## 2 Methods  
The simulations of near-field nano-manipulation were preformed based on 3D FDTD. Figure 1 shows the configurations of single probe and dual probe irradiated by polarized near-field laser coming from an SNOM probe. According to commercial probes, the model of the SNOM probe consists of the waveguide region and tip region. Both regions have a SiO\({}_{2}\) core and an aluminum shell with thickness \(h=80\) nm. The maximum and minimum diameters of SiO\({}_{2}\) core in tip region are \(\phi_{1}=700\) nm and \(\phi_{2}=200\) nm, and the length is \(L=600\) nm. Likewise, the model of an AFM probe was simplified to a gold cone with angle \(\theta=36^{\circ}\) and a spherical tip (the radius is set to 10 nm). According to previous simulations, the enhancement of near-field and trapping force are stronger when the incident angle is \(90^{\circ}\), with lower distance between the AFM and SNOM probes [26]. So, considering experimental feasibility, the incident angle and distance along \(x\) direction of two probes were set to \(90^{\circ}\) and \(100\) nm, respectively. For dual probes, two configurations with angles \(90^{\circ}\) and \(180^{\circ}\) of axes of two AFM probes in \(xz\) plane, named as orthogonal dual probes (ODPs) and coaxial dual probes (CDPs) respectively, were simulated in this paper, as shown in Figure 1c,d. A silver NP (the diameter is \(d\)) was located under the AFM tip with distances \(D_{1}\) (single probe) or \(D_{2}\) (dual probe, the distance between two tips was marked as D\({}_{3}\)) to study the near-field enhancement and trapping force. Vertical polarized (VP, along axis \(z\)) and horizon polarized (HP, along axis \(y\)) lasers were coupled with the SNOM probe. In consideration of calculation efficiency, the incident laser was set to a normalized signal, as Figure 1e showed. Since the maximum amplitude was 1, the electric field intensity \(E\) and trapping force \(F\) calculated in this paper were both relative values. In this paper, relative values are enough to discuss the influence of the polarized laser and the configurations of probes on the trapping ability. Wavelengths ranging from 300 nm-1200 nm were adopted to explore the effect on manipulation. For global FDTD calculations, the simulation temperature was 300 K. The size of the simulation area was 2 \(\upmu\)m (\(x\)) \(\times\) 1.1 \(\upmu\)m (\(y\)) \(\times\) 1.1 \(\upmu\)m (\(z\)) with mesh size 2 nm, as the origin dash lines showed in Figure 1a,b. Moreover, to guarantee calculation accuracy, the area of AFM tips and NP (red dash lines) had a fine mesh with a 1 nm resolution. A perfectly matched layer with a minimum layer number of 12 was used to avoid energy reflection in the boundary. The simulation time 300 fs was more than enough to achieve desirable results.  
To calculate the trapping forces, the method based on Maxwell stress tensor combined with FDTD was applied. Given that the volume \(v\) and corresponding external boundary \(S\) contain NP, the force exerted on the NP can be written as:  
\[\mathbf{F}=\int\limits_{v}\mathbf{f}\cdot dv=\oint\limits_{S}\overleftarrow{\mathbf{T}} \cdot ndS-\epsilon_{0}\mu_{0}\int\limits_{v}\frac{\partial\mathbf{S}}{\partial t}dv \tag{1}\]  
where \(\mathbf{S}\) is Poynting vector. Here, for static magnetic field, \(\partial\mathbf{S}/\partial t=0\). \(\overleftarrow{\mathbf{T}}\) is Maxwell stress tensor, which was defined as  
\[\overleftarrow{\mathbf{T}}=\epsilon_{0}(\mathbf{E}\mathbf{E})-\frac{\epsilon_{0}}{2} \bigg{(}\overleftarrow{\mathbf{I}}\mathbf{E}^{2}\bigg{)}+\frac{1}{\mu_{0}}(\mathbf{B}\mathbf{B })-\frac{1}{2\mu_{0}}\bigg{(}\overleftarrow{\mathbf{I}}\mathbf{B}^{2}\bigg{)} \tag{2}\]  
where \(\epsilon_{0}\) and \(\mu_{0}\) are vacuum electric constant and magnetic constant, respectively. \(\overleftarrow{\mathbf{I}}\) is Kronecker's delta. \(\mathbf{E}\) and \(\mathbf{B}\) are electric and magnetic field respectively, which can be calculated by FDTD. So, Maxwell stress tensor and FDTD can be used to calculate the optical forces around NPs.