2.1.

Optical Pulling by a Single-Structured Beam

Lee et al.²⁴ found a diffraction-free solution for the Helmholtz equation, i.e., the optical solenoid beam, in which the principal intensity peak spirals around the optical axis and its wavefront are characterized by an independent helical pitch, as shown in Fig. 2(a1). In this beam, the intensity gradient force traps a small object to the maximum intensity, whereas the scattering force can drive the object around the spiral, which is determined by the phase gradient force.⁴¹ When the wavefront’s pitch relative to propagation direction is tuned from forward to retrograde, OPF will be obtained. Finally, the combination of those two forces pulls the object to the source direction, as indicated by the experimental demonstration depicted in Fig. 2(a3). Another similar helical tractor beam was also reported by Carretero et al.⁴²

Fig. 2

OPF by structured light beams: (a) experimental demonstration of OPF by a solenoid beam:²⁴ (a1) the spiral intensity peak pattern in experiment, (a2) wave vector back down the spiral, and (a3) experimental measurement of the pushing and pulling trace. (b) Theoretical proposal of OPF achieved by the excitation of multipoles in the object,^27,40 and (c1) theoretical proposal of OPF by a Bessel beam with a cone angle of $\alpha$. (c2) and (c3) OPF changes with the relative permittivity and permeability of the object.²⁸

Another kind of structured beam for generating an OPF is the Bessel beam,^27,28,43 which is also diffraction-free within a limited region in propagation. Different from the solenoid beam, the intensity peak in this beam is along a straight line, which can be obtained using an axicon⁴⁴ or an optical metasurface.^45,46 When a properly designed particle is illuminated by such a beam, multipoles (not only the dipole) may be excited simultaneously, and the interferences between them can maximize the forward scattering while suppressing the backward scattering.^27,40 Finally, the net linear momentum propagation along the direction of propagation is increased, and in turn the object is recoiled toward the light source, as shown in Fig. 2(b). For some values of particle size, the pulling is stable for a transparent and lossy object, marked by the thick black curve in Fig. 2(b).

The pulling force can also be understood from the direction of the wavevector of Bessel beams.²⁸ A Bessel beam can be regarded as the superposition of a series of plane waves, in which the wavevectors lie on a cone with an apex angle $\alpha$ relative to the propagation direction (such as the $z$ axis), as shown in Fig. 2(c1). When scattered by a carefully designed object, the wavevector $\mathbf{k}$ may be realigned forcedly to the $z$ direction due to the scattering of the object. Since the amplitude of $\mathbf{k}$ is related to the linear momentum of photons, the momentum projection along the $z$ direction is enhanced, and the extra momentums are balanced by the backward force on the object, as shown in Figs. 2(c2) and 2(c3). What is more, the transverse stability is also guaranteed in this configuration, due to the restoring intensity gradient force provided by the Bessel beam.

One disadvantage of using a Bessel beam to generate OPF is the sensitivity to the object size and optical parameters of the object, perturbation of which may disturb or even destroy the optical traction completely. (Certainly, this feature is also quite useful, such as in particle sorting, since the pulling force is size and optical parameters dependent.^27,28) In order to overcome this shortcoming, Novitsky et al.⁴³ proposed universal criteria for the material-independent or size-independent OTB and found that the nonparaxial Bessel beam is an excellent candidate for this kind of robust tractor beam. Pfeiffer and Grbic⁴⁷ reported an interesting method to realize the needed Bessel beam using a practical metasurface. The designed silicon metasurface can convert the linearly polarized Gaussian beam into the superposition of transverse-electric and transverse-magnetic polarized Bessel beams, which can stably pull a polystyrene sphere within the diffraction-free range. Recently, the core–shell structure was proved to have potential power in the tailoring of light scattering,^48,49 and thus it is also used in the generation of enhanced OPF, even in the Rayleigh region using Bessel beams, which is also transversely stable.⁵⁰ Also a cylindrical shape of dielectric particles can effectively mitigate the nonparaxiality requirements to the Bessel beam.⁵¹ A more comprehensive analysis of stable pulling by a Bessel beam is provided in Ref. 52. Interestingly, except for cylindrical objects, other kinds of elongated objects, such as optically bound particles, can also be used to enhance the OPF.^53,54

2.2.

Optical Pulling by Interferences of Multiple Beams

Although a single-structured beam can act as a tractor beam, multiple beams cooperating together can make the OPF more flexible. In the supplementary materials of Ref. 27, Chen et al. theoretically proposed that two plane waves are possible to pull some small object. Later, Sukhov and Dogariu⁵⁵ theoretically proposed the general mechanism to realize optical pulling for arbitrary objects using multiple plane waves. The method is to launch a series of plane waves (such as 24 waves) propagating along a cone surface (with the apex angle of $\theta =84\mathrm{deg}$) with the same amplitude but optimized relative polarizations and phases, as shown in Fig. 3(a). In this scheme, all the plane wave components have the same longitudinal wavevector, thus form a diffraction-free beam suitable for long-range pulling. For the optimized incident waves, a pulling force of $-0.24\mathrm{pN}$ is obtained. Actually, this latter method based on multiple-plane wave interfaces is more powerful and could be optimized to get an almost arbitrary scattering pattern and various kinds of optical forces, such as the transverse optical force.

Fig. 3

OPF by the interference of multiple beams. (a) Using the interference of a series of plane waves,⁵⁵ and (b) using the interference of two Gaussian beams:⁵⁴ (b1) schematic illustration of the configuration, (b2) the s-polarization can get forward scattering enhancement, and thus a pulling force, and (b3) the p-polarization cannot. (c) OPF using the interference of two codirectional Bessel beams:⁵⁶ (c1) schematic illustration of the Bessel beam by an SLM and a lens, (c2) volumetric reconstruction of the Bessel beam, (c3) phase hologram encoding the optical conveyor, and (c4) volumetric reconstruction of the beam projected by the hologram in (c3).

While the multiple-plane wave interference method is powerful, handling so many waves is not easy in practice. Is it possible to get an OPF using fewer waves? Brzobohatý et al.⁵⁴ theoretically proposed and experimentally demonstrated the OPF on a polystyrene particle with a radius of 410 nm using two Gaussian beams, or using one Gaussian beam and a reflection mirror, as shown in Fig. 3(b). The key point in this geometry is the large angle between the two incident waves (the angle between wavevector ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ ranges from 152 deg to 180 deg), which makes the majority of the beam scatter in a forward direction for s-polarization. When the transverse forces are cancelled with each other, a final net force is left reverse to the direction of ${\mathbf{k}}_1+{\mathbf{k}}_2$, i.e., an OPF. For the p-polarization, however, the force is always pushing. Since the direction of the force is size-dependent and polarization-dependent, this method is also efficient in particle sorting and can be switched by polarization.

Another mechanism for OPF achieved by Ruffner et al.⁵⁴ seems more flexible, as shown in Fig. 3(c). The directions of the particle could be tuned actively and it behaves as an optical conveyor.^57,58 They launched two coherent Bessel beams along the same direction with slightly different longitudinal wavevectors and tunable relative phases, and an active tractor beam was obtained. Since the wavevectors of the two beams are different, a series of intensity peaks are obtained, and thus a particle could be trapped by the intensity gradient force. Then by tuning the relative phases $\varphi (t)$ of the two beams, the positions of the peaks could shift forward or backward, and the trapped object can be transported upstream or downstream. Actually, this principle has been investigated by Čižmár⁵⁹ for submicron particle organization and bidirectional shift. Basically, this method is a little different from those mentioned above, because the scattering is not the key issue in this case, but the shift of the trapping center.^57,60

3.1.

Optical Pulling Force by Optical Gain and Loss

Except for using one or more structured light beams, OPF is also possible to achieve using an object with proper optical features. The first one that comes to mind is the optical gain object.^25,61–65 For example, Mizrahi and Fainman²⁵ reported the idea of negative radiation pressure using gain media, such as slabs, spheres, and deep subwavelength structures, as shown in Fig. 4(a). The underlying physics is not difficult to understand. Since the object is with optical gain, the incident photon number (the total light momentum) may be amplified by the gain object when stimulated emission occurs. According to the principle of linear momentum conservation, the object will get a pulling force.

Fig. 4.

Theoretical proposals of OPF by objects with exotic optical parameters: (a) OPF on an object with optical gain,²⁵ (b) OPF on an extremely anisotropic lossy object,⁶⁶ and (c) OPF on a PT-bilayer object with loss and gain.⁶⁷

According to the analysis above, it can be understood easily that a lossy object is not likely to be pulled. However, Novitsky and Qiu⁶⁶ found that the pulling force is still possible in case of lossy object, as shown in Fig. 4(b). Using a metal-dielectric multilayer, a hyperbolic object with loss is fabricated. When illuminating a dipole sphere made of the hyperbolic material using a nonparaxial beam, OPF can be obtained when the loss is relatively small. More interestingly, Alaee et al.⁶⁷ recently reported the optical pulling on a parity-time-symmetric bilayer, which is the combination of gain and loss. The pushing or pulling is dependent on the direction of the light incident on this structure. Moreover, light can exert asymmetric pulling, pushing, or zero forces on parity-time-symmetric metasurfaces, which are composed of arrays of meta-atoms (coupled spheres), by balanced loss and gain constituents.⁶⁸

3.2.

Optical Pulling Force Related to Chirality

Another theoretical proposal to get an OPF is using the interaction of a chiral object with chiral light.^61,69–72 Comparing with the achiral medium, there is a chiral-dependent optical force, which provides extra freedom to realize the OPF, by coupling the linear momentum of a chiral object with the spin angular momentum of light.

The chirality-related OPF was first explored by Ding et al.⁶⁹ The chirality of light is defined by the handedness of the circular polarization. For the left- and right-circularly polarized beams, the chirality is opposite. The chirality object used here is made of a series of metallic spheres ($\epsilon =-5+0.13i$ of gold at wavelength $\lambda =337\mathrm{nm}$) arranged on a left-handed spiral, as shown in Fig. 5(a1). Using two incoherent plane waves with counter propagation, the force components unrelated to chirality are cancelled out. Results show that both the positive and negative spin angular density fluxes can generate a pulling force (dependent on the size of the particles), as shown in Fig. 5(a2).

Fig. 5

OPF related to chirality. (a) OPF on a chiral structure formed by metallic spheres aligned on a spiral line (black curve):⁶⁹ (a1) the schematic structure and (a2) optical force versus the diameter of the spheres. (b) OPF on a chiral slab with the assistance of reflection mirror:⁷⁰ (b1) the chiral slab is transparent for the incident handedness of light, but absorptive for the reflection beams; due to the way handedness is reversed by the mirror, the total force is pulling; (b2) when incident handedness is reversed, the slab is pushing forward.

Another novel scheme for OPF using chirality is proposed by Fernandes et al., as depicted in Fig. 5(b1). The most interesting thing found by the authors is that the optical force on the chiral object can be independent of its location relative to the mirror, and the basic reason is the chirality-dependent transmission of the chiral slab. Using some optimized metallic units, the authors proposed a chiral slab, as shown in Fig. 5(b1). For the incident chiral beam, the slab is transparent and no momentum is transferred from the light to the slab. When reflected by the mirror, the handedness of the reflection beam is reversed and absorbed by the slab. Thus the slab experiences a pulling force. On the other hand, if the handedness of the incident beam is reversed, the slab will be pushed forward, as shown in Fig. 5(b2).⁷⁰ Although light absorption plays a key role in this configuration, no photophoretic force appears here because it is analyzed in vacuum; the photophoretic force will be discussed in the following sections.

4.1.

Interfacial Tractor Beam

The background medium is extremely important in the interaction of light and matter,⁷³ which may also greatly modulate the scattering properties, and thus the optical force behavior. Basically, a structured background provides richer properties for light and matter interaction, which ensures more channels for the tuning of light momentum, including both the amplitude and directions.

The simplest form of structured background beyond a homogeneous one is an interface of two homogeneous backgrounds. Kajorndejnukul³⁸ experimentally demonstrated a scheme for realizing OPF on an air–water interface,⁷⁴ as shown in Fig. 6(a). In this scheme, the object floats on the air–water interface (half-immersed in water and half-immersed in air). The incident beam (with a wavelength of 532 nm, which is transparent for water, air, and the object) is launched from air and impinges on the object and then transmits into the water. According to the Minkowski formula, the momentum per photon is proportional to the refractive index of the background medium. This may result in an increase of the light linear momentum since the refractive index of water is 1.33 times larger than that of air. This idea is verified both by numerical simulation and experiments.^38,74,76 In this configuration, the structured background, i.e., the air–water interface, provides an extra channel to amplify the forward momentum of light, which is absent in a homogeneous background. Apart from obtaining the OPF, this scheme is also valuable for distinguishing the validation of different forms of stress tensor and force density and helps to illuminate the Abraham–Minkowski debate (the detailed analysis of this debate is out of the scope of our topic; readers may refer to papers by Pfeifer et al.,³⁹ Milonni and Boyd,⁷⁷ and Barnett⁷⁸).

Fig. 6

OPF realized on an interface. (a) Optical pulling on an air–water interface, which is realized by the linear momentum increase when the incident light is scattered from air to water through the object.³⁸ (b) Optical pulling on a plasmonic surface, which is realized by the directional excitation of the SPP on the air–silver interface.⁷⁵

Another interesting theoretical proposal on an interface is using the plasmonic interface by Petrov et al.,⁷⁵ as shown in Fig. 6(b). In this scheme, two key factors result in the extremely asymmetric excitation of the surface plasmon polariton (SPP) on the plasmonic interface. The first one is the excitation of a rotating dipole in the particle due to the interference of incident and reflected fields. The second one is the strongly asymmetric directional excitation of the SPP wave by the spin–orbital coupling of the rotating dipole and the SPP wave, which increases the linear momentum of the scattered wave along the interface direction. As a result, the object on the plasmonic interface is recoiled in a backward direction.

4.2.

Optical Pulling Force in Waveguide Channels

Another kind of structured background is the optical channels, which support various modes. Intaraprasonk and Fan⁷⁹ reported the pulling force in a ring-waveguide system, as shown in Fig. 7(a). In this scheme, the optical waveguide supports both the zeroth-order and first-order modes, and the zeroth mode has a larger effective forward momentum (i.e., the larger propagation constant) than the first mode. When the waveguide is excited using the first mode, and scattered resonantly by the ring resonator, part of the energy will transfer to the zeroth mode adiabatically without energy reflection, thus the momentum of guiding mode is increased. As a result, the object (i.e., the ring resonator) experiences an OPF. In this scheme, the transverse stability is also possible when the incident frequency is carefully tuned. Similarly, the pulling force can also be obtained via a multimode fiber and particle system.⁸²

Fig. 7

OPF realized in waveguide channels: (a) optical resonant pulling of a ring resonator by a dual-mode optical waveguide,⁷⁹ (b) OPF in a hollow core photonic crystal waveguide,⁸⁰ and (c) tunable optical pushing and pulling using a waveguide made of phase change material of ${\mathrm{Ge}}_{\mathrm{2}}{\mathrm{Sb}}_{\mathrm{2}}{\mathrm{Te}}_{\mathrm{5}}$ (GST).⁸¹

Since the ring (as particles) is outside the multimode waveguide (fiber) and couples with the guiding mode through the evanescent wave only, the scattering efficiency is low. Due to this reason, another different configuration has been proposed by Zhu et al.,⁸⁰ where a hollow core photonic crystal (PC) waveguide is used, and the object is set inside the waveguide just at the intensity peak locations. The PC waveguide also supports both the zeroth and first modes. When the first-order mode is launched, part of it is scattered into the forward zeroth-order mode (the reflections can be neglected by the optimization of the objects). Since the effective linear momentum of the guiding mode is increased, the object is recoiled naturally by the conservation law of linear momentum. Actually, OPF is generic in a large class of systems where more than one mode with different momentum densities exists, even in the scattering of heavy baryons into light leptons on cosmic strings.⁸³

Except for the optical guiding mode, it is also possible to achieve the OPF using the mode near the cutting frequency.^81,84,85 For an optical waveguide, it is known that the mode will be cut when the frequency is less than some critical value. For those frequencies below the cutting point, the launched source cannot propagate but decays exponentially along the waveguide. This decaying feature generates an intensity gradient force toward the source direction. Thus an object will be pulled in such a mode. On the other hand, when the frequency is tuned slightly, the mode can be switched between guiding and decaying, and the force can be switched accordingly between pushing and pulling.

More interestingly, Zhang et al.⁸¹ proposed a new method to dynamically tune the direction of the optical force at the same incident frequency, as shown in Fig. 7(c). The key point is that the waveguide is made of a phase change material of ${\mathrm{Ge}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_5$ (GST), which can change between the amorphous phase and crystalline phase.⁸¹ When the waveguide is in the amorphous phase, a guiding mode with negligible decaying is obtained, which pushes the object along the waveguide. When the waveguide is in the crystalline phase (illuminated by a pump light), the loss of GST increases greatly, the guiding mode becomes exponentially decaying, and the intensity gradient force will pull the object to the source direction. Using a similar decaying feature, atoms around a hot object can be pulled due to the blackbody radiation.⁸⁶

4.3.

Optical Pulling Force in Waveguides with Negative Mode Index

Metamaterial with negative refractive index (NRI) is an interesting artificial optical material that has a refractive index of $<0$.⁸⁷ The most interesting feature in this kind of material is the antiparallel between the energy flow and the wavevector direction. Due to this property, it seems possible to generate OPF using NRI material. However, an ideal NRI metamaterial exists only in theory (at least at the present time), and it is also inconvenient to pull an object in a solid NRI background made of subwavelength units. In this circumstance, researchers^88–90 found that the effective mode indices of some wave guiding modes are negative and can be used to achieve OPF.

Salandrino and Christodoulides⁸⁸ proposed a method to get an effective NRI background using a $2\times 3$ dielectric waveguide array, as shown in Fig. 8(a). The waveguide array is translation invariant along the $z$ axis (out-of-plane-direction); the waveguide boundaries are shown by the six smaller squares. This waveguide array is made of germanium square rods in air and is designed to mimic the Clarricoats-Waldron waveguide.⁹¹ The refractive index of germanium is 4, and the side length and period are 600 and 850 nm, respectively. The effective mode index of about $-0.27$ could be achieved at the wavelength of $2\mu \mathrm{m}$. In this case, the object (not shown) is placed inside the empty region between the squares and can be pulled to the source direction (out of the plane direction) continuously.

Fig. 8

OPF in waveguide channels with effective negative mode index: (a) a square dielectric waveguide array, which mimics the Clarricoats-Waldron waveguide with negative mode index;⁸⁸ (b1) and (b2) a plasmonic film in vacuum, which supports backward wave and can resonantly pull a dielectric sphere above it with very high-momentum-to-force efficiency;⁸⁹ and (c) optical pulling in a biaxial slab layered structure.⁹⁰

In fact, the proposal reported in Ref. 88 is not easy to achieve in practice, because infinitely periodic waveguide arrays or a perfect electrical conductor outer boundary is needed. Is it possible to get an effective NRI mode in a more flexible structure? Using a plasmonic film embedded in air, the backward guiding mode with the phase velocity and energy velocity (the Poynting vector) antiparallel can be obtained at proper frequencies,^89,92,93 as shown in Fig. 8(b). For the scattering by a dielectric sphere of the backward wave, efficient momentum-to-force conversion appears, and pulling force reverse to the power flow is obtained.⁹⁴ The peaks in Fig. 8(b) denote the resonant whispering gallery mode of the spherical object. Also, an effective NRI can be acquired by two biaxial dielectric slabs with a hollow layer (for the setting of guiding particles) between them,⁹⁰ as shown in Fig. 8(c). It is noted that the effective negative refractive mode in a PC does not guarantee a pulling force, at least for a dipole object.⁹⁵

4.4.

Optical Pulling by the Self-Collimation Mode in Photonic Crystals

Most recently, another quite interesting scheme for achieving OPF is theoretically proposed in a periodic PC.⁹⁶ It is known that a PC can support different kinds of Bloch modes, which provide more possibilities for tailoring the interaction of light and matter.^97–99 The self-collimation (SC) mode is a unique Bloch mode that can propagate infinitely long without diffraction with a finite-transverse size, due to the coherent interaction of light with the periodic background.¹⁰⁰ When an object is embedded in an SC mode, a continuous and robust OPF may be exerted on it.

Fig. 9

OPF in a PC structure by the SC mode.⁹⁶ (a) Scattering of the SC mode by an embedded object. (b) Intensity profile along the beam symmetry axis; a negative intensity gradient across the object can be observed clearly, which is the physical origin for the OPF. (c) and (d) Intensity profile of the beam around the object at two different positions.

For an elongated object introduced into the SC beam, it scatters the SC mode adiabatically and forms a local intensity gradient on the object itself, as shown in Figs. 9(a) and 9(b). This means that the light intensities in the fore part and rear part of the elongated object are sharply different, which generates an obvious negative intensity gradient force and contributes to the pulling force greatly, as shown in Figs. 9(c) and 9(d). On the other hand, the scattering force component is extremely small in this case since the SC mode almost keeps its original shape when scattered by the object. Actually, the SC mode can recover itself within some distance after the scattering, which ensures the pulling ability of multiple objects.

Essentially, the intensity gradient pulling force here is the same as those in traditional tweezers when the object is behind the beam focus. However, the details and the final result are quite different. Here the intensity gradient field is generated by the scattering of the object itself and shifts with the object synchronously, which is the reason for the continuous pulling force over an infinitely long range. In optical tweezers, the intensity gradient is formed by an external focusing lens, which is independent of the object. Thus an object will be trapped near the focus and can only be pulled when the external focusing lens is shifted mechanically. In other words, for the pulling in this method, the object acts as the focusing lens as well as the target to be pulled. In this sense, the pulling force is self-induced, and the object is “pulled” by itself.