Introduction

Since its inception in 19701, optical trapping has developed into a large field of investigation and application, catalyzing substantial advances in diverse disciplines such as biophysics2,3,4, material science5,6,7,8, and macroscopic quantum physics9,10,11. Even after five decades of research, new topics in optical trapping continue to emerge. These include fascinating physical phenomena such as optical matter formation12,13,14,15,16,17, many-body photonic interactions and symmetry breaking-induced dynamics leading to optical matter machines18,19, symmetry-breaking electrodynamics20,21,22,23,24,25,26,27,28, and mesoscopic dipole interactions29,30,31.

All of the aforementioned studies and essentially all of optical trapping heretofore utilized the optical force associated with the electric fields and gradients in the point dipole32,33,

$$\langle {{{{\bf{F}}}}}_{{{{\rm{e}}}}}^{{{{\rm{PD}}}}}\rangle=\frac{{\alpha }_{{{{\rm{e}}}}}^{{\prime} }}{4}\nabla | {{{\bf{E}}}}{| }^{2}+\frac{{\alpha }_{{{{\rm{e}}}}}^{{\prime\prime} }}{2}| {{{\bf{E}}}}{| }^{2}\nabla \Phi$$
(1)

while neglecting the (possible) magnetic counterpart (see “Methods” for a more general form for Eq. (1)). The first and second terms in Eq. (1) represent: (i) the intensity gradient force associated with the incoming electric field E and the real part of the electric polarizability \({\alpha }_{{{{\rm{e}}}}}^{{\prime} }\); and (ii) the radiation pressure associated with the gradient of the phase, Φ, of the electric field (i.e., the local tilt or curvature of the wavefront) and the imaginary part of the electric polarizability \({\alpha }_{{{{\rm{e}}}}}^{{\prime\prime} }\). Although optical beams are electro-magnetic fields, where both their electric and magnetic components can potentially interact with matter, near universal adoption of Eq. (1) seems reasonable since induced magnetizations are practically zero at optical frequencies32.

Still, if one considers the possibility of optical magnetic forces, the most simple expectation is based on a point-dipole model34 that the magnetic field-mediated force behaves analogous to the electric version. That is,

$$\langle {{{{\bf{F}}}}}_{{{{\rm{m}}}}}^{{{{\rm{PD}}}}}\rangle=\frac{{\alpha }_{{{{\rm{m}}}}}^{{\prime} }}{4}\nabla | {{{\bf{H}}}}{| }^{2}+\frac{{\alpha }_{{{{\rm{m}}}}}^{{\prime\prime} }}{2}| {{{\bf{H}}}}{| }^{2}\nabla \Phi,$$
(2)

where all terms are the magnetic (field) analogs of those in Eq. (1). This expression is analogous to what is given in ref. 35; see the “Methods” for a generalized form of Eq. (2). The magnetic field-associated gradient force (first term in Eq. 2) says that positive optical magnetization would give rise to an attractive force. Yet, despite some recent theoretical work34,36 suggesting possible routes for experimental confirmation using magnetodielectric nanoparticles with their inducible magnetizations, experimental evidence of optical magnetic forces being important in optical trapping is still lacking. One exception is a recent experimental study of photoinduced magnetic dipolar forces using (atomic) force microscopy35.

There are also unresolved fundamental questions concerning induced optical magnetic forces. In particular, Novotny predicted32 that induced magnetization in optical trapping would behave opposite to what induced electric polarizations do; i.e., a particle with a negative induced magnetization will be attracted to a magnetic field maximum. This suggestion contradicts the expectation based on Eq. (2). These conflicting perspectives urgently demand confirmation of the underlying mechanism of optical magnetic field trapping specifically and light-matter interactions generally.

Although Eqs. (1) and (2) are suitable for point particles (and, as we affirm below, for off-resonance conditions), a correct description of matter-radiation interactions for finite-size particles involves higher-order electric and magnetic multipole modes (and cross-terms), resulting in complete yet more complicated theoretical expressions that necessitate careful design and the interpretation of experiments. Moreover, if the wavelength of the optical trapping beam is on-resonance, these modes will dominate the interaction. In the present study, the electric dipole (ED) and magnetic dipole (MD) modes are on-resonance, and only Eq. (3) (given below) suffices for quantitative interpretation of our experimental findings.

In this paper, we experimentally demonstrate optical magnetic field associated trapping of nanoparticles with a large refractive index and provide a quantitatively accurate theoretical description thereof. Large index of refraction Silicon (Si) nanoparticles of various sizes were investigated, exploring the spectral detuning of their Mie optical magnetic dipole resonance relative to the fixed trapping laser wavelength. We observed that when optically excited at their magnetic dipole resonance, single Si nanoparticles selectively (thermally) sample the most intense magnetic field regions of focused vector beams37. However, application of the point dipole optical magnetic (magnetization) model of Eq. (2) cannot explain all of our observations. In particular, when the Si nanoparticle diameter is large (i.e.,  >200 nm dia.), it has negative optical magnetic polarizability at the 770 nm trapping laser wavelength. Therefore, by Eq. (2), it should be repelled from regions of strong optical magnetic fields, but it is not.

We show that the full range of optical trapping observations we report can be explained by including transverse scattering forces (TSF) arising from the Photonic Hall Effect (PHE)38,39,40,41 (see additional details in Supplementary Note 16). The PHE we introduce is distinct from the magnetically induced transverse diffusion of light, which is more analogous to the Beenakker–Senftleben effect42,43, despite its recent designation as PHE44. We develop a scattering-corrected model (SCM) using generalized Lorentz-Mie theory (GLMT)45,46,47 that includes the PHE-associated optical force. The latter is a nonconservative scattering force that manifests an anisotropy associated with the polarization of the trapping laser field. The optical magnetism and transverse scattering forces are validated by Maxwell stress tensor (MST) calculations. Extension to other high index material-based nanoparticles is demonstrated (see Supplementary Note 17).

Results

Optical beams and materials with strong optical magnetic properties

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. Figure 1a shows the result of an electrodynamics simulation of a focused azimuthally polarized vector beam37. The spatially segregated magnetic field32,37,48 is primarily a longitudinally polarized component, Hz, at the beam center (where the electric field vanishes) together with the azimuthally polarized electric component, E. There is an additional magnetic component of light with radial polarization, Hρ, that shares the annular region with the electric component. Note that we use Hρ to designate the magnetic field transverse to the Hz component, but also to indicate that this field is radially polarized for focused azimuthally polarized vector beams. Figure 1b shows that the intensity of the longitudinally polarized magnetic field, Hz, increases with the tightness of the beam focus; the ratios ∣ηHz∣2/∣E∣2 and ∣Hz∣2/∣Hρ∣2 increase with the numerical aperture (NA) of the objective (shown from NA = 0.4 to 1.29)32,37. When the azimuthally polarized beam is tightly focused with a NA of 1.27 (our experimental condition), the longitudinal magnetic field Hz attains an intensity 1.5 and 3.4 times greater than the transverse electric E and magnetic Hρ 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 ϵw and μw are water’s permittivity and permeability. This focused trapping beam would enhance the Hz 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.

Fig. 1: Optical magnetic field-associated trapping of Si nanoparticles with an azimuthally polarized beam.
figure 1

a The intensity profile of a tightly focused azimuthally polarized beam decomposed into three constituent fields in the focal volume: E (reddish), Hρ (reddish), and Hz (blue), with black arrows representing their polarization directions. b The ratios of Hz to E and Hρ intensities as a function of numerical aperture (NA) of the focusing lens. The black dots in the plot indicate the ratios for an NA of 1.27, which corresponds to the microscope objective used for trapping Si nanoparticles in water (n = 1.33). c The scattering cross sections of Mie multipole resonances of Si nanoparticles are calculated at the 770 nm trapping laser wavelength. The first intersection between the MD and ED scattering cross-section curves occurs at Si nanoparticle dia. =182 nm. This nanoparticle size separates the optical trapping into E-field- and H-field-dominated trapping domains. The hatched and grey regions represent the E-H transition (182–190 nm) and higher order multipole dominated (>250 nm) regions, respectively. Si nanoparticles with sizes from about 180 to 250 nm in dia. manifest trapping forces with strong optical magnetic contributions. d The calculated potential energy profiles across the focused azimuthally polarized vector beam illustrate the expected (energetic) tendencies of Si nanoparticles to trap in the E- and H-field dominated portions of the optical beam.

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Demonstration of optical magnetic trapping also requires a material with a strong optical magnetic dipole resonance. Si nanoparticles satisfy this condition49 (see Fig. 2d) and are small enough that they can interact dominantly with the longitudinally polarized magnetic field Hz via their Mie magnetic dipole (MD) resonance. This electromagnetic field-nanoparticle interaction is expected to create optical magnetic field-associated 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) resonance49 that can be excited by the annularly polarized and distributed electric field E. Therefore, electric field-associated forces (Eq. (1)) can also act on the Si nanoparticles. The magnitude of these two types of forces depends 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 scattering cross-section of the MD mode dominates over the ED mode only in a narrow size range about 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; i.e., that only Si nanoparticles in a certain size range (as shown in Fig. 1c) will manifest optical magnetic field trapping.

Fig. 2: Experimental evidence of optical magnetic field-associated trapping of Si nanoparticles.
figure 2

a Schematic representation of the microscope objective that is employed to generate a tightly focused azimuthally polarized beam for trapping Si nanoparticles in a water filled chamber. The closeup schematic of the focused beam (at the top) shows a Si nanoparticle at the trapping plane of the azimuthally polarized beam, with arrows indicating the longitudinal and azimuthal polarization of Hz (blue) and E (red), respectively. b SEM micrograph of a drop-cast Si nanoparticle that coincidentally has the same dimensions as the trapped one shown in (c). c Dark-field microscopy image of a representative Si nanoparticle with a diameter of 205 nm trapped at the center of the focused azimuthally polarized beam delineated by the white dashed circle. d The size of the Si nanoparticle shown in (c) is determined by comparing the particle’s in-situ measured scattering spectrum (upper panel) with calculated Mie scattering spectra (lower panel; calculated in a water environment) and adjusting the Si nanoparticle size in the simulations until their scattering spectral signatures align, including the multi-pole modes. The red dashed vertical line indicates the trapping laser wavelength. The inset is a color dark-field microscopy image of the specific 205 nm dia. Si nanoparticle. e Nanoscale localizations of three representative Si nanoparticles with different sizes overlaid with their measured trapping beam profiles as gray-scale background. The upper and lower rows exhibit the experimental and GLMT-Langevin dynamics simulation results, respectively. The scale bars in the experimental images are 720 nm. f The radial position distributions associated with the experimental (upper) and simulation (lower) results of (e).

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Optical trapping of Si nanoparticles

Testing the aforementioned hypotheses and verifying optical magnetic trapping is accomplished by obtaining the Si nanoparticle’s position distribution functions to determine where they 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 and hence allow determining the nature of matter-radiation interactions in the inhomogeneous trapping field (Fig. 1a). The calculated 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 largest size of Si nanoparticles that we consider in the experiments is limited by when their magnetic quadruple (MQ) mode in Fig. 1c begins to affect the scattering (at dia. >250 nm). As a result, optical forces due to higher-order modes become dominant for larger Si nanoparticle sizes (see Supplementary Fig. 5) and fall outside the scope of this study. Furthermore, we do not consider thermophoresis50 due to the small absorbance of Si nanoparticles at 770 nm (see Supplementary Fig. 4) and negligible heat transfer to solvent.

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 μm) water-filled chamber containing colloidal Si nanoparticles suspended in water. 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 Hz (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 Supplementary Notes 4 and  6.

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  ±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 calibration is provided in Supplementary Fig. 6.

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 Supplementary Fig. 3). 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 localization51 of single Si nanoparticles and the associated probability density functions (PDF) of different trapped Si nanoparticles along the radial direction are shown as red, blue, and green colors in Fig. 2e, 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 Hz 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 electromagnetic field of the trapping laser via Eq (1). The result for the smaller (162 nm dia.) Si nanoparticle is consistent with the well-established electric point-dipole model of electric-field light-matter interactions34.

Surprisingly, Fig. 2e, f also show that a large, 239 nm dia., Si nanoparticle (green) is strongly trapped at the center of the beam, with an even narrower distribution in the radial position PDF! This result is unexpected from either Eqs. (1) or (2). First, 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, adding to the surprise, Supplementary Fig. 9 shows that 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 Supplementary Fig. 7), 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 discussed here in detail; Si nanoparticles smaller than  ~185 nm dia. were trapped in the annular region of the focused azimuthally polarized beam, while all larger nanoparticles were trapped in the center region. See the Supplementary Notes 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 Eqs (1) and (2)) and that this so far unaccounted for interaction dominates the electrodynamic forces acting on large Si nanoparticles.

Transverse scattering (TS) force and scattering corrected model (SCM)

To resolve the seemingly conflicting trapping behaviors discussed above, we introduce the transverse scattering forces associated with the photonic Hall effect. We first establish their unique properties using a simple model employing a linearly polarized Gaussian beam.

Figure 3 qualitatively and quantitatively demonstrates the directional forces associated with the transverse directional scattering of light. 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), and the directions of the E and H fields for a linearly polarized Gaussian beam result 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 transverse scattering (TS) force is generated, as shown schematically in Fig. 3b. For this size nanoparticle, the scattering of 770 nm light originates from both the ED and MD modes. The excited dipole moments align with their respective exciting field polarizations 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 it makes a negligible contribution to the total force on that and similarly “small” Si nanoparticles. Therefore, the point-dipole model of Eq. (1) accurately describes the electrodynamic forces in this size regime (see Supplementary Fig. 8).

Fig. 3: Transverse scattering force(s) on Si nanoparticles due to the photonic Hall effect (PHE).
figure 3

a The PHE for the case of the 205 nm dia. Si nanoparticle is demonstrated by characterizing its scattering properties at three designated positions (labeled 1, 2, and 3) in a linearly-polarized, tightly-focused Gaussian beam. b–d Normalized far-field radiation patterns (left) of the Si nanoparticle at the three locations in (a) show that light scatters differently depending upon the polarization state of the optical beam and the local intensity gradients of the exciting electric and magnetic fields. These results define the force contributions from the PHE on the Si nanoparticle described in Eq. (3). The schematic (right) describes the Mie transverse scattering (TS) from a Si nanoparticle optically excited near its electric or magnetic dipolar resonances. A particle at the center of the optical trapping beam manifests symmetric scattering that becomes asymmetric when the particle is away from the center. A net force \({F}_{{{{\rm{net}}}}}^{{{{\rm{TS}}}}}\) arises to balance the momentum carried by the directional scattering. e Optical forces exerted on the 205 nm dia. Si nanoparticle calculated at various positions along the (left) x- and (right) y-directions in the Gaussian beam shown in (a). The forces are calculated by four methods: transverse scattering (TS; orange dashed); point-dipole (PD; purple dashed) approximation; Maxwell stress tensor (MST; black dashed), and our scattering-corrected model (SCM; green dashed). Note, all forces are normalized to the maximum of the MST-calculated force.

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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 linearly polarized 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. Therefore, the resulting transverse scattering forces must be added to the gradient force model described in Eqs. (1), (2), and (4) (in “Methods”), as reflected in Eq. (9). A more readily appreciated analytical approximation of the transverse scattering force given (or Eq. (15)):

$${F}^{{{{\rm{TS}}}}-{{{\rm{ana}}}}}={F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}+{F}_{{{{\rm{ED}}}}}^{{{{\rm{TS}}}}}\,\cong \,-\frac{{k}^{3}{\mu }_{{{{\rm{m}}}}}}{12\pi {\epsilon }_{{{{\rm{m}}}}}}{{{\rm{Im}}}}\left[{\alpha }_{{{{\rm{e}}}}}{\alpha }_{{{{\rm{m}}}}}^{*}\right]\left(\nabla {U}_{B}{\cos }^{2}\theta -\nabla {U}_{E}{\sin }^{2}\theta \right),$$
(3)

where k denotes the wavevector in the surrounding medium with permittivity ϵm and permeability μm; UB and UE represent the energy densities of the magnetic and electric fields of light, respectively. The parameter θ is the angle between the electric field vector (i.e., polarization direction) and the direction corresponding to the gradient of the respective field’s energy densities. See the derivation in Eqs. (12–15) in “Methods” and the quantification in Supplementary Notes 8–15. The force depends on both the electric and magnetic polarizabilities, αe and αm. A general rule can be deduced from Eq. (3): MD-associated TS forces (\({F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\)) attract particles to magnetic field maxima, while ED-associated TS forces (\({F}_{{{{\rm{ED}}}}}^{{{{\rm{TS}}}}}\)) repel particles away from electric field maxima. This polarization-dependent directional scattering is the photonic Hall effect38,39,40 and is essential for understanding the trapping behavior of Si nanoparticles in the focused optical vector beam optical trap.

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. We determined the TS forces and their directional properties as indicated in Eq. (3) by numerically solving for the net momentum carried by the (anisotropically) scattered radiation and thus quantify the TS forces with Eqs. (10) and (11) in Methods. These results are compared to rigorous numerical MST (FMST) calculations. Figure 3e 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. Specifically, Fig. 3e shows the TS forces (\({F}_{{{{\rm{ED}}}}/{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\); orange curves) 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 angle-dependent 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)32 or point-dipole model (PD, in green dash)32, 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 (also see Supplementary Fig. 8). 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 (i.e., from Eqs. (1) and (2)) highlight the inadequacy of the conventional point-dipole model in describing the forces acting on the particle. In fact, the TS forces (FTS) quantitatively account for the forces omitted in the PD model. The resulting “Scattering Corrected Model” (SCM; green circles) \({F}_{{{{\rm{total}}}}}^{{{{\rm{SCM}}}}}={F}^{{{{\rm{PD}}}}}+{F}^{{{{\rm{TS}}}}}\) is appropriate for quantitatively understanding the forces of resonant optical trapping and optical magnetic contributions to the forces and dynamics (see “Methods” and Supplementary Note 11).

Size-dependence of SCM optical trapping forces of Si nanoparticles

The scattering-corrected model (SCM) of optical trapping forces explains the trapping behavior of Si nanoparticles in various optical beams, including the focused azimuthally polarized beam employed in our experiments and the intuitively more simple case of a (linearly polarized) focused Gaussian beam considered in Figs. 3 and 4. Figure 4 shows the trapping stiffness coefficients along the electric and magnetic field axes (as defined in Fig. 3a) as a function of particle size. The stiffness coefficient, κ, of an optical trap is defined as κ = F/r, where the F and r represent the optical force acting on the trapped particle and the displacement of the particle from its equilibrium position, respectively. The encoded colors indicate the dominant contribution to the force on the Si nanoparticle and how these contributions change with Si nanoparticle diameter; i.e., from dipole force FPD dominated (gray) to the ED/MD-associated TS forces \({F}_{{{{\rm{ED}}}}}^{{{{\rm{TS}}}}}\)/\({F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\) dominated (red/blue). The changes occur because the resonances shift to longer wavelengths with increasing Si nanoparticle size thereby tuning increasingly on-resonance with the ED and MD modes (as seen in Fig. 1c).

Fig. 4: Computed optical trap stiffness and force contributions as a function of Si NP diameter.
figure 4

Trapping stiffness coefficients of the Si nanoparticle in the x- and y-directions in the Gaussian beam shown in Fig. 3a as a function of nanoparticle diameter. The color is encoded to indicate the changing contributions transitioning from dipole force of Eqs. (1), and (2) to transverse scattering force dominated case as the particle’s size increases. Positive values of κy mean repulsion that pushes the particle away from the central region of the Gaussian beam in the y-direction.

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Two key observations arise from the results shown in Fig. 4. First, ∣κx∣ reaches its minimum value while ∣κy∣ 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-associated TS force. Second, ∣κx∣ and ∣κy∣ 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-associated TS forces, respectively, in orthogonal directions.

These two properties of the TS forces reveal the mechanism of optical magnetic field associated trapping. Figure 5a schematically decomposes the \({F}_{{{{\rm{ED}}}}}^{{{{\rm{TS}}}}}\) and \({F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\) optical forces exerted on a Si nanoparticle illuminated by an azimuthally polarized beam. Note that the electric field associated force is either radially inward or outward depending on whether the Si nanoparticle is closer or further than the E-field intensity maximum, respectively.

Fig. 5: Synopsis of experimental and simulated results of Si nanoparticle trapping in an azimuthally polarized beam.
figure 5

a The mechanism for an azimuthally polarized beam stabilizing Si nanoparticles at its center principally relies on the magnetic field associated scattering force \({F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\) actuated by the Hz component of the magnetic field. This inward-directed optical force drives Si nanoparticles to the most intense Hz region (left panel). The electric field associated scattering force \({F}_{{{{\rm{ED}}}}}^{{{{\rm{TS}}}}}\) actuated by the electric field E can enhance the trapping stability by radial repulsion (right panel). Note, this “pushing force” is bi-directional with respect to the radial gradient of the annular electric field profile. Therefore, the radially directed scattering force will help to confine Si nanoparticles with the magnetic field scattering force \({F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\) only near the beam axis. b Radial distribution of Si nanoparticles trapped in the azimuthally polarized beam plotted as a function of their diameters, ranging from 100 nm to 250 nm in steps of 5 nm (gray scale). Experimental radial distributions (red points with standard deviation error bars) of Si nanoparticles with selected dimensions are superimposed for comparison. c Comparison of the trap stiffness coefficient of different sized Si nanoparticles trapped in azimuthally, κAZM, and circularly polarized Gaussian, \({\kappa }_{{{{\rm{CP}}}}}^{{{{\rm{Gaus}}}}}\), beams provide guidance for choosing beams for trapping Si nanoparticles in the two distinct particle size regimes. All simulations and experiments are for 770 nm (vacuum) optical beams.

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The gray-scale results in Fig. 5b show the radial position probability density function (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 Supplementary Note 2). 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 of the Si nanoparticle (dia. = 200 nm). Moreover, the decrease in the standard deviations of the distributions with increasing particle size shows that the trapping efficacy is enhanced.

When the particle is excited near its optical magnetic dipole resonance, it is pushed toward the magnetic field maximum due to \({F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\) (Fig. 5a left panel). The optical MD mode of larger Si nanoparticles shifts to longer wavelengths relative to the (770 nm) trapping laser. Therefore, Si nanoparticles > 200 nm dia. experience radial inward directed forces due to not only optical magnetic field (Hz) through the magnetic TS force (\({F}_{{{{\rm{MD}}}}}^{{{{\rm{TS}}}}}\)) but also optical electric field (E) through electric TS force \({F}_{{{{\rm{ED}}}}}^{{{{\rm{TS}}}}}\) from the radial intensity gradient of E(r, Φ) 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, Hz. Moreover, the 239 nm dia. Si nanoparticle shows greater confinement in Fig. 2 because \({F}_{{{{\rm{ED}}}}}^{{{{\rm{TS}}}}}\), the transverse scattering force becomes larger for larger Si nanoparticles. More quantitative details of the effects of the transverse scattering forces and intensity gradient forces are shown in Supplementary Notes 15–16.

Discussion

We have demonstrated optical magnetic field trapping of Si nanoparticles in experiments using azimuthally polarized beams. The failure to accurately explain the experimentally observed electrodynamic forces on the Si nanoparticles with a simple extension of the standard electric point-dipole model (Eq. (1)) to describe magnetic dipole gradient forces (Eqs. (2) and (6) in “Methods”) led us to identify the ED- and MD-mediated transverse scattering forces and quantify their unique properties. This realization required the development of the scattering (force) corrected model (Eqs. (3) and (15) in “Methods”), which determines and includes the TS forces quantitatively (via numerical computation). The TS forces (FTS) provide a key missing element of the traditional point-dipole model, enable uncovering optical magnetic aspects of optical trapping, and give a more comprehensive description of electromagnetic trapping.

The manner in which the nanoparticles scatter light, especially for near- or on-resonance optical trapping, gives rise to an important and perhaps dominant force acting on the nanoparticles. Polarization-dependent directional light scattering from nanoparticles is already known in terms of 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 materials40,41,52. In contrast, in the present case, the existence of both MD and ED resonances and their interaction as indicated in Eq. (3) give rise to the photonic Hall effect (PHE) associated transverse scattering forces acting on the Si nanoparticles. The PHE and associated transverse scattering forces depend on: (i) the incident light’s linear polarization state; and (ii) the intensity gradients of the electric and magnetic fields. See Supplementary Note 16 for more details.

The resulting generalized description of optical trapping, including the importance of optical magnetic forces that this paper introduces through experiments, their simulation, and modeling, suggests opportunities to utilize the optical resonances of the (nanoparticle) sample. First, our results extend to other high-index-of-refraction materials, as shown in Supplementary Figs. 16–18, for both visible and mid-infrared wavelengths. Second, our results suggest that despite its generally divergenceless nature53, scattering forces can manifest stable optical trapping, in at least two dimensions. As indicated in Eq. (3), the TS forces resemble the well-known dipole forces; both rely on gradients of field energy density. This similarity allows the TS force fields, generated in azimuthally polarized beams as shown schematically in Fig. 5a, to create a well-defined potential solely from scattering forces. This is demonstrated in the electrodynamics-Langevin dynamics simulation results shown in Fig. 5c. These simulations, which compare the trapping stiffnesses of different size Si nanoparticles trapped in an azimuthally polarized beam (κAzm) and a circularly polarized Gaussian beam (\({\kappa }_{{{{\rm{cp}}}}}^{{{{\rm{Gaus}}}}}\)), reveal that while the Gaussian beam provides stable trapping of Si nanoparticles with small sizes (whose optical magnetic dipole is off-resonance with the trapping beam), its trapping force degrades dramatically as the particle becomes larger. In contrast, the optical magnetic field-mediated scattering is maximized in an azimuthally polarized beam when the trapping laser is maximally on-resonance with the MD mode.

This insight can empower new investigations of mesoscopic quantum optomechanical systems that are currently constrained by the use of red-detuned dipole traps that limit particle selection to those with small diameters and low refractive index materials54. Moreover, the trapping configuration we demonstrate, where particles are stabilized in the region of minimal electric field intensity of an azimuthally polarized beam, significantly reduces exposure to the high intensity of light (in the annular region or in a focused Gaussian beam), thereby substantially suppressing thermal perturbations that generally degrade quantum coherence54.

The dipole-governed anisotropic transverse trapping forces that arise under illumination of a nonuniform (intensity) linearly polarized beam can be leveraged to design optical-driven mechanical systems for a variety of applications, such as particle sorting and optical conveyors, as well as optofluidic wells55. Furthermore, the present work also creates opportunities to study N-body effects56 and the formation and mechanical dynamics of optical matter materials, such as nonreciprocal optical forces and negative optical torques18, arising from magnetic (vs. electric) light-matter interactions.

Finally, although this paper extends the domain of optical trapping appreciations by demonstrating optical magnetic field-associated trapping with transverse scattering forces, there remains an open question: Can particles be trapped in the longitudinally polarized magnetic field Hz solely by magnetic gradient forces? Answering this question is important since it will serve as direct evidence for the possibility of constructing red-detuned optical magnetic traps where only magnetic dipole forces are considered without scattering forces involved. This demonstration requires the electric and magnetic dipole resonances of the trapped particles to be spectrally well-separated so that the strength of MD-associated forces are large relative to ED forces and even larger than that of the Si nanoparticle we reported in this paper. We believe that this proof-of-principle experiment could be accomplished by utilizing meta-atoms48 and/or high-index materials with MD resonances in the mid-infrared region (see Supplementary Fig. 16).

Methods

Point-dipole (PD) approximation

In the PD model, the investigated nanoparticles (e.g., Si) are assumed to be Rayleigh particles that simultaneously exhibit electric and magnetic dipole resonances that allow interaction with the incident trapping laser’s electromagnetic field and, consequently, manifest two field-induced dipole moments. Both the electric and magnetic components of the illuminating light thereby have contributions to the overall mechanical force exerted by the electromagnetic field on the Si nanoparticles. Therefore, one can cast the mechanical force as

$$\langle {{{{\bf{F}}}}}^{{{{\rm{PD}}}}}\rangle=\langle {{{{\bf{F}}}}}_{{{{\rm{e}}}}}^{{{{\rm{PD}}}}}\rangle+\langle {{{{\bf{F}}}}}_{{{{\rm{m}}}}}^{{{{\rm{PD}}}}}\rangle .$$
(4)

The analytical expressions for each individual field-determined force can be expressed as32:

$$\langle {{{{\bf{F}}}}}_{{{{\rm{e}}}}}^{{{{\rm{PD}}}}}\rangle=\frac{{\alpha }_{{{{\rm{e}}}}}^{{\prime} }}{4}\nabla | {{{{\bf{E}}}}}_{{{{\rm{inc}}}}}{| }^{2}+\frac{{\alpha }_{{{{\rm{e}}}}}^{{\prime\prime} }}{2}| {{{{\bf{E}}}}}_{{{{\rm{inc}}}}}{| }^{2}\nabla \Phi,$$
(5)

and

$$\langle {{{{\bf{F}}}}}_{{{{\rm{m}}}}}^{{{{\rm{PD}}}}}\rangle=\frac{{\alpha }_{{{{\rm{m}}}}}^{{\prime} }}{4}\nabla | {{{{\bf{H}}}}}_{{{{\rm{inc}}}}}{| }^{2}+\frac{{\alpha }_{{{{\rm{m}}}}}^{{\prime\prime} }}{2}| {{{{\bf{H}}}}}_{{{{\rm{inc}}}}}{| }^{2}\nabla \Phi,$$
(6)

where the subscripts e and m refer to the electric- and magnetic-field related parameters, respectively, and \({\alpha }_{j}={\alpha }_{j}^{{\prime} }+i{\alpha }_{j}^{{\prime\prime} }\) denotes the particle’s complex electric (j = e) and magnetic (j = m) polarizabilities. Note that an alternative form of the PD model that is particularly suitable for trapping light with arbitrary polarizations (including circular polarization) can be found elsewhere57,58. The dipole forces only consider incident fields as indicated in Eqs. (5) and (6) (and implicit in Eqs. (1) and (2)). The first terms in Eqs. (5) and (6) are the intensity gradient forces (IGF), and the second terms are the radiation pressures that involve the phase gradient, ∇ Φ. As indicated by Eqs. (5) and (6), only the IGF influences the particle’s motion at the focal plane (where ∇ Φ = 0) in the PD model, while the radiation pressure is a longitudinal force driving a particle along the incident beam’s propagation direction.

Since the investigated Si nanoparticles have dimensions beyond (larger than) the Rayleigh regime, the PD theory needs to be extended by defining the particle’s polarizabilities in terms of the Mie scattering coefficients59,60,61,

$${\alpha }_{{{{\rm{e}}}}}=i\frac{6\pi {\epsilon }_{{{{\rm{NP}}}}}}{{k}^{3}}{a}_{1},$$
(7)

and

$${\alpha }_{{{{\rm{m}}}}}=i\frac{6\pi }{{\mu }_{{{{\rm{NP}}}}}{k}^{3}}{b}_{1},$$
(8)

where k is the wave vector, and ϵNP and μNP are the permittivity and permeability of the particle, respectively. The first-order Mie scattering coefficients, a1 and b1, are associated with the electric and magnetic dipole resonances, respectively (see Supplementary Eq. 1).

However, the Mie-extended PD treatment is not sufficient to describe forces on Si nanoparticles when excited at their ED and MD resonances. Furthermore, the interaction between the MD and ED becomes non-negligible and results in momentum transfer from the incident to the resultant scattering fields, which is not captured within the point-dipole approximation as expressed in Eqs. (5) and (6).

Scattering-corrected model (SCM)

We developed the SCM to capture the transverse scattering force acting on Si nanoparticles, but it applies generally. The SCM is particularly important when the particles’ dipolar scattering strength is very significant, i.e., when the trapping laser is on-resonance with the MD and ED modes. The SCM-calculated force, FSCM, is constituted of two forces: the dipole force and the transverse scattering (TS) force

$$\langle {{{{\bf{F}}}}}^{{{{\rm{SCM}}}}}\rangle=\langle {{{{\bf{F}}}}}^{{{{\rm{PD}}}}}\rangle+\langle {{{{\bf{F}}}}}^{{{{\rm{TS}}}}}\rangle .$$
(9)

In the present study, the electromagnetic field-driven motion of the Si nanoparticles is restricted to the focal plane, and, therefore, only the transverse component of the scattering force is taken into account. The dipole force in Eq. (9) determines the intensity gradient force; i.e., the first terms in Eqs. (5) and (6).

The TS force is non-conservative and created by the transverse component of the field scattered by a Si nanoparticle. In the scattering process, a portion of the incident beam’s energy (and momentum) is transferred to the Si nanoparticle, exciting its dipole mode. The absorbed energy is subsequently dissipated  through radiative emission and/or relaxation into phonons, producing heat in solution. Momentum conservation requires that light scattered by the Si nanoparticle should result in a recoil (reaction) of the particle in an opposite direction, and the recoil force should be proportional to the intensity of the scattered field. Therefore, if there is a symmetry breaking of this particle-field system, then a net transverse component is manifest in the scattered field and its opposite on the nanoparticle.

We derive an expression to quantify the net TS force. Our derivation builds on theoretical studies on optical scattering forces62,63,64,65. The TS force in a specific direction \(\hat{{{{\boldsymbol{r}}}}}\) with orientation (0∘, φ) is obtained by integrating the intensity of the scattered field, Iscat(θ, ϕ), over polar θ and azimuthal ϕ angles:

$${{{{\bf{F}}}}}^{{{{\rm{TS}}}}}(\varphi )=\hat{{{{\boldsymbol{r}}}}}\int\limits_{0}^{2\pi }\int\limits_{0}^{\pi }\frac{2\pi {\epsilon }_{0}}{{k}^{2}}{I}_{scat}(\theta,\phi )\cos (\phi -\varphi )\sin \theta \,{{{\rm{d}}}}\theta \,{{{\rm{d}}}}\phi,$$
(10)

where Iscat(θ, ϕ) = ∣S1(θ, ϕ)∣2 + ∣S2(θ, ϕ)∣2. The amplitude functions \(\left[{S}_{1}(\theta,\phi ),{S}_{2}(\theta,\phi )\right]\) define the amplitudes of the fields scattered in any direction. Based on GLMT theory and notation, they are expressed as:

$${S}_{1}(\theta,\phi )=\, {\sum}_{n=1}^{{L}_{\max }} {\sum}_{m=-n}^{n}{E}_{mn}\left[{a}_{n}{p}_{mn}{\pi }_{mn}(\cos \theta )+{b}_{n}{q}_{mn}{\tau }_{mn}(\cos \theta )\right]{e}^{im\phi }\\ {S}_{2}(\theta,\phi )=\, {\sum}_{n=1}^{{L}_{\max }} {\sum}_{m=-n}^{n}{E}_{mn}\left[{a}_{n}{p}_{mn}{\tau }_{mn}(\cos \theta )+{b}_{n}{q}_{mn}{\pi }_{mn}(\cos \theta )\right]{e}^{im\phi },$$
(11)

where \({\pi }_{mn}(\cos \theta )\) and \({\tau }_{mn}(\cos \theta )\) are generalized Legendre functions66. The rest of the parameters (Emn, an, bn, pmn, qmn) are introduced in Supplementary Eqs. 1–3.

As shown in Fig. 3b–d of the main text, when a Si nanoparticle is placed at the center of a focused linearly polarized beam (e.g., Gaussian beam), its dipole radiation maintains mirror symmetry; the exciting field does not exert a net TS force on the particle. However, when mirror symmetry is broken by displacing the nanoparticle from the center of the beam, a net TS force correspondingly arises on the particle that can be determined quantitatively using Eq. (10).

Analytical approximation of the transverse scattering force

Numerical computation of Eq. (10) allows for precise quantification of TS forces. Yet, a deeper physical insight into the TS force arising from the magnetic field Hz and would benefit from an analytical expression. Moreover, the photonic Hall effect (PHE) for Si nanoparticles that we have introduced can then also be better understood. Given the dual-dipole (i.e., ED and MD) resonance of Si nanoparticles, their scattering behavior should be governed by an effective polarizability accounting for both the electric (αe) and magnetic (αm) polarizabilities. This process is analogous to two-body systems, where two optically bound nanoparticles can alter the effective polarizability of dipole-dipole structures26 through the interplay between the induced electric dipole responses. This dipolar interaction between the induced electric and magnetic modes has been theoretically proposed by M. Nieto-Vesperinas et al. in regard to optical forces acting on small magnetodielectric particles34.

Based on the physical parameter defined here, we reproduce the essential part of the expression for the TS force derived by M. Nieto-Vesperinas et al.34:

$${F}^{{{{\rm{TS}}}}-{{{\rm{ana}}}}}=\, -\frac{{k}^{3}{\mu }_{{{{\rm{m}}}}}}{6\pi {\epsilon }_{{{{\rm{m}}}}}}{{{\rm{Im}}}}\left[{\alpha }_{{{{\rm{e}}}}}{\alpha }_{{{{\rm{m}}}}}^{*}\right]{{{\rm{Im}}}}\left[{{{\bf{E}}}}\times {{{{\bf{B}}}}}^{*}\right]\\=\, -\frac{{k}^{3}{\mu }_{{{{\rm{m}}}}}}{12\pi {\epsilon }_{{{{\rm{m}}}}}}{{{\rm{Im}}}}\left[{\alpha }_{{{{\rm{e}}}}}{\alpha }_{{{{\rm{m}}}}}^{*}\right]\left[{{{{\bf{E}}}}}^{*}\times \left(\nabla \times {{{\bf{E}}}}\right)+{{{{\bf{B}}}}}^{*}\times \left(\nabla \times {{{\bf{B}}}}\right)\right].$$
(12)

In the first line, we intentionally omit a term associated with \({{{\rm{Re}}}}\left[{{{\bf{E}}}}\times {{{{\bf{B}}}}}^{*}\right]\) in the original expression, Eq. 44 of ref. 34, since it represents the longitudinal force along the laser beam propagation direction. Obtaining the second line from the first line of Eq. (12) requires the identity:

$${{{\rm{Im}}}}\left[{{{\bf{E}}}}\times {{{{\bf{B}}}}}^{*}\right]=\frac{1}{2}\left[{{{{\bf{E}}}}}^{*}\times \left(\nabla \times {{{\bf{E}}}}\right)+{c}^{2}{{{{\bf{B}}}}}^{*}\times \left(\nabla \times {{{\bf{B}}}}\right)\right].$$
(13)

Using the paraxial approximation and cylindrical coordinate system convention (with unit direction vectors (\({\hat{{{{\rm{e}}}}}}_{\rho }\), \({\hat{{{{\rm{e}}}}}}_{\theta }\), \({\hat{{{{\rm{e}}}}}}_{z}\))), the electric and magnetic fields of a linearly polarized Gaussian beam are:

$${{{\bf{E}}}}=\, {u}_{E}\cos \theta {\hat{{{{\rm{e}}}}}}_{\rho }-{u}_{E}\sin \theta {\hat{{{{\rm{e}}}}}}_{\theta };\\ {{{\bf{B}}}}=\, -\frac{{u}_{B}}{c}\sin \theta {\hat{{{{\rm{e}}}}}}_{\rho }-\frac{{u}_{B}}{c}\cos \theta {\hat{{{{\rm{e}}}}}}_{\theta }.$$
(14)

where uE and uB represent the fundamental transverse Gaussian mode (including phasor \({e}^{\left[i\left(kz-\omega t\right)\right]}\)) describing the electric and magnetic field amplitudes. The parameter θ denotes the electric field polarization direction that ranges from 0∘ (x-polarization) to 90∘ (y-polarization). Substituting Eq. (14) into the second line of Eq. (12) yields the TS force:

$${F}^{{{{\rm{TS}}}}-{{{\rm{ana}}}}}=-\frac{{k}^{3}{\mu }_{{{{\rm{m}}}}}}{12\pi {\epsilon }_{{{{\rm{m}}}}}}{{{\rm{Im}}}}\left[{\alpha }_{{{{\rm{e}}}}}{\alpha }_{{{{\rm{m}}}}}^{*}\right]\left(\nabla {U}_{B}{\cos }^{2}\theta -\nabla {U}_{E}{\sin }^{2}\theta \right),$$
(15)

where UE and UB denote the energy densities of the electric and magnetic fields of light, respectively. This derivation of Eq. (15), which is equivalent to Eq. (3) in the main text, emphasizes that the (mechanism of) the TS force depends on both the electric field polarization and the magnetic field polarization directions.

For example, assume a common trapping condition where a Si nanoparticle is exposed to a paraxial Gaussian beam with x-polarization (electric field). Equation. (15) implies that when the particle’s position is displaced along the polarization direction of the electric field (θ = 0∘), the TS force is mainly associated with the gradient of the magnetic field, UB. Moreover, the TS force then acts perpendicular to the magnetic field polarization direction. Conversely, when the particle’s position is displaced along the polarization direction of the magnetic field (θ = 90∘), the TS force is solely determined by the gradient of energy density of the electric field, UE, and acts perpendicular to the electric field polarization direction. Moreover, the opposite signs before the trigonometric functions exemplify the anisotropy of dipole-mediated TS forces. Therefore, the electric and magnetic field-associated TS forces are always orthogonal to each other and exhibit opposite behaviors; i.e., if one is attractive, then the other must be repulsive. The final determination of attraction or repulsion requires considering the sign of the polarizability cross term \({{{\rm{Im}}}}\left[{\alpha }_{{{{\rm{e}}}}}{\alpha }_{{{{\rm{m}}}}}^{*}\right]\).

Equation. (15) can be simplified by unifying the energy densities \(\left({U}_{E}={U}_{B}=U\right)\) since the electric and magnetic fields of light share the same spatial profile in the paraxial approximation. Hence, Eq. (15) can be rewritten:

$${F}^{{{{\rm{TS}}}}-{{{\rm{ana}}}}}=-\frac{{k}^{3}{\mu }_{{{{\rm{m}}}}}}{12\pi {\epsilon }_{{{{\rm{m}}}}}}{{{\rm{Im}}}}\left[{\alpha }_{{{{\rm{e}}}}}{\alpha }_{{{{\rm{m}}}}}^{*}\right]\cos 2\theta \nabla U.$$
(16)

Equation (16) shares the exact same mathematical structure as one reported by X. Xu et al.36. Once again, Eq. (15) reveals physical and intuitive insights that allow distinguishing the contributions of both the electric field and the magnetic field to the TS forces.

Although this analytical equation (Eq. (15)) is derived assuming a paraxial Gaussian beam, it can still provide a meaningful approximation for tightly focused beams of light as well as structured electromagnetic fields, thus complementing the rigorous numerical approach represented by Eq. (10) with a physical interpretation. Its applicability in tightly focused Gaussian and azimuthal beams is investigated in Supplementary Notes 12–15.