Analytical description of lateral binding force exerted on bi-sphere induced by high-order Bessel beams

Journal of Quantitative Spectroscopy & Radiative Transfer 214 (2018) 71–81

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Analytical description of lateral binding force exerted on bi-sphere induced by high-order Bessel beams J. Bai a, Z.S. Wu a,∗, C.X. Ge a, Z.J. Li a, T. Qu b, Q.C. Shang a a b

School of Physics and Optoelectronic Engineering, Xidian University, Xi’an, Shaanxi 710071, China School of Electronic Engineering, Xidian University, Xi’an, Shaanxi 710071, China

a r t i c l e

i n f o

Article history: Received 9 January 2018 Revised 20 April 2018 Accepted 26 April 2018 Available online 1 May 2018 Keywords: Lateral binding force Bi-sphere High-order Bessel beam Spherical vector wave functions

a b s t r a c t Based on the generalized multi-particle Mie equation (GMM) and Electromagnetic Momentum (EM) theory, the lateral binding force (BF) exerted on bi-sphere induced by an arbitrary polarized high-order Bessel beam (HOBB) is investigated with particular emphasis on the half-conical angle of the wave number components and the order (or topological charge) of the beam. The illuminating HOBB with arbitrary polarization angle is described in terms of beam shape coefficients (BSCs) within the framework of generalized Lorenz-Mie theories (GLMT). Utilizing the vector addition theorem of the spherical vector wave functions (SVWFs), the interactive scattering coefficients are derived through the continuous boundary conditions on which the interaction of the bi-sphere is considered. Numerical effects of various parameters such as beam polarization angles, incident wavelengths, particle sizes, material losses and the refractive index, including the cases of weak, moderate, and strong than the surrounding medium are numerically analyzed in detail. The observed dependence of the separation of optically bound particles on the incidence of HOBB is in agreement with earlier theoretical prediction. Accurate investigation of BF induced by HOBB could provide an effective test for further research on BF between more complex particles, which plays an important role in using optical manipulation on particle self-assembly. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Optical binding denotes a significant phenomenon of lightmatter interaction which can lead to the self-arrangement of particles into optically conjunct states [1–3]. This intriguing selfarrangement is based upon the delicate equilibrium between the optical forces resulting both from the incident beam and from the light re-scattered by the other objects. Accurate prediction of the optical binding force (BF) enables better understanding of the physical mechanism of self-organization, and may offer important applications towards contact-free storage of biological cells [4,5] and ion traps for quantum computing [6]. The research on BF was first demonstrated experimentally in early 1989s, Burns et al [7–9] found a series of bound states for two polystyrene particles and produced a stream of remarkable work that laid the foundations for the field. After that, different approaches have been developed for the theoretical prediction of BF exerted on bi-sphere system. Geometrical optics (GO) [10,11] can be employed to the prediction of BF in the ray optics regime, which requires the particles be much larger than the inci-

∗

Corresponding author. E-mail address: [email protected] (Z.S. Wu).

https://doi.org/10.1016/j.jqsrt.2018.04.031 0022-4073/© 2018 Elsevier Ltd. All rights reserved.

dent wavelength i.e. d  10λ with d the diameter of particles and λ the wavelength. Conversely, the Rayleigh dipole approximation (RDA) [12,13] can be employed to the prediction of BF exerted on particles which are much smaller than the wavelength, i.e. d  λ. For particles whose sizes are of incident wavelength order, both GO and RDA are inapplicable, because diffraction phenomenon cannot be neglected in this case. In order to cover the whole d/λ range, some researchers have been devoted to the rigorous prediction of BF stemming from the solution of Maxwell’s equations which is suitable for modeling arbitrary number of particles of arbitrary sizes without additional approximation. For example, based on the Lorenz Mie theory [14] and the Maxwell Stress Tensor approach, Jack investigated BF between bi-sphere cluster of arbitrary size under the illumination of a plane wave [15,16]. Xu introduced the additional theorem to explore the interaction of collective homogeneous spheres [17,18]. Besides, the GMM equation between an incident beam with arbitrary profile and an assembly of spheres had already been given by Gouesbet et al. [19,20], with taking advantage of many ingredients developed for the spherical GLMT [21– 23]. Following this work, Xu and Kall [24,25] put forward the extended Mie theory to calculate BF between closely spaced silver nanoparticle aggregates. Chvatal et al. presented the binding selfarrangement of a pair of Au particles in a wide Gaussian standing

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wave [26]. Despite the great wealth of knowledge obtained from these works, the investigations on BF in the previous studies are mainly focused on plane wave or Gaussian beam incidence. Recently, due to the special characteristics of non-diffraction and self-reconstruction, Bessel beams have attracted growing attention since its naissance by Durnin [27] and have been widely applied in various fields, including optical entrapment and manipulation, optical acceleration, particle sizing and nonlinear optics [28–31]. Motivated by the features and applications of such a beam, analytical and numerical analysis are undertaken to investigate the beam expansion, field description, scattering and radiation in acoustic, optics, and microwave. Accurate description of a Bessel beam can be obtained by the beam shape coefficients as a double quadrature over spherical coordinates [32], which is the original method used in the GLMTs [33,34]. In order to overcome the time-consuming and complexity in the numerical calculation [35–37], many researchers have devoted to the analytical description of BSCs. Lock [38] analyzed the BSCs of general zeroorder Bessel beams based on the angular spectrum representation (ASR). The similar procedure was also extended by Ma et al. [39] to investigate the scattering of un-polarized HOBB by spheres. Besides, Gouesbet and Lock [40,41] established the dark theorem in terms of BSCs and predicted the existence of high-order non-vortex Bessel beams. Wang derived the general description of circularly symmetric Bessel beams of arbitrary order [42,43]. Based on these representations, the scattering problem of Bessel beams by a dielectric sphere [44,45], a uniaxial anisotropic sphere [46] as well as a concentric sphere [47] has been investigated extensively by using the analytical approach. In addition, some studies have also been carried out on the trapping force induced by Bessel beams using the Rayleigh model [48], the geometrical optics [49,50] or the rigorous electromagnetic theory [51–53]. Nevertheless, the published work to which we have referred mainly focused on cases of single spherical particle. Manipulation of multiple particles simultaneously is both very different and much less mastered than that of singular sphere. Accurate prediction of BF induced by HOBB with arbitrary polarization angle is of great help for the efficient generation of optical manipulation system operating with non-diffracting beam. In this paper, we will rely on the general description of HOBB derived by Wang et al. [54], who succeeds in dealing analytically with BSCs by using quadrature expressions in the classical framework of GLMT [21,55], with using GMM equations and EM theories to analyze lateral BF exerted on bi-sphere induced by HOBB in detail. The remainder of this paper is organized as follows. In Section 2, two kinds of descriptions on the profile of a HOBB are given. Moreover, the expansion expression and coefficients of the arbitrary polarized incident field in terms of SVWFs are given within the framework of GLMTs. Based on the GMM equation, Section 3 derives the analytical solutions to the scattering problem of a HOBB by two homogeneous spherical particles. Section 4 investigates the theoretical expressions of lateral BF between two homogeneous spheres induced by a HOBB using the EM theory. Section 5 establishes the discussions for numerical effects of various parameters and comparisons of our numerical results with earlier theoretical prediction. Finally, a conclusion is shown in Section 6. 2. Theoretical analysis A Cartesian coordinate system Oxyz is built with a fixed global coordinate system to indicate the randomness of the polarized direction of incident Bessel beam and the configuration of the bisphere system [Fig. 1(a)]. Considering two homogeneous spheres with radius aj (j = 1, 2) and refractive index nj (j = 1, 2) embedded in the dielectric medium with refractive index nm . The particle co-

Fig. 1. Configuration of bi-sphere induced by a HOBB.d: Inter-particle distance. θ : Angle between the polarization direction and bi-sphere orientation. The set displaying is the intensity of a first-order x-polarized Bessel beam.

ordinate system Oj xj yj zj is established parallel to the primary system Oxyz, and the center of the jth sphere Oj is located at (xj , yj , zj ). Without loss of generality, the bi-sphere central line is along x axis and the inter-particle distance is denoted by d. The particles are vertically illuminated by a polarized HOBB that propagates in the z -direction in the Cartesian coordinate system O x y z , which is known as the beam coordinate system. The coordinates of beam center O in Oxyz are (x , y , z ), and the angle between the polarization direction of HOBB and the bi-sphere central line is represented by β . This pseudo-polarization angle can then determine the polarization mode of the wave and may be regarded as a real polarization angle. For the vertical incidence, the beam is in transverse magnetic mode (TM) when β = 00 , which corresponds to the case in which the electric vector vibrates in the incident plane (i.e. the x O z -plane). Then, the beam is in the transverse electric mode (TE) when β = 900 , which corresponds to the case where the magnetic vector vibrates in the incident plane (i.e. the y O z -plane). When β presents other values, it represents another polarization mode. The Bessel beam is assumed to propagate in an isotropic homogeneous medium and is scattered by a bi-sphere system. Electromagnetic fields outside and inside the particles must satisfy these vector wave equations (or Helmholtz equations):

 + k2 H  =0 ∇ 2 E + k2 E = 0, ∇ 2 H

(1)

where k is the wavenumber. The solutions can be derived by introducing the SVWFs, whose expressions used here are the same as those used in Ref. [56] Eqs. (1) and ((2) there).





(l )  mn M (kr, θ , φ ) = (−1 )m imπnm (cos θ )iˆθ − τnm (cos θ )iˆφ zn(l ) (kr )eimφ



n ( n + 1 ) (l ) (1 )  mn N (kr, θ , φ ) = (−1 )m zn (kr )Pnm (cos θ )iˆr kr





(l ) 1 d rzn (kr ) + kr dr



 τnm (cos θ )iˆθ + imπnm (cos θ )iˆφ eimφ

(2)

where zn(l ) (kr ) represents an appropriate kind of spherical Bessel functions: the first kind jn , the second kind yn , or the third kind hn(1 ) and hn(2 ) , denoted by l = 1, 2, 3 or 4 respectively. Pnm (cos θ ) is the associated Legendre Function of the first kind. Then the incident, scattered, and internal fields can be expressed as an infinite

J. Bai et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 214 (2018) 71–81

series of these vector functions. Solutions of the binding problem could be derived utilizing the boundary condition. 2.1. Description of the high-order Bessel beam To solve the present scattering problem, the incident Bessel beam should be expanded in terms of the SVWFs in every particle coordinate system Oj xj yj zj . So far, in the analytical description of an ideal Bessel beam, two different procedures are commonly applied to obtain the fields of an l-order Bessel beam: (a) the ASR procedure which obtains the fields by a superposition of partial plane wave, and (b) the Davis procedure which obtains the fields from a polarized vector potential. Although the two different approaches give two seemingly different answers for the fields, it turns out that the functional dependence of the two answers is identical for HOBB. A general description for HOBB has been derived by Wang et al. [42], which can provide fairly good results for making the Davis type Bessel beam and the ASR type Bessel beam merely the two simplest cases of an infinite number of possible HOBBs, corresponding to different values of arbitrary function g(α 0 ). This generalization of the description can provide fairly good results for satisfying Maxwell’s equations. Thus this mode could be conveniently expanded in terms of SVWFs. According to this work, Electromagnetic fields of the original x-polarized HOBB shown in Fig. (1) in a Cartesian coordinate system (Oxyz) can be written as [42,53]:



Ex = E0 g(α0 )(−i )l eil ϕ exp[−ikz (z − z )] (1 + cos α0 )Jl (kt ρ ) 1 − cos α0 2iϕ + [e Jl+2 (kt ρ )] + e−2iϕ Jl−2 (kt ρ )] 2



(3)

(4)

Ez = iE0 g(α0 )(−i )l eil ϕ exp[−ikz (z − z )] sin α0 [eiϕ Jl+1 (kt ρ )] − e−iϕ Jl−1 (kt ρ )]

Hx = H0 g(α0 )(−i )l eil ϕ exp[−ikz (z − z )]

2.2. Expansion of the arbitrary polarized high-order Bessel beam In practice, we need the incident beam with arbitrary polarization angles expressed in the particle coordinate system (Oj xj yj zj ). As shown in Fig. 1, temporary coordinate system Oj xj  yj  zj  (j = 1, 2) is established parallel to O x y z and is known as the beam system. As the axes of the coordinate system Oj xj yj zj are parallel with the axes of primary coordinate system Oxyz, after the system Oj xj yj zj rotates β (00 ≤ β ≤ 1800 ) anticlockwise with Oj xj -axis, the system Oj xj  yj  zj  is obtained. If the coordinate of the sphere center Oj at the primary coordinate system Oxyz is (xj ,yj ,zj ), the coordinate (xj  , yj  , zj  ) of the beam center O at the coordinate system Oj xj  yj  zj  can be gained by

  xj y j z j

(5)



sinβ cos β 0

0 0 1

x − x j y − y j z − z j



(9)

Since the general mathematical description of the HOBB in Eqs. (3)–(8) stemming from Maxwell equations, the electric and magnetic fields of HOBB with arbitrary polarization angles can be expanded in terms of SVWFs in the temporary coordinate system Oj xj  yj  zj  as [54]:

Einc j = E0

∞

n











(1 ) (1 ) Cnm −igmjn,T E Mmn r j , k + gmjn,T M Nmn r j , k

n=1 m=−n

Hinc j = E0

∞ n k0

ω μ0



n=1 m=−n

(1 ) m jn,T M Mmn







(1 ) Cnm gmjn,T E Nmn r j , k



r j , k

(10)

where rj  is the position vector from sphere center Oj , the superscript “inc” indicates the relative parameters of the incident fields, and the subscript indicates the relative parameters of the jth sphere.

Cnm = kCnPW (−1 )(m−|m|)/2

1 − cos α0 2i

1 (n − m )! PW C = (−i )n+1 · 2n + kn(n + 1 ) ( n + m )! n

Following the quadrature formulation in the GLMT [21], the beam shape coefficients can be written as [54]:

(6)

( n − m )! exp(ikz z j ) (n + |m|)!  l−m+1 (l−m+1)φ  jJ i e l−m+1 kρ j sin α0

(m/|m|)/2 m gjn,T M = − g(α0 ) (−1 )

Hy = H0 g(α0 )(−i )l eil ϕ exp[−ikz (z − z )]{(1 + cos α0 )Jl (kt ρ ) 1 − cos α0 2iϕ [e Jl+2 (kt ρ )] + e−2iϕ Jl−2 (kt ρ )] 2



[τnm (cos α0 ) + mπnm (cos α0 )]



(7)

+ il−m+1 e(l−m−1)φ j Jl−m−1 kρ j sin α0



[τnm (cos α0 ) − mπnm (cos α0 )]

Hz = H0 g(α0 )(−i )l eil ϕ exp[−ikz (z − z )] sin α0 [eiϕ Jl+1 (kt ρ )] + e−iϕ Jl−1 (kt ρ )]

=

cos β −sinβ 0

(11)

[e2iϕ Jl+2 (kt ρ )] − e−2iϕ Jl−2 (kt ρ )]

−



+ ig

1 − cos α0 l Ey = E0 g(α0 )(−i ) eil ϕ exp[−ikz (z − z )] 2i [e2iϕ Jl+2 (kt ρ )] − e−2iϕ Jl−2 (kt ρ )]

73

(8)

In which E0 is the electric field strength, α 0 is the socalled half-conical angle of HOBB, kz = kcos α 0 is longitudinal component of the wave number k, with kt = ksin α 0 being the transverse component, and ρ = [(x − x )2 + (y − y )2 ]1/2 , ϕ = tan − 1 [(y − y )/(x − x )]. The function g(α 0 ) is the generalizing function. In particular, when g(α 0 ) = (1 + cos α 0 )/4, the expression in Eqs. (3)–(8) reduce to those of a Davis HOBB used in [57,58]. When g(α 0 ) = 1/2, they reduce to those of an ASR Bessel beam used in [59]. To give readers a better understanding, we give in Fig. 2 the magnitude plots for the components of the electric fields for a zero-order (n = 0) Bessel beam (ZOBB), as well as the first-order (n = 1) and second-order Bessel beam (n = 2). All the beams have the half-cone angle α 0 = 300 and the wavelength λ = 1064nm.

(12)

(n − m )! exp(ikz z j ) (n + |m|)!  l−m+1 (l−m+1)φ  jJ i e l−m+1 kρ j sin α0

(m/|m| )/2 m gjn,T E = ig(α0 ) (−1 )

[τnm (cos α0 ) + mπnm (cos α0 )]



− il−m+1 e(l−m−1)φ j Jl−m−1 kρ j sin α0



[τnm (cos α0 ) − mπnm (cos α0 )] In which:

ρ j =

 2  xj

 2 1/2

+ y j

πnm (cos α0 ) =

φ j  = tan−1



(13) y j x j



Pnm (cos α0 ) m dP m (cos α0 ) τn (cos α0 ) = n sin α0 dα0

(14)

(15)

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J. Bai et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 214 (2018) 71–81

Fig. 2. Theoretical magnitude cross-sectional profiles for the electric field components of the zero-order Bessel beam and the HOBB with different orders. The units in the (x, y) plane are in λ. The half-cone angle of the Bessel beam is set toα 0 = 300 . Fig. 1(a)−(c) Zero-order. Fig. 1(d)−(f) First-order. Fig. 1(g)−(i) Second-order.

Because of the rotation relation between the coordinate systems Oj xj  yj  zj  and Oj xj yj zj , the SVWFs of the two coordinate systems have the following relation [60]: n  

 (1 ) (1 ) rj , k = χ (m, s, n)(M, N)sn r j, k (M, N)mn

(16)

where



ainc jmn binc jmn

 =

n

χ (s, m, n) = (−1 )m+s

s=−n

Substituting (16) into (10), the expansion fields of the HOBB with arbitrary polarization angles can be expressed in the jth sphere coordinate system Oj xj yj zj as:

Einc j = E0

∞

n



 (1 )



 (1 )

m m χ (s, m, n )Cns (−ign,T E , gn,T M )

s=−n



(n + m)!(n − m )! (n + s)!(n − s )!

inc ainc jmn Mmn r j , k + b jmn Nmn r j , k

1 / 2

(n ) ums (−α )

(19)

1 / 2   

n+s (n + m)!(n − m )! n−s ums (−α ) = · n−m−σ σ (n + s)!(n − s )! σ  α 2σ +m+s  −α 2n−2σ −m−s (−1 )n−m−σ cos sin (20) (n )



2



(18)

2

3. Multi-scattering theory

n=1 m=−n

Hinc j = E0

∞ n k0

ω μ0

n=1 m=−n











(1 ) (1 ) inc ainc jmn Nmn r j , k + b jmn Mmn r j , k

(17)

Based on generalized multi-particle Mie equation (GMM) and GLMT [20,61], the interactional scattering between two isotropic

J. Bai et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 214 (2018) 71–81

75

Fig. 3. Comparisons of our results on lateral binding force in thexdirection by a degenerated HOBB with that of a plane wave incidence using the coupled dipole method:(a) θ = 00 polarization; (b): θ = 900 polarization.

spheres and an arbitrary polarized HOBB is considered now. In accordance with the general description of the incident HOBB in Eq. (17), the scattered and internal fields of the jth (j = 1, 2) sphere can also be expanded in terms of SVWFs in the jth sphere coordinate system Oj xj yj zj as:

Esj =

∞

n





n=1 m=−n

Hsj =



 E0 asjmn Nmn (r j , k ) + bsjmn Mmn (r j , k )

ω μ0

E1j = E0

(3 )

(21)

n=1 m=−n

∞

n



(1 ) A1jmn Nmn (r j , k ) +

n=1 m=−n

H1j = E0

(3 )

∞ n k

ωμ n=1 m=−n



k kj

 μj 1 (1 ) B jmn Mmn (r j , k ) μ

k (1 ) B1jmn Nmn (r j , k ) + j k

(23)

Hit

where j and j represent the total electromagnetic fields induced on sphere j. They generally consist of the initial incident fields and the scattered fields radiated from the other sphere p:

=

Einc j

+

Esp, j Hitj

=

Hinc j

+

Esp, j =



E0 aspmn

n=1 m=−n

∞

ν

ν =1 μ=−ν





(1 ) mn (1 ) Amn μν Nμν (r j , k ) + Bμν Mμν (r j , k )

 ∞

υ 

 s mn (1 ) mn (1 ) + b pmn Aμν Mμν (r j , k ) + Bμν Nμν (r j , k ) ν =1 μ=−υ

∞

ν





ν =1 μ=−ν



ν =1 μ=−ν

(1 ) Amn μν Nμν (r j , k )





(1 ) (1 ) Aμν Mμν (r j , k ) + Bmn μν Nμν (r j , k ) mn

(25)

mn where Amn μv and Bμv are the so-called addition theorem coefficients, which are discussed by Xu [62] in detail. After substituting Eqs. (17) and (25) into Eq. (24), the total incident fields for the jth sphere can be derived as follows: ∞

n





(1 ) (1 ) aitjmn Mmn (r j , k ) + bitjmn Nmn (r j , k )

∞ n k



ωμ n=1 m=−n



(1 ) (1 ) aitjmn Nmn (r j , k ) + bitjmn Mmn (r j , k )

(26)

The corresponding expansion coefficients are:

aitjmn = aijmn + bitjmn = bijmn +

∞

v

v=1 μ=−v ∞

v

v=1 μ=−v



μv

μv 

μv

μv 

aspμv Amn + bspμv Bmn ( p = j )



aspμv Bmn + bspμv Amn ( p = j )

(27)

Substituting Eqs. (21), (22) and (26) into the boundary conditions in Eq. (23), and utilizing the expressions of the SVWFs [63], the interactional scattering coefficients of the jth (j = 1, 2) sphere can be obtained by:



(24)

incident fields denoted by M(1) : n

(1 ) + Bμν Mμν (r j , k )

∞

ν

asjmn = a jn aitjmn

Hsp, j

where j = 1, 2, p = 1, 2, p = j, and Einc and Hinc are the initial j j incident electromagnetic fields, Esp, j and Hsp, j are the incident electromagnetic fields at sphere j coordinate system which are transformed from the scattered fields at sphere p coordinate system. On basis of the addition theorem of SVWFs [18], the scattered fields Esp, j and Hsp, j denoted by M(3) can be converted to the relative ∞

n=1 m=−n

mn

Hitj = E0

where the superscript “s” and “1” respectively indicate the relative parameters of the scattered fields and the internal fields, kj and μj are the wave number and permeability of the jth sphere. For the jth sphere, the tangential boundary conditions at rj = aj are written as:

Eitj



E0 bspmn

n=1 m=−n

(22)

Eit

ω μ0

Eitj = E0

 μ 1 (1 ) A M (r , k ) μ j jmn mn j

      E1j  = Eitj  + Esj  , H1j  = Hitj  + Hsj  t t t t t t

∞ n k0

+ a spmn

(3 ) (3 ) E0 asjmn Mmn (r j , k ) + bsjmn Nmn (r j , k )

∞ n k0

Hsp, j =

= a jn aijmn +

∞

v

v=1 μ=−v



μv 

μv

aspμv Amn + bspμv Bmn ( p = j )

bsjmn = b jn bitjmn



= b jn

bijmn

+

∞

v

v=1 μ=−v





μv aspμv Bmn

+

μv bspμv Amn



 ( p = j )

(28)

where ajn and bjn are the scattering coefficients of a sphere to the plane wave, namely Mie scattering coefficients [64]. 4. Derivation of binding force According to the classical electromagnetic field theory, laser beams carry energy and momentum. Assuming a steady-state con-

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dition, when a strongly convergent laser beam impinges on a bisphere system, part of the momentum will be transferred from the laser beam to the particle due to the process of scattering. It will generate a force (BF) that results from the exchange of momentum between the light and objects and between the objects mutually via re-scattered light, which can be represented through BF received by the particles [21,65].

F=

1 Re 2

 2π  π 0

0

 1 ε0 Er E + μ0 Hr H − (ε0 E 2 + μ0 H 2 )rˆ dS 2

(29)

where ε 0 and μ0 pertain to the permittivity and permeability of the surrounding medium respectively, dS denotes the surface element enclosing the sphere. The electromagnetic fields E and H indicate the superposition of the incident and scattered electromagnetic fields:E = Eitj + Esj , H = Hitj + Hsj . The numerical calculation of the duplicate integral in Eq. (29) can be avoided by substituting the series expressions of electromagnetic field components of Eqs. (21) and (26), which process used here is the same as those used in Ref. [32]. After making use of the orthotropic relations of associated Legendre functions and trigonometric functions [66], the lateral BF between bi-sphere can be expressed as a series over the expansion coefficients of the totally incident and scattered fields:

with the wavelength λ = 0.488 μm in Fig. 3(a) and (b) respectively. The two isotropic spheres (radiusλ/50) are assumed the same with n = 1.414 in the air, one of which is located at the origin of the primary coordinate system Oxyz, the other is separated from it by a distance of d along x-axis. If the beam order r equals zero and the conical angle α 0 decreases, then the radius of the Bessel beam increases and, in the limit kz → k0 , the incident HOBB will be degenerated to a plane wave which propagates along z-axis. On the plots below, the red and black curves are associated with particles 1 and 2 induced by a HOBB respectively; the red and black solid dots denote the relevant results of a plane wave incidence in the reference [67]. As expected, our results in the case of a degenerated HOBB beam incidence are in excellent agreement with that of a plane wave incidence. This consistency can verify the validity of our theory and codes. It is obvious that the forces on sphere 1 and sphere 2 are equal but of opposite sign to each other because of the symmetry of structures. Moreover, depending on the polarizations, the force between small particles at small distances changes its sign. For θ = 00 polarization, the short range optical force is attractive whereas it is repulsive for θ = 900 polarization. This is determined by the interaction between the dipoles related to one sphere and to the variation of the field created by the other

⎧ ⎫  ∞  n −1 −1 ⎪ ⎪ [ (n − m )(n + m + 1 )Nmn Nm+1n (aitjmn bsjm∗ +1n + bitjmn asjm∗ +1n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n=1 m=−n ⎪ ⎪ s it∗ s it∗ s s∗ s s∗ ⎪ ⎪ + a a + a b + 2 a b + 2 b a ) ⎪ ⎪ jmn jm+1n jmn jm+1n jmn jm+1n jmn jm+1n ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( n − m − 1 )( n − m ) ⎨ −i ⎬ −1 −1 it s∗ (n − 1 )(n + 1 )Nmn Nm+1n−1 (a jmn a jm+1n−1 n0 I0 ( 2 n − 1 )( 2 n + 1 ) Re Fx = π ck20 ⎪ ⎪ ⎪ +bitjmn bsjm∗ +1n−1 + asjmn ait∗ + bsjmn bit∗ + 2asjmn asjm∗ +1n−1 + 2bsjmn bsjm∗ +1n−1 ) ⎪ ⎪ ⎪ jm+1n−1 jm+1n−1  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ( n + m + 1 )( n + m + 2 ) ⎪ ⎪ −1 −1 it s ∗ ⎪ −i ⎪ n(n + 2 )Nmn Nm+1n+1 (a jmn a jm+1n+1 ⎪ ⎪ ⎪ ⎪ ( 2n + 1 )(2n + 3 ) ⎪ ⎪ ⎩ it s∗ ⎭ s it∗ s s∗ s s∗ +b jmn b jm+1n+1 + asjmn ait∗ + b b + 2 a a + 2 b b ) ] jmn jm+1n+1 jmn jm+1n+1 jmn jm+1n+1 jm+1n+1 where c is the speed of light in vacuum, k = 2π /λ is the wave number, I0 = k0 π E02 /(4ωμ0 ) denotes the intensity of the incident HOBB, and



Nmn =

(2n + 1 )(n − m )! ( m = 0, ± 1, · · · , ± n ) 4 π ( n + m )!

(31)

We also remark that, in what follows, when we represent the normalized force, that means F/(4π ε 0 |E0 |2 ), where ε 0 is the permittivity of vacuum and |E0 | denotes the electric field strength. 5. Numerical analysis In order to illustrate our methods, we consider two identical isotropic spheres with the same size and material composition induced by a HOBB of arbitrary order and polarization. The beam propagates along + z axis and is focused at the position(0, 0, 0). In all the calculations, we set the intensity of HOBB I0 going through the focal plane to be unit. Numerical effects of various parameters with respect to beam conical angle, incident wavelengths, particle sizes, material losses and the complex index of refractive are numerically analyzed in detail. Some numerical results for the BF experienced by each particle as a function of the distance between them are shown. To validate the accuracy of the computations, some results are selected to compare with those of a plane wave incidence provided in [67], which put forward the coupled dipole method to calculate BF between two small particles illuminated by a plane wave whose wave vector is normal to the axis of particles. Considering the same case as shown in Fig. 3, the distributions of normalized lateral BF experienced by each particle with the varying inter-particle separation induced by a HOBB are given. We calculate θ = 00 and θ = 900 polarizations of the incident beam

(30)

sphere. Chaumet [68] has geometrically explained that such binding and anti-binding states can be interpreted as associated with the same phase and the opposite phase between the field due to one sphere at the position of the other sphere and the dipole of the other sphere. The similar effect can also be observed in an atom mirror, or on a small silver particle in an evanescent field [69]. Besides, we also notice that the force in θ = 00 polarization decreases faster than that in θ = 900 polarization, this is attributed to the absence of a propagating field along the x-axis in the far field. Fig. 4 presents the lateral BF exerted on two identical particles of r1 = r2 = 250nm and n1 = n2 = 1.59 with various polarization angles θ (00 , 450 , 900 ) in water irradiated by a focused first-order Bessel beam at some specific wavelengths. In this case, the two identical particles are assumed to be located at the central zone of beam symmetrically. Since the forces on sphere 1 are equal but of opposite sign to the sphere 2 because of the symmetry of structures, we only consider the results of lateral BF Fx exerted on the sphere 2. In the case of 00 , 900 polarization, the electric field is oriented parallel, perpendicular to the connecting line between particles, respectively. As shown in Fig. 4, the stable equilibrium positions of bi-sphere occur at the inter-particle distances corresponding to null force with negative slopes. And the first stable equilibrium position is the most stable one, since it has the largest negative slope under the same influence of irradiation. The optical BF can be regarded as the restoring force of a virtual spring, which performs attractive and repulsive forces as particles moving away or closer to from each other around these stable equilibrium positions [70]. On the contrary, the other null force distances with a positive slope are the unstable equilibrium positions, since they equivalent to a spring with negative spring constants. For the con-

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77

Fig. 4. Lateral binding force acting on particle 2 depending on the polarizations and the incident wavelengths induced by a first-order Bessel beam. (a): Incident wavelength λ = 1064nm (b): Incident wavelengthλ = 808nm (c): Incident wavelength λ = 632.8nm.

sidered wavelengths, it is interesting to find that the smaller the wavelength, the faster the oscillation. This is due to the fast oscillations with decreasing amplitude correspond to the direct lateral binding between particles and their period equals approximately to the wavelength of the incident beam in water (λ/n). Besides, the amplitude of these stable oscillations is higher for 900 polarized incident first-order Bessel beam because the induced dipoles in both small particles are oriented such that they scatter the wave strongly towards the other particle. This gives stronger optical interaction between them, resulting in greater lateral BF, indicating that the 900 polarized HOBB can trap and manipulate bi-sphere system faster and stabler than other polarization angles do. Fig. 5 plots the lateral BF exerted on two identical particles (r = 250nm, n = 1.59) embedded in water by a focused 1064-nm HOBB, where the polarization angle is θ = 900 . The influence of beam orders on lateral BF exerted on sphere 2 versus various interparticle distances d are given in Fig. 5(a), and (b) depicts the corresponding intensity profiles. For this case, we fix α 0 = 300 and change the orders of HOBB to n = 0, 1, 2, 3 respectively. In theory, if we fix α 0 as a constant, radius of the first bright ring increases with the increment of the beam order [71]. As shown in Fig. 5(a), When order ν increases (for ν > 0), the interval of equilibrium points becomes larger, and the number of points will decrease, which indicates that the maximum number of particles that can

be bounded will decrease. This is the result of the interval of intensity peaks increases due to the increasing of ν as in Fig. 5(b). In addition, a zero-order Bessel beam has central spot surrounded by concentric rings, as in Fig. 2(a), while all high-order Bessel beams have zero central amplitude, as shown in Fig. 2(d) and (g), causing the effect of the transverse gradient force induced by a zeroorder Bessel beam to be much greater than that of the HOBB [72], since the gradient force towards the center of Bessel beam pushes the bi-sphere system to the center, while the resorting behavior of the lateral BF exhibits the repulsive one, which may break the stability of the bi-sphere system. Hence, the lateral BF induced by a zero-order Bessel beam is less than that by a HOBB. Besides, it’s clear from Fig. 5(b) that the larger orders ν will decrease the intensity peaks, causing the amplitude and the stable negative slopes of the BF decreased with the increasing of ν . Indicating the firstorder Bessel beam can trap and manipulate bi-sphere system faster and stabler than the other HOBBs do. The angle of conical α 0 is a key parameter for producing Bessel beams, and will greatly affect the BF. Similar to Fig. 6, the effects of the varying beam conical angles on lateral BF induced by a focused 1064 nm first-order Bessel beam are shown in Fig. 6, where the polarization angle is θ = 900 . Two identical particles of r = 250nm with n = 1.59 embedded in water are taken into account. It is clear that a zero-order Bessel beam is termed as a non-vortex

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Fig. 5. (a): Lateral binding force acting on sphere 2 induced by a high-order Bessel beam with orderν as parameter. (b): Intensity profile of a high-order Bessel beam with orderν as parameter.

Fig. 6. (a). Lateral binding force induced by a first-order Bessel beam exerted on sphere 2 depending on the beam conical angles. (b): Intensity profile of a first-order Bessel beam with conical angle α 0 as parameter.

beam and it has a very narrow non-diffracting central core (bright central spot) and the central peak of the intensity maintains, as in Fig. 2(a). However, a HOBB propagates over a characteristic length without spreading (dark central region) and the doughnut shape is conserved, as shown in Fig. 2(d) and (g). The size of the central spot highly depends on the half-cone angle. The smaller half-cone angle, the larger the size of the central spot (bright for zero order and dark for first order and vice versa). To study the influence of the half-cone angle, we set α 0 increasing from 150 to 400 , and the other parameters remain unchanged. The computed BF components are shown in Fig. 6(a). And the corresponding intensity profiles are given in Fig. 6(b). Again the equilibrium positions with respect to the zero force and negative derivatives of the BF are observed in all the cases, as in Fig. 6(a), the corresponding spring constants with regard to small conical angles are larger than those of large conical angles, especially for the first and second stable equilibrium positions. Since it’s clear from Fig. 6(b) that the larger conical angles α 0 will decrease the intensity peaks, causing the amplitude and the stable negative slopes of the BF decreased with the increasing of α 0 . Besides, Fig. 6(a) also demonstrates that the stable equilibrium positions decrease as the conical angle of the beam decreases. These results show that particles with larger conical angles incidence are much more localized. This is due to the interval of intensity peaks decreases due to the increasing of α 0

Fig. 7. Lateral binding force induced by a first-order Bessel beam exerted on sphere 2 with the particle size r as parameter.

as in Fig. 6(b). Hence, a smaller conical angle is necessary to provide sufficient lateral BF overcoming the gradient force to stabilize the stable equilibrium, while a larger conical angle will make the maximum number of particles that can be bounded increase.

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Fig. 8. Lateral binding force induced by a first-order Bessel beam on sphere 2 depending on the spherical refractive index including the cases of weak, moderate, and strong than the surrounding medium.

Fig. 7 plots the results for varying particle sizes induced by a focused 1064 nm first-order Bessel beam, where the polarization angle is θ = 900 . Again, the results of lateral BF exerted on sphere 2 with n = 1.59 for various inter-particle distances d are considered. For all particle sizes we observe the usual oscillations of the force that result from the interference between the fields scattered by the particles (retardation effect) as the distance between them is varied, while the amplitude of the oscillation is directly related to the strength of the binding. It is clear from the insets that the larger radius can increase the short-range multistability (discussion on multi-stability can be found in [73,74], i.e., increasing the number of stable configurations. One bottleneck of optical binding is that it is difficult to bind a large number of particles; instability seems to emerge if the number of optically bounded particles increases beyond a certain threshold. We anticipate that they can also increase the maximum number of particles that can be bounded. Besides, the corresponding spring constants of the smaller bi-sphere at the first stable equilibrium position are less than those of a bigger bi-sphere, as shown in Fig. 7. We can conclude that the closer the distance between the spheres, the stronger the interactions between them in reality. Moreover, the amplitude and the spring constants of a bigger bi-sphere system are always higher and larger than those of a smaller one, especially for the first stable equilibrium distance. It means that the stability of the former is better than the latter. This is due to the two identical particles trapped inside the line tweezer undergo thermally induced Brownian motion, which is more vigorous for the smaller particles. Hence, a larger-sized bi-sphere system is necessary to provide sufficient lateral BF overcoming the gradient forces of HOBB to stabilize the stable equilibrium. Fig. 8 illustrates the lateral BF produced by highly focused firstorder Bessel beam with the polarization angle θ = 900 on highindex and low-index particles. We study theoretically how the distance between spheres in the stable configuration depends on their refractive index. Selected results presented in Fig. 8 show the BF acting on sphere 2 as a function of the inter-particle distance. We choose the radius of the sphere as r = 250nm, the refractive index of the ambient as nm = 1.33 (water). Three refractive indices nh are taken for high-index particles whose refractive indices are stronger than the surrounding medium, while three low-index refractive indices nl are chosen weaker than the ambient. One can see that the larger contrast between the refractive index of particles with respect to the refractive index of background, the stronger the BF between particles and the larger the corresponding spring constants. This effect can be directly understood from the Born ap-

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Fig. 9. Lateral binding force induced by a first-order Bessel beam exerted on sphere 2 depending on the material losses.

proximation: the scattered field is directly proportional to |n − nm | and is therefore reduced in a low-refractive contrast system. Besides, when the same contrast between the particles with respect to the background, for high-refractive index particles, the interval of the equilibrium inter-particle separation decrease and the number of stable points will increase, this indicates that the maximum number of particles that can be bounded will increase. And the corresponding spring constants of the stable equilibriums for highrefractive particles are larger than those of the low-refractive particles under the same contrastive effect. Hence, a higher refractive index compared with the surrounding medium is necessary to provide sufficient lateral BF overcoming the Brownian motion to stabilize the steady equilibrium. Fig. 9 presents the effects of the varying material losses on lateral BF induced by a focused 1064-nm first-order Bessel beam, where the polarization angle is θ = 90◦. Two identical particles of r = 250 nm embedded in water are taken into account. In addition, the real part of the spherical refractive index is Re(n ) = 1.59. To study the influence of the material losses, we set the losses increasing from 0.05 to 0.5, and the other parameters remain unchanged. We also consider the lateral BF exerted on sphere 2 for various inter-particle distancesd. As shown in Fig. 9, when the material loss increases, the interval of the equilibrium points decrease, and the number of points will increase, which indicates that the maximum number of particles that can be bounded will increase. Besides, the corresponding spring constants of the stable equilibriums for large material losses are larger than those for cases of small material losses, and the overall effect of losses is to increase the magnitude of the BF. This is due to the increase of absorbed photons as the spherical material loss increases, causing the effect of scattering contributions to the binding force [75]. 6. Conclusion In summary, an analytical solution of the lateral BF exerted on bi-sphere induced by a normal illuminating HOBB is derived. The accuracy of the theory is verified by comparing our numerical results reduced to the special case of a plane wave incidence with the theoretical treatment developed under dipole approximation given by references. The effects of various parameters such as beam conical angle, beam polarization state, incident wavelength, particle size, material loss and the refractive index, including the cases of weak, moderate, and strong than the surrounding medium are numerically analyzed in detail. Results show that there are sev-

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