Energy and momentum are the two fundamental dynamical quantities of light, each manifested through the respective conservation laws.1^,2 Among these, momentum has attracted considerable attention in elucidating fundamental photonic interactions. The components of momentum—linear momentum, spin angular momentum (SAM), orbital angular momentum (OAM), and their interconversions, known as the spin–orbit interaction (SOI) of light3^–5—play a crucial role in all light–matter interactions. The SOI of light is manifested in numerous optical elements encompassing the in-plane and out-of-plane Goos–Hänchen and Imbert–Fedorov shifts,6 akin to the spin Hall effect (SHE) of bounded beams undergoing reflection or transmission from planer interfaces,7^–9 spin-dependent optical vortex generation in tight focusing10 as well as in the subwavelength epsilon-near-zero slabs,11 SHEs in scattering5 and tight focusing,12 etc. Some of the recent SOI-driven phenomena include unusual transverse spin angular momentum (TSAM) both in evanescent wave13^,14 and tight focusing of circularly polarized (CP) light,15^–18 simultaneous spin-dependent directional guiding,5 and wave-vector-dependent spin acquisition (spin-direction-spin coupling) in plasmonic crystals,19 rotational SHE in structured Gaussian beams,12 etc.

Apart from these manifestations, several other crucial aspects emerged in tightly focused fields, particularly in particle manipulation using optical tweezers.20^–23 The controllable spinning of microparticles employing orbital Hall effect,7^–9^,24^–26 spin-momentum locking due to tight focusing of CP light,5^,27 orbital motion around the beam axis due to transfer of OAM28 as well as due to enhanced SOI in a stratified medium,20^,29 and Larmor-like concurrent precessional and partial orbital motion30 are a few instances of SOI-driven optomechanical effects. It is, therefore, quite clear that SOI can be efficiently utilized to facilitate complex light-driven motion of microparticles by precisely tailoring the focused field to meet specific requirements.17^,31 The necessary field can be crafted by modifying the surrounding propagating mediums near the focus or by choosing beams with diverse field parameters, such as SAM, OAM, polarization, and intensity distribution.4^,28^,32^,33 Among the various types of beams, cylindrical vector beams have attracted significant attention due to the cylindrical or axial symmetry around the beam axis34^–36 and the inhomogeneous polarization across the beam cross section.26^,37^–39 These vector beams are higher-order solutions of the paraxial vector Helmholtz equation and can be represented as superpositions of Hermite–Gaussian or Laguerre–Gaussian modes.40^,41 For the last two decades, these vector beams have been employed in a wide variety of both fundamental and application-based research topics, including multiplexing,42 quantum sensing,43^,44 quantum information,45^,46 optical communication,47 particle trapping,48^,49 optical encryption,50 quantum memory,51 and superresolution microscopy.52

Among numerous vector beams, radially and azimuthally polarized beams are the ones frequently encountered.4^,34^,53 By combining these two beams, a spirally polarized vector (SPV) beam can be generated, exhibiting either clockwise (CW) or anticlockwise (ACW) spiral polarization distribution depending on the superposition type.54^,55 As the constituent radial and azimuthal beams lack OAM or SAM, the resultant SPV beams also possess a net zero angular momentum, with individual momenta being zero separately.56^–58 These SPV beams exhibit cylindrical symmetry about the propagation axis z, akin to their constituent beams.34^,38 One particularly intriguing application of vector beams has been demonstrated recently. Wu et al.26 explored the controlled spinning of asymmetric birefringent particles using vector vortex beams with zero SAM, utilizing the SOI effects of tight focusing. Although their results are compelling, their approach of employing an OAM superposition state (l≠0) with a radially polarized vortex beam to achieve the separation of σ=−1 and σ=+1 helicities of the SAM is inherently dependent on the value of l. As a result, their method would not work for beams possessing zero input angular momentum (SAM + OAM). Furthermore, studies, such as those in Refs. 54, 55, and 59 provide, both theoretically and numerically, insights into the transverse, longitudinal, and total intensity distributions of focused SPV beams, as well as experimental methods for synthesizing these beams.60 Motivated by these foundational studies and the unique polarization properties of SPV beams, we sought to investigate a novel application. The effects of SOI, known to generate spin from input spinless beams after tight focusing,26^,29^,61 raise an intriguing question: can SPV beams be effectively coupled into optical tweezers to create interesting rotational or spin dynamics of trapped birefringent microparticles? Such microparticles can exchange SAM with light, offering a platform to explore these interactions further. Specifically, could one engineer an optical trap with spatially varying regions of spin polarization, enabling particles trapped in these regions to exhibit differential spinning behaviors? This is the question we address in this paper, where we tightly focus SPV beams through a high numerical aperture (NA) objective lens into a refractive index (RI) stratified medium. Our study not only reveals that these “spin-less” beams are capable of driving rotational motion in birefringent microparticles but also showcases site and size-specific control of such spinning motions—hitherto unfeasible or inaccessible for other types of beams, to the best of our knowledge. Specifically, both CW and ACW spinning motions, depending on the size of the particles, are achieved at different spatially separated regions.

The underlying reason is found to stem from the distribution of the longitudinal spin angular momentum (LSAM) density caused by tight focusing into the RI-stratified medium depicting a distinctive form of the SHE. Indeed, the RI-stratified medium also broadens the permissible parameter space suitable for practical usage of the novel particle manipulation technique. To gain a deeper understanding of the observed phenomena, it can be intuitively explained that the SPV beam, after tight focusing, can be expressed as a superposition of right- and left-circular polarizations with different field amplitudes. These amplitudes change with the axial propagation of the focused light, leading to an axial variation in the resultant SAM. This interplay of polarization components along the propagation axis is key to the observed dynamics. Our experimental results are substantiated by the simulations performed under the light of full vectorial Debye–Wolf diffraction theory for high NA focusing. The structure of this paper is as follows: Sections 2 and 3 describe the experimental methods and findings. Section 4 outlines the theoretical framework for the focused field of the incident SPV beams, which is further analyzed through numerically simulated field quantities and discussions in Sec. 5. Finally, an overall summary of the study is provided in Sec. 6.

In our experiments using optical tweezers, two aspects—(a) generating the SPV beams and (b) a stratified medium featuring varied RI layers to control the nature of the SOI—are of primary importance. The schematic and details of our optical tweezer setup are shown in Figs. 1(a)–1(c). A polarizer, a half-wave plate (HWP), and a zero-order vortex plate (q-plate) designed for a wavelength of 671 nm [arranged in the sequence shown in Fig. 1(c)] were utilized to convert the incident Gaussian laser beam of wavelength 671 nm (Lasever, 350 mW) into SPV beams. The polarizer’s axis was fixed at 0 deg, and the fast axis of the HWP was set at 22.5 deg and 67.5 deg with respect to the polarizer axis, so that 45 deg (Ein=E02[11]T) and 135 deg (Ein=E02[1−1]T) linearly polarized lights were produced, respectively. The q-plate’s axis was then aligned at 0 deg (horizontal direction) to convert the 45 deg linearly polarized beam into an ACW-SPV beam (Erad+Eazi; ESPVACW=E0′2[cosϕ+sinϕsinϕ−cosϕ]T) and the 135deg linearly polarized beam into a CW-SPV beam (Erad−Eazi ; ESPVCW=E0′2[cosϕ−sinϕsinϕ+cosϕ]T), where E0 and E0′ represent the Gaussian and doughnut-shaped amplitude profiles, respectively.60 The resulting spiral polarization distributions of the ACW-SPV and CW-SPV beams, plotted over the doughnut-shaped intensity pattern before focusing, are shown in Fig. 1(b).

These beams were then guided into the back port of the inverted microscope (Carl Zeiss Axio vert.A1) equipped with an objective lens with NA 1.4. The RI-stratified medium, comprising four layers—immersion oil (RI 1.516), a coverslip (RI 1.814), the sample dispersed in deionized (DI) water (RI 1.33), and a top glass slide (RI 1.516)—constitutes our sample-mounting 1D inhomogeneous structure, as shown in Fig. 1(a). Although it is expected to use a coverslip with RI matching that of the immersion oil, employing one with a mismatched RI is particularly efficacious for our purposes due to the increased RI contrast in the stratified medium; we refer to this scenario as the “mismatched case,” while the former is called the “matched case.”20^,29^,62

The experimental results showcased involved a coverslip with an RI of 1.814, deliberately chosen for its substantial mismatch with that of the immersion oil. It is noteworthy that as the immersion oil (first layer) remains the same throughout the experiment, the RI of the coverslip (second layer) becomes pivotal in altering the behavior of light within the sample chamber (third layer), potentially increasing the capabilities of microparticle manipulation as desired. The effects of the focused field within the sample chamber were visualized using spin-responsive microparticles derived from an anisotropic liquid crystal (LC) called 5CB (4′-pentyl-4-biphenylcarbonitrile from Sigma-Aldrich).63^,64 A dispersion of these microparticles was prepared by mixing 5μl of 5CB, 100μl of polyvinyl alcohol, and 200 ml of deionized water (DI) in a microcentrifuge tube, vigorously shaken for several minutes. The particles/droplets had radii ranging from 0.5 to 10μm and predominantly exhibited bipolar behavior, as evident from the intensity patterns observed across the cross section of the particles due to the cross-polarization technique used for imaging.65^,66

The unconventional trapping mechanism utilizing the RI stratified medium focuses the field into an off-axis, spherically aberrated, ring-shaped region while also concentrating a significant amount of energy at the beam center (as detailed in Sec. 5). As a result, both the off-axis intensity annular ring and the on-axis trap center are suitable for trapping microparticles. Depending on the position where the LC particles are trapped, we can probe the spatially resolved both (σ−=−1 and σ+=+1) helicity of LSAM, which we describe in detail below.

The schematic detailing the spatially resolved σ− and σ+ helicity of LSAM is shown in Fig. 1(a), whereas the experimental results are presented in Figs. 2(a) and 2(b) and Figs. 3(a) and 3(b). First, particles of radius ranging from 3 to 4μm and trapped at the beam center exhibited ACW rotational motion for the ACW-SPV beam, and CW rotation for the CW-SPV beam, respectively, as comprehended from the rotation of the crossed patterns on the particles (see Videos 1 and 2 in the Supplementary Material). Expectedly, at the beam center, the observed motion aligned with the inherent ACW and CW spiral patterns of the electric fields (or polarization) at the respective beams’ cross sections [see Figs. 2(a) and 2(b)]. Second, in the case of smaller particles, typically with radii ranging from 0.5 to 1μm trapped in an off-axial location due to the radial intensity gradient, unexpected events were observed: CW rotations were noted for the ACW-SPV beam, whereas ACW rotations were observed for the CW-SPV beam (see Videos 3 and 4 in the Supplementary Material). Thus, for the same incident SPV beam, either ACW-SPV or CW-SPV, a reversal in directions was observed when comparing the rotations of the larger particles (at the axial location) with those of the smaller particles (at the off-axial location). The time-lapse frames for two different-sized particles, one with diameters of ∼7 to 8μm and another of ∼2μm, display these size-dependent opposite rotations in Figs. 2 and 3, respectively. Indeed, for small-sized particles ∼1 to 2μm in diameter, we could switch the rotation from CW to ACW by adjusting the focus of the microscope objective lens. As observed in Video 5, the particle initially spins in a CW direction at the center of the beam. However, as we change the focus, the particle shifts its position from the on-axis (beam center) to an off-axis position in the trap (due to the radial intensity gradient). It starts rotating in an ACW direction because the helicity of the LSAM at the off-axis position is opposite that at the beam center for an input CW-SPV beam. Notably, the rotation frequency of the off-axis trapped particle is approximately 4 times less than that of the on-axis trapped same-sized particle, which is expected due to the highly divergent nature of the beam. Such focusing-dependent change of the rotation direction (CW or ACW) offers considerable flexibility in controlling the spin of particles by adjusting the focusing objective lens during an experiment. To conclude that the rotational motion we observed was only due to SAM (and not due to OAM), we conducted the experiment with polystyrene spheres of diameter 3μm and did not observe any rotation. Note that, in order to experience OAM, a particle need not be birefringent, so that polystyrene spheres are adequate to perceive the effects of OAM.

Additionally, some particles were observed to periodically exhibit subtle focusing and defocusing effects while rotating in either CW or ACW directions. Typically, the CW/ACW rotations stem from the longitudinal SAM, while the focusing and defocusing effects result from the transverse spinning of the particles due to the transverse SAM (see Video 3 in the Supplementary Material). Nevertheless, this motion was observed to be significantly less frequent compared to the size-dependent rotational motion. Indeed, such size-dependent opposite rotational motion exhibited by bipolar particles, when trapped by a single spinless SPV beam at the beam center (on-axis) and off-axis intensity annular ring, stands as the key experimental discovery of this investigation. The ability of SPV beams to impart CW and counterclockwise torques prompts a close examination of the intensity and spin distribution near the focal region of the tightly focused polarized light, as we now proceed to carry out using the Debye–Wolf integrals described below.

Details of our theoretical formalism are provided in the Supplementary Material—here we provide the basic outline. As mentioned previously, radial, azimuthal, and SPV beams are solutions of the paraxial vector Helmholtz equation ∇×∇×E−k2E=0, which can be expressed as34^,56Erad/azi(ρ,z)=(E0/w0)ρexp(−ρ2w2)ei,i=ρ,ϕ.(1)

Here, i=ρ and i=ϕ represent the radial (eρ=[cosϕsinϕ]T) and azimuthal (eϕ=[sinϕ−cosϕ]T) polarizations of the beam, respectively. The SPV beam before focusing can be expressed as a linear superposition of radially and azimuthally polarized vector beams, ESPVACW/CW(ρ,ϕ,z)=(E0/w0)ρexp(−ρ2w2)(cosϕ±sinϕsinϕ∓cosϕ).(2)

Here, E0 is the overall amplitude factor, w is the size parameter of the beam, and ρ2=x2+y2 is the radial distance from the central wave vector k (or beam center). The terms ESPVACW=Erad+Eazi and ESPVCW=Erad−Eazi describe the ACW and CW spiral polarization directions of the SPV beam, respectively. It is important to note that only the first-order vector beams are considered throughout this study. The expressions in Eqs. (1) and (2) follow the paraxial approximations and can be directly integrated with the angular spectrum-based Debye–Wolf formalism for tight focusing by a high NA objective lens.41^,67

The Debye–Wolf formalism portrays the refracted, highly nonparaxial spherical wavefront emanating from an aplanatic lens into infinite plane waves commonly depicted as spatial harmonics. The transformations of the electric and magnetic field components are mimicked by the lens’s transfer function, which relies on the decomposition of the incoming fields into TE (s-polarization) and TM (p-polarization) components at the lens’s surface, as shown in Fig. 4.41^,62 Subsequent modifications of the field components due to the interfaces formed by the stratification of different layers in an experimental environment are also incorporated in a similar decomposition manner at each interface. Hence, the lens itself and Fresnel’s transmission and reflection coefficients are the two key ingredients in desirably tailoring the focus field. The resultant field is finally obtained by the linear superposition of the TE and TM components of all the plane waves, thus retaining the full vectorial details of the field in the image plane. According to this formalism, the time-independent monochromatic focused electric field is given by the angular spectrum integral41^,67E(ρ,ψ,z)=∫0θmax∫02πE∞(θ,ϕ)eikzcosθeikρsinθcos(ϕ−ψ)sinθdθdϕ.(3)

Here, we neglected the evanescent fields and considered only the far-field component, denoted as E∞(θ,ϕ). The wave vector in the medium is k, θmax=sin−1(NA/n) is the maximum angle determined by the NA of the objective lens, while n denotes the RI of the medium. E∞(θ,ϕ) for the ACW-SPV/CW-SPV beam is calculated using a coordinate transformation through the aplanatic lens (or the lens’s transfer function) and substituted into Eq. (3), after which the integration over θ and ϕ is performed.41^,67 The detailed calculations are provided in the Supplementary Material. It is important to note that the magnetic field can be derived similarly. For the injected ACW-SPV/CW-SPV beam, the focused electric and magnetic field components, expressed in Cartesian coordinates, are given by [ExEyEz]SPVACW/CW=[i(I1cosψ±I2sinψ)i(I1sinψ∓I2cosψ)−I0],(4)[HxHyHz]SPVACW/CW=[±i(I1cosψ∓I2sinψ)±i(I1sinψ±I2cosψ)∓I0],(5)where ESPVACW/CW=Erad±Eazi and HSPVACW/CW=Hrad±Hazi are the electric and magnetic fields of the focused light, respectively. Note that the magnetic field is CW in nature for the input ACW-SPV beam; however, it is ACW in nature for the input CW-SPV beam, with equal and opposite nonzero z components in both cases. The Debye–Wolf diffraction integrals for the transmitted and reflected waves, I0=I0t(ρ)+I0r(ρ), I1=I1t(ρ)+I1r(ρ), and I2=I2t(ρ)+I2r(ρ), are determined by the polar angles of incidence (θ) of the plane waves, and by Fresnel’s transmission (ts,tp) and reflection (rs,rp) coefficients. The strength of the spin–orbit conversion is encapsulated by these integrals, as detailed in Eq. (13) of the Supplementary Material.29^,41^,62

Due to the linearity of the system, the focused SPV beams in a nonmagnetic medium (i.e., μ=μ0) are a linear superposition of the focused radially and azimuthally polarized vector beams. Furthermore, to understand the deep origin of spatially resolved SAM, we first examine the focusing characteristics of SPV beams. Accordingly, the focused electric and magnetic fields of SPV beams can be decomposed into right- and left-circularly polarized (RCP/LCP) components as [ExEyEz]SPV(ACW/CW)=i(I1∓iI2)2eiψ[1−i0]+i(I1±iI2)2e−iψ[1+i0]−I0[001].(6)

From Eq. (6), it is evident that the coefficients of the left-circularly polarized (LCP) component (I1∓iI2) differ from those of the right-circularly polarized (RCP) component, (I1±iI2). Consequently, the transverse magnitudes are unequal during propagation, causing the superimposed light to exhibit different helicities in distinct spatial zones. Following this, we calculate the SAM density S, which is one of the key electromagnetic dynamical quantities and can be explicitly written as13^,14^,27S=14ωn2Im[ϵ(E*×E)+μ(H*×H)].(7)

Here, ω denotes the angular frequency, ϵ is the permittivity, μ is the permeability, and n is the RI of the focused medium. The electric SAM density of the focused ACW-SPV/CW-SPV beam in Cartesian coordinates can be recast as $$\begin{gathered} Sxe=i(I1′sinψ∓I2′cosψ), \\ Sye=−i(I1′cosψ±I2′sinψ), \\ Sze=∓(I1*I2−I1I2*), \end{gathered}$$(8)where I1′=(I0I1*+I0*I1) and I2′=(I0I2*+I0*I2). Now, we carry out numerical simulations of these theoretical expressions to gain a clear understanding of our experimental results.

The integral formalism to determine the focused field is utilized in a MATLAB code with parameters from the real experimental setup. As stated earlier, the four-layer stratified medium incorporates (a) a microscope immersion oil, (b) a coverslip with an RI of 1.814, (c) a dispersion of sample in DI, and (d) a top cover glass that is considered semi-infinite, as shown in Fig. 1(a). In the simulation, the origin of the coordinates is located inside the sample chamber, at an axial distance of 5μm from the interface between the sample and the coverslip. The interfaces are positioned as follows: the objective–oil interface is at −170μm, the oil–coverslip interface is at −165μm, the coverslip–sample chamber interface is at −5 μm, and the sample chamber–glass slide interface is at +30μm. Thus, the layer thicknesses are 5μm for the immersion oil, 160μm for the coverslip, and 35μm for the water (which holds LC particles), respectively. With this simulation setup, the intensity distribution is first examined, followed by an investigation of the spin distributions.

In this simulation setup, the first step involves examining the transverse intensity distribution of both SPV beams before and after focusing. Both SPV beams exhibit a polarization (or intensity) singularity at the beam center before focusing, as shown in Figs. 5(e) and 5(f). The quiver plots over the intensity distribution illustrate the polarization direction of the SPV beams. However, after tight focusing of the SPV beams, the contributions from the radial and azimuthal components need to be evaluated separately. It has been demonstrated earlier20^,29 that while the singularity in a radially polarized beam at the beam center vanishes after tight focusing, an azimuthally polarized beam maintains its singularity even after focusing. Thus, due to the linear nature of the fields, the radial component within both the SPV beams consistently contributes to the energy at the center of the focused light, facilitating particle trapping therein.

This is apparent from Eqs. (4) and (5), where the longitudinal component (or the z component) for both beams is solely influenced by the I0 diffraction integrals. In Figs. 5(a)–5(d), we show the plots of the transverse (xy) intensity distribution at the focus, and at 1, 2, and 3μm away from the focus, respectively. These plots reveal the spatial extent of intensity at both on-axis and off-axis positions for the input SPV beams. As we move away from the focus, an off-axis annular intensity ring is formed, as shown in Figs. 5(c) and 5(d), which is crucial for observing spatially resolved helicities of light with σ=+1 and σ=−1. In Fig. 5(g), we compare the intensity distributions at 2 and 3μm away from the focus. This comparison illustrates that, beyond 2μm, the intensity at the beam center decreases significantly. Therefore, z=2μm is the optimized distance from the focus to probe the spatially resolved helicities of light. The increased RI contrast of the second layer within the stratified medium enhances spherical aberration [see Figs. 1(a), 5(c), and 5(d)], leading to the formation of these high-intensity rings. The integrals listed in Eq. (13) of the Supplementary Material provide an intuitive understanding of the distribution. Among these integrals, I0, which involves the zeroth-order Bessel function J0, and I1 and I2, which involve the first-order Bessel function J1, govern the field distributions at the central and off-axis regions, respectively. The component-wise electric field intensity distribution of the SPV beam at z=2μm from the focus for an RI of 1.814 is plotted in Figs. 2(a)–2(c) of the Supplementary Material to visually illustrate the contributions of the J0 and J1 Bessel functions to the total intensity distributions. In Fig. 2(d) of the Supplementary Material, a comparison of the transverse and longitudinal intensity components with respect to the total electric field intensity is shown. A comparative analysis of the highest intensity values in both the central region and the off-axis rings of the intensity profile, across various commercially available coverslips [as shown in Fig. 5(h)], indicates that the coverslip used in our experiment, with an RI of 1.814, is well suited for trapping large particles (size ∼1 to 10μm in diameter) at the center and smaller particles (size ∼1 to 2μm in diameter) at off-axis positions. This suitability is reflected in the central intensity being ∼50% to 60% of the highest off-axis intensity.

The transverse intensity distributions of both off-axis and along the axis suggest similar interesting patterns in other dynamic electromagnetic quantities. Of these, our focus, understandably, would be on investigating the LSAM density—since this is behind the rotation events we report in this work [see Figs. 6(a)–6(f)]. Note that here we show results for axial distances beyond (or after) the focus since that is the region where we performed our experiments. The SAM distributions before the focus are provided in detail in the Supplementary Material. It is clear that the LSAM aggregates in concentric rings in the transverse plane, similar to the intensity rings we depicted in Figs. 5(c) and 5(g). In the case of the ACW-SPV beam, positive helicity components of the LSAM primarily accumulate in the central rings [Fig. 6(a)], whereas negative helicity components accumulate further outward (off-axis). A similar distribution with flipped helicities is observed for the CW-SPV beam [Fig. 6(b)]. In Fig. 6(c), we demonstrate a line plot that illustrates the range of radii for positive (negative) helicity, extending from the center (radius, r=0) to around 1.5μm, whereas the negative (positive) helicity component spans from around 1.5 to 2μm for an ACW-SPV (CW-SPV) vector beam. In Fig. 6(d), we present the distributions of TSAM at z=2μm from the focal point for both SPV beams. The TSAM distribution is in the form of concentric rings, similar to the LSAM pattern. As mentioned earlier, subtle focusing and defocusing effects were observed in the experiment as the particle rotated either CW or ACW due to the effect of TSAM (see Video 3 in the Supplementary Material). In Fig. 6(e), we present the line plots of LSAM density (light blue) and intensity (orange) at z=2μm from the focal point, illustrating the overlap of spatially resolved LSAM and intensity peaks at both on-axis and off-axis positions. Notably, the high-intensity annular ring at the off-axis position overlaps with a negative LSAM density peak for an input ACW-SPV beam, ensuring that a particle trapped in this region will spin in the CW direction, as observed in our experiments.

Typically, the angular momentum experienced by a particle can be assumed to be influenced by the net LSAM within a circular area covered by the particle. Here the net LSAM is the integrated value of the local LSAM density multiplied by the area covered. To perceive the size dependence of the particle’s rotational motion, the net LSAM within a circular region centered at the different locations in the transverse plane is investigated. This net LSAM predicts the direction of the particle’s rotation. If it is positive, the particle will rotate counterclockwise and if negative, the particle will rotate CW. The radii of the circular areas are increased gradually to consider particles of diverse sizes trapped at on-axis and off-axis spatial locations across the transverse plane, as shown in Videos 6 and 7, which display simulations of our system. Thus, two particles—one centered at the trap center (simulating an on-axis trap) and another off-center (simulating an off-axis trap)—exhibit distinctly different spin dynamics. A particle trapped at the beam center will rotate counterclockwise/CW under the influence of the incident ACW-SPV/CW-SPV beam, regardless of its size, since the SAM at the center is much higher in value compared to that off-axis [see Figs. 1(a) and 6(a)–6(c)]. Conversely, a particle trapped in a high-intensity off-axis ring will rotate in a direction determined by its size. Smaller particles, with a radius around 0.5 to 1μm, will rotate CW/ACW for the ACW-SPV/CW-SPV beam, whereas larger particles of radius more than 2μm will rotate counterclockwise/CW.

In Fig. 6(f), we show the results for an input ACW-SPV beam. The reverse pattern can be observed for the CW-SPV beam (not depicted here). Thus, the critical radius of the particle required to demonstrate a transition from CW/ACW to ACW/CW rotation at a trapping plane z=2μm away from the focus is numerically found to be 1μm, matching very well with the experiment. Also, the value of the off-axis net LSAM is much less than the on-axis net LSAM, which explains the lower frequency of rotation for small particles trapped at the off-axis intensity ring as observed in the experiment (see Video 5 in the Supplementary Material). Note that the optimized distance from the focus is z=2μm, where the net LSAM density at off-axis positions exhibits the extremum of opposite helicity (negative/positive value of LSAM for ACW-SPV/CW-SPV input beam). Before and after z=2μm, the off-axis net LSAM density decreases. As a result, the critical radius at which rotational reversal occurs may vary depending on the depth of trapping and the RI of the coverslips used in the experiment. Changes in RI affect spherical aberration, which in turn influences both the intensity and LSAM ring diameters in the focused light. Furthermore, the characteristics of LSAM distributions differ significantly before and after the focus. Before the focus, the separation of σ=+1 and σ=−1 helicities occurs not only in the transverse plane but also along the axial direction (z axis), as illustrated in Figs. 3(a)–3(f) of the Supplementary Material. This indicates the presence of a three-dimensional SHE before the focus. In contrast, after the focus, the SHE is confined to the transverse plane.68 Additional details are provided in Sec. 2.1 in the Supplementary Material.

The SOI of light due to the tight focusing of truly spinless (zero SAM and OAM) structured SPV beams is utilized in spherically aberrated optical tweezers to engineer particle-sized dependent spin dynamics of birefringent microparticles at different spatial locations. Due to the tight focusing of such a beam, the underlying components of opposite helicities—whose amplitudes were the same prior to focusing—become different. The helicity components also evolve differently along the axial direction and are, therefore, observed to be spatially resolved into concentric regions in the transverse plane. This leads to the generation of opposite helicities at different spatial locations, the distinct signatures of which are observed from the spin dynamics of spin-responsive LC microspheres. Particles trapped along the beam propagation axis will always rotate following the direction of the electric field’s rotation (or spiral polarization) across the beam cross section—rotating CW for the CW-SPV beam and counterclockwise for the ACW-SPV beam. Conversely, particles trapped at off-axis locations exhibit size-dependent directional rotation, with smaller particles rotating counter to the rotation of the electric field in the beam cross section, whereas larger particles will follow the rotation of the electric field. Most importantly, smaller particles can be made to spin in opposite directions by trapping them either on-axis or off-axis by changing the focus of the trapping objective lens. Consequently, this work presents an experimentally viable strategy for engineering optical traps capable of inducing controlled and specific spin motions of trapped particles using a single-trapping spinless SPV beam. Importantly, these effects are a beautiful manifestation of the diverse possibilities offered by spin–orbit optomechanics, which precludes the need to structure complex beam profiles using advanced algorithms involving adaptive optics for the manipulation of particles in optical tweezers. In the future, this work can be extended to demonstrate the three-dimensional SHE before the focus using SPV beams. Additionally, we aim to develop a new class of optical micromachines driven by spin–orbit optomechanics employing similar mechanisms.

Ram Nandan Kumar completed his BS degree from Banaras Hindu University and joined the Indian Institute of Science Education and Research (IISER) Kolkata in 2017 as an Integrated PhD student. Currently as a Senior Research Fellow at the ‘Light Matter Lab’, IISER Kolkata, he specializes in theoretical and experimental studies of Spin-Orbit Interaction (SOI) of light in Optical Tweezers.

Ayan Banerjee is presently a professor of physics at the IISER Kolkata, and has been working in the field of precision optics and spectroscopy for nearly 30 years. At the IISER Kolkata, he has set up the “Light Matter Lab” which focuses on the physics of light and mesoscopic matter employing optical tweezers. He is a SPIE Senior Member, with about a hundred published papers and patents, and has co-authored a text book in Classical Optics.

Biographies of the other authors are not available.