Optical Bessel beam illumination of a subwavelength prolate gold (Au) spheroid coated by a layer of plasmonic material: radiation force, spin and orbital torques

The optical radiation force, spin and orbital torques exerted on a subwavelength prolate gold spheroid coated by a layer of plasmonic material with negative permittivity and illuminated by either a zeroth-order (non-vortex) or a first-order vector Bessel (vortex) beam are computed in the framework of the electric dipole approximation method. Calculations for the Cartesian components of the optical radiation force on a subwavelength spheroid with arbitrary orientation in space are performed, with emphasis on the order (or topological charge), half-cone angle of the beam, and the plasmonic layer thickness on- and off-resonance. A repulsive (pushing) force is predicted for the layered subwavelength prolate spheroid, on- and off-resonance along the direction of wave propagation. Moreover, the Cartesian components of the spin radiation torque are computed where a negative longitudinal spin torque component can arise, suggesting a rotational twist of the spheroid around its center of mass in either the counter-clockwise or the clockwise (negative) direction of spinning. In addition, the longitudinal component of the orbital radiation torque exhibits sign reversal, indicating a revolution around the beam axis in either the counter-clockwise or the clockwise directions. The results show that the plasmonic resonance strongly alters the force, spin and orbital torque components, causing major amplitude enhancements, signs twists, and complex distributions in the transverse plane.


Introduction
Elongated gold (Au) spheroidal nanoparticles coated by a metallic plasmonic layer (such as silver, copper, or other artificial materials etc) are particularly useful in various inorganic chemistry, metamaterials, drug delivery, biosensing [1] and biophysics applications to name a few examples. A carefully-designed conformal plasmonic layer coating of such particles provides the capability of detecting single-molecule events [2] by monitoring the shift of the plasmonic resonance. In practice, these particles can be rotated and transported before they get released to the desired site using optical tweezers [3] and tractor beams [4,5], due to the transfer of linear and angular momenta carried by light waves [6].
Optimal design often relies on the computational modeling [7,8] of the induced optical force, spin and orbital torques [9][10][11][12][13][14] as a priori information used for experimental guidance. Thus, numerical simulations constitute indispensable tools for guiding and improving the functionalities of the technological devices and experimental setups.
Several investigations considered computational analyses for the optical force and torques on spheroidal particles in optical tweezers [15][16][17][18][19][20][21], including the long-wavelength (Rayleigh) limit in Gaussian [22] and focused beams [23], Bessel beams [24,25], and light-sheets [26] as well as the geometrical optics regime [27,28]. Tailoring the incident wave field and beam profile to enhance or reduce light scattering from the particle can create a paradigm shift in optical manipulation procedures, such that negative pulling or positive pushing forces Figure 1. The schematic describing the interaction of an optical high-order Bessel vortex beam with a coated prolate spheroid with arbitrary orientation in space. The internal/external major and minor semi-axes are a i , a e and b i , b e , respectively, where the relative thickness factor is e=b e /b i 1.

Method
A dielectric subwavelength coated spheroid subjected to an incident optical beam develops an electric dipole moment, such that the resulting electric radiation force F in a medium of wave propagation that is nonmagnetic and nonelectric (with a unit index of refraction) such as vacuum, is expressed as [60], { } R is the real part of a complex number, the symbol Ä denotes a tensor product, p E, a = represents the electric dipole moment, a is the modified electric polarizability, which accounts for the radiative nature of the field, the double over-bar denotes a tensor of rank two [61], E is the incident electric field, and the superscript * denotes the conjugate of a complex number. The modified electric polarizability tensor is generally a complex number, given as [62], where the matrix representing 0 a for a coated spheroid is expressed as, The coefficients 0, a^and 0, a || are the components of the static electric molecular polarizability tensor, which are expressed as, where c e and s e are the permittivities of the coating and the bare spheroid, respectively. Notice that for opticallyabsorptive (or active) coating and/or core materials, c e and s e are generally complex numbers. The parameter where the interior and exterior semi-axes satisfy the confocal condition The rotation of the spheroid with respect to its center of mass is described by two additional angular parameters (θ 0 , f 0 ), where θ 0 and f 0 are the polar and azimuthal angles of the major axis, respectively (figure 1). The rotation of the spheroid in space affects the modified and static polarizability tensors a and , 0 a respectively, such that in matrix notation, where the rotation matrix is given by [28], In addition to the radiation force, the layered spheroid experiences a spin radiation torque vector T s causing its spinning around its center mass, as well as a longitudinal orbital torque component T T e , (known also as the torque of revolution [10]) causing its rotation around the main axis of the incident beam, in the plane perpendicular to the direction of wave propagation, where e z is the unit vector along the direction of wave propagation. Unlike the case of a (coated) sphere [64] for which energy absorption in its coating/core materials is required in order that it experiences a spin torque causing its rotation around its center of mass [10], the (layered) spheroid considered here, which is made of lossless coating/core materials, is subjected to a non-zero spin torque depending on its orientation in space. Furthermore, the orbital torque only has a longitudinal component T , z o as the orbiting of the spheroid around the beam axis occurs solely in the transversal plane (x, y). The total radiation torque is expressed as, where the spin torque [65,66] is expressed as The longitudinal orbital torque component (along the direction of wave propagation z) is calculated from the transverse components of the force given by equation (1), such that where r is the vector position. Equations (1), (9), and (10) show that the radiation force and torque expressions depend directly on the incident electric E=(E x , E y , E z ) of the HOBVB. For an unpolarized HOBVB [56,57] with a harmonic timedependence in the form of t exp iw -( )omitted from the equations for convenience, the expressions for the Cartesian components of the electric field are given by, is the radial distance to a point in the transverse plane (x, y), β is the half-cone angle of the beam, and J m ⋅ ( ) is the cylindrical Bessel function of the first kind of order m, which determines also the topological charge (or order) of the HOBVB.

Numerical results and discussions
The analysis is started by developing a MATLAB numerical code to implement equations (1)- (13) and compute the optical radiation force, spin and orbital torque components for a gold (Au) spheroid in air with a permittivity coefficient ε s =−50, coated by a plasmonic layer of material with a permittivity coefficient ε c . The Cartesian components of the optical radiation force and radiation spin and orbital torques are computed in the ranges −10(kx, ky)10.
Initially, a Rayleigh prolate spheroid is considered, with ka e =0.25>kb e =0.1. To characterize the properties of the layered spheroid, computational plots for the components of the modified electric polarizability given by equation (2) are evaluated, by varying the relative thickness (b e /b i 1) of the layer as well as its permittivity. The chosen bandwidths are 1(b e /b i )10 and −5ε c 0. Panels (a)-(d) of figure 2 illustrate the results for this example where the real part of the transverse a^and longitudinal a || modified polarizability components (i.e., panels (a) and (c)) display positive and negative values while the imaginary part (shown in panels (b) and (d)) only show positive values. Those plots show that each of the modified polarizability component for the subwavelength spheroid coated by a layer of a plasmonic material, exhibits a particular resonance determined by the values of the layer thickness and its permittivity. It is anticipated that these effects significantly modify the radiation force, spin and orbital torques, as will be further examined in the following.
The first example considers the case of a gold prolate spheroid with ka e =0. 25, kb e =0.1 and ε s =−50, coated by a layer of a plasmonic material such that (b e /b i )=1.2 and ε c =−4.134. The layer material properties are selected based upon the plots of figure 2 (denoted by mark A in figure2(c)), such that no resonance occurs for this set of parameters. The subwavelength spheroid in air is illuminated by an optical zeroth-order (i.e., m=0) Bessel beam with β=35°, and its orientation angles in space are (θ 0 , f 0 )=(45°, 0°) chosen to illustrate the analysis. Panel   figure 4 displays the longitudinal force component at resonance, which is about five orders of magnitudes larger than its counterpart, away from the plasmonic resonance (i.e., panel (a) of figure 3). Moreover, the transverse vector force field denoted by the arrows converges no longer to the main central region as observed in panel (a) of figure 3, rather it has a tendency to push the prolate spheroid along the negative kx-direction. In addition, a quasi-vortex behavior of the transverse vector field is manifested from either side of the main axis (kx, ky)=(0, 0), such that for ky>0, a clockwise sense of rotation is observed, whereas for ky<0, a counter-clockwise sense of rotation is exhibited. The plots for the longitudinal spin and orbital torque components displayed in panels (b) and (c) of figure 4 also show an amplitude increase of about two orders of magnitudes compared to those of figure 3, and negative longitudinal spin and orbital torques can still occur at the plasmonic resonance. Moreover, panel (b) shows that the transverse spin torque vector field is oriented towards the positive ky direction for kx>0, while the sign of the vector arrows is reversed for kx<0.
The effect of increasing the order of the beam to m=1 is further investigated for the prolate spheroid having ka e =0.25, kb e =0.1 and ε s =−50, and coated by a layer of a plasmonic material such that (b e /b i )=1.2 and  the spin radiation torque components where T z s can alternate between positive and negative values, while the transverse spin torque vector field behaves similarly to its counterpart for the zeroth-order Bessel beam at the plasmonic resonance (i.e., panel (b) of figure 4). Interestingly, panel (c) of figure 6 for the longitudinal orbital torque component T z o shows a behavior comparable to that of F z of panel (a) with the emergence of a crescent region of large amplitude, nonetheless, it reverses sign depending on the shift from the center of the beam (where it vanishes therein) in the transverse plane. The transverse radiation force vector field is superimposed and denoted by the arrows that display the same quasi-vortex-like structure in the counter-clockwise direction, shown in panel (a).
The effect of increasing the half-cone angle to β=85°is further investigated, and the corresponding radiation force and torque results for the layered spheroid for which ka e =0.25, kb e =0.1, ε s =−50, (b e /b i )=4 and ε c =−4.134, in a first-order Bessel vortex beam, are shown in the panels of figure 7. The oneto-one comparison of each of the panels with those of figure 6 shows that the longitudinal force, spin and orbital torque components are reduced, and the amplitude maxima are narrower for panels (a) and (b). Complex patterns arise as the half-cone angle of the beam increases.
Finally, the variations of the radiation force, spin and orbital torque components versus the orientation angles have been computed in the ranges 0°θ 0 180°and 0°f 0 360°. The same layered prolate spheroid parameters considered previously for figure 6 are used, such that ka e =0.25, kb e =0.1, ε s =−50, (b e /b i )=4 and ε c =−4.134, considering a zeroth-order Bessel beam with β=85°. The shift from the center of the beam is (kx, ky)=(0.1, 2.5) chosen as an example to illustrate the analysis. The corresponding results are displayed in panels (a)-(g) of figure 8 for the Cartesian components of the force and torques. As the shift from the center of the beam is non-zero, the transverse components of the optical force shown in panels (a) and (b) do not vanish. Nonetheless, along some specific polar and azimuthal orientation directions, the transverse components vanish due to symmetry consideration. Notice also that the longitudinal force component shown in The effect of increasing the order of the beam to m=1 and keeping all the other parameters the same, is also examined, and the corresponding results are displayed in the panels of figure 9. The one-to-one comparison of each of the panels with those of figure 8 shows that the force, spin and orbital torque components are significantly altered as more complex variations arise as m increases.
One aspect in which the calculations for the optical radiation force, spin and orbital torque components obtained here can be beneficial is the prediction of the translational and rotational dynamics of the subwavelength layered spheroid in a HOBVB. Depending on the incident beam parameters and field intensity as well as the physical properties of the spheroid, Brownian motion [67], viscous drag [68][69][70], optical streaming [71], a nearby boundary/wall, buoyancy and other elements can influence the translational and rotational dynamics. Typically, pre-calculated force, spin and orbital torque vector components at any position in space are used in applying Newton's second law of motion to compute the trajectories. Examples can be found for an infinite cylinder of circular cross-section in Hermite-Gaussian light-sheets [72], a sphere in Airy [73] or a coated sphere in Bessel-pincers light-sheets [64], large ellipsoids in collimated and focused beams [74], and multiple spheroidal particles in a circularly polarized dual beam trap [21] to mention a few. The scope of the present analysis can be further extended to include particle dynamics computations, and this work should assist along that direction of research. It should be also noted here that the generation of negative optical spin and orbital torques is in complete agreement with energy conservation since computations (not shown here) for the rates of energy extinction and scattering [75] by the lossless spheroid coated by a non-absorptive plasmonic layer of material have shown equal results. Notice that the rate of energy absorption by the subwavelength spheroid considered here is zero. It must be recalled here that the spin torque reversal has been observed initially in the context of a scalar (acoustical) HOBVB incident upon a viscous sphere located arbitrarily in space (figure 4 in [47]), as well as vector (optical) HOBVBs on dielectric absorptive and semiconducting spheres [45,48,76] satisfying energy conservation. Furthermore, the orbital torque sign reversal, initially observed in the context of a circularly polarized Gaussian laser beam incident upon an absorptive sphere (figure 7 in [10]), and predicted previously for vector HOBVBs [46] on a semiconducting lossy sphere, is anticipated here on a subwavelength layered lossless spheroid as shown by the results of the present study. Notice, on the other hand, that for a scalar (acoustical) HOBVB incident on a subwavelength (Rayleigh) viscous fluid sphere, the orbital torque sign reversal does not occur [77].

Conclusion
Within the framework of the dielectric dipole approximation method, analytical solutions for the optical radiation force, spin and orbital torques are used to compute the mechanical effects of optical Bessel beams of zeroth and first orders on a non-absorptive prolate subwavelength spheroid, coated by a plasmonic layer of lossless material. In contrast to the cases dealing with Bessel beam illumination of an optically-active [24] or a semiconducting [25] subwavelength spheroid where negative pulling longitudinal forces arise (i.e. displaying a tractor beam effect), the lossless dielectric subwavelength spheroid coated by a plasmonic layer of nonabsorptive material considered here always experiences a pushing longitudinal force component directed in the forward direction of wave propagation, and the transverse forces induce a collimating effect toward the central Figure 7. The same as is figure 6, but the half-cone angle of the beam is increased to β=85°at the plasmonic resonance. axis of the beam, off the plasmonic resonance. Nevertheless, this effect does not occur at the plasmonic resonance, where large amplitude enhancements of the positive longitudinal force component occur. Moreover, the layered spheroid is shown to experience negative spin and orbital torques causing rotation around its center of mass and revolution around the beam axis, respectively, in either the counter-clockwise or the clockwise directions. Unlike the case of a sphere, the subwavelength spheroid experiences a rotation around its center of mass in the ideal case of no-absorption inside its coating/core materials when located arbitrarily in the field of optical Bessel beams, depending on the polar and azimuthal orientation angles. The results of this analysis find potential applications in optical tweezers, tractor beams and other related topics in particle manipulation and handling. Figure 8. The plots for the Cartesian components of the radiation force, spin and orbital torque vectors for a layered subwavelength prolate spheroid at the plasmonic resonance, placed in the field of an optical zeroth-order Bessel beam with β=85°, with a shift (kx, ky)=(0.1, 2.5) from the center of the beam, versus the orientation angles θ 0 and f 0 .