0ptical trapping with optical magnetic field and photonic Hall effect forces

Our operational hypothesis for demonstrating optical magnetic field trapping requires: (i) a focused electromagnetic beam that allows distinguishing the electric and magnetic fields; and (ii) nanoscale size materials that (strongly) interact with optical magnetic fields. Fig. 1a shows the result of an electrodynamics simulation of a focused azimuthally polarized vector beam${}^{\mbox{\scriptsize\immediate{?}}}$. The segregated magnetic field is primarily a longitudinally polarized component, $H_{z}$, at the beam center (where the electric field vanishes) together with the azimuthally polarized electric component, $E$${}^{\mbox{\scriptsize\immediate{?},\penalty 1000\thinspace{?},\penalty 1000\thinspace{?}}}$. There is an additional magnetic component of light with radial polarization, $H_{\rho}$, that shares the annular region with the electric component. Figure 1b shows that the intensity of the longitudinally polarized magnetic field, $H_{z}$, increases with the tightness of the beam focus; the ratios $|\eta H_{z}|^{2}$/$|E|^{2}$ and $|H_{z}|^{2}$/$|H_{\rho}|^{2}$ increase with the numerical aperture (NA) of the objective (shown from NA = 0.4 to 1.29)${}^{\mbox{\scriptsize\immediate{?}}}$. When the azimuthally polarized beam is tightly focused with a NA of 1.27 (our experimental condition) the longitudinal magnetic field $H_{z}$ attains an intensity 1.5 and 3.4 times greater than the transverse electric $E$ and magnetic $H_{\rho}$ field intensities, respectively. The calculation assumes a water environment (n=1.33) and considers the wave impedance $\eta=\sqrt{\epsilon_{w}/\mu_{w}}$ for the magnetic-to-electric intensity ratio where $\epsilon_{w}$ and $\mu_{w}$ are water’s permittivity and permeability. This focused trapping beam would enhance the $H_{z}$ associated optical force and reduce any competing forces due to other fields at the beam center thereby satisfying one condition for selective optical magnetic trapping.

Our demonstration of optical magnetic trapping also requires a material with a strong optical magnetic dipole resonance. Si nanoparticles satisfy this condition${}^{\mbox{\scriptsize\immediate{?}}}$ (see Fig. 2d) and are small enough that they can interact dominantly with the longitudinally polarized magnetic field $H_{z}$ via their Mie magnetic dipole (MD) resonance. This electromagnetic field-nanoparticle interaction is expected to create optical magnetic forces as described by Eq. 2. However, as suggested in Fig. 1c and shown explicitly in Fig. 2d, Si nanoparticles also have a strong (and spectrally broad) electric dipole (ED) resonance${}^{\mbox{\scriptsize\immediate{?}}}$ that can be excited by the annularly distributed electric field $E$. Therefore, electric field-mediated forces (Eq. 1) can also act on the Si nanoparticles. The magnitude of these two types of forces depend not only on the intensity of the respective fields, but also on the strength of the field-particle interactions that follow from the single Si nanoparticle ED and MD scattering cross-sections at the trapping laser wavelength (see Fig. 2d). Figure 1c presents a GLMT calculated scattering spectrum (in black) of Si nanoparticles as a function of diameter for the 770 nm trapping laser wavelength. The total scattering is decomposed into the MD (blue) and ED (red) mode contributions. The MD mode dominates over the ED mode only in a narrow size range centered at a diameter of 200 nm, which corresponds to the peak of the MD resonance (for 770 nm trapping laser wavelength). This diameter-dependent scattering suggests another operational hypothesis that only Si nanoparticles in a certain size range (as shown in Fig. 1c) will manifest optical magnetic field trapping.

Testing the aforementioned hypothesis and verifying optical magnetic trapping is accomplished by observing where the Si nanoparticles are most probable to be localized in the focused azimuthally polarized trapping beam. Since the experiments are conducted in room temperature solution, the random Brownian forces enable statistical (thermal) sampling of force interactions in the inhomogeneous trapping field (Fig. 1a). The potential energy profiles of Fig. 1d for Si nanoparticles in the beam show that the energy minimum changes from the azimuthally polarized beam’s annular E-field region to its center when the dominant electromagnetic interaction (and trapping force) changes from ED mode dominated to MD mode dominated with increasing Si nanoparticle size. Note that the upper size boundary that we consider in the experiments is when the magnetic quadruple (MQ) mode in Fig. 1c begins to dominate the scattering (at dia.$>$250 nm). Optical forces due to higher-order modes become pronounced (eventually dominant) beyond this Si nanoparticle size (see Fig. S5) and fall outside the scope of this study. Furthermore, we do not consider thermophoresis${}^{\mbox{\scriptsize\immediate{?}}}$ due to the small absorbance of Si nanoparticles at 770 nm (see Fig. S4).

Figure 2a presents a schematic of optical trapping where a water-immersion microscope objective (Nikon CFI SR Plan Apo IR 60X) focuses an azimuthally polarized beam into a thin (120 $\mu$m) water-filled chamber with colloidal Si nanoparticles. The expanded view depicts the azimuthally polarized beam focused near the top water-glass interface, with arrows indicating the (local) polarization directions and spatial distributions of both the longitudinal magnetic $H_{z}$ (blue) and electric $E$ (red) fields. The Si nanoparticle located at the maximum of the longitudinal magnetic field (i.e., at the center of the beam) provides a visual representation of magnetic field trapping. Details about Si nanoparticle synthesis and experimental implementation are given in the Supplementary Information.

Scanning electron microscopy (SEM) was employed for ex-situ characterization of particle size and shape. Figure 2b shows a SEM micrograph of a representative Si nanoparticle that was prepared on a substrate using a standard drop-casting technique. The optical scattering spectrum of this Si nanoparticle (not shown) is very similar to the one shown in Fig. 2d. The SEM measurement confirms that the comparison of experimental and calculated optical scattering spectra give an accurate determination of nanoparticle size with tolerance of $\pm$2 nm. Furthermore, well-resolved peaks in the measured scattering spectra indicate spherical nanoparticle shapes. A detailed description of the particle size determination method and calibrations are provided in Fig. S6.

Figure 2c shows a darkfield microscopy image of a single Si nanoparticle stably trapped in the central region of the focused azimuthally polarized beam; the dashed circle outlines the trapping beam (its optical image was blocked with a notch filter as shown in Fig. S3). An in-situ measurement of the single Si nanoparticle’s scattering spectrum was performed (upper panel of Fig. 2d) to allow size determination of this and each single Si nanoparticle studied. The GLMT-calculated spectrum (lower panel of Fig. 2d) that matches the measured spectral features (peaks) establishes this particle’s diameter to be 205 nm. This particle’s MD resonance, measured and calculated in a water environment, is nearly maximally on-resonance with the 770 nm trapping laser wavelength, and the ratio of the optical magnetic dipole to electric dipole scattering is nearly maximized.

The nanoscale localization${}^{\mbox{\scriptsize\immediate{?}}}$ of single Si nanoparticles and the associated probability density functions (PDF) of different trapped Si nanoparticles along the radial direction (different colors) are shown in Figs. 2e and f, respectively. Particles of different sizes localize in different regions of the focused azimuthally polarized beam. Statistical analysis of experimental (upper) and simulation (lower) results demonstrate that the 205 nm dia. Si nanoparticle (blue points) is trapped at the beam center where the maximum longitudinally polarized magnetic field $H_{z}$ exists. This result is consistent with optical magnetic field trapping with a trapping force given by Eq. 2. By contrast, a 162 nm dia. Si nanoparticle (red points) is localized in the annular region of the focused azimuthally polarized beam, which is consistent with electric field trapping. This latter result is consistent with our expectation because the ED mode dominates the interaction with the electric field of the trapping laser via Eq 1, which is the well-established electric point-dipole model of electric light-matter interactions${}^{\mbox{\scriptsize\immediate{?}}}$.

Surprisingly, Figs. 2e and f also show that a large, 239 nm dia., Si nanoparticle (green) is strongly trapped at the center of the beam, with a narrower distribution in the radial position PDF! This result is unexpected since it starkly contrasts with that for the 162 nm dia. nanoparticle even though in both cases the interaction with their ED mode dominates the MD mode for 770 nm trapping laser light. Furthermore, as shown in Fig. S9, the magnetic polarizability ( and therefore induced magnetization ) is negative for the 239 nm dia. nanoparticle at 770 nm trapping laser wavelength. Therefore, the optical magnetic dipole gradient force of Eq. 2 would be repulsive (see Fig. S7), and the 239 nm dia. Si nanoparticle should be expelled from the center of the trap! Other Si nanoparticles of various sizes that we trapped and characterized showed results consistent with the three nanoparticles we discussed here in detail. See the Supplementary Information and Methods sections for more details about the experiments and simulations. These results clearly indicate that an additional, unaccounted-for optical force must exist to resolve the contradiction with expectations from a simple point-dipole model (of Eqs1 and 2) and that this force dominates the electrodynamic forces acting on large Si nanoparticles.

Figure 3 qualitatively and quantitatively demonstrates the directional forces associated with the transverse directional scattering of light. Figure 3b shows the results of GLMT simulations of a Si nanoparticle interacting with a focused Gaussian beam where the electric and magnetic fields are polarized along the in x- and y-directions, respectively. Placing the 205 nm dia. Si nanoparticle at three representative positions in the trapping beam, as shown in Fig. 3a (labelled 1, 2, and 3), results in the three far-field scattering patterns shown in Fig. 3b-d. The scattering pattern for the particle at the origin (position 1) displays mirror (and inversion) symmetry in the transverse plane so no net TS force is generated, as shown schematically in Fig. 3b. Note that for this size nanoparticle, the light scattering at 770 nm originates from both the ED and MD modes. The excited dipole moments align with their respective exciting fields and radiate electromagnetic fields in the transverse plane (see Fig. 3b). Therefore, the 205 nm dia. Si nanoparticle scatters more intensely along the x-axis because of the large scattering cross-section associated with the optical magnetic dipole (see Fig. 2d). It is important to note that this additional optical force also exists for small particles (such as the 162 nm dia. Si nanoparticle), but its trivial contribution to the total force on the Si nanoparticle allows the point-dipole model to accurately describe the electrodynamic forces in this size regime (see Fig. S8).

The mirror symmetry of the scattering is lost when the particle is displaced from the central axis of the trapping beam. The x- and y-axes are the (reflection) symmetry axes for the electric and magnetic components of the Gaussian beam, respectively. When the particle is placed at position 2, its deviation from the (y-oriented) symmetry axis for the magnetic field leads to symmetry breaking of the MD-mediated scattering along the x-direction, as shown Fig. 3c. Similarly, the asymmetric ED-mediated scattering at position 3 occurs because of the particle’s displacement from the (x-oriented) symmetry axis for the electric field, as shown Fig. 3c. However, these two configurations show different biases in scattering, with the ED mode scattering light along the electric field intensity gradient and the MD mode scattering light in the direction opposite to the magnetic field intensity gradient. A Si nanoparticle off-center in an electromagnetic beam with a (typical) Gaussian transverse intensity profile, where its electric and magnetic fields are not spatially segregated, scatters the incident photons into the opposite direction (inward or outward) depending upon the photons’ linear polarization state (i.e., along the x- or y-directions) . This contrarian behavior is attributed to the interference between the MD and the ED modes, as reflected in Eq. 3. This polarization-dependent directional scattering is the photonic Hall effect and is essential for understanding the trapping behavior of Si nanoparticles.

Following this reasoning, we propose that a transverse scattering force must be added to the gradient force model described in Eqs. 1, 2, and 13 (in Methods). Figure 3 shows that the TS force results from directionally biased scattering; the particle’s “reaction” balances the momentum carried by the scattered radiation, satisfying conservation of momentum. Therefore, how the nanoparticles scatter light, especially for near- or on-resonance optical trapping, is an important and perhaps dominant force acting on the nanoparticles. As we show below, this scattering force on the Si nanoparticle arises from the PHE${}^{\mbox{\scriptsize\immediate{?},\penalty 1000\thinspace{?},\penalty 1000\thinspace{?}}}$. Polarization-dependent directional light scattering from nanoparticles is already known in the photonic spin Hall effect (PSHE), which relies on: (i) the light’s circular polarization state; and (ii) gradients in the (local) refractive index of materials${}^{\mbox{\scriptsize\immediate{?},\penalty 1000\thinspace{?},\penalty 1000\thinspace{?}}}$. In contrast, in the present case, due to the existence of both MD and ED resonances and their interaction as indicated in Eq. 3, the PHE acting on the Si nanoparticles depends on: (i) the incident light’s linear polarization state; and (ii) the intensity gradients of the electric and magnetic fields (more details in Sect. S13 in Supplementary Information) .

The TS forces are a consequence of the biased scattering just described, and have magnetic and electric dipolar contributions, as evidenced by an analytical approximation:

We verified the TS forces and their properties indicated in Eq. 3 by numerically solving for the net momentum carried by the (unevenly) scattered radiation and precisely quantify the TS forces with Eqs. 19 and 20 in Methods. These results are benchmarked against rigorous numerical MST ($F^{\text{MST}}$) calculations. Figure 3e shows that the TS forces ($F_{\text{ED/MD}}^{\text{TS}}$; orange curves) were numerically calculated for the particle at various positions along the electric (left, x) and magnetic (right, y) field axes in the Gaussian beam shown Fig. 3a. The behavior of the scattering force in Fig. 3e highlights the anisotropic nature of the dipole-based scattering forces—i.e., being attractive and repulsive along the orthogonal axes.

The optical forces computed using the Maxwell stress tensor (MST, in black)${}^{\mbox{\scriptsize\immediate{?}}}$ or point-dipole model (PD, in green dash)${}^{\mbox{\scriptsize\immediate{?}}}$, are also presented in addition to the numerically calculated TS forces. Comparing these forces demonstrates the significance of the TS forces when optical trapping occurs in the optical resonant regime (see Fig. S8). The MST-computed forces are the most accurate determination of the actual optical forces experienced by the Si nanoparticle because the MST rigorously represents the particle’s electromagnetic response by fully solving Maxwell’s equations. The substantial discrepancies shown in Fig. 3e between the MST-computed forces and the PD-computed forces highlight the inadequacy of the conventional point-dipole model in describing the forces acting on the particle. In fact, the TS forces ($F^{\text{TS}}$) account for the forces omitted in the PD model. The resulting “Scattering Corrected Model” (SCM; green circles) $F^{\text{SCM}}_{\text{total}}=F^{\text{PD}}+F^{\text{TS}}$ is suitable for understanding the electrodynamics of resonant optical trapping and optical magnetic contributions (see Methods and Supplementary Information).

The SCM explains the trapping behavior of Si nanoparticles in various optical beams including the focused azimuthally polarized beam employed here. Figure 4 shows the trapping stiffness coefficients along the electric and magnetic field axes (as defined in Fig. 3a) as a function of particle dimensions. The encoded colors indicate the dominant contribution to the force on the Si nanoparticle and how these contributions change with Si nanoparticle diameter, from the dipole force ($F^{\text{PD}}$: gray) to the ED/MD-mediated TS forces ($F^{\text{TS}}_{\text{ED}}$/$F^{\text{TS}}_{\text{MD}}$: red/blue) in red/blue. The changes occur because the resonances shift to longer wavelengths with Si nanoparticle size thereby tuning on-resonance with the ED and MD modes (as seen in Fig. 1c). Two key observations arise from the results shown in Fig. 4. First, $|\kappa_{x}|$ reaches its minimum value while $|\kappa_{y}|$ is nearly zero near the optical magnetic dipole resonance (vertical black dashed line), indicating that particles in this size range are primarily influenced by the MD-mediated TS force. Second, $|\kappa_{x}|$ and $|\kappa_{y}|$ become comparable in magnitude but with opposite sign for larger particles. This means that large Si nanoparticles will simultaneously experience attraction and repulsion from the optical magnetic dipole and electric dipole-mediated TS forces in orthogonal directions.

These two properties of the TS forces reveal the mechanism of optical magnetic field trapping. Figure 5a schematically decomposes the optical forces exerted on a Si nanoparticle under illumination from an azimuthally polarized beam. When the particle is excited near its optical magnetic dipole resonance, it is attracted toward the magnetic field maximum due to $F^{\text{TS}}_{\text{MD}}$ (left panel). The optical MD mode of larger Si nanoparticles is detuned to longer wavelengths relative to the (770 nm) trapping laser. Therefore, the Si nanoparticles ($>$200 nm dia.) experience not only attractive forces due to optical magnetic fields through the magnetic TS force, but also a repulsion from $F^{\text{TS}}_{\text{ED}}$ from the radial intensity gradient of $E(r,\Phi)$ in azimuthally polarized optical beams. The latter pushes the Si nanoparticle towards the beam center (Fig. 5a right panel). This is why both the 205 nm and 239 nm Si nanoparticles (and others shown in Fig. 5b) were trapped at the location of the longitudinally polarized magnetic field, $H_{z}$. The 239 nm dia. Si nanoparticle shows greater confinement because $F^{\text{TS}}_{\text{ED}}$, the transverse scattering force, becomes larger for larger Si nanoparticles. More quantitative details of the transverse scattering force effects as well as intensity gradient forces are shown in Figs. S9-S13.

Figure 5b shows the radial position distribution (PDF) calculated as a function of particle diameter demonstrating how the optical trapping of Si nanoparticles depends on particle size. These results were obtained using GLMT-Langevin dynamics simulations (see Methods). The simulation (gray) and experimental (red points) results show an abrupt transition from electric field trapping to optical magnetic field trapping when the trapping laser is near/on-resonance with the magnetic dipole mode (Dia.=200 nm). Moreover, the decrease in the standard deviations of the distributions with increasing particle size shows that the trapping efficacy is enhanced. This result further confirms the validity of our force analysis and the key role of the photonic Hall effect.

The procedure used for Si nanoparticle synthesis shown in Fig. S1a follows a previously reported method\citesupplsugimoto2020mie. Lumps of 99.99% purity 3-10 mm in size SiO (Sigma Aldrich) were crushed to powder and annealed at 1500 °C in an inert gas atmosphere for 30 minutes using an MTI GSL-1700X tube furnace to promote the solid-state disproportionation of SiO into Si and SiO₂. This was followed by etching the SiO₂ matrices in a concentrated hydrofluoric acid (HF) solution (48% from Sigma Aldrich) for one hour, which resulted in the extraction of free Si nanoparticles. The extracted particles were transferred to water after several rinses through centrifugal filtration (AccuSpin 8C centrifuge by Fisher Scientific), followed by 2-minutes of ultrasonication using a Fisher Scientific FB505 sonic dismembrator.

Dynamic light scattering measurements (DLS) were performed to characterize the polydispersity of the as-synthesized Si nanoparticles. In the measurements, an aqueous colloidal suspension of Si nanoparticles in water was sonicated for 3 minutes; a 1 mL aliquot was extracted immediately after sonication and transferred to a cuvette that was loaded into a Malvern Zetasizer Nano ZS for DLS measurement. Multiple DLS measurements were conducted. A representative outcome shown in Fig. S2 demonstrates the particle sizes are in the range of 100 nm to 300 nm dia. The size distribution exhibits a log-normal profile with its peak at 200 nm diameter. The log-normal distribution means that the preponderance of particles are smaller than 200 nm in diameter.

In addition, zeta potential measurements were undertaken to assess the stability of the Si nanoparticles in the colloidal suspension. A 1 mL sample was created in the same manner as just described and measured using the Malvern Zetasizer Nano ZS. The resulting representative zeta potential was -32.0 mV.

Figures S1b and c as well as Fig. 2b of the main text demonstrate the high morphological quality of the synthesized Si nanoparticles as determined with scanning electron microscopy (SEM, Carl Zeiss Merlin) measurements. The sample was prepared for measurement by drop-casting that created a sparse distribution of Si nanoparticles on an indium tin oxide (ITO) coated coverslip (SPI Supplies^®). For higher resolution measurements, the Si nanoparticles were dispersed on a glass coverslip by drop-casting and transferred into a high resolution sputter coater (Cressington 208HR) for the deposition of a uniform Pt/Pd layer. A thin metal layer (thickness 5$\pm$0.2 nm) was deposited on the Si nanoparticles (and glass substrate) by carefully controlling the vacuum pressure (0.08 mbar) of the coater.

Relative to the scattering cross-section, the calculated absorption cross-section of 770 nm light by Si nanoparticles remains consistently close to zero throughout the nanoparticle size range of interest (within the white-colored region of Fig. S4), which reflects a minuscule absorbance of a Si nanoparticle illuminated at 770 nm. Even with the almost imperceptible increase in absorbance associated with the magnetic dipole resonance for a 200 nm dia. Si nanoparticle, we conclude that thermal heating of the aqueous solution will be inconsequential based on our prior experience with metallic (Ag and Au) nanoparticles\citesupplfigliozzi2017driven. Therefore, local temperature changes are not considered in the present study.

As illustrated by the curve, the optical force in a Gaussian beam due exclusively to higher-order (mainly from the magnetic quadrupole) resonances gradually becomes important with increasing size of the Si nanoparticle. This force becomes pronounced for sizes larger than 250 nm. In order to concentrate on dipole-governed electrodynamics, our study is limited to Si nanoparticles smaller than 250 nm diameter.

The trapping behavior of a single Si nanoparticle under illumination of an azimuthally polarized beam depends on the particle’s size since it directly determines both the magnitude and relative dominance of the electric and magnetic dipole resonances excited. Understanding the size dependence requires an accurate determination of each Si nanoparticle’s size while being held by the electromagnetic field of the trapping laser. In-situ scattering by the Si nanoparticles is most suitable and not disruptive of optical trapping. We calibrated the accuracy of this approach by comparing experimental scattering spectra and scanning electron microscopy (SEM) measurements of single Si nanoparticles we intentionally deposited on the glass slide as shown in Fig. S6a.

More than 40 Si nanoparticles with a large range of sizes were dispersed and stabilized on the surface of a microscope slide using a drop-casting method. The prepared sample was initially assessed with spectral scattering measurements, where it was incorporated into a water-filled chamber to mimic the conditions of the optical trapping environment. Therefore, the scattering spectra of the particles are chosen with typical dimensions as shown in Fig. S6a that are representative of the ones obtained in the laser trapping experiments.

The experimental scattering spectra (green dashed curves in Fig. S6b) were compared with LMT-based calculations to establish the particles’ sizes. The resonance features (peaks) observed in the experimental spectra (e.g., the magnetic dipole and quadrupole resonance peaks marked by the vertical black dashed lines) are well reproduced at the same spectral locations in the calculated spectra (orange solid curves in Fig. S6b). In addition, a fitting model composed of a series of Lorentz and Gaussian functions was developed to facilitate the identification of these experimentally acquired scattering signatures. The spectral resonance fitting functions, expressed in terms of angular frequency $\omega$, are:

Figure S6c shows a detailed example of curve fitting that identifies five resonances for the Si nanoparticle with a diameter of 177 nm. Here, $\omega_{0}$ serves as the spectral position of each peak. Specifically, Lor_1, Gau_2, and Gau_3 are mainly associated with the magnetic dipole, magnetic quadrupole, and electric dipole, respectively, while Gau_4 and Gau_5 are associated with higher-order resonances of the Si nanoparticle. It should be noted that the fitting functions used here are only for the purpose of determining the spectral positions of these resonance features, rather than ascertaining the true dielectric tensors of the Si material.

The fitting process employs the Levenberg-Marguardt algorithm that minimizes the mean square error (MSE). The MSE is defined in terms of the deviations of the fitted $SC_{i}^{\text{fit}}$ and measured $SC_{i}^{\text{Exp}}$ data sets as given by:

After the particle sizes were characterized via the optical scattering measurement, the same particles were subsequently measured by SEM and visualized in Fig. S6a. It is important to note that the small Si nanoparticles accumulated near the principle ones (for 177 nm, 188 nm, and 191 nm) when the sample was removed from the water-filled chamber and dried in a vacuum chamber. The surface tenson of the evaporating water could easily drag smaller particles, which usually have weaker adhesion to the surface, to the proximity of the larger ones. Therefore, the presence of these satellites in the SEM micrographs did not alter the scattering spectra and affect the sizes of the principle Si nanoparticles measured by SEM and the validation of particle size determination by comparing the measured and LMT-calculated scattering spectra . This agreement demonstrates the reliability and high precision of the size characterization with the in-situ scattering measurements of the Si nanoparticles in an optical trap.

The magnetic polarizability $\alpha^{\prime}_{\text{m}}$ of 162 nm dia. Si nanoparticles is positive yet smaller than its electric counterpart (refer to Fig. S9b). $F_{\text{MD}}^{\text{PD}}$ and $F_{\text{ED}}^{\text{PD}}$ are calculated using Eqs. 14 and 15 and shown in Fig. S7c. The magnetic dipole force on the Si nanoparticle is toward the beam center. However, this is opposed by the electric dipole force that is directed outward and has a greater magnitude. Consequently, the particle is trapped within the annular region of the azimuthally polarized beam where the electric field amplitude is maximum. This conclusion agrees with the experimentally measured and ED-LD simulated dynamics of small-sized Si nanoparticles (see Fig. 5b in the main text).

Conversely, the 770 nm trapping laser is nearly maximally on-resonance with the magnetic dipole resonance of the 205 nm dia. Si nanoparticle as seen in the scattering spectrum shown in Fig. S7a. This could cause a strong interaction with the longitudinal magnetic field $H_{z}$. Therefore, one might expect the magnetic dipole force to play the decisive role in the trapping behavior as compared to the electric counterpart. However, this expectation is not substantiated by the force spectra plotted in Fig. S7c, where a lightly repulsive magnetic dipole force is found instead. This unexpected behavior can be understood by inspecting the magnetic polarizability shown in Fig. S7a. Near the magnetic dipole resonance, $\alpha^{\prime}_{\text{m}}$ is close to but less than zero for a 205 nm dia. Si nanoparticle resulting in the unexpected direction of the net force. A significant repulsive magnetic dipole force is found for the 239 nm Si nanoparticle since the polarizability is now strongly negative.

The calculated repulsive nature of the optical magnetic dipole force seems to be in contradiction with the strong trapping of larger Si nanoparticles at the center of the azimuthally polarized beam ( where $H_{z}$ is maximal) that is observed in the experiments. Hence, it is evident from the contradictory experimental analyses that another type of force must be responsible for the trapping behaviors of larger Si nanoparticles.

In order to probe the behavior of the dipole-governed transverse scattering (TS) force and validate our proposed SCM, force analyses are carried out for different size Si nanoparticles that span the trapping situations of interest: off-, near- and on-resonance with the optical magnetic dipole mode. These forces are evaluated with the same procedure and method used in Fig. 3D; hence, the data should be read in the identical manner. Noting that all forces are normalized by the maximum value of the MST-calculated forces for ease of comparison.

Two remarkable conclusions can be drawn from the results shown in Fig. S8, one of which is obvious while the other requires further scrutiny in conjunction with the parameters shown in Fig. S9. The first is how well the SCM accounts for the electrodynamic forces that are absent from the point dipole model, particularly in the size region associated with on-resonance trapping . The curves of both the $F^{\text{SCM}}$ and $F^{\text{MST}}$ are in perfect agreement for the entire range of particle sizes. When the particle size is small (Dia.$\leq$180 nm), the scattering forces $F_{\text{ED/MD}}^{\text{TS}}$ are negligible, meaning that the point dipole model is accurate in capturing the optical forces. This is evident from the good agreement between the curves of $F^{\text{PD}}$ and $F^{\text{MST}}$. Increasing the particle size transitions the system into the on-resonance regime (for the 770 nm trapping laser) where the contributions from the transverse scattering force $F_{\text{ED/MD}}^{\text{TS}}$ becomes pronounced. As a result, the accuracy of the point dipole model deteriorates as seen in the growing discrepancy between $F^{\text{PD}}$ and $F^{\text{MST}}$. However, including $F_{\text{ED/MD}}^{\text{TS}}$ allows agreement with $F^{\text{MST}}$, thus affirming that the TS forces are the missing feature of the point dipole model.

The second conclusion concerns the spatial anisotropy of the dipole-governed TS forces; that is, the scattering forces opposite signs along the orthogonal (x- and y-axis) directions for a Gaussian trapping beam linearly polarized along the x-axis (see Fig. S10). This finding holds even for the small Si nanoparticles. The root cause of this anisotropic phenomenon is discussed in Methods “Analytical approximation of transverse scattering force” in association with Eq 24.  The interesting finding revealed from numerical calculations is the reversal of the scattering force that occurs between the small- (dia.$\leq$180 nm) and large-sized (dia.$\geq$190 nm) Si nanoparticles. The small-sized Si nanoparticles experience a repulsive magnetic field-associated TS force ($F_{\text{MD}}^{\text{TS}}$) along the x-axis but an attractive electric field-actuated TS force ($F_{\text{ED}}^{\text{TS}}$) along the y-axis. The anisotropic behaviors of these two field-associated TS forces are reversed for the large Si nanoparticles. According to Eq. 24 and shown in Fig. S9b, this reversal is attributed to the Im$\left[\alpha_{\text{e}}\alpha_{\text{m}}^{*}\right]$ term, which changes sign from positive to negative at the size labeled as the first type Kerker condition. Based on the convention of the Kerker conditions\citesupplkerker1983electromagnetic, and illustrated by the plots in Fig. S9a, the first type Kerker condition requires the equality of both the magnitude and phase of the electric and magnetic polarizabilities, giving rise to an in-phase interference of the electric and magnetic dipolar fields. In contrast, the second Kerker condition occurs with a $\pi/2$ phase shift between the polarizabilities, causing the respective fields to manifest $\pi$ out-of-phase destructive interference. It is apparent from Fig. S9a that only the first Kerker condition affects the TS forces’ change of sign with size.

Finally, we note that Fig. S9b shows that the Im$\left[\alpha_{\text{e}}\alpha_{\text{m}}^{*}\right]$ term goes to zero for small Si nanoparticles (dia.$<$130 nm). Therefore, the transverse scattering forces vanish for these small nanoparticles and conventional electric dipole trapping as described in the Mie-corrected point dipole approximation are obtained. therefore, the transverse scattering forces associated with the optical magnetic dipole resonance and described by Eq. 24 can only be obtained for large Si nanoparticles.

Figure S10 shows the performance of the analytical approximation of the transverse scattering (TS) force (Eq. 24) in both tightly and loosely focused Gaussian beams with linear polarization (marked by the arrows in the insets) as assessed by comparing it with the numerical method (Eq. 19). The high accuracy of the latter was validated in Sect. S8. When the Gaussian beam is loosely focused and the paraxial approximation is valid, the energy densities of the electric and magnetic fields shown in the insets of Fig. S10b are spatially indistinguishable, indicating that Eq. 25 and 24 are equivalently same. The excellent agreement with the numerical result (of Eq. 19) confirms the accuracy of the analytical approximation in a paraxial beam.

For a tightly focused Gaussian beam as shown in Fig. S10a, the energy densities of the electric and magnetic fields start to be spatially distinct from each other. The energy densities become elongated in orthogonal directions in relation to the respective electric and magnetic field polarization directions. The anisotropy in the spatial distributions indicates an inaccuracy of Eq. 25, which assumes uniform energy densities to obtain the TS forces. By contrast, Eq. 24 accounts for the anisotropy of the fields, resulting in a satisfactory approximation relative to the numerically calculated TS forces. These comparisons affirm the discussion in Sect. S8 that the applicability of the analytical approximation for a tightly focused beam, and hence its utility in unraveling the nature of the dipole-governed TS forces.

Figure S11 shows the TS forces acting on a 200 nm Si nanoparticle obtained from simulations in a tightly focused azimuthally polarized beam. The TS forces experienced by the Si nanoparticles are for the case of on-resonance trapping. Equation 19 (solid black curve) is used to accurately determine the TS force acting on the particle. The analytical approximation (Eq. 24; solid red curve) is used to determine the TS force’s constituents attributed to the azimuthally polarized electric $E$ field in the annular region and the longitudinally polarized magnetic field $H_{z}$ at the center as shown in Fig. S11c and b, respectively. Comparing the results of these two methods, their similarity, particularly in the central region of the beam, affirms the validity of the analytical approximation (Eq. 24) as well as its representation of the electric and magnetic field contributions. As seen in Fig. S11, the $H_{z}$ associated TS force ($F_{H_{z}}^{\text{TS-ana}}$) drives the 200 nm dia. Si nanoparticle toward the magnetic field maximum, while the $E$ associated TS force ($F_{E}^{\text{TS-ana}}$) simultaneously repels the particle away from annular region where the electric field is maximized. This repulsion causes the confinement of the particle in the central region of the beam. In the case of the 200 nm dia. Si nanoparticle, the contribution from the radially polarized magnetic field $H_{\rho}$ along the radial direction is negligibly small, in part because the magnitude of the $H_{\rho}$ is relatively small in the tightly focused beam (refer to Fig. 1b in the main text). Therefore, The TS force actuated by $H_{\rho}$ is not important in the present work.

Figure S12 shows the optical force ($F^{\text{Tot}}$) exerted on the 205 nm dia. Si nanoparticle trapped in the azimuthally polarized beam. The total force is decomposed into individual components delineated by the constituent fields (i.e., $E$, $H_{\rho}$, and $H_{z}$) of the beam and associated categories of forces (i.e., dipole $F^{\text{PD}}$ and scattering $F^{\text{TS}}$ forces). The scattering forces are evaluated with the analytical method (Eq. 24) that was quantitatively examined and validated in Sect. S10.

According to the force plots of Fig. S12, the electric field $E$ produces a balanced situation where the dipole and scattering forces are nearly equal in strength but opposite in direction, causing their combined trivial influence on the particle’s dynamics. Likewise, the impact of the radially polarized magnetic field $H_{\rho}$ on the particle’s trapping behavior is minimal, as reflected in its very small dipole force. This $H_{\rho}$-associated force is very small because of the near-zero magnetic polarizability of particles of this size (see Fig. S7a). In fact, the force is repulsive (from the beam center) due to the negative magnetic polarizability of the large 205 nm dia. particle size and associated resonance peak of the magnetic dipole mode. On its own, this $H_{\rho}$-associated force would push the particle away from the annular region with the $H_{\rho}$ towards the central region. The associated scattering force has intentionally been excluded from consideration, mainly because this force has no contribution along the radial direction, as explained in Sect. S10.

In contrast with the above-mentioned two fields, the longitudinal magnetic field $H_{z}$ exerts the decisive influence on the electrodynamics of the 205 nm dia. nanoparticle, as evident from its associated scattering force that is dominant over other forces and pushes the particle toward the central region where the induced magnetic field is maximum. In addition, the longitudinal magnetic field $H_{z}$ also exerts a repulsive dipole force on the particle because of the negative magnetic polarizability $\alpha_{\text{m}}^{{}^{\prime}}$ of the Si nanoparticle (see Sect. S7). However, the magnitude of $\alpha_{\text{m}}^{{}^{\prime}}$ is close to zero, leading to the repulsive dipole force that is sufficiently small to be negligible.

Figure S13 shows the optical force and its decomposition for a large 239 nm dia. Si nanoparticle. Compared to the 205 nm Si nanoparticle, the optical force associated with the electric field $E$ becomes important in addition to the optical force associated with and magnetic field $H_{z}$. Given the electric and magnetic polarizability of the particle (see Fig. S7a and Fig. S8b and c), the scattering and dipole forces associated with the $H_{z}$ are closely matched in strength (see lower panel in Fig. S13), which results in a cancellation that essentially eliminates the field’s impact on the particle’s dynamics. By contrast, the forces resulting from the $E$ and $H_{\rho}$ (see upper panel in Fig. S13) become the primary factors that constrain the particle in the central region through repulsive forces. However, the underlying mechanisms behind the $E$- and $H_{\rho}$-associated forces are different. The repulsion due to $H_{\rho}$ originates from the magnetic dipole, while the repulsion due to the $E$-field stems from transverse scattering force. The origin of the repulsive magnetic dipole force has been discussed in Fig. S12. As for the $E$-associated scattering force, its rapid growth is attributed to the particle becoming on-resonance with the electric dipole mode, which significantly increase the electric dipole radiation. Thus, in addition to the optical forces mediated by the magnetic fields, the scattering force associated with the electric field acts as an additional contribution, enhancing the effectiveness of the azimuthally polarized beam in confining the particle to the center of the beam.

Since its inception in electronic systems, the Hall effect has been routinely used to describe transport of charge carriers driven by applied electric fields in conductors/semiconductors or photons propagating through media with refractive index variations. Light has been proven to exhibit spin Hall effect, known as spin Hall effect of light\citesupplonoda2004hall,hosten2008observation. This effect occurs when light crosses a planar or structured optical interface, leading to a transverse splitting of light into components carrying opposite spin angular momentum (i.e., right- and left-handed circular polarizations) because of spin-orbit interactions\citesupplonoda2004hall,hosten2008observation. This principle extends to nanoparticles. Probing the energy flux in proximity to a nanoparticle illuminated by a beam reveals that the particle modulates its near-fields with opposite directions that depend on the incident beam’s circular polarization (spin) states\citesupplhaefner2009spin. In addition, recent work has shown the possibility of enhancing the photonic spin Hall effect by breaking the parity symmetry of the systems\citesupplrodriguez2010optical,li2019photonic,neugebauer2018weak. For instance, parity-symmetry breaking, induced by either the anisotropic geometry or spatial displacement of single nanoparticles from the beam center, allows dramatically altering the directionality of the particles’ far-field scattering light by switching between the excitation beam’s two opposite spin states\citesupplli2019photonic,neugebauer2018weak.

Our definition of the photonic Hall effect (PHE) builds upon this previous work. In the present study, we identified a linear polarization-dependent directionality of far-field electromagnetic radiation scattered by single Si nanoparticles for an illumination condition that breaks parity-symmetry (mirror symmetry). Since this far-field scattering is associated with the excitation light’s linear polarization state rather than its spin state, we refer to this phenomenon as the “photonic Hall effect” to differentiate it from its spin counterpart. How this effect take places is demonstrated in Figs. S14 and S15, where the far-field of scattering light from a 205 nm dia. Si nanoparticle is calculated for two illumination conditions: (1) with the nanoparticle centered transversely in the Gaussian beam as shown in Fig. S14; and (2) with the Si nanoparticle shifted away from the center along the x-axis by a distance $\Delta x$ as shown in Fig. S15. Note that the calculations are based on the GLMT as detailed in the Methods section.

Figure S14 shows that when the Si nanoparticle is positioned precisely at the center of a symmetric focused Gaussian beam, its far-field scattering intensity distribution, $I_{\text{scat}}^{\text{far}}$, is computed as a function of the polar angle, $\theta$. We consider the scattered intensity in the x-z plane and $\theta$ is relative to the z-axis (or incident light propagation direction, $\vec{k}$). The plot illustrates that the particle predominantly scatters light along and symmetrically about the z-axis. This behavior persists when switching between the two orthogonal linear polarization states of the incident light: (1) the incident electric field $E_{\text{in}}\perp$ y-axis and (2) the incident magnetic field $H_{\text{in}}\perp$ y-axis (equivalently, $E_{\text{in}}\parallel$ y-axis). However, as shown in Fig. S15, when the particle is displaced from the center by a small distance, $\Delta x$, the angular distribution of the scattering is no longer independent of the linear polarization state of the incident beam. The displacement-induced (parity) symmetry breaking leads to a shift of the peak (and mean) of the angular scattering distributions (as indicated by the dashed vertical lines) in opposite directions relative to the z-axis. Each distribution, with opposite angular shifts, corresponds to one linear polarization state of the incident light. This dependence of the scattering directionality on the incident light’s linear polarization state manifests the proposed photonic Hall effect. This PHE results in what we have termed ”transverse scattering forces,” which, according to Eqs. 3 and 24, are linked to the exciting electric and magnetic fields and demonstrate their quasi-anisotropy. These characteristics have not been discovered by any previous studies concerning the optical trapping behaviors influenced by interactions between light’s magnetic component with matter\citesupplchen2011optical,duan2022asymmetric,duan2021transverse.

In this section, we show that the magnetic field-mediated optical trapping and its associated transverse scattering optical forces that are demonstrated in this paper for Si nanoparticle is applicable to a wide range of materials. These nanoparticles and materials may have other useful functionalities. Furthermore, nanoparticles made of other materials are candidates for nano-/macro-particle manipulations across the visible and mid-infrared spectral regions. To demonstrate this, Figure S16 shows the scattering cross-sections for nanoparticles made of four commonly used semiconductors (i.e., SiC\citesupplsingh1971nonlinear, CdS\citesupplgomez1985weighted, PbTe\citesupplweiting1990temperature, and AlAs\citesupplfern1971refractive, ) characterized by high refractive indices, $n$, and low extinction coefficients, $\kappa$, for the effective field confinement and suppression of thermal effects. Analogous to Si, nanoparticles of these materials exhibit modified optical Mie scattering spectra (black curves) featured by the ED (red curves) and MD (blue curves) components due to their individual complex indices of refraction and boundary conditions. With appropriately engineered sizes for each material, they all exhibit pronounced MD modes, with the resonance peaks spectrally separated from those of the ED modes, implying their suitability for magnetic field-mediated optical trapping. The efficacy of the MD modes can be further enhanced if the ambient condition is switched to air or vacuum (n=1.0) due to the increased refractive index contrast. Figure S17 shows the Mie scattering cross-section of a SiC nanoparticle in air, where its MD’s increases as the resonance narrows giving a higher Q-factor compared to its MD mode when the same size SiC nanoparticle is in aqueous solution (shown in Fig. S16).

Magnetic field-mediated optical trapping is demonstrated by the ED-LD simulations for AlAs shown in Fig. S18. This example illustrates the dynamics of the AlAs nanoparticle within an azimuthally polarized beam. The two distinct laser wavelength excitations underscore the two types of trapping mechanisms that are predominantly driven by the magnetic (left panel) and electric components (right panel) of light, respectively. Specifically, When the trapping laser wavelength is set at 510 nm, strongly on-resonance with the MD mode (see Fig. S16), the AlAs nanoparticle is trapped in the central region of the azimuthally polarized beam, a region dominantly occupied by the longitudinally polarized magnetic field $H_{z}$ (refer to Fig. 1 in the main text). Tuning the trapping laser to 600 nm makes the particle’s ED mode dominant, which causes a shift of the trapping from the beam’s center to its annular region due to overwhelming electric field-matter interactions. The trapping dynamics shown in Fig.  S18, which are consistent with those observed experimentally for Si nanoparticles, also apply to the other three materials (SiC, CdS, and PbTe). These simulation results affirm the broad applicability of optical magnetic field associated trapping.

naturemag \bibliographysupplscibib