Light–matter interaction has a long history in both physics and astronomy. About 400 years ago, Kepler observed the deflection of a comet’s tail away from the Sun, which may constitute the first reported conjecture of the radiation force.^1–3 In the 1700s, John Michell attempted to measure radiation pressure,⁴ while Euler hypothesized that light beams induce pressure on illuminated bodies.⁵ In the 1800s, Maxwell predicted that light was an electromagnetic wave,⁶ which was confirmed by the first demonstration of a radiation force originating from thermal light sources by Lebedev^7,8 and Nichols and Hull⁹ in 1901. We now know that light beams can be considered as a large collection of photons, each carrying a quantized amount of momentum, which can be transferred to matter. However, as derived by Poynting in 1906, the radiation pressure is so minute that it only affects small bodies.¹⁰ Shortly afterward, Mie¹¹ and Debye¹² proposed exact physical models to calculate scattering force and radiation pressure of light in 1908 and 1909, respectively. At that stage, no one could imagine any practical value of radiation pressure with it being too weak to overcome frictional forces in most circumstances. For a considerable time, researchers focused their attention on the use of radiation pressure in space, e.g., solar sail propulsion systems,¹³ due to the absence of appreciable friction in space.

This situation did not change significantly until the invention of the laser in 1960. In the years following this discovery, Ashkin demonstrated,¹⁴ for the first time, laser trapping of micrometer-sized dielectric particles with two counterpropagating beams. To get stable three-dimensional (3D) confinement of particles, the scheme of two weakly focused beams with opposing radiation pressure was adopted. Optical trapping underwent a revolution after Ashkin et al.¹⁵ found that even a single, tightly focused laser beam can form a 3D stable optical trap—optical tweezers. Since then, the broad field of optical manipulation or so-called “micromanipulation” has found a tremendous range of applications in many fields, such as biomedicine,^16–18 physics,^19–21 and chemistry,²² and has been noted in three Nobel Prizes in physics: in 1997, for the development of methods to laser cooling and trapping atoms; in 2001, for the achievement of Bose–Einstein condensation; and in 2018, for optical tweezers and their application to biological systems.

It is noted that although the annular (high-order LG) beam had been used to study the self-focusing and self-trapping of a laser beam in artificial Kerr media in 1981,^23,24 the fundamental (or TEM00) transverse Gaussian mode of a laser beam was exclusively used for optical trapping experiments in the early decades of laser development. It is only in the last two decades that structured light beams have been adopted widely in optical tweezers. Structured light beams have added new dimensions and functionalities to optical manipulation, as well as provided new insights into light–matter interactions, where trapped particles can act as probes of structured light fields. Optical trapping and structured beams, therefore, help each other understand and improve these two important areas. Structured light beams can be of two distinct forms, either scalar or vector. In the scalar form, a structured light beam can be tailored by its amplitude and phase while polarization is only modified in the vectorial form of the beam. Structured light beams have been widely used in optical manipulation, due to their unique properties, such as optical vortices carrying orbital angular momentum²⁵ (OAM) and propagation invariant beams.²⁶

Undoubtedly, the ability to tailor the optical properties of a trapping beam is crucial in the development of novel optical trapping techniques. Thus far, structured light beams with customized phase and amplitude have been successfully applied to drive the optical transport of particles in 3D trajectories by exerting optical forces arising from high intensity and phase gradients. Recently, there are several excellent review articles on this topic,^27–33 including optical pulling force,²⁸ optical transport of small particles,²⁹ optomechanics with levitated particles,³⁰ acoustic and optical trapping for biomedical research,³¹ and so on.³² More recently, advanced optical manipulation using structured light was reviewed,³³ which focused on the manipulation of transparent dielectric particles. In this paper, we review the breadth of structured beams and discuss the recent advances in optical manipulation employing structured beams, notably for both scalar and vectorial forms. We provide an overview of seminal contributions that have changed the landscape of optical tweezers, with an extensive reference list. This emerging field is being continuously and rapidly reshaped by new approaches, and we hope this paper will appeal to a broad audience with an interest in optical manipulation techniques. Our aim is to offer an up-to-date status of the field of optical manipulation with structured light.

Optical tweezers are a powerful technique to hold and move microscopic particles or biological specimens with a single tightly focused laser beam, akin to normal tweezers.¹⁵ Here, we briefly describe the principle of optical tweezers. A more detailed discussion can be found in Refs. 2, 21, and 34–39 and references therein.

Depending on the relative size of spherical particles to the laser wavelength, optical forces can be described in three regimes: the Rayleigh regime,⁴⁰ the intermediate regime,⁴¹ and the ray optics regime.⁴² The size parameter of the particle ξ is defined as ξ=kma, where km=2πnm/λ0, λ0 is the wavelength of the trapping beam in vacuum, and a is the radius of the spherical particle. The refractive indices of the particle and the surrounding medium are np and nm, respectively. When ξ≫1, it is in the ray optics regime where the force can be described by a ray optics model. When ξ≪1 and ξ·np/nm≪1, it corresponds to the Rayleigh regime where the particle can be approximated as a dipole. For particles of size between the above two, this is the intermediate regime where the Lorenz–Mie theory can be used to investigate the optical force.

First, we use the simple ray optics model to explain how optical tweezers work. Figure 1(a) shows a laser beam with two light rays (white arrows a and b) passing through a dielectric spherical particle located off-axis of the laser beam. The light rays will change their propagation directions due to the refraction, resulting in a change in their momentum. As shown in Fig. 1(a), the central portion of the beam with higher intensity (indicated by a thick arrow a) is refracted to the left, which means a change in the laser’s momentum to the left. Based on the conservation of momentum at the particle boundary, the particle will feel a momentum kick to the right and therefore has a force Fa toward the center of the beam (dotted line). Analogously, the light ray b with lower intensity will change its momentum to the right and exert an optical force Fb on the particle away from the beam center. Since the light ray represented by a is much stronger than the ray represented by b, the net force will push the particle to the right. Conversely, if the particle is located on the right side of the beam axis, the optical force will push it to the left. As such, the light field intensity gradient always causes a gradient force (Fgrad) on the particle toward the maximum intensity of the beam. Besides, the rays reflected from the particle surface can produce forward scattering forces (Fscat) along the beam propagation direction. Figure 1(b) shows the longitudinal gradient force pushing the particle down toward the focal plane, and vice versa, in a highly focused laser beam. Therefore, the net force pushes the particle to the focus of the beam, as shown in Fig. 1(c). In the case of Fgrad≫Fscat, a stable 3D trap can be formed in a tightly focused laser beam spot.

In the Rayleigh regime, ray optics are not applicable to explain the optical forces, as the particle size is smaller than the wavelength of light. Here, the particle needs to be considered as a point electric dipole, and the optical forces can be written as⁴²⟨F⟩=14Re(αp)∇|E|2+σ(αp)2cRe(E×H*)+σ(αp)c∇×(ϵ04ωiE×E*),(1)where αp is the polarizability of the particle, and E, H are the electric field and magnetic field, respectively. σ(αp) denotes the total particle cross-section. The first term in Eq. (1) indicates the gradient force (Fgrad), and the last two terms represent the scattering force (Fscat). The second term in Eq. (1) is normally called the scattering force, and the third term is the so-called spin curl force.⁴²

Equation (1) indicates that each of these three optical forces on a particle depends on the polarizability, which can be expressed as⁴²αp=α01−iα0k3/6πϵm,(2)where α0=4πnm2r3ϵ0(η2−1η2+2) is the Clausius–Mossotti relation, r is the particle radius, and η=np/nm is the relative refractive index of the particle to the surrounding medium. Therefore, we can play with the polarizability to optimize the optical traps. Then, the optical gradient force and optical scattering force can be rewritten as^2,43Fgrad=−2πnmr3c(η2−1η2+2)∇I(r),(3)and Fscat=8πnmk4r63c(η2−1η2+2)2I(r)z^,(4)where c is the speed of light, z^ is the unit vector along the z direction, I is the intensity of light, and k is the wavenumber.

Both the ray optics and the dipole theory are powerful tools to study the optical forces in their respective size regimes, which have all given good physical pictures of the trapping mechanism. However, particles with their size lying between these two regimes make these approaches no longer valid. Instead, the Lorenz–Mie theory can be used to calculate the optical forces for such particle sizes, which are the exact solutions of the Helmholtz equations.^41,44 If the particles are not spherical, or the incident beam is not a plane wave, the generalized Lorenz–Mie theory can be used.⁴⁵ There are also many other fully numerical methods to calculate the optical force, such as finite-difference time-domain (FDTD) and finite element method. For instance, the FDTD analysis allows the numerical simulation of the scattered light field over an arbitrary particle. The optical force would be the surface integral of the Maxwell stress tensor. It should be noted that nondielectric materials, e.g., metals, semiconductors, and nonspherical particles, require different approaches. Proper choice of the force evaluation method relies on particle size, geometry, and the structure of the light field.^45,46

A conventional optical tweezers system is schematically shown in Fig. 2. First, a collimated laser beam is expanded to overfill the back aperture of a microscope objective. A dichroic mirror (DM1) reflects the laser beam to the high numerical aperture (NA) objective lens, which focuses the beam to a tightly focused diffraction limited spot inside a sample chamber for trapping. To image trapped particles on a CCD camera through DM1, an LED white light source is typically used to illuminate the sample. A condenser lens collects forward scattered light from the trapped particles and projects an image onto a quadrant photodetector (QPD) using back focal plane interferometry.³⁴ The balanced photodetection provided by the QPD allows the precise measurement of the motion of the trapped particles. The axial position z can be measured by the sum of the intensity of the four quadrants.³⁴

The motion of a Brownian particle in a fluid can be described by the Langevin equation.⁴⁷ In the overdamped case, where particles are immersed in a viscous medium, e.g., water, the one-dimensional motion (x direction) of an optically trapped Brownian particle will be described by the equation Γ0x˙=−κ0x+Fth,(5)where the thermal random force Fth drives the Brownian motion through collisions with surrounding molecules of the fluid, and Γ0=6πηa is the Stokes drag coefficient (η is the viscosity of the fluid, and a is the radius of the particle). The trap stiffness κ0=mΩ02, where m is the mass of the particle, and ω0 is the trap frequency, determines the magnitude of the optical restoring force depending on its position relative to the trap center. For a silica particle of radius a=1μm (m≈1×10−14kg) in water at room temperature, the resonant frequency of the harmonic oscillator is 103≤1/(2π)κ0/m≤104Hz for typical stiffnesses in the range of 0.5≤κ0≤50pN/μm.

The power spectral density (PSD) of the particle’s motion can be used to characterize the trap. At equilibrium, the PSD is Lorentzian.⁴⁸ For the particle’s x displacement, Sx(f)=4Γ0kBT/κ021+f2/fc2,(6)where fc=κ0/(2πΓ0) is the corner or roll-off frequency that verifies the trap stiffness. The integral of the PSD in units of m/Hz yields the position variance or the mean square displacement (MSD) of the particle, which verifies the equilibrium temperature of the particle: ∫0∞Sx(f)df=⟨x2⟩=kBTκ0.(7)The position variance ⟨x2⟩ directly measured by a CCD or a QPD (see Sec. 2.1) can also verify the trap stiffness, which benchmarks the PSD method.

Rotation of micro- and nano-objects, caused by the transfer of angular momentum from light beams, is of great interest due to its potential applications in optically driven micromachines, motors, actuators, or biological specimen. A beam of circularly polarized light carries spin angular momentum (SAM), which was derived by Poynting⁴⁹ in the early 1900s. The first experimental observation of SAM was performed by Beth⁵⁰ in 1936. In this experiment, he observed a mechanical torque on a double refracting slab due to a change in the circular polarization based on the conservation of angular momentum. It is now well understood that SAM of ±ℏ per photon is associated with the circular polarization of light, where the sign depends on its handedness. SAM of light has been used for rotation of both elongated, birefringent, and absorbing particles as well as particle cluster in optical tweezers.³⁴ Notably, with birefringent particles,⁵¹ a maximal torque efficiency of 2ℏ per photon can be achieved, contrary to the 0.05ℏ per photon achieved with elongated particles. In addition, optically trapped micron-sized birefringent particles were rotated by a circularly polarized beam at rotating rates as high as 350 Hz in water³⁹ and 10 MHz in vacuum.⁴⁰ Crucially, in recent time, it was demonstrated that SAM can also be used for the selective 3D trapping of chiral micro- and nanoparticles.^52,53 It was shown that under appropriate conditions, a light beam with SAM can induce nonrestoring or restoring forces on chiral microparticles.⁵² Similar results have been also reported on the interaction of chiral nanoparticles with chiral optical fields.⁵³

In 1992, Allen et al.²⁵ introduced the concept of a light beam possessing OAM, which is in addition to any SAM and characterized by an azimuthally varying phase of exp(iℓφ), where ℓ is an integer value, termed the topological charge, and φ is the azimuthal angle. The first experimental demonstration of OAM transfer from light to matter, in the context of optical tweezers, was performed by the group of Rubinsztein–Dunlop,^54,55 who demonstrated the optically induced rotation of absorptive particles. This was followed by the demonstration of the simultaneous transfer of both SAM and OAM to the same absorptive particles with circularly polarized LG0ℓ modes.⁵⁶ In these experiments, the rotation rate was shown to be proportional to the sum of SAM and OAM, i.e., the total angular momentum.⁵⁷ Other pioneering experiments include the transfer of SAM and OAM to a birefringent particle, which causes the particle to spin about its own axis as well as to rotate about the beam axis,^58,59 as shown in Fig. 3. While these previous demonstrations were performed with particles confined in two-dimensions, with the particle being pushed against a microscope slide, a key experiment that demonstrated the transfer of SAM and OAM to particles in 3D was performed by Simpson et al.⁶⁰ Besides the high-index particles, the 3D optical trapping of low-index particles has been studied as well,^61,62 and it was shown that the low-index particles can be trapped stably on the axis and slightly above the focal plane of a strongly focused optical vortex beam.

Light is an electromagnetic wave; as such, it can be characterized by its wavelength (color), amplitude, and phase or polarization, the later associated with the direction of oscillation of the electromagnetic field in space, transverse to the direction of propagation. For unpolarized light, the direction of oscillation is random. On the contrary, for polarized light, this can take distinctive forms including linear, circular, or elliptical, which are the preferred configurations for beam shaping. In what follows, we will restrict our review to the case of polarized light, first discussing the case of homogeneous polarization (scalar fields), followed by the nonhomogeneous case (vector beams). Inevitably, any discussion about structured light fields involves the Helmholtz equation, either in its exact or paraxial forms. As such, we will start our discussion by reviewing some of the most relevant solutions to the Helmholtz equation that have played a crucial role in the development of optical tweezers with structured light as we know them today.

Laguerre–Gaussian (LG) modes are a set of solutions to the paraxial wave equation in cylindrical coordinates. Their normalized transverse profile can be defined as²⁵$$\begin{gathered} LGpℓ(ρ,φ,z)=ω0ω(z)2p!π(|ℓ|+p)!(2ρω(z))|ℓ|Lp|ℓ|[2(ρω(z))2]×exp[i(2p+|ℓ|+1)ξ(z)] \\ ×exp[−(ρω(z))2]exp[−ikρ22R(z)]exp(iℓφ), \end{gathered}$$(8)where Lp|ℓ| (x) is the associated Laguerre polynomials with ℓ∈Z and p∈N as the azimuthal and radial indices, respectively. A set of the parameters ξ(z), ω(z), and R(z) is defined as ω(z)=ω01+(z/zR)2, R(z)=z1+(z/zR)2, and ξ(z)=arctan(z/zR), respectively. Here, zR is the Rayleigh range, and ω0 is the beam waist. Importantly, LG beams carry OAM ℓℏ per photon, associated with the phase term exp(iℓϕ). This phase term results in a spiral azimuthal phase that creates an inclined wavefront. The Poynting vector has, therefore, a nonzero azimuthal component that is at the origin of the angular momentum. Figure 4 shows four examples of the transverse intensity profiles of LG modes.

Another set of solutions to the Helmholtz equation in free space, when solved in cylindrical coordinates (ρ,φ,z), is the Bessel modes,^63–65B(ρ,φ,z)=E0Jℓ(ktρ)exp(ikzz)exp(ℓφ),(9)where Jℓ(x) represents the ℓ’th order Bessel polynomial, and ℓ∈Z is associated with OAM ℓℏ per photon carried by the beam. Moreover, kt and kz are the transverse and longitudinal components of the wave vector k, respectively, obeying the relations |k|=k=2π/λ and |k|2=kt2+kz2.

A more intuitive way of describing a Bessel beam is by considering these as the result of a superposition of plane waves propagating on a cone, where each of them undergoes identical phase shifts, kzΔz over a distance Δz. This interpretation can be observed in the angular spectrum of the Bessel beam, which takes the form of a ring or radius kr in the k-space. Therefore, the optical Fourier transform of the Bessel beams is a ring, and, vice versa, the optical Fourier transform of a ring will result in a Bessel beam. Therefore, Bessel beams can be generated using a ring-slit aperture.⁶⁴ Using this method, Durnin⁶³ produced a Bessel beam experimentally for the first time. Figure 5 shows this concept schematically; in Fig. 5(a), we show the transverse intensity profiles of a Bessel beam with topological charge ℓ=0 and its Fourier transform, while in Fig. 5(b) we show that of a Bessel beam of topological charge ℓ=1 along with its Fourier transform.

Two of the most prominent properties of Bessel modes are, on one hand, their tendency to maintain an invariant intensity profile, namely, IB(ρ,φ,z≥0)=IB(ρ,φ,0) and, on the other, their tendency to recover its original form when an opaque obstruction is placed in its path.⁶⁴ Such a property can be explained by invoking again the plane waves propagating on a cone approach as detailed next. When the opaque object or radius a is placed in the center of the Bessel beam, some of the waves that create the beam are blocked by this object, creating a shadow region. Nonetheless, some other plane waves can pass the object unaffected, which ultimately are the ones that reconstruct the intensity profile of the beam after a certain propagation distance.^65,66 As mentioned earlier, Bessel beams can be generated by placing a slit aperture in front of a ring-slit aperture; none the less, this is a very inefficient way since most of incident beam’s intensity is blocked. A far more efficient way to produce a Bessel beam is using an axicon.⁶⁴ The on-axis intensity is formed by conical wavevectors that propagate on the surface of a cone.

As we will discuss later, the zeroth-order Bessel beam has demonstrated its great relevance in optical trapping for the study of multiparticle arrangements along the beam axis.^67,68 Since Eq. (9) shows that these modes have infinitely extending sidelobes and carry an infinite amount of energy, their experimental realization is approximated by adding a Gaussian envelope. These modes are known as Bessel–Gauss modes, which at z=0 take the form^69,70B(ρ,φ,0)=Jℓ(ktρ)exp[−(ρω0)2]exp(−iℓφ).(10)

The concept of the “perfect vortex beam” was proposed by Ostrovsky et al.,⁷¹ whose intensity profile is independent of its topological charge. Its complex amplitude at a given propagation distance can be expressed as⁷¹Vℓ(ρ,θ)=exp(−ρ−ρ0Δρ2)eiℓθ,(11)where ρ0 is the radius of the bright ring, Δρ is the width of the ring, and, in general, Δρ≪ρ0. Contrary to LG modes, in which the radius of the ring-like transverse intensity profile scales up with the topological charge, in perfect vortex modes, it remains constant. Even so, recent reports suggest that the width of such modes experiences a small scaling with the topological charge.⁷² This is shown in Fig. 6, where the transverse intensity profiles of a set of perfect vortex beams with topological charges ℓ=1, 4, 10, and 15 are shown. Vaity and Rusch⁷³ pointed out that a perfect vortex beam is actually a Fourier transform of a Bessel beam, and it can be generated by employing a phase hologram whose transmittance is the phase of a Bessel beam.^71–74 It is worth noting that although the concept of “perfect vortex beam” was proposed in 2013, such a beam had been introduced and used for 3D optical trapping and transport of particles by Roichman and Grier,⁷⁵ Roichman et al.⁷⁶ had pointed out that the radius of this 3D ring trap is independent of the topological charge, and the first experimental demonstration of optical trapping using the “perfect vortex beam” has been reported under the name holographic ring trap. The perfect vortex has been used by the optical trapping community as a particular case of the structured light beam to study the dynamics of driven particles in the form of optical matter.^77–80

Another interesting set of vector modes is the Mathieu–Gauss beams, which are obtained when the Helmholtz equation is solved in elliptical cylindrical coordinates.⁵⁹ In such coordinates, defined by x=fcoshξcosη, y=fsinhξsinη, and z=z, where ξ∈[0,∞) is the radial coordinate, and η∈[0,2π) is the angular coordinate, the Helmholtz equation can be separated into a longitudinal part and a transverse part. The former is easily solved by having a solution of the form exp(−ikzz), and the latter is found as a solution of the equation^81,82[∂2∂ξ2+∂2∂η2+f2kt22(cosh2ξ−cos2η)]uT(ξ,η)=0.(12)

Here, the semifocal distance is represented by f, given in terms of the major and minor as f2=a2−b2a rand b, and related to the eccentricity e as e=f/a. Further, the parameter kt is the transverse component of the wave vector k. Equation (12) can be split into the radial and angular Mathieu equations, using the well-known separation of variables technique to obtain the nondiffracting Mathieu beams:⁸³Mme(ξ,η,z;q)=CmJem(ξ;q)cem(η;q)exp(ikzz),(13)Mmo(ξ,η,z;q)=SmJom(ξ;q)sem(η;q)exp(ikzz).(14)

In the above, Cm and Sm are normalization constants, whereas Jem and Jom are the even and odd radial Mathieu functions, respectively, and cem and sem are the even and odd angular Mathieu functions. For even modes, the subindex m=0,2,3,…, while for odd modes it takes the values m=1,2,3,4,…. Again, the nondiffracting Mathieu beams carry an infinite amount of energy and cannot be realized experimentally. Nonetheless, a finite-energy version can be realized using a Gaussian envelope, which is known as the Mathieu–Gauss beam. Such modes retain the nondiffracting properties of the ideal Mathieu beams over a finite propagation distance. The Mathieu–Gauss modes are described mathematically as⁸³MGme(ξ˜,η˜,z;q)=exp(−ikt22kzμ)Mme(ξ˜,η˜,z;q)⁢exp(−r2μω02)exp(ikz)μ,(15)MGmo(ξ˜,η˜,z;q)=exp(−ikt22kzμ)Mmo(ξ˜,η˜,z;q)⁢exp(−r2μω02)exp(ikz)μ.(16)

Using the new definitions, x=fo(1+iz/zR)coshξ˜cosη˜ and y=fo(1+iz/zR)sinhξ˜sinη˜, where now f0 is the semifocal separation at z=0. The parameter μ is defined in terms of the Rayleigh range zR=kω02/2 as, μ=1+iz/zR, where ω0 is the waist radius of a Gaussian beam. Figure 7(a) shows the transverse intensity distribution of a set of even, while Fig. 7(b) shows those of odd Mathieu–Gauss beams, given by the parameters z=0, kt=6, e=0.9, and f0=0.9. Importantly, Mathieu–Gauss beams are another class of “nondiffracting” optical fields, which are a variant of superposition of uniform conical waves, i.e., Bessel beams. Therefore, they have a similar capability of self-reconstruction after an opaque finite obstruction.

Another important solution to the paraxial wave equation is the Airy beam, which is given in Ai(x),^84,85A(sx,sy,ξ)=Ai[sx−(ξ2)2]Ai[sy−(ξ2)2]⁢exp[iξ2(sx+sy−ξ33)],(17)where sx=x/x0 and sy=y/y0 are dimensionless coordinates in the transverse plane and set by the scale parameters x0 and y0. Moreover, ξ=z/(kx02) represents a normalized propagation distance. Similar to the Bessel modes, Airy beams exhibit unique properties of self-acceleration, “nondiffraction,” and self-reconstruction. Among these, its tendency to accelerate in the transverse plane following a parabolic trajectory has attracted considerable interest [see Figs. 8(a) and 8(b)]. Since Airy beams also carry an infinite amount of energy, its experimental realization with finite-energy can be approximated by Airy–Gaussian modes [see Figs. 8(c) and 8(d)] and given as $$\begin{gathered} A0(sx,sy,ξ)=Ai[sx−(ξ2)2+ibξ]Ai[sy−(ξ2)2+ibξ] \\ ×exp[b(sx+sy)−bξ2+ibξ2−ξ36+iξ(sx+sy)2], \end{gathered}$$(18)where b<1 is a negative parameter that limits the energy of the Airy beam. As a result, the nondiffracting property can only persist for a finite distance. The inverse Fourier transform of A0 at ξ=0 yields a product of a Gaussian beam and a cubic phase,⁸⁴F−1{A0(sx,sy)}∝exp[−b(kx2+ky2)]exp[i(kx3+ky3)3],(19)where kx and ky are the transverse components of the inverse Fourier transform. Therefore, we can experimentally generate an Airy beam by modulating a Gaussian beam with a cubic phase in the Fourier domain.

The Ince–Gaussian (IG) modes are another important family of orthogonal solutions to the paraxial wave equation, which can be described as^86,87IGp,me(r,ϵ)=Cω0ω(z)Cpm(iξ,ϵ)Cpm(η,ϵ)exp(−r2ω2(z))⁢exp{i[kz+k22R(z)−(p+1)ξ(z)]},(20)IGp,mo(r,ϵ)=Sω0ω(z)Spm(iξ,ϵ)Spm(η,ϵ)exp(−r2ω2(z))⁢exp{i[kz+k22R(z)−(p+1)ξ(z)]},(21)where IGp,me and IGp,mo represent the even and odd solutions of order p and degree m, with C and S being normalization constants, and Cpm and Spm being even and odd Ince polynomials, respectively. ϵ=2f02/ω02 together with f0, ω0 are the scale parameters related to the geometric size of the mode. We note that IG modes are a continuous transition from LG to Hermite–Gaussian modes. Figure 9 shows the intensity profiles of even ^[Fig. 9(a)] and odd ^[Fig. 9(b)] IG modes for ϵ=2.

Helico-conical (HC) beams, contrary to the above optical modes, have radial phase dependence.^88,89 The phase in HC beams is unique, and it is given by a product of both the radial and the azimuthal coordinates,⁶⁶ψ(r,φ)=ℓφ(K−r/ro),(22)where ℓ is the topological charge, K is either 0 or 1, and ro is a normalization constant in the radial coordinates. As a result of this phase dependence, upon propagation, the optical field exhibits a helical geometry with anomalies in both phase and amplitude. Figure 10 shows the transverse intensity profiles of the HC beam comparing experimental and simulation results. Interestingly, these beams also exhibit a self-healing property, as demonstrated recently.⁶⁷

In the previous sections, we considered a set of structured light fields, in which the polarization distribution in the transverse plane was homogeneous. Here, we will now look into a more general class of light fields with spatially inhomogeneous polarization, commonly known as vector beams, which arise naturally as solutions to the vectorial Helmholtz equation.^90,91 These modes are commonly regarded as nonseparable superpositions of spatial modes and polarization,⁹²U(r)=u1(r)eiδ1e^R+u2(r)eiδ2e^L,(23)where the spatial degree of freedom is represented by the orthogonal functions u1(r) and u2(r), and the polarization degree of freedom is represented by the orthogonal unitary vectors e^R and e^L in the circular polarization basis. Moreover, the parameters δ1 and δ2 are intermodal phases that introduce a phase delay between both polarization components. Note that the spatial degree of freedom spans an infinite space, and the number of combinations between the spatial and polarization degrees of freedom is also infinite, giving rise to an infinite set of vector modes. In principle, the spatial degree of freedom can be any of the scalar modes described above. In the LGpℓ basis, Eq. (23) can be rewritten as^91,92U(r)=12(LGp1ℓ1eiδ1e^R+LGp2ℓ2eiδ2e^L).(24)

In principle, ℓ and p can take any integer value. Here, we will consider the simplified case of ℓ1=−ℓ2=ℓ, p1=p2=0 and δ1=0, δ2=δ, in which case, Eq. (24) can be expressed as U(r)=12(LG0ℓe^R+LG0−ℓeiδe^L).(25)

By substituting ℓ=1, the following set of orthogonal vector modes can be obtained: UTE(r)=12(LG01e^R+LG0−1e^L),(26)UTM(r)=12(LG01e^R−LG0−1e^L),(27)UHEo(r)=12(LG01e^L+LG0−1e^R),(28)UHEe(r)=12(LG01e^L−LG0−1e^R).(29)

These cylindrical vector modes are commonly known as Bell states, which are eigenmodes of both free-space and optical fibers.⁹³ Both the UTE(r) and UTM(r) modes with radial ^{[Fig. 11(a)]} and azimuthal ^{[Fig. 11(c)]} polarizations have been used in optical tweezers due to their unique transverse and longitudinal force components in the trapping plane. This will be discussed in more detail in a later section.

Experimentally, the complex beams described above can be generated in a wide variety of ways, which include interferometric arrays,^94–96 glass cones,^97,98 liquid crystal wave plates,^99,100 metamaterials,¹⁰¹ spatial light modulators (SLMs),^102–105 and, in recent time, digital micromirror devices (DMDs).^106–108 Noteworthily, SLMs enable the generation of complex 2D and 3D shaped light beam patterns, which have significantly advanced the configurable optical trapping of particles. Since these devices are typically utilized in the Fourier plane of an optical system, optical trapping systems using holographically generated light beams are known as holographic optical tweezers (HOTs). In this section, we overview the use of complex light fields in optical tweezers.

In this section, we will discuss some of the key demonstrations of HOTs within the domain of single optical traps.

As discussed in Sec. 2.2, the LG beam is one of the most common beams used for optical trapping. Since the high-order LG beams can carry OAM, which can be used to achieve the orbital rotation of particles. Pioneering experiments demonstrated a full 3D rotational control of microspheres and biological specimens using LG0ℓ modes.^109,110 The LG0ℓ modes were interferometrically combined with a reference LG00 mode (Gaussian mode) to produce a spiral structure. While the phase profile of the LG00 mode can be considered to be flat (spherical more precisely), the LG0ℓ modes are featured with a more intricate structure, resembling a helical staircase. Hence, the interference of an LG0ℓ mode with a Gaussian mode produces a new structured azimuthal phase with an intensity pattern containing ℓ spiral arms. Importantly, the spiral structure can be rotated in any direction and at arbitrary rates by introducing phase delays in one of the arms of the interferometer. In this way, particles trapped in the bright lobes of the interference pattern can be rotated in any direction around the optical axis of the beam. By simply changing the azimuthal index ℓ, it is possible to manipulate objects with different shapes or to trap many objects simultaneously.¹⁰⁹ Importantly, this pioneering experiment showed for the first time the potential application for the control of biological organisms, in which case, the authors trapped and rotated a Chinese hamster chromosome.

Another experiment¹¹⁰ was performed using a collinear interferometric superposition of LG0ℓ modes, with opposite topological charges ℓ and −ℓ, to produce the intensity patterns with petal-like structures of 2ℓ bright lobes. Similarly, the structure could be rotated by introducing phase delays in one of the arms. Figures 12(a) and 12(b) show 3D manipulation and rotation of particles trapped in intensity patterns produced by the superposition of LG0ℓ and LG0−ℓ. Figure 12(c) shows the rotation of the 3D microstructure of particles. Finally, Fig. 12(d) shows a schematic representation of the 3D structures that were created in this experiment by stacking more particles on each bright spot.

More recently, LG beams were used to explore the manipulation of single or multiple silicon nanowires.¹¹¹ It demonstrated the orbiting of silicon nanowires around the optical vortex aligned parallel to the propagation axis of the beams. Figure 13(a) shows the position of the particles as a function of time, evincing a clear orbital motion; the right inset shows a schematic representation of the trapped nanowire overlapped with the intensity pattern of the trapping beam. In addition, the author demonstrated that silicon nanorods oriented perpendicular to the propagation axis of the beam can be used as light-driven nanorotors, resulting from the transfer of SAM.

It is noted that the small particles can be trapped in the bright ring of LG vortex beams, while if the size of the particle is comparable to the waist size of the incident structured beam, the particle is predicted to be trapped at the center of the beam with the proper choice of the beam parameters such as vortex charge and polarization.¹¹² Moreover, a very recent study showed that when the LG beam is strongly focused, the rotation direction of the trapped particles might be in the opposite direction,¹¹³ which is much different for the case of paraxial LG beams. Figure 14 shows the anomalous motion of a particle trapped in a strongly focused LG beam. From Fig. 14(c), we can see that in the region β, the direction of the optical torque is opposite that of OAM of the LG beam.

The trapping geometry and particle shape can have a great influence on the angular momentum transfer. Jesacher et al.¹¹⁴ reported an observation of particles orbiting in a reverse direction with respect to the OAM of the incident light field. Irregular-shaped glass microparticles were trapped at an air–water interface with an LG mode of topological charge ℓ=±30 based on the holographic tweezers setup. Figure 15 shows the rotation of a polystyrene bead and a glass sliver depending on the sign of the topological charge. Intriguingly, only the polystyrene bead changes its direction of rotation by inverting the topological charge.

A simple ray optics model for a wedge-shaped glass particle placed at the air–water interface indicates that reflection and refraction of light rays on the particle surfaces can cause the asymmetric object to move against the direction of the OAM of the incident light field. Importantly, this demonstration suggests that the shape-anisotropy of the particles could be optimized for optical momentum transfer and thus for efficient optical nanotransport techniques. Other demonstrations of negative optical torques include reversed orbiting and spinning with a Bessel light beam,¹¹⁵ elliptically polarized beams,¹¹⁶ and circularly polarized Gaussian light beams.¹¹⁷

With the nondiffracting property, Bessel beams have great advantages for optical manipulation. The first experiment using zeroth-order Bessel beams demonstrated the trapping of multiple particles along the beam axis, as well as the transport of these particles over long distances (∼5mm).²⁶ Subsequent experiments⁶⁸ showed the manipulation of microparticles in multiple optical planes along the beam axis, using two sample cells (100μm in thickness) separated by 3 mm, as shown in Fig. 16(a). Crucially, even though the Bessel–Gaussian beam is distorted by the particles in the first cell, its self-healing properties allow recovery of its original intensity profile after a certain propagation distance, which coincides with the separation distance of the cells. Hence, the beam is capable of trapping again microparticles in its multiple rings inside the second cell. We note that in this experiment, the radius and nondiffracting propagation distance of the Bessel–Gaussian beam are 5μm and ∼4mm, respectively. It is worth mentioning that such a beam can also align multiple rods along the beam propagation direction.⁶⁸ Along the same line, counterpropagating Bessel beams have also been used to manipulate submicron particles in three dimensions. This optical conveyor belt is based on the generation of a standing wave, created from the on-axis superposition of two counterpropagating Bessel beams, which allows us to confine and deliver over long distances (hundreds of micrometers) and with high precision multiple submicron particles ^{[Fig. 16(b)]}. The delivering of particles was achieved by dynamically changing the phase of one of the beams, which in turn causes the whole structure of nodes and antinodes to shift along the optical axis of the standing wave.¹¹⁸ In 2008, a practical application was demonstrated, where microspheres trapped with Bessel beams served as objective lenses to focus a laser onto a surface to enable near-field direct writing with nanometer resolution.¹¹⁹ This technique was used to demonstrate nanopatterning of arbitrary patterns with dimensions in the order of 100 nm and positioning accuracy in the order of 40 nm.

Very recently, a superposition of multiple copropagating Bessel beams, so-called frozen waves, was used to build stable optical trapping as well.¹²⁰ Besides demonstrating that such beams possess greater optical trapping stabilities, the authors also demonstrated their capabilities of trapping in multiple parallel planes along the propagation direction, as schematically shown in Fig. 17(a). Here, to observe the motion along the propagation direction, the authors implemented a digital propagation, in which the observation plane is fixed, and different propagation planes are simulated holographically by changing the hologram that generates the frozen waves. An experimental sequence of frames of the beam at different propagation planes is shown in Figs. 17(a1)–17(d1) and 17(a2)–17(d2) for the case on one and two trapped particles, respectively.

If we replace the zeroth-order Bessel beams with higher-order Bessel beams, namely, Bessel vortex beams, the situation will change significantly. That is because the center of the Bessel vortex beam has zero intensity, due to the phase singularity on the axis. Therefore, the trapping of multiple transparent particles in multiple rings of intensity maxima of higher-order Bessel beams was experimentally demonstrated,⁷⁰ and the rotation of the trapped particles is observed as well. It is worth noting that the optical trapping behavior is dependent on the size of the particles.¹²¹ If the size of the particle is much smaller than the radius of the vortex bright ring, the particle will be trapped in the bright ring; meanwhile, the trapped particle will orbit along the bright ring, as shown in Fig. 18(a). If the size of the particle increases and is close to the radius of the vortex ring, then the particle will be trapped stably, without rotation, at certain off-axis positions of the inner bright ring, as shown in Fig. 18(b). If the particle is large enough to cover the whole bright ring of the vortex beam, then the particle will be trapped in the center of the vortex beam, and the particle does not rotate ^{[Fig. 18(c)]}. As a comparison, Fig. 18(d) shows the optical trapping of a silver nanowire with an LG vortex beam, and it was shown that the nanowire rotates only if its length is longer than the size of the bright ring of vortex beam.¹²²

Optical manipulation using perfect vortex modes possessing annular intensity profiles independent of topological charges has been of great interest in recent years.¹²³ The motion of trapped particles is determined by the gradient and scattering forces resulting from the perfect vortex beam. The particles continuously move along the annulus due to the scattering force from the inclined wavefront, as shown in Figs. 19(a) and 19(b). As the OAM density is well-defined in the perfect vortex beam, a linear relationship between the rotation rate and the OAM is guaranteed for both negative and positive topological charges, as shown in Fig. 19(c). The experimental realization of perfect vortex beams is rather challenging, as they tend to exhibit azimuthal intensity variations due to optical aberrations. As a result, the rotation of a single particle is difficult to maintain at a constant velocity around the trap due to local intensity hot spots. Nonetheless, with intensity profile correction techniques, it is possible to eliminate these hot spots,¹²³ in which a single trapped particle can continuously move along the vortex ring, as schematically illustrated in Fig. 19(d). However, the trapped particle still exhibits a nonuniform angular velocity due to local variations of the OAM density,¹²⁴ as shown in Fig. 19(e). Based on the linear dependence of the particle rotation rate upon the OAM density, the angular velocity can be adjusted in situ by correcting the local OAM density with a phase correction mask. It is worth noting that experimentally the perfect vortex beam can induce rotation of the trapped particle at a constant velocity with integer topological charges, while fundamentally this cannot be achieved with fractional topological charges.⁷⁴

Mathieu beams are a family of propagation invariant beams with self-healing properties. Compared to Bessel–Gaussian beams, they are especially well-suited for the 3D arrangement of particles due to their rich variety of complex transverse mode distributions.¹²⁵ Notably, Mathieu beams allow us to trap and orient elongated objects in the transverse x−y plane to the beam (z) axis, which is not normally possible with single optical tweezers. Figure 20 shows optical trapping of nonspherical particles with a fourth order (m=4) even Mathieu beam ^{[Figs. 20(a)–20(c)]}, which was rotated within the x−y plane ^{[Figs. 20(d) and 20(e)]}. Elongated particles (∼3×5μm) were arranged with their long axis in the transversal plane ^{[Figs. 20(f) and 20(g)]}, and their orientation depends on the beam orientation.

We note that it is possible to assemble multiple particles in the z direction because of the self-healing property of Mathieu beams. Furthermore, Mathieu beams also offer superpositions of even and odd modes, resulting in helical Mathieu beams that provide continuous phase variations. Thus, these beams were also used to investigate the transfer of OAM¹²⁶ to particles in optical micromanipulation.¹²⁷

One of the most prominent properties of Airy beams is their ability to freely accelerate in the transverse plane, resulting in a parabolic trajectory upon propagation. Exploiting this property, Baumgartl et al. transported particles along parabolic trajectories, as schematically represented in Fig. 21(a).^128,129 An Airy beam was shaped from a 25-mW argon-ion laser using an SLM and focused down to a size of 10μm into a chamber containing an aqueous suspension of colloidal glass microspheres. The optical gradient forces attracted particles toward the main lobe of the Airy beam, where they were transported along the parabolic trajectory due to radiation pressure. The Airy beam shifted laterally over a distance of ∼10μm after only a propagation distance of 75μm. Figures 21(b) and 21(c) show experimental results of the so-called “snowblowing” effect. The field of view was divided in four quadrants marked with different colors to see this effect clearly. Figure 21(b) shows that an Airy beam launched into the second (green) quadrant cleared particles in this section by carrying them into the third (purple) quadrant along the 3D parabolic trajectory. This process is reversible if the orientation of the Airy beam is inverted, as shown in Fig. 21(c). Recently, a circular Airy vortex beam has attracted interest due to its autofocusing property,^130,131 which is a hollow beam carrying OAM, as shown in Fig. 21(d). Such a beam carries OAM, which can be used to trap and rotate particles as well, as shown in Fig. 21(e). Moreover, it is shown that the rotational velocity of the trapped microparticles first increases with the increasing topological charge and then decreases after it reaches a maximum.¹³⁰

As previously mentioned, IG modes appear as natural solutions to the paraxial wave equation in elliptical coordinates (see the previous section on IG modes). A rich variety of transverse intensity profiles given by high-order assemblies of this family of modes allow micromanipulation of particles at each bright spot. Woerdemann et al.¹³² utilized these IG modes to create 3D structures of trapped microparticles. Figure 22 shows the IG modes generated by an SLM (top row) and particles trapped in the bright parts of these IG modes (bottom row). Notably, microbeads can be trapped in 3D according to the 3D structures of these beam profiles.

Another important class of OAM beams is the HC beams with spiral phase and intensity profiles.^88,89 A schematic representation of beam generation and its 3D beam profile upon propagation are shown in Fig. 23(a). With this unique beam property, both linear momentum and OAM are to be transferred to microparticles.¹³³ In the experiment, 2-μm-diameter silica beads are dispersed in water in a chamber with a thickness of 100μm. Figure 23(b) shows time-lapse images of a trapped bead moving upward from the bottom along a spiral trajectory around the axis of the HC beam (ℓ=20).

As we have shown in previous sections, SLMs have provided the ability to create 2D and 3D optical traps for guiding or transporting microparticles. Nonetheless, most of the previously discussed techniques rely on the use of common light beams that are solutions to the wave equation, except for HC light beams. In what follows, we will discuss a more general class of optical traps, which rely on arbitrary customized 2D and 3D light beams.

An immediate and almost obvious step forward to achieve the simultaneous trapping of many particles is through the use of galvo-scanning mirrors or acousto-optic beam deflectors,¹³⁴ in which a single beam can be time-shared at multiple locations to simultaneously trap micron-sized polymer spheres. This approach can be used in the study of the single molecule H-NS protein, for example.¹³⁴ An alternative approach to generate arrays of optical traps relies on the use of diffractive optical elements (DOEs),¹³⁵ as shown in Fig. 24. The key idea in this approach is based on the deflection of an input beam to a different angle defined by period of a diffraction grating. Therefore, the generation of multiple trapping focal spots can be achieved by superimposing multiple diffraction gratings. A typical binary version of the DOE pattern for a hexagonal tweezers array is shown in Fig. 24(b). The trapped microspheres in the array are shown in Fig. 24(c).

SLMs can facilitate this task by adding more flexibility and new features to this field. Not only can the wavefront of the incident beam be modulated to desired optical traps, but also the wavefront distortion (aberration) and attenuation can be corrected or compensated by SLMs.^136–138 In addition, SLMs enable dynamic and interactive control of holographic optical traps (HOTs). Figure 24(d) schematically illustrates a typical HOT setup that incorporates the use of an SLM.^136,139 The phase pattern for generating an array of 20 by 20 traps (bottom-right inset with a zoom-in image for the trapped microspheres) is shown on the bottom left inset. Importantly, such HOTs can be brought to a third dimension for the simultaneous manipulation of multiple particles in 3D, which is useful for creating, for example, microcrystal structures. Proposed techniques can create structures of several tens of microns capable of rotating dynamically about arbitrary axis.^137,140,141

A step forward in the generalization of customized light shapes to manipulate microparticles consist of the computation of the phase, in a reverse-engineered process, of the desired light beam, which we briefly explain next. In essence, a 2D phase distribution Φin(r) of the beam wavefront is calculated from the desired intensity pattern at the trapping plane. Figure 24(a) shows the relationship of the electric field between the input and the focal planes.¹³⁵ The monochromatic plane wave is modulated by a phase Φin(r) at the input leading to Ein(r)=Ain(r)exp[Φin(r)], where the amplitude and imposed phase are real-valued functions. The electric field at the focal plane yields Ef(ρ)=Af(ρ)exp[Φf(ρ)], which constitutes a Fourier transform pair with Ein(r) and can be written as¹³⁵Ef(ρ)=k2πfeiθ(ρ)∫d2rEin(r)e−ikr·ρ/f,(30)where f represents the focal length of the lens and k=2π/λ the wavenumber. Since there is no analytical solution for the phase distribution, usually an iterative algorithm, e.g., an adaptive-additive algorithm, is applied to search for an optimized phase.¹³⁵

So far, we have discussed the creation of static arrays of optical traps using an SLM. In the following sections, we will discuss approaches for the creation of complex light fields for micromanipulation.

Optical trapping along arbitrary trajectories was first demonstrated in 2003.¹⁴² The idea behind this technique is the fact that, for the family of LG transverse modes, the mode profile and radius of peak intensity vary with the topological charge ℓ of the beam:¹⁴²R(θ)=aλNA[1+1ℓ0dφ(θ)dθ],(31)Rℓ≈aλ/NA(1+ℓ/ℓ0),(32)where NA is the numerical aperture of the focusing lens, and a and ℓ0 are constants related to the radial amplitude profile of the beam. It provides a way to tailor the radius of the vortex beam via the function of θ. For example, Lissajous patterns can be directly obtained with φ(θ)=ℓ[θ+αsin(mθ+β)], where the constants α and β are used to control the depth of modulation. Figure 25(a) shows the intensity patterns for a constant α and a varying m (top row) and a constant m and a varying α (bottom row). Optical tweezers with beams generated in this way can drive nanoparticles along with closed intricate circuits. Figure 25(b) shows two polystyrene nanospheres (400 nm in radius) suspended in water are transported along with a modulated optical vortex with l=60, m=3, and α=0.1, in which each particle completes a cycle within 2 s. It is worth noting that the optical forces driving the particles are larger when R(θ) is smaller, when the light is most intense.

Recently, a new class of optical tweezers along with 2D and 3D trajectories was demonstrated, driving particles based on phase gradients of the light field. This method employs a technique to create light fields with an arbitrary (and programmable) transverse phase profile ϕ(r). A particular case⁷⁶ of such a transverse phase gradient is the azimuthally varying phase ϕ(r)=ℓφ. Similar phase profiles have been used in the field of laser remote sensing to directly measure the velocity component along the transverse plane.^143–145 When trapped in a beam with a phase gradient, particles can move along the beam. In the case of linear phase gradient ϕ(x)=|q|x, where |q|=12rad/μm, colloidal silica microspheres (1.53μm in diameter) dispersed in water were transported along with a transverse linear trap of 5μm in length at a speed of 2μm/s. The second case of a parabolic gradient ϕ(x)=±(qx)2 is more interesting.⁷⁶Figures 26(a1) and 26(b1) show that two colloidal spheres were trapped with negative ϕ(x)=−(qx)2 and positive ϕ(x)=+(qx)2 parabolic phase profiles, respectively. In the negative case, the particles were pushed toward the ends of the linear trap, whereas in the positive case they are pulled toward the center of the trap. Figures 26(a2) and 26(b2) show the linear intensity profiles at the trapping plane. A cross-section of the trapping beams in the x−z plane shows the phase gradient, which originates from either light rays diverging or converging toward the optical trapping plane, as shown in Figs. 26(a3) and 26(b3).

In 2010, a new class of beams was demonstrated, the so-called optical solenoid, which is nondiffracting solutions of the Helmholtz equation in cylindrical coordinates.¹⁴⁶ Perhaps the most notable property of these beams is their 3D spiralling intensity profiles, whose wavefronts carry an independent helical pitch. Moreover, their radial intensity distribution remains invariant in the spiralling frame of reference. These beams are capable of trapping microparticles along with the helical beam intensity profile and transporting them along the spiral trajectory via a phase gradient. Mathematically, these beams can be synthesized as a superposition of the m’th Bessel beam as uγ,ℓ(r,z)=∑m=[ℓ−γk]|ℓ|ℓ−mγ2Jm(qmR)exp[i(ℓ−m)γz]⁢exp(imθ)Jm(qmr),(33)where qm2=k2−(ℓ−m)2/γ2 and [ℓ−γk] represents the integer part of ℓ−γk. Figure 27(a) shows the 3D intensity profile of Iγ,ℓ(r,z)=|uγ,ℓ(r,z)|2 for kR=10, θ=30deg, and ℓ=10. To generate such an optical solenoid beam, the required hologram was created on a phase-only liquid crystal SLM and projected into a sample cell containing colloidal silica beads (1.5μm in diameter) immersed in water, through a 100× microscope objective. The transmitted beam was reflected by a mirror mounted on a translation stage and imaged onto a CCD camera. Figure 27(b) shows the experimental 3D intensity profile of the beam, where the radius of the helical trajectory transverse to the beam axis was R=5μm. Microspheres can be trapped in the solenoid beam and transported along its helical path ^{[Fig. 27(c)]}. Importantly, the direction of their motion upward or downward along the beam axis can be controlled by alternating the sign of the topological charge. The gray-scale image in Fig. 27(c) was created by the superposition of six video frames capturing the same microsphere at different time instances. This experiment demonstrates that a suitable combination of phase- and intensity-gradients can exert retrograde forces on microparticles, which can be transported against the beam propagation direction.

The use of LG0ℓ modes to orbit particles confines the motion of the trapped particles to the plane perpendicular to the propagation direction moving around the annular intensity profile, where the azimuthal phase variation drives the particles via scattering forces. Therefore, the trajectory of trapped particles is determined by the intensity and phase profiles of the beam, both of which are always interlinked. Newer approaches proposed advanced techniques, in which the intensity and phase profiles of the beam are independent from each other. These allow the creation of 3D parametrized curves (trajectories) of trapped particles in the form of R0(s)=(x0(s),y0(s),z0(s)) as a function of the arc length s.¹⁴⁷ To achieve the movement of particles along 3D curves, a hologram that produces the 3D parametrized curve at the focal plane is required,¹⁴⁷R0(s)=R(cos(sR)cosβ,sin(sR),cos(sR)sinβ),(34)where R is the radius of the curve projected on the x−y plane, rotated by an angle β about the y axis with s∈[0,2πR]. Figure 28(a) shows an experimental example of nine microspheres (5.17μm in diameter) trapped along the curve of radius R=9μm tilted at an angle β=π/4 rad. Figure 27(b) shows the corresponding schematic representation in 3D. More complex structures of light can be generated by the same principle. For example, two knotted rings of light, tilted at opposite angles β=±π/8, are shown in Fig. 28(c). Here, the radius of the ring is R=12.9μm, and their centers are separated by R/2. Figure 28(d) is an experimental demonstration showing that the rings act as two 3D optical traps and are capable of organizing colloidal silica spheres. Interestingly, the trapped spheres could freely move between these knotted rings, which are schematically shown in Fig. 28(e).

More recently, alternative techniques have been proposed for the generation of arbitrary 3D parametrized curves of the beam with high intensity and independent control of the phase, also known as freestyle optical traps.^148–152 The main idea behind the generation of a light beam that focuses into a parametric curve of the form c(t)=(R(t),uz(t)), with R(t)=(R(t)cost,R(t)sint), is to display on an SLM a hologram that generates the polymorphic beam,¹⁵¹E(r0)=∫0Tg(t)exp[−ik2f2uz(t)r02]exp[ikfr0R(t)]dt,(35)where r0=(x0,y0) represent the transverse spatial coordinates, and the parameter T represents the maximum value the azimuthal angle can take. Further, the function g(t) is a complex value function, which, as we will show later, plays a major role in the design of the curved laser trap. The desired optical field is generated in the far field, which is achieved by means of a lens of focal length f, acquiring the specific complex amplitude distribution E˜(r,z=uz(t))=−iλexp[ikuz(t)]f∫0Tg(t)δ[1f(R(t)−r)]dt.(36)

Here, the function g(t)=|g(t)|exp[iΦ(t)] accounts for the phase of the trapping beam, which provides the mechanism behind the propulsion of the trapped particles along the 3D parametric curve. More precisely, the variation of the phase of E˜(t) is controlled by the function Ψ(t)=Φ(t)+kuz(t), which for convenience can be expressed as Ψ(t)=2πℓS(T)S(t),(37)where S(T) is a real function. Further, the parameter ℓ is associated to the accumulation of the phase along the entire parametric curve and, for closed curves, coincides with the topological charge of the beam. Crucially, since the force exerted on the particles is proportional to the phase gradient, their speed can be controlled as-desired by means of the function S(t). For example, to achieve a uniform propelling force and therefore a uniform speed, this function can be set to S(t)=∫0t|c′(τ)|dτ.(38)

Here, |c′(τ)|=R(t)2+R′(t)2+uz′(t)2 and c′(t)=dc/dt. The direction of motion of the particles can also be controlled by simply reversing the sign of the topological charge ℓ. See Refs. 149 to 151 for a detailed description of the mechanism behind the optical trapping and the propulsion forces involved in this technique. By way of example, let us consider the case E(r)=1T∫0TΦ(r,t)φ(r,t)|c2′(t)|dt(39)to be encoded on an SLM,¹⁵⁰ where t∈[0,T], c2(t)=(x0(t),y0(t),z0(t)) is a 3D parametrized curve, and r=(x,y). Further, the derivative c2′(t)=dc2(t)/dt is associated with the length L=∫0t|c2′(t)|dt, where |c2′(t)|=[x0′(t)2+y0′(t)2]12 of the 3D curve. The function Φ(r,t) can be written in the form of¹⁵⁰Φ(r,t)=exp{iρ2[yx0(t)−xy0(t)]}exp[i2πmS(T)S(t)],(40)where S(t)=∫0t[x0(τ)y0′(τ)−y0(τ)x0′(τ)]dτ,(41)with m and ρ being the free parameters to control the phase gradient and the ring radius R=λf/2πρ at the focal plane, respectively. The term φ(r,t) is a quadratic phase term of the form φ(r,t)=exp{iπ[x−x0(t)]2+[y−y0(t)]2λf2z0(t)},(42)where z0(t) is a defocusing parameter defined along c2(t). Figure 25 shows an example of the phase and intensity distributions that are obtained. More precisely, Fig. 29(a) shows the phase profile of a ring trap with a radius of R=5μm with topological charge m=30. The corresponding intensity profile is shown on the right column. Figure 29(b) shows the phase profile of a hypocycloid-shaped trap, parametrized as x0(t)+iy0(t)=ρ[2exp(it)+exp(−i2t)/2], where the phase gradient varies along with the trap, as shown in Fig. 29(b1).

Figure 30 shows the complex motion of particles propelled along a helically modulated toroidal surface.¹⁴⁹Figure 30(a) shows the time-lapse images of trapped particles over a period of 7 s along the decagon trajectory in Fig. 30(b), which is a projection of the 3D toroidal beam [see Fig. 30(c)] onto the x−y plane. Figure 30(d) shows experimental intensity profiles of the 3D toroidal beam at two different axial planes of z=±1μm.

The general approach to determine the 3D parametric curve consists of first defining a set of m points P(n), n=1,2,3,…,m, through which it must pass. The problem then reduces to finding a piecewise parametric curve that passes through these points. This problem was solved by Rodrigo et al., using a set of parametric curves, known in computer graphics theory as Beziér Splines bn(τ), that join the points P(n). As explained in Ref. 151, each Beziér spline is defined by four points, two knot points, and two associated points, the former denoted by Ps(n) and Pe(n) and the later by Ts(n) and Te(n), where the subscripts s and e denote the starting and ending points. Each Beziér spline is defined in terms of these points as bn(τ)=(1−τ)3Ps(n)+3τ(1−τ)2Ts(n)+3τ2(1−τ)2Te(n)+τ3Pe(n),(43)where τ∈[0,1]. The parametric curve is then given as c(t)={b1(τ),b2(τ),…,bm(τ)},(44)which has to be continuous and differentiable; see Ref. 151 for further details. An example of a parametric curve constructed in this way is shown in the left panel of Fig. 31(a), where six Beziér splines, represented by different colors, were used. The intensity and phase profile of a laser beam focused along this curve are shown in the middle and left panels, respectively, of the same figure. Crucially, this construction method allows the real-time reconfiguration of c(t) by simply shifting the knots points of the Beziér splines. By way of example, the left panel of Fig. 31(b) shows the progressive change of c1(t) into cN(t) by shifting one knot point. Here, four microparticles, labeled as A, B, C, and D, trapped along the curve, are illustrated, where the arrows show their motion direction. The actual performance of such a real-time reconfigurable optical trap was demonstrated experimentally. To illustrate this, the middle panel of Fig. 31(b) shows an example of the transition from c1(t) into c22(t) in an actual optical trap experiment, where multiple microparticles were set to move along the different curves. Here, only the cases defined by the curves c1(t), c12(t), c16(t), and c22(t) are shown. A time lapse of the whole transition is shown in the right panel of the same figure, where the motion direction of the particles is indicated by the direction of the arrows.

Controlling the polarization state of light is of paramount importance in many fields of pure and applied sciences; however, it is only in recent years that increasing attention has been given to complex vector fields, often referred to as classically entangled beams.^153,154 In optical tweezers, the degree of polarization plays a crucial role in vector beams. For example, when tightly focused, different states of polarization of the input beams can lead to distinctive optical forces at the trapping plane. Vector beams with radial and azimuthal polarizations are two special cases. Radial polarization possesses a strong field component along the propagation direction, while this is absent from azimuthal polarization.^155–158Figure 32(a) schematically illustrates this behavior. In a radially polarized vector beam, the electric field oscillates radially in the transverse plane, with no longitudinal component. However, when tightly focused through a high numerical aperture lens, the beam is strongly refracted toward the focus, giving rise to longitudinal electric field components, as shown in Fig. 32(b). In the case of an azimuthally polarized beam [see Fig. 32(c)], the electric field oscillates circularly along a transverse plane to the beam propagation that cannot generate a longitudinal electric field component under tight focusing conditions. Indeed, radially polarized vector beams are known for their ability to produce the smallest focal spot size among the family of vector beams.^159,160

Crucially, the use of complex states of polarization in tightly focusing systems allows engineering of light fields, such as polarization and intensity. For example, the shape of the focused beam can be designed as flat-top by creating appropriate superpositions of modes with specific polarization states.¹⁶¹ The addition of DOEs such as SLMs at the input plane allows for the generation of exotic beam shapes, such as chains of light fields containing 3D dark volumes (low intensity spots) along the light field distributions.^162,163 Precise control over the beam shape to create a perfect spherical spot can be achieved using counterpropagating vector beams, which are constructively interfering at the common focal plane.^164–166 In this configuration, radially polarized vector beams can produce the spherical focal spot containing solely longitudinal electric fields, in contrast to the use of azimuthally polarized vector beams, which only contain the transverse components in the electric field.¹⁶⁷ It is worth noting also that arbitrary control of the polarization state of a focused field can be achieved by creating coaxial superpositions of two vector beams with radial and azimuthal polarization, each of which has a different weighting factor.¹⁶⁸

In recent years, it has been widely reported that the state of polarization plays a crucial role in optical tweezers, which was ignored in the early days of the study. Recent reports indicate that radially polarized beams can enhance the axial trapping efficiency up to twice that of linearly polarized light beams at the expense of reducing the transverse trapping efficiency by up to a half.¹⁶⁹ Furthermore, azimuthally polarized vector beams have demonstrated higher transverse trapping efficiencies than those of radially polarized ones.¹⁷⁰ These features of cylindrical vector beams are particularly useful for trapping metallic particles exhibiting high levels of scattering and absorption of light. For example, the enhanced axial trapping efficiency in radially polarized beams due to the strong axial field component with a null axial Poynting vector allows the trapping of metallic particles.^171–173 In addition, the concurrent use of π-phase radially and azimuthally polarized vector beams can enhance optical forces to a trapped particle by tuning the relative phase between the eigenmodes comprising the beams.¹⁷⁴

Recently, the generation of multiple vector beams from a multiplexed hologram displayed on a single SLM was demonstrated.¹⁷⁵ Each vector beam was generated by a pair of independent holograms superimposed on an SLM. As they propagate along independent optical paths using different diffraction gratings, their polarization states can be modulated independently prior to a coaxial recombination through a polarizing beam splitter. In this way, an array of multiple vector beams can be generated from multiple pairs of multiplexed holograms with independent control over the transverse spatial positions of the beams and their polarization states. Figure 33(a) shows the intensity profile of nine vector beams with their polarization distribution in Fig. 33(b). This approach allows the manipulation of multiple particles at different vector beams possessing different polarization states. To experimentally deliver a desired array of vector beams, a 4f lens system relays the SLM plane to the entrance pupil of the microscope objective, as shown in Fig. 33(c). With appropriate diffraction angles, the generated beam pairs can be projected to the same position in the trapping plane.¹⁷⁵Figure 33(c) inset 2 shows an experimental image of four vector beams at the trapping plane, after passing through the linear polarizer.

In contrast to the above discussed optical tweezers, where the optical force exerted on the micro- and nanoparticles produces an acceleration along the same direction of the photon flow in an optical tractor beam, the optical forces drag small microparticles against this flow, i.e., toward the photons source.^28,176–180 In contrast to a conveyor belt that requires gradient forces to pull microparticles toward the source of the photons,¹¹⁸ the pulling forces of a tractor beam originate from a momentum conservation law. In essence, when the forward scattered light is more collimated than the incident light, and if the backward scattering is weak enough, the momentum conservation law predicts the existence of a pulling force.^177,181 This can be generated, for example, by the interference of two plane waves [see Fig. 34(a)]. As illustrated here, each beam propagates along the wave vector k1 and k2, whereas the resulting beam propagates in the forward direction along k1+k2. Under these conditions, the majority of the scattering is produced in the forward direction, which causes a pushing force Fz in the opposite direction, as also indicated here. Crucially, this polarization-dependent effect allows us to switch between a pushing and a pulling force, by simply changing the polarization of the incident waves, as shown in Figs. 34(b) and 34(c), respectively. Crucially, the pulling effect has been also explained in terms of the optical singularity of the pointing vector around the scatterer.¹⁸²

The optical pulling force can be achieved by utilizing structured beams, objects with specific opical parameters, structured background media, and photophoresis effect as well.¹⁸⁰ The tractor beam relying on photophoretic force is shown in Fig. 35. A long-distance, stable and switchable optical transport was achieved by the use of cylindrical vector beams.¹⁸² Here, thin-walled spherical glass shells with a radius from 25 to 35μm and a thickness from 200 to 400 nm were used, in which light can be absorbed either in the vicinity of the front side (outside) or in the rear side (inside) of the particle. In the former case, the photophoretic force pushes the particle along the direction of light propagation, whereas in the latter case the particle may be pulled against the beam propagation direction. Importantly, these two situations could be exchanged by switching from a radially polarized beam to an azimuthally polarized one or vice versa.

Figure 35(a) schematically illustrates a part of the experimental setup for this effect. To enhance the photophoretic forces, the glass shells were coated with a thin layer of Au (7 to 15 nm) and placed inside a rotating cylindrical cuvette, in which the rotating axis was aligned to be parallel to the beam axis. Figure 35(a) shows that the glass shells falling by gravity are trapped and pushed outside the rotating cuvette by the vector beam. Here, two half-wave plates allowed us to switch the vector beam from radial to azimuthal polarization to generate a pushing or a pulling force. Figure 35(c) shows the particle velocity depending on the radius of the shell, for both azimuthally (pulling) and radially (pushing) polarized beams. Figure 35(d) schematically shows the experimental observation of a particle pulled by an azimuthally polarized beam, whereas Fig. 35(e) shows the case of a particle pushed by a radially polarized beam.

We know that the refraction and the dominant gradient force are the bases for stable optical trapping of dielectric particles, while the laser beam can induce relatively large scattering and absorption forces on metal particles close to their localized surface plasmon resonances. Therefore, in the early days, it was generally accepted that stable optical trapping of metal nanoparticles cannot be obtained readily.¹⁸³ However, Svoboda and Block¹⁸⁴ demonstrated that metal Rayleigh particles can be trapped stably in 3D. In their experiment, the near-infrared laser beam was used to trap the gold nanoparticle, viz., the optical trapping was achieved off-resonance. Interestingly, in 2008, Dienerowitz et al.¹⁸⁵ experimentally demonstrated that metal nanoparticles can be trapped by laser beams close to their plasmonic resonance. They showed that a vortex beam (LG beam with annular profile) can confine the metal nanoparticles in the dark region of the beam center. Since the vortex beam carries OAM, they observed the rotation of particles as well ^{[Fig. 36(a)]}. Later, Lehmuskero et al.¹⁸⁶ optically trapped gold nanoparticles and set them into orbital rotation with an orbiting frequency of 86 Hz using an LG vortex beam carrying OAM. In their experiment, to achieve a stable trap, a q-plate ^{[Fig. 36(b)]} was adopted to produce a vortex beam with uniform intensity distribution, and a circularly polarized laser beam was used to induce circular symmetric optical gradient forces. Recently, the dynamics of electrodynamically coupled metal nanoparticles in an optical ring vortex trap were studied.⁷⁷Figure 36(c) shows the schematic of the focused Bessel–Gauss optical vortex beam and ring trap over a gold nanoplate. They used a retroreflection geometry with a gold nanoplate mirror to generate a constant-radius optical vortex. They demonstrated that, compared with a glass coverslip, the retroreflection geometry can significantly increase the spatial confinement and optical drive force, and a superior trap can be created accordingly.

Moreover, similar to the case of trapped dielectric particles, the trapped metal particles can move along the beam when trapped in a structured laser beam with a phase gradient as well. Figure 37(a) shows the intensity profile of a Gaussian beam with two different phase masks for producing the phase gradient with opposite signs, namely, the type I and type II denote that the phases are modulated by a convex and concave cylindrical lens, respectively.¹⁸⁷Figure 37(b) shows that for a line optical trap of the type I phase structure, the single silver nanoparticle moves from the end to the center of the line, and, for a type II trap, the single silver nanoparticle moves from the center to the end. Such line optical traps were implemented for the assembly of multiple silver nanoparticles. Besides, it was demonstrated that these nanoparticle arrays could be assembled and disassembled as desired by simply changing the sign of the phase gradient. Recently, the pairwise interactions between metal nanoparticles in a vortex ring trap with transverse phase gradients were studied.¹⁸⁹ This study focused on the deep understanding of multiparticle dynamics during the self-assembly of optical matter. They revealed that for small phase gradients, the total force is modulated by a separation-dependent interference effect. On the contrary, for strong phase gradients, the symmetry of the interaction between two nanoparticles breaks, resulting in a change of the distance, for which stable binding can be achieved. More recently, Rodrigo et al.¹⁸⁸ studied how to control the speed of metal nanoparticles by tailored phase-gradient propulsion force. Figures 37(c) and 37(d) show two ring vortex traps with a uniform phase gradient and a tailored nonuniform phase gradient, respectively. From Figs. 37(c) and 37(d), we can see that the metal nanoparticle in the second trap rotates much faster than that in the first trap.

It is worth noting that the aforementioned works are limited to 2D particle rotation in contact with a surface of the coverslip, and the 3D trapping and motion control of metal nanoparticles are not achieved until 2021. Rodrigo et al.¹⁵² experimentally demonstrated the transport of metal nanoparticles in an optical vortex ring trap along the 3D trajectory on demand. To obtain 3D optical manipulation of metal nanoparticles along an arbitrary curve, they adopted a freestyle laser trap,¹⁵¹ which can induce a propulsion optical force due to the phase gradient of the beam. The 3D ring trap of metal nanoparticles with uniform phase gradient is shown in Fig. 38(a), which indicates that the propulsion force is constant along the ring, while Fig. 38(b) shows that the propulsion force increases with the increasing angle. Besides, the tilted ring optical trapping and transport of metal nanoparticles were demonstrated as well.¹⁵²

Structured light has demonstrated its extensive applications in optical tweezers for micromanipulation in a liquid environment. Only recently there has been a surge of interest in levitated optomechanics using optical tweezers in air/vacuum.^30,190–216 The dynamics of a levitated particle in a diluted gas or vacuum environment (underdamped) is fundamentally different from those trapped in a viscous medium like water (overdamped) due to the particle’s finite inertia, which is typically ignored in overdamped systems (see Sec. 2.2). Including the inertial term, the equation for the center-of-mass motion of an optically trapped Brownian particle in one dimension (x direction) is described by the Langevin equation: mx¨=−Γ0x˙−κ0x+Fth,(45)where x is the particle position, m is its mass, Γ0 is the friction coefficient, and κ0=mΩ02 is the trap stiffness at the trap frequency Ω0/(2π). The fluctuating force Fth is due to the random collisions from the surrounding fluid molecules. The corresponding form of the PSD of the oscillation from Eq. (45) yields the Lorentzian response Sx(ω)=kBTπmΓ0(ω2−Ω02)2+ω2Γ02,(46)where kB is Boltzmann’s constant, and T is the absolute temperature. In the underdamped regime (Γ0≪Ω0), the particle undergoes high-quality mechanical oscillations with a quality factor Q=Ω0/Γ0 at the oscillation frequency Ω0. In the experiment, the damping Γ0 can be controlled by changing the pressure inside the vacuum chamber, while Ω0 is determined by κ0/m, i.e., dependent on the optical power for trapping and mass of the particle. At atmospheric pressures, Γ0(≫2Ω0) is typically larger than the critical damping (Γ0=2Ω0), and the PSD is similar to those observed in liquids, exhibiting a roll-off frequency at Ω0. At a gas pressure <10mbar, the levitated particle is typically underdamped (Γ0≪Ω0), where a strong resonant peak evolves at Ω0, which is the signature of high-quality mechanical oscillations. Thus, levitated nanomechanical oscillators with high quality factors have been identified as promising candidates for ultrasensitive force and acceleration detectors,^217,218 achieving a force sensitivity in the order of zepto-Newton in high vacuum.²¹⁹