Optical torque on a magneto-dielectric Rayleigh absorptive sphere by a vector Bessel (vortex) beam

Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

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Optical torque on a magneto-dielectric Rayleigh absorptive sphere by a vector Bessel (vortex) beam Renxian Li a,b,n, Ruiping Yang a, Chunying Ding a, F.G. Mitri c a b c

School of Physics and Optoelectronic Engineering, Xidian University, Xi'an 710071, China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an 710071, China Chevron, Area 52 Technology-ETC, 5 Bisbee Ct., Santa Fe, NM 87508, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 13 January 2017 Received in revised form 4 February 2017 Accepted 6 February 2017 Available online 9 February 2017

The optical torque exerted on an absorptive megneto-dielectric sphere by an axicon-generated vector Bessel (vortex) beam with selected polarizations is investigated in the framework of the dipole approximation. The total optical torque is expressed as the sum of orbital and spin torques. The axial orbital torque component is calculated from the z-component of the cross-product of the vector position r and the optical force exerted on the sphere F . Depending on the beam characteristics (such as the half-cone angle and polarization type) and the physical properties of the sphere, it is shown here that the axial orbital torque vanishes before reversing sign, indicating a counter-intuitive orbital motion in opposite handedness of the angular momentum carried by the incident waves. Moreover, analytical formulas for the spin torque, which is divided into spin torques induced by electric and magnetic dipoles, are derived. The corresponding components of both the optical spin and orbital torques are numerically calculated, and the effects of polarization, the order of the beam, and half-cone angle are discussed in detail. The left-handed (i.e., negative) optical torque is discussed, and the conditions for generating optical spin and orbital torque sign reversal are numerically investigated. The transverse optical spin torque has a vortex-like character, whose direction depends on the polarization, the half-cone angle, and the order of the beam. Numerical results also show that the vortex direction depends on the radial position of the particle in the transverse plane. This means that a sphere may rotate with different directions when it moves radially. Potential applications are in particle manipulation and rotation, single beam optical tweezers, and other emergent technologies using vector Bessel beams on a small magneto-dielectric (nano) particle. & 2017 Elsevier Ltd. All rights reserved.

Keywords: Optical torque Vector Bessel beam Generalized Lorenz-Mie theory Magneto-dielectric Rayleigh sphere Polarization

1. Introduction In addition to the optical force [1–3], which has the ability to trap and manipulate microscopic particles, the optical torque [4–11] offers another mechanical degree of freedom in order to manipulate and rotate small particles. The optical torque arises from the transfer of angular momentum (AM) to a particle, and has received significant attention in various fields, including optical levitation [12], atomic physics, cell biology, etc. The optical torque can be divided into spin and orbital torques [6]. An optical spin torque (OST) can cause the particles to rotate around its center of mass, and an optical orbital torque (OOT) induces the particle rotation around the beam axis. Various investigations have been devoted to the theoretical prediction of the optical torque induced by various kinds of beams, and different approaches have been developed. The geometrical optics [13] and Rayleigh models [14,15] are two approximate approaches, and can be used to predict numerically the optical torques exerted on a particle which is much larger or smaller than the wavelength of the incident beam, respectively. For rigorous predictions of optical torques, the electromagnetic theory [16–21,7,22] based on Maxwell's equations is introduced. For the case of particles with arbitrary shape, numerical methods, including the T-matrix method [23], the discrete dipole approximation (DDA) [24–27], the finite-difference time-domain (FDTD) [21,28], and the finite element [29,30], are used. On the basis of such theories, counterintuitive mechanical effects of light have emerged. The first concerns the negative optical force generation [31–38] and the second is related to the emergence of a left-handed optical torque [31,39–42]. It n

Corresponding author at: School of Physics and Optoelectronic Engineering, Xidian University, Xi'an 710071, China. E-mail address: [email protected] (R. Li).

http://dx.doi.org/10.1016/j.jqsrt.2017.02.003 0022-4073/& 2017 Elsevier Ltd. All rights reserved.

R. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

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is Newton's third law of motion that explains the emergence of the negative pulling force. It is generally associated with a reduction in the backscattering direction and a forward scattering enhancement. The induced reaction of this effect causes the emergence of a pulling force acting in opposite direction of the wave propagation (i.e., action-reaction law). This reaction nonconservative force (which is distinct from the conservative gradient force) forms the basis of the optical tractor beam concept. Left-handed optical torque means that the direction of the optical torque is reversed with respect to the direction of the angular momentum carried by the incident beam, as a consequence of light absorption and re-scattering by the particle. Recently, tailoring and engineering the wavefronts [43–45] to enhance or reduce the scattering from a particle in order to achieve effective rotation and manipulation and to realize novel mechanical effects including left-handed optical torque, has been investigated. A particular kind of beams that can induce an optical torque is the Bessel vortex beam [46–51], which is one particular kind of nondiffracting beams [52–54]. Bessel beams have two special features. One is limited-diffraction, meaning that their transverse intensity is propagation-invariant over a particular distance of many wavelengths. Another one is self-healing or self-reconstructing [55,56], which means that the beam can re-establish its transverse intensity profile after being disturbed by objects. An ideal vector Bessel beam is a solution of the vector Helmholtz equation ∇E2 +

ω2 c2

E = 0 and Maxwell's equations, with

ω and c being respectively the angular fre-

quency of the beam and light speed. In cylindrical coordinates, the electric field vector of an l − order vector Bessel beam [57–61] can ^ uE J (k ρ) eilϕeikz ze−iωt . Where J (·) is the cylindrical Bessel function of order l. The parameters be mathematically expressed as E (ρ , ϕ, z ) = e ρ 0 l l

x2 + y2 , and ϕ = arctan (y/x ). kρ = k sin α0 and k z = k cos α0 are respectively the radial and axial components of wavenumber k = ω/c . ^ u represents the polarization state, and the superscript u indicates the kind of polarization. is the half-cone angle of the Bessel beam. e

ρ=

α0

Though an ideal vector Bessel beam can not be experimentally realized, a vector quasi-Bessel beam can be synthesized using an axicon [62,63] or a spatial light modulator (SLM) [64–66]. The electromagnetic field of a vector Bessel beam can be determined by the decomposition into transverse electric (TE) and magnetic (TM) modes [67], or by using the angular spectrum decomposition (ASD) [68,60,69,70]. The former approach has the advantage of generating an optical pulling force, while, additionally, the latter emphasizes the effect of polarization. An axicon-generated vector Bessel beam (AGVBB), whose electromagnetic field components have been derived using the angular spectrum decomposition (ASD) [70] and are appended in Appendix A, can induce an optical pulling force on a dielectric [33] or a magneto-dielectric [34] sphere depending on the order, the half-cone angle, and the polarization of the plane wave component forming the incident Bessel beam. Moreover, the direction of the axial OST on a Rayleigh absorptive sphere by an AGVBB can be reversed by choosing an appropriate polarization, order, and half-cone angle, and the transverse OST will manifest vortex-like behaviors [71]. Nonetheless, the influence of the orbital torque axial component induced by an AGVBB is yet to be considered in order to analyze thoroughly and completely the generation of a total lefthanded torque. This paper is therefore devoted to fill this gap, with particular emphasis on polarization. The rest of the paper is organized as follows. In Section 2, the optical torque including the OOT and the OSTs induced by both electric and magnetic dipoles are derived on the basis of the dipole approximation. Various polarizations of the plane wave components forming the Bessel (vortex) beam, including linear, circular, azimuthal, and radial polarizations, are considered. Section 3 presents and discusses the numerical results, where the corresponding components of the OST and OOT are analyzed, respectively. Section 4 is devoted to the conclusions.

2. Theoretical background In this section, the optical torque exerted on a magneto-dielectric sphere by an AGVBB is derived. In the dipole approximation, the total optical torque is divided into optical orbital torque (OOT), which makes the particle orbit around the beam axis, and an optical spin torque (OST), which causes the particle to rotate around its center of mass. The spin torque is induced by both the electric and magnetic dipoles. Subsequently, the (total) optical torque is expressed as spin T = T orbit + T espin + T m

(1)

where, the superscripts orbit and spin respectively represent the orbit and spin torques, and the subscripts e and m respectively denote the OST spin ) are derived induced by the electric and magnetic dipoles. In this section, the OOT (denoted by T orbit ) and the OST (denoted by Tespin and Tm separately. 2.1. Optical orbital torque The optical orbital torque (OOT) is defined by [6]

T orbit = r × F

(2)

where r is the position vector. F is the total electromagnetic force exerted on the particle, and includes the optical forces induced on the electric dipole Fe and on the magnetic dipole Fm , and the force due to the interaction between both dipoles Fem [34]. The analytical expressions of these forces with selected polarizations have been derived using the dipole approximation in our previous paper [34], and for completeness, they are provided in the Appendices B, C, and D, respectively. Substituting the forces into Eq. (2) and after some algebra, we can obtain the analytical expressions of the OOT by an AGVBB with selected polarizations. Note that in our calculations, the axial component of the orbital torque is calculated directly by performing the cross product of r and F using Eq. (2) along the axial z-direction. Notice that the orbital torque causing the rotation of the sphere around the beam axis z is based on the transverse components of the optical force, in the plane perpendicular to light propagation. 2.2. Optical spin torque induced by electric dipole The optical spin torque induced by the electric dipole (OSTE) is given by [72,10,8]

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R. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

T espin =

⎛ 1 ⎞ 1 αe 2 Re ⎜ E × E*⎟ * 2 α ⎝ e,0 ⎠

(3)

where E is the external electric field exerted on the particle, and its components are given in Appendix A, and the superscript * denotes a conjugate. αe is the modified electric polarizability defined by

αe =

αe,0 1 1− ik3αe,0 6πε0

(4)

with

αe,0 = 4πε0 a3

ε−1 ε+2

(5)

where, k = 2π /λ is the wavenumber with λ being the wavelength of incident beam. a is the radius of the sphere, and ε is the complex permittivity coefficient of the particle with relative to that of the surrounding medium with permittivity ε0. The substitution of the electric field given in Appendix A into Eq. (3) will result in the OSTE by AGVBBs with various polarizations. For an AGVBB with x polarization of the individual plane wave components, the OSTE Cartesian components can be obtained as:

T espin, x

⎡ ⎧P P ⎡ J J − J J ⎤ ⎫⎤ l − 1 l − 2 ) sin ϕ + ( Jl + 1 Jl − 2 − Jl + 2 Jl − 1 ) sin 3ϕ⎦ ex ⎪ ⎥ ⎢ ⎪ ∥ ⊥ ⎣ ( l+ 2 l+ 1 ⎢ ⎪⎥ ⎪ ⎡ 2J ( J − J ) cos ϕ + ( J J − J J ) cos ϕ ⎤ l l− 1 l+ 1 l− 2 l− 1 l+ 2 l+ 1 ⎢ i ⎪ ⎪⎥ ⎥ ⎢ 1 e P P + 2 y ⎬⎥ ∥ ⊥ ⎨ = αe Re ⎢ ⎥ ⎢ cos 3 J J J J + − ϕ * ( ) 2 ⎦ ⎣ l+ 2 l− 1 l− 2 l+ 1 ⎪⎥ ⎢ αe,0 ⎪ ⎪⎥ ⎪ ⎢ 1 ⎡ ⎤ 2 2 ⎪⎥ ⎪ ⎢ + P⊥ ⎣ 2Jl ( Jl + 2 − Jl − 2 ) cos 2ϕ + Jl + 2 − Jl − 2 P⊥ ⎦ ez ⎭⎦ ⎩ ⎣ 2

(

)

(6)

where, ex , ey , and ez are the unit vectors in the Cartesian system, and the superscript x denotes the polarization. Jl is the cylindrical Bessel function of first kind of order l, and its argument σ = kρ ρ = kρ sin α0 is omitted for the sake of brevity. Other parameters are:

P⊥ =

1 −cos α0 , 1 +cos α0

P∥ =

sin α0 , 1 +cos α0

ϕ = arctan (y/x ), ρ =

x2 + y2 , and EB0 = πEpw0 (1 + cos α0 ), where Epw0 is the amplitude of the individual plane

wave component. Note that for a non-absorptive sphere, both αe,0 and all the terms in braces {·} are real, and then Tespin, x expressed by Eq. (6) is zero for any order l and any half-cone angle α0. Similar conclusions can be obtained for AGVBBs of other polarizations discussed below. The Cartesian components of OSTE by an AGVBB with y polarization of the individual plane wave components can be written as

T espin, y

⎡ ⎫⎤ ⎧ ⎡ 2 sin ϕJ ( J − J ) P + ( J J − J J ) sin ϕP P ⎤ ∥ ⊥ l l+ 1 l− 1 ∥ l+ 2 l+ 1 l− 2 l− 1 ⎢ ⎪ ⎥ ex ⎪ ⎥ ⎢ ⎢ ⎪⎥ ⎪ ⎥⎦ ⎢⎣ +( Jl + 2 Jl − 1 − Jl − 2 Jl + 1 ) sin 3ϕP∥ P⊥ ⎢ i ⎪ ⎪⎥ 1 ⎬⎥ ⎨ ⎡ = αe 2 Re ⎢ ⎤ 2 ⎢ αe*,0 ⎪ − ⎣ ( Jl + 1 Jl + 2 − Jl − 1 Jl − 2 ) cos ϕ + ( Jl − 1 Jl + 2 − Jl + 1 Jl − 2 ) cos 3ϕ⎦ P∥ P⊥ ey ⎪ ⎥ ⎪⎥ ⎪ ⎢ 1 ⎪⎥ ⎪ ⎢ + ⎡⎣ Jl2+ 2 − Jl2− 2 P⊥ − Jl ( Jl + 2 − Jl − 2 ) 2 cos 2ϕ⎤⎦ P⊥ ez ⎭⎦ ⎩ ⎣ 2

(

)

(7)

For an AGVBB with circular polarization of the individual plane wave components, the Cartesian components of the OSTE can be expressed as

T espin, lc

⎡ ⎧ 4J ( J + J P ) P sin ϕe ⎫ ⎤ x l+ 1 l l+ 2 ⊥ ∥ ⎥ ⎢ ⎪ ⎪ ⎪ ⎪ 1 i ⎢ ⎨ − 4Jl + 1 ( Jl + Jl + 2 P⊥ ) P∥ cos ϕey ⎬ ⎥ = αe 2 Re ⎢ 2 αe*,0 ⎪ ⎪⎥ 2 2 2 ⎥ ⎢ ⎪ ⎪ 2 J J P e − − z ⎩ ⎭⎦ l l+ 2 ⊥ ⎣

(8)

⎡ ⎧ 4J ( −J − J P ) P sin ϕe ⎫ ⎤ x ⎥ l l− 2 ⊥ ∥ ⎢ ⎪ ⎪ l− 1 ⎪ 1 i ⎪ ⎢ 2 ⎨ + 4Jl − 1 ( Jl + Jl − 2 P⊥ ) P∥ cos ϕey ⎬ ⎥ = αe Re ⎢ αe*,0 ⎪ 2 ⎪⎥ ⎥ ⎢ ⎪ ⎪ − 2 −Jl2 + Jl2− 2 P⊥2 ez ⎭⎦ ⎩ ⎣

(9)

(

T espin, rc

(

)

)

where, the superscripts lc and rc denotes left and right circular polarizations, respectively. The Cartesian components of the OSTE for an AGVBB with radial polarization of the individual plane wave components can be given by

⎡ ⎧ 4J ( J − J ) sin ϕ sin α cos α e ⎫ ⎤ 0 0 x l l− 1 l+ 1 ⎥ ⎢ ⎪ ⎪ ⎪ ⎪ 1 1 i ⎢ spin, rp 2 4 J J J cos sin cos − − + ϕ α α 0 ey ⎬ ⎥ ⎨ Te Re = αe ( ) 0 l l 1 l 1 + − 2 2 ⎪⎥ ( 1 + cos α0 ) ⎢⎢ αe*,0 ⎪⎪ ⎥ ⎪ − 2 Jl2− 1 − Jl2+ 1 cos2 α0 ez ⎩ ⎭⎦ ⎣

(

)

(10)

For an AGVBB with azimuthal polarization of the individual plane wave components, the Cartesian components of the OSTE become

R. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

T espin, ap = −

1 2

( 1 + cos α0 )

⎧ 1 ⎫ i Jl2− 1 − Jl2+ 1 ez ⎬ αe 2 Re ⎨ * α ⎩ e,0 ⎭

(

)

99

(11)

2.3. Optical spin torque induced by magnetic dipole The optical spin torque induced by the magnetic dipole (OSTM) is defined by [72,10] spin Tm =

⎛ 1 ⎞ 1 αm 2 Re ⎜ H × H*⎟ 2 ⎝ αm*,0 ⎠

(12)

where H is the external magnetic field illuminating the particle, and is also given in Appendix A. ability:

αm =

αm is the modified magnetic polariz-

αm,0 1−

1 ik3αm,0 6πμ0

(13)

with

αm,0 = 4πμ 0 a3

μ−1 μ+2

(14)

where, μ is the complex permeability of the particle with relative to that of the surrounding medium with permeability μ0. Substituting the magnetic fields given in Appendix A into Eq. (12) and after some algebra, we can obtain the Cartesian components of OSTM for AGVBBs with various polarizations (linear, circular, radial, and azimuthal polarizations) of the individual plane wave components. They can be expressed as:

spin, x Tm

⎡ ⎫⎤ ⎧⎡ ⎤ ⎢ ⎪ ⎢ 2Jl ( Jl + 1 − Jl − 1 ) sin ϕ + ( Jl + 1 Jl + 2 − Jl − 2 Jl − 1 ) P⊥ sin ϕ ⎥ P e ⎪ ⎥ x ∥ ⎢ ⎪⎥ ⎪⎢ ⎥⎦ +( Jl − 1 Jl + 2 − Jl − 2 Jl + 1 ) P⊥ sin 3ϕ ⎢ ⎪⎥ ⎪⎣ ⎢ i ⎪ ⎪⎥ ⎡ ( J J − J J ) cos ϕ ⎤ 1 ⎬⎥ ⎨ = + + − − αm 2 Re ⎢ l 2 l 1 l 2 l 1 ⎥ P∥ P⊥ ey −⎢ 2 ⎪⎥ ⎢ αm*,0 ⎪ ⎢⎣ +( J J − J J ) cos 3ϕ ⎥⎦ l+ 2 l− 1 l− 2 l+ 1 ⎪⎥ ⎪ ⎢ ⎪⎥ ⎪ ⎢ 1 ⎪ + ⎡ − 2J ( J ⎢ − Jl − 2 ) cos 2ϕ + Jl2+ 2 − Jl2− 2 P⊥ ⎤⎦ P⊥ ez ⎪ ⎥ ⎣ + l l 2 ⎭⎦ ⎩ ⎣ 2

(15)

⎡ ⎫⎤ ⎧ ⎡ ( J J − J J ) sin ϕ ⎤ l− 2 l− 1 ⎢ ⎪⎥ ⎪ ⎢ l+ 2 l+ 1 ⎥ P∥ P⊥ ex ⎢ ⎪⎥ ⎪ ⎢⎣ −( J J − J J ) sin 3ϕ ⎥⎦ l+ 2 l− 1 l− 2 l+ 1 ⎢ ⎪⎥ ⎪ ⎢ i ⎪ ⎡ ⎪⎥ ⎤ 1 ⎬⎥ ⎨ ⎢ 2Jl ( −Jl + 1 + Jl − 1 ) cos ϕ + ( −Jl + 1 Jl + 2 + Jl − 2 Jl − 1 ) P⊥ cos ϕ ⎥ = αm 2 Re ⎢ P∥ ey ⎪ ⎥ 2 ⎢ αm*,0 ⎪ + ⎢ ⎥ J J J J P cos 3 ϕ + − ( l− 1 l+ 2 l− 2 l+ 1 ) ⊥ ⎦ ⎪⎥ ⎪ ⎣ ⎢ ⎪⎥ ⎪ ⎢ 1 ⎪⎥ ⎪ ⎢ + ⎡⎣ 2Jl ( Jl + 2 − Jl − 2 ) cos 2ϕ + Jl2+ 2 − Jl2− 2 P⊥ ⎤⎦ P⊥ ez ⎭⎦ ⎩ ⎣ 2

(16)

⎡ ⎧ 2J ( J + J P ) P sin ϕe ⎫ ⎤ x l+ 1 l l+ 2 ⊥ ∥ ⎥ ⎢ ⎪ ⎪ ⎪ i* ⎪ ⎨ − 2Jl + 1 ( Jl + Jl + 2 P⊥ ) P∥ cos ϕey ⎬ ⎥ = αm 2 Re ⎢ ⎢ α m,0 ⎪ ⎪⎥ 2 2 2 ⎥ ⎢ ⎪ ⎪ J J P e − + z ⊥ ⎩ ⎭⎦ l l+ 2 ⎣

(17)

⎡ ⎧ − 2J ( J + J P ) P sin ϕe ⎫ ⎤ x ⎥ l− 1 l l− 2 ⊥ ∥ ⎢ ⎪ ⎪ ⎪ i ⎪ ⎨ 2Jl − 1 ( Jl + Jl − 2 P⊥ ) P∥ cos ϕey ⎬ ⎥ = αm 2 Re ⎢ ⎢ αm*,0 ⎪ ⎪⎥ 2 2 2 ⎥ ⎢ ⎪ ⎪ + − J J P e z ⊥ ⎭⎦ ⎩ − l l 2 ⎣

(18)

(

spin, y Tm

)

(

spin, lc Tm

(

spin, rc Tm

)

(

spin, rp Tm =

spin, ap Tm =

1 2

( 1 + cos α0 )

cos α0

( 1 + cos α0 )2

)

⎡ i αm 2 Re ⎢ ⎣ αm*,0

)

{ (J

2 l+ 1

⎤ − Jl2− 1 ez ⎥ ⎦

) }

(19)

⎡ ⎧ ⎡ − 2J J − J ⎤ ⎫⎤ l ( l+ 1 l − 1 ) sin α sin ϕ⎦ ex ⎪ ⎥ ⎢ ⎪⎣ ⎪⎥ ⎢ i ⎪ ⎡ ⎨ + ⎣ 2Jl ( Jl + 1 − Jl − 1 ) sin α cos ϕ⎤⎦ ey ⎬ ⎥ αm 2 Re ⎢ *,0 ⎪ α ⎪⎥ m ⎢ ⎪ ⎪⎥ + ⎡⎣ Jl2+ 1 − Jl2− 1 cos α⎤⎦ ez ⎢⎣ ⎩ ⎭⎦

(

)

(20)

100

R. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

Fig. 1. The axial OST induced by l ¼ 0 vector BBs with α0 = 20° .

3. Numerical results and discussions In this section, the optical torque exerted on an absorptive magneto-dielectric Rayleigh sphere induced by AGVBBs is analysed. In the calculation, the non-dimensional size parameter of the particle is ka¼0.25, with k and a being wavenumber of incident beam and radius of particle respectively. The complex permittivity and permeability of the particle are ε = 4.0 + 10−7i and μ = 3.0 + 10−7i , respectively. The wavelength of the incident beam is λ = 532 nm , and the amplitude of electric field of each plane wave component is Epw0 = 1.96 × 106 V/m . The orbital and spin optical torque are investigated separately, and the effects of beam parameters, including order l, half-cone angle α0, and polarization, are mainly discussed. The units of the optical torque are N.m (in SI system). 3.1. Optical spin torque The OST will make a particle rotating around its center of mass, and now the axial and transverse components of OST are numerically investigated separately. For the axial OST component, we focus on the generation of a left-handed OST. For the transverse OST component, the vortex-like character, especially its direction, will be mainly discussed. 3.1.1. Axial OST ,u , where u represents the polarization state, is initially analysed, and the OST reversal is mainly discussed. As The axial OST T spin z ,u spin, u and OSTM Tm previously discussed in Section 2, the axial OST can be divided into axial OSTE Tespin ,z , z , which are induced by electric and magnetic dipoles, respectively. Initially, the axial OSTs for an absorptive magneto-dielectric sphere in the field of a zeroth-order (l ¼0) Bessel beam are investigated. The effects of polarization and half-cone angle α0 are analysed. Linear (x and y), circular (left and right), radial, and azimuthal polarizations are considered. From Eqs. (6)–(20), it is noticed that the axial OSTEs induced by Bessel beams with x, y, radial, and azimuthal polarizations of the individual plane wave components are zero, since Jl + 2 = Jl − 2 and Jl2− 1 = Jl2+ 1 for l ¼0. The axial OSTs for the cases of these polarizations are not given here for brevity. The axial OSTs by Bessel beams with circular polarizations of the individual plane wave components are not zero, and their directions are sensitive to the half-cone angle α0. Figs. 1 and 2 give the axial OSTs induced by Bessel beams with half-cone angles α0 = 20° and 85°, respectively. The upper and lower rows of each figure correspond to the case of left and right circular polarizations, respectively. The first, second and third columns are axial OST by electric dipole (axial OSTE), by magnetic dipole (axial OSTM), and total axial OST (OSTE þOSTM), respectively. For smaller half-cone angle ( α0 = 20°), the OSTE, OSTM and total OST for left circular polarization are all positive, while for right circular polarization, they are all negative. So for smaller α0, left-handed optical torque can not be realized. This is due to the fact that the axial optical spin torque (Tzspin) is very sensitive to the non-paraxiality of the beam, which is denoted by the term P⊥. Taking OSTE with left circular polarization as an example (Eq. (8)), the z − component of the OSTE has a term Jl − Jl2+ 2 P⊥2, and its direction changes with P⊥, which increases with the increasing of the half-cone angle α0. For smaller α0, P⊥ → 0 and the

term Jl − Jl2+ 2 P⊥2 tends to Jl. While for larger α0 and larger P⊥, the sign of the term Jl − Jl2+ 2 P⊥2 changes with α0 and it can be negative. So, for a

larger half-cone angle ( α0 = 85°), left-handed OST can be observed, as shown in Fig. 2. Note that left-handed OST means Tzspin, lc < 0 for left circular polarization but Tzspin, rc > 0 for right circular polarization. Panels (a)–(c) show that for left-circular polarization, the axial spin torque components OSTE, OSTM and total OST are maximal at the center of the beam. As the sphere departs from the center, the OSTE, the OSTM and the total OST cross zero before reversing their sign. This means that the sphere yields total neutrality and remains unaffected by the OST. The central axis region is surrounded by an annular region. For enhanced visualization, the fourth column of Fig. 2 only displays

R. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

101

the regions where left-hand OST occurs ( Tzspin, lc < 0). For right-circular polarization, the axial spin torque components have similar contributions, while they have opposite sign compared to the case left-circular polarization. Next, the axial OST for an absorptive megneto-dielectric sphere in the field of vortex Bessel beams ( l > 0) is investigated. More attention is paid to the left-handed axial OST, and the effect of polarization.

Fig. 2. The axial OST induced by l ¼0 vector BBs with α0 = 85° .

Fig. 3. The axial OST induced by l ¼1 vector BBs with α0 = 20° .

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Fig. 3. Continued.

Fig. 3 displays the axial OST induced by a first-order (l ¼ 1) vortex Bessel beam with α0 = 20°. The rows from top to bottom correspond to x, y, left and right circular, radial, and azimuthal polarizations, respectively. The columns from left to right give the OSTE, OSTM, total OST, and left-hand OST, respectively. In general, a vortex Bessel beam with selected polarization will exert non-zero ,x ,x , Tmspin or Tzspin, x ) vanishes axial spin torque on a spherical absorptive particle, as shown in Fig. 3. For x -polarization, the axial OST (Tespin ,z ,z at the beam center. When the sphere departs from the beam center, the sign of the axial OST is decided by the position of the sphere in the field. Typically, the axial OST is divided into four parts. In the ranges ϕ ∈ (315°, 45°) and ϕ ∈ (135°, 225°), the axial OSTE is negative, while the OSTM is positive. On the contrary, in the ranges ϕ ∈ (45°, 135°) and ϕ ∈ (225°, 315°), the axial OSTE is positive, and the OSTM is negative. First, for smaller α0 the z -components of the OSTE (Eq. (6)) and the OSTM (Eq. (15)) are both dominated by the first term in the square brackets, which are related to the angle ϕ. So the directions of the OSTE and the OSTM depend on the angular position ϕ of the particle in the transverse plane. Second, from Eqs. (3) and (12) the axial components of the OSTE and the OSTM can spin 2 2 ⎡ ⎡ ⎤ * * * ⎤ * * * be expressed by Tespin , z = 1/2 αe Re ⎣ 1/ αe,0 (Ex E y − Ex Ey ) ⎦ and Tm, z = 1/2 αm Re ⎣ 1/ αm,0 (Hx H y − H x Hy ) ⎦, respectively. For a Bessel beam with x -polarization of the individual plane wave component and smaller half-cone angle α0, its electric and magnetic fields (given by Eq. (A1)) are dominated by the x - component Ex and y -component Hy, respectively, so the corresponding axial OSTE and OSTM are dominated by Ex and H *y . This leads to opposite directions of the axial OSTE and OSTM. Since the axial OSTM is larger than the axial OSTE, the total OST is dominated by the axial OSTM, and has similar distribution with the axial OSTM. When an absorptive magnetodielectric sphere is placed in the field, it will experience left-hand OST, as shown in panel (d). Note that when ϕ = 45°, 135°, 225°, and 315°, the OST is zero. This suggests that at these places the absorptive sphere may not spin around its center of mass. The same situation also occurs for the axial OST by a first-order Bessel beam with y -polarization, except that the axial OST represents a 90° rotational shift of the axial OST by a Bessel beam with x -polarization. For the circular polarization cases, the axial OST is circularly symmetrical, and remains positive (for left circular polarization) or negative (for right circular polarization). Such beam with circular , rp , rp polarization can not generate left-hand axial OST on an absorptive megneto-dielectric sphere. The axial OSTs ( Tespin , Tmspin , Tzspin, rp , ,z ,z , ap , ap , Tmspin , and Tzspin, ap ) by Bessel beams with radial and azimuthal polarizations have central maximum. As the sphere departs from Tespin ,z ,z the center, the axial OST crosses zero before reversing its sign over an annular region around the central region. Panels (p) and (t) displays the regions where Tzspin, rp < 0 and Tzspin, ap < 0, respectively. The effect of increasing the half-cone angle to α0 = 85° on the axial OST induced by a first-order Bessel beam is shown in Fig. 4. Here we pay a closer attention to the left-handed axial OSTs, which are given by the last column. For linear polarizations (x and y), the total axial OST vanishes at the beam center. Compared to the case of smaller α0, a series of extra concentric rings exist. This means that when a sphere departs from the beam center, the sign reversal of its experienced axial OST occurs, and it will spin around its

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center of mass in the anti-clockwise or the clockwise direction depending on its position in the beam. The axial OST by a Bessel beam with y -polarization represents a 90° rotational shift compared with the x polarization configuration. Similar to the case of smaller α0, the left-handed axial OST by a first-order Bessel beam with right circular polarization vanishes. However, the sphere will experience left-handed axial OST when it is placed in a first-order Bessel beam with left circular polarization. This is quite different from the case of smaller α0. The extra concentric rings can also be observed for the case of radial and azimuthal polarizations, which is given by panels (p) and (t). 3.1.2. Transverse OST components Next the focus is on analyzing the transverse OST, namely the x - (Txspin, u ) and y (T yspin, u ) components of the OST. The vortex-like structure of the transverse OST is mainly discussed, and the condition for changing the direction of such vortex-like structure is analysed. First, we explore the effect of polarization on the total transverse OST, namely Txspin, u + T yspin, u with u representing the polarization of the plane wave component forming the incident beam. Fig. 5 displays the transverse OST exerted on an absorptive magneto-dielectric sphere placed in a zero-order Bessel beam with half-cone angle α0 = 20°. Panels (a) - (f) correspond to the cases of linear (x and y), circular (left and right), azimuthal and radial polarizations. The direction of the arrows gives the direction of the transverse OST. For a Bessel beam with linear (x and y polarizations) and left circular polarization, the transverse OST vanishes at the beam center. When the particle departs from the beam center, the vortex-like structure, which forms a series of concentric rings, can be observed, and the vortex can be oriented in the clockwise or the anti-clockwise direction depending on the position of the sphere in the transverse plane. For the central ring ( r ∈ (0, 0.6 μm)), the vortex direction is anti-clockwise. The vortex direction becomes clockwise for the second ring ( r ∈ (0.6 μm, 0.9 μm)). The vortex will be oriented in the anti-clockwise direction for the third ring ( r > 0.9 μm ). The transverse OST by a Bessel beam with y -polarization has a vortex-like structure with the same direction as that by a Bessel beam with x -polarization, and the transverse OST by a Bessel beam with y -polarization represents a 90° rotational shift of that of a Bessel beam with x -polarization. However, for right circular polarization, the vortex-like structure vanishes. A Bessel beam with azimuthal or radial polarization can also produce a vortex-like transverse OST, which has an opposite spinning direction compared to the case of linear and left circular polarizations. The effects of increasing the half-cone angle to α0 = 85° on the transverse OST are shown in Fig. 6. In general, the amplitude of the transverse OST is mainly concentrated around the beam axis. For right circular polarization, the transverse OST can only display local quasi-vortex structures in the vicinity of the beam center. For other polarizations, the transverse OSTs have well-defined vortex characters around the beam center. For linear and left circular polarizations, the vortex direction is anti-clockwise, while it is clockwise for the azimuthal and radial polarizations.

Fig. 4. The axial OST induced by l ¼1 vector BBs with α0 = 85° .

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Fig. 4. Continued.

Fig. 5. The transverse OST induced by l ¼0 vector BBs with α0 = 20° .

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Fig. 6. The transverse OST induced by l ¼ 0 vector BBs with α0 = 85° .

Fig. 7. The transverse OST induced by l ¼ 1 vector BBs with α0 = 20° .

Next the transverse OST by a high-order vortex Bessel beam is investigated, and the effect of order l to the direction of the vortex is mainly discussed. Figs. 7 and 8 display the transverse OST by a first- (l ¼1) and a second-order (l¼ 2) vector Bessel beams with half-cone angle α0 = 20°, respectively. For linear (x and y) polarizations, the vortex-like structures can be observed for l = 0, 1 and 2, and they form a series of concentric rings. The increasing of the order from l ¼0 to l¼ 1 and l ¼2 changes the direction of the vortex on each ring. For instance, the vortex direction of the central ring is anti-clockwise for l ¼0 (Figs. 5(a) and (b)), while it is clockwise for l ¼1 (Figs. 7(a) and (b)) or l ¼2 (Figs. 8(a) and (b)). The vortex direction on the other rings will also change. Note that the radii of all rings also change with the increasing of the order l. For left circular polarization, the vortex direction remains unchanged when the order l increases. For right circular polarization, the vortex vanishes for l¼ 0, while the vortex-like structures can be observed around the beam axis for high-order Bessel

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Fig. 8. The transverse OST induced by l ¼2 vector BBs with α0 = 20° .

Fig. 9. The transverse OST induced by l ¼ 1 vector BBs with α0 = 85° .

beam (l¼ 1 or 2). For azimuthal and radial polarizations, vortices can be seen, and they have opposite directions for zeroth- and nonzeroth-order Bessel beams. For larger half-cone angle ( α0 = 85°), the transverse OSTs by first- (l ¼1) and second-order (l ¼2) Bessel beams are given by Figs. 9 and 10, respectively. It is obvious that the vortex direction changes with the order l. However, such change is different from the case of smaller α0. Here we show this difference by discussing the transverse OST around the beam. For linear (x and y) polarizations, the vortex directions is the same for l¼ 0 (Figs. 6(a) and (b)) and l ¼1 (Figs. 9(a) and (b)), while it changes when l increases to l¼2 (Figs. 10(a) and (b)). For left circular polarization, the vortices for all orders (l = 0, 1 and 2) have the same rotation directions. The transverse OSTs for zeroth- (l ¼0) and first-order (l¼ 1) Bessel beams with right circular polarization do not form vortices, while that of a second-order (l ¼2) Bessel beam has a

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107

Fig. 10. The transverse OST induced by l ¼2 vector BBs with α0 = 85° .

Fig. 11. The axial OOT induced by l ¼ 0 vector BBs with α0 = 20° .

vortex-like structure around the beam axis. For azimuthal and radial polarizations, the vortex direction of the transverse OST around the beam axis changes from clockwise to anticlockwise when the order increases from l ¼0 to l ¼1 and 2. 3.2. Optical orbital torque In this part, we will investigate the optical orbital torque (OOT), which will make the particle orbit around the beam axis. Therefore, the OOT, which is given by Eq. (2), has an axial component T∥orbit , such that:

T orbit = T∥orbit

(21)

In a system of cylindrical coordinates, the expression for the axial component is given as,

T∥orbit = ρFϕ z^,

= − zFϕ ρ^ + (zFρ − ρFz ) ϕ^

(22)

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Fig. 12. The axial OOT induced by l ¼0 vector BBs with α0 = 85° .

Fig. 13. The axial OOT induced by l ¼1 vector BBs with α0 = 20° .

where, ρ = x2 + y2 , ϕ = arctan (y/x ), and z are the three components of r in the cylindrical coordinates, and ρ^ , ϕ^ , and z^ are their corresponding unit vectors. The three components ( Fρ, Fϕ , and Fz) of the force F are given in Appendix A. In the following, the axial OOT

T∥orbit will be discussed separately. According to Eq. (22), the axial OOT is given by T∥orbit = T zorbit , u z^ = ρFϕ z^ with T zorbit , u being the amplitude of T∥orbit , u , so the axial OOT vanishes at the center of the beam ( ρ = 0) for any case, and the direction of T∥orbit , u (or the sign of T zorbit , u ) is determined by the azimuthal component of the optical force Fϕ. Fig. 11 displays the plots for the axial OOT exerted on an absorptive magneto-dielectric sphere by zeroth-order (l ¼0) Bessel beams with half-cone angle α0 = 20°. Panels (a)–(f) correspond to x, y, left circular, right circular, azimuthal, and radial polarizations, respectively. For x − polarization, the axial OOT can be positive or negative, depending on the position of the particle in the transverse plane. At the beam center, the axial OOT vanishes since ρ = 0. As the sphere radially departs from the center of the beam, it will experience positive ( ϕ ∈ (90, 180°) and (270°, 360°)) or negative ( ϕ ∈ (0, 90°) and (180°, 270°)) axial OOT. The axial OOT vanishes at about ρ = 0.6 μm , after which the axial OOT reverses sign. Note that along x (y¼0) and y (x¼ 0) axes, the axial OOTs vanish because of the null azimuthal optical force Fϕ. The same situation occurs for the axial OOT by Bessel beam with y polarization, except that the axial OOT represents a 90°

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109

Fig. 14. The axial OOT induced by l ¼1 vector BBs with α0 = 85° .

rotational of the axial OOT by Bessel beam with x polarization. For circular polarizations, the axial OOTs have circular symmetry, and remain positive (for left circular polarization) or negative (for right circular polarization). A sphere will only experience a radial optical force (namely null azimuthal optical force) in the transverse plane when placed in a Bessel beam with azimuthal or radial polarization, thus no axial OOT will be exerted on it. The effects of increasing the half-cone angle to α0 = 85° on the axial OOT are shown in Fig. 12. For linear polarizations, the particle can experience positive or negative axial OOT depending on its position in the transverse plane. The azimuthal variation of the axial OOT is very similar to the case of small half-cone angle, whereas its sign changes more frequently along radial direction. The axial OOT is positive (for left circular polarization) or negative (for right circular polarization). Similar to the case of smaller half-cone angle, the axial OOT by a Bessel beam with azimuthal or radial polarization vanishes. Next the axial OOT by a first-order (l ¼1) Bessel beam is investigated. Figs. 13 and 14 display the plots for the cases of half-cone angles α0 = 20° and α0 = 85°, respectively. Compared to the case of a zeroth-order Bessel beams incidence, the axial OOTs by first-order Bessel beams have the following typical differences. First, the axial OOTs by a first-order Bessel beam with linear polarizations have opposite sign compared to that of a zeroth-order Bessel beam. Second, the axial OOTs induced by a first-order Bessel beam with right circular polarization is positive, while that of a zeroth-order Bessel beam is always negative. Finally, a first-order Bessel beam with azimuthal or radial polarization will induce axial OOTs, while for a zeroth-order Bessel beam the axial OOTs vanish. This is caused by the fact that a first-order Bessel beam with azimuthal or radial polarization has a non-zero azimuthal component of the optical force, while such an azimuthal component of the optical force vanishes for a zeroth-order Bessel beam. As shown in Figs. 13 and 14(e), (f), the axial OOTs are always positive, and they have circular symmetry about the beam axis.

4. Conclusion This investigation presents numerical predictions for the optical torque acting on a Rayleigh (small) magneto-dielectric absorptive sphere by Bessel (vortex) beams, with emphasis on the generation of a left-hand optical torque. The axial component of the optical orbital torque (OOT) causing the rotation of the sphere around the axis of the beam is calculated directly from the transverse optical force components and the optical spin torque (OST) is derived in the framework of the dipole approximation. Both the OOT and OST are numerically investigated by considering their corresponding components separately. A zeroth-order Bessel beam with a smaller half-cone angle ( α0 = 20°) can not generate a left-handed OST for any polarization. Increasing the half-cone angle to α0 = 85° as well as the order of the beam to l ¼1 can induce a left-handed OST, which depends also on polarization. The beam parameters can also change the direction of the axial OOT. The transverse OST has a vortex-like character, whose direction will change with the polarization, half-cone angle, and the order of the beam. Note that the vortex direction also depends on the radial position of the sphere in the transverse plane. For a first-order Bessel beam with left circular and azimuthal polarizations and half-cone angle α0 = 85°, the vortices in the central region are directed anticlockwise, while in the first annular region they display a clockwise sense of rotation. These results find applications in the manipulation of small particles with a single device, and the present research emphasizing beam polarization effects allows adequate experimental design for optical rotation, cell sorting, and other emergent areas in optofluidics research and optical tweezers.

Appendix A. Electric and magnetic fields

 x polarization

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⎧ ⎧ ⎫ 1 Jl + 2 e2iϕ + Jl − 2 e−2iϕ P⊥ ⎬ ⎪ Exx = EB0 eik z z eilϕil ⎨ Jl + ⎩ ⎭ 2 ⎪ ⎪ 1 ⎪ Eyx = EB0 eik z z eilϕil − 1 Jl + 2 e2iϕ − Jl − 2 e−2iϕ P⊥ 2 ⎪ ⎪ E x = E eik z z eilϕil − 1 J eiϕ − J e−iϕ P ⎪ z ∥ B0 l+ 1 l− 1 ⎨ ⎪ x 1 ϕ − ϕ − ik z il l 1 2 i 2 z Jl + 2 e − Jl − 2 e iϕ P⊥ ⎪ Bx = 2 BB0 e e i ⎪ ⎫ 1 ⎪ B x = B eik z z eilϕil ⎧ ⎨ Jl − J e2iϕ + Jl − 2 e−2iϕ P⊥ ⎬ B0 ⎪ y ⎩ ⎭ 2 l+ 2 ⎪ x − ϕ ϕ ϕ ik z il l i i z ⎪ P∥ ⎩ Bz = BB0 e e i −Jl + 1 e − Jl − 1 e

(

)

(

)

(

)

(

)

(

)

(

)

(A1)

 y polarization ⎧ y 1 ⎪ Ex = EB0 eik z z eilϕil − 1 Jl + 2 e2iϕ − Jl − 2 e−2iϕ P⊥ 2 ⎪ ⎪ y ⎧ ⎫ −2iϕ J ik z ilϕ l J − 1 e2iϕ J ⎪ Ey = EB0 e z e i ⎨ l l+ 2 + e l − 2 P⊥ ⎬ ⎩ ⎭ 2 ⎪ ⎪ E y = E ik z z eilϕil − 2 J eiϕ + J e−iϕ P ⎪ z ∥ l+ 1 l− 1 B0 ⎨ ⎧ ⎫ ⎪ y ik z z ilϕ l − 2 J + 1 J e2iϕ + Jl − 2 e−2iϕ P⊥ ⎬ ⎪ Bx = BB0 e e i ⎨ ⎩l ⎭ 2 l+ 2 ⎪ ⎪ y 1 −2iϕ J ik z z ilϕ l − 1 −e2iϕ J l+ 2 + e l − 2 P⊥ ⎪ By = 2 BB0 e e i ⎪ y −iϕ P ik z z ilϕ l − 1 −J iϕ ⎪ ∥ l + 1 e + Jl − 1 e ⎩ Bz = BB0 e e i

(

)

(

)

(

)

(

)

(

)

(

)

(A2)

 left circular polarization ⎧ E lc = E eik z z eilϕil J + J e2iϕP ⊥ B0 l l+ 2 ⎪ x ⎪ lc ik z z eilϕ il − 1 −J + e2iϕ J E = E e B 0 l l + 2 P⊥ ⎪ y ⎪ lc + − ϕ ik z i l l 1 1 ⎪ Ez = 2EB0 e z e ( ) i Jl + 1 P∥ ⎨ ⎪ Bxlc = BB0 eik z z eilϕil − 1 Jl + Jl + 2 e2iϕP⊥ ⎪ ik z z eilϕ il J − e2iϕ J ⎪ Blc y = BB0 e l l + 2 P⊥ ⎪ lc + ϕ ik z i l 1 l ⎪ ( ) z i Jl + 1 P∥ ⎩ Bz = − 2BB0 e e

(

)

(

(

)

)

(

)

(A3)

 right circular polarization ⎧ E rc = − E eik z z eilϕil − 2 J + J e−2iϕP ⊥ B0 l l− 2 ⎪ x ⎪ rc −2iϕ P ik z z eilϕ il − 1 J − J E = E e e ⊥ B0 l l− 2 ⎪ y ⎪ rc ⎪ Ez = − 2EB0 eik z z ei ( l − 1) ϕil − 1Jl − 1 P∥ ⎨ ⎪ Bxrc = − BB0 eik z z eilϕil − 1 Jl + Jl − 2 e−2iϕP⊥ ⎪ ⎪ Byrc = BB0 eik z z eilϕil − 2 −Jl + Jl − 2 e−2iϕP⊥ ⎪ rc ⎪ ⎩ Bz = − 2BB0 eik z z ilei ( l − 1) ϕJl − 1 P∥

(

(

)

)

(

(

)

)

(A4)

 radial polarization ⎧ rp 1 −Jl + 1 eiϕ + Jl − 1 e−iϕ cos α0 ⎪ Ex = EB0 eilϕeik z z il − 1 1 + cos α0 ⎪ ⎪ rp 1 J eiϕ + Jl − 1 e−iϕ cos α0 ⎪ E y = EB0 eik z z eilϕil 1 + cos α0 l + 1 ⎪ ⎪ rp 2 J sin α0 ⎪ Ez = − EB0 eik z z eilϕil ⎨ 1 + cos α0 l ⎪ 1 ⎪ Bxrp = BB0 eik z z eilϕil − 2 J eiϕ + Jl − 1 e−iϕ ⎪ 1 + cos α0 l + 1 ⎪ 1 ⎪ Byrp = BB0 eik z z eilϕil − 1 −Jl + 1 eiϕ + Jl − 1 e−iϕ 1 + cos α0 ⎪ ⎪ rp ⎩ Bz = 0

(

(

)

(

(

 azimuthal polarization

)

)

)

(A5)

R. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

⎧ ap 1 J eiϕ + Jl − 1 e−iϕ ⎪ Ex = EB0 eik z z eilϕil − 2 α0 l + 1 1 cos + ⎪ ⎪ ap 1 −Jl + 1 eiϕ + Jl − 1 e−iϕ ⎪ E y = EB0 eik z z eilϕil − 1 1 + cos α0 ⎪ ⎪ Ezap = 0 ⎪ ⎨ ap 1 J eiϕ − Jl − 1 e−iϕ cos α0 ⎪ Bx = BB0 eik z z eilϕil − 1 1 + cos α0 l + 1 ⎪ ⎪ ap 1 J eiϕ + Jl − 1 e−iϕ cos α0 ⎪ By = BB0 eik z z eilϕil − 2 1 + cos α0 l + 1 ⎪ ⎪ ap 2 eik z z eilϕilJl sin α0 ⎪ Bz = BB0 1 + cos α0 ⎩

(

111

)

(

)

(

)

(

)

(A6)

Appendix B. Force on induced electric dipole

 x polarization ⎧ ⎧ ⎫⎫ Jl ( Jl − 1 − Jl + 1 ) ( 1 − P⊥ ) − ( Jl + 1 Jl + 2 − Jl − 1 Jl − 2 ) P⊥ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ 1⎡ 2 ⎤ J J J J J J P + − + − ⎪ ⎪ ( ⎣ ⎪⎪ l+ 3 ) l− 2 ( l− 3 l− 1 ) ⎦ ⊥ 2 l+ 2 l+ 1 ⎬⎪ ⎪ α′e ⎨ 1 ⎪ ⎪ ⎪⎪ x ⎤ ⎡ ⎬ Fe, ρ = k⊥ ⎨ ⎪ 1 Jl + 1 ( 3Jl − 3Jl − 2 − Jl + 2 ) + Jl − 1 ( 3Jl + 2 − 3Jl + Jl − 2 ) ⎥ P⊥ cos ( 2ϕ)⎪⎪ 4 ⎪ + ⎢ ⎪ ⎪ ⎥ ⎢ +Jl ( Jl − 3 − Jl + 3 ) ⎪ ⎩ 2⎣ ⎦ ⎭⎪ ⎪ ⎪ ⎪ ⎪ 1 − α″e ⎡⎣ ( Jl − 1 + Jl + 1 )( Jl − Jl + 2 − Jl − 2 ) + Jl ( Jl + 3 + Jl − 3 ) ⎤⎦ P⊥ sin ( 2ϕ) ⎪ ⎪ ⎩ ⎭ 2 ⎧ ⎧ 2 ⎫⎫ 1 2 2 2 2 2 ⎪ ⎪ lJl + ⎡⎣ (l + 1) Jl + 1 + (l − 1) Jl − 1 ⎤⎦ P⊥ + ⎡⎣ (l + 2) Jl + 2 + (l − 2) Jl − 2 ⎤⎦ P⊥ ⎪⎪ 2 ⎪ ⎨ ⎬⎪ ⎪ 1 1 ⎪ α″e ⎪+ J ⎡ l + 1 J ⎪⎬ ⎨ Fex, φ = ⎤ l J lJ J P + − − ϕ 1 2 cos 2 ) l+ 2 ( ) l− 2 ⎦ ( ) ⊥ l⎣( l+ 1 l− 1 2 ρ⎪ ⎩ ⎭⎪ ⎪ ⎪ ⎪ ⎪ − α′e ⎡⎣ Jl ( Jl + 2 + Jl − 2 ) − 2Jl + 1 Jl − 1 ⎤⎦ P⊥ sin ( 2ϕ) ⎩ ⎭

{

Fex, z =

}

⎫ ⎧ J2 + J2 + J2 P + ⎡ J J ⎣ l ( l + 2 + Jl − 2 ) − 2Jl + 1 Jl − 1 ⎤⎦ P⊥ cos ( 2ϕ)⎪ l+ 1 l− 1 ⊥ ⎪ l 1 ⎬ α″e k z ⎨ 1 2 2 ⎪ ⎪ Jl + 2 + Jl2− 2 P⊥2 + ⎭ ⎩ 2

(

)

(

)

(B1)

where, α′e and α″e are respectively, the real and imaginary parts of electric polarizability given by Eq. (4), namely αe = α′e + iα″e .

 y polarization

⎧ ⎧ ⎫⎫ 1 ⎪ ⎪ Jl ( Jl − 1 − Jl + 1 ) ( 1 − P⊥ ) + ⎡⎣ Jl + 2 ( Jl + 1 − Jl + 3 ) + Jl − 2 ( Jl − 3 − Jl − 1 ) ⎤⎦ P⊥2 ⎪⎪ 2 ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎤ ⎡ J ( 3J J J − + 3 ) l l+ 2 ⎥ ⎬⎪ ⎪ α′e ⎨ ⎢ l+ 1 l− 2 ⎪ ⎪ 1 1⎢ ⎥ P⊥ cos 2ϕ ⎪⎬ Fey, ρ = kρ ⎨ ⎪ J J J J P J J J J + − − + − − 3 3 ( ) ( ) ⊥ l − l − l + l + l − l l + l − 1 2 1 2 1 2 2 ⎪⎪ ⎥ 4 ⎪ ⎪ 2⎢ ⎪⎪ ⎥⎦ ⎢⎣ −Jl ( Jl − 3 − Jl + 3 ) ⎪ ⎪ ⎭ ⎩ ⎪ ⎪ ⎪ ⎪ 1 ⎡ ⎤ − α″e ⎣ ( Jl + 1 + Jl − 1 )( Jl − 2 + Jl + 2 − Jl ) − Jl ( Jl − 3 + Jl + 3 ) ⎦ P⊥ sin 2ϕ ⎪ ⎪ ⎭ ⎩ 2

Fey, φ

⎧ ⎧ 2 ⎫⎫ 1 2 2 2 2 2 ⎪ ⎪ lJl + ⎡⎣ ( l + 1) Jl + 1 + ( l − 1) Jl − 1 ⎤⎦ P⊥ + ⎡⎣ ( l + 2) Jl + 2 + ( l − 2) Jl − 2 ⎤⎦ P⊥ ⎪⎪ 2 ⎪ ⎬⎪ ⎪ 1 1 ⎪ α″e ⎨ ⎪ ⎪⎬ ⎨ = ⎡⎣ ( l + 1) J ⎤⎦ − 2 lJ J ϕ J l J P − + − 1 cos 2 ( ) ⊥ + − − + l l l l l 2 2 1 1 ⎩ ⎭⎪ 2 ρ⎪ ⎪ ⎪ ⎪ ⎪ + α′e ⎡⎣ Jl ( Jl + 2 + Jl − 2 ) − 2Jl − 1 Jl + 1 ⎤⎦ P⊥ sin 2ϕ ⎩ ⎭

{

Fey, z =

}

⎧ ⎫ 1 1 2 α″e k z ⎨ Jl2 + Jl2+ 1 + Jl2− 1 P⊥ + J + Jl2− 2 P⊥2 − ⎡⎣ Jl ( Jl − 2 + Jl + 2 ) − 2Jl + 1 Jl − 1 ⎤⎦ P⊥ cos 2ϕ⎬ ⎩ ⎭ 2 2 l+ 2

(

)

(

)

(B2)

 left circular polarization 1 α′e kρ ⎡⎣ Jl ( Jl − 1 − Jl + 1 ) + Jl + 2 ( Jl + 1 − Jl + 3 ) p⊥2 + 2Jl + 1 ( Jl − Jl + 2 ) P⊥ ⎤⎦ 2 α″ = e ⎡⎣ lJl2 + ( l + 2) Jl2+ 2 P⊥2 + 2 ( l + 1) Jl2+ 1 P⊥ ⎤⎦ ρ

Felc, ρ = Felc, φ

(

Felc, z = α″e k z Jl2 + Jl2+ 2 P⊥2 + 2Jl2+ 1 P⊥

 right circular polarization

)

(B3)

112

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1 α′e kρ ⎡⎣ Jl ( Jl − 1 − Jl + 1 ) + Jl − 2 ( Jl − 3 − Jl − 1 ) P⊥2 + 2Jl − 1 ( Jl − 2 − Jl ) P⊥ ⎤⎦ 2 α″ Ferc, φ = e ⎡⎣ lJl2 + ( l − 2) Jl2− 2 P⊥2 + 2 ( l − 1) Jl2− 1 P⊥ ⎤⎦ ρ Ferc, z = α″e k z ⎡⎣ Jl2 + Jl2− 2 P⊥2 + 2Jl2− 1 P⊥ ⎤⎦ Ferc, ρ =

(B4)

 radial polarization Ferp, ρ =

1 1 kρ α′e cos2 α0 ⎡⎣ Jl + 1 ( Jl − Jl + 2 ) + Jl − 1 ( Jl − 2 − Jl ) ⎤⎦ + 2Jl ( Jl − 1 − Jl + 1 ) sin2 α0 2 ( 1 + cos α0 )2

Ferp, φ =

1 α″e cos2 α0 ⎡⎣ ( l + 1) Jl2+ 1 + ( l − 1) Jl2− 1 ⎤⎦ + 2lJl2 sin2 α0 ρ ( 1 + cos α0 )2

{

{

Ferp, z = α″e k z

1 2

( 1 + cos α0 )

{ (J

2 l+ 1

)

+ Jl2− 1 cos2 α0 + 2Jl2 sin2 α0

}

}

}

(B5)

 azimuthal polarization Feap ,ρ =

1 1 α′e kρ ⎡⎣ Jl + 1 ( Jl − Jl + 2 ) + Jl − 1 ( Jl − 2 − Jl ) ⎤⎦ 2 ( 1 + cos α0 )2

Feap ,φ =

1 α″ ⎡⎣ l + 1 J 2 + l − 1 J 2 ⎤⎦ l+ 1 l− 1 ρ ( 1 + cos α0 )2

Feap , z = α″e k z

(

1

( 1 + cos α0 )2

)

(

)

( Jl2+ 1 + Jl2− 1 )

(B6)

Appendix C. Force on induced magnetic dipole

 x polarization ⎧ ⎫ 1 1 2 α″m k z ⎨ Jl2 + J + Jl2− 2 P⊥2 + Jl2+ 1 + Jl2− 1 P∥2 − ⎡⎣ Jl ( Jl + 2 + Jl − 2 ) − 2Jl − 1 Jl + 1 ⎤⎦ P⊥ cos 2ϕ⎬ ⎩ ⎭ 2 2 l+ 2 ⎧ ⎫ ⎫ ⎧ Jl ( Jl − 1 − Jl + 1 ) ( 1 − P⊥ ) + ( Jl − 1 Jl − 2 − Jl + 1 Jl + 2 ) P⊥ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ 1⎡ 2 ⎤ + ⎣ Jl + 2 ( Jl + 1 − Jl + 3 ) + Jl − 2 ( Jl − 3 − Jl − 1 ) ⎦ P⊥ ⎪ ⎪⎪ ⎪ 2 ⎬⎪ ⎪ α′m ⎨ ⎪ 1 ⎪⎪ ⎪ ⎫ ⎧ ⎬ Fmx, ρ = kρ ⎨ ⎪ Jl + 1 ( 3Jl − 3Jl − 2 − Jl + 2 ) + Jl − 1 ( 3Jl + 2 − 3Jl + Jl − 2 )⎪ 1 ⎪ 4 ⎪ ⎬ P⊥ cos 2ϕ ⎪⎪ − ⎨ ⎪ ⎪ ⎪ ⎪ 2 ⎩ +Jl ( Jl − 3 − Jl + 3 ) ⎪ ⎭ ⎭⎪ ⎩ ⎪ ⎪ ⎪ ⎪ 1 ⎡ ⎤ sin 2 − α ″ J + J J + J − J − J J + J P ϕ ⎪ ⎪ m ⎣ ( l+ 1 l − 1 )( l + 2 l− 2 l) l ( l+ 3 l− 3 )⎦ ⊥ ⎩ ⎭ 2 ⎧ ⎫ ⎡ ⎤ α′m ⎣ Jl ( Jl + 2 + Jl − 2 ) − 2Jl − 1 Jl + 1 ⎦ P⊥ sin 2ϕ ⎪ ⎪ ⎪ ⎧ 2 ⎫⎪ ⎪ 1⎡ 1 1⎪ 2 2 ⎤ 2 2 ⎤ 2 x ⎡ ⎨ Fm, φ = ⎪ lJ + ⎣ ( l − 1) Jl − 1 + ( l + 1) Jl + 1 ⎦ P⊥ + ⎣ ( l − 2) Jl − 2 + ( l + 2) Jl + 2 ⎦ P⊥ ⎪⎬ 2 2 ρ ⎪ + α″m ⎨ l ⎬⎪ ⎪ ⎪ ⎪⎪ − Jl ⎡⎣ ( l + 1) Jl + 2 + ( l − 1) Jl − 2 ⎤⎦ − 2lJl − 1Jl + 1 P⊥ cos 2ϕ ⎪ ⎩ ⎭⎪ ⎩ ⎭ Fmx , z =

(

)

{

(

)

}

(C1)

where, α′m and α″m are respectively, the real and imaginary parts of magnetic polarizability given by Eq. (13), namely αm = α′m + iα″m .

 y polarization

R. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 191 (2017) 96–115

⎧ ⎫ 1 1 2 α″m k z ⎨ Jl2 + J + Jl2− 2 P⊥2 + Jl2+ 1 + Jl2− 1 P∥2 + ⎡⎣ Jl ( Jl + 2 + Jl − 2 ) − 2Jl − 1 Jl + 1 ⎤⎦ P⊥ cos 2ϕ⎬ ⎩ ⎭ 2 2 l+ 2 ⎫ ⎧ ⎫ ⎧ 1 2 ⎪ ⎪ Jl ( Jl − 1 − Jl + 1 ) ( 1 − P⊥ ) + ⎡⎣ Jl + 2 ( Jl + 1 − Jl + 3 ) + Jl − 2 ( Jl − 3 − Jl − 1 ) ⎤⎦ P⊥ ⎪⎪ 2 ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎡ J ( 3J − 3J − Jl + 2 ) ⎤ + − l l l 1 2 ⎬⎪ ⎪ α′m ⎨ ⎥ ⎢ 1 ⎪ 1⎢ ⎪⎪ ⎪ y ⎥ ⎬ Fm, ρ = k ⎨ J J J J P J 3 J 3 J J P cos 2 ϕ + − + + − + ( ) ( ) ⊥ l− 1 l− 2 l+ 1 l+ 2 ⊥ l l− 2 ⎪⎪ ⎪ ⎥ 4 ⎪ 2 ⎢ l− 1 l+ 2 ⎪ ⎪ ⎥⎦ ⎢⎣ +Jl ( Jl − 3 − Jl + 3 ) ⎪ ⎭⎪ ⎩ ⎪ ⎪ ⎪ ⎪ 1 − α″m ⎣⎡ ( Jl + 1 + Jl − 1 )( Jl − Jl + 2 − Jl − 2 ) + Jl ( Jl + 3 + Jl − 3 ) ⎤⎦ P⊥ sin 2ϕ ⎪ ⎪ ⎭ ⎩ 2 ⎧ ⎫ α′m ⎡⎣ Jl ( Jl + 2 + Jl − 2 ) − 2Jl − 1 Jl + 1 ⎤⎦ P⊥ sin 2ϕ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ 1 1 1 2 2 2 2 2 2 ⎨ Fmy, φ = ⎪ lJl + ⎡⎣ ( l − 1) Jl − 1 + ( l + 1) Jl + 1 ⎤⎦ P⊥ + ⎡⎣ ( l − 2) Jl − 2 + ( l + 2) Jl + 2 ⎤⎦ P⊥ ⎪⎬ 2 2 ρ ⎪ + α″m ⎨ ⎪ ⎬ ⎪ ⎪ ⎪⎪ − Jl ⎡⎣ ( l + 1) Jl + 2 + ( l − 1) Jl − 2 ⎤⎦ − 2lJl − 1Jl + 1 P⊥ cos 2ϕ ⎪ ⎩ ⎭⎪ ⎩ ⎭

(

Fmy , z =

)

(

113

)

{

}

(C2)

 left circular polarization ⎡ 2 lc 2 2 2 2⎤ Fm , z = α″m k z ⎣ Jl + Jl + 2 P⊥ + 2Jl + 1 P∥ ⎦

(

1 α′m kρ ⎡⎣ Jl ( Jl − 1 − Jl + 1 ) + Jl + 2 ( Jl + 1 − Jl + 3 ) P⊥2 ⎤⎦ + 2Jl + 1 ( Jl − Jl + 2 ) P∥2 2 1 ⎡ 2 2 2 2 2 = α″m ⎣ lJl + ( l + 2) Jl + 2 P⊥ ⎤⎦ + 2 ( l + 1) Jl + 1 P∥ ρ

{

lc Fm ,ρ = lc Fm ,φ

)

{

}

}

(C3)

 right circular polarization ⎧ rc ⎡ 2 2 2 2 2⎤ ⎪ Fm, z = α″m k z ⎣ Jl + Jl − 2 P⊥ + 2Jl − 1 P∥ ⎦ ⎪ 1 ⎪ rc 2⎤ 2 ⎡ ⎨ Fm, ρ = α′m kρ ⎣ Jl ( Jl − 1 − Jl + 1 ) + Jl − 2 ( Jl − 3 − Jl − 1 ) P⊥ ⎦ + 2Jl − 1 ( Jl − 2 − Jl ) P∥ 2 ⎪ 1 ⎡ 2 ⎪ rc 2 2 2 2 Fm, φ = α″m ⎣ lJl + ( l − 2) Jl − 2 P⊥ ⎤⎦ + 2 ( l − 1) Jl − 1 P∥ ⎪ ρ ⎩

(

)

{

{

}

}

(C4)

 radial polarization ⎧ rp 1 J2 + J2 ⎪ Fm, z = α″m k z ( 1 + cos α0 )2 l+ 1 l− 1 ⎪ ⎪ ⎪ rp 1 1 ⎡⎣ J ( J − J ) + J ( J ⎤ ⎨ Fm, ρ = α′m kρ l+ 1 l l+ 2 l − 1 l − 2 − Jl ) ⎦ 2 ( 1 + cos α0 )2 ⎪ ⎪ 1 1 rp ⎡ l + 1) J 2 + ( l − 1) J 2 ⎤⎦ ⎪ Fm , φ = α″m l+ 1 l− 1 2⎣( ⎪ ρ 1 + cos α ( ) 0 ⎩

(

)

(C5)

 azimuthal polarization ⎧ ap 1 ⎡ J 2 + J 2 cos2 α + 2J 2 sin2 α ⎤ ⎪ Fm, z = k z α″m 0 0⎦ ⎣ l ( 1 + cos α0 )2 l+ 1 l− 1 ⎪ ⎪ ⎪ ap 1 1 2 ⎡⎣ J ( J − J ) + J ( J ⎤ 2 ⎨ Fm, ρ = α′m kρ l+ 1 l l+ 2 l − 1 l − 2 − Jl ) ⎦cos α 0 + 2Jl ( Jl − 1 − Jl + 1 )sin α 0 2 2 1 + cos α0 ) ( ⎪ ⎪ 1 1 ap ⎡⎣ ( l + 1) J 2 + ( l − 1) J 2 ⎤⎦ cos2 α0 + 2lJ 2 sin2 α0 ⎪ Fm , φ = α″m l+ 1 l− 1 l ⎪ ρ ( 1 + cos α0 )2 ⎩

(

)

{

{

Appendix D. Force due to the interaction between both dipoles

 x polarization

}

} (C6)

114

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⎧ ⎫ ⎧P i J J ⎫ ⎪ ⊥ ( l + 1 l + 2 + Jl − 1 Jl − 2 ) sin ϕ + ( Jl − 1 Jl + 2 − Jl + 1 Jl − 2 ) cos 3ϕ ⎪ ⎪ ⎪ ⎨ ⎬ i P e α α e m ∥ x ⎪ ⎪ ⎡⎣ ( −J + J ) cos ϕ − i sin ϕ ( J + J ) ⎤⎦ ⎪ ⎪ J − ⎪ ⎪ l+ 1 l− 1 l+ 1 l− 1 ⎩ l ⎭ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ k4 ⎡ ⎤ Jl ⎣ ( Jl + 1 + Jl − 1 ) cos ϕ + i ( −Jl + 1 + Jl − 1 ) sin ϕ⎦ ⎪ ⎪ ⎬ = − Re ⎨ ⎬ P∥ ey ⎪ 3 ⎪ + αe αm ⎨ ⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎩ +⎣ ( Jl + 1 Jl + 2 + Jl − 1 Jl − 2 ) cos ϕ + i ( Jl − 1 Jl + 2 − Jl + 1 Jl − 2 ) sin 3ϕ⎦ P⊥ ⎭ ⎪ ⎪ ⎧ 2 ⎫ 1 2 ⎪ ⎪ 2 2⎬ ⎨ J iJ J J P J J P e − α α − + − ϕ + + sin 2 ( ) ⊥ e m z ⊥ l l− 2 l+ 2 ⎪ ⎪ l− 2 ⎩ l ⎭ 2 l+ 2 ⎩ ⎭

{

x F em

}

(

)

(D1)

 y polarization

y Fem

⎧ ⎫ ⎧ ⎫ −Jl ⎡⎣ ( −Jl + 1 + Jl − 1 ) cos ϕ + ( Jl + 1 + Jl − 1 ) i sin ϕ⎤⎦ ⎪ ⎪ ⎪ ⎪ ⎬ P∥ ex ⎪ ⎪ − iαe αm ⎨ ⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎩ +⎣ −i sin ϕ ( Jl + 1 Jl + 2 + Jl − 1 Jl − 2 ) + ( Jl − 1 Jl + 2 − Jl + 1 Jl − 2 ) cos 3ϕ⎦ P⊥ ⎭ ⎪ ⎪ 4 ⎪ ⎪ ⎧ ⎫ k −Jl ⎡⎣ ( Jl + 1 + Jl − 1 ) cos ϕ + ( Jl + 1 − Jl − 1 ) i sin ϕ⎤⎦ ⎪ ⎪ ⎬ = − Re ⎨ ⎨ ⎬ P e − α α 3 ∥ y ⎪ e m ⎪ ⎡ ⎤ ⎪ ⎪ J J J J i J J J J P + − + ϕ + − ϕ cos sin 3 ( l− 1 l+ 2 l+ 1 l− 2 ) ⎦ ⊥⎭ ⎪ ⎪ l− 1 l− 2 ) ⎩ ⎣ ( l+ 1 l+ 2 ⎪ ⎪ ⎧ ⎫ 1 2 ⎪ ⎪ Jl + 2 + Jl2− 2 P⊥2 ⎬ ez − αe αm ⎨ −Jl2 − iJl ( Jl − 2 − Jl + 2 ) P⊥ sin 2ϕ + ⎪ ⎪ ⎩ ⎭ 2 ⎩ ⎭

(

)

(D2)

 left circular polarization F lcem

⎧ 2iα α J ⎫ e m l + 1 ( Jl + Jl + 2 P⊥ ) 2i sin ϕ P∥ ex ⎪ ⎪ ⎪ ⎪ Re ⎨ + 4αe αm Jl + 1 ( Jl + Jl + 2 P⊥ ) cos ϕey ⎬ = − 3 ⎪ ⎪ ⎪ ⎪ − 2αe αm −Jl2 + Jl2+ 2 P⊥2 ez ⎩ ⎭ k4

{

}

(

)

(D3)

 right circular polarization F rc em

⎧ − 4α α J ( J + J P ) P sin ϕe ⎫ e m l− 1 l x l− 2 ⊥ ∥ ⎪ ⎪ ⎪ ⎪ k4 α α ϕ J J J P P + + e 4 cos = − Re ⎨ e m l− 1 ( l y⎬ l− 2 ⊥ ) ∥ 3 ⎪ ⎪ ⎪ ⎪ − 2αe αm −Jl2 + Jl2− 2 P⊥2 ez ⎭ ⎩

(

)

(D4)

 radial polarization

F rp em

⎧ ⎫ 1 J ⎡ −Jl + 1 + Jl − 1 ) cos ϕ + i ( Jl + 1 + Jl − 1 ) sin ϕ⎤⎦ sin α0 ex ⎪ ⎪ i2π 2αe αm 2 l⎣( + α 1 cos ( ⎪ ⎪ 0) ⎪ ⎪ ⎪ ⎪ 1 k4 2 ⎡ ⎤ J J + Jl − 1 ) cos ϕ + i ( −Jl + 1 + Jl − 1 ) sin ϕ⎦ sin α0 ey ⎬ = − Re ⎨ + 2π αe αm 2 l ⎣ ( l+ 1 3 ( 1 + cos α0 ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ + 2π 2αe αm Jl2+ 1 + Jl2− 1 cos α0 ez 2 ⎪ ⎪ ( 1 + cos α0 ) ⎩ ⎭

(

)

(D5)

 azimuthal polarization

F ap em

⎧ ⎫ 1 J ⎡ −Jl + 1 + Jl − 1 ) cos ϕ − i ( Jl + 1 + Jl − 1 ) sin ϕ⎤⎦ sin α0 ex ⎪ ⎪ − 2iαe αm 2 l⎣( ( 1 + cos α0 ) ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ 1 k 2 J ⎡ J + Jl − 1 ) cos ϕ + i ( Jl + 1 − Jl − 1 ) sin ϕ⎤⎦ sin α0 ey ⎬ = − Re ⎨ + 2π αe αm 2 l ⎣ ( l+ 1 + α 1 cos 3 ( ⎪ ⎪ 0) ⎪ ⎪ 1 2 2 2 ⎪ ⎪ + 2π αe αm J + Jl − 1 cos α0 ez 2 l+ 1 ⎪ ⎪ ( 1 + cos α0 ) ⎩ ⎭

(

)

(D6)

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