General theory of optical force for particles of arbitrary morphology

For an arbitrary particle, the optical force can be rigorously calculated using the formula ${\mathbf{F}}_{opt}={∯}_S{\stackrel{⃡}{\mathbf{T}}}^{\max }\cdot \widehat{n}dS$, where ${\stackrel{⃡}{\mathbf{T}}}^{\max }$ is the Maxwell stress tensor, S is any surface in the air that encloses the particle, and $\widehat{n}$ is the unit outward normal vector on S. Here, we consider S as the combination of S₁ and S₂, where S₂ is a planar surface immediately above the metamaterial substrate, and S₁ is a cap that encompasses the particle when combined with S₁, as shown in Fig. 1b. Given that the outward normal vector of S₂ points in the negative-z direction, the longitudinal optical force can be alternatively expressed as follows:

Given that, in air, the Maxwell stress tensor is the same as the Minkowski stress tensor ${\stackrel{⃡}{\mathbf{T}}}^{\min }$, we can substitute ${\stackrel{⃡}{\mathbf{T}}}^{\max }$ with ${\stackrel{⃡}{\mathbf{T}}}^{\min }$ in Eq. (2), and the last term of Eq. (2) can be rewritten as $-{\iint }_{S_2}{T}_{xz}^{\max }dS=-{\iint }_{S_2}{T}_{xz}^{\min }dS$, where $T_{xz}^{\min }=\widehat{x}\cdot {\stackrel{⃡}{\mathbf{T}}}^{\min }\cdot \widehat{z}=\frac{1}{2}\mathrm{Re}\left(E_x{D}_z^{*}+H_x{B}_z^{*}\right)$. Because $E_x,D_z,H_x,B_z$ are continuous across the interface, $T_{xz}^{\min }$ is also continuous, yielding the following equation:

where S₃ is immediately below the interface and mirrors S2 symmetrically; see Fig. 1b. In the case of a lossless homogeneous metamaterial, the integration of ${\stackrel{⃡}{\mathbf{T}}}^{\min }$ over any closed surface within the metamaterial is zero, for example, the closed surface formed by S₃+S_4, as shown in Fig. 1b. As a part of the closed surface S₃+S₄, the outward normal of S₃ points in the positive-z direction. Therefore,

Combining Eqs. (2)–(4), the longitudinal optical force can be calculated as follows

Here, S₁+S₄ form an arbitrary closed surface enclosing the particle. Thus, according to Eq. (5), the longitudinal optical force can be calculated by integrating the Minkowski stress tensor over any closed surface enclosing the particle.

Demonstrating the equivalence between Eqs. (2) and (5) constitutes a crucial step. The deliberate selection of the closed surface for integration greatly simplifies the subsequent derivation. Notably, opting for a surface situated at a considerable distance from the particle enables us to circumvent the intricacies associated with the complex near field. Because the Minkowski stress tensor is connected to the canonical momentum flux density of light, Eq. (5) establishes a connection between the longitudinal optical force and canonical momentum. We emphasize that the same treatment can be applied to the lateral optical force $F_y$ but not to the transverse optical force $F_z$ because $T_{xy}^{\min }$ is generally discontinuous across the planar interface.

By rigorous derivation (see Supplementary Note ¹ for details), Eq. (5) can be further transformed into the following:

where $W_{ext}$ represents the extinction rate of the particle in the system; $W_{sca}^{\left(s\right)}$, $W_{sca}^{\left(f,a\right)}$, and $W_{sca}^{\left(f,m\right)}$ represent the scattering rates to the SWs, the FPWs in air, and the FPWs in the metamaterial, respectively; cos ϕ, cos ₁, and cos ₂ represent the corresponding averaged cosines of the scattering angles with respect to the x direction; c represents the light speed in air; and $v_p$ and $v_f$ represent the phase speeds of the SW and FPW in the metamaterial, respectively. Equation (6) is a key result of our work. It can be regarded as an extension of the optical force by a plane wave in free space expressed as follows^1,58,59:

Equation (6) reduces to Eq. (7) when the incident beam is a plane wave and is only scattered into the FPWs within the air. Compared with Eq. (7), in the context of free space, the optical force described by Eq. (6) can be further adjusted by manipulating the phase speed, in addition to the scattering angle. This introduces a different approach to achieving optical pulling. Given that the same treatment can be applied to the lateral optical force, we can obtain a similar expression to Eq. (6) for $F_y$. This enables us to comprehend and explore the lateral optical force acting on the particle near an interface^60–64.

On the basis of the relationships among the canonical momenta, energy densities, and energy fluxes of SWs and FPWs, Eq. (6) also reveals that the longitudinal optical force is equivalent to the change in the total canonical momentum of light per unit time (see details in Supplementary Note ²). Like the optical force in free space^53,65, the first term in Eq. (6) corresponds to the momentum transfer resulting from directly capturing the incident photons, representing the incident force, whereas the remaining terms correspond to the recoil forces induced by reemitting the captured photons.

In the absence of gain, the extinction rate is given as $W_{ext}=W_{sca}^{\left(s\right)}+W_{sca}^{\left(f,a\right)}+W_{sca}^{\left(f,m\right)}+W_{abs}$, where $W_{abs}$ is the absorption rate of the particle in the system. Importantly, all these rates are nonnegative. When $k_p$ is negative, the phase speed of the SW, denoted as $v_p=\omega /k_p$, is also negative. Note that $\mid v_p\mid >\max \left\{c,\mid v_f\mid \right\}$ (refer to the confirmation in the following section), and the longitudinal optical force satisfies the following equation:

Accordingly, any SW with a negative $k_p$ will exert a pulling force on any passive particle, thus qualifying it as a general-purpose OSTB. The magnitude of the incident force is clearly always greater than the magnitude of the recoil force. Therefore, when $k_p<0$, the incident force is negative, ensuring that the total longitudinal optical force is pulling. This stands in stark contrast to traditional tractor beams, where optical pulling arises from a stronger recoil force. Similarly, the longitudinal optical force is always pushing when $k_p>0$, as indicated in Eq. (8).

Metamaterial supporting the OSTB

We demonstrate the feasibility of supporting the OSTB on the surface of a specific type of double-negative index metamaterial. We focus on the TE-polarized OSTB, which is characterized by an electric field polarized in the xoz plane. The corresponding TM-polarized OSTB can be achieved using a duality transformation $\epsilon \leftrightarrow \mu$.

For a TE-polarized surface mode supported on the interface between the metamaterial and air (the interface is at z=0), considering that the tangential component should be continuous, the magnetic fields at two domains are expressed as follows:

where $h_y$ is the complex amplitude of the magnetic field at the interface, and ${\kappa }_0$ and $\kappa$ are the decay constants in the air and metamaterial, respectively. In the lossless limit, the propagating wavenumber $k_p$ must be real, and the decay constants must be both real and positive. In the air and metamaterial,

where $\epsilon ,\mu$ are the relative permittivity and permeability of the metamaterial, respectively. According to Maxwell’s equation $\nabla \times \mathbf{H}=-i\omega \epsilon {\epsilon }_0\mathbf{E}$, the electric fields are expressed as follows:

The continuity of electric field along the x direction across the interface z=0 requires

Combining Eqs. (10) and (12), we obtain the following:

Here, we have discarded the solutions in which $\kappa ,{\kappa }_0$ are negative. Because both ${\kappa }_0$ and $\kappa$ are positive, $\epsilon$ must be negative, according to the second of Eq. (13). Since $k_p$ and $\kappa$ must be real, according to Eq. (13), one can obtain the following:

As previously discussed, the OSTB possesses a negative canonical momentum when it propagates forward. In the context of light propagation, the direction of the total energy flux serves as an indicator. Consequently, the OSTB can be regarded as an SW with a positive total energy flux but a negative propagating wave number $k_p$. Using the expressions of Eqs. (9) and (11), the total energy flux of the TE-polarized SW is given as follows:

Given that ${\kappa }_0$ is a positive real value, for $k_p<0$, the total energy flux is positive only when $\frac{{\epsilon }^2-1}{{\epsilon }^2}<0,$ which leads to the following:

Substituting Eq. (16) into Eq. (14), we easily obtain $\epsilon \mu >1$. Therefore, we can establish both the necessary and sufficient conditions for supporting the TE-polarized OSTB as follows:

Given that the OSTB is characterized by a negative $k_p$, our selection for the propagating wavenumber of the TE-polarized OSTB is solely the negative solution found in the first part of Eq. (13), resulting in the following:

Conversely, if the SW exhibits a positive canonical momentum, we opt for the positive solution. By duality, the conditions for supporting the TM polarized OSTB are expressed as follows:

The propagating wavenumber of a TM-polarized OSTB is expressed as follows:

Using Eq. (17) and Eq. (13), it is easy to show that $k_p^2=\frac{{\omega }^2}{v_p^2}>k_0^2,k_p^2>\epsilon \mu k_0^2=\frac{{\omega }^2}{v_f^2}$, confirming that $\mid v_p\mid >\max \left\{c,\mid v_f\mid \right\}$. As an example, Fig. 2a depicts the excitation of a TE-polarized SW with a negative $k_p$ on the surface of a metamaterial with ε=−0.9 and μ=−1.2. Applying the Fourier transform to the rightward-propagating SW yields the propagating wavenumber $k_p=-1.19k_0$, which is consistent with Eq. (18), as shown in Fig. 2b.

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Fig. 2

Excitation of a TE-polarized OSTB on a homogeneous metamaterial surface.

a OSTB and particle. The OSTB is excited by five line magnetic currents with a uniform amplitude of 1V and position-dependent phases of $e^{i{k}_p{x}_i}$, where $k_p=-1.19k_0$ and x_i are the coordinates marking a point on the line currents along the x direction, as illustrated in the inset. The particle, with a radius of r, is positioned above the surface at a distance of h (from the center of the particle to the interface). The line sources operate at the frequency f=100THz (the vacuum wavelength is 3μm). b Magnitude of the Fourier transform of the z-component electric field $E_z$ of the incident OSTB as a function of the wavenumber. The Fourier transform is calculated as follows: $E_z\left(k\right)=\frac{1}{x_d-x_s}{\int }_{x_s}^{x_d}{E}_z\left(x,z_0\right)e^{-ikx}dx$, where $z_0=0$ denotes the position of the interface and [$x_s,x_d$] represents the interval used for the Fourier transform. (Source data are provided as a source data file.).

Robust and loss-enhanced optical pulling using the OSTB

We now demonstrate the OSTB using numerical simulations. To simplify the computational process, let us consider an infinite cylinder oriented along the y direction as the particle of interest. The longitudinal forces acting on this circular cylinder with a dielectric constant ${\epsilon }_r$ and a radius r are plotted in Fig. 3a. As expected from Eq. (8), these forces are consistently negative. Figure 3b further illustrates that the forces remain negative for transparent and lossy cylinders with circular, triangular, or square cross-sections. Notably, traditional optical tractor beams face significant challenges in pulling lossy particles. However, in our approach, we successfully overcome this limitation, enabling unconditional pulling regardless of the particle size, composition, or shape. Given that the longitudinal optical force is solely due to the transfer of canonical momentum, for an OSTB carrying a negative canonical momentum, any absorption will indeed intensify the optical pulling force. On the other hand, as indicated by Eq. (8), loss actually enhances the optical pulling force by increasing the absorption rate, except near resonances where loss can reduce the extinction and scattering rates, as depicted in Fig. 3b.

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Fig. 3

Optical forces exerted by the OSTB.

a The longitudinal optical force, $F_x$, is plotted as a function of the dielectric constant, ${\epsilon }_r$, and the radius, r, of the particle. The optical force is expressed in units of μN/m. The distance of the particle is h=1μm. b The longitudinal optical force, $F_x$, is plotted against the real part of the dielectric constant, $\mathrm{Re}\left({\epsilon }_r\right)$, for particles of different geometries. The black dashed line corresponds to the dashed line in a (r=0.7μm, $\text{Im}\left({\epsilon }_r\right)=0$). The red solid line denotes the longitudinal optical force when the particle experiences losses, with $\text{Im}\left({\epsilon }_r\right)=0.5$. The blue dotted and magenta dash-dotted lines correspond to transparent ($\text{Im}\left({\epsilon }_r\right)=0$) particles with triangular (side length equals $\sqrt{3}r$) and square (side length equals $\sqrt{2}r$) shapes, respectively. c The transverse optical force, $F_z$, is plotted against the distance, h, of a circular particle with a radius of r=0.7μm when its dielectric constant is set to 3 (black), 9 (red), and 15 (blue). d The trajectory (black dashed line) of the particle ($r=0.7\mathrm{\mu }\mathrm{m},{\epsilon }_r=3$) near the metamaterial is calculated by solving the motion equation. The arrow indicates the direction of movement. (Source data are provided as a source data file.).

In Fig. 3c, we present the transverse optical force $F_z$ acting on the particle as a function of the height h (distance from the particle center to the interface) for different dielectric constants of the particle. The transverse optical force is governed primarily by the optical gradient force, which attracts the particle toward regions of high light intensity where the longitudinal pulling force is strong. Notably, for certain particles (e.g., r=0.7μm, ε_r=9, as indicated by the red line in Fig. 3c), equilibrium positions exist along the y direction. These particles are suspended above the interface, avoiding contact with the surface and associated friction forces. As a result, the particles are initially drawn toward the interface by the gradient force and then pulled toward the light source, as observed in the particle trajectory shown in Fig. 3d and Supplementary Note ³. This feature provides a significant advantage over other optical pulling schemes in waveguides^39–43 and metamaterials^44–46, where the particle must be located inside the waveguides or metamaterial structures using additional means. The method of simulating the particle trajectory is introduced in the “Methods” section.

This work is supported by the Key Project of the National Key Research and Development Program of China (2022YFA1404500), the National Natural Science Foundation of China (12074267, 12174263, 12074169) and the Guangdong Province Talent Recruitment Program (2021QN02C103).