Optical trapping with optical magnetic field and photonic Hall effect forces

Domain: physics
Published: 2025-11-24
Editorial outcome: minor_revision

Abstract
Optical trapping offers robust nanoscale control of matter but, to date, has been dominated by the interaction between a material’s electric polarizability, αe, and the electric part of light, therefore defined by electric-field intensity-gradient forces. Magnetic light-matter interactions, despite their potential to reshape optical trapping research, have remained experimentally unrealized. This paper addresses this long-standing deficiency by realizing optical magnetic field-associated trapping of high-index (i.e., Si) nanoparticles. Experiments, validated by our theoretical framework and Maxwell stress tensor calculations, reveal the essential role of a material’s magnetic polarizability, αm, and electric-magnetic scattering forces arising from the photonic Hall effect. This magnetic contribution allows exploration of stable trapping, distinct from purely electric-field control. Our findings open avenues for nanoparticle manipulation beyond conventional paradigms, enable previously unexamined optical matter formation driven by magnetic interactions, and suggest unexplored N-body effects and symmetry-breaking dynamics in optical matter systems.

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 biophysics2–4, material science5–8, and macroscopic quantum physics9–11. Even after five decades of research, new topics in optical trapping continue to emerge. These include fascinating physical phenomena such as optical matter formation12–17, many-body photonic interactions and symmetry breaking-induced dynamics leading to optical matter machines18,19, symmetry-breaking electrodynamics20–28, and mesoscopic dipole interactions29–31.

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,

hFPDi= α0e/4 ∇|E|2+ α00e/2 |E|2∇Φ (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 α0e; 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 α00e. 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,

hFPDi= α0m/4 ∇|H|2+ α00m/2 |H|2∇Φ, (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)38–41 (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)45–47 that includes the PHE-associated optical force. The latter is a non-conservative 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 η = √ϵw/μw 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.

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.

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 indicates 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 nanoparticles 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).

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)):

FTS,ana = - k3μ m / 12πϵm Im [αe α*m] ∇UB cos2θ - ∇UE sin2θ , (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 (FTS,MD) attract particles to magnetic field maxima, while ED-associated TS forces (FTS,ED) repel particles away from electric field maxima. This polarization-dependent directional scattering is the photonic Hall effect38–40 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 (FTS,ED=MD; 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) Ftotal,SCM = FPD + FTS 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 FTS,ED/FTS,MD 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).

Two key observations arise from the results shown in Fig. 4. First, |κ x| reaches its minimum value while |κ 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-associated TS force. Second, |κ x| and |κ 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-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 FTS,ED and FTS,MD 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.

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 FTS,MD (Fig. 5a left panel). The optical MD mode of larger Si nanoparticles shift 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 (FTS,MD) but also optical electric field (E) through electric TS force FTS,ED 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 FTS,ED, 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.

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

hFPDi = hFPDi,e + hFPDi,m: (4)

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

hFPDi,e = α0e/4 ∇|E|2inc + α00e/2 |E|2inc ∇Φ, (5)

and

hFPDi,m = α0m/4 ∇|H|2inc + α00m/2 |H|2inc ∇Φ, (6)

where the subscripts e and m refer to the electric- and magnetic-field related parameters, respectively, and αj = α0j + iα00j 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 need to be extended by defining the particle’s polarizabilities in terms of the Mie scattering coefficients59–61,

αe = i 6πϵNP / k3 a1, (7)

and

αm = i 6πμNP / k3 b1, (8)

where k is the wavevector, 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

hFSCMi = hFPDi + hFTSi: (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 forces62–65. The TS force in a specific direction ^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:

FTS(φ) = ^r k2 / 2πϵ0 ∫2π0 ∫π0 Iscat(θ, ϕ) cos(ϕ-φ) sinθ dθ dϕ, (10)

where Iscat(θ, ϕ) = |S1(θ, ϕ)|2 + |S2(θ, ϕ)|2. The amplitude functions S1(θ, ϕ), S2(θ, ϕ) define the amplitudes of the fields scattered in any direction. Based on GLMT theory and notation, they are expressed as:

S1(θ, ϕ) = ∑Lmax n=1 ∑m=-n n Emn [an πmn(cosθ) + bn τmn(cosθ)] eimϕ

S2(θ, ϕ) = ∑Lmax n=1 ∑m=-n n Emn [an τmn(cosθ) + bn πmn(cosθ)] eimϕ, (11)

where πmn(cosθ) and τmn(cosθ) 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:

FTS,ana = - k3μ m / 6πϵm Im [αe α*m] Im <E×B*> = - k3μ m / 12πϵm Im [αe α*m] <E*×(∇×E) + B*×(∇×B)> (12)

In the first line, we intentionally omit a term associated with Re <E×B*> 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:

Im <E×B*> = 1/2 <E*×(∇×E) + c2B*×(∇×B)> (13)

Using the paraxial approximation and cylindrical coordinate system convention (with unit direction vectors (eρ, eθ, ez)), the electric and magnetic fields of a linearly polarized Gaussian beam are:

E = uE cosθ eρ - uE sinθ eθ;
B = - c B sinθ eρ - c B cosθ eθ (14)

where uE and uB represent the fundamental transverse Gaussian mode (including phasor e[i(kz-ωt)]) describing the electric and magnetic field amplitudes. The parameter θ denotes the electric field polarization direction. Substituting Eq. (14) into the expression for the TS force:

FTS,ana = - k3μ m / 12πϵm Im [αe α*m] ∇UB cos2θ - ∇UE sin2θ , (15)

where UB and UE 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 Im [αe α*m].

Equation (15) can be simplified by unifying the energy densities UE = UB = U since the electric and magnetic fields of light share the same spatial profile in the paraxial approximation. Hence, Eq. (15) can be rewritten:

FTS,ana = - k3μ m / 12πϵm Im [αe α*m] cos2θ ∇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.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "I appreciate the efforts from the authors in improving the manuscript. The authors have addressed most of my concerns. However, I am still not fully convinced and would like to invite the authors to give more details. I would say that the experimental part is good, but the theoretical part is obscured, or even confusing at some places. I would recommend the publications of the concerned manuscript in Nature Communications if the authors can address the following issues properly.\n(1) For equation (1) and (2), I appreciate the clarification from the authors. Nevertheless, they are not alternatives to the correct equations in general. If the authors insist to use the current form of equation (1) and (2), please justify their applicability. Also, were these equations applied in the manuscript or supplementary material? Could this introduce errors or uncertainty?\n(2) I strongly oppose about the discussion on Optical Earnshaw Theorem. To my knowledge, beyond electric dipole approximation, there are no universally agreed definition on the scattering and gradient force. Some considers the conservative part of the force as gradient force, and the nonconservative part of the force as scattering force. In this definition, Ashkin’s theorem is always correct. On the other hand, Ashkin’s theorem will be incorrect for all other definitions. If the authors insist on naming the recoil force as the scattering force, please justify. Even if the authors can justify such a claim, the “breaking” of the Optical Earnshaw Theorem is not something interesting scientifically.\n(3) The authors called Eq. (3) the transverse scattering force, and it has an electric and a magnetic part. However, as is clear from Equation (3), such force is proportional to the product of alpha_e and alpha_m. It seems to be that this is the interference term between electric and magnetic dipoles. It can be proved that such term is always given by pxm. Thus the transverse force is still the recoil force.\n(4) The term local curvature has special meaning in differential geometry. I would suggest avoiding this term.", "reviewer_2": "The manuscript “Optical trapping with optical magnetic field and photonic Hall effect forces “ by Yanzeng Li et al presents a study that deals with optical trapping of the high refractive index nanoparticles that dominantly interact with the electromagnetic fields through the magnetic dipolar resonance. The paper presents both the experimental results of trapping silicon nanoparticles trapped in a focused azimuthally polarized vector beam as well as the simple theoretical model which explains observed experimental behavior. In principle, neither of the results in completely new: high refractive index particle had been trapped, azimuthally polarized beams had been used for optical manipulation and the analytical theory beyond simple Rayleigh approximation had been developed. However, the authors combine these results in quite novel way and clearly demonstrate validity and usefulness of their approach as well ass the novel “Photonic Hall Effect (PHE).”\nThe manuscript is clearly written and mostly quite easy to follow, methodology is sound and all details required to reproduce the results are provide.\nIn my opinion the weakest point of the study is the motivation, i.e. how this effect would be useful for the other researchers, when this methodology would help or set a direction for followup works. I may vaguely see some applications in the area of levitated optomechanics or tractor beams but it would really help if the authors considered one example in greater detail and wrote a few sentences about this topic, e.g. in conclusion close to line 345. It may also be worth to mention if the presented methodology can be useful for trapping of particles with lower refractive index or in structured trapping fields.\nThe second issue is that PHE is discussed only in the direction perpendicular to the trapping beam propagation axis. This is fine in the context of current experiments but the axial trapping stability is a issue that majority of the other researches would definitely look into as well. On lines 602-604 authors state “In the first line, we intentionally omit a term associated with Re [E × B*] in the original expression, Eq. 44 of Ref 36 , since it represents the longitudinal force along the laser beam propagation direction.” The main result of the paper is indeed the “transverse scattering force” but it seems (based on Ref 36) that there should be an axial term as well. I think the the authors should show this in the methods or provide an arguments why this term is zero.\nThere are also other issues that will be mentioned bellow.\n1. It seems that a term “Photonic Hall Effect” had been “claimed” by a different effect about 2 months ago, see Observation of the photonic Hall effect and photonic magnetoresistance in random lasers by Wenyu Du et al. Nature Communications 15 (2024) 4589, https://www.nature.com/articles/s41467-024-48942-6\n2. I think that paper Equivalent multipole-moment theory for dielectrophoresis and electrorotation in electromagnetic field by Masao Washizu Journal of Electrostatics 62 (2004) 15–33 may contain similar theoretical results as Ref 36 and current manuscript.\n3. Slightly similar experimental result had been observed in higher order Bessel beam where particle jumped between bright ring of maximal electric field intensity and the beam axis, see Optical forces induced behavior of a particle in a non- diffracting vortex beam by Martin Siler Optics Express 20 (2012) 24304. Although only GLMT model only was considered there.\n4. lines 193 and bellow: text suddenly jumps from vector azimuthally polarized beam to Gaussian beam. I guess that this is to show that the PHE is more general effect but this sudden change is quite hard to understand upon the first read.\n5. Equation 3 and 24: I am not fully sure if the energy densities |U_E| and |U_B| should be squared. Based on Eqs 21-23 I would expect squared amplitudes |u_E| or |u_B| but not energy densities.\n6. lines 273-275: in text vector beam is implied while figure 4 shows results for Gaussian beam\n7. figure 4, 5c, text on page 17 and further on: trap stiffness is discussed but it is not clearly defined. It seems that authors consider force = stiffness*displacement, i.e. negative stiffness gives stable trapping. This is the opposite sign that is usually considered when force = - stiffness*displacement, see e.g. P. Jones Optical Tweezers:Principles and applications book\n8. figure 5c, lines 335-344: both curves in fig 5c have the same label. The discussion of Fig. 5c in the text does not correspond to figure. I would actually say that the red curve assumes F=-kx, while blue curve F=kx.\n9. Many of the results presented in the Supplementary material are not mentioned in the main text or the Methods. E.g. supplementary figure S8 could be mentioned on line 553 or section S14 in the conclusions. On the other hand, some Supplementary figures are not referenced in supplement text, e.g. figure S5 in section S5.\n10. Equation S1: I would expect that the denominator should be in absolute value.\n11. Figure S9: 1st Kerker condition is mentioned. This would mean minimal back-scattering of photons and one may expect the strong pulling force along z direction which may lead to the tractor-beam like effect, similar to A long-range polarization-controlled optical tractor beam by Vladlen Shvedov Nature Photonics volume 8, pages 846–850 (2014). I think that this may be worth potential future study.\n12. Figure S10 and text in section S9. In the text figure S10b is discussed before figure S10a. I recommend to switch the order of panels in figure or order of paragraphs in the text\n13. line S297: typo in “delineated”\n14. figures S12 and S13: probably compare final curve in the 4th column with GLMT"}, "author_response": "Response to reviewer comments for manuscript entitled “Optical trapping with\noptical magnetic field and photonic Hall effect forces.”\nThe authors extend their gratitude to the reviewers for their valuable time and insightful comments\nand suggestions. The following is a point-by-point response that addresses the reviewers’\ncomments and concerns. The changes in response to the comments have been made to both the\nmain text and Supplementary Information (SI). The manuscript and SI have been revised to\nenhance clarity and completeness. The revised manuscript with all edits and changes highlighted\nis provided.\nReviewer #1 (Remarks to the Author):\nComment #1: I appreciate the efforts from the authors in improving the manuscript. The authors\nhave addressed most of my concerns. However, I am still not fully convinced and would like to\ninvite the authors to give more details. I would say that the experimental part is good, but the\ntheoretical part is obscured, or even confusing at some places. I would recommend the publications\nof the concerned manuscript in Nature Communications if the authors can address the following\nissues properly.\nResponse #1: We thank the review’s continued feedback on our manuscript and recognition of\nimprovements made, and also appreciate reviewer’s detailed comments on the theoretic aspects of\nour work.\nComment #2: For equation (1) and (2), I appreciate the clarification from the authors. Nevertheless,\nthey are not alternatives to the correct equations in general. If the authors insist to use the current\nform of equation (1) and (2), please justify their applicability. Also, were these equations applied\nin the manuscript or supplementary material? Could this introduce errors or uncertainty?\nResponse #2: We appreciate the reviewer's careful examination of the accuracy and applicability\nof Eqs. 1 and 2 in the main text. Equation 1, derived from Novotny’s textbook “Principles of Nano-\nOptics,” presents a general model of electric field-mediated optical forces experienced by a particle\nwhen excited off-resonance. This equation succinctly captures both the intensity gradient (i.e.,\ndipole) and phase gradient (i.e., radiation pressure) forces. Although there are alternative\nformulations of this equation (such as one mentioned in reference (1)) they are physically\nequivalent. Similarly, Eq. 2, which deals with the magnetic counterpart, follows the same\nderivation process as Eq. 1 and shares an analogous structure (2).\nThe validity and applicability of these equations are thoroughly analyzed in Sect. 8 of the\nSupplementary Information. Here, the optical forces for a single Si nanoparticle of various sizes,\nranging from 160 nm to 240 nm, were calculated to demonstrate the validity of these equations for\nparticles with scattering features transitioning from the Rayleigh to Mie regimes relative to the\nexcitation wavelength of 770 nm. These calculations, represented by the purple dashed curves in\nFig. S8, were cross-verified with optical forces computed using the Maxwell Stress Tensor (MST),\nwhich quantifies the actual optical forces experienced by the particle. Notably, in cases where the\nparticle size is small (dia. < 180 nm) and intensity gradient forces predominate, the excellent\nagreement between the force curves calculated by Eqs. 1 and 2 and those calculated by MST\nvalidates the use of these equations for computing optical forces in the Rayleigh regime.\nComment #3: I strongly oppose about the discussion on Optical Earnshaw Theorem. To my\nknowledge, beyond electric dipole approximation, there are no universally agreed definition on\nthe scattering and gradient force. Some considers the conservative part of the force as gradient\nforce, and the nonconservative part of the force as scattering force. In this definition,\nAshkin’s theorem is always correct. On the other hand, Ashkin’s theorem will be incorrect for all other\ndefinitions. If the authors insist on naming the recoil force as the scattering force, please justify.\nEven if the authors can justify such a claim, the “breaking” of the Optical Earnshaw\nTheorem is not something interesting scientifically.\nResponse #3: We thank the reviewer’s insightful comment on the applicability of our findings to\nthe optical Earnshaw theorem. We appreciate the perspective the reviewer brought to our attention\nand recognize the reviewer’s points in certain contexts. However, we have some reservations\nregarding the validity of the comment as it applies to our case.\nAs clearly states in the reference (3), scattering force is nonconservative force resulting from\nthe removal of momentum from the incident light. In the context of optical physics, when light\ninteracts with a particle, some of the light's momentum is transferred to the particle, typically\nthrough scattering processes. The scattered light carries away momentum, leading to a change in\nthe momentum of the incident beam, which corresponds to the force exerted on the particle. This\nforce, termed a scattering or recoil force, is inherently nonconservative because it does not\nconserve the mechanical energy within the system, generally resulting in energy dissipation\nthrough mechanisms like scattering radiation.\nFurthermore, the optical Earnshaw theorem (3) asserts that \" scattering force by itself cannot\nform a trap.\" Apparently, our findings relax this assertion by demonstrating that solely utilizing\nscattering (nonconservative) forces can indeed create a stable optical trap. This challenges the\ntraditional understanding of the theorem and presents significant implications, particularly in the\nstudy of optical levitation for exploring mesoscopic quantum mechanisms.\nWhile we are confident in the scientific relevance of our results, which brings new perspective\nto advance the understanding of the optical Earnshaw theorem, we also acknowledge the\nreviewer’s concerns about potential confusion arising from differing definitions of conservative\nand nonconservative optical forces. Given that this debate is not central to the manuscript, we have\ndecided to remove the related discussion to maintain clarity and focus. Several paragraphs have\nbeen revised accordingly and now read as follows.\nIn lines 20-23 on page 1 (abstract section):\n“Our findings bring new opportunities for nanoparticle manipulation, motivate optical matter\nformation14–19 by optical magnetic interactions, and suggest new N-body effects20, 21 and symmetry\nbreaking22 to drive dynamics of optical matter systems.”\nIn lines 335-336 on page 19 (conclusion section)\n“Second, our results suggest that scattering forces by itself can form a stable optical trapping\neven though its divergence-free nature53.”\nComment #4: The authors called Eq. (3) the transverse scattering force, and it has an electric and\na magnetic part. However, as is clear from Equation (3), such force is proportional to the product\nof alpha_e and alpha_m. It seems to be that this is the interference term between electric and\nmagnetic dipoles. It can be proved that such term is always given by pxm. Thus the\ntransverse force is still the recoil force.\nResponse #4: We agree with review’s insight on the nature of the transverse scattering force\ndescribed in Eq. 3. This force is a recoil force, arising from unbalanced transverse scattering\nradiations due to the interaction between the induced electric and magnetic dipole modes. However,\nit is important to also recognize, as noted in Eq. 3 and demonstrated in the main text, that the\ntransverse scattering forces depend separately on the energy densities of the electric and magnetic\nfields, indicating the possibility of separate modulations of the transverse scattering forces by the\nelectric and magnetic parts of light.\nComment #5: The term local curvature has special meaning in differential geometry. I would\nsuggest avoiding this term.\nResponse #5: We appreciate the reviewer's suggestion regarding the terminology that may cause\nconfusion when considered from a differential geometry perspective. However, in the field of\noptical physics, the term \"local curvature\" is widely used to describe the profile or deformation of\nthe light wavefront at specific points in free space, as referenced in (4). Therefore, after careful\nconsideration, we have decided to retain this terminology to maintain consistency within the\noptical physics community.\nReviewer #2 (Remarks to the Author):\nComment #1: The manuscript “Optical trapping with optical magnetic field and photonic Hall\neffect forces “ by Yanzeng Li et al presents a study that deals with optical trapping of the high\nrefractive index nanoparticles that dominantly interact with the electromagnetic fields through the\nmagnetic dipolar resonance. The paper presents both the experimental results of trapping silicon\nnanoparticles trapped in a focused azimuthally polarized vector beam as well as the simple\ntheoretical model which explains observed experimental behavior. In principle, neither of the\nresults in completely new: high refractive index particle had been trapped, azimuthally polarized\nbeams had been used for optical manipulation and the analytical theory beyond simple Rayleigh\napproximation had been developed. However, the authors combine these results in quite novel way\nand clearly demonstrate validity and usefulness of their approach as well ass the novel “Photonic\nHall Effect (PHE).”\nThe manuscript is clearly written and mostly quite easy to follow, methodology is sound and all\ndetails required to reproduce the results are provide.\nResponse #1: We appreciate the reviewer’s acknowledgement of the novel aspects of our work and\nare also pleased to hear that our methodology is sound and comprehensive.\nComment #2: In my opinion the weakest point of the study is the motivation, i.e. how this effect\nwould be useful for the other researchers, when this methodology would help or set a direction for\nfollow up works. I may vaguely see some applications in the area of levitated optomechanics or\ntractor beams but it would really help if the authors considered one example in greater detail and\nwrote a few sentences about this topic, e.g. in conclusion close to line 345. It may also be worth\nto mention if the presented methodology can be useful for trapping of particles with lower\nrefractive index or in structured trapping fields.\nResponse #2: We thank the reviewer’s thoughtful suggestions to incorporate “more detailed”\nexamples of potential applications of our findings to enhance our manuscript. We have accordingly\nupdated portions of the conclusion section. However, we want to reiterate certain points in this\nresponse to further clarify the significance of our findings in the field indicated.\nAs it is known, scattering forces and thermal perturbations are the primary effects that disrupt\nthe trapping stability of a macroscopic quantum optomechanical system (using a single trapping\nlaser beam). These effects can degrade the system’s quantum coherence, consequently causing\nchallenges in probing the quantum states of macroscopic mechanical oscillators. Encouragingly,\nour methodology, which employs an azimuthally polarized beam for optical levitation of high-\nindex nanoparticles, demonstrate the potential to mitigate these challenges. In our system, the\nscattering forces are converted into forces that enhance optical trapping stability, while thermal\nperturbations can be significantly suppressed, as the particle stabilizes in the region of the laser\nbeam where the intensity is minimal. Given these advantages, we believe that our findings provide\na new tool to investigate quantum phenomena in macroscopic world.\nIn addition, the study of optical matter materials can also benefit from our findings. Previously,\nthe formation (i.e., optical binding and lattice arrangement) and dynamics (i.e., novel mechanical\nproperties arising from electrodynamics of two- and N-body effects) of optical matter structures\nhave been understood primarily through electric field-matter interactions, without the need for\nconsidering their magnetic counterparts. Our findings, which initially unravel the role of the\nmagnetic part of light in optical manipulations from both theoretical and experimental perspectives,\ncan spark a series of research interests, with a focus on how magnetic field-matter interactions\nmight challenge the existing physical models, particularly when matter is able to strongly interact\nwith the magnetic part of light. This critical area of inquiry remains largely unexplored and thereby\nwarrants immediate attention and innovative perspectives to fully comprehend the complexities of\nlight-matter interactions in the realm of nanoparticle manipulation.\nAnother insight arising from the reviewer’s comment concerns the applicability of our findings\nto dielectric particles of low refractive index. While it is feasible to induce magnetic dipole mode\nwith such dielectric particle, their low index contrast with the surrounding medium would require\nthe particle to be relatively large to exhibit magnetic dipolar resonance comparable to its electric\ncounterpart. This size requirement is linked to the magnetic dipole-associated scattering coefficient\n(denoted as b in Eq. 4 of the Methods section), which becomes significant as the value of mkr\n1\napproaches 1 (5), where n, k, r representing relative refractive index, wave vector, and the radius\nof particles, respectively. Furthermore, the small refractive index contrast tends to result in a broad\nbandwidth of each optical resonant modes, which spreads across the spectral range of interest. This\nbroadening causes considerable overlap among the optical resonant modes, dramatically\ncomplicating the analysis of optical forces and challenging the identification of the physical origins\ncontributing to the resultant optical forces.\nThe following shows the modified conclusion in accordance with the reviewer’s suggestions.\nIn lines 348-357, on page 21 of the main text:\n“This new insight paves the way for exploring mesoscopic quantum optomechanical systems,\nwhich are currently constrained by the use of red-detuned dipole traps that limit particle selection\nto those with small diameters and low refractive index materials54. Moreover, the unique trapping\nconfiguration we proposed, where probing particles stabilize in the minimal intensity region of an\nazimuthally polarized beam, significantly reduce exposure to the high intensity of light, thereby\nsubstantially suppressing thermal perturbations, which generally degrade quantum coherence54.\nThe present work also creates new opportunities to study N-body effects and the formation and\nmechanical dynamics of optical matter materials, such as nonreciprocal optical forces and\nnegative optical torques, arising from magnetic (vs. electric) light-matter interactions.”\nComment #3: The second issue is that PHE is discussed only in the direction perpendicular to the\ntrapping beam propagation axis. This is fine in the context of current experiments but the axial\ntrapping stability is a issue that majority of the other researches would definitely look into as\nwell. On lines 602-604 authors state “In the first line, we intentionally omit a term associated with\nRe [E × B*] in the original expression, Eq. 44 of Ref 36 , since it represents the longitudinal force\nalong the laser beam propagation direction.” The main result of the paper is indeed the “transverse\nscattering force” but it seems (based on Ref 36) that there should be an axial term as well. I think\nthe the authors should show this in the methods or provide an arguments why this term is zero.\nResponse #3: We thank the reviewer’s suggestion to elaborate on the radiation pressure, as\nrepresented by the term Re[𝑬×𝑩∗] in Eq. 21 of the Methods section. We want to clarify that we\ndid not state this term is zero. The reason of the trapped Si nanoparticles can be stabilized in the\naxial direction results from a balance between the radiation pressure and the electrostatic repulsive\nforce exerted by the electrostatically charged glass slide, as detailed in Sect. 3 of the\nSupplementary Information.\nComment #4: There are also other issues that will be mentioned bellow.\nIt seems that a term “Photonic Hall Effect” had been “claimed” by a different effect about 2 months\nago, see Observation of the photonic Hall effect and photonic magnetoresistance in random lasers\nby Wenyu Du et al. Nature Communications 15 (2024) 4589,\nhttps://www.nature.com/articles/s41467-024-48942-6\nResponse #4: We appreciate the reviewer’s concern regarding potential confusion stemming from\nour use of the term “photonic Hall effect,” given its application in earlier published work (6).\nHowever, after thorough examination of this reference (6), we found that the core phenomenon\nreported there ⎯ the magnetically induced transverse diffusion of light ⎯ is more closely related\nto the Beenakker-Senftleben effect based on the works (7, 8) cited in the reference (6). In contrast,\nour definition of the photonic Hall effect is aligned with the photonic spin Hall effect, which\nclosely parallels the fundamental mechanisms of the spin Hall effect observed in electron systems.\nIn light of this, and to maintain consistency with the previous works that study the Hall effect\nin photonics, we have decided to retain the term in our manuscript. To aid readers and address the\nconcerns raised, we have included a note in the main text clarifying how the photonic Hall effect\nas defined in our study differs from that in the reference (6), thereby mitigating potential confusion.\nThe statement in lines 78-80 on page 5 of the main text now reads:\n“Here, we clarify that the PHE we introduce is distinct from the magnetically induced\ntransverse diffusion of light, which is more analogous to the Beenakker-Senftleben effect42, 43,\ndespite its recent designation as PHE44.”\nComment #5: I think that paper Equivalent multipole-moment theory for dielectrophoresis and\nelectrorotation in electromagnetic field by Masao Washizu Journal of Electrostatics 62 (2004) 15–\n33 may contain similar theoretical results as Ref 36 and current manuscript.\nResponse #5: We understand the reviewer expressing what seems to be similarities between our\nwork and others from a theoretical perspective. Such an impression comes from the fact that multi-\npolar expansions (from low to high order, also including their interferences) form the mathematical\nbasis to interpret optical forces acting on small particles across Rayleigh to Mie regimes. However,\nthe review’s recognition is actually not correct. Different derivations provide unique insights,\nleading to distinct interpretations and applications of these forces.\nUnlike the references mentioned by the reviewer, our theoretical derivation specifically\nidentifies the transverse scattering forces associated with the intensity gradients of both electric\nand magnetic fields light ⎯ a property reminiscent of the conventional dipole forces. This insight\nhas enabled us to unravel the photonic Hall effect and establish a generalized dipole model that\ncomprehensively quantifies optical forces in cases where trapped nanoparticles exhibit both\ninduced electric and magnetic dipolar resonances. Therefore, we respectfully disagree that the\nuniqueness of our theoretical findings is similar to any existing works.\nComment #6: Slightly similar experimental result had been observed in higher order Bessel beam\nwhere particle jumped between bright ring of maximal electric field intensity and the beam axis,\nsee Optical forces induced behavior of a particle in a non-diffracting vortex beam by Martin Siler\nOptics Express 20 (2012) 24304. Although only GLMT model only was considered there.\nResponse #6: We thank the reviewer for mentioning the reference (9) that seems to have reported\nexperimental observations similar to our findings. In this reference (9), the positioning of a\npolystyrene microparticle trapped in a Bessel beam is determined by the ratio of the particle size\nto the radius of the first-order high-intensity ring. The particle stabilizes at the beam’s center (dark\nregion) only when its size is comparable to or larger than the ring dimension, meaning that the\nhigh-intensity ring needs to be entirely confined within the particle’s boundary. In other words, the\ntrapping trajectories of the microparticle is still subject to the high intensity part of light. In our\nmanuscript, we explicitly demonstrate that the tendency of Si nanoparticles to be trapped at the\nbeam center (where is electric field is vanishing) is driven by magnetic field-matter interactions,\nwhich depends solely on the trapping laser wavelength and are independent of microparticle size\nrelative to the trapping beam radius. This fundamental difference in the mechanisms of trapping\nparticles at the dark region of a laser beam clearly demonstrate that the experimental results in our\nstudy and those in the reference (9) are distinctly different.\nComment #7: lines 193 and bellow: text suddenly jumps from vector azimuthally polarized beam\nto Gaussian beam. I guess that this is to show that the PHE is more general effect but this sudden\nchange is quite hard to understand upon the first read.\nResponse #7: We thank the reviewer for pointing out the issue with the readability of our\nmanuscript that may hinder reader comprehension. In response, we have revised the pertinent\nsection, which in lines 195-197 on page 12 of the main text now reads:\n“To clarify the conflicting trapping behaviors discussed above, we need to pause here and\nintroduce the transverse scattering forces determined by the photonic Hall effect, along with their\nunique properties, using a simple optical model employing a linear polarized Gaussian beam.”\nComment #8: Equation 3 and 24: I am not fully sure if the energy densities |U_E| and |U_B| should\nbe squared. Based on Eqs 21-23 I would expect squared amplitudes |u_E| or |u_B| but not energy\ndensities.\nResponse #8: We are grateful to the reviewer for meticulously examining our mathematical\nexpressions. The power of two was mistakenly added to the energy densities. This has now been\ncorrected, and the revised expression in line 246 now reads:\n“𝐹𝑇𝑆−𝑎𝑛𝑎 = 𝐹𝑇𝑆 +𝐹𝑇𝑆 ≅ −\n𝑘3𝜇𝑚\nIm[𝛼 𝛼∗ ](∇𝑈 cos2𝜃−∇𝑈 sin2𝜃)”\n𝑀𝐷 𝐸𝐷 𝑒 𝑚 B E\n12𝜋𝜖𝑚\nComment #9: lines 273-275: in text vector beam is implied while figure 4 shows results for\nGaussian beam.\nResponse #9: We thank the review for drawing our attention on the statement that could cause\npotential confusion. We have recapitulated the statement in line 277-279 on page 17 of the main\ntext as follows:\n“The SCM explains the trapping behavior of Si nanoparticles in various optical beams\nincluding the focused azimuthally polarized beam employed in our experiments and the focused\nGaussian beam exemplified here.”\nComment #10: figure 4, 5c, text on page 17 and further on: trap stiffness is discussed but it is not\nclearly defined. It seems that authors consider force = stiffness*displacement, i.e. negative stiffness\ngives stable trapping. This is the opposite sign that is usually considered when force = -\nstiffness*displacement, see e.g. P. Jones Optical Tweezers: Principles and applications book.\nResponse #10: We appreciate the reviewer for highlighting that the sign of the stiffness coefficient\nmight cause confusion without a clear definition. We have added a sentence defining the stiffness\ncoefficient as used in this manuscript has been added. The sentence in lines 280-283 on page 17\nreads:\n“Note, the stiffness coefficient, κ, of an optical trap is defined as κ=F/r, where the F and r\nrepresent the optical force acting on the trapped particle and the displacement of the particle from\nits equilibrium position, respectively.”\nComment #11: figure 5c, lines 335-344: both curves in fig 5c have the same label. The discussion\nof Fig. 5c in the text does not correspond to figure. I would actually say that the red curve assumes\nF=-kx, while blue curve F=kx.\nResponse #12: We are grateful to the reviewer for identifying the mislabeling in Fig. 5 of the main\ntext, which has been corrected. We would like to emphasize that the calculations for the stiffness\ncoefficient curves remains consistent with the definition ⎯ 𝜅 = 𝜅/𝑟, as detailed in the revised\nmanuscript. The positive portion of the orange curve, which represents the azimuthally polarized\nbeam, indicates that the beam cannot stabilize small-sized Si nanoparticle at its center.\nThe corrected figure is shown below:\nComment #12: Many of the results presented in the Supplementary material are not mentioned in\nthe main text or the Methods. E.g. supplementary figure S8 could be mentioned on line 553 or\nsection S14 in the conclusions. On the other hand, some Supplementary figures are not referenced\nin supplement text, e.g. figure S5 in section S5.\nResponse #12: We appreciate the reviewer’s comment. We have verified that all supplementary\nfigures are appropriately referenced in the main text, the Methods section, and the Supplementary\ninformation. For example. Fig. S8 is mentioned in line 213 on page 13 of the main text, and Sect.\nS14 is mentioned in line 85 on page 5 of the main text. Additionally, in some sections of the\nSupplementary Information, such as Sect. S5, where the figure and its corresponding text are\norganized on a single page, we believe that additional referencing is unnecessary. This layout\nensures clarity for the reader without causing any confusion about associating the text with the\nfigure.\nComment #13: Equation S1: I would expect that the denominator should be in absolute value.\nResponse #13: We thank the reviewer’s reminder. We utilized the imaginary part of Eq. S1 to fit\nthe experimental data. We have clarified this in the text. The revised statement in lines 161-163 on\npage13 of the Supplementary Information now reads:\n“In addition, a fitting model composed of a series of Lorentz (only utilizing its imaginary part)\nand Gaussian functions was developed to facilitate the identification of these experimentally\nacquired scattering signatures.”\nComment #14: Figure S9: 1st Kerker condition is mentioned. This would mean minimal back-\nscattering of photons and one may expect the strong pulling force along z direction which may\nlead to the tractor-beam like effect, similar to A long-range polarization-controlled optical tractor\nbeam by Vladlen Shvedov Nature Photonics volume 8, pages 846–850 (2014). I think that this\nmay be worth potential future study.\nResponse #14: We thank the reviewer’s insights on the potential research direction based on the\n1st Kerker condition discussed here. We agree that the development of novel optical tractor beams\ncapable of attracting objects from a distance could benefit from our findings. We will consider this\nsuggestion in our future research endeavors.\nComment #15: Figure S10 and text in section S9. In the text figure S10b is discussed before figure\nS10a. I recommend to switch the order of panels in figure or order of paragraphs in the text\nResponse #15: We thank the reviewer’s suggestion. The figure is now displayed below:\nComment #16: line S297: typo in “delineated”\nResponse #16: The sentence has been rewritten in lines 298-300 on page 25 of the Supplementary\nInformation and now reads:\n“The total force is decomposed into individual components described by the constituent fields\n(i.e., E, H , and H ) of the beam and associated categories of forces (i.e., dipole FPD and scattering\nρ z\nFTS forces).”\nComment #17: figures S12 and S13: probably compare final curve in the 4th column with GLMT.\nResponse #17: The curves shown in the 4th columns of Figs. S12 and S13 are the GLMT-calculated\ntotal optical forces the particles experience, which can be compared directly.\nReviewer #2 (Remarks on code availability):\nComment #18: The code is hosted on github with good instructions describing the installation\nprocess. I was able to install the code and run some of the examples. I briefly checked a few of\nsource code files and they seem to be reasonable well documented with an understandable structure.\nI am not a Python expert so I was not able to perform any extensive tests. However, in my\nunderstanding the source code is not intended to be part of this manuscript so I have not specifically\nlooed for a code that produces current manuscript figures.\nResponse #18: We thank the reviewer’s effort on the brief examination of our code and confirm\nthe accessibility of our home-built software, MiePy, used for the simulations in this manuscript.\nReferences\n[1] S. Albaladejo, M. I. Marqués, M. Laroche, J. J. Sáenz, “Scattering forces from the curl of\nthe spin angular momentum of a light field,” Phys. Rev. Lett. 102, 1 (2009).\n[2] J. Zeng, M. Albooyeh, M. Rajaei, A. A. Sifat, E. O. Potma, H. K. Wickramasinghe, F.\nCapolino, “Direct detection of photoinduced magnetic force at the nanoscale reveals\nmagnetic nearfield of structured light,” Sci. Adv. 8, (2022).\n[3] A. Ashkin, J. P. Gordon, “Stability of radiation-pressure particle traps: an optical\nEarnshaw theorem,” Opt. Lett. 8, 511 (1983).\n[4] J. Hernández-Delgado, Z. Malacara-Hernández, D. Malacara-Doblado, B. Vázquez-\nDorrío, D. Malacara-Hernández, “Local curvatures and its measurements of an optical\nsurface or a wavefront: a review,” Opt. Eng. 61, 1 (2022).\n[5] W. V. Gmbh, C. Kgaa, “Absorption and Scattering of Light by Small Particles Absorption\nand Scattering of Light by Small Particles,” J. Appliied Phys. 13, 12 (2004).\n[6] W. Du, L. Hu, J. Xia, L. Zhang, S. Li, Y. Kuai, Z. Cao, F. Xu, Y. Liu, K. Zhou, K. Xie, B.\nYu, E. P. Raposo, A. S. L. Gomes, Z. Hu, “Observation of the photonic Hall effect and\nphotonic magnetoresistance in random lasers,” Nat. Commun. 15, 1 (2024).\n[7] B. A. van Tiggelen, “Transverse Diffusion of Light in Faraday-Active Media,” Phys. Rev.\nLett. 75, 422 (1995).\n[8] G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of magnetically induced transverse\ndiffusion of light,” Nature 381, 54 (1996).\n[9] M. Šiler, P. Jákl, O. Brzobohatý, P. Zemánek, “Optical forces induced behavior of a\nparticle in a non-diffracting vortex beam,” Opt. Express 20, 24304 (2012)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "After reading the response, my stance remained unchanged. To me, the experimental part is fine. For the theory part, I can agree with the authors in several issues, but it is still inappropriate at a couple of places. The unaddressed concerns left over from previous round of reviews are listed below.\n1. Equations (1) and (2) work only when there is no spin-curl force (see Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field, PRL 102, 113602 (2009). It is true that linear polarization has no spin curl force, so the calculation in Fig. S8 may be justified, but the equations are not the general one that works for all incident field. It would be better to clarify this point in the manuscript. Also, Equations (1) and (2) have nothing to do with off or on resonance.\n2. To me, the so call breaking of the optical Earnshaw theorem is more a gimmick than anything else. Also, statement like “our results suggest that scattering forces by itself can form a stable optical trapping even though its divergence-free nature” is incorrect. Any divergence-free force field cannot trap a particle by itself, see any text on electrostatic Earnshaw theorem.", "reviewer_2": "Dear Editor,\nI appreciate the authors’ thorough response to my previous review. In my opinion all issues raised have been adequately resolved, and I now find the manuscript suitable for publication in Nature Communications.\nI have just a tiny note regarding supplementary equation Eq (S1). In my opinion it may be better to explicitly include \\Im\\left{ .... \\right} in the definition of the fitted Lorentz function. The imaginary part is indeed mentioned in the text but the current form is (in my opinion) still confusing.\nI reviewed the code during the previous round of the review process. The reply to comments of all reviewers did not imply any significant changes in the code that would require another check."}, "author_response": "Response to reviewer comments for our manuscript\n“Optical trapping with optical magnetic field and photonic Hall effect forces.”\nWe sincerely thank the reviewers for their continued engagement, time, and effort in evaluating our\nmanuscript through this final round of review. Their constructive feedback and insightful suggestions have\nbeen invaluable in further refining our work. Below, we provide a detailed, point-by-point response\naddressing each comment and concern. Corresponding revisions have been incorporated into both the main\ntext and the Supplementary Information (SI) to enhance clarity, accuracy, and completeness. A revised\nversion of the manuscript, with all edits and modifications clearly highlighted (in red text), is provided.\nReviewer #1 (Remarks to the Author):\nComment #1: After reading the response, my stance remained unchanged. To me, the experimental part is\nfine. For the theory part, I can agree with the authors in several issues, but it is still inappropriate at a couple\nof places. The unaddressed concerns left over from previous round of reviews are listed below.\nResponse #1: We thank the reviewer for continuing engagement with our work and for providing valuable\nfeedback. We have carefully considered the remaining concerns and have revised the manuscript to address\nthem in detail.\nComment #2: Equations (1) and (2) work only when there is no spin-curl force (see Scattering Forces from\nthe Curl of the Spin Angular Momentum of a Light Field, PRL 102, 113602 (2009). It is true that linear\npolarization has no spin curl force, so the calculation in Fig. S8 may be justified, but the equations are not\nthe general one that works for all incident field. It would be better to clarify this point in the manuscript.\nAlso, Equations (1) and (2) have nothing to do with off or on resonance.\nResponse #2: We appreciate the reviewer’s insight on the exclusion of the spin-curl force term in Eqs. (1)\nand (2), which is included and discussed in the reference cited by the reviewer. However, Eq. (1) is the\nwidely accepted formula for optical forces within the dipole approximation― and is even valid for light\nwith circular polarizations [1]. Therefore, we feel that Eqs (1) and (2) are sufficient for describing optical\nforces on small (Rayleigh) particles [1]. However, we do acknowledge that optical forces arising from the\nspin of incident photons become nontrivial when the interacting particle size approaches the wavelength of\nthe trapping light, a condition that is relevant to the case studied in our current manuscript. To clarify this\npoint as suggested by the reviewer, we have made the following modifications.\nIn line 395 on page 23:\n“Note that an alternative form of the PD model that is particularly suitable for trapping light with\narbitrary polarizations (including circular polarization) can be found elsewhere56, 57.”\nRegarding the reviewer’s comment on Eqs (1) and (2) being independent of optical resonance, we want\nto emphasize the fact that the accuracy of the equations in describing the optical forces is influenced by\nbeing near or on-resonance. As demonstrated in Supplementary Fig. 8, when the trapping laser wavelength\nis close to optical resonance(s) (i.e., the electric and magnetic dipole resonances), the dipole force\napproximation in Eqs (1) and (2) becomes insufficient to fully capture the electrodynamic behavior of a Si\nnanoparticle in the given trapping configuration. This is an important result reported in our manuscript. We\nbelieve that we are largely in agreement with the essence of the Reviewer’s intent.\nComment #3: To me, the so call breaking of the optical Earnshaw theorem is more a gimmick than anything\nelse. Also, statement like “our results suggest that scattering forces by itself can form a stable optical\ntrapping even though its divergence-free nature” is incorrect. Any divergence-free force field cannot trap a\nparticle by itself, see any text on electrostatic Earnshaw theorem.\nResponse #3: We thank the reviewer for the comment. Although we maintain some reservations on this\npoint, as outlined in our previous response, we have revised the corresponding section to mitigate this\npotentially contentious argument.\nIn line 345 on page 21:\n“Second, our results suggest that despite its generally divergenceless nature53, scattering forces can\nmanifest stable optical trapping, in at least two dimensions.”\nReviewer #2 (Remarks to the Author):\nComment #1: I appreciate the authors’ thorough response to my previous review. In my opinion all issues\nraised have been adequately resolved, and I now find the manuscript suitable for publication in Nature\nCommunications.\nResponse #1: We appreciate the reviewer’s acknowledgement of our work and our efforts to refine the\nmanuscript to meet the standards of Nat. Commun. We thank the reviewer for helping to improve the\nmanuscript.\nComment #2: I have just a tiny note regarding supplementary equation Eq (S1). In my opinion it may be\nbetter to explicitly include \\Im\\left{ .... \\right} in the definition of the fitted Lorentz function. The imaginary part is indeed mentioned in the text but the current form is (in my opinion) still confusing.\nResponse #2: We thank the reviewer’s comment on the clarification of the Lorentz function by directly\npresenting the imaginary part of Eq. (S1). The equation has been changed accordingly, which now reads:\nNow, the equation (in line 271 on page 20 in the SI) reads:\n𝐴𝜔 𝛾2𝜔\n0 𝑏\n𝐿𝑜𝑟(𝜔)=\n(𝜔2−𝜔2)2+𝜔2𝜔2\n0 𝑏\nComment #3: I reviewed the code during the previous round of the review process. The reply to comments\nof all reviewers did not imply any significant changes in the code that would require another check.\nThe manuscript is clearly written and mostly quite easy to follow, methodology is sound and all details\nrequired to reproduce the results are provide.\nResponse #3: We thank the reviewer for careful assessment and recognizing the clarity of our manuscript,\nthe soundness of the methodology, and the sufficiency of the details provided for reproducibility. We also\nappreciate the reviewer’s confirmation that no further code review is necessary in this round.\nReferences\n[1] Ruffner, D.B. and Grier, D.G. Comment on “Scattering forces from the curl of the spin angular\nmomentum of a light field,” Phys. Rev. Lett. 111, 059301 (2013)."}]

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