Light can exert forces (torques) on objects during the light–matter interaction and therefore is used as an optical manipulation tool for micro/nano-objects. As early as 1619, the concept of “force of light” was first proposed by Johannes Kepler in an attempt to explain the phenomenon that when a comet enters the solar system, its tail is always deflected away from the sun^[1]. The underlying mechanism was later summarized by Maxwell’s electromagnetic theory, which states that light, though electromagnetic waves, carries momentum^[2]. Accounting for forces that stem from the momentum exchange between the radiation field and the interactive matter, the force of light belongs to a general phenomenon known as the “radiation force”^[3–5]. For the sake of brevity, the electromagnetic radiation force has now been more frequently addressed as the “optical force.” Specifically, the most vivid picture of optical force should be the case that when a beam of light is fired at a reflecting mirror, a pushing force is generated as the consequence of the momentum transfer from photons to the mirror, as the direction of light momentum is reversed upon reflection. However, due to the “extreme minuteness” of the optical force, John Henry Poynting deemed its application untenable in driving mechanical locomotion in terrestrial scenarios^[6]. The potential of light momentum, or rather, the optical force, was not truly appreciated until the advent of the laser and the landmark invention of optical tweezers by Arthur Ashkin, who demonstrated optical trapping and manipulation of micro/nano particles, living cells, and molecules using optical force in fluidic environments^[7,8]. By virtue of his remarkable work, Ashkin was awarded the Nobel Prize in Physics in 2018. His experiments also formed the basis of another Nobel Prize in Physics in 1997 for Steven Chu’s work on the optical cooling of atoms, showing that optical manipulation is a fascinating field in fostering scientific explorations at the “bottom” (quoting Richard Feynman’s speech^[9]).

In the macro regime, a semi-quantitative estimation of the optical force exerted on a reflective surface is $F_{\text{opt}}=2rP/c$, where $P$ is the optical power, $r$ the reflectance, and $c$ the speed of light^[7,10]. The expression of optical force shares the general traits of radiation force, proportional to the incident power divided by the wave velocity^[3,4]. Correspondingly, for a light beam of power $P=150\mathrm{mW}$, the optical force is 1 nN, and that is still under the assumption of perfect reflection ($r=1$). A force of such order of magnitude can easily be eclipsed by gravitation or even environmental perturbations at the macroscale, which somehow confirms the assertion of Poynting^[10,11]. In the micro regime, considering that not all incident photons would fall within the target volume, it is the local light intensity (i.e., the Poynting power density) rather than the optical power that is of more immediate concern. Hence, the notion of “pressure” should be adopted instead, as in “radiation pressure” in early literatures^[7,11,12], given by $P_{\text{opt}}=2rI/c$, where $I$ denotes the light field intensity^[11,13].

For micrometer or sub-wavelength objects, a $\sim \mathrm{pN}$-scale optical force would become relevant in that it generates roughly 10²−10⁵ times the gravitational acceleration, which thus sets it apart from that at the macroscale. To ensure optical force reaches piconewton, which, from an empirical standpoint, has become the basic standard to implement stable trapping in optical tweezers, coherent light sources and high-numerical-aperture (NA) objectives should be employed that condense the incident light power within diffraction-limited spots and maintain the local light intensity as high as ${10}^5–{10}^8\mathrm{W}/{\mathrm{cm}}^2$, and that is where the role of lasers and Ashkin’s design of the optical trap apparatus should be appreciated. Over the past few decades, continuous efforts have been made in enhancing the optical force attainable at micro/nano scales, successful examples including the incorporation of plasmonic^[14,15] and resonance dielectric structures^[16,17]. Yet the diffraction and speed of light (or the dispersion relation of photons) have fundamentally set the limit of optical force, which typically could not surpass $\sim 1000\mathrm{pN}$ even with rather strong field localization. Because of its magnitude, optical force has mainly been exploited in fluidic environments, where the significance of optical force is still prominent against countering effects such as Brownian diffusion and viscous drag.

To implement optical manipulation in the fluidic domain, aside from acquiring mechanical momentum directly from light momentum, an indirect route can be taken, which requires making use of the hydrodynamic surroundings. A representative example is the photophoretic force, denoting the migration of light-absorptive particles in gaseous suspensions^[18,19]. Specifically, the generation of photophoretic force demands the existence of gas molecules, the collisions between which and the particle create a net force pointing opposite to the particle’s surface temperature gradient. Assuming a particle with zero thermal conductivity, the relevance of the gas pressure greatly diminishes, and the photophoretic force can be estimated as $F_p=\alpha P/3v_m$, where $\alpha$ denotes the conversion efficiency from light energy to the thermal energy of surrounding air molecules, $P$ the accessible optical power in the target region, and $v_m$ the average gas molecular velocity^[20–22]. The Maxwell’s law of velocity distribution has determined $v_m$ to be in the range of $\sim 100\mathrm{m}/\mathrm{s}$ at room temperature, considerably smaller than the traveling speed of light. Consequently, for strongly absorbing and poorly thermal conducting particles, the effects of optical force could be overwhelmed by photophoretic force, rendering the latter a destructive factor in optical-force-based manipulation; while from another perspective, by interfacing the energy channel of light, the photophoretic force, together with other heat-mediated and fluidic-assisted effects, e.g., the Ludwig–Soret effects^[23] and electrothermoplasmonic flow^[24,25], provides alternative options to enable robust, long-distance and multi-mode optical manipulation with relaxed requirements upon local light intensity. Notably, in these schemes, instead of imparting forces directly on target objects, light, through photothermal effects, will first induce flows of hydrodynamic environments by imposing specific temperature profiles, which then give rise to the concomitant locomotion of suspended particles.

Fluidic environments have proved to be golden testing grounds for optical manipulation, while an inevitable trend is to further extend its capability to the solid domain, the exploration of which is doomed to be difficult because of two major challenges: (1) the adhesive and friction forces reach the order of $\sim \mathrm{\mu N}$, causing a tremendous scale gap with the optical force^[26], and (2) the fluid-assisted effects are inaccessible on dry surfaces due to the lack of fluidity and the no-slip condition along solid boundaries^[27].

To meet the challenges, a tactful resolution is to “inflict fluidity” in solid environments through photothermal effects, which bypasses the adhesion and additionally creates auxiliary hydrodynamic flows for mass transportation. In the meantime, it is vital that proper drives should be found that are both adequate in magnitude and compatible with solid environments. Still interfacing the energy channel of light, researchers in the early 1990s demonstrated the detachment and propulsion of adhering particulates via pulsed light illumination on absorbing substrates, the technique of which has been widely exploited in semiconductor surface cleaning^[28,29]. As the result of opto-thermo-mechanical muliphysics coupling, the impulsive thermal expansion/compression of the substrate translates into surface acoustic waves, and the particulates immersed in the acoustic momentum field experience the acoustic radiation force calculated as $F_{\text{acou}}=2\beta P_{\text{tran}}/v_s$, where $\beta$ denotes the photoacoustic conversion efficiency, $P_{\text{tran}}$ the transient optical power of the pulsed incident light, and $v_s$ the sound speed in elastic media (e.g., $v_s$ of bulk silicon is $\sim 2200\mathrm{m}/\mathrm{s}$)^[30,31]. Denoting the momentum exchange between the acoustic field and the particle, the acoustic radiation force is the acoustic counterpart of the optical force (i.e., electromagnetic radiation force) and possesses the same fundamental expression, which, owing to the moderate sound speed as opposed to that of light, is of considerably larger magnitude at the same input power. Typically, acoustic waves are excited by applying electric signals on piezoelectric substrates and have been utilized for the construction of acoustic tweezers and enabling acoustic sorting and assembly of particles^[3,32,33]. The fact that acoustic waves can also be excited via pulsed light illumination provides opportunity for optical manipulation extending to the solid domain^[34]. Specifically, apart from its large photoacoustic conversion efficiency, the photothermal process of pulsed light features enormous transient energy deposition (large $P_{\text{tran}}$). At a moderate pulse energy of $\sim \mathrm{nJ}$, it is capable of generating transient acceleration of adhering particulates sufficient in escaping from the adhesive force^[28,29].

Inspired by the working principles of machineries, a series of intriguing work has reported multi-degree-of-freedom locomotion of micrometer-sized actuators in dry adhesive environments based on an internal force-driven mechanism^[5,35–38]. Interestingly, instead of endeavoring to find the proper driving forces comparable to the $\sim \mathrm{\mu N}$-scale adhesive/friction forces, which is admittedly difficult when the force of interest belongs to the “external force,” researchers took a different approach by inducing the impulsive deformation of actuators via pulsed light illumination. The deformation, though also in the form of acoustic waves, functions effectively as the internal force in facilitating the locomotion on frictional substrates, in analogy to the operation of machineries or the crawling of earthworms via internal coordination rather than external assistances. Owing to its internal force nature, light-induced deformation effects may not be explicitly expressed as in the form of optical force, photophoretic force, or acoustic radiation force, which are all external forces (see Table 1 for comparison). Following the same mechanism, flexible polymers or hydrogels could be constructed into light-addressable soft robotics with delicate design and assembly, which take on more familiar modalities outside the framework of acoustic waves. Comparatively, their deformations are more profound and should be completed within considerably longer time windows in a semi-steady state, which are typically driven by modulated CW light.Table 1

Properties of Different Light-Induced Forces/Effects.

As the opening salvo of this review, the above discussion is aimed to introduce the topic of optical manipulation and provide some general ideas about its actual implementation in fluidic and solid domains from the perspective of different light-induced forces/effects (Table 1). A more comprehensive overview will be provided in the following content, which is also summarized in Fig. 1. The organization of this review is as follows. In Sec. 2, we introduce the physical mechanisms and theories of several light-induced forces involved in optical manipulation processes, including the optical force/torque and the thermophoretic force in fluidic environments and light-induced deformation effects in solid environments. Next, revolving around the fluid as the operational environment, we revisit representative optical manipulation techniques in Sec. 3 and categorize them by the locomotion degree of freedom. In Sec. 4, techniques adapted to the solid domain optical manipulation are presented, which are classified with respect to the working mechanisms. In Sec. 5, we selectively introduce several applications regarding historically important or emerging topics. Finally, we conclude the main contribution of this review, and envision future directions in the field of optical manipulation.

In Sec. 1, we introduce four types of light-induced forces (effects) that can be exploited in optical manipulation, namely, the optical force, photophoretic force, light-induced acoustic radiation force, and deformation effects, the latter two both originating from opto-thermo-mechanical coupling. For the sake of clarity, we denote that in what follows in this review, optical force (also optical torque) refers exclusively to the force (or torque) arising from momentum transfer between the light field and matter, i.e., electromagnetic radiation force (torque), not to be confused as the general term for all light-induced forces. Moreover, the photophoretic force is a sub-branch under a larger category termed “thermophoretic force,” describing the transmission of small particles in both air (i.e., photophoretic force) and liquid (i.e., Ludwig–Soret effects) media, the differentiation between which will be made clear in Sec. 2.2. Acoustic-wave-related forces, on the other hand, consist of both the external force, as in acoustic radiation force, which stems from acoustic waves excited in substrates, and internal force in the form of light-induced deformation effects (acoustic waves) in actuators.

In this section, physical mechanisms and theories are presented about three representative forces (effects). In Sec. 2.1, the origin and theoretical derivation of optical force and torque will be first provided (from Secs. 2.1.1–2.1.4), followed by a brief introduction of the measurement methods (Sec. 2.1.5) of optical force, which is of great practical significance in optical tweezer experiments. Section 2.2 is devoted to introducing the thermophoretic force in air (Sec. 2.2.1) and liquid suspensions (Sec. 2.2.2), which are associated with different interpretations and analytical treatments. Given that the acoustic radiation force is adequately illustrated in Sec. 1, Sec. 2.3 mainly focuses on the part of the internal force, that is, specifically, the light-induced deformation effects. Note that we have left out “force” in addressing these effects to avoid their being miscomprehended as external forces.

Due to the fundamental homogeneity and isotropy of space, closed physical systems carry two conserved quantities termed as linear and angular momenta^[39]. Unsurprisingly, light also has linear and angular momenta; thus, it could exert force and torque on physical objects via light–matter interactions, such as reflection, refraction, scattering, and absorbing processes. The earliest realization of the existence of the linear momentum of light can be tracked back to 1619, when Kepler speculated that the pressure of sunlight pushes comet tails away from the Sun. Two centuries later, after the establishment of his famous electromagnetic theory, Maxwell correctly calculated the pressure of solar radiation on the Earth’s surface^[40], the similar physics of which confirms Kepler’s speculation. By analog with a point particle in classical mechanics, it is straightforward to argue that a light beam with a linear momentum $\mathit{p}$ should carry an angular momentum $\mathit{r}\times \mathit{P}$, the so-called external angular momentum, which depends on the choice of the origin of the coordinate system. However, light is beyond this expectation. Besides the external angular momentum, light possibly carries intrinsic angular momentum independent of the choice of the coordinate system. In 1909, Poynting first pointed out that circularly polarized light of angular frequency $\omega$ carries angular momentum, the ratio between which and light energy is ${\sigma }_z/\omega$, where ${\sigma }_z=1$ or $-1$ for left- and right-circular polarization, respectively^[41]. This polarization associated angular momentum is nowadays called spin angular momentum (SAM) of light. In addition to SAM, light can also carry orbital angular momentum (OAM), the discovery of which was in 1992 by Allen et al., who recognized that a Laguerre–Gaussian mode with a twist phase surface $\exp (il\phi )$ has OAM equal to $(l/\omega )\times \mathrm{light}\mathrm{energy}$^[42].

At the fundamental micro-level, the force and the torque exerted by light can be computed by summarizing the Lorentz force ($\mathit{F}=q\mathit{E}+q\mathit{v}\times \mathit{B}$) on individual atoms without referring to the concept of light momentum. However, this approach is formally inconvenient because electromagnetic fields at atoms can be precisely obtained only by using microscopic electromagnetism, in which bound atomic charge densities and convective free-carrier currents are treated as elementary sources in free-space Maxwell equations. Instead, a convenient, widely accepted way is to apply macroscopic electromagnetism^[43], which neglects atomic features and considers average electromagnetic fields at scales well beyond atomic sizes. The linear and angular momenta of light are then prioritized as the basic concepts, the dynamic evolutions of which directly give the force and the torque just as in classical mechanics.

In macroscopic electromagnetism, the electromagnetic energy–momentum tensor, a $4\times 4$ matrix concerning the densities and fluxes of the electromagnetic energy and momentum, is a tool to characterize the energy–momentum dynamics and derive the optical force and torque. However, even though macroscopic electromagnetism is generally considered well developed, rival expressions of the electromagnetic energy–momentum tensor surprisingly exist, each supported with compelling evidences and arguments^[44]. Among the various expressions, two of the most famous are arguably the so-called Minkowski^[45] and Abraham tensors^[46], which were both introduced in the first decade of the 20th century. Specifically, the Minkowski tensor, denoted by ${\overline{\overline{T}}}_{\text{em}}^M$, is given by $${\overline{\overline{T}}}_{\text{em}}^M=\left(\begin{array}{cc}{u}_{\text{em}}& c{\mathit{g}}_{\text{em}}\\ c{\mathit{p}}_{\text{em}}^M& \mathit{T}\end{array}\right).$$(1)Here, $c$ is the velocity of light in vacuum; $u_{\text{em}}$ and ${\mathit{g}}_{\text{em}}{c}^2$ denote the electromagnetic (EM) energy density and energy flux density, respectively; ${\mathit{p}}_{\text{em}}^M$ and $\mathit{T}$ represent the linear momentum density and flux density tensor, respectively. Note that the negative of the momentum flux density tensor, $-\mathit{T}$, is also called the Maxwell stress tensor. In a lossless, non-dispersive and reciprocal medium, the expressions of $u_{\text{em}}$, ${\mathit{g}}_{\text{em}}$, ${\mathit{p}}_{\text{em}}^M$, and $\mathit{T}$ are given by $$u_{\text{em}}=\frac{1}{2}\mathit{E}·\mathit{D}+\frac{1}{2}\mathit{H}·\mathit{B},$$(2.a)$${\mathit{g}}_{\text{em}}=\frac{1}{c^2}\mathit{E}\times \mathit{H},$$(2.b)$${\mathit{p}}_{\text{em}}^M=\mathit{D}\times \mathit{B},$$(2.c)$$\mathit{T}=\frac{1}{2}(\mathit{D}·\mathit{E}+\mathit{B}·\mathit{H})\overline{\overline{I}}-\mathit{D}\otimes \mathit{E}-\mathit{B}\otimes \mathit{H},$$(2.d)where $\mathit{x}\otimes \mathit{y}$ denotes the outer product, and $\overline{\overline{I}}$ is the identity tensor. The momentum flux density tensor, $\mathit{T}$, is symmetric due to the reciprocity with $\mathit{\varepsilon }={\mathit{\varepsilon }}^{\mathrm{T}}$ and $\mathit{\mu }={\mathit{\mu }}^{\mathrm{T}}$ ($\mathit{\varepsilon }$ and $\mathit{\mu }$ are material permittivity and permeability tensors, respectively, and ${\mathit{A}}^{\mathrm{T}}$ the transpose operation applied to a tensor $\mathit{A}$). The energy–momentum tensor ${\overline{\overline{T}}}_{\text{em}}^M$ is, however, asymmetric since ${\mathit{g}}_{\text{em}}\ne {\mathit{p}}_{\text{em}}^M$ unless in free space. The asymmetry of the Minkowski tensor was criticized considerably, since it violates the conservation of angular momentum^[2]. To fix this issue, the Abraham tensor was proposed with the momentum density modified to $${\mathit{p}}_{\text{em}}^A=\frac{1}{c^2}\mathit{E}\times \mathit{H},$$(3)which equals ${\mathit{g}}_{\text{em}}$, thus recovering the symmetry of the energy–momentum tensor.

The contradiction between the Minkowski and Abraham energy–momentum tensors [or more precisely, the Minkowski and Abraham momentum densities, cf. Eqs. (2.c) and (3)] is confusing, since it leads to the indefiniteness of the light momentum that is, however, to have a unique expression due to its physical reality. To explicitly illustrate the predictive difference between the Minkowski and Abraham formulations, we consider that a light wave packet with volume $V$ and a linear momentum ${\mathit{P}}_0$ transmits from free space into a dielectric medium with refractive index $n$. Further, we assume that the dielectric medium is transparent, so that the reflection is negligible, and, thus, the power fluxes of the incident and transmitted light are equal. As a result, the Minkowski and Abraham momentum densities in the dielectric medium are ${\mathit{p}}_{\text{em}}^M=n^2{\mathit{p}}_0$ and ${\mathit{p}}_{\text{em}}^A={\mathit{p}}_0$, differently, where ${\mathit{p}}_0$ denotes the light momentum density in free space. By integrating the momentum density of the wave packet that occupies a reduced volume of $V/n$ in the dielectric medium, the linear momentum of transmitted light is ${\mathit{P}}^M=n{\mathit{P}}_0$ or ${\mathit{P}}^A={\mathit{P}}_0/n$. Therefore, the Minkowski expression predicts that, on entering a dielectric medium ($n>1$) from free space, light increases its momentum, while the Abraham one claims the opposite. With this dramatic contrast, one naturally expects that the correct formulation should be undoubtedly identified in principle by measuring the mechanical deformation of the dielectric interface induced by the recoiling force due to the increment or decrement of the light momentum. In 1973, Ashkin and Dziedzic performed such a measurement by sending a laser beam from air into water, and observed that the water surface rises^[47]. The observation at first sight seems to support Minkowski’s prediction. However, later serious calculations^[48], by taking into account the nonuniformity of laser illumination, revealed that it is the Abraham momentum acting in this type of experiment. Moreover, before Ashkin and Dziedzic’s experiment, Jones and Richards measured the light pressure exerted on a mirror suspended in water in the 1950s^[49,50]. They observed that light pressure linearly increases with the refractive index of water, thus supporting the Minkowski momentum. The experimental evidences did not settle the arguments^[51–53].

This Minkowski–Abraham dilemma has attracted considerable theoretical efforts since the late 1960s. The solution that emerged is a bit unexpected: both the Minkowski and Abraham energy–momentum tensors are physically “acceptable,” yet “flawed,” because they alone are incomplete in describing a closed light–matter system. A complete energy–momentum tensor should include both electromagnetic and material parts. Therefore, if appropriate material counterparts are supplemented, the summarized energy–momentum tensor should always be unique. However, as pointed out by Brevik in 1979 in his comprehensive paper, there unfortunately exists no unique solution partitioning the energy–momentum tensor into electromagnetic and material parts^[54], the reason behind which partly relates to the ambiguous definition of momentum densities. Notably, as derived by Barnett in 2010, the Abraham and Minkowski momentum densities [cf. Eqs. (2.c) and (3)] correspond to kinetic and canonical momentum densities, respectively^[55]. The difference between such two types of momentum densities can be understood by specifically considering a charged particle in electromagnetic fields. In this case, the kinetic momentum density of the particle is simply the product of the mass density and velocity, ${\rho }_m\mathit{v}$, which describes the motion of the particle. Differently, the canonical momentum density is ${\rho }_m\mathit{v}+{\rho }_c\mathit{A}$ (${\rho }_c$, particle charge density; $\mathit{A}$, electromagnetic vector potential), which is the conjugate variable of the position and is the translation generator in quantum theory. Similarly, light also has its kinetic and canonical momentum. In this sense, the Minkowski and Abraham momentum densities are both meaningful, but have different physical meanings. The preferred choice of one over the other is simply a matter of convenience for interpreting the physics without referring to the full light–matter expression. As argued by Barnett, in most optical experiments that mainly focus on measuring displacements of micro-objects in host fluidic media, the Minkowski form is preferable since the canonical momentum intimately relates to the translation operation^[55].

For the Minkowski tensor [Eq. (1)], we denote its accompanied material tensor by ${\overline{\overline{T}}}_{\text{mat}}^M$, and, thus, the complete light–mater tensor is $\overline{\overline{T}}={\overline{\overline{T}}}_{\text{em}}^M+{\overline{\overline{T}}}_{\text{mat}}^M$. ${\overline{\overline{T}}}_{\text{mat}}^M$ is material dependent. For a non-viscous, nondispersive, isotropic fluid, Mikura in 1976, derived that^[56]$${\overline{\overline{T}}}_{\text{mat}}^M=\left(\begin{array}{cc}{\rho }_0{c}^2+u_i& {\rho }_{0}c\mathit{v}\\ {\rho }_{0}c\mathit{v}-c\mathit{D}\times \mathit{B}+\frac{1}{c}\mathit{E}\times \mathit{H}& {\rho }_0\mathit{v}\otimes \mathit{v}+\phi \overline{\overline{I}}\end{array}\right),$$(4)where ${\rho }_0$ is the material density, $u_i$ is the internal energy of non-electromagnetic nature, and $\phi$ is fluidic pressure. With the unique light–matter tensor $\overline{\overline{T}}$, the energy and momentum conservation laws are given by ${\partial }_j{\overline{\overline{T}}}_{ij}=0$, where $i=0,1,2,3$ correspond to 4D space–time coordinates $\{ct,x,y,z\}$. Specifically, the momentum conservation law is expressed with $$\frac{\partial {\rho }_0\mathit{v}}{\partial t}+\nabla ·{\rho }_0\mathit{v}\otimes \mathit{v}=-\nabla \phi -\nabla ·\mathit{T}-\frac{1}{c^2}\frac{\partial \mathit{E}\times \mathit{H}}{\partial t}.$$(5)

The right-hand side of Eq. (5) gives the force density exerted on the fluid, the optical part of which is ${\mathit{f}}_{\text{em}}=-\nabla ·\mathit{T}-\frac{1}{c^2}\partial \mathit{E}\times \mathit{H}/\partial t$. Generally, optical manipulations with optical force are performed with continuous laser light, that is, electromagnetic fields have a harmonic time dependence $\exp (-i\omega t)$. In this case, the time-averaged optical force density, denoted by $\langle {\mathit{f}}_{\text{em}}\rangle$, is concerned, which is given by $$\langle {\mathit{f}}_{\text{em}}\rangle =-\nabla ·\langle \mathit{T}\rangle ,$$(6)with $\langle \mathit{T}\rangle =\frac{1}{2}\mathrm{Re}(\frac{1}{2}({\mathit{D}}^{*}·\mathit{E}+{\mathit{B}}^{*}·\mathit{H})\overline{\overline{I}}-{\mathit{D}}^{*}\otimes \mathit{E}-{\mathit{B}}^{*}\otimes \mathit{H})$. Accordingly, the time-averaged optical force exerted on a closed domain enclosed by the boundary ${\mathrm{\Omega }}_s$ is $$\langle {\mathit{F}}_{\text{em}}\rangle =-{\oint }_{{\mathrm{\Omega }}_s}\widehat{\mathit{n}}·\langle \mathit{T}\rangle \mathrm{d}s,$$(7)where $\widehat{\mathit{n}}$ denotes the outward unit normal vector on ${\mathrm{\Omega }}_s$. Note that in some literature, the optical force is expressed differently in terms of the Maxwell stress tensor, which is the negative of the momentum flux density, so that there exists a sign difference compared with Eq. (7).

The time-averaged optical force exerted on a solid particle embedded in a fluid, a prototype problem in optical manipulation, can be calculated by performing the surface integral of the momentum flux density with Eq. (7). Note that, even though this formulation is derived in the fluidic case, its application also extends to other materials as long as electrostriction and magnetostriction effects are negligible^[54]. Departing from Eq. (7), further analysis of the optical force can be conducted either through examining the linear momentum transfer of light and representing the optical force in terms of the scattering and absorption quantifiers (e.g., time-averaged scattering and absorption power), or by expressing the optical force in terms of the particle polarizabilities and the electromagnetic fields acting on the particle. The former approach adopts a light wave perspective, while the latter is more oriented to a particle perspective. They together provide complementary insights.

An elementary plane wave with wave vector $\mathit{k}$ carries a time-averaged momentum flux density equal to $\langle \mathit{T}\rangle =\frac{n}{c}\langle {\mathit{S}}_{\mathit{k}}\rangle \otimes \widehat{\mathit{k}}$, where $n$ is the refractive index of the background medium, $\langle {\mathit{S}}_{\mathit{k}}\rangle \equiv (1/2)\mathrm{Re}(\mathit{E}\times {\mathit{H}}^{*})$ is the time-averaged power flux density, and $\widehat{\mathit{k}}\equiv \mathit{k}/k$. A generalized incident light beam could be expressed with a linear superposition of plane waves of different $\mathit{k}$: $${\mathit{E}}_{\text{in}}(\mathit{r})=\sum _{j=1,2}\iint a_j(\mathit{k})\exp (i\mathit{k}·\mathit{r}){\mathit{e}}_{\mathit{k};j}\mathrm{d}\mathrm{\Omega },$$(8.a)where ${\mathit{e}}_{\mathit{k};1}$ and ${\mathit{e}}_{\mathit{k};2}$ are two orthogonal polarization vectors with ${\mathit{e}}_{\mathit{k};1}·{\mathit{e}}_{\mathit{k};2}=0$ and ${\mathit{e}}_{\mathit{k};j}·\mathit{k}=0$, $\mathrm{d}\mathrm{\Omega }\equiv \sin \theta \mathrm{d}\theta \mathrm{d}\phi$ with spherical angular coordinates $\theta \in [0,\pi ]$ and $\phi \in [0,2\pi ]$ that specify the plane wave vector $\mathit{k}=k\sin \theta \cos \phi \widehat{x}+k\sin \theta \sin \phi \widehat{y}+k\cos \theta \widehat{z}$, and $a_j$ is the amplitude of each plane wave component. Consider that a light beam interacts with a micro-object. Generally speaking, part of incident waves is absorbed and part is scattered. The scattered electric fields are expressed with $${\mathit{E}}_{\text{sca}}(\mathit{r})=\sum _{j=1,2}\iint b_j(\mathit{k})\exp (i\mathit{k}·\mathit{r}){\mathit{e}}_{\mathit{k};j}\mathrm{d}\mathrm{\Omega },$$(8.b)where $b_j$ denotes the amplitude of a scattered plane wave. Note that in Eq. (8.b), at positions far away from the object, e.g., at $z\to \infty$ and $z\to -\infty$, integration is restricted to angular ranges $\theta \in [0,\pi /2]$ and $\theta \in [\pi /2,\pi ]$ to respect the Sommerfeld outgoing-wave conditions. The total electric field is the summation of incident and scattered fields, ${\mathit{E}}_{\text{in}}$ and ${\mathit{E}}_{\text{sca}}$, respectively. Then, by integrating the momentum flux density on the boundary $z\to \pm \infty$ with Eq. (7), the time-averaged optical force is obtained with $$\begin{gathered} \langle {\mathit{F}}_{\text{em}}\rangle =\frac{n}{c}\iint \frac{S_{\mathrm{ext}}(\mathit{k})}{k^2}\widehat{\mathit{k}}\mathrm{d}\mathrm{\Omega }-\frac{n}{c}\iint \frac{S_{\mathrm{sca}}(\mathit{k})}{k^2}\widehat{\mathit{k}}\mathrm{d}\mathrm{\Omega } \\ \equiv \frac{n}{c}{P}_{\text{ext}}\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle -\frac{n}{c}{P}_{\text{sca}}\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle . \end{gathered}$$(9)

Here, $P_{\mathrm{sca}}\equiv \iint S_{\mathrm{sca}}(\mathit{k})/k^2\mathrm{d}\mathrm{\Omega }$ with $S_{\mathrm{sca}}(\mathit{k})=2{\pi }^2\sqrt{\varepsilon /\mu }·\sum _{j=1,2}|b_j(\mathit{k}){|}^2$ denotes the time-averaged optical scattering power, while $P_{\mathrm{ext}}\equiv \iint S_{\mathrm{ext}}(\mathit{k})/k^2\mathrm{d}\mathrm{\Omega }$ with $S_{\mathrm{ext}}(\mathit{k})=-4{\pi }^2\sqrt{\varepsilon /\mu }\sum _{j=1,2}\mathrm{Re}(a_j(\mathit{k})b_j(\mathit{k}))$ denotes the time-averaged extinction power, which sums up the scattering and absorption power. Also, $\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle$ and $\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle$ are defined as $\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle \equiv \iint S_{\mathrm{ext}}(\mathit{k})/(k^2{P}_{\text{ext}})\widehat{\mathit{k}}\mathrm{d}\mathrm{\Omega }$ and $\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle \equiv \iint S_{\mathrm{sca}}(\mathit{k})/(k^2{P}_{\text{sca}})\widehat{\mathit{k}}\mathrm{d}\mathrm{\Omega }$, respectively. The physical meaning of $\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle$ is straightforward from its definition, and its direction is interpreted as the average direction of the scattered waves. However, the physical meaning of $\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle$ seems to be ambiguous. Nevertheless, by referring to a specific case in which the incident wave has only a single plane wave component, i.e., $a_j(\mathit{k})\sim \delta (\mathit{k}-{\mathit{k}}_{\text{in}})$ (${\mathit{k}}_{\text{in}}$, the wave vector of the incident plane wave), we notice $\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle ={\mathit{k}}_{\text{in}}/|{\mathit{k}}_{\text{in}}|$. In this sense, we interpret $\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle$ as the average direction of the incident waves sensed by the micro-object. Note that in Ref. [4], an expression similar to Eq. (9) is presented, which, however, is limited to propagation-invariant incident waves (such as a single plane wave, and Bessel beams) and to the lossless particle, and Eq. (9) is free of these limitations.

Equation (9) well interprets the origin of the optical force from the perspective of light scattering and absorption. First, this equation is reformulated to $\langle {\mathit{F}}_{\text{em}}\rangle =\frac{n}{c}{P}_{\mathrm{abs}}\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle +\frac{n}{c}{P}_{\mathrm{sca}}(\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle -\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle )$, where $P_{\mathrm{abs}}\equiv P_{\mathrm{ext}}-P_{\mathrm{sca}}$ is the time-averaged absorption power. Then, we notice that the first term on the right-hand side of the above expression gives the optical force due to the absorption of the light beam with the linear momentum in the direction of $\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle$, while the second term describes the recoiling force due to the change of the direction of the light momentum. Thus, Eq. (9) well characterizes that the optical force is generated due to the linear momentum transfer from light to matter. Moreover, with Eq. (9), the order of the optical force can be immediately estimated. For instance, when $P_{\mathrm{ext}}$, $P_{\text{sca}}\sim \mathrm{mW}$, which is often the case in typical experiments of optical manipulation, we have $\langle {\mathit{F}}_{\text{em}}\rangle \sim \mathrm{pN}$.

When the incident wave is a single plane wave component with wave vector ${\mathit{k}}_{\text{in}}$, Eq. (9) is simplified to $\langle {\mathit{F}}_{\text{em}}\rangle =\frac{n}{c}{P}_{\text{ext}}{\widehat{\mathit{k}}}_{\text{in}}-\frac{n}{c}{P}_{\text{sca}}\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle$. Especially, when the target object happens to be a reflecting mirror with a planar surface and incident light is directed vertically to it, from a geometrical optics point of view, $\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle =-{\widehat{\mathit{k}}}_{\text{in}}$, and Eq. (9) will be reduced to the intuitive expression provided in Sec. 1 (Table 1). In a more general sense, for a passive particle, we have $P_{\text{ext}}\ge P_{\text{sca}}>0$, so that $\langle {\mathit{F}}_{\text{em}}\rangle ·{\widehat{\mathit{k}}}_{\text{in}}$ is positive, that is, the optical force exerted by a plane wave always pushes the particle in the same direction as its propagation. Therefore, to enable a pulling force on the particle, a non-plane wave, such as a Bessel beam as suggested by Chen et al.^[4], is indispensable.

The light wave perspective is advantageous in providing a clear physical picture of the optical force in terms of the linear momentum exchange through scattering and absorption processes. However, it does not bring concrete insights on how one could engineer light beam profiles or electromagnetic responses of micro-objects to generate a desired optical force. Moreover, conventional classifications of the optical force into scattering, gradient forces, and other types, do not follow from the light wave perspective.

Here lies the worth of the particle perspective, in which the electromagnetic responses of the micro-object are parameterized by a series of induced electric and magnetic multipoles. Particularly, for the object size much smaller than light wavelength, i.e., in the so-called Rayleigh-limit regime, it suffices to consider only the electric and magnetic dipole moments, denoted by $\mathit{p}$ and $\mathit{m}$, respectively, which relate to the incident electromagnetic fields by $\mathit{p}={\alpha }_e{\mathit{E}}_{\text{in}}$ and $\mathit{m}={\alpha }_m{\mathit{B}}_{\text{in}}$, where ${\alpha }_e$ and ${\alpha }_m$ are the electric and magnetic polarizabilities of the object, respectively. The scattered electromagnetic fields ${\mathit{E}}_{\text{sca}}$ can be expressed in terms of $\mathit{p}$ and $\mathit{m}$ by employing the Green’s function technique^[2]. Then, after knowing ${\mathit{E}}_{\text{sca}}$ in terms of $\mathit{p}$ and $\mathit{m}$, the optical force can be calculated with Eq. (9), and its expression is given by^[4,57]$$\langle {\mathit{F}}_{\text{em}}\rangle =\langle {\mathit{F}}_{\text{grad}}\rangle +\langle {\mathit{F}}_{\text{sca}}\rangle +\langle {\mathit{F}}_{\text{pm}}\rangle ,$$(10)with $$\langle {\mathit{F}}_{\text{grad}}\rangle =\frac{1}{4}\mathrm{Re}({\alpha }_e)\nabla {|{\mathit{E}}_{\text{in}}|}^2+\frac{1}{4}\mathrm{Re}({\alpha }_m)\nabla {|{\mathit{B}}_{\text{in}}|}^2,$$(11.a)$$\langle {\mathit{F}}_{\text{sca}}\rangle =-\frac{1}{2}\mathrm{Im}({\alpha }_e)\mathrm{Im}(\nabla {\mathit{E}}_{\text{in}}^{*}·{\mathit{E}}_{\text{in}})-\frac{1}{2}\mathrm{Im}({\alpha }_m)(\nabla {\mathit{B}}_{\text{in}}^{*}·{\mathit{B}}_{\text{in}}),$$(11.b)$$\langle {\mathit{F}}_{\text{pm}}\rangle =-\frac{k^4}{12\pi }\sqrt{\frac{\mu }{\varepsilon }}\mathrm{Re}({\alpha }_e{\alpha }_m{\mathit{E}}_{\text{in}}\times {\mathit{B}}_{\text{in}}^{*}).$$(11.c)

Here, $\langle {\mathit{F}}_{\text{grad}}\rangle$ is the gradient force, which describes a type of optical force that points towards the hotspot of the light beam. $\langle {\mathit{F}}_{\text{sca}}\rangle$ is usually called the scattering force in the existing literatures. This is because, under the incidence of a plane wave with wave vector ${\mathit{k}}_{\text{in}}$, $\langle {\mathit{F}}_{\text{grad}}\rangle =0$, while $\langle {\mathit{F}}_{\text{sca}}\rangle =\frac{1}{2}\mathrm{Im}({\alpha }_e){|{\mathit{E}}_{\text{in}}|}^2{\mathit{k}}_{\text{in}}+\frac{1}{2}\mathrm{Im}({\alpha }_m){|{\mathit{B}}_{\text{in}}|}^2{\mathit{k}}_{\text{in}}=\frac{n}{c}{P}_{\text{ext}}{\widehat{\mathit{k}}}_{\text{in}}$, which, besides the optical absorption, intimately relates to wave scattering. Therefore, in a rigorous sense, it is more proper to call $\langle {\mathit{F}}_{\text{sca}}\rangle$ the extinction force. Both $\langle {\mathit{F}}_{\text{grad}}\rangle$ and $\langle {\mathit{F}}_{\text{sca}}\rangle$ are contained in the first term on the right-hand side of Eq. (9), i.e., $\frac{n}{c}{P}_{\text{ext}}\langle {\widehat{\mathit{k}}}_{\text{in}}\rangle$. $\langle {\mathit{F}}_{\text{pm}}\rangle$ describes the optical force due to the joint contribution from electric and magnetic dipoles, corresponding to the second term in Eq. (9) that relates only to scattered fields, i.e., $-\frac{n}{c}{P}_{\text{sca}}\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle$. The non-vanishing of $\langle {\mathit{F}}_{\text{pm}}\rangle$ necessarily requires that both electric and magnetic dipoles exist. Otherwise, if only the electric or magnetic dipole exists, the scattered fields will distribute symmetrically around the axis of the dipole orientation, thus leading to $\langle {\widehat{\mathit{k}}}_{\text{sca}}\rangle =0$, which makes the second term in Eq. (9) vanish.

The optical gradient and scattering forces are widely explored for various types of optical manipulation, the detailed review of which is provided in Sec. 3. Specifically, the most straightforward way to generate the gradient force is to use a focused Gaussian beam, such that the force points towards the beam center. This gradient force makes it possible to trap the particle against Brownian motion, of which the “optical tweezer” is perhaps the most famous application. Nowadays, due to new insights emerging from nanophotonics, the use of a focused beam is no longer a mandatory condition for generation of the gradient force. The steep hotspots can be induced with an unfocused beam by unitizing plasmonic near-field effects and dielectric resonances. On the other hand, as long as a propagating beam is scattered or absorbed by a micro-object, the scattering force always exists, which can be used to drive the motion of the object. In this regard, to maximize the scattering force, the electric and magnetic resonances, manifesting in the spectral peaks of $\mathrm{Im}({\alpha }_e)$ and $\mathrm{Im}({\alpha }_m)$, can be utilized. Even though the gradient and scattering forces are derived in the Rayleigh-limit regime, their existence is independent of the particle size. Particularly, when the particle size is much larger than light wavelength, an intuitive way to understand the gradient and scattering forces is to use ray optics, where optical forces can be regarded as a recoiling force originating from the momentum direction change of the rays due to refraction, as shown in Fig. 2, which intentionally highlights the application of optical tweezers.

The transfer of the angular momentum of light generates an optical torque on the object. Under the Minkowski energy–momentum tensor, the angular momentum density and its accompanied flux density tensor are expressed by $${\mathit{j}}_{\text{em}}^M=\mathit{r}\times (\mathit{D}\times \mathit{B}),$$(12.a)$$\mathit{M}=-\mathit{T}\times \mathit{r}.$$(12.b)Here, ${\mathit{j}}_{\text{em}}^M$ directly follows from the definition of the angular momentum in classical mechanics, while $\mathit{M}$ is derived from the conservation law of the angular momentum^[58]. The torque density is given by $-\nabla ·\mathit{M}$. Then, considering that a light beam with an angular frequency of $\omega$ interacts with a micro-object, the induced time-averaged optical torque, denoted by $\langle {\mathit{\tau }}_{\text{em}}\rangle$, is given by $$\langle {\mathit{\tau }}_{\text{em}}\rangle =-{\oint }_{{\mathrm{\Omega }}_s}\widehat{\mathit{n}}·\langle \mathit{M}\rangle \mathrm{d}s,$$(13)similar to the formulation of the optical force in Eq. (7). Here, $\langle \mathit{M}\rangle =-\langle \mathit{T}\rangle \times \mathit{r}$ is the time average of $\mathit{M}$ with $\langle \mathit{T}\rangle$ given below Eq. (6).

The angular momentum of light includes OAM and SAM. In quantum mechanics, the angular momentum operators are generators of simple rotations, which here rotate both the amplitudes and the polarization orientations of electromagnetic fields. Intuitively, the rotation of the field amplitudes associates with the OAM, while the rotation of the polarization orientations relates to the SAM. Despite this clear physical picture, neither OAM nor SAM is true angular momentum as pointed out by van Enk and Nienhuis, since their respective rotation operators violate the transversality of electromagnetic waves^[59,60]. Nevertheless, both OAM and SAM are physically meaningful, as have been measured by a number of experiments.

The angular momentum density can be decomposed into OAM and SAM components. Specifically, for harmonic electromagnetic fields, the time-averaged OAM and SAM densities, denoted by $\langle {\mathit{l}}_{\text{em}}^M\rangle$ and $\langle {\mathit{s}}_{\text{em}}^M\rangle$, respectively, are given by^[61]$$\langle {\mathit{l}}_{\text{em}}^M\rangle =\frac{1}{4\omega }\mathrm{Im}(\varepsilon {\mathit{E}}^{*}·(\mathit{r}\times \nabla )\mathit{E}+\mu {\mathit{H}}^{*}·(\mathit{r}\times \nabla )\mathit{H}),$$(14.a)$$\langle {\mathit{s}}_{\text{em}}^M\rangle =\frac{1}{4\omega }\mathrm{Im}(\varepsilon {\mathit{E}}^{*}\times \mathit{E}+\mu {\mathit{H}}^{*}\times \mathit{H}).$$(14.b)

Using Maxwell’s equations, it can be directly checked that $\langle {\mathit{j}}_{\text{em}}^M\rangle =\langle {\mathit{l}}_{\text{em}}^M\rangle +\langle {\mathit{s}}_{\text{em}}^M\rangle$. Note that the term $\mathit{r}\times \nabla$ appearing in $\langle {\mathit{l}}_{\text{em}}^M\rangle$ resembles the OAM operator in quantum mechanics. Moreover, the non-vanishing of $\langle {\mathit{s}}_{\text{em}}^M\rangle$ raises specific requirements on light polarization; for instance, for linearly polarized light, $\langle {\mathit{s}}_{\text{em}}^M\rangle =0$. The similar separation of the angular momentum flux density tensor is mainly discussed in the context of paraxial optics or cylindrically symmetrical light beams^[58]. Later, by referring to a concrete light beam that carries both OAM and SAM, we will provide more discussion on the OAM and SAM flux densities.

Historically, the intensive study of the angular momentum of light appeared in the 1990s, when Allen explicitly showed that a Laguerre–Gaussian beam carries a well-defined angular momentum^[42], which could be decomposed into an OAM part associated with the azimuthal phase $\exp (il\phi )$ ($l$ is an integer) and an SAM part relating to light polarization. After Allen’s milestone discovery, researchers found that, besides the Laguerre–Gaussian beam, myriad beams, such as Bessel beams^[62], perfect vortex beams^[63], and higher-order Poincaré sphere beams^[64], all carry angular momentum.

To pedagogically elucidate Allen’s discovery, we here consider a light beam propagating in the $z$ direction, the electric fields of which are expressed in the cylindrical coordinate system $\{\rho ,\phi ,z\}$ by $$\mathit{E}=u(\rho ,z)({\alpha }_L{\mathit{c}}_L+{\alpha }_R{\mathit{c}}_R)\exp (il\phi )\exp (ikz).$$(15)

For simplicity of mathematical derivations, the $z$ component of $\mathit{E}$ is set to zero without loss of generality. The complex numbers ${\alpha }_{L,R}$ are normalized coefficients for left- and right-circularly polarized light with ${\mathit{c}}_L=\sqrt{\frac{1}{2}}(\widehat{x}+i\widehat{y})$ and ${\mathit{c}}_R=\sqrt{\frac{1}{2}}(\widehat{x}-i\widehat{y})$, respectively, and satisfy ${|{\alpha }_L|}^2+{|{\alpha }_R|}^2=1$. The overall polarization state of the light beam, quantified by ${\sigma }_z$, is defined by $${\sigma }_z\equiv {|{\alpha }_L|}^2-{|{\alpha }_R|}^2.$$(16)

Apparently, ${\sigma }_z\in [-1,1]$, ${\sigma }_z=1$ and ${\sigma }_z=-1$ correspond to left- and right-circular polarizations, respectively, while ${\sigma }_z=0$ characterizes the linear polarization. Since the considered light beam is cylindrically symmetric, only the $z$-component angular momentum exists, the evaluation of which involves the $z$-component magnetic field $H_z$, as indicated in Eq. (12.a). From Maxwell’s equations, we derive that $H_{\text{in};z}={\alpha }_L(-lu/\rho +\partial u/\partial \rho )\exp (il\phi +i\phi )\exp (ikz)/\omega \mu +{\alpha }_R(-lu/\rho -\partial u/\partial \rho )\exp (il\phi -i\phi )\exp (ikz)/\omega \mu$. Then, applying Eq. (13), the $z$ component of the time-averaged angular momentum density is obtained with $\langle j_{\text{em};z}^M\rangle =(1/2)\mathrm{Re}(\widehat{z}·(\mathit{r}\times ({\mathit{D}}_{\text{in}}^{*}\times {\mathit{B}}_{\text{in}})))=\frac{1}{2}\varepsilon ({|u|}^{2}l-{\sigma }_z/2\partial {|u|}^2/\partial \rho )/\omega$. Therefore, the time-averaged $z$-component angular momentum is derived to be $$\langle J_{\text{em};z}^M\rangle =(l+{\sigma }_z)\int \frac{\varepsilon {|{\mathit{E}}_{\text{in}}|}^2}{2\omega }{\mathrm{d}}^3\mathit{r}\equiv \langle S_{\text{em};z}^M\rangle +\langle L_{\text{em};z}^M\rangle ,$$(17.a)with $$\langle S_{\text{em};z}^M\rangle \equiv \frac{{\sigma }_z{U}_{\text{em}}}{\omega }\text{and}\langle L_{\text{em};z}^M\rangle \equiv \frac{l{U}_{\text{em}}}{\omega }.$$(17.b)

Here, $U_{\text{em}}\equiv \int \frac{\varepsilon |{\mathit{E}}_{\text{in}}{|}^2}{2}{\mathrm{d}}^3\mathit{r}$ is the total electromagnetic energy of the incident fields. $\langle S_{\text{em};z}^M\rangle$ and $\langle L_{\text{em};z}^M\rangle$ characterize the time-averaged $z$-component SAM and OAM, respectively, which can also be computed by performing volume integrations to the SAM and OAM densities expressed in Eqs. (14.a) and (14.b).

Concerning the angular momentum flux density of the cylindrically symmetric beam as discussed above, Barnett derived the expressions of the $zz$-component flux density that explicitly separate the OAM and SAM contributions^[58]: $$\langle M_{zz}^L\rangle =\frac{1}{4\omega }\mathrm{Im}(-\frac{\partial E_y}{\partial \phi }{H}_x^{*}+E_y\frac{\partial H_x^{*}}{\partial \phi }+\frac{\partial E_x}{\partial \phi }{H}_y^{*}-E_x\frac{\partial H_y^{*}}{\partial \phi }),$$(18.a)$$\langle M_{zz}^S\rangle =\frac{1}{2\omega }\mathrm{Im}(E_x{H}_x^{*}+E_y{H}_y^{*}),$$(18.b)where $\langle M_{zz}^L\rangle$ and $\langle M_{zz}^S\rangle$ denote the time-averaged OAM and SAM flux densities through the x−y plane in the $z$ direction, respectively. The time-averaged OAM and SAM fluxes through the $x–y$ plane, denoted by $\langle {\mathcal{M}}^L\rangle$ and $\langle {\mathcal{M}}^S\rangle$, respectively, are contributed only from $\langle M_{zz}^L\rangle$ and $\langle M_{zz}^S\rangle$, and their expressions are $$\langle {\mathcal{M}}^L\rangle =\iint \rho \mathrm{d}\rho \mathrm{d}\phi \langle M_{zz}^L\rangle \widehat{z}\widehat{z}=\langle \mathit{P}\rangle \frac{l}{\omega }\widehat{z},$$(19.a)$$\langle {\mathcal{M}}^S\rangle =\iint \rho \mathrm{d}\rho \mathrm{d}\phi \langle M_{zz}^S\rangle \widehat{z}=\langle \mathit{P}\rangle \frac{{\sigma }_z}{\omega }\widehat{z},$$(19.b)where $\langle \mathit{P}\rangle =\frac{1}{2}\iint \rho \mathrm{d}\rho \mathrm{d}\phi \mathrm{Re}(\mathit{E}\times {\mathit{H}}^{*})$ denotes the time-averaged power flux through the $x–y$ plane, which for the light beam expressed in Eq. (16) is along the $\widehat{z}$ direction.

Apparently, if the light beam is absorbed by a micro-object, part of its OAM and SAM can be transferred to the object. The generated optical torque can then be calculated by integrating the flux density over a closed surface enclosing the object (e.g., $z=\pm \infty$) with Eq. (13). In this way, there is $\langle {\mathit{\tau }}_{\text{em}}\rangle =\langle P_{\text{abs}}\rangle (l+{\sigma }_z)/\omega \widehat{z}$, where $\langle P_{\text{abs}}\rangle$ denotes the time-averaged optical power absorbed by the object, which will induce the rotation of the micro-object. It is interesting to point out that absorption also leads to an optical force ${\mathit{F}}_{\text{em}}=\frac{n\langle P_{\text{abs}}\rangle }{c}\frac{\mathit{k}}{k}$, as suggested by Eq. (9). As a result, in this case, the magnitude of ${\mathit{\tau }}_{\text{em}}$ is related to that of ${\mathit{F}}_{\text{em}}$ by a simple relationship: $$\frac{|{\mathit{\tau }}_{\text{em}}|}{|{\mathit{F}}_{\text{em}|}}=\frac{\lambda (l+{\sigma }_z)}{2\pi },$$(20)where $\lambda =\frac{2\pi c}{n\omega }$ is the wavelength of the background medium. As discussed above, $|{\mathit{F}}_{\text{em}}|$ is typically of the order of pN when $P_{\text{abs}}\sim \mathrm{mW}$. Besides assuming that the light wavelength is of the order of µm and the quantum number of the exchanged angular momentum $l+{\sigma }_z$ is of the order of one, we estimate that the magnitude of $|{\mathit{\tau }}_{\text{em}}|$ is of the order of $\mathrm{pN}\times \mathrm{\mu m}$. In Eq. (20), the explicit value of the quantum number of the exchanged angular momentum is unknown, and it assumes that the momentum transfer is through absorption processes. It is proposed mainly to gain a concrete appreciation of the order of the magnitude of the optical torque in certain physical scenarios. Consequently, to precisely estimate the optical torque, electromagnetic simulations should be indispensable.

If the object is transparent and birefringent and does not alter the wavefront of the light beam, only the SAM can be transferred to the object. More precisely, denoting that the time-averaged optical power associated with the polarization change due to the birefringent effects by $\langle P_{\text{conv}}\rangle$, the torque is calculated to be $\langle {\mathit{\tau }}_{\text{em}}\rangle =\langle P_{\text{conv}}\rangle \mathrm{\Delta }{\sigma }_z/\omega \widehat{z}$ from Eq. (13), where $\mathrm{\Delta }{\sigma }_z$ quantifies the polarization change of the light beam. This polarization-conversion-induced torque was first demonstrated in 1936 by Beth who used a birefringent wave plate to enable conversion between left- and right-circularly polarized light, thus inducing the rotation of the wave plate^[65]. The usages of light beams with angular momentum and birefringent effects of target objects are not the only approaches to induce optical torque. Many other approaches exist, e.g., by taking advantage of anisotropic electric responses in non-spherical particles^[66,67], and special beam shapes^[68,69]. Their underlying physical mechanisms are very dispersive^[70], the review of which is discussed in Sec. 3 by referring to concrete experimental demonstrations.

Optical force can be measured or calibrated in optical tweezer systems. With directly accessible data being the recorded particle displacements, the force status of the trapped particle is obtained indirectly by correlating the two sets of data, namely, “x” and “F,” through the Langevin equation (Newtonian laws of motion) in fluidic environments. Hence, deduction of the optical force is a two-step process: (1) collecting the temporal trajectories of the trapped particle, and (2) converting “x” data to “F” data. The corresponding experimental setup and the data processing methods are introduced below. On the other hand, optical torques always associate with the change of light beam polarization, the measurement of which is relatively uncomplex compared to that of the optical force, and relevant content is covered in Sec. 3.3.1.

Typically, an optical tweezer is built on top of a commercial confocal microscope. At the heart of the trapping system is a high-NA objective lens that produces diffraction-limited focal spots. To avoid water absorptions, trapping lasers with wavelengths in the visible and near-infrared regimes are favored, which reduces the heating effects and mitigates photodamage to fragile biological samples. Figure 3 displays an instance of ordinary optical tweezer configurations. Using a beam expander, the trapping laser is expanded to either slightly underfill or overfill the objective, with the aim of optimizing the trapping efficiency by weighing the relative effects of the size of the laser focus against the light power truncation^[71]. The optical path of the trapping laser is then coordinated by two dichroic mirrors (${\mathrm{DM}}_1$ and ${\mathrm{DM}}_2$), while a white light source is aligned with it for sample illumination and direct observation through the charged couple device (CCD camera).

In situations that require high-precision measurements of the optical force and torque, or alternatively, when optical tweezers are calibrated for accurately sensing external stimuli, simply visualizing trapped particles using imaging devices is no longer effective. Instead, it is imperative to “track” the moving trajectories of the particles at high sampling rates, by which the trap stiffness can be deduced with particles’ Brownian motion as the reference signal^[72,73]. Video recordings could encode the temporal positions of the trapped particle. However, even for high-speed cameras, their sampling rate is ultimately limited by the exposure time and imaging processing technique, which compromises the measurement accuracy considerably^[72]. In this context, quadrant photodetectors (QPDs), bearing the advantage of high-bandwidth recording, enter the picture, and they have now been widely adopted in state-of-the-art research for position detection.

Specifically, the QPD should be placed at the conjugate plane to the back focal plane of the condenser, where the collected light pattern unveils the interference between incident light and scattered light from the sample (see Fig. 3). Each of the four quadrants of the QPD produces a voltage signal, denoted by $A$, $B$, $C$, $D$, and the lateral displacement $(X,Y)$ of a spherical object (the origin is chosen to coincide with the trap center) can be calculated from the normalized differential outputs as^[10]$$X=\frac{(A+D)-(B+C)}{A+B+C+D},$$(21.a)$$Y=\frac{(A+B)-(C+D)}{A+B+C+D}.$$(21.b)

The Langevin equation describes the stochastic behaviors in fluidic suspensions, and it bridges the two sets of information (x and F) with the following formula^[10,74]: $$m\ddot{x}+\beta \dot{x}+kx=\sqrt{2D}\beta \xi (t).$$(22)

In the 1D Langevin equation (readily extended to other dimensions), the left three terms denote the inertial force, friction force, and optical restoring force in the optical trap, respectively, and on the right-hand side is the contribution from random thermal fluctuations that arise from particle colliding with the surrounding fluids. Note that for spherical particles, the Stokes drag coefficient is a known quantity with $\beta =6\pi \eta a$ ($\eta$ is the viscosity of the fluid and $a$ the radius of the particle), and so is the Brownian diffusion constant derived from the Stokes–Einstein relation $D=k_{B}T/\beta$, where $k_{B}T$ stands for the product of the Boltzmann constant and the absolute temperature. Moreover, to account for the Gaussian randomness of the collision events, the function $\xi (t)$ is determined as $$\langle \xi (t)\rangle =0;\langle \xi (t)\xi (t^{\prime })\rangle =\delta (t-t^{\prime }).$$

In an overdamped system, as is always the case in fluidic environments (except in high vacuum), the inertia term is dwarfed by both the viscous drag and the optical force, and therefore can be dropped for simplicity. Consequently, taking the Fourier transformation of the times series ${|x(t)|}^2$, the as-obtained power spectrum $S_{xx}(f)$ would possess the feature that its expectation value corresponds to a Lorentzian^[73,75]: $$\langle S_{xx}(f)\rangle =\frac{1}{2{\pi }^2}\frac{D}{f^2+f_c^2}.$$(23)

In the above formula, $f_c=k/2\pi \beta$ is the cutoff frequency of the damped oscillator, from which the optical force, or rather, the trap stiffness, can be determined by optimizing the fitting parameters of the Lorentzian. Apparently, faithfully recording the power spectrum is key to precise measurements, which is possible only when the bandwidth of the detector is adequate to avoid aliasing and the loss of high-frequency signals.

Another method for optical force deduction is based on the energy equipartition theorem, which statistically relates the thermal fluctuation to the averaged energy as $$\frac{1}{2}k\langle x^2\rangle =\frac{1}{2}{k}_{B}T.$$(24)

The equipartition theorem applies to all three dimensions. For 3D optical trapping using a single Gaussian beam, the lateral and axial trapping stiffness will differ, which leads to unequal mean square displacement (different level of Brownian diffusion) of the trapped particle along the two directions. Though not explicitly dependent on the viscous drag, this method is intrinsically relevant to the former approach in that the mean square displacement of the particle $\langle x^2\rangle$ corresponds to the integral of the power spectrum $S_{xx}(f)$^[72]. In addition to the requirements regarding the bandwidth of position detection, extra care should be taken to calibrate the origin of particle displacement (trap center).

Thermophoresis is particle motion in fluidic suspensions driven by the temperature gradient, and the ultimate energy source, as far as this section is concerned, is from light absorption. Intrinsically, thermophoresis in air and liquid environments has the same mechanism, that is, to be specific, modification of the particle–medium interface by the spatially varying temperature field^[76]. Nonetheless, for historical reasons, scientists took different routes in pursuit of proper descriptions of thermophoretic forces involved in the two fluidic environments, i.e., the photophoretic force in air and the Soret effects in liquids, with the kinetics model specialized for the former and the hydrodynamic treatment favored for the latter^[23], which we will address concretely and separately below.

Photophoretic force originates from nonuniform absorption and the thermal process of particles suspended in gaseous environments (aerosols), which can push, pull, or drive complex motions of light-absorbing particles, depending on specific physical conditions. Different from the optical force, photophoretic force is based on the momentum transfer between gas molecules and the target particle, where light functions as the energy pump instead of the momentum carrier. Specifically, when an absorbing particle is subject to light irradiation, the scheme of momentum transfer occurs via nonelastic collisions of gas molecules that are unbalanced between hot and cool sides^[77].

Obeying Maxwell’s law of velocity distribution, gas molecules bounced off the particle acquire thermal energy related to a statistical mean velocity of $\overline{v}\propto \sqrt{T_r}$, where $T_r$ denotes the temperature of the leaving molecules after collision, as opposed to the initial temperature of the gas surroundings $T_0$ and the surface temperature of the particle $T_s$ ($T_s>T_0$ in this discussion)^[78]. Note that $T_r$ does not necessarily coincide with $T_s$. Indeed, particles differ in their capabilities of endowing colliding gas molecules with thermal energy, which is quantified by the thermal accommodation coefficient, calculated as^[79]$$\alpha =\frac{T_r-T_0}{T_s-T_0}$$(25)[see Fig. 4(c)]. When $\alpha =0$, $T_r=T_0$, gas molecules gain no additional thermal energy from the particle, while when $\alpha =1$, $T_r=T_s$, full accommodation occurs, and gas molecules grab the largest share of thermal energy possible, which is manifested in their collective bounce-off velocities^[78]. The accommodation coefficient is affected by both the composition and surface topology of particles. For instance, glazed platinum has an $\alpha$ of 0.315, whereas platinum black with a very structured surface has an $\alpha$ of 0.72^[80]. This phenomenon can be understood by the multiple reflections of the molecules in platinum black, which have a greater chance to pick up the surface temperature.

In the spirit of momentum conservation, the particle receives a recoil kick from each individual gas molecule through nonelastic collisions, or rather, a local pressure is exerted on the particle pointing opposite to the molecule’s bounce-off direction. The net impulsive force is then obtained by integrating the local pressure over the entire surface, which would be nullified unless asymmetrically heated, with the particle featuring two separate “hot” and “cold” regions, or alternatively, the particle is intrinsically nonuniform in its thermal accommodation coefficient $\alpha$^[19,81]. In the two schemes that result in either $\mathrm{\Delta }T$- or $\mathrm{\Delta }\alpha$-force, gas molecules favorably acquire more thermal energy (higher $\overline{v}\propto \sqrt{T_r}$) at the side of the particle featuring higher temperature or larger accommodation coefficient. Though intertwined in experiments, $\mathrm{\Delta }T$ and $\mathrm{\Delta }\alpha$ forces are often treated independently for numerical and theoretical clarity.

Assuming a homogenous distribution of the thermal accommodation coefficient across the particle, the photophoretic motion of the particle is purely driven by the $\mathrm{\Delta }T$-force. Given that the $\mathrm{\Delta }T$-force is directed from the hot to cool side, in the geometric-optics regime, it generally acts as the repulsive force for strongly absorptive particles, where the optical near side (illuminated side) is the thermally hot side, and reversely, for weakly absorptive particles, as the attraction force towards the light source, where the optical rear side (unilluminated side) generates more heat due to the focusing effect of the particles’ convex surfaces or when they serve as ball lenses [see Figs. 4(a) and 4(b)]^{[19,22,82,83]}. For particles in the Mie or Rayleigh regime, situations become more complex in that multipole interference and resonances might be involved and cause significant directionality of the light field not necessarily aligned with light propagation. More rigorous analyses of wave optics are needed for these situations.

Quantitatively, the expression of the $\mathrm{\Delta }T$-photophoretic force differs for different values of the Knudsen number, which is the ratio of the mean free path of the surrounding gas molecules to the radius of the target particle, $K_n=l/R$. In the continuum and slip flow regimes of fluid mechanics, that is, $K_n\ll 1$, Yalamov et al. constructed a theoretical framework that coupled the electromagnetic field, heat transfer, and hydrodynamics governing equations and have deduced the $\mathrm{\Delta }T$-force as^[77,84]$$F_{\text{ph}}=\frac{-4\pi RK{\mu }_m^{2}I{J}_1}{{\rho }_m{T}_m{k}_p},$$(26)where $K$ denotes the thermal slip factor, $I$ the radiation intensity, $k_p$ the particle thermal conductivity, ${\mu }_m$, ${\rho }_m$, $T_m$ the viscosity, mass density, and temperature of the surrounding media, respectively, and most importantly, $J_1$ as the symmetrical factor that accounts for the electromagnetic field distribution in the target particle, in that the positive value of $J_1$ signifies photophoretic pulling, and the negative value suggests pushing. Note that $J_1$ is a highly packaged item depending on myriad factors, the determination of which requires meticulously solving the electromagnetic equations with assigned boundary conditions, material and geometric parameters, etc., and could be intricate when dealing with Mie or Rayleigh particles, as stated before. Being case dependent, the $J_1$ factor may alter considerably with a slight change of parameters, the engineering of which adds to the degree of freedom of optical manipulation in determining the direction of the $\mathrm{\Delta }T$-force (detailed discussion provided in Secs. 3.1.5 and 3.2.2).

Contrastingly, in the free molecule regime ($K_n\gg 1$), Hidy et al. obtained the expression of the $\mathrm{\Delta }T$-force in the case of a completely opaque sphere, and the resultant formula is more explicit in its physics in that the involved physical quantities carry more direct meanings as compared to Eq. (26), which is approximated as^[20]$$F_{\text{ph}}\cong \frac{-\pi R^{3}Ip}{6(\frac{1}{2}p{v}_{m}R+k_p{T}_m)},$$(27)where $R$ denotes the particle radius, $I$ the radiation intensity, $p$ the gas pressure, $v_m$ the average speed of the gas molecules, $k_p$ the particle thermal conductivity, and $T_m$ the ambient gas temperature. Note that Eq. (27) keeps only the linear term in the original solution, which is expressed by the Legendre polynomial. Despite this approximation, it is rather instructive in manifesting the evolution of the $\mathrm{\Delta }T$-force in relation to the dominant physical quantities such as the atmospheric pressure, ambient pressure, and thermal conductivity of the particle of interest. Notably, if $k_p=0$, Eq. (27) will be reduced to the intuitive expression in Sec. 1 (Table 1).

The nonuniformly distributed thermal accommodation coefficient leads to asymmetry in the momentum transfer between the particle and gas molecules, and hence would result in an unbalanced $\mathrm{\Delta }\alpha$-force, as shown in Fig. 4(c). The $\alpha$-variation inside a single particle might be caused by a difference in surface roughness or material composition, and the latter relates to the use of Janus particles for asymmetry-induced optical manipulation (see Secs. 3.2.2 and 3.3.2). Since a larger accommodation coefficient corresponds to a faster mean velocity of the reflected gas molecules, the law of momentum conservation determines that the $\mathrm{\Delta }\alpha$-force, as the recoil force, points in the descending direction of $\alpha$, which is directly relevant to the particle orientation while irrespective of (or less affected by) the illumination directions [see the lower panel in Fig. 4(c)]. Concrete deduction of the $\mathrm{\Delta }\alpha$-force is still lacking, whereas readers can refer to Refs. [19,85] for more insight.

Thermophoresis in liquid environments, or rather, the Soret effects, may not be easily embodied as a concrete “force,” since the kinetic theory loses its validity, and what proves convincing instead is the hydrodynamic picture. In general, scientists and researchers would preferably use the concept of “thermal diffusion” and “mass flux” to describe and quantify, respectively, the Soret effects in liquid suspensions^[23].

Specifically, a temperature gradient should exist before the Soret effects can take place. Different from photophoresis where this very thermal nonuniformity is assured by the particle absorption, in liquids, the long-range temperature field is more often established through substrate absorption (except for Janus particles), whereas the absorptivity of the particles is not quite relevant. For instance, even transparent particles can be manipulated via Soret effects, provided that the substrate is light-absorptive and the particles possess a nonzero Soret coefficient (which will be introduced below).

With the presence of a spatially varying temperature field, an extra drift of particles could occur on top of Brownian diffusion, typically from the hot to cold side, and the total mass flow can be expressed as^[23,86]$$\mathit{j}=-c{D}_T\nabla T-D\nabla c,$$(28)where $c$ is the concentration of the target objects (e.g., particles, molecules), $D_T$ is the thermophoretic mobility (or thermal diffusion coefficient, for historical reasons), $\nabla T$ is the temperature gradient, and $D$ is the Brownian diffusion coefficient. In particular, the first term represents the mass flow due to the “extra drift” (drift velocity $\mathit{u}=-D_T\nabla T$), which is proportional to the temperature gradient. In the steady state, with the net mass flow vanishing, a nonuniform concentration profile would be established, written as $$\nabla c=-c\frac{D_T}{D}\nabla T=-c{S}_T\nabla T,$$(29)where the Soret coefficient $S_T$ is defined as $S_T=D_T/D$. The magnitude of $S_T$ measures the strength of thermophoretic flow relative to the tendency of random Brownian diffusion, and its sign specifies the direction of thermophoretic flow. For $S_T>0$, which is the most common situation, objects move from the hot to cold region and exhibit a thermophobic property, and vice versa for $S_T<0$. Note that the Soret coefficient $S_T$ is essentially related to the detailed configuration of the particle–medium interface. Hence, the thermal diffusive behavior of different particles would vary significantly, which can also be tuned by adding surfactants that effectively modify the interfacial property, or by adjusting the pH or setting up the temperature range ($S_T$ is a function of $T$) of the solvents^[87,88].

Recently, by utilizing the Soret effects in electrolytic solutions, researchers have introduced a real “force” upon suspended particles, which is based on the opto-thermally induced electric field^[87,89]. To go from the light signal to the electric field, light–thermal conversion is again the necessary intermediate to imprint the temperature field in liquid suspensions. Afterwards, the trick is to first decorate the target particle with charged micelles, and then trigger the spatial segregation between positive and negative ionic species, which is fostered by their difference in thermophoretic mobilities. Indeed, it is a sub-branch of the Seebeck effects in the liquid domain^[88]. More details on the opto-thermoelectric force are provided in Secs. 3.1.6 and 3.2.3.

The aforementioned optical and photophoretic forces are generally applied to manipulate micro-objects in low-adhesive fluidic environments, where adhesion/friction forces are tiny, typically of the order of pN or even smaller. Their use on dry solid interfaces is, however, expected to fail, because the adhesion/friction forces therein are way too strong, easily reaching $\sim \mathrm{\mu N}$^[90], which is several orders of magnitude greater than optical and photophoretic forces. Consequently, other physical mechanisms/effects are required to achieve optical manipulation on solid interfaces.

Among various proposals for optical manipulation on solid interactions, one group of explorations based on light-induced deformation effects attracts significant interest due to its rich physics at interfaces among nanophotonics, nanomaterials, and solid mechanics and to its promising technological applications. Roughly, light-induced deformation effects are a simplified term that covers a type of phenomenon—an object deforms its shape under the irradiance of light. They necessarily need to convert light energy into mechanical energy. The energy conversion could be mediated by thermal effects, which change the temperature of the object and then induce lattice oscillations (i.e., elastic waves). Alternatively, it can also take advantage of phase transition effects, so that the lattice reconfigurations generate strong stresses to enable shape deformation. During the shape deformation of the object, the adhesion/friction forces act as resistance, which, however, generally cannot overwhelm the deformation, as long as the latter is sufficiently intensive.

Light-induced deformation effects themselves do not automatically render the desired optical manipulation. The realizations of optical manipulation and motion control additionally demand elaborate structural designs and material choices. The research on this topic is diverse and multidisciplinary^[81]. Notable exemplifications include the use of a bimorph structure composed of two stacked thin films with large contrasting coefficients of thermal expansion, liquid crystal elastomers and networks, hydrogels, and so on, a detailed review of which is provided in Sec. 4.2. Even though these examples demonstrate versatile motions, such as vibrations, translations, and rotations, the target objects are largely limited to macroscopic dimensions ($\sim \mathrm{mm}$, cm).

Recently, a new solution to manipulate micro-objects on solid interfaces based on using elastic waves induced by pulsed light was reported. Compared with the conventional approaches that are similarly based on light-induced elastic deformations, the essential technical ingredient of this new solution lies in the use of nanosecond pulsed light rather than continuous light, thereby transforming the physical picture from “quasi-static” elastic deformation conceptually based on, e.g., thermal expansion/contraction, to the dynamic deformation picture that requires to take temporal elastic wave evolutions into account. Using this technique, researchers have successfully achieved nearly a full degree of freedom actuation of micro-sized objects on micro-fibers, as reviewed in Sec. 4.1.2. Below, we intentionally highlight the physical mechanisms behind this technique, since they have not been comprehensively reviewed before, and, moreover, the authors of this review have been intensively working on the technique for the last five years. Highlighting it here is a matter of personal taste, and we hope that readers will allow this choice.

To pedagogically clarify the principle of optical manipulation on solid interfaces driven by pulsed-light-induced elastic waves, we here refer to a concrete 2D physical model, as shown in Fig. 5. A thin microplate is placed on a substrate. The friction force, which is simplified to be a point force, is exerted on the plate when the latter moves on the substrate. Under irradiance of pulsed light, elastic waves are excited due to temperature rising via optical absorption. Note that in the existing literatures on this topic, some specific names for excited mechanical waves, such as acoustic waves, Lamb waves, and Rayleigh waves, can be found, which, nevertheless, can all be grouped under the same name—elastic waves—for clarification. Along with the excitations of elastic waves, the plate deforms its shape and induces the friction force. In such physical processes, to better guide our discussion, we raise two questions to the reader: (i) whether the plate could be driven in the sense that the whole plate translates a distance on the substrate, and (ii) if it could be driven, what are the essential physical mechanisms.

To approach the proposed questions, we first write the linear elastic equation that describes the elastic deformation of the plate due to the temperature variation $\delta T$ and the induced friction force $F_f\delta (\mathit{r}-{\mathit{r}}_0)\widehat{x}$ [see Fig. 5]: $$\begin{gathered} \nabla \times \nabla \times \mathit{u}(\mathit{r};t)-\frac{2(1-\sigma )}{1-2\sigma }\nabla \nabla ·\mathit{u}(\mathit{r};t)+\frac{2\rho (1+\sigma )}{E}\frac{{\partial }^2\mathit{u}(\mathit{r};t)}{\partial t^2} \\ =-{\alpha }_{\mathrm{th}}\frac{2(1+\sigma )}{1-2\sigma }\nabla \delta T(\mathit{r};t)+2\frac{(1+\sigma )}{E}{F}_f\delta (\mathit{r}-{\mathit{r}}_0)\widehat{x}, \end{gathered}$$(30)where $\mathit{u}$ denotes the displacement fields of the excited elastic waves, and $\sigma$, $E$, $\rho$, and ${\alpha }_{\mathrm{th}}$ denote Poisson’s ratio, Young’s modulus, mass density, and linear coefficient of thermal expansion of the plate, respectively. Then, to explicitly reveal the contribution from the friction force, we decompose $\mathit{u}$ into $\mathit{u}={\mathit{u}}^{\text{th}}+{\mathit{u}}^f$, where ${\mathit{u}}^{\text{th}}$ is the solution of Eq. (30) with $F_f=0$, i.e., quantifying the sole contribution from the temperature variation $\delta T$, and, thus, the friction contribution is left in ${\mathit{u}}^f$.

We first point out that, in the absence of the friction force, the excited elastic waves can deform only the shape of the plate, but are unable to drive the plate translation. This is because, without the friction force, the external force in the translation direction vanishes, thus leaving the center of mass of the plate unchanged, that is, the spatial average of ${\mathit{u}}^{\text{th}}$ is zero. Consequently, to enable the translation motion, an external friction force is indispensable. This deduction is not surprising, and, actually, the same principle is unconsciously exploited by ourselves every day when walking.

We analyze the friction-induced elastic displacement ${\mathit{u}}^f$ by expanding it with elastic modes. Assuming that the plate thickness is much smaller than the wavelength of the elastic waves, the fundamental longitudinal elastic modes are dominantly excited with wavenumber $k_L$ and velocity $v_L$ at angular frequency $\omega$ given by $$k_L=\omega \sqrt{\frac{\rho (1-{\sigma }^2)}{E}}\text{and}{v}_L=\sqrt{\frac{E}{\rho (1-{\sigma }^2)}}.$$(31)

The longitudinal modes have the displacement fields almost parallel to their propagation direction (i.e., $\widehat{x}$ direction) and are uniformly distributed over the plate thickness. The dominant $x$-component of ${\mathit{u}}^f$, denoted by $u_x^f$, is approximated by $$u_x^f(x,t)={\int }_{-\infty }^t{G}_{\mathrm{el}}(x,x_0;t-t^{\prime })F_f(t^{\prime })\mathrm{d}{t}^{\prime },$$(32.a)with $$G_{\mathrm{el}}(x,x_0;t-t^{\prime })\simeq \frac{t_{\mathit{R}}}{4M_p}\sum _{s=0}^3\sum _{n=0}^{\infty }H(t-t^{\prime }-t_s-n{t}_R).$$(32.b)Here, $t_{\mathit{R}}=2L/v_L$ is one-round-trip travel time for the elastic waves propagating in the plate with $L$ denoting the plate length in the $\widehat{x}$ direction; $t_s$ ($s=0,1,2,3$) denote the travel time for the elastic waves traveling from the friction point $x_0$ to the observation point four times within one round trip, with $t_0=|x-x_0|/v_{\mathit{L}}$, $t_1=t_{\mathit{R}}-|x-x_0|/v_{\mathit{L}}$, $t_2=|L-x-x_0|/v_{\mathit{L}}$ and $t_3=|L+x+x_0|/v_{\mathit{L}}$; $H(t)$ is the Heaviside step function with $H(t)=1$ for $t\ge 0$ and $H(t)=0$ otherwise; $n=0,1,2,3\cdots$ label the round trips of the elastic waves bouncing back and forth inside the plate; $M_p$ is the mass of the plate.

The friction force is the parallel component of the surface adhesive force. It includes the contribution mainly from van der Waals forces, and could also be affected by a variety of surface factors, such as roughness and possibly accumulated surface charges. Therefore, a precise estimation of the friction force from the first principles seems to be impractical. To bypass this difficulty, Tang and co-authors in Ref. [37] introduced a phenomenological model to determine $F_f$. In their model, the key quantity is the sliding resistance $F_f^s$, which is the maximum allowable static friction, and the dynamic friction force (when the plate is in motion) is set to equal $F_f^s$ for analysis simplicity. Their strategy to determine $F_f$ is based on an intuitive physical argument: $F_f$ is induced to mitigate the elastic deformation by letting the magnitude of the deformation velocity ($\partial u_x/\partial t$) at the friction point be as small as possible. Along this line, it is found that the friction force $F_f$ depends not only on ${\mathit{u}}^{\text{th}}$, but also on itself at the previous time. Here, for intuitive demonstrations of the essential physics, we sacrifice the exactness and neglect the less important latter dependence; more precisely, this approximation amounts to retaining the leading-order term ($s=0$ and $n=0$) in Eq. (32.b). As a result, the friction force is approximately given as follows: when the friction point is still, which requires $$|v_x^{\text{th}}|\le v^s,$$where $v_x^{\text{th}}=\partial u_x^{\text{th}}(x_0;t)/\partial t$ denotes the deformation velocity of the friction point due to the temperature variation $\delta T$, there is $$F_f(t)\simeq -\frac{4M_p{v}_x^{\text{th}}(t)}{t_{\mathit{R}}};$$(33.a)otherwise, when the friction is in motion, which occurs when $$|v_x^{\text{th}}|>v^s,$$there is $$F_f(t)\simeq -\mathrm{sgn}[v_x^{\text{th}}(t)]F_f^{\mathrm{s}},$$(33.b)where $\mathrm{sgn}[x]=1$ for $x>0$ and $\mathrm{sgn}[x]=-1$ for $x<0$. In Eqs. (33.a) and (33.b), the threshold velocity $v^s$—the relationship between it and $v_x^{\text{th}}$ defining the motion state of the friction point—is given by $$v^s=\frac{F_f^s{t}_R}{4M_p}.$$(33.c)

Summarizing $u_x^{\text{th}}$ and $u_x^f$, the $x$ component of the displacement fields of the friction point can be approximately formulated: $$u_x(x_0;t)={\int }_{-\infty }^t(v_x^{\text{th}}(t^{\prime })-\mathrm{sgn}[v_x^{\text{th}}(t^{\prime })]v^s)H(|v_x^{\text{th}}(t^{\prime })|-v^s)\mathrm{d}{t}^{\prime },$$(34)where $H(x)$ is the Heaviside step function defined below Eq. (32.b). We note that the $x$-component displacement of the friction point, $u_x(x_0;t)$, is generally different from the displacement of the center of the mass due to inhomogeneous deformation. Nevertheless, it still offers a proper reference to infer the motion state of the plate. This is similar to the deduction of someone’s movement by observing the positions of his/her feet. Moreover, as $t\to \infty$ with $\delta T\to 0$, the plate returns to its initial non-deformed shape, where the displacement of the friction point is the true displacement of the whole plate. Therefore, it is meaningful to use Eq. (34) to clarify the motion physics of the plate, which is summarized as follows.

First, as straightforwardly indicated by the presence of the Heaviside step function in Eq. (34), the friction point moves only in the period when the deformation velocity due to the temperature variation $\delta T$ is sufficiently large with $|v_x^{\text{th}}|>v^s$. In this regard, the friction force, which determines $v^s$ in Eq. (33.c), plays a negative role in preventing the friction point from moving. However, this picture is incomplete: if the friction force completely vanishes, we have $u_x$ approaching zero as $t\to \infty$ due to the absence of the external force. In this sense, the friction force is also indispensable for enabling the translation of the plate.

Second, Eq. (34) implicitly suggests the use of pulsed light to achieve a large displacement distance (as $t\to \infty$). To elucidate the benefits from the use of pulsed light, we recall that, under injection of a light pulse, the plate undergoes thermal heating and cooling phases successively, during which the temperature rises and falls, respectively. Specifically, the elastic deformations in the heating and cooling phases expand and contract the plate volume differently. As a result, in the corresponding two thermal phases, $v_x^{\text{th}}$ shows opposite signs, tending to cancel each other in Eq. (34) and reduce the displacement value. To realize a large displacement, it is thus favorable to enlarge the asymmetry in the heating and cooling time scales, so that the magnitude of $v_x^{\text{th}}$ in one thermal phase significantly exceeds the other. For instance, considering that the heating phase occurs much faster than the cooling phase with $t_{\text{heat}}\ll t_{\text{cool}}$, an ideal scenario is that $|v_x^{\text{th}}|>v^s$ in the heating phase, while $|v_x^{\text{th}}|<v^s$ in the cooling phase. In this way, the friction point moves only in the heating phase, while being kept still by the friction force in the cooling phase, which helps the accumulation of a large displacement. Apparently, this suggested fast heating and slow cooling can be conveniently implemented by using pulsed light with temporal duration much smaller than the cooling time.