• Photonics Research
  • Vol. 13, Issue 6, 1756 (2025)
Lü Feng1, Ruohu Zhang1, Zhigang Li1, Bingjue Li2..., Huajin Chen3,4,5,* and Guanghao Rui1,6,*|Show fewer author(s)
Author Affiliations
  • 1Department of Optical Engineering, School of Electronic Science and Engineering, Southeast University, Nanjing 211189, China
  • 2School of Mechanical Engineering, Southeast University, Nanjing 211189, China
  • 3School of Electronic Engineering, Guangxi University of Science and Technology, Liuzhou 545006, China
  • 4Guangxi Key Laboratory of Multidimensional Information Fusion for Intelligent Vehicles, Liuzhou 545006, China
  • 5e-mail: huajinchen13@fudan.edu.cn
  • 6e-mail: ghrui@seu.edu.cn
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    DOI: 10.1364/PRJ.561607 Cite this Article Set citation alerts
    Lü Feng, Ruohu Zhang, Zhigang Li, Bingjue Li, Huajin Chen, Guanghao Rui, "Chirality-assisted local transverse spin angular momentum transfer for enantiospecific detection at the nanoscale," Photonics Res. 13, 1756 (2025) Copy Citation Text show less

    Abstract

    The enantiospecific detection of the chirality of substances at the nanoscale has attracted significant attention due to its importance in materials science, chemistry, and biology. This study presents, to our knowledge, a novel method for chirality detection based on transverse optical torque (OT), which leverages the transverse rotation of achiral particles induced by the transfer of chirality from the chiral particle within interference fields formed by the incident light without spin angular momentum (SAM). We demonstrate, both numerically and analytically, that by modulating the chirality of the chiral particle within a dimer system, it is possible to achieve the transfer of local SAM to the gold particle, thereby generating a transverse OT perpendicular to the light propagation direction. Furthermore, by adjusting the orientation of linear polarization in the excitation field, the respective contributions of electric and magnetic responses to the chirality-transfer-induced transverse OT can be exclusively observed separately, providing deeper insights into the underlying physical mechanisms. More importantly, the transverse OT exhibits an approximately linear dependence on the chirality parameter of the chiral particle, enabling enantiospecific detection of nanosamples. By replacing gold nanoparticles with suitable high-refractive-index dielectric materials such as germanium, the induced transverse magnetic dipolar OT can be further enhanced by more than two orders of magnitude, significantly improving the sensitivity of chirality detection and making it possible to detect weak chiral signals with exceptional precision. This work broadens the application scope of OTs in chirality detection and highlights the potential of chirality transfer mechanisms for advanced optical manipulation and the identification and analysis of chiral substances.

    1. INTRODUCTION

    When light interacts with matter, the absorption and scattering of light by the material result in the exchange of angular momentum, which includes orbital angular momentum (OAM) and spin angular momentum (SAM) [14]. Both types of angular momentum are fundamental dynamic properties of light [5]. Due to the conservation of total angular momentum in closed systems, the angular momentum carried by light is transferred to matter, inducing optical torque (OT) that causes the rotation of particles [68]. This phenomenon has diverse applications in quantum mechanics [9,10], biomedicine [1113], and physics [1416]. In 1936, Beth’s experiment first demonstrated that circularly polarized light can cause an absorbing object to rotate about the photon’s spin axis [17], a result of spin OT induced by SAM transfer [1820]. In contrast, OAM transfer generates orbital OT, which drives the particle to orbit around the OAM axis [2123]. Generally, OT can be classified into longitudinal and transverse components, inducing rotation around directions parallel and perpendicular to the light’s propagation direction, respectively [2426]. Recent studies have shown that the imaginary part of the Poynting vector in cylindrical vector beams can generate local angular momentum even in the absence of SAM and OAM, enabling isotropic particles to undergo orbital motion [27,28]. Moreover, a tightly focused radially polarized vortex beam can induce spin OT via local SAM transfer from the optical Hall effect, causing trapped birefringent particles to spin [29]. For asymmetric particles, linearly polarized light can exert spin OT by exciting multipolar pseudo-chiral responses [3032]. In addition, the longitudinal OT can also manipulate the rotation of other types of particles such as triangular prisms [33], twisted metal nanorods [34], and chiral nanocylinders [35]. Transverse OT, on the other hand, can be engineered by breaking the symmetry of the scatterer’s shape [36] or using light fields carrying transverse SAM, such as evanescent waves [37], tightly focused beams [38], and multi-wave interference [39,40], causing transverse spinning motion. However, it remains an open question whether isotropic spherical particles can exhibit transverse spinning in the absence of SAM.

    In recent years, the application of optical forces in chiral substance detection has gained significant interest. As the cornerstone of life, chiral materials are ubiquitous at both macroscopic and microscopic scales. Distinct enantiomers often exhibit vastly different biological activities and toxicities, making chiral detection and separation crucial in fields such as medicine and biochemistry [41]. Although traditional spectroscopic techniques, such as optical rotation [42], circular dichroism [43], and Raman optical activity [44,45], are widely used to detect the structure of chiral samples like proteins, the size mismatch between the light wavelength and microscopic samples poses significant challenges for detecting micro- and nanoscale materials. The cross-polarization characteristics of chiral substances make optical tweezers a key technology in overcoming these challenges [4648]. For instance, enhanced enantioselective optical forces based on quasi-bound states in the continuum have been employed for chiral separation [49], while transverse chiral optical forces can drive particles with different handedness to distinct transverse positions [5052]. Additionally, strong intensity gradients from surface plasmons can further amplify transverse chiral optical forces, as demonstrated with elliptical nanopores [53], double split-ring resonators [54], and coaxial nanoapertures [55]. Optical forces also hold promise for nanoscale chiral information measurement [5658], where optical fields carrying chiral dipole moments enable the detection of different chiral parameters in anisotropic particles [58]. In contrast, while the OT is crucial in optical tweezers, its role in enantioselective detection and characterization remains underexplored.

    In this study, we demonstrate the feasibility of detecting nanoscale chiral objects by utilizing the transverse spinning of achiral particles in an interference field. Firstly, the phenomenon of chirality transfer in a dimer system composed of a gold nanoparticle and a chiral nanoparticle is investigated. When the excitation field is composed of two counter-propagating linearly polarized plane waves, it is shown that the localized SAM density can be transferred from the chiral nanoparticle to the gold nanoparticle by tuning the chiral parameter of the former. This transfer results in the generation of a transverse OT on the gold nanoparticle, with its orientation determined by the chirality of the chiral particle. Numerical results demonstrate that this conclusion is universally applicable to spherical particles of arbitrary composition, spanning from the Rayleigh to the Mie scattering regimes. To provide a theoretical foundation for this phenomenon, a dual-dipole model based on the dipole approximation theory is established. Through this model, it is revealed that the chirality-transfer-induced transverse OT arises primarily from the independent control of the electric or magnetic components of the local SAM, depending on the polarization state of the excitation field. Within this framework, specific detection of molecular chirality is achieved by exploiting the approximately linear relationship between the transverse spin OT and the chiral parameter. Finally, the applicability of this mechanism to various types of achiral particles is confirmed, with certain dielectric materials being shown to exhibit the magnitude of transverse magnetic dipolar OT that exceeds that of plasmonic materials (e.g., gold) by two orders of magnitude. These findings provide insight into the physical mechanisms underlying chirality-transfer-induced OT on particles and suggest new possibilities for optical manipulation in the exploration of sample chirality and molecular concentration.

    2. THEORY

    The OTs on spherical particles can be accurately calculated numerically by a full-wave calculation method based on the multiple scattering theory [5961] and the generalized Lorentz-Mie theory [62]. The electric E and magnetic H fields acting on the j-th sphere are composed of the incident field and the scattered field from another sphere, and are expressed using the vector spherical wave function Nmn(J)(k,r) and Mmn(J)(k,r), viz. [60,61], E=n=1m=nniEm,n[am,nNm,n(3)(k,r)+bm,nMm,n(3)(k,r)pm,nNm,n(1)(k,r)qm,nMm,n(1)(k,r)],H=kωμn=1m=nnEm,n[bm,nNm,n(3)(k,r)+am,nMm,n(3)(k,r)qm,nNm,n(1)(k,r)pm,nMm,n(1)(k,r)],with Em,n=|E0|in[2n+1n(n+1)(nm)!(n+m)!]1/2,where k, ε, and μ are the wave number, permittivity, and permeability in the background medium. ω indicates the circular frequency. The scattering coefficients am,n and bm,n of the j-th sphere are related to the expansion coefficients pm,n and qm,n of the field and are denoted as am,n=an(1)pm,n+an(2)qm,n,bm,n=bn(1)qm,n+bn(2)pm,n,with pm,n=pm,n(j,j)ljv,u[au,v(l)Amnuv(l,j)+bu,v(l)Bmnuv(l,j)],qm,n=qm,n(j,j)ljv,u[au,v(l)Bmnuv(l,j)+bu,v(l)Amnuv(l,j)],where the translation coefficients Amnuv(l,j) and Bmnuv(l,j) are given in Ref. [60]. For chiral spheres, the Mie coefficients an(1), an(2), bn(1), and bn(2) are expressed as [48,59] an(1)=[An(2)Vn(1)+An(1)Vn(2)]Qn,an(2)=[An(1)Wn(2)An(2)Wn(1)]Qn,bn(1)=[Bn(1)Wn(2)+Bn(2)Wn(1)]Qn,bn(2)=an(2),with An(j)=ZsDn(1)(xj)Dn(1)(x0),Bn(j)=Dn(1)(xj)ZsDn(1)(x0),Wn(j)=ZsDn(1)(xj)Dn(3)(x0),Vn(j)=Dn(1)(xj)ZsDn(3)(x0),Qn=ψn(x0)/ξn(x0)Vn(1)Wn(2)+Vn(2)Wn(1),where Zs=μs/εs is the wave impedance of particles. κ, εs, and μs represent the chiral parameter, permittivity, and permeability of the particles, respectively. Dn(1)(x)=ψn(x)/ψn(x) and Dn(3)(x)=ξn(x)/ξn(x), with ψn(x) and ξn(x) denoting the Riccati-Bessel functions of the first and third kinds, respectively, are the corresponding logarithmic derivatives. By integrating the time-averaged Maxwell stress tensor on the closed surface around the particle, the time-averaged OT exerted on the spherical particle can be accurately calculated [16], viz., T=Sn·[T×r]dσ=Sr×[T·n]dσ,with the time-averaged Maxwell stress tensor I being expressed as [63] T=12Re[ε0EE*+μ0HH*12(ε0E·E*+μ0H·H*)I],with the I denoting the unit dyad and the superscript symbol * being the complex conjugate. Owing to the conservation of momentum, for chiral particles immersed in a lossless background medium, the integration can be realized at infinity. Consequently, the three Cartesian components of the OT can be reduced to [40] Tx=Re(τ1),Ty=Im(τ1),Tz=Re(τ2),with τ1=2πε0E02k3n,m[(nm)(n+m+1)n2(n+1)2]1/2t1,τ2=2πε0E02k3n,mmt2,where t1 and t2 can be expressed as t1=am,nam+1,n*+bm,nbm+1,n*12(am,npm+1,n*+pm,nam+1,n*+bm,nqm+1,n*+qm,nbm+1,n*),t2=am,nam,n*+bm,nbm,n*12(am,npm,n*+pm,nam,n*+bm,nqm,n*+qm,nbm,n*).

    3. RESULTS AND DISCUSSION

    A. Generation of Transverse OT by Chirality Transfer

    As illustrated in Fig. 1(a), a dimer structure consisting of chiral and gold dipole particles with the surface-to-surface separation of g is excited by two counter-propagating linearly polarized plane waves, where the chiral particle is located at the origin of the Cartesian coordinate system, and the gold particle is located along the x axis. The electric field of the dual-beam interference field can be expressed as E=E0E1eik1·r+E0E2eik2·r, where kj=k0cos(θj)z^ is the wave vector of the j-th plane wave with wave number k0, and the position vector is given by r=xx^+yy^+zz^. The complex amplitude vector Ej of each wave is described as Ej=pjcos(θj)x^+qjy^+pjsin(θj)z^, with θj the angle between the wave vector kj and the z axis, where complex numbers pj and qj are the polarization parameters that satisfy the normalization condition |pj|2+|qj|2=1. It is well known that this type of excitation field does not exert the OT on achiral isotropic spherical particles, but only applies a longitudinal torque on a single chiral particle [64]. Thus, generating a transverse spin OT on an achiral particle using an optical field without SAM remains a significant challenge.

    (a) Schematic illustration of the anomalous transverse spin OT induced by chirality transfer (depicted by the sky-blue curved arrow) from a chiral particle to a gold sphere. The dimer system is excited by a pair of counter-propagating linearly polarized plane waves with identical polarization states. The green double arrow represents an electric field polarized along the y-direction, while the dark blue double arrow represents polarization along the x-direction. It observes the variation of the transverse spin OT as a function of the position xAu of the gold sphere when it is placed to the (b) left and (c) right of the chiral particle.

    Figure 1.(a) Schematic illustration of the anomalous transverse spin OT induced by chirality transfer (depicted by the sky-blue curved arrow) from a chiral particle to a gold sphere. The dimer system is excited by a pair of counter-propagating linearly polarized plane waves with identical polarization states. The green double arrow represents an electric field polarized along the y-direction, while the dark blue double arrow represents polarization along the x-direction. It observes the variation of the transverse spin OT as a function of the position xAu of the gold sphere when it is placed to the (b) left and (c) right of the chiral particle.

    To address the challenge of generating transverse spin OT on an achiral isotropic particle, we propose a strategy based on chirality transfer. In this approach, the presence of a nearby chiral particle induces a transverse spin OT on the gold sphere, resulting in its transverse spinning motion. By utilizing multiple scattering theory [5961] and generalized Lorenz-Mie theory [62], we investigate the relationship among the induced transverse spin OT, the chirality, and the position of the gold particle. This analysis aims to intuitively understand how chirality transfer influences the transverse spin OT. In the calculations, we assume a light field with a wavelength of λ=808  nm and an amplitude of E0=8.68×105  V/m. The polarization vectors are set as (p1,q1)=(p2,q2)=(1,0). The radii of both particles, RAu (gold particle) and Rc (chiral particle), are 80 nm. Unless otherwise specified, the permittivity εc and permeability μc of the chiral particle are set to εc=2.53 and μc=1, respectively. Using Drude’s dielectric function model [65,66], the dielectric constant of the gold particles is given by εAu=25.6926+1.551i. For isotropic chiral particles, the constitutive relations [62,67] are expressed as D=ε0εcE+iκε0μ0H, B=μ0μcHiκε0μ0E, where κ is the chiral parameter constrained by κ<εcμc [67]. As shown in Figs. 1(b) and 1(c), the full-wave calculation results indicate that the gold particle experiences a transverse spin OT when the adjacent particle has nonzero chirality (κ0). Fixing the chirality at κ=0.5, as the separation distance g increases, the direction of the transverse spin OT on the gold particle exhibits alternating changes and its absolute magnitude diminishes, eventually approaching zero. Additionally, reversing the chirality of the chiral particle or changing the location of the gold particle alters the direction of the transverse spin OT, i.e., Tx(xAu,κ)=Tx(xAu,κ) and Tx(xAu,κ)=Tx(xAu,κ). These findings clearly demonstrate that chirality transfer is the physical origin of the transverse spin OT exerted on isotropic spherical gold particles.

    To gain a deeper understanding of the underlying physical mechanism, the time-averaged OT exerted on an isotropic spherical particle can be derived using the dipole approximation [16] T=Te+Tee+Tm+Tmm,with Te=12Re(p×E*),Tee=k0312πε0Im(p×p*),Tm=12Re(m×B*),Tmm=μ0k0312πIm(m×m*),where Te (Tm) comes from the interaction between the incident light field and the electric (magnetic) dipole induced in the particle, while Tee (Tmm) arises from the coupling between electric (magnetic) dipoles. For a chiral particle, the electric dipole moment p and magnetic dipole moment m are expressed as follows [16,68,69]: p=αeeE+αemB,m=αemE+αmmB,where αee, αmm, and αem describe the electric, magnetic, and magnetoelectric polarizability of the chiral particle. These polarizability elements for the dipolar chiral particle are directly related to the Mie scattering coefficients [16] αee=i6πε0k03a1,αmm=i6πμ0k03b1,αem=6πZ0k03c1,where Z0 is the wave impedance in vacuum, and a1, b1, and c1 are the Mie coefficients [59]. For a dimer system, the optical field experienced by each particle includes both the incident field and the scattered field from the other particle. Using the coupled dipole approximation [70,71], the electric field and magnetic field acting on each particle can be expressed as Ec=Einc,c+Ge(d)pac+1cGm(d)mac,Hc=Hinc,c+1Z0[Gm(d)pac+1cGe(d)mac],Eac=Einc,ac+Ge(d)pc+1cGm(d)mc,Hac=Hinc,ac+1Z0[Gm(d)pc+1cGe(d)mc],with pac=αee,acEac,pc=αee,cEc+αem,cμ0Hc,mac=αmm,acμ0Hac,mc=αem,cEc+αmm,cμ0Hc,where the subscripts ac and c correspond to the achiral and chiral particles, respectively. The dyadic Green’s functions Ge(d) and Gm(d) are given as [63] Ge(d)=eik0d4πε0d3[(k02d33ik0d+3)d(d)Td2+(k02d2+ik0d1)I],Gm(d)=ik0(ik0d1)eik0d4πε0d3d×,with I denoting the unit dyad, the superscript symbol T being the transpose, and d is the center-to-center separation between the two spheres.

    The effects of the gold sphere’s position on the origin of transverse spin OT are investigated for x-polarized (p1=p2=1,q1=q2=0) and y-polarized (p1=p2=0,q1=q2=1) incident light. As shown in Fig. 2, the results of full-wave calculations (black circles) are in perfect agreement with the dipole approximation (red solid lines) in Eqs. (12) and (13), verifying the correctness of the derived expressions, in which the chirality parameter of the chiral particle is assumed to be 0.5. Generally, for particles much smaller than the excitation wavelength, the contribution of the magnetic dipole is often neglected due to the significantly stronger electric response of gold nanoparticles compared to their magnetic response [56,57]. However, in the proposed dimer system, it is observed that for x-polarized light excitation [Fig. 2(a)], the coupling terms associated with the magnetic dipole (Txm and Txmm) contribute more to the transverse spin OT than the terms associated with the electric dipole (Txe and Txee). Notably, when the location of the gold sphere satisfies xAu>0.5  μm, the electric dipole coupling terms cancel out due to their nearly equal amplitudes but opposite directions, leaving the transverse spin OT primarily driven by the magnetic dipole terms. In contrast, for y-polarized light excitation [Fig. 2(c)], the contribution of the magnetic dipole becomes negligible regardless of the separation distance. Here, the transverse spin OT is predominantly caused by the interaction between the electric dipoles and the incident light field (Txe), as well as the coupling between electric dipoles (Txee). These findings highlight that by tuning the polarization orientation and the gold sphere’s position, the contributions of the electric and magnetic components of the excitation field to the transverse spin OT can be selectively controlled.

    The physical origin of chirality-transfer-induced transverse spinning motions of achiral particles. The relationship between the location xAu of gold sphere and the exerted transverse spin OT, along with its components associated with the electric and magnetic dipole moments as described in Eq. (12) (a), (c), or the electric and magnetic SAM densities as described in Eq. (19) (b), (d). The system is excited by counter-propagating fields with (a), (b) x-polarization and (c), (d) y-polarization. The transverse spin OT TxFW based on full-wave calculation (black circle) is also plotted for comparison. The symbol ×0.1 indicates that the respective values have been scaled by a factor of 10 for better visualization.

    Figure 2.The physical origin of chirality-transfer-induced transverse spinning motions of achiral particles. The relationship between the location xAu of gold sphere and the exerted transverse spin OT, along with its components associated with the electric and magnetic dipole moments as described in Eq. (12) (a), (c), or the electric and magnetic SAM densities as described in Eq. (19) (b), (d). The system is excited by counter-propagating fields with (a), (b) x-polarization and (c), (d) y-polarization. The transverse spin OT TxFW based on full-wave calculation (black circle) is also plotted for comparison. The symbol ×0.1 indicates that the respective values have been scaled by a factor of 10 for better visualization.

    From the perspective of the optical field, the generation of spin torques is typically associated with the SAM of the illuminating field. To investigate the effect of localized SAM on the spin OT, substituting Eq. (14) into Eqs. (12) and (13) for a gold particle with κ=0, viz., αem=0, the exerted OT only has the achiral components that can be expressed as [16] T=Tse+Tsm,with Tse=[2ωε0Im(αee)ωk033πε02αeeαee*]Se,Tsm=[2ωμ0Im(αmm)ωμ02k033παmmαmm*]Sm,where Tse and Tsm represent the OT components independent of the particle’s chirality, arising from the electric contributions Se=ε04ωIm(E*×E) and the magnetic contributions Sm=μ04ωIm(H*×H) of the time-averaged SAM density, respectively. The total spin OT can be separated into the electric dipole torque (Tse) and the magnetic dipole torque (Tsm) in Eq. (19). The former arises from the interaction between the particle’s electric response and the electric component of SAM, while the latter results from the interaction between the particle’s magnetic response and the magnetic component of SAM. Using Eq. (20), the connection between transverse spin OT and local SAM becomes clearer. As shown in Figs. 2(b) and 2(d), for x-polarized (y-polarized) incident plane waves, the transverse spin OT primarily originates from the magnetic dipole torque Txsm (electric dipole torque Txse), corresponding to the magnetic (electric) contribution of SAM density. Based on Eqs. (15) and (20), we show that the coefficients associated with the Mie terms remain constant with changes in position, resulting in the transverse OT being primarily governed by the local SAM. Consequently, when the spacing between the gold nanoparticle and the chiral particle is altered, the sign of the local SAM density reverses with the changing distance, leading to a directional reversal of the transverse OT. Additionally, when the polarization state of the illuminating field is switched from the x-direction to the y-direction, the dominant contribution to the local SAM transitions from the magnetic to the electric component. This transition induces a corresponding change in the transverse OT exerted on the gold nanoparticle, which serves as a macroscopic manifestation of the particle’s distinct electric and magnetic responses, as quantitatively described by Eq. (20). The numerical results show consistent trends when either magnetic or electric dipolar torque dominates the transverse spinning of particles. Although the magnetic response of the metallic sphere is weaker than its electric response, the magnetic spin OT generated is only slightly smaller than the electric spin OT, highlighting the promising role of the magnetic response in optical tweezers. Moreover, the contributions of the electric and magnetic components of the local SAM density to the transverse spin OT can be effectively controlled.

    B. Local Transverse SAM due to Chirality Transfer

    To understand the origin of the generated SAM, the local transverse SAM densities at the position of the gold particle are calculated. Since the local field (E and H) comprises both the incident field (Einc and Hinc) and the scattered field (Esca and Hsca), the SAM density of the local field can be expressed as Sloc=Sinc+Smix+Ssca,with Sinc=14ωIm(ε0Einc*×Einc+μ0Hinc*×Hinc)=Since+Sincm,Smix=12ωIm(ε0Einc*×Esca+μ0Hinc*×Hsca)=Smixe+Smixm,Ssca=14ωIm(ε0Esca*×Esca+μ0Hsca*×Hsca)=Sscae+Sscam,where Sinc, Smix, and Ssca represent the SAM density originating from the incident field, the coupling between the incident and scattered fields, and the scattered field, respectively. In Fig. 3, the vertical axis depicts the ratio of each SAM density component to the maximum value of the total SAM density, evaluated at x=0.25  μm. As shown in Fig. 3(a), the local SAM density is almost entirely derived from the magnetic contribution Slocxm(Slocxm=Sincxm+Smixxm+Sscaxm) for illumination with x-polarization. Furthermore, it can be observed that Slocxm mainly originates from the magnetic contribution Smixxm in the mixed term [shown in Fig. 3(b)]. However, the case is entirely different for illumination with y-polarization, in which the electric contribution Slocxe(Slocxe=Sincxe+Smixxe+Sscaxe) dominates the local SAM density, and it comes from the electric contribution in the mixed term Smixxe, as well as the electric contribution Sscaxe and the magnetic contribution Sscaxm of the scattered field [shown in Figs. 3(c) and 3(d)]. The local transverse SAM density and its decomposed terms show an approximately linear relationship with the chirality of the chiral particle, and the changes in configuration of the particle can alter the sign of SAM density. Furthermore, the polarization of the illumination affects not only the ratio of electric and magnetic contributions to the local SAM but also the contribution of the scattered field to the local SAM, offering a new strategy for polarization-controlled electric and magnetic responses of the particles. Since the incident field itself does not carry SAM, it is the chirality of the particle that determines the transfer of local transverse SAM, which in turn leads to the generation of transverse spin OT on the gold particle.

    Local transverse SAM arises from the contribution of chirality transfer. For excitation light with (a), (b) x-polarization and (c), (d) y-polarization, the normalized x-component of the SAM density and its electric and magnetic contributions, as well as the components based on Eqs. (21) and (22) as a function of the chirality parameter κ of the particle.

    Figure 3.Local transverse SAM arises from the contribution of chirality transfer. For excitation light with (a), (b) x-polarization and (c), (d) y-polarization, the normalized x-component of the SAM density and its electric and magnetic contributions, as well as the components based on Eqs. (21) and (22) as a function of the chirality parameter κ of the particle.

    C. Enantiospecific Detection of the Chiral Nanoparticle Using the Transverse Spin OT

    Based on the physical mechanism of chirality-transfer-induced transverse spin OT, we propose a novel method for qualitative and quantitative chirality detection at the nanoscale. Although chirality-transfer-induced transverse OT can still be observed under single-plane-wave illumination, the simultaneous presence of multiple OT components leads to complex rotational motion of the achiral particle. To address this issue, we primarily employ two counter-propagating linearly polarized plane waves. Under x-polarized illumination, the transverse spin OT predominantly arises from the interaction between localized SAM and the magnetic dipole response of the particle. By neglecting the electric dipole contributions, the expression for OT can be simplified to TxbC1Im[(N1μ0αmm,c+N4)αem,c*+(N2μ0αmm,c*+N3)αem,c],where C1=E02ε0μ08π2d5[Im(αmm,ac)k03μ06πRe(αmm,acαmm,ac*)],and N1=k0Z0(i+k03d3),N3=4πd3k0Z0(i+k0d)eik0d,N2=k0Z0(ik03d3),N4=4πd3k0Z0(ik0d)eik0d.Similarly, for y-polarized illumination, the transverse spin OT primarily stems from the interaction between the localized SAM and the electric dipole response of the particle. By neglecting magnetic dipole contributions, the simplified expression becomes TxaC2Im[(N1αee,c+N4ε0)αem,c*+(N2αee,c*+N3ε0)αem,c],where C2=E028π2ε0d5[Im(αee,ac)k036πε0Re(αee,acαee,ac*)]. Note that C1 and C2 are constants that are tightly related to the magnetic and electric responses of achiral particles.

    These equations highlight the direct relationship between the transverse spin OT and the intrinsic properties of the particles. Specifically, the magnetic and electric responses of the particles contribute separately to the torque, with chirality serving as the decisive factor in generating transverse spin OT. By observing the transverse OT exerted on gold nanoparticles, the chirality of nearby particles can be detected. Since the sign of the magneto-electric polarizability depends on the chirality configuration, enantiomers experience transverse OTs of equal magnitude but opposite directions. To further elucidate the relationship between chirality and OT, the Taylor expansion of magneto-electric polarizability under size parameter x0=k0Rc can be expressed as αem,c=12πZ0k03(iκx036+3εcκ2+if2x05+f3x06).Here, f2 and f3 represent a complex expression closely related to the particle’s permittivity and chirality parameter. When the size of the lossless chiral particle is much smaller than the incident wavelength, the last two terms of the formula can be neglected. Notably, the transverse spin OT acting on the gold particle is proportional to the chirality parameter of the chiral particle. This indicates that chirality detection of nanoscale samples can be achieved by measuring the OT. Figures 4(a) and 4(b) depict the linear relationship between transverse spin OT and the chirality parameter κ under x- and y-polarized illumination, respectively. It can be seen that the results from the simplified expression [Eqs. (23) and (26)] agree well with that from the rigorous expression [Eq. (20)]. As expected, the transverse spin OT increases linearly with the chirality parameter of the chiral particles, confirming its potential in chirality detection applications. Additionally, as shown in Figs. 4(c) and 4(d), increasing the radius of the gold particle (RAu) enhances the OT, while increasing the particle separation reduces its magnitude and reverses its direction. The peak value of transverse electric spin OT is approximately three times higher than its magnetic counterpart due to the localized surface plasmon effect, although polarization modulation can induce significant magnetic dipole responses.

    Effects of chirality-transfer-induced transverse spin OT on achiral particles under excitation light with (a), (c) x-polarization and (b), (d) y-polarization. (a), (b) The transverse spin OT and its decomposition terms as functions of the chirality parameter κ of the particle. The transverse spin OT TxFW based on full-wave calculations (circle) and Txb (triangle) and Txa (diamond) based on simplified derivation are also plotted for comparison. (c) The transverse magnetic spin OT and (d) electric spin OT as functions of the separation distance g of the dimer system and the radius of the gold particle.

    Figure 4.Effects of chirality-transfer-induced transverse spin OT on achiral particles under excitation light with (a), (c) x-polarization and (b), (d) y-polarization. (a), (b) The transverse spin OT and its decomposition terms as functions of the chirality parameter κ of the particle. The transverse spin OT TxFW based on full-wave calculations (circle) and Txb (triangle) and Txa (diamond) based on simplified derivation are also plotted for comparison. (c) The transverse magnetic spin OT and (d) electric spin OT as functions of the separation distance g of the dimer system and the radius of the gold particle.

    For particles beyond the dipolar regime, the physical characteristics governing the transverse OT can be comprehensively attributed to the mechanism of chirality transfer when chiral particles are involved, which are closely linked to the illumination field. Based on the generalized Lorenz-Mie theory [62] and multiple scattering theory [5961], our numerical results confirm that this conclusion is universally applicable to spherical particles of arbitrary composition, spanning both the Rayleigh and Mie scattering regimes. To further examine the impact of a chiral particle on a nearby achiral counterpart, we consider a representative example involving a gold nanoparticle with a radius of 80 nm, positioned at xAu=3.5  μm. As illustrated in Fig. 5(a), the transverse OT acting on the gold particle, plotted as a function of the radius of the nearby chiral particle under x-polarized excitation, exhibits vanishing torque at κ=0, while satisfying the antisymmetric relation Tx(κ)=Tx(κ), thus reaffirming the role of chirality in breaking the mirror symmetry of the system. Moreover, as the size of the chiral particle increases beyond the validity regime of the dipole approximation, the optical torque induced by chirality transfer no longer displays an approximately linear dependence on the chiral parameter κ, as clearly demonstrated by the black curve in Fig. 5(b). However, within the moderate chirality range of 0.15<κ<0.15, the transverse OT induced by large chiral particles still exhibits a nearly linear relationship with κ, and its magnitude significantly surpasses that induced by chirality in smaller particles, thereby enhancing the sensitivity of chiral detection. Additionally, Fig. 5(b) shows that, even when κ reaches its extreme values, the torque in the dipolar regime maintains an approximately linear dependence on κ, and interestingly, the detectable range of chirality for smaller particles is broader than that for larger ones. These findings collectively suggest that chirality-transfer-induced OT is not only applicable to small particles in the dipolar regime, but also remains effective in the case of large particles, thereby offering a promising strategy for improving the detection and manipulation of chirality across a wide range of particle sizes.

    (a) Transverse OT exerted on a gold nanoparticle with a fixed radius of RAu=80 nm, plotted as a function of the radius of a nearby chiral particle. The gold nanoparticle is positioned at xAu=3.5 μm. (b) Transverse OT on the same gold nanoparticle as a function of the chirality parameter κ of the chiral particle, whose radius is either 500 nm (black dashed line) or 80 nm (red line). The permittivity and permeability of the chiral particle are fixed at εc=2.53 and μc=1, respectively. The surface-to-surface separation between chiral and gold dipole particles is 90 nm.

    Figure 5.(a) Transverse OT exerted on a gold nanoparticle with a fixed radius of RAu=80  nm, plotted as a function of the radius of a nearby chiral particle. The gold nanoparticle is positioned at xAu=3.5  μm. (b) Transverse OT on the same gold nanoparticle as a function of the chirality parameter κ of the chiral particle, whose radius is either 500 nm (black dashed line) or 80 nm (red line). The permittivity and permeability of the chiral particle are fixed at εc=2.53 and μc=1, respectively. The surface-to-surface separation between chiral and gold dipole particles is 90 nm.

    On the other hand, the transverse spin OT induced by chiral transfer is a phenomenon strongly dependent on both material properties and polarization, and it is not limited to gold nanoparticles but can manifest more prominently in high-refractive-index materials. As shown in Fig. 6, germanium (Ge) [72] nanoparticles exhibit a magnetic transverse spin OT under x-polarized light that is two orders of magnitude stronger than that of gold, silver [66], or silicon [72] nanoparticles, while y-polarized light significantly weakens and reverses the transverse spin OT. Specifically, the condition |Txsm|>|Txse| holds when xAu=0.25  μm, with both components having opposite signs. In this case, Txsm exhibits a negative value, which ultimately leads to a reversal of the directional transitions of the total OT. Under y-polarized illumination, this phenomenon arises primarily due to the fact that the varying positions of the Ge nanoparticles influence the reversal mechanism of the transverse OT. Under strong near-field coupling, the electric and magnetic components of the transverse OT work synergistically, collectively contributing to the total torque. However, as the distance between the dimers increases, the electric component of the local SAM becomes significantly larger than its magnetic counterpart, causing the transverse OT to ultimately be dominated by the electric part. Moreover, the selection of material [66,7275] plays a critical role in optimizing spin OT: Ge and gallium antimonide (GaSb) [72] nanoparticles are suitable for inducing strong magnetic spin OT, while aluminum (Al) [73] and indium phosphide (InP) [72] nanoparticles are better suited for generating strong electric spin OT.

    The chirality-transfer-induced transverse spin OT on various types of achiral particles under excitation light with (a) x-polarization and (b) y-polarization. The symbol ×100 (×10) indicates that the respective values have been magnified by a factor of 100 (10) for better visualization.

    Figure 6.The chirality-transfer-induced transverse spin OT on various types of achiral particles under excitation light with (a) x-polarization and (b) y-polarization. The symbol ×100  (×10) indicates that the respective values have been magnified by a factor of 100 (10) for better visualization.

    To elucidate the physical mechanism behind this enhancement, we calculated the transverse electric spin OT (under y-polarization) and the transverse magnetic spin OT (under x-polarization) in the visible spectrum for chiral and achiral particles with a radius of 50 nm. The analysis reveals that the electric spin OT is mainly observed in Au nanoparticles [Fig. 7(a)], while the magnetic spin OT is more pronounced in Si [Fig. 7(b)] and Ge [Fig. 7(c)] nanoparticles. Specifically, for Si particles, the combination of the magnetic Mie coefficient and the local magnetic SAM density enhances the magnetic spin OT at the resonance peak (λ=475  nm), which surpasses the electric spin OT of gold particles (λ=467  nm). For Ge particles, the resonance of its magnetic response at λ=613  nm generates an exceptionally strong transverse magnetic spin OT, which is two orders of magnitude higher than the magnetic spin OT of gold nanoparticles and one order of magnitude higher than its electric spin OT.

    Electric and magnetic contributions to chirality-transfer-induced transverse spin OTs for Au (left column), Si (middle column), and Ge (right column) spheres. The spectrum of (a)–(c) the transverse spin OT, (d)–(f) the parameters related to the Mie coefficients, and (g)–(i) the normalized x-component of the local SAM density at the location of the achiral sphere. The red solid lines and blue dashed lines represent electric and magnetic quantities, respectively. Resonances facilitating electric or magnetic chirality transfer are marked by magenta and green dashed lines.

    Figure 7.Electric and magnetic contributions to chirality-transfer-induced transverse spin OTs for Au (left column), Si (middle column), and Ge (right column) spheres. The spectrum of (a)–(c) the transverse spin OT, (d)–(f) the parameters related to the Mie coefficients, and (g)–(i) the normalized x-component of the local SAM density at the location of the achiral sphere. The red solid lines and blue dashed lines represent electric and magnetic quantities, respectively. Resonances facilitating electric or magnetic chirality transfer are marked by magenta and green dashed lines.

    According to Eqs. (15) and (20), the magnitude of the transverse spin OT is closely related to the Mie coefficients and the local SAM density, where the electric spin OT, TxseRe(a1)|a1|2 or Sxe, and the magnetic spin OT, TxsmRe(b1)|b1|2orSxm. Based on this, we present the relevant parameters of the Mie coefficients in Figs. 7(d)–7(f) and the components of the local SAM density in Figs. 7(g)–7(i). For gold nanoparticles, the peak of the electric Mie coefficient couples with the local electric SAM density, leading to a significant transverse electric spin OT at λ=467  nm. However, due to the small difference between the local magnetic and electric SAM densities and the absence of magnetic Mie resonance, the magnetic spin OT is relatively weak. In contrast, for silicon particles, the magnetic Mie coefficient and the local magnetic SAM density combine at the resonance peak (λ=475  nm), significantly enhancing the transverse magnetic spin optical torque.

    From Eqs. (23) and (26), it is evident that the local transverse electric or magnetic SAM density primarily originates from the electric or magnetic response of chiral particles, resulting in similar local SAM densities for different achiral materials. However, because the magnetic response of silicon particles at the resonance peak is stronger, the transverse magnetic spin OT for silicon is significantly larger than the transverse electric spin OT for gold. As the wavelength increases, the spin OT acting on gold and silicon particles decays rapidly, weakening the effect of chirality transfer. For Ge particles, although the peak of the local SAM density combined with the electromagnetic response does not significantly enhance ordinary spin OT, the magnetic response resonance at λ=613  nm generates an exceptionally strong transverse magnetic spin OT, which is two orders of magnitude higher than the magnetic spin OT of gold nanoparticles and one order of magnitude higher than its electric spin OT. Consequently, the transverse spin OT in the dipolar regime induced by chirality transfer is mainly determined by the electric and magnetic responses of achiral particles. By adjusting the polarization state of the incident light and appropriately utilizing electric and magnetic resonances, the chirality transfer effect can be maximized, providing an optimized pathway for spin optical applications.

    4. CONCLUSION

    In conclusion, this study proposes a novel method for chirality transfer in nanophotonics and demonstrates its potential application in detecting the chirality of nanoparticles. By illuminating a dimer system composed of a gold nanoparticle and a chiral nanoparticle with two counter-propagating linearly polarized beams, chirality is transferred from the chiral nanoparticle to the gold nanoparticle, inducing a transverse spin OT on the latter. This conclusion is universally applicable to spherical particles of arbitrary composition, spanning both Rayleigh and Mie scattering regimes. Notably, this effect does not depend on the SAM of the excitation light field but arises from the electric or magnetic responses of the particles to the light field within the dipolar regime, which can be controlled by adjusting the polarization direction of the incident light. Moreover, leveraging the approximately linear relationship between the transverse spin OT and the chirality parameter, a novel approach is proposed for indirect molecular chirality detection by measuring the OT on the achiral nanoparticle. Additionally, replacing gold nanoparticles with suitable dielectric materials, such as germanium, significantly enhances the induced transverse magnetic dipolar OT by more than two orders of magnitude, leading to the improved sensitivity of chirality detection. These findings deepen our understanding of unconventional transverse OT, establish a basis for achieving controllable transverse spinning motion through chirality transfer, and pave the way for utilizing OT in the precise detection and characterization of a wide range of chiral particles.

    Acknowledgment

    Acknowledgment. L.F. acknowledges the support by the Postgraduate Research & Practice Innovation Program of Jiangsu Province.

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