Optical radiation force on a dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet

Optical radiation force on a dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet

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Optical radiation force on a dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet Ningning Song, Renxian Li, Han Sun, Jiaming Zhang, Bojian Wei, Shu Zhang, F.G. Mitri PII: DOI: Reference:

S0022-4073(19)30801-5 https://doi.org/10.1016/j.jqsrt.2020.106853 JQSRT 106853

To appear in:

Journal of Quantitative Spectroscopy & Radiative Transfer

Received date: Revised date: Accepted date:

27 October 2019 22 January 2020 23 January 2020

Please cite this article as: Ningning Song, Renxian Li, Han Sun, Jiaming Zhang, Bojian Wei, Shu Zhang, F.G. Mitri, Optical radiation force on a dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet, Journal of Quantitative Spectroscopy & Radiative Transfer (2020), doi: https://doi.org/10.1016/j.jqsrt.2020.106853

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Highlights 

Optical force on a sphere of arbitrary size illuminated by an Airy lightsheet is investigated using GLMT.



The BSCs of the Airy light-sheet are calculated using ASDM and VSWFs.



The negative longitudinal optical force is particularly emphasized



The two kinds of polarizations (TE and TM) of Airy light-sheet are discussed.



The influences of the transverse scale parameter ω 0 and attenuation parameter γ of the Airy light-sheet are discussed.

Optical radiation force on a dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet Ningning Songa , Renxian Lia,b,∗, Han Suna , Jiaming Zhanga , Bojian Weia , Shu Zhanga , F. G. Mitric a School b Collaborative

of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China c Santa Fe, NM 87508, USA

Abstract Based on the generalized Lorenz-Mie theory (GLMT) and the angular spectrum decomposition method (ASDM), we calculate the optical radiation force exerted on a lossless dielectric sphere of arbitrary size illuminated by an Airy light-sheet. The beam shape coefficients (BSCs) of the Airy light-sheet are calculated using the vector angular spectrum decomposition and vector spherical wave functions methods. The optical radiation force acting on the spherical particle is obtained by the integral of Maxwell’s stress tensor. The transverse (Fy ) and longitudinal (Fz ) forces are numerically computed. Two kinds of polarization (TE and TM) are considered for the Airy light-sheet, and the negative longitudinal optical (pulling) force is particularly emphasized. The influence of the transverse scale parameter w0 and attenuation parameter γ of the Airy light-sheet on the force is discussed. The results of the present theory are verified using the dipole approximation method in which the gradient force has been also computed for a Rayleigh sphere. The numerical results show that when the transverse scale parameter w0 and attenuation parameter γ increase, the transverse and longitudinal forces decrease. Furthermore, the force caustic (i.e., maximum) shifts to the direction of y < 0 as the transverse scale parameter w0 increases. As the dimensionless size parameter of the sphere ka increases (where k is the wavenumber and a is the radius), the resonance peaks of the optical forces become larger. The results of this paper are of practical significance for the development of Airy light-sheet based optical manipulation technologies. Keywords: Optical force, Airy light-sheet, GLMT, optical tweezers, TE and TM polarization

1. Introduction The development of optical manipulation technologies include optical trapping and immobilization, optical binding and attraction/repulsion, rotation and particle transport. Among such different methods, light trapping is used most widely, namely by the means of optical tweezers. In 1966, the angular stabilization of dust particles in a He-Ne laser cavity was observed, suggesting that the term ”photodynamic stability” is appropriate [1]. Subsequently, it was verified that photophoresis is the source of driving forces and stabilizing torques [2]. In 1970, dielectric particles were manipulated in water with two focusing laser beams propagating in opposite directions, proving that there is indeed a force exerted from the laser on the particles [3]. In 1986, the stable capture of dielectric particles in water by using a single beam of strongly focused laser was achieved, by a gradient force trap, marking the birth of optical tweezers technology [4]. Optical tweezers technology is a invention of great significance, which enriches and promotes the development of the optical field further, making optics one of the most cutting-edge and most potential scientific fields in the 21st century. The advantage of optical tweezers is the contact free mode, high precision and the ability to measure the magnitude of forces in a scale of 10−10 N. It is widely used in manipulation of particles ranging ∗ Corresponding author at: School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China. E-mail address: [email protected] (R. Li).

Preprint submitted to Journal of Quantitative Spectroscopy and Radiative Transfer

January 30, 2020

from atomic size to hundreds of micrometers order. It has been applied in cell biology, single molecular biology, soft material colloid science, physics and other fields [5–16]. The theoretical basis of optical manipulation is to study the interaction between different specific light fields and particles by various calculation methods. Traditional optical tweezers used the focused Gaussian beam as a specific structured light field [17–20]. On the other hand, laser-sheets (i.e. light-sheets ) [21–26] have been developing in recent years, which have important applications including particle manipulation [27, 28], evaluation of a particle sizing [29], laser microsurgery [30], light-sheet microscopy [31], fluorescence microscopy [32], optical tomography sectioning [33, 34], flow visualization [35], etc, where Hermite-Gaussian light-sheet [27], Bessel light-sheet [36], Bessel-Gauss light-sheet [37], bottle light-sheets [38] and Airy lightsheet [28, 31] are suggested. The optical radiation force (particularly negative optical radiation force) and torques are mainly computed in the previous analysis. Research shows that a plasmonic layer of optimal thickness coating a small sphere can boost the longitudinal and transverse force components as well as the axial torque induced by Bessel pincers light-sheets [36]. The generation of negative optical forces and torques depend on the beam parameters, derivative order of the beam, the polarization of the electric/magnetic vector potentials, the orientation of subwavelength spheroid in space and its aspect radio [37, 39]. Negative or positive axial torques can arise depending on the choice of the size and the location of the particle in the field [27]. The Airy light-sheet is a kind of non-diffracting beam, which has the characteristics of self-healing [40], non-diffraction [41], and self-acceleration [42, 43], providing a possibility for the special and complex manipulation of particles. These intrinsic features have made the Airy light-sheet the focus of several works in the interaction of light-sheet with particles. The subwavelength sphere [28, 39] and non-spherical particle [44] in the fields of Airy light-sheet have been studied. The purpose of this investigation is to extend the scope of the previous studies for the case of a lossless dielectric sphere of arbitrary size illuminated by an Airy light-sheet, using the GLMT to investigate the optical radiation force (particularly the generation of a negative longitudinal optical force), including the transverse (Fy ) and longitudinal (Fz ) forces, with emphasis on two kinds of polarization (TE and TM). The influences of transverse scale parameter w0 and attenuation parameter γ of the Airy light-sheet are also examined. The rest of this paper is organized as follows. The beam shape coefficients (BCSs) for the Airy light-sheet with TE and TM polarizations are derived using the angular spectrum decomposition method (ASDM) in section 2. The general expressions for the components of the optical radiation force exerted on a sphere of arbitrary size/radius involving the BSCs of an Airy light-sheet are provided. Section 3 discusses numerical results of the optical force exerted on a sphere of arbitrary size, with particular emphasis on polarization, the transverse scale parameter w0 and attenuation factor γ of the polarized Airy light-sheet. The negative longitudinal optical forces are emphasized. Finally, a conclusion of the present work is given in Section 4. 2. Theory Consider a polarized Airy light-sheet propagating toward the half free space z > z0 , with the z−axis taken to be the axis of wave propagation, and focused on point (y0 , z0 ) (Fig.1). When the polarization is TE, the incident electric field is polarized along the x−direction, propagating in the yz plane. Thus, we have Eiy = Eiz = Hix = 0. On the other hand, when the polarization is TM, its incident magnetic field is polarized along the x−direction. Thus, we have Hiy = Hiz = Eix = 0. The incident electric field of an Airy light-sheet can be expressed using the ASDM [41, 45–48]. 2.1. Airy light-sheet Assuming a TE-polarized Airy light-sheet, the only non-zero component of the incident electric field propagating along the z-axis can be expanded into an angular spectrum of plane waves as [45] Z +∞ Z +∞ i[ky y+kz (z−z0 )] ik[qy+p(z−z0 )] Eix (y, z) = E0 AE dky = E0 k AE dq (1) x (ky ; z0 )e x (q)e −∞

−∞

1 2

where q = sin α sin β, p = cos α, k = (µ) ω/c, µ and ε are the permeability and permittivity of the surrounding medium, respectively. The parameters (ky , kz ) are the transverse and axial components of the 2

Figure 1: Definition of k, r, and the corresponding angles

Figure 2: The graphical representation of the interaction of an Airy light-sheet with a sphere of arbitrary size located arbitrarily in space

wave vector (Fig.2), ky = kq = k sin α sin β, and kz = kp = k cos α. AE x (q) is the angular spectrum, given by the Fourier transform of the initial electric field in the z = z0 plane. Assuming TE-polarization, the incident electric field vector of the Airy light-sheet is described as [41, 46, 47]     y − y0 γ(y − y0 ) Ei (y, z0 ) = ex Eix (y, z0 ) = ex Ai exp (2) w0 w0 where w0 is the transverse scale, Ai(·) is the Airy function, (y0 , z0 ) are the coordinates of a point in the transverse plane (yz) of the incident field, and γ is the attenuation parameter. Using the Fourier transformation, the angular spectrum at the source can be obtained as    Z +∞  y − y0 1 γ(y − y0 ) −ikqy AE (q) = e Ai exp e dy x x 2π −∞ w0 w0   (γ − ikw0 q)3 −iky y0 w0 exp e (3) = ex 2π 3 Considering Eqs. (1) and (3) and ignoring the contribution of evanescent waves as they decay away from the source and do not contribute to the radiated field, the incident electric field vector is expressed as Z +∞ ik[qy+p(z−z0 )] Ei (y, z) = E0 k e x AE dq x (q)e −∞

=

E0 k

Z

0

π 2

ik·r −ikpz0 ex AE e cos αdα x (α)e

(4)

The term ex eik·r can be expanded using vector spherical wave functions as[49] ex eik·r =

∞ X n X

n=1 m=−n

where Dmn = p0mn = −in+1 e−ipβ



h i 0 (1) Dmn p0mn N(1) mn + qmn Mmn

(5)

(2n + 1)(n − m)! n(n + 1)(n + m)!

(6)

τmn (cos α)eθ (α) − iπmn (cos α)eφ (α) 3



· ex

(7)

0 qmn = −in+1 e−ipβ



πmn (cos α)eθ (α) − iτmn (cos α)eφ (α)



· ex

M(1) mn = [iπmn (cos θ)eθ − τmn (cos θ)eφ ] jn (kr) exp(imφ)

(8) (9)

1 d [rjn (kr)] exp(imφ) kr dr jn (kr) +er n(n + 1)Pnm (cos θ) exp(imφ) kr

N(1) mn = [τmn (cos θ)eθ + iπmn (cos θ)eφ ]

(10)

with

Pnm (cos α) (11) sin α dPnm (cos α) τmn (cos α) = (12) dα where jn (kr) is the spherical Bessel function of the first kind, Pnm (cos α) represents the associated Legendre functions of degree n and order m, and e(r,θ,φ) are the radial, polar, and azimuthal unit vectors. Substituting Eq.(5) into Eq.(4), the incident electric field of the Airy light-sheet is expanded in terms of vector spherical wave functions (VSWFs) as πmn (cos α) = m

Ei (r, θ, φ) = −

n ∞ X X

n=1 m=−n

h i (1) pol (1) iEmn ppol m,n Nmn + qm,n Mmn

(13)

where the superscript pol corresponds to the type of polarization, i.e., T E or T M , respectively, and Emn is p (14) Emn = E0 in Dmn pol The following coefficients ppol m,n and qm,n are obtained as[50]

ppol mn =

i1−n p kr Dmn 4πE0 jn (kr)

pol qmn =−

Z

kr i−n Z p Dmn 4πE0 jn (kr)

π

θ=0

Z

Z

π

θ=0

2π

φ=0

Z

[er · Ei (r, θ, φ)] Pnm (cos θ)e−imφ sin θdθdφ

(15)

[er · Hi (r, θ, φ)] Pnm (cos θ)e−imφ sin θdθdφ

(16)

2π

φ=0

p where Z = µ/ε is the impedance of the medium of wave propagation surrounding the sphere. For a pol TE-polarized field, the coefficients ppol m,n and qm,n become E pTm,n

TE qm,n

√

Z π2 Dmn −ikpz0 = k n+1 p0mn AE cos αdα x (α)e −i 0 √ Z π2 Dmn 0 −ikpz0 = k n+1 qmn AE cos αdα x (α)e −i 0

(17)

(18)

Since the electric field of the Airy light-sheet is polarized along the x−axis, ex = −eφ (α), and the E TE expressions for pTm,n and qm,n become p0mn

−in+1 e−ipβ



τmn (cos α)eθ (α) − iπmn (cos α)eφ (α)

πmn (cos α)



· ex

= −in+1 e−ipβ



πmn (cos α)eθ (α) − iτmn (cos α)eφ (α)



· ex

= =

0 qmn

n −ipβ

i e

n −ipβ

= i e

τmn (cos α)

4

(19)

(20)

The substitution of Eqs. (19) and (20) into Eqs. (17) and (18) leads to E pTm,n = ik

TE qm,n

Z p Dmn

= ik

π 2

0

p

Dmn

Z

−ikpz0 e−ipβ πmn (cos α)AE cos αdα x (α)e

(21)

−ikpz0 cos αdα e−ipβ τmn (cos α)AE x (α)e

(22)

π 2

0

E TE The final expressions of pTm,n and qm,n are given as E pTm,n = ik

Z p π Dmn e−ip 2

TE qm,n = ik

π 2

0

p

π

Dmn e−ip 2

Z

π 2

0

−ikpz0 cos αdα πmn (cos α)AE x (α)e

(23)

−ikpz0 cos αdα τmn (cos α)AE x (α)e

(24)

For a TM-polarized Airy light-sheet, the direction of propagation of the incident electric field is the z−axis, and its incident magnetic field is polarized along the x−direction. Based on Maxwells equations M TM and the duality principle, we acquire the expressions of pTm,n and qm,n as M pTm,n = −k TM qm,n = −k

Z p π Dmn e−ip 2

π 2

0

p

π

Dmn e−ip 2

Z

0

π 2

−ikpz0 τmn (cos α)AH cos αdα x (α)e

(25)

−ikpz0 πmn (cos α)AH cos αdα x (α)e

(26)

where AH x (α) is the angular spectrum of the magnetic field polarized along the x-axis of the TM-polarixed Airy light-sheet. 2.2. Optical Force In GLMT, the incident, internal and scattered electromagnetic fields can be expended, respectively, using VSWFs as n ∞ X h i X (1) pol (1) iEmn ppol (27) Ei (r, θ, φ) = − m,n Nmn (kr) + qm,n Mmn (kr) n=1 m=−n

Hi (r, θ, φ) =

∞ n h i ik X X pol pol (1) iEmn qm,n N(1) mn (kr) + pm,n Mmn (kr) ωµ n=1 m=−n

El (r, θ, φ) = − Hl (r, θ, φ) =

∞ X n X

n=1 m=−n

h i (1) pol (1) iEmn dpol m,n Nmn (mkr) + cm,n Mmn (mkr)

∞ n h i imk X X (1) pol (1) iEmn cpol m,n Nmn (mkr) + dm,n Mmn (mkr) ωµ n=1 m=−n

Es (r, θ, φ) =

∞ X n X

n=1 m=−n

Hs (r, θ, φ) = −

h i (3) pol (3) iEmn apol m,n Nmn (kr) + bm,n Mmn (kr)

∞ n h i ik X X (3) pol (3) iEmn bpol m,n Nmn (kr) + am,n Mmn (kr) ωµ n=1 m=−n

(28)

(29)

(30)

(31)

(32) (3)

where m in Eqs.(29) and (30) is the index of refraction of the sphere. The VSWFs of the third kind (Nmn (3) (1) (1) and Mmn ) are described by replacing jn (kr) by hn (kr) in Eqs.(9) and (10), where hn (kr) is the spherical 5

pol Hankel function of the first kind. The expansion coefficients of the internal and scattered field apol m,n , bm,n , pol pol cm,n and dm,n are obtained via the Mie coefficients an , bn , cn and dn [51, 52], namely pol apol m,n = an pm,n ,

pol bpol m,n = bn qm,n

pol cpol m,n = cn qm,n ,

pol dpol m,n = dn pm,n

(33)

According to GLMT, the optical radiation force acting on the sphere can be obtained by integrating the time-average Maxwell’s stress tensor as [53] I D↔E b · T dS F= n (34) S

bEis the unit normal vector on a surface enclosing the sphere, dS is the differential surface element, where D ↔n and T is the time-averaged Maxwell stress tensor, defined as [53]  D↔E 1  ↔ 1 T = Re εE ⊗ E∗ + µH ⊗ H∗ − (εE · E∗ + µH · H∗ ) I 2 2

(35) ↔

where the symbol ⊗ denotes a tensor product, the superscript ∗ denotes a complex conjugate and I is the unit tensor. E and H are the sum of incident and scattered fields, respectively. Considering the lossless background medium, the optical force vector components (Fx ,Fy ,Fz ) can be expressed by substituting the total (incident + scattered) electromagnetic field into Eq.(35) as shown in [54]. Therefore, the expressions for the force components where Re and Im denote, respectively, the real and imaginary parts of a complex number, become, Fx = Re [F1 ] , Fy = Im [F1 ] , Fz = Re [F2 ] (36) where ! ( 1 ∞ X n X ˜∗pol ˜pol ˜∗pol 2πε [(n − m)(n + m + 1)] 2 a ˜pol 2 m,n bm+1,n + bm,n a m+1,n F1 = 2 |E0 | × ∗pol pol ∗pol k n(n + 1) p˜m+1,n ˜m+1,n − q˜m,n −˜ ppol m,n q n=1 m=−n " # 12 ! ˜pol ˜∗pol n(n + 2)(n + m + 1)(n + m + 2) a ˜pol ˜∗pol m,n a m+1,n+1 + bm,n bm+1,n+1 − × 2 pol ∗pol −˜ ppol ˜∗pol ˜m,n q˜m+1,n+1 (n + 1) (2n + 1) (2n + 3) m,n p m+1,n+1 − q # 12 " !  ∗pol ˜bpol ˜b∗pol n(n + 2)(n − m)(n − m + 1) a ˜pol a ˜ + m,n+1 m+1,n m,n+1 m+1,n × + pol ∗pol pol ∗pol 2 −˜ pm,n+1 p˜m+1,n − q˜m,n+1 q˜m+1,n  (n + 1) (2n + 1) (2n + 3)  ∞ X n   X m 4πε 2 pol ˜∗pol pol ∗pol |E | × a ˜ b − p ˜ q ˜ 0 m,n m,n m,n m,n k2 n(n + 1) n=1 m=−n " ! # 12  ∗pol ˜bpol ˜b∗pol n(n + 2)(n − m + 1)(n + m + 1) a ˜pol a ˜ + m,n m,n+1 m,n m,n+1 + × 2 pol ∗pol pol ∗pol −˜ pm,n p˜m,n+1 − q˜m,n q˜m,n+1  (n + 1) (2n + 1) (2n + 3)

(37)

F2 = −

and

1 pol pol a ˜pol m,n = am,n − pm,n , 2 ˜bpol = bpol − 1 q pol , m,n m,n 2 m,n 6

1 pol p 2 m,n 1 pol = qm,n 2

p˜pol m,n = pol q˜m,n

(38)

(39)

3. Numerical results and discussions Initially, in the first subsection, the theory and numerical code are verified. We first verify the BSCs by comparing the original field to the reconstructed field. Also, the differential scattering cross section (DSCS) calculated using this theory is compared to a previous result available in the literature [55]. In addition, the optical forces calculated using the dipole approximation method and the gradient force for a Rayleigh sphere are compared to verify our theory and code. In the second subsection, the optical radiation force components are evaluated based on the GLMT for a spherical particle with different radii. 3.1. Verification and validation 2ULJLQDO 

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Figure 3: Comparision of the original incident electric field intensity and its reconstructed counterpart 2

Initially, a comparison of the original relative incident electric field intensity |Ei /E0 | and its reconstructed form using the BSCs is performed, based on Eqs.(13), and (23)-(26). The original field is expressed by Eqs.(3) and (4). The wavelength of the incident Airy light-sheet is λ = 632.8 nm, and its amplitude is E0 = 1 × 106 V/m, y varies in the range −20 6 y 6 20 µm, and z = 10 µm. The TE and TM-polarized incident fields are considered. The transverse scale parameter of the Airy light-sheet is w0 =1.1 µm, and the attenuation parameter is γ=0.1. Fig. 3 displays the results, where a complete agreement is observed between the original and reconstructed data. Then, the differential scattering cross section is calculated. According to Refs. [55] and [56], the wavelength of the Airy light-sheet is λ = 633 nm, the radius of the sphere is 1 µm, and the scattering angle varies in the range 0 6 θ 6 180 ◦ . Results in Fig. 4 agree with the plots shown in panel (b) of Fig. 4 previously presented in [55] . Finally, the optical forces calculated using the exact GLMT are compared to the results obtained by two theoretical models for a relatively small particle. In the first theoretical model, the optical radiation force using the dipole approximation method is expressed as [57]   3 X  1 Fζ = εRe αEij ∇ζ Eij ∗ (40)   2 j=1 where ε is the permittivity of the medium of wave motion. ∇ζ ≡ ∂r∂ζ , ζ and j denote the three different directions, and Fζ is the ζ-th component of the optical radiation force vector, Eij is the j-th component of the incident electric vector field, where Ei means the incident electric field, and α is the polarizability of the object given by [57] α0 α= (41) 3 1 − 32 k4πα0 with

α0 = 4πa3 7

m2 − 1 m2 + 2

(42)



ka=5 ka=10

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Figure 5: Optical radiation force calculated by Rayleigh model and GLMT model

8



where a is the radius of the sphere. Eq. (40) gives the total optical force on a small sphere which is the sum of a gradient force and scattering force. The optical gradient force is isolated and is expressed as [4]   n3 a3 m2 − 1 Fgrad = − m ∇E2i (43) 2 m2 − 2 where nm is the refractive index of the medium of wave motion. Using the expressions for the forces in the GLMT given by Eqs.(36) - (38), numerical calculations are performed for a small dielectric sphere , and the results are shown in Fig. 5. The results are compared to those obtained by Eqs. (40) and (43). In the calculations, the sphere of radius a = 10 nm and its center is located at z = 10 µm. The transverse Fy and longitudinal Fz forces are calculated. Fig. 5 displays the results using GLMT and the approximate methods. Panels (a) and (b) show the optical force when the incident Airy light-sheet is TE-polarized while panels (c) and (d) correspond to the TM-polarization case. Clearly, the results using the approximate methods are in agreement with those obtained by the GLMT. The fact that the gradient force component agrees well with the total radiation force means that when the radius of the dielectric sphere is much smaller than the wavelength of the incident field, the gradient force dominates. The results show also that the longitudinal gradient force is about two orders of magnitude smaller than the transverse gradient force. 3.2. Optical force In this subsection, the optical forces on the spherical particle illuminated by an Airy light sheet are calculated. Both TE and TM polarizations are considered, and both Rayleigh (a = 10 nm) and Mie (a = 700 nm) spheres are chosen to illustrate the analysis. The longitudinal and transverse optical forces are calculated, respectively, in which the negative longitudinal optical forces are emphasized. In addition, the influences of the transverse scale parameter w0 and attenuation parameter γ of the Airy light-sheet on the optical forces are mainly discussed. 3.2.1. Optical force on Rayleigh particle The optical force on a Rayleigh particle is now considered. We assume that the dielectric particle is made of water (m = 1.33) in vacuum, with an incident wave length λ = 632.8 nm, the radius of the particle a = 10 nm and the amplitude of the incident field E0 is 1 × 106 V/m. Computations for the optical force vector acting on a sphere in the (yz) plane, are performed in the ranges 0 6 z 6 100 and −20 6 y 6 20 (µm). First, the effect of the transverse scale w0 on the optical force components is discussed. Here the attenuation parameter is fixed (i.e., γ = 0.1), and the incident Airy light-sheet is TE-polarized. Fig.6 shows the numerical results for the transverse and longitudinal radiation force components in the (yz) plane, when the transverse scale w0 changes. From top to bottom, the transverse scales are w0 = 0.8 µm, 1.1 µm, and 1.4 µm, respectively. The first column displays the plots for the transverse force Fy , and the second column shows those of the longitudinal force Fz . The negative longitudinal forces (i.e., the regions for which Fz < 0) are emphasized, and are depicted in the third column for improved visualization. In panels (a), (d) and (g) of Fig.6, the transverse optical force Fy appears along the direction of y < 0 or y > 0. In panels (b), (e) and (h) of Fig.6, a negative or a positive longitudinal force arises depending on the position in the (yz) plane. The negative longitudinal forces in panels (c), (f), and (i), are in the region around the caustic (i.e., main lobe) of the incident Airy light-sheet. The existence of negative longitudinal optical force acting in the direction of z < 0 makes it possible that the sphere can be pulled in the opposite direction of wave propagation. Moreover, when transverse scale w0 increases, the force caustic diminishes accordingly in the first and second columns. Concurrently, the main lobe of the optical force broadens. However, the regions of negative optical force are enlarged, which are shown in the third column. As w0 increases further, the force caustic becomes straight gradually. In addition, the maximum of the optical radiation force decreases when w0 increases. The maximum value of the transverse optical force Fy = 1.41 × 10−3 fN along the direction of y > 0 in panel (a) of Fig.6, and the maximum negative value of the longitudinal force is 3.03 × 10−4 fN along the direction of z < 0 in panels (b) and (c) of Fig.6. However, in panel (g) the maximum value of the 9



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transverse optical force is 8.18 × 10−4 fN along the direction of y > 0, and in panels (h) and (i) the negative longitudinal force caustic is 7.99×10−5 fN along the direction of z < 0. As a result, we can draw a conclusion that when a dielectric sphere is in the electromagnetic field of an Airy light-sheet, the longitudinal force can pull/push it along the z-axis and at the same time, the transverse optical force may trap the particle in the main lobe or side-lobes. Subsequently, we discuss the effect of the attenuation parameter γ on the optical force components. In these computations, the transverse scale w0 of the incident TE-polarized Airy light-sheet is 1.1 µm. From top to bottom, the attenuation parameter takes the values, respectively, as γ = 0.01, 0.1 and 0.4, when γ > 0.4 the diffraction of Airy light-sheet can be very severe. Fig.7 shows the variation of the optical force in the (yz) plane with the change of the attenuation parameter γ. Panels (a), (d) and (g) of Fig.7 display the computations of the transverse optical force Fy , and panels (b), (e) and (h) of Fig.7 display the computations of the longitudinal optical force. Similarly to Fig.6, panels (c), (f), and (i) display the regions in the plane (yz) where Fz < 0. By comparison, the transverse optical force is generally larger than the longitudinal force, while the maximum value of the transverse optical force is 3.50 × 10−3 fN in panel (a), but the maximum positive value of the longitudinal optical force is 2.17 × 10−3 fN in panel (b) along the direction of y > 0. Furthermore, the maximum value of the optical force decreases as the attenuation parameter γ increases. As the attenuation parameter γ increases, the main lobe of the optical force broadens and the side-lobes decrease. The regions of the negative longitudinal optical force also reduce in size. At the same time, the curved force caustics become straight gradually. Therefore, we can conclude that both the increase of transverse scale w0 and attenuation parameter γ will lead to a decrease of the force components. Moreover, the force caustic broadens, and curved lines of forces decrease and become straight. Also, the attenuation parameter γ has a stronger influence on the curved force caustics and their distribution in the (yz) plane. Fig.8 displays numerical results similar to those shown in Fig.6 but the Airy light-sheet is TM-polarized. Compared with Fig.6, the optical force components in Fig.8 have a comparable variation tendency such that the lines of force decrease and the maximum value becomes smaller while the transverse scale parameter w0 increases. Also, the curved lines of force become straight. The increase of the attenuation parameter γ of the TM-polarized Airy light-sheet is also investigated, and the results are displayed in Fig.9, for the transverse and longitudinal optical force components, respectively. Similarly, compared with Fig.7, different polarizations have minor influences on the lines of optical force, and the variation trend in Fig.9 is somewhat similar to that shown in Fig.7. 3.2.2. Optical force on Mie particle A larger dielectric (water) sphere is now considered, such that its radius is a = 700 nm, which is comparable to the wavelength of the incident field. In this part, we compute the variation of the optical force in the ranges z = 10 µm and −20 6 y 6 20 (µm), corresponding to one-line profile plots. Panels (a) and (b) of Fig.10 display the longitudinal and transverse optical force respectively, assuming the attenuation parameter γ is 0.1, but the transverse scale parameter w0 is 1.1 µm, 2.1 µm and 3.1 µm, respectively, for the TE-polarized Airy light-sheet. Accordingly, panels (c) and (d) display the transverse and longitudinal optical force components respectively, while transverse scale parameter w0 is 1.1 µm, but the attenuation parameter γ is 0.01, 0.1 and 0.4. The plots in panel (a), show that when the transverse scale parameter w0 increases, the magnitude of the transverse optical force decreases gradually. The maximum of the transverse optical force is 2.95 × 10−1 pN along the direction y > 0, and 1.36 × 10−1 pN along the direction of y > 0, as the transverse scale parameter w0 takes the values of 1.1 µm and 3.1 µm, respectively. In addition, the side-lobes of the transverse and longitudinal optical forces decrease. At the same time, the main lobe of the transverse optical force moves toward the negative transverse direction (y < 0) in panel (a). Panel (b) in Fig.10 displays the longitudinal optical force for a different transverse scale parameter w0 . In panel (b), the main lobe of the longitudinal optical force also moves toward the negative transverse direction, too. However, the magnitude of the force caustic increases, while the transverse scale w0 increases as shown in panel (b). 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15



transverse optical force decrease, while the maximum of the transverse optical force decreases. Panel (d) is similar to panel (c) showing that as the attenuation parameter γ increases the magnitude of longitudinal force decreases. In addition, when the attenuation parameter γ is 0.01, the longitudinal optical force fluctuates around 0.1 pN in the region where y < 0. The case of a TM-polarized Airy light-sheet is now considered, and the results are displayed in the panels of Fig. 11. The variation tendency of the optical force components in Fig.11 is similar to that of Fig.10, but the magnitude of the transverse optical force has some difference. In general, the transverse optical forces in Fig.11 are large. It means that when the incident field is a TM-polarized Airy light-sheet, the transverse optical force will be larger compared with the incident field of a TE-polarized Airy light-sheet, when the radius of the sphere is 700 nm. 3.2.3. Optical force versus the size parameter ka 1e−13 

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16

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17

The optical force components are now varied versus the dimensionless size parameter ka = πd/λ (d is the diameter) of the sphere with different attenuation and transverse scale parameters, considering TE-polarized and TM-polarized Airy light-sheets. The sphere is located at the central point of main lobe when z = 10 µm in the transverse plane (yz) in the different attenuation and transverse scale parameters. Assuming the ka is varied between 0.6 and 16, corresponding to a sphere radius varying between 6.328×10−2 µm and 1.528 µm. When the optical force changes with ka, resonance peaks appear, which are related to the scattering coefficients an and bn . Fig.12 shows the evolution of the optical force components Fy and Fz , whose attenuation parameters γ=0.1. The effect of the transverse scale parameters w0 on the optical force components is discussed. From top to bottom, the transverse scales are w0 = 1.1 µm, 2.1 µm, and 3.1 µm, respectively. The first column displays the transverse force Fy and the second column shows the longitudinal force Fz . In panels (a), (c) and (e) of Fig.12, the transverse optical force component Fy decreases as the w0 increases. When the Airy light-sheet is TM-polarized, the transverse force component Fy is larger than for the case where the Airy light-sheet is TE-polarized. However, in panels (b), (d) and (f) of Fig.12, the longitudinal optical force Fz increases as w0 increases. Moreover, changing the polarization of the light sheet has a minor impact on the amplitude of the longitudinal force component. As the dimensionless size parameter ka increases, which means the larger the radius of the particle, the greater the optical force. In panels (a), (c), and (e), some resonance peaks for the transverse optical force Fy appear in the region of ka > 6, ka > 7, ka > 9 , respectively. As ka increases, the resonance peaks become denser and fluctuations become larger. Notice that the fluctuations of the resonance peaks assuming a TE-polarized light sheet are larger than those predicted for a TM-polarized light sheet. In panels (b), (d) and (f) resonance peaks appear in ka > 9, and changing the polarization of the incident light sheet has a minor impact on the resonance peaks of the longitudinal force component as well. Fig.13 shows the evolution of the optical force components Fy and Fz , while the transverse scale parameter w0 =1.1 µm. From top to bottom, the attenuation parameters take the values, respectively, of γ=0.01, 0.1, 0.4. When γ increases, the amplitude of the transverse optical force Fy and the longitudinal force Fz both become smaller. When the Airy light-sheet is TM-polarized, the transverse force component Fy is larger than when the Airy light-sheet is TE-polarized. Similar to Fig.12, different polarizations have little impact on the amplitude of the longitudinal force component. In panels (a), (c) and (e), the resonance peaks appear for ka > 6. In panels (b), (d) and (f), the resonance peaks appear for ka > 8 clearl. The resonance peaks generally appear when the radius of the sphere is approximately equal to the wavelength (λ = 632.8 nm) of the incident field. 4. Conclusion In this paper, the BCSs of Airy light-sheets with two different polarizations (TE and TM) are derived and used to compute the optical radiation force components for a lossless dielectric water sphere. The effects of varying the transverse scale w0 and attenuation parameter γ of the polarized Airy light sheets on the optical radiation force components are examined based on the GLMT. The results show that the transverse scale w0 and attenuation parameter γ of the incident Airy light-sheet influence the amplitude and spatial distributions of the force components in the transverse plane (yz). When w0 and γ increase, the amplitude of the force components decrease. Furthermore, the attenuation parameter γ has a stronger impact reducing the amplitudes of the optical force components than the transverse scale w0 . In addition, as w0 increases, the main lobe of optical force moves to the direction of y < 0, when the sphere radius is 700 nm, which is comparable to the wavelength of the incident field. Different polarizations of the Airy light-sheet have similar effects for the longitudinal radiation force component suggesting that retrograde motion can be achieved regardless of polarization. Nonetheless, the transverse force component changes. We have also investigated the changes of the optical radiation force components versus ka ranging from 0.6 to 16, which shows that the resonance peaks always appear where the radius of particle is comparable the wavelength of incident field. As ka increases, the resonance peaks become denser and fluctuations become larger. Finally, the numerical results reveal the generation of negative longitudinal forces for a lossless Mie 18

sphere, which provides the impetus to use polarized Airy light-sheets in experimental setups in the realm of optical tweezers. ACKNOWLEDGEMENTS The authors acknowledge support from the 111 Project (B17035).

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Conflict of Interest and Authorship Conformation Form Please check the following as appropriate: o

All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

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This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

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The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

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The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:

Author’s name Ningning Song Renxian Li Han Sun Jiaming Zhang Bojian Wei Shu Zhang F.G. Mitri

Affiliation Xidian University Xidian University Xidian University Xidian University Xidian University Xidian University Santa Fe, NM

Author Statement Ningning Song: Conceptualization, Methodology, Software Renxian Li: Supervision Han Sun: Data curation Jiaming Zhang: Software, Validation Bojian Wei: Visualization, Writing Shu Zhang: Writing- Reviewing and Editing F.G. Mitri: Supervision