution. For calculating the tangential and normal components of the electromagnetic field on a preset curved surface we propose an asymptotic method, using which we derive equations for calculating stationary points and asymptotic relations for the electromagnetic field components in the form of one-dimensional (1D) integrals over a radial component. The results obtained through the asymptotic approach and the direct calculation of the Kirchhoff integral are identical. For a particular case of focusing into a ring, an analytical relation for stationary points is derived. Based on the electromagnetic theory, we design and numerically model the performance of diffractive optical elements (DOEs) to generate field distributions shaped as two-dimensional (2D) and 3D light spirals with the variable angular momentum. We reveal that under certain conditions, there is an effect of splitting the longitudinal electromagnetic field component. Experimental results obtained with the use of a spatial light modulator are in good agreement with the modeling results.

Controllable propagation and transformation of chiral intensity field at focus

As an intrinsic feature of the optical field, chirality could induce many novel phenomena due to the interaction between chiral light and matter. Thus, the generation of optical fields possessing 2D or 3D chiral intensity patterns, called chiral intensity fields, has been widely studied. However, the control of chiral intensity field along the optical axis is still a challenge. Here, we propose a method to manipulate the axial propagation property of a focused chiral intensity field. Two modulation effects are realized: extended chiral intensity field with a focal depth >2λ at 90% mode correlation and tunable transformation of chirality during the axial propagation. This method is simple, stable, and easy to perform and therefore offers broad applications especially in optical tweezers and metamaterial fabrication.

Designing the phase and amplitude of scalar optical fields in three dimensions

The ability to generate any arbitrarily chosen optical field in a three-dimensional (3D) space, in the absence of any sources, without modifying the index of refraction, remains an elusive but much-desired capability with applications in various fields such as optical micromanipulation, imaging, and data communications, to name a few. In this work, we show analytically that it is possible to generate any desired scalar optical field with predefined amplitude and phase in 3D space, where the generated field is an exact duplicate of the desired field in case it is a solution of Helmholtz wave equation, or if the existence of such field is strictly forbidden, the generated field is the closest possible rendition of the desired field in amplitude and phase. The developed analytical approach is further supported via experimental demonstration of optical beams with exotic trajectories and can have a significant impact on the aforementioned application areas.

Shaping caustics into propagation-invariant light

Structured light has revolutionized optical particle manipulation, nano-scaled material processing, and high-resolution imaging. In particular, propagation-invariant light fields such as Bessel, Airy, or Mathieu beams show high robustness and have a self-healing nature. To generalize such beneficial features, these light fields can be understood in terms of caustics. However, only simple caustics have found applications in material processing, optical trapping, or cell microscopy. Thus, these technologies would greatly benefit from methods to engineer arbitrary intensity shapes well beyond the standard families of caustics. We introduce a general approach to arbitrarily shape propagation-invariant beams by smart beam design based on caustics. We develop two complementary methods, and demonstrate various propagation-invariant beams experimentally, ranging from simple geometric shapes to complex image configurations such as words. Our approach generalizes caustic light from the currently known small subset to a complete set of tailored propagation-invariant caustics with intensities concentrated around any desired curve.

Rapidly and accurately shaping the intensity and phase of light for optical nano-manipulation

Holographic optical tweezers can be applied to manipulate microscopic particles in various optical patterns, which classical optical tweezers cannot do. This ability relies on accurate computer-generated holography (CGH), yet most CGH techniques can only shape the intensity profiles while the phase distributions remain poor. Here, we introduce a new method for fast generation of holograms that allows for accurately shaping both the intensity and phase distributions of light. The method uses a discrete inverse Fourier transform formula to directly calculate a hologram in one step, in which a random phase factor is introduced into the formula to enable complete control of intensity and phase. Various optical patterns can be created, as demonstrated by the experimentally measured intensity and phase profiles projected from the holograms. The high-quality shaping of intensity and phase of light provides new opportunities for optical trapping and manipulation, such as controllable transportation of nanoparticles in optical trap networks with variable phase profiles.

Synergy of Intensity, Phase, and Polarization Enables Versatile Optical Nanomanipulation

Micromanipulation by optical tweezers mainly relies on the trapping force derived from the intensity gradient of light. Here we show that the synergy of intensity, phase, and polarization in structured light allows versatile optical manipulation of nanostructures. When a metal nanoparticle is confined by a linearly polarized laser field, the sign of optical force depends on the particle shape and the laser intensity, phase, and polarization profiles. By tuning these parameters in optical line traps, optical trapping, transporting, and sorting of silver nanostructures have been demonstrated. These findings inspired us to control the motion of nanostructures with designed intensity, phase, and polarization of light using holographic optical tweezers with advanced beam shaping techniques. This work provides a new perspective on active colloidal nanomanipulation in fully controlled optical landscapes, which largely expands the existing optical manipulation toolbox.

Refractive twisted microaxicons

Complex-shaped light fields with specially designed intensity, phase, and polarization distributions are highly demanded for various applications including optical tweezers, laser material processing, and lithography. Here, we propose a novel (to the best of our knowledge) optical element formed by the twisting of a conic surface, a twisted microaxicon, allowing us to controllably generate high-quality spiral-shaped intensity patterns. Performance of the proposed element was analyzed both analytically and numerically using ray approximation and the rigorous finite difference time domain (FDTD) solution of Maxwell's equation. The main geometric parameters, an apex cone angle and a degree of twisting, were considered to control and optimize the generated spiral-shaped intensity patterns. The three-dimensional structure of such a microaxicon cannot be described by an unambiguous height function; therefore, it has no diffraction analogue in the form of a thin optical element. Such an element can be produced via direct laser ablation of transparent targets with structured laser beams or direct laser writing via two-photon photopolymerization and can be used in various micro- and nano-optical applications.

Above and beyond: holographic tracking of axial displacements in holographic optical tweezers

How far a particle moves along the optical axis in a holographic optical trap is not simply dictated by the programmed motion of the trap, but rather depends on an interplay of the trap's changing shape and the particle's material properties. For the particular case of colloidal spheres in optical tweezers, holographic video microscopy reveals that trapped particles tend to move farther along the axial direction than the traps that are moving them and that different kinds of particles move by different amounts. These surprising and sizeable variations in axial placement can be explained by a dipole-order theory for optical forces. Their discovery highlights the need for real-time feedback to achieve precise control of colloidal assemblies in three dimensions and demonstrates that holographic microscopy can meet that need.

Accurate and rapid measurement of optical vortex links and knots

Optical vortices can evolve in light fields, of which the singularity evolution forms dark lines with complex topological structures, knotted or linked. We propose a method to more accurately and rapidly measure the topology of optical vortex fields. To accurately locate the phase singular points, phase measurement based on digital holography and, further, a numerical search algorithm, are utilized. A motor-driven right-angle prism enables the implementation of a single exposure of hologram for each measurement along the propagation direction, greatly improving the measurement speed. The three-dimensional (3D) spatial distributions of several typical vortex links and knots are experimentally reconstructed. The proposed method is expected to rapidly observe the 3D evolution of other complicated, or even vector, fields.

Multiple-twisted spiral beams

We propose a method to construct paraxial light fields whose transverse intensity rotates as a whole during the field propagation (spiral beams), and the total rotation angle is an integer multiple of π/2 (multiple-twisted beams). We derive analytical expressions for the field complex amplitude by utilizing an integral description of Laguerre-Gaussian beams. This method provides a straightforward way to obtain multiple-twisted spiral beams of various intensity shapes and rotation rates.

Generation of spirally accelerating optical beams

We developed a generalized spectral phase superposition approach for generating accelerating optical beams along arbitrary trajectories. Such beams can be customized by predefining an appropriate superimposed phase pattern that consists of multiple sub-phases. We generated a spirally accelerating beam in a three-dimensional space and developed an algorithm to improve the uniformity of the intensity along the trajectory by introducing phase-shift factors. We also experimentally verified our numerical simulations. The proposed approach breaks the conventional convex trajectory restrictions. These various accelerating beams would pave the way for optically moving particles along a desired trajectory. The generation of such arbitrary accelerating beams is likely to give rise to new applications in flexible optical manipulation, wave front control, and optical transportation and guidance of particles.

Centrosymmetric Optical Vortex

We report on a novel optical vortex, named as centrosymmetric optical vortex (CSOV), which is constructed via four conventional optical vortices (OVs) with different topological charges (TCs). The orbital angular momentum (OAM) density satisfies centrosymmetric distribution. Meanwhile, it is confined within a single ring whose radius is determined by the cone angle of an axicon. Furthermore, its magnitude and distribution are modulated by a parameter determined via the TCs of the four OVs, named as phase reconstruction factor. Our work provides a novel detached asymmetric light field, which possesses the potential application in macro-particle manipulation, especially separating cells.

Simultaneous Generation of Multiple Three-Dimensional Tractor Curve Beams

A tractor beam, which has the ability to attract objects, is a class of special optical beams. Currently, people are using the holographic technology to shape complex optical tractor beams for both fundamental research and practical applications. However, most of the work reported is focusing on generating two-dimensional (2D) tractor beams and simple three-dimensional (3D) tractor beams, which has limitations in the further development on mechanism and application of beam shaping. In the present work, we are introducing our study in designing multiple 3D tractor beams with spatial location regulated independently. Meanwhile, each individual beam could be prescribed along arbitrary geometric curve and twisted at arbitrary angles as desired. In our method, the computer-generated hologram (CGH) of each curve is calculated, and all the CGHs are multiplexed and encoded into one phase-only hologram by adding respective linear phase grating such that different 3D curves appeared in the different positions of the focal regions. We experimentally prove that the generation of optical tractor beams at 3D configuration can be readily achieved. The generated beams in the present study are especially useful for applications such as multiple micro-machining optical trapping and complex 3D manipulation.

Simultaneous Generation of Complex Structured Curve Beam

At present, people are using holographic technologies to shape complex optical beams for both fundamental research and practical applications. However, most of the reported works are focusing on the generation of a single beam pattern based on the computer-generated hologram (CGH). In this paper, we present a method for simultaneously shaping the multiple beam lattice where the intensity and phase of each individual beam can be prescribed along an arbitrary geometric curve. The CGH that is responsible for each individual beam is calculated by using the holographic beam shaping technique, afterwards all the CGHs are multiplexed and encoded into one phase-only hologram by adding respective linear phase grating such that different curves are appeared in different positions of the focal regions. We experimentally prove that the simultaneous generation of multiple beams can be readily achieved. The generated beams are especially useful for applications such as multitasking micro-machining and optical trapping.

Chiral optical field generated by an annular subzone vortex phase plate

We proposed and experimentally demonstrated a novel method for generating a chiral beam with controllable intensity twist lobes and direction by using annular subzone (AS) vortex phase plates, which is composed of different ASs and different vortex phases. The phase distribution continuity between two adjacent ASs determines the intensity distribution of the light field. The rotated direction of the optical filed is determined by the topological charge sign. The number of intensity twist lobes is determined by the topological charge gradient between adjacent subzones. The experimental results show that this method is effective and practical, which offers broad potential applications in particle manipulation, chiral microstructure fabrication, and optical tweezers.

Vector polymorphic beam

A scalar polymorphic beam is designed with independent control of its intensity and phase along a strongly focused laser curve of arbitrary shape. This kind of beam has been found crucial in the creation of freestyle laser traps able to confine and drive the motion of micro/nano-particles along reconfigurable 3D trajectories in real time. Here, we present and experimentally prove the concept of vector polymorphic beam adding the benefit of independent design of the light polarization along arbitrary curves. In particular, we consider polarization shaped tangential and orthogonal to the curve that are of high interest in optical manipulation and laser micromachining. The vector polymorphic beam is described by a surprisingly simple closed-form expression and can be easily generated by using a computer generated hologram.

Controlled Mechanical Motions of Microparticles in Optical Tweezers

Optical tweezers, formed by a highly focused laser beam, have intriguing applications in biology and physics. Inspired by molecular rotors, numerous optical beams and artificial particles have been proposed to build optical tweezers trapping microparticles, and extensive experiences have been learned towards constructing precise, stable, flexible and controllable micromachines. The mechanism of interaction between particles and localized light fields is quite different for different types of particles, such as metal particles, dielectric particles and Janus particles. In this article, we present a comprehensive overview of the latest development on the fundamental and application of optical trapping. The emphasis is placed on controllable mechanical motions of particles, including rotation, translation and their mutual coupling under the optical forces and torques created by a wide variety of optical tweezers operating on different particles. Finally, we conclude by proposing promising directions for future research.

Generation of optical vortex array along arbitrary curvilinear arrangement

We propose an approach for creating optical vortex array (OVA) arranged along arbitrary curvilinear path, based on the coaxial interference of two width-controllable component curves calculated by modified holographic beam shaping technique. The two component curve beams have different radial dimensions as well as phase gradients along each beam such that the number of phase singularity in the curvilinear arranged optical vortex array (CA-OVA) is freely tunable on demand. Hybrid CA-OVA that comprises of multiple OVA structures along different respective curves is also discussed and demonstrated. Furthermore, we study the conversion of CA-OVA into vector mode that comprises of polarization vortex array with varied polarization state distribution. Both simulation and experimental results prove the performance of the proposed method of generating a complex structured vortex array, which is of significance for potential applications including multiple trapping of micro-sized particles.

Monte Carlo simulations of three-dimensional electromagnetic Gaussian Schell-model sources

This article presents a method to simulate a three-dimensional (3D) electromagnetic Gaussian-Schell model (EGSM) source with desired characteristics. Using the complex screen method, originally developed for the synthesis of two-dimensional stochastic electromagnetic fields, a set of equations is derived which relate the desired 3D source characteristics to those of the statistics of the random complex screen. From these equations and the 3D EGSM source realizability conditions, a single criterion is derived, which when satisfied guarantees both the realizability and simulatability of the desired 3D EGSM source. Lastly, a 3D EGSM source, with specified properties, is simulated; the Monte Carlo simulation results are compared to the theoretical expressions to validate the method.

Shaping of optical vector beams in three dimensions

We present a method of shaping three-dimensional (3D) vector beams with prescribed intensity distribution and controllable polarization state variation along arbitrary curves in three dimensions. By employing a non-iterative 3D beam-shaping method developed for the scalar field, we use two curved laser beams with mutually orthogonal polarization serving as base vector components with a high-intensity gradient and controllable phase variation, so that they are collinearly superposed to produce a 3D vector beam. We experimentally demonstrate the generation of 3D vector beams that have a polarization gradient (spatially continuous variant polarization state) along 3D curves, which may find applications in polarization-mediated processes, such as to drive the motion of micro-particles.

3D shaping of electron beams using amplitude masks

Shaping the electron wavefunction in three dimensions may prove to be an indispensable tool for research involving atomic-sized particle trapping, manipulation, and synthesis. We utilize computer-generated holograms to sculpt electron wavefunctions in a standard transmission electron microscope in 3D, and demonstrate the formation of electron beams exhibiting high intensity along specific trajectories as well as shaping the beam into a 3D lattice of hot-spots. The concepts presented here are similar to those used in light optics for trapping and tweezing of particles, but at atomic scale resolutions.

Accelerating incoherent hollow beams beyond the paraxial regime

We propose a non-paraxial hollow accelerating beam, which is formed by incoherently superposing two well-designed coherent accelerating beams. Very interestingly, this incoherent superposition does not hamper the acceleration dynamics pertaining to the coherent ones, but results in a hollow intensity pattern in the cross section transverse to the circular accelerating trajectory. By a simple optimization, this hollow cross section pattern can be effectively extended to an angle close to 90°. The magnitude and the phase of the angular spectrum of the beam are given followed by a suggested scheme to generate the beam in practice. Such highly self-bending hollow beams may find applications in some fields such as optical manipulation.

Polymorphic beams and Nature inspired circuits for optical current

Laser radiation pressure is a basis of numerous applications in science and technology such as atom cooling, particle manipulation, material processing, etc. This light force for the case of scalar beams is proportional to the intensity-weighted wavevector known as optical current. The ability to design the optical current according to the considered application brings new promising perspectives to exploit the radiation pressure. However, this is a challenging problem because it often requires confinement of the optical current within tight light curves (circuits) and adapting its local value for a particular task. Here, we present a formalism to handle this problem including its experimental demonstration. It consists of a Nature-inspired circuit shaping with independent control of the optical current provided by a new kind of beam referred to as polymorphic beam. This finding is highly relevant to diverse optical technologies and can be easily extended to electron and x-ray coherent beams.

Light-driven transport of plasmonic nanoparticles on demand

Laser traps provide contactless manipulation of plasmonic nanoparticles (NPs) boosting the development of numerous applications in science and technology. The known trapping configurations allow immobilizing and moving single NPs or assembling them, but they are not suitable for massive optical transport of NPs along arbitrary trajectories. Here, we address this challenging problem and demonstrate that it can be handled by exploiting phase gradients forces to both confine and propel the NPs. The developed optical manipulation tool allows for programmable transport routing of NPs to around, surround or impact on objects in the host environment. An additional advantage is that the proposed confinement mechanism works for off-resonant but also resonant NPs paving the way for transport with simultaneous heating, which is of interest for targeted drug delivery and nanolithography. These findings are highly relevant to many technological applications including micro/nano-fabrication, micro-robotics and biomedicine.

Inverted Gabor holography principle for tailoring arbitrary shaped three-dimensional beams

It is well known that by modifying the wavefront in a certain manner, the light intensity can be turned into a certain shape. However, all known light modulation techniques allow for limited light modifications only: focusing within a restricted region in space, shaping into a certain class of parametric curves along the optical axis or bending described by a quadratic-dependent deflection as in the case of Airy beams. We show a general case of classical light wavefront shaping that allows for intensity and phase redistribution into an arbitrary profile including pre-determined switching-off of the intensity. To create an arbitrary three-dimensional path of intensity, we represent the path as a sequence of closely packed individual point-like absorbers and simulate the in-line hologram of the created object set; when such a hologram is contrast inverted, thus giving rise to a diffractor, it creates the pre-determined three-dimensional path of intensity behind the diffractor under illumination. The crucial parameter for a smooth optical path is the sampling of the predetermined curves, which is given by the lateral and axial resolution of the optical system. We provide both, simulated and experimental results to demonstrate the power of this novel method.

Simultaneous shaping of amplitude and phase of light in the entire output plane with a phase-only hologram

An iterative beam shaping algorithm is proposed to simultaneously shape the amplitude and phase of an optical beam. The proposed algorithm consists of one input plane and two completely overlapped output planes which refer to the output plane in real space. The two output planes are imposed with both amplitude and phase constraints, and the constrained areas in the two output planes are complementary. As a result, both the amplitude and phase in the entire output plane are controllable and arbitrary target complex amplitudes can be achieved with the proposed algorithm. The computing result of the proposed algorithm is a phase-only distribution which can be conveniently realized with a spatial light modulator or a fabricated diffractive optical element. Both simulations and experiments have verified the high performance of the proposed algorithm.

Curved optical tubes in a 4Pi focusing system

We demonstrate the possibility of creating curved optical tubes in a 4Pi focusing system. The focal fields of such optical tubes have interesting properties: the energy is concentered in the neighborhood of a prescribed three-dimensional (3D) curve while the cross section is of hollow shape. The creation of these optical tubes is based on the annular focal spot of a vortex beam, which is employed as a building block. An optical tube is thus obtained by covering the central-axis curve of the tube by various such building blocks. Each building block has a certain orientation and position, realized by a rotation plus a certain translation. The spatial spectrum (the input field as well) of the optical tube is obtained by linearly superposing the spectrum of each transformed building block. The curve is rather arbitrary. Three examples of optical tubes: a torus, a solenoid and a trefoil knot are given, showing a good agreement with the expected results.

Curved singular beams for three-dimensional particle manipulation

For decades, singular beams carrying angular momentum have been a topic of considerable interest. Their intriguing applications are ubiquitous in a variety of fields, ranging from optical manipulation to photon entanglement, and from microscopy and coronagraphy to free-space communications, detection of rotating black holes, and even relativistic electrons and strong-field physics. In most applications, however, singular beams travel naturally along a straight line, expanding during linear propagation or breaking up in nonlinear media. Here, we design and demonstrate diffraction-resisting singular beams that travel along arbitrary trajectories in space. These curved beams not only maintain an invariant dark "hole" in the center but also preserve their angular momentum, exhibiting combined features of optical vortex, Bessel, and Airy beams. Furthermore, we observe three-dimensional spiraling of microparticles driven by such fine-shaped dynamical beams. Our findings may open up new avenues for shaped light in various applications.