1. Introduction

The stable optical trapping of nanoparticles (NPs) and the subsequent control of the trap position has remained a challenge since the invention of optical tweezers in 1970 [1]. This is because conservative gradient forces arising in Gaussian beams scale with the third power of the particle size, and thus they are not effective to trap nanoparticles [2–4]. During the past decades, the use of plasmonic nanostructures [5,6] such as metallic nanotips [7], nanoapertures [8,9], nanocavities [10–12], or periodic inclusions [13–17] have addressed this issue by locally enhancing gradient forces [18], and have led to numerous applications in biology [19] and chemistry [20,21]. The main challenge of these configurations is that they depend on the structure geometry and thus cannot be employed to manipulate the trap position in real time. As a potential solution, the use of tunable plasmonic materials such as graphene [22–27] have been proposed to engineer nanostructures able to dynamically manipulate optical traps. In such devices, the position of the maximum field intensity and the response of gradient trapping forces are controlled by changing the carrier density of graphene using a gate bias [26]. However, all these nanostructures exhibit a strong resonant and narrowband behavior and require the use of precise fabrication processes as well as lasers with a specific operation wavelength, which limit their application in practice.

More recently, several studies have focused on nonconservative spin-orbit recoil forces acting on nanoparticles located near plasmonic structures [28–31]. The underlying physics of this force is quite different than the mechanism that governs conservative gradient forces [32], and relies on the polarization spin acquired by the nanoparticle and the subsequent excitation of directional surface plasmons polaritons [33]. The surge of momentum exerts a recoil force on the particle acting in the direction opposite of the plasmon wavevector, thus satisfying momentum conservation [28,29]. This phenomenon has been exploited to drastically enhance optical trapping forces acting on nanoparticles located near plasmonic and anisotropic surfaces that are illuminated by a Gaussian beam [34,35]. There, the particle acquires a polarization spin rotating against the beam axis, and the resulting recoil forces originates an optical trap at the beam axis [35]. Again, this type of plasmonic systems depend on the geometry of the nanostructures and thus are unable to manipulate the trap position in real time.

In a related context, nonreciprocal plasmonic platforms biased with an external in-plane momentum bias, such as magneto-optical systems [36,37] and drifted metals and 2D materials [38–44], have been proposed to obtain significant and tunable recoil optical forces acting on nanoparticles [45,46]. Such surfaces exhibit a dispersion relation with broken symmetry in the momentum space, and support nonreciprocal surface plasmons. Recently, several works have explored nonreciprocity-induced optical forces that arise on nanoparticles located near these systems when they are illuminated by plane waves [45,46]. It has been found that the strength of this recoil force component depends on the broken dispersion relation of the system [46] and thus it can be manipulated by changing the external bias.

In this work, we propose to construct and manipulate at the nanoscale optical traps for dipolar, spherical and Rayleigh nanoparticles located near nonreciprocal plasmonic surfaces by illuminating them with a Gaussian beam. The trap mechanism is governed by the interplay between nonreciprocity-induced and spin-orbit recoil forces. On one end, spin-orbit recoil forces depend on the polarization spin acquired by the nanoparticle due to its position within the laser beam, and act toward the beam axis. On the other, nonreciprocity-induced forces only depend on the broken dispersion relation of the system, and act along the applied bias. Balancing these two forces leads to an optical trap whose position can be controlled by adjusting the bias. To characterize this platform in terms of optical forces exerted on the nanoparticle and trapping potentials, a detailed numerical and analytical formalism is derived. As a practical example, we propose the use of drift-biased graphene as a nonreciprocal surface and explore its performance to trap and move nanoparticles. We also study the different mechanisms that allow to control the tunning range of the engineered optical traps, including the laser beamwidth, graphene’s Fermi level, and the nanoparticle size. Results show that drift-biased graphene provides stable optical traps using a laser beam with a low power density $\ge 6\mathrm{mW\mu }{\textrm{m}^{ - 2}}$ oscillating at any frequency within a broad range in the mid-infrared (IR), and that the trap position can be widely and continuously manipulated by adjusting the applied bias.

2. Theoretical framework

This section describes a numerical and an analytical formalism to calculate the optical forces acting on a nonmagnetic, isotropic, spherical, and dipolar Rayleigh nanoparticle of radius a located at a position ${{\boldsymbol r}_0} = \{{{x_0},{y_0},{z_0}} \}$ above a nonreciprocal plasmonic metasurface. The system nonreciprocity is obtained by applying an in-plane momentum bias, such as magnetic field [36] or drift-current bias [39–43]. Figure 1(a) shows the schematic of a specific platform: drift-biased graphene [42,43]. The relative permittivity of the media above and below the system are denoted by ${\varepsilon _1}$ and ${\varepsilon _2}$, respectively, and the device is illuminated with a normally incident transverse magnetic (TM)-polarized Gaussian laser beam oscillating in the IR with width ${w_0}$ and focused at a distance ${f_0}$ from the surface [34,35]. For simplicity, the center axis of the beam is aligned with $\hat{z}$-axis of the reference coordinate system [34,35].

 

Fig. 1. Engineering and manipulating optical traps on drift-biased graphene illuminated with a Gaussian beam. (a) Schematic of the configuration. The interplay between nonreciprocal ($F_y^{nr}$) and spin-orbit ($F_y^s$) recoil forces generates an optical trap away from the beam axis. (b) Isofrequency contour of the modes supported by the device for two different drift velocities. (c) Normalized $z$-component of SPPs excited by a gold nanoparticle when it is located at the beam axis (left panel) and at the trap position (right panel). Inset shows the particle polarization state and the direction of the arising recoil forces. (d) Similar as (c) but without applying a drift-bias to the graphene sheet. The particle has radius $a = 15nm$ and vertical position ${z_0} = a + 1[{nm} ]$; laser operation wavelength is ${\lambda _0} = 10\mu m$ with a beam width ${w_0} = {\lambda _0}$ and focus position ${f_0} = 0$; graphene is transferred onto a 10nm-thick h-BN layer (${\varepsilon _2} = 3.8$) deposited over SiO₂ and has a chemical potential and relaxation time of ${\mu _c} = 0.2eV$ and $\tau = 0.3ps$, respectively.

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In this platform, the laser beam is employed to polarize the particle in such a way that it acquires a polarization spin with rotation handedness against the beam axis and experiences spin-orbit recoil forces directed toward the beam axis, as described elsewhere [34,35]. The radially symmetric electric field vector components of such a Gaussian beam can be modelled as [47]

Here, ‘$\textrm{Re}$’ is the real part of a complex number; $\boldsymbol{p}$ is the effective dipole moment induced on the particle [46]; ${\boldsymbol{E}^{GW}}$ is electric field of the stationary wave above the metasurface, composed of the superposition of the incident laser beam and the one reflected from the surface [35,46]; ${\boldsymbol{E}^s} = {\omega ^2}{\mu _0}{\bar{\boldsymbol{G}}^s} \cdot \boldsymbol{p}$ is electric field scattered by the particle calculated from the scattered dyadic Green’s function ${\bar{\boldsymbol{G}}^s}$ of the system [46,47]; $\omega = 2\pi f$ is the angular frequency of operation; ${\mu _0}$ is the free space permeability; and ‘$\mathrm{\ast }$’ is the complex conjugate.

In general, optical forces described by Eq. (2) can be decomposed in conservative and nonconservative components [47]. However, conservative forces arising from Gaussian beams are proportional to the third power of the particle size [47] and thus negligible to trap nanoparticles. Therefore, in this nonreciprocal plasmonic system, the dominant optical forces are of nonconservative nature. As recently described in the literature [46], nonconservative forces arise from two main mechanisms: (i) spin-orbit effects that originates due to the dipole polarization spin [28], leading to a recoil force ${\boldsymbol{F}^s}$; and (ii) the asymmetrical excitation of surface plasmons due to the nonreciprocal response of the surface, leading to a recoil force ${\boldsymbol{F}^{nr}}$ [46]. Considering that an external momentum bias is applied along the $\hat{y}$-direction of the nonreciprocal system (see Fig. 1(a)), the lateral optical forces induced on a nanoparticle can be calculated as

Here, $k_y^ + $, $k_y^ - $ and ${k_x}$ are the plasmon wavenumbers supported by the platform along, against and in the orthogonal lateral direction of the external momentum bias, respectively; ${c_0}$ is speed of light in free space; and ${\varepsilon _r} = \frac{{{\varepsilon _1} + {\varepsilon _2}}}{2}$. Equation (4) shows that the direction of the spin-orbit force is solely determined by the dipole spin handedness (i.e., sign of helicity), whereas the direction of $F_y^{nr}$ depends on the momentum difference of the plasmons excited along and against the bias. Assuming that the particle is located close to the surface (i.e., ${z_0} \to 0$), then $F_y^{nr}$ acts along the bias direction [46] because $|{k_y^ - } |> |{k_y^ + } |$ (see Fig. 1(b)). When the forces $F_y^s$ and $F_y^{nr}$ are equal in magnitude and act in opposite directions, an optical trap is conformed. Such trap position can be dynamically manipulated: the direction of $F_y^s$ depends on the polarization spin acquired by the nanoparticle due to its position within the Gaussian beam, and $F_y^{nr}$ strength depends on the applied momentum bias. Therefore, changing the applied bias permits moving the optical trap position through the y-axis of the Gaussian beam footprint.

It is instructive to analyze the polarization acquired by the nanoparticle during the light scattering process to understand the trapping mechanism. To this purpose, and without loss of generality, let us consider that the optical trap is set at $({0,{y_T},{z_T}} )$, a position in which $F_x^s = 0$ and $F_y^s + F_y^{nr} = 0$. From Eq. (4), the dipole helicity at the trap position can be computed as

Equation (5) shows that for reciprocal systems, i.e., $k_y^ +{=} k_y^ - $, a trapped nanoparticle does not acquire any polarization spin and behaves as a linearly polarized dipole [34,35]. In this scenario, the particle excites symmetrical surface plasmons propagating against the beam axis. In nonreciprocal systems, the trapped nanoparticle acquires a polarization with a spin rotating along the axis perpendicular to the applied bias (${\eta _x}$ in our specific example). The acquired polarization spin generates the exact amount of spin-orbit recoil forces $F_y^s$ acting against the beam axis that compensates the nonreciprocal force $F_y^{nr}$ acting towards the bias direction. Modifying the applied bias breaks this delicate balance, shifting the optical trap to a different location within the Gaussian beam footprint in which the particle acquires the polarization spin described by Eq. (5b) and ensures that $F_y^s + F_y^{nr} = 0$ is satisfied.

In the following, we employ our formalism to explore lateral trapping forces acting on a gold nanoparticle located near drift-biased graphene supported by hexagonal boron nitride – a platform that has been demonstrated experimentally [39,40]. Exploiting the Helmholtz decomposition method [34,35], we calculate the trapping potential and study the maximum tunability of the trap position versus applied bias. In our analysis, we compute the potential energy and the required trap depth for stable trapping aiming to overcome the thermal fluctuations [48] and Brownian motion [5]. We also study the mechanism that controls the tunning range of the engineered traps and provide examples of realistic laser sources that can be readily applied to achieve such response in practice.

3. Nanoscale manipulation of optical traps

Let us consider a gold nanoparticle with radius $a = 15nm$ and located at $({{x_0},{y_0},{z_0}} )= ({0,{y_0},a + 1nm} )$ above a longitudinally DC-biased graphene layer. The bias generates drifting electrons along graphene with velocity ${v_d} = 0.5{v_F}$, where ${v_F} = {10^6}m/s$ is the Fermi velocity. The platform is illuminated with a normally incident TM-polarized Gaussian beam at ${\lambda _0} = 10\mu m$ that is focused on the surface (${f_0} = 0$) and has a width ${w_0} = {\lambda _0}$. Figure 1(c) shows the $z$-component of the electric field excited on the surface when the particle is located at two different positions. Specifically, the left panel considers that the nanoparticle is located at the beam axis (i.e., ${y_0} = 0$), where it acquires a linear polarization and excites all modes supported by the surface. Note that unidirectional modes are not excited due to the lack of spin-orbit effects [33], and thus the spin-orbit forces $\boldsymbol{F}_t^s$ are strictly zero in this case. However, the particle experiences a nonreciprocal force $F_y^{nr}$ that pushes it along the bias direction. That forces appear due to the momentum imbalance in the excitation of surface plasmons propagating toward the positive and negative y-semi planes. The right panel of the figure shows the fields of the waves excited when the particle is located exactly at the trap position, where it acquires a polarization spin rotating around the $\hat{x}$-axis. On one hand, a certain degree of unidirectional excitation of SPPs appears due to the emitter spin, which leads to a force $F_y^s$ acting toward the beam axis. On the other, there is a still an imbalance in the momentum of the excited plasmons that leads to a nonreciprocal force $F_y^{nr}$ acting against the beam axis. These two forces exactly compensate at this position, thus engineering an optical trap. In all cases, optical forces directed toward the direction perpendicular to the applied bias are negligible (i.e., ${F_x} \to 0$) because the platform response is symmetrical along that direction (see Fig. 1(b)). Figure 1(d) explores the formation of optical traps when no longitudinal bias has been applied on the graphene sheet. As expected [34,35], the optical trap is created now at the center of the Gaussian beam footprint. This is because the nanoparticle acquires there a linear polarization that leads to symmetrical excitation of SPPs within the plane (left panel) and spin-orbit forces $\boldsymbol{F}_t^s$ vanish. When the nanoparticle is moved away from the beam center, it acquires an out-of-plane polarization spin that leads to the unidirectional excitation of SPPs (right panel) and to spin-orbit forces $\boldsymbol{F}_t^s$ acting toward the trap position. In this scenario, the lack of nonreciprocal forces prevents the dynamic control of the trap position.

Figure 2(a) explores the response of the lateral force components versus the particle position, ${y_0}$, along the drift axis using the numerical approach described by Eqs. (2)–(3). As expected, the gradient force, $F_y^g$, is very weak and becomes negligible. The spin-orbit component $F_y^s$ exhibits a well-known odd symmetry versus ${y_0}$ and is zero at the beam axis, i.e., when ${y_0} = 0$ [34,35]. The nonreciprocity-induced force $F_y^{nr}$ acts along the drift direction and exhibits maximum strength near the beam axis -the particle scatters the larger amount of power $P_{rad}^{yz}$ there- and progressively lessens away from it. The balance between $F_y^{nr}$ and $F_y^s$ generates an optical trap at ${y_0} \approx 4\mu m$, indicated by a gold sphere in Fig. 2(b) and in agreement with our field study shown in Fig. 1(c). In the absence of applied bias, the optical trap appears at the beam axis as recently explored in the case of reciprocal structures [34,35]. Such response is robust versus small and moderate variations (up to ∼$\lambda $) of the beam focal distance ${f_0}.$

 

Fig. 2. Lateral recoil forces induced on a nanoparticle located in the platform show in Fig. 1. (a) Nonreciprocity-induced (red) and spin-orbit (magenta) recoil forces as well as gradient forces (yellow) plotted versus the particle position ${y_0}$ along the y-axis keeping ${x_0} = 0$. (b) Total lateral forces induced on the particle along the y-direction when it is located on a biased (blue) and unbiased (green) graphene platform. Gold spheres indicate the trap locations. Results are obtained using the proposed numerical (solid) and analytical (markers) formalism. (c) Total lateral forces acting along the y-direction versus drift velocity ${v_d}$ and particle position ${y_0}$ computed using the proposed numerical (left) and analytical (right) formalism. Solid black line indicates the trap position. Results are normalized with respect to the laser power density.

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Figure 2(c) studies the force strength versus the particle position, ${y_0}$, and the carriers’ velocity, ${v_d}$, and explores the tunability of the trap location (black line where ${F_y} \to 0$). Results are computed numerically solving Eqs. (2)–(3) [left panel] and using our analytical formulation shown in Eq. (4) [right panel], showing an excellent agreement. The trap position can be determined by enforcing ${F_y} = F_y^s + F_y^{nr} = 0$ versus the velocity of drifting electrons. Using this approach, Fig. 3(a) reveals that the platform is capable to continuously shift the trap position up to a distance of ${\sim}{\pm} 5\mu m$ with deeply subwavelength resolution by adequately tuning the applied DC bias. It also shows the dipole helicity along $\hat{x}$-direction acquired by the nanoparticle when it is located the trap position. Numerical and analytical [Eq. (5)] results show again good agreement. The figure insets illustrate how the polarization spin changes sign when the trap is engineered in the positive or negative $y$-half spaces of the Gaussian beam footprint. This response is similar to the one obtained above reciprocal surfaces [34,35]: the dipole always acquires a polarization spin with rotation handedness against the beam axis independent to its position within the beam, whereas it acquires a linear polarization at the beam axis. The proposed platform exploits the resulting spin-orbit forces to counteract nonreciprocity-induced forces and manipulate the trap position in a unique way. In practice, the resolution of the system to position the trap at a desired location will be limited by several factors, including the properties (material, size) of the nanoparticle, the spatial stability of the incoming Gaussian beam, and potential heating effects.

 

Fig. 3. Tunable optical traps over the nonreciprocal plasmonic surfaces shown in Fig. 1. (a) Trap position (left axis) and particle helicity ${\eta _x}$ (right axis) versus the velocity of drifting electrons. Insets show the polarization spin acquired by the particle so that it can be trapped it in the negative/positive $y$-half spaces. Trap position versus the velocity of drifting electrons for various Gaussian beamwidth ${w_0}$ (b), graphene’s Fermi levels (c), and size of the nanoparticle (d). When not varied, the platform parameters are as described in Fig. 2.

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The tunable range of the engineered trap is mostly determined by three factors: the beamwidth ${w_0}$ of the incoming Gaussian beam, graphene’s Fermi level, and the nanoparticle size. The influence of the laser beamwidth is explored in Fig. 3(b). On one end, nonreciprocal forces $F_y^{nr}$ are proportional to the metasurface nonreciprocity and barely depend on the particle polarization, and thus their strength is independent of the particle lateral position within the surface (see Eq. (4c)). On the other, spin-orbit recoil forces $F_y^s$ originate from the polarization acquired by the nanoparticle through the incoming light and thus depend on the relatively position between the nanoparticle and the Gaussian beam footprint [35] (see Eq. (4b)). The position in which a nanoparticle will acquire a polarization ${\eta _x}$ associated with a force $F_y^s ={-} F_y^{nr}$ will shift away from the beam center as the width of the Gaussian beam increases. Consequently, illuminating the platform with larger/smaller beamwidths will increase/decrease the tunable range in which optical traps can be constructed. Graphene’s Fermi level also controls the strength of nonreciprocal forces for a fixed drift velocity [39–43]. In practice, this response yields additional flexibility to electrically manipulate the position of the optical trap: lower (higher) Fermi levels lead to extended (contracted) tunning range. Figure 3(c) numerically explores this scenario. Finally, the size of the nanoparticle also affects the tunable range of the trap, as shown in Fig. 3(d). Such response appears because the nonreciprocal recoil force $F_y^{nr}$ depends on the effective distance between the particle and the surface $({{z_0}} )$, a response analytically captured in Eq. (4c). On one end, the nanoparticle will scatter more power as its size increases, leading to a larger term $P_{rad}^{yz}$ in the equation. On the other, the effective distance ${z_0}\; $ between the nanoparticle and the metasurface increases with the nanoparticle size. As a function of this distance, the fields scattered by the nanoparticle may be filtered out by free-space and will be unable to couple to the surface. Drift-biased graphene supports SPPs that are strongly/weakly confined in the directions against/along the applied bias, as illustrated in the isofrequency contours of Fig. 1(b). When the nanoparticle size increases, evanescent scattered fields will progressively lessen their coupling to confined SPPs traveling against the drift while still effectively coupling to weakly confined SPPs propagating along the drift. Consequently, the overall strength of $F_y^{nr}$ will progressively diminish with the particle size and eventually change direction once the coupling to weakly confined SPP dominates (see Eq. (4c)). This translates into the size-dependent tunning range for optical traps illustrated in Fig. 3(d): the range is larger for smaller nanoparticles and progressively decreases as the nanoparticle size increases. Very specific nanoparticles will not experience any nonreciprocal recoil forces because they excite SPPs with identical momentum along and against the drift. These particles will only be trapped at the center of the Gaussian beam footprint. Larger particles will still be trapped with a reduced tunable range. In such cases, the direction of nonreciprocal forces is reversed leading to an inverted position of the optical trap versus applied bias. It should be noted that the strength of the recoil forces that conform the optical trap decreases as the particle size increases thus requiring higher beam intensities to achieve stable trapping.

To further investigate the performance of the proposed platform, it is important to analyze the potential energy and depth of the engineered traps. It should be noted that the correspondence between trapping potential and nonconservative forces arising in this platform is not straightforward. This is because nonconservative forces possess a solenoidal component and thus the trapping potential ${U_c}(\boldsymbol{r} )$ at a position $\boldsymbol{r}$ cannot be obtained using the common ${U_c}(\boldsymbol{r} )={-} \smallint_{ - \infty }^r {\boldsymbol{F}({\boldsymbol{r}\mathrm{^{\prime}}} )\cdot \partial \boldsymbol{r}} \mathrm{^{\prime}}$ approach because the integral becomes path dependent [47]. To overcome this challenge, we combine the optical force formulation described in Section 2 with the Helmholtz decomposition method [35] to calculate the potential energy of optical traps engineered using nonconservative forces. Using this approach, Figs. 4(a)-(b) show the distribution of the trapping potential versus the particle position $({{x_0},{y_0}} )$ in the proposed platform, considering a reciprocal case (i.e., not applied bias) and when the velocity of drifting electrons is set to ${v_d} = 0.5{v_F}$. The former case shows a rotationally symmetric distribution of energy, with a minimum located at the beam axis, i.e., $({{x_0},{y_0}} )\approx ({0,0} )$. Note that the potential depth is weak and would require a laser beam with a power density of ${\sim} 90mW\mu {m^{ - 2}}$ to conform a stable optical trap with a potential >$10{k_B}T$. In the presence of the drift, the potential distribution is drastically modified by: (i) boosting the absolute potential over an order of magnitude; (ii) shifting the potential minima to $({{x_0},{y_0}} )\approx ({0,4\mu m} )$, in agreement with results shown in Figs. 2 and 3; and (iii) exhibiting an asymmetrical distribution of energy along the $\hat{y}$-axis originated from the nonreciprocal plasmon excitation along that direction. Note that the potential distribution remains symmetrical along the $\hat{x}$-axis because the in-plane bias is applied along the orthogonal direction, $\hat{y}$ (see Fig. 1(b)). Figure 4(c) shows the potential energy in the directions along (magenta line) and orthogonal to (blue line) the applied bias. Results shows that the nanoparticle experiences a local energy maximum in the $y < {y_T}$ half space, which is associated to the local enhancement of the forces that appears in this region (see Fig. 2(b)). As a result, a local potential barrier and a larger trap depth $\nabla {U_y}$ appears in the direction against the applied bias with respect to the ones arising along it. If a particle acquires sufficient energy to escape from the trap, it will probably avoid this local potential barrier and follow a path in the ${y_0} > {y_T}$ semi plane. Figure 4(d) shows a similar response for different drift velocities, confirming that the minimum trap potential progressively moves along the bias direction as the drift velocity increases. For stable trapping, able to overcome thermal fluctuations and Brownian motion [49], the absolute value of potential minima should be larger than $10{\textrm{k}_\textrm{B}}\textrm{T}$. This requires a laser power density $\ge 6\sim 10\mathrm{mW\;\ \mu }{\textrm{m}^{ - 2}}$, which can be obtained in practice using femtosecond pulses [50], CO₂ lasers [51,52], or an optical parametric oscillator [53]. Moreover, it should be emphasized that drift-biased graphene is a broadband nonreciprocal platform [42] in the sense that it supports nonreciprocal surface plasmons over a wide frequency band in the IR. As a result, the proposed platform can operate with IR beams oscillating within a large bandwidth. Such bandwidth can be calculated by determining the beam frequencies for which the strength of the spin-orbit and nonreciprocal recoil forces induced on the nanoparticle are comparable, and thus stable optical traps can be engineered. In the platform described in Figs. 1 and 2, that bandwidth is roughly $8\mu m - 12\mu m$ which allows using various types of IR sources and lasers to trap and manipulate nanoparticles.

 

Fig. 4. Normalized trap potential versus the particle position $({{x_0},{y_0}} )$ within the nonreciprocal platform shown in Fig. 1. 3D-distribution of the trap potential when the velocity of the drifting electrons in graphene is set to (a) ${v_d} = 0$; and (b) ${v_d} = 0.5{v_F}$. (c) Trap potential versus the particle position along the $\hat{x}$ and $\hat{y}$-axes going through the optical trap (blue and magenta lines in panel b). (d) Trap potential versus ${y_0}$ keeping ${x_0} = 0$ for different drift velocities ${v_d}$, showing control of the trap position.

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Moving forward, the flexibility of the proposed platform can be enhanced by applying additional bias voltages. For instance, two metallic contact pads can be added along the x-direction of the platform as suggested in [42] in the context of nonreciprocal plasmonics. The resulting device will support two orthogonal set of drifting electrons paths, creating an effective biasing current at any targeted direction radial to the beam center. Optical traps along such radial directions can be engineered with a tunable range and performance like the one described above. As a result, the platform will engineer potential wells at any desired location within the surface. Additionally, another metal contact can be added to the platform in such a way that a gate voltage can be applied to control graphene’s Fermi level. On top of adjusting the tunable range (see Fig. 3(c)), the applied gate voltage controls the platform central operation wavelength and bandwidth. This is because the electromagnetic frequency region in which graphene exhibits drift-induced nonreciprocal responses strongly depends on its Fermi level [39–44]. Decreasing (increasing) graphene’s Fermi level shifts the operational bandwidth down (up) to far-infrared (near-infrared) frequencies. Chemical potentials of 0.1 eV, 0.2 eV and 0.5 eV enable the proposed platform to operate with IR light oscillating in the spectral regions $8\mu m - 20\mu m$, $8\mu m - 12\mu m$, and $3\mu m - 10\mu m$, respectively, opening the door to use this technology with several type of IR sources.

4. Conclusions

In summary, we have investigated stable optical trapping of a dipolar nanoparticle located near nonreciprocal metasurfaces when illuminated by a laser beam, and explored the possibility to dynamically manipulate the trap location. The trap mechanism is governed by the interplay between spin-orbit and nonreciprocity-induced lateral recoil forces. To analyze this trapping platform, we have first developed a numerical framework based on the Lorentz force merged with anisotropic and Green’s functions and then obtained an insightful analytical formalism. The resulting expressions link the optical forces with the dispersion relation of the system, and allow to predict and monitor the trap position versus the properties of the laser beam and the induced nonreciprocity. The proposed framework is general in the sense that it can be applied to any nonreciprocal system provided that its dispersion relation is available, including topological materials [36], magneto-optical surface [36,37], and drift-biased thin metals [38] or graphene [39,40], among others [54]. As an example, we have proposed the use of drift-biased graphene illuminated by an IR laser beam laser. We have assessed the stability of the resulting optical traps as well as the ability of the platform to manipulate them with subwavelength resolution by adjusting a DC bias. Compared to the state of the art [55–57], this approach permits to achieve tunable and high-precision trapping position by simply applying a voltage bias over an unpatterned surface. For instance, Ref. [55] use radially polarized fields and a thin gold film to create a virtual probe to trap and manipulate nanoparticles. However, the nanoparticles are effectively routed within the plane by either accurately controlling the direction of the laser beams or using a moving stage with limited resolution. Reference [56] achieved nanometer-precision sorting of nanoparticles using a complex system based on synchronized optofluidic dual barriers, while Ref. [57] employed an array of optical waveguides excited with different modes and optical power. In this context, nonreciprocal platforms have the potential to significantly advance the field of nano-optical plasmonic tweezers in terms of dynamic and precise routing of nanoparticles using simple DC-biased plasmonic devices.