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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–65

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Light scattering of a non-diffracting zero-order Bessel beam byuniaxial anisotropic bispheres

Z.J. Li a,n, Z.S. Wu a, T. Qu a, H.Y. Li a, L. Bai a, L. Gong b

a School of Physics Optoelectronic Engineering, Xidian University, Xi'an, Shaanxi 710071, Chinab School of Photoelectric Engineering, Xi'an Technological University, Xi'an 710021, China

a r t i c l e i n f o

Article history:Received 21 September 2014Received in revised form19 January 2015Accepted 30 January 2015Available online 9 February 2015

Keywords:Bessel beamScatteringAnisotropic mediumSpherical vector wave function

x.doi.org/10.1016/j.jqsrt.2015.01.02673/& 2015 Elsevier Ltd. All rights reserved.

esponding author.ail address: lizj@xidian.edu.cn (Z.J. Li).

a b s t r a c t

Based on the generalized multi-particle Mie theory and the Fourier transformationapproach, light scattering of two interacting homogeneous uniaxial anisotropic sphereswith parallel primary optical axes illuminated by a zero-order Bessel beam (ZOBB) isinvestigated. The size and configuration of the particles are arbitrary. The expansionexpressions of the ZOBB are given in terms of the spherical vector wave functions (SVWFs)and the expansion coefficients are derived. Utilizing the vector addition theorem of theSVWFs, the interactive scattering coefficients are derived through the continuousboundary conditions on which the interaction of the bispheres is considered. The effectsof the conical angle, beam centre position, sphere separation distance, and anisotropicparameters on the far-region field distributions are numerically analyzed in detail. Someresults are compared with those results for a Gaussian beam incidence. Selected results ofbispheres consisting of typical medium such as TiO2, SiO2, Silicon, water are exhibited.This investigation could provide an effective test for further research on the scatteringcharacteristic of an aggregate of anisotropic spheres by a high-order Bessel vortex beamand radiation forces, which are important in optical tweezers and particle manipulationapplications.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Because of its non-diffraction and self-reconstructionproperty, Bessel beam has gained growing attentions sinceits naissance by Durnin [1] and has widely applied tovarious fields, such as optical trapping and manipulation,particle sizing, optical guiding and alignment [2–5]. As alaser beam, the profile and representation in spherical andcylindrical coordinate system have been studied by manyscholars [6–11]. Based on the representation, the scatter-ing problem of a Bessel beam by spherical particles hasbeen investigated by many scholars. Marston [12] has

studied the exact scattering of a Bessel beam by a sphericalparticle using plane-wave decomposition. Ma et al. [13]investigated an unpolarized Bessel beam scattering byspherical particle. The incident Bessel beam is regardedas superposition of plane waves proposed by Cizmar et al.[8] and expanded in terms of SVWFs. It is actually anaxicon-generated Bessel beam. Based on Generalized Lor-enz–Mie theory (GLMT) [14], Li et al. [15] studied thescattering of a spherical particle illuminated by thisaxicon-generated Bessel bam. The incident beam isexpanded using an integral localized approximation(ILA). Applying the ILA, Ambrosio [10] also gave anexpansion coefficient in terms of SVWFs. However, theBessel beam is described by a scalar wave theory, whichprovides satisfactory results only if the conical angle isvery small. Mishra [11] derived a vector wave theory to

Fig. 1. Geometry for light scattering of a Bessel beam by uniaxialanisotropic bispheres, and α¼51.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–65 57

describe the Bessel beam for arbitrary conical angle. Usingthis vector description, Mitri investigated scattering of aZOBB [16] and a high-order Bessel vortex beam [17] by adielectric sphere, and the radial electric fields decaying inproportion to (1/r)2 are evaluated by plotting a 3D-directivity patterns. Using an analytical method, the scat-tering of an on-axis incident ZOBB by a concentric sphereis investigated by Chen et al. [18]. Utilizing a numericalmethod, Cui [19] studied the scattering of a ZOBB by anarbitrarily shaped homogeneous dielectric particle.Recently, Qu et al. [20] investigated scattering of an off-axis ZOBB by a uniaxial anisotropic sphere. Numericalsimulations show that there also are extremum points inthe direction or neighboring direction of the conical anglefor uniaxial anisotropic sphere on the beam axis, and theexistence of the extremum points depends on the ratio ofsphere radius to the size of the central spot of the Besselbeam. However, for most papers to which we havereferred, the investigations focus on the scattering ofBessel beam by single isotropic sphere. The study onanisotropic particle is exiguous.

Due to recent advances in material science and tech-nology etc., a great interest in the interaction between EMwave and anisotropic media is being grown. Using differ-ential theory, Brian [21] gave the solution of the scatteringof an arbitrary-shaped body made of arbitrary anisotropicmedium. Wong [22] and Qiu [23] devoted their endeavorsto the scattering of a uniaxial anisotropic sphere. Byintroducing the Fourier transformation method, Geng[24] studied the scattering of a uniaxial anisotropic sphere.Recently, Wu et al. [25,26] studied the scattering charac-teristic of a uniaxial anisotropic sphere illuminated by on-axis, off-axis, and arbitrarily incident Gaussian beams.However, the scatterer in all these studies is limited to asingle anisotropic object.

Since the first presentation of a comprehensive solutionof scattering by a two-sphere chain by Bruning and Lo [27],the study of scattering of multiple isotropic spheres hasbeen developed quickly. Fuller and Kattawar [28] introducedthe order-of-scattering technique to obtain the consummatesolution of electromagnetic (EM) scattering by a cluster ofspheres. The T-Matrix approach is also a very effectivemethod and has been applied to the study of this problemby many researchers [29–31]. Xu introduced the generalizedmulti-particle Mie theory (GMM) to explore EM scatteringby an aggregate of isotropic spheres [32,33]. However, inmost of the papers that we have referred, the investigationsfocus on a plane wave scattering by multiple isotropicspheres. Recently, the authors [34,35] have studied the EMscattering multiple uniaxial anisotropic spheres. The scatter-ing of uniaxial anisotropic bispheres located in a focusedGaussian beam has also been studied [36]. Since the Besselbeam has a very special trait comparing with plane waveand Gaussian beam, the interaction of a Bessel beam withmultiple anisotropic spheres will show different and inter-esting characteristics and is one of considerable merit.Inasmuch as the ZOBB can be easily realized in the labora-tory; here we only consider the scatting problem of a ZOBBby uniaxial bispheres.

In this paper, we focus on the interaction of a ZOBB anduniaxial anisotropic bispheres. In Section 2, two kinds of

descriptions on the profile of a ZOBB are presented.Section 3 derived the analytical solution of the scatteringproblem of a ZOBB by two uniaxial anisotropic sphericalparticles. Section 4 is devoted to numerical results anddiscussions. Finally, a conclusion is shown in Section 5. Inaddition, a time dependence of the form exp(� iωt) isassumed and suppressed in the subsequent depiction,where ω is the circular frequency.

2. Describe of the Bessel beam

Considering two different uniaxial anisotropic sphereswith radius aj (j¼1, 2) and the primary optical axescoincident with the z-axis in a global coordinate systemOxyz, which is also the beam coordinate system, as shownin Fig. 1, the centres of the spheres O1 and O2 are located at(x10,y10,z10) and (x20,y20,z20), respectively. The particlecoordinate systems O1x1y1z1 and O2x2y2z2 are parallel tobeam system Oxyz. The particles are illuminated by a ZOBBpropagating along the z-axis.

In an isotropic, dielectric, non-magnetic homogeneousmedium, a scalar wave equation in a cylindrical coordinatesystem can describe the electric field of a ZOBB:

∇2Eðr; tÞ� 1c2

∂2

∂t2Eðr; tÞ ¼ 0 ð1Þ

Solving Eq. (1) and using the property of cylindricalBessel function, the scalar approximation of the electricfield of the ZOBB in Oxyz can be written as

Eðr; tÞ ¼ E0J0ðkRRÞeikzze� iωt ð2Þwhere J0( � ) is the cylindrical Bessel function of the firstkind of the zero order, E0 is the characteristic amplitude,kR ¼ k0 sin α is the transverse wave number, kz ¼ k0 cos αis the longitudinal wave number, α is so-called the conicalangle of the ZOBB, R¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þy2

p, and k0 is the wave number

in the medium.Generally, the scalar expression of the ZOBB described

by Eq. (2) can provide fairly good results only if the conicalangle is very small, i.e. αr101. In other words, when theconical angle is very large, such as when kREk0, the scalarexpressions prove to be inaccurate to describe such a non-diffracting Bessel beam. However, Mishra [11] was

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–6558

cognizant of this problem and gave a very significantcorrection by taking the vector nature of EM wave propa-gation into account.

Therefore, for an EM wave propagating in an isotropic,dielectric, nonmagnetic homogeneous medium, a vectorpotential A may be defined to describe the EM fields:

Hffiffiffiffiffiffiffiffiffiffiffiffiffiμ0=ε0

p¼∇� A

E¼ ik0½Aþ∇ð∇UAÞ=k20� ð3Þwhere ε0 and μ0 are the permittivity and permeability ofthe medium, respectively, and A satisfies the Helmholtzvector equation. After some arithmetic manipulations, theEM components of the ZOBB propagating along the z-axisalong its own Cartesian coordinate system Oxyz can bederived [11,16]:

E¼ 12E0eikzz

(1þkz

k0�k2Rx

2

k20R2

!J0ðkRRÞ� J1ðkRRÞ

kRðy2�x2Þk20R

3

" #:ex

þxy J1ðkRRÞ2kRk20R

3� k2Rk20R

2J0ðkRRÞ

" #ey

þ xik0R

1þkzk0

� �kRJ1ðkRRÞez

)ð4Þ

H¼ 12E0

ffiffiffiffiffiffiffiffiffiffiffiffiffiε0=μ0

qeikzz

(xy

2kRk20R

3J1ðkRRÞ�

k2Rk20R

2J0ðkRRÞ

" #:ex

� J1ðkRRÞkRx2�y2

k20R3

þ 1þkzk0

� y2

k20R2k2R

!J0ðkRRÞ

" #ey

þkRyikR

1þkzk0

� �J1ðkRRÞez

)ð5Þ

where ex, ey and ez are the unite vector in Oxyz. Eqs. (4)and (5) are the EM fields in the beam coordinate systemOxyz. According to the relationship between the Oxyz andthe particle coordinate system Ojxjyjzj (j¼1, 2):

xjþxj0yjþyj0zjþzj0

264

375¼

x

y

z

264375 ð6Þ

The EM fields can be transformed to the particle systemOjxjyjzj (j¼1, 2).

3. Multiscattering theory

Now the interaction between uniaxial anisotropicbispheres and a ZOBB is considered. It begins from theexpansion of the incident fields. Since the vector descrip-tion of the ZOBB in Eqs. (4) and (5) stem from Maxwellequations, the ZOBB can be expanded in terms of SVWFs inthe particle coordinate system Ojxjyjzj (j¼1, 2) as

Eincj ¼

X1n ¼ 1

Xnm ¼ �n

Emn aincjmnMð1Þmn rj; k0� �þbincjmnN

ð1Þmn rj; k0� �h i

Hincj ¼ k0

iωμ0

X1n ¼ 1

Xnm ¼ �n

Emn bincjmnMð1Þmn rj; k0� �þaincjmnN

ð1Þmn rj; k0� �h i

ð7Þwhere rj is the position vector from sphere centre Oj, thesuperscript “inc” indicates the relative parameters of the

incident fields, and the subscript indicates the relativeparameters of the jth sphere. MðlÞ

mnðr; kÞ and NðlÞmnðr; kÞ are

the SVWFs, and l¼1, 2, 3, or 4 represent four kinds ofspherical Bessel functions in the SVWFs, and

Emn ¼ E0in ð2nþ1Þðn�mÞ!nðnþ1ÞðnþmÞ!

� �1=2ð8Þ

By virtue of Eq. (7), the radial components of theincident EM fields can be written as

Eincr ¼X1n ¼ 1

Xnm ¼ �n

Emnbincjmnnðnþ1ÞjnðkrÞ

krPmn ð cos θÞeimϕ

Hincr ¼ k0

iωμ0

X1n ¼ 1

Xnm ¼ �n

Emnaincjmnnðnþ1ÞjnðkrÞkr

Pmn ð cos θÞeimϕ

ð9ÞAccording to the orthogonality of the associated

Legendre function and exponential function:

R 2π0 exp imϕ

� �exp � im0ϕ

� �dϕ¼

2π; m¼m0

0; mam0

(Z π

0Pmn cos θ� �

Pml cos θ� �

sin θ dθ¼ 22nþ1

nþmð Þ!n�mð Þ!δnl ð10Þ

and assuming the incident EM fields for which the radialcomponents of the electric and magnetic fields are knownover a spherical surface of radius aj. The expansioncoefficients are derived as [37]

aincjmn

bincjmn

0@

1A¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiμ0=ε0

p1

!ð�1ÞnEmnðkajÞ2

4πψnðkajÞZ 2π

0

Z π

0

iHincrj ¼ aj

Eincrj ¼ aj

0@

1APm

n cos θ� �

e� imϕ sin θ dθ dϕ ð11Þ

where Eincr and Hincr are the radial EM fields of the original

ZOBB in particle coordinate system, which can be obtainedfrom Eqs. (4) to (6) by utilizing the transformation rela-tionship er ¼ sin θ cos ϕexþ sin θ sin ϕeyþ cos θez.

The scattered fields of the jth (j¼1, 2) sphere can alsobe expanded in terms of SVWFs in the jth sphere coordi-nate system Ojxjyjzj as

Esj ¼

X1n ¼ 1

Xnm ¼ �n

Emn asjmnMð3Þmnðrj; k0ÞþbsjmnN

ð3Þmnðrj; k0Þ

h i

Hsj ¼

k0iωμ0

X1n ¼ 1

Xnm ¼ �n

Emn asjmnNð3Þmnðrj; k0ÞþbsjmnM

ð3Þmnðrj; k0Þ

h ið12Þ

where the superscript “s” indicates the relative parametersof the scattered fields. For uniaxial anisotropic medium,the permittivity and permeability should be characterizedby tensors εj and μj, expressed as

εj ¼ ε0

εjt 0 00 εjt 00 0 εjz

264

375; μj ¼ μ0

μjt 0 00 μjt 00 0 μjz

264

375: ð13Þ

Since the permittivity and permeability are tensor, theelectric field and magnetic field are coupled. Thus it is verydifficult to expand the EM field in anisotropic mediumcompared with those in isotropic medium. Fourier

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–65 59

transformation method, due to its briefness for a uniaxialanisotropic medium, is introduced to expand the internalfields in the jth sphere in terms of the SVWFs as [35,36]

EIj ¼

X2q ¼ 1

X1n ¼ 1

Xnm ¼ �n

X1n0 ¼ 1

2πGjmn0q

Z π

0½Ae

jmnqMð1Þmnðrj; kjqÞþBe

jmnqNð1Þmnðrj; kjqÞ

þCejmnqL

ð1Þmnðrj; kjqÞ�Pm

n0 ð cos θkj Þk2jq sin θkj dθkj ;

HIj ¼

X2q ¼ 1

X1n ¼ 1

Xnm ¼ �n

X1n0 ¼ 1

2πGjmn0q

Z π

0½Ah

jmnqMð1Þmnðrj; kjqÞþBh

jmnqNð1Þmnðrj; kjqÞ

þChjmnqL

ð1Þmnðrj; kjqÞ�Pm

n0 ð cos θkj Þk2jq sin θkj dθkj : ð14Þ

where Aejmnq; Be

jmnq; Cejmnq; Ah

jmnq; Bhjmnq and Ch

jmnq are theexpansion coefficients and their expressions can be foundin [24], but the expressions should be coincident with thecorresponding permittivity tensor εj and permeability μjfor each uniaxial anisotropic sphere. Gjmn0q is the unknownexpansion coefficient relative to the internal fields and isdetermined by the boundary conditions. Similar as Mð1Þ

mnand Nð1Þ

mn, Lð1Þmn is also the SVWFs:

Lð1ÞmnðkrÞ ¼djnðkrÞdr

Pmn ð cos θÞeimϕerþ

jnðkrÞr

dPmn ð cos θÞdθ

eimϕeθ

þ imjnðkrÞr

Pmn ð cos θÞsin θ

eimϕeϕ ð15Þ

Applying the boundary conditions and the additiontheorem of the SVWFs, the interacting scattering coeffi-cients can be derived as [35,36]

asjmn ¼1

hð1Þn ðk0ajÞ1

Emn

X2q ¼ 1

X1n0 ¼ 1

2πGjmn0q

"

Z π

0AejmnqjnðkjqrjÞPm

n0 ð cos θjkÞk2jq sin θjkdθjk� f itjmnjnðk0ajÞ#

ð16Þ

bsjmn ¼1

hð1Þn ðk0ajÞiωμ0

k0

1Emn

X2q ¼ 1

X1n0 ¼ 1

2πGjmn0q

"

Z π

0AhjmnqjnðkqrjÞPm

n0 ð cos θjkÞk2jq sin θjk dθjk�gitjmnjnðk0ajÞ#

ð17Þwhere f itjmn and gitjmn are the expansion coefficients of thetotal incident field illuminating the jth uniaxial anisotropicsphere [35].

Knowing the scattering coefficients, the total scatteredfields, which are the vector superposition of the scatteredfield of every uniaxial anisotropic sphere, can be derivedby applying the addition theorem of the SVWFs again. Thetotal scattered fields in the particle coordinate systemO1x1y1z1, which would be assumed as the primary coordi-nate system here, can be derived:

Est ¼X1n ¼ 1

Xnm ¼ �n

Emn astmnMð3ÞmnþbstmnN

ð3Þmn

h i

Hst ¼ k0iωμ0

X1n ¼ 1

Xnm ¼ �n

Emn astmnNð3ÞmnþbstmnM

ð3Þmn

h ið18Þ

where astmn and bstmn are the total scattering coefficients.Utilizing the approximate forms of SVWF's when r-1,

the total scattering coefficients can be written simplyas [33]

astmn ¼ as1mnexpð� ik0Δ1Þþas2mnexpð� ik0Δ2Þbstmn ¼ bs1mnexpð� ik0Δ1Þþbs2mnexpð� ik0Δ2Þ ð19ÞwhereΔj ¼ xj sin θ cos ϕþyj sin θ sin ϕþzj cos θ ðj¼ 1; 2Þ.

With these solved coefficients, the field components ofthe total scattered, transmitted, and incident fields can beobtained by corresponding substitutions. Then we canpresent some numerical results for the scattering of theuniaxial anisotropic bispheres illuminated by a ZOBB. Theresults to be presented are in terms of radar cross section(RCS), which is defined as

σ ¼ limr-1

4πr2 Est 2= Ei 2� �

¼ 4πk20

X1n ¼ 1

Xnm ¼ �n

Emnð� iÞneimϕ mastmnPmn ð cos θÞsin θ

þbstmndPm

n ð cos θÞdθ

� �2

8<:

þX1n ¼ 1

Xnm ¼ �n

Emnð� iÞnþ1eimϕ astmndPm

n ð cos θÞdθ

þmbstmnPmn ð cos θÞsin θ

� �29=;

ð20Þwhere Εi indicates the initial incident electric field. With-out loss of generality, the amplitude of the electric field forthe incident beam is assumed to be unity.

4. Numerical analysis

In this section, some numerical results for the scatter-ing of a ZOBB by uniaxial anisotropic bispheres are shown.The E-plane corresponds to the xOz-plane and the H-planecorresponds to the yOz-plane. The double integration iscalculated through a Gaussian integration method. Andthe iterations are carried out until the residual error isreduced to below 0.000001. In our code, we adopt theformula Nstop¼xþ4x1/3þ2 to terminate the infinite ser-ies. This truncation formula and conclusion can be foundin [38], where x is the particle size of every anisotropicsphere. When the particle sizes of the two spheres are notsame, we adopt the maximal Nstop to terminate theinfinite series. Generally, it is enough to assure the con-vergence of the infinite series.

To validate the accuracy of the computations, someresults are selected to compare with the results in the caseof a plane wave incidence provided in [35]. As shown inFig. 2, the angular distributions of the RCS of two close-packed uniaxial anisotropic spheres along the z-axis withboth electric and magnetic anisotropy for different conicalangles are calculated. The two spheres are assumed thesame with εt¼2.4, εz¼2.8, μt¼2.0, μz¼2.8, a¼1λ andλ¼1 μm. The coordinates of the sphere centers are(x10¼y10¼z10¼0) and (x10¼y10¼0, z10¼2λ). When theconical angel α¼01, the ZOBB will be reduced to a TMmode plane wave, which propagates along the z-axis andpolarizes along the x-axis. It can be easily obtained fromEqs. (4) and (5) if the trait of the zero-order and one-ordercylindrical Bessel function: J0(0)¼1 and J1(0)¼0 areapplied. As expected, our results in the case of ZOBBincidence for α¼01 are in excellent agreement with the

Fig. 2. Angular distribution of the RCS of uniaxial anisotropic bispheresplaced along the z-axis illuminated by a ZOBB with different conicalangles.

Fig. 3. Normalized RCS versus the scattering angle for isotropic sphere,isotropic and anisotropic bispheres illuminated by a ZOBB (α¼201).

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–6560

results in the case of a plane wave incidence. This agree-ment can verify the validity and correctness of our theoryand codes. It is obvious that as the conical angle of thebeam increases, the RCS becomes smaller than that for aplane wave incidence due to the influence of the beamshape coefficients. Moreover, the maximum RCS appears inthe direction or neighboring direction of the conical anglewhen α¼301, which is very different from the case of aGaussian beam or a plane wave, in which the maximumRCS occurs at the propagation direction of the incidentwave [36]. This is determined by the structural character-istics of the ZOBB and also a very particular property. Manyscholars discussed the relationship between the extremumpoint and the radius of spheres to the central spot size ofthe ZOBB. Marston [12] has geometrically explained thatsuch extremum points can be interpreted as associatedwith the superposition of the local forward scattering ofthe individual plane-wave components of the incidentwave since the radii of particles are larger than the centralspot size of the ZOBB, which is defined as ρ¼2.405/(k0 sin α). Marston also illustrated examples of such max-ima in acoustic situations when the sphere radius is morethan the central spot size of the ZOBB [12], while suchmaxima fails to exist when the sphere radius is less than

the central spot size [3,39]. Chen [18] gave similar discus-sion for an isotropic concentric sphere illuminated by aZOBB. Ma [13] and Qu [20] gave a more idiographicexplanation for an isotropic sphere and uniaxial asnitoro-pic sphere. Such extremum points occur only when theratio of the radius of spheres to the central spot size of theZOBB δ¼aj/ρ45/4.

Since the numerical results on the interaction betweenZOBB with isotropic bispheres in published works areexiguous, the uniaxial anisotropic sphere will be reducedto an isotropic sphere when εt¼εz. The angular distribu-tions of the RCS of isotropic sphere, isotropic bispheres andanisotropic bispheres illuminated by a ZOBB with a conicalangle of α¼201 are shown in Fig. 3. Note that in the figure“1IS” indicates one isotropic sphere; “2IS-z” and “2IS-x”indicate two isotropic spheres are placed along the z-axisand x-axis, respectively; “2AS-x” indicates two anisotropicspheres are placed along the x-axis. The parameters forthese curves are given in Table 1.

The radii of the spheres and permeability are equalto one here. It can be found that the results denoted by“1IS-literature” provided by [19] are in accordance with

Table 1Parameters relative to Fig. 3.

1IS 2IS-z 2IS-x 2AS-x 1IS-literature

ε1 εt1¼εz1¼4 εt1¼εz1¼4 εt1¼εz1¼4 εt1¼4, εz1¼2 εt1¼εz1¼4ε2 – εt2¼εz2¼4 εt2¼εz2¼4 εt2¼4, εz2¼2 –

(x10, y10, z10) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0)(x20, y20, z20) – (0, 0, 2.5λ) (2.5λ, 0, 0) (2.5λ, 0, 0) –

Fig. 4. Effects of anisotropy on the RCS, the bispheres are placed alongthe z-axis illuminated by a ZOBB with α¼101.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–65 61

the results denoted by “1IS” when anisotropic bispheresare reduced to an isotropic sphere, which can verify theaccuracy of our computations further. Comparing with theresults of one sphere, the angular distributions of the RCSof bispheres placed along the z-axis vibrate more drasti-cally and the forward RCSs in both E-plane and H-planebecome large because of the interaction of these twospheres. When the isotropic bispheres are placed alongthe x-axis, the symmetry of the angular distribution of theRCS in E-plane is broken due to the shift of the secondsphere off the axis of the propagating direction of thebeam, while the symmetry of that in H-plane still holds.The angular distributions of the RCS of uniaxial aniso-tropic bispheres are very distinct from those of isotropicbispheres especially the forward and backward RCS for the

reason that two kinds of EM waves are produced inuniaxial anisotropic medium resulting from difference ofthe εt and εz.

Since α¼201, namely δ¼0.89o5/4, there is no extre-mum point in the direction or neighboring direction ofthe conical angle for single sphere or bispheres along thez-axis. However, there is a very obvious extremum point inthe direction or neighboring direction of the conical anglefor bispheres along the x-axis. It may be attributed to thefact that the second sphere is illuminated by the sidelobeof the ZOBB.

Angular distributions of the RCS of two differentspheres along the z-axis illuminated by a ZOBB withα¼101 are calculated in Fig. 4. Here, we assume that theincident wavelength is 0.6328 μm, and the radii of thesetwo spheres are 1 μm. The characters “IS” and “AS” inFig. 4 indicate isotropic sphere and anisotropic sphere,respectively. The other parameters are given in Table 2.

Thus, the curve denoted by “IS–IS” indicates the angu-lar distribution of two isotropic spheres (in fact two waterdroplets with refractive index n¼ ffiffiffiffiffi

εtp ¼ ffiffiffiffiffi

εzp ¼ 1:33) along

the z-axis illuminated by a ZOBB. When one of the sphereis anisotropic, the RCS in E-plane at lateral direction suchas 301oθo1501 is different from the RCS of isotropicbispheres. The difference is more obvious when both thespheres are anisotropic. This may be attributed to the factthat the transverse permeability tensor element εtbecomes large. While the RCS in E-plane oscillates moredrastically when εt becomes large.

TiO2 (titanium dioxide) characterized by εt¼5.913 andεz¼7.197 is a typical uniaxial anisotropic medium. Fig. 5calculates the angular distributions of the RCS of twoclose-packed TiO2 spheres with different radii along thez-axis. The first sphere centre O1 is located at the origin ofthe beam system with x10¼y10¼z10¼0. When α¼301, thecentral spot size of the ZOBB is ρ¼0.76λ. It can be foundthat the larger the radii of bispheres, the more adjacent tothe direction of the conical angle the position of themaximum RCS, and the smaller the oscillation of theangular distributions as it should be. When both the radiiof the bispheres are less than ρ, the extremum point is stillin the neighboring direction of the conical angle but not inforward. For instance, the extremum point occurs atθ¼231 when a1¼0.76λ and a2¼0.5λ, and it occurs atθ¼151 when a1¼0.76λ and a2¼0.5λ. This property is verydifferent from that of single sphere shown in [13,20]because of the interaction of bispheres is considered.

In Fig. 6, the angular distributions of the RCS of twoTiO2 spheres placed along the z-axis with different sphereseparation distances are shown. The radius of the sphere isaj¼1λ and the conical angle of the incident ZOBB is 301.

Fig. 6. Effects of sphere separation distances on the normalized RCS; thebispheres are placed along z-axis illuminated by a ZOBB with α¼301. (Forinterpretation of the references to color in this figure, the reader isreferred to the web version of this article.)

Table 2Parameters relative to Fig. 4.

IS–IS IS–AS AS–IS AS–AS

ε1 εt1¼εz1¼1.7689 εt1¼εz1¼1.7689 εt1¼3, εz1¼1.7689 εt1¼3, εz1¼1.7689ε2 εt2¼εz2¼1.7689 εt2¼3, εz2¼1.7689 εt2¼εz2¼1.7689 εt2¼3, εz2¼1.7689(x10, y10, z10) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0)(x20, y20, z20) (0, 0, 2 μm) (0, 0, 2 μm) (0, 0, 2 μm) (0, 0, 2 μm)

Fig. 5. Effects of sphere radii on the normalized RCS; the bispheres areplaced along the z-axis illuminated by a ZOBB with α¼301.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–6562

Note that the character “1sphere” in Fig. 6 indicates singleTiO2 sphere, which is illuminated by a ZOBB. Since theZOBB has a non-diffracting characteristic, the first spherecentre O1 is also located at the origin of the beam systemwith x10¼y10¼z10¼0, while the second sphere movesaway from beam centre along the z-axis. It can be foundthat the angular distribution of the RCS of bispheresscattering oscillates sharper than that of a single spherescattering. As the sphere separation distance increases, theRCS oscillates sharper and sharper and the extremumpoint is more and more adjacent to the direction of theconical angle. For instance, the extremum point occurs atθ¼281 when d¼z20–z10¼2.1λ, and at θ¼291 whend¼z20–z10¼4λ, while the extremum point occurs atθ¼301 when d¼z20–z10¼10λ, which is indicated by a

red line in the figure. Because both the interacting scatter-ing and the interference scattering are considered, theinteracting effect of the anisotropic bispheres is decreased,while the interference effect is more visible with theincrease of the sphere separation distance; thus, the RCSwill oscillate sharper and sharper until it arrives at theresonance zoom.

Similar to Fig. 6, the effects of the sphere separationdistance on the RCS of two TiO2 spheres placed along thex-axis are shown in Fig. 7. Here both the radii of thebispheres are 0.76λ, which equals the central spot size ofthe ZOBB with α¼301. The other parameters are the sameas those shown in Fig. 6. Since the anisotropic bispheresare placed along the x-axis, the symmetry of the RCS inE-plane is broken and the effect of the sphere separation

Fig. 8. Effects of beam centre position; the bispheres are along or parallelthe z-axis illuminated by a ZOBB with α¼301.

Table 3Parameters relative to Fig. 8.

x10¼x20¼0 x10¼x20¼0.76λ x10¼x20¼1.52λ

(x10, y10, z10) (0, 0, 0) (0.76λ, 0, 0) (1.52λ, 0, 0)(x20, y20, z20) (0, 0, 1.52λ) (0.76λ, 0, 1.52λ) (1.52λ, 0, 1.52λ)

Fig. 7. Effects of sphere separation distances on the normalized RCS; thebispheres are placed along the x-axis illuminated by a ZOBB with α¼301.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–65 63

distance on the RCS in E-plane is sensitive than that inH-plane. As sphere separation distance increases, the RCSin E-plane oscillates more and more drastically. However,the RCS of bispheres in E-plane and H-plane is not same asthose of single sphere even though the sphere separationdistance is much larger than the central spot size of ZOBB,such as x20¼20λ. The intensity of ZOBB versus x isexhibited by a small figure in Fig. 7(a). It can be foundthat the intensity of the ZOBB is very small but notcompletely zero when x20¼20 μm. Thus, the secondsphere scattering illuminated by the ZOBB is not negligibleyet even though it is very far from beam centre. This traitis very different from that of bispheres illuminated by aGaussian beam shown in our other paper [36]. For aGaussian beam incidence, the RCS in H-plane tends to beinvariable when the sphere separation distance is largerthan 5λ resulting from that the interaction can be negli-gible since the second sphere cannot irradiate by theGaussian beam.

SiO2 (silicon dioxide) characterized by εt¼2.3 andεz¼2.25 is also a typical uniaxial anisotropic medium. InFig. 8, the angular distributions of the RCS of two close-packed SiO2 spheres illuminated by a ZOBB with α¼301are calculated. The radii of the bispheres equal the central

spot size of the ZOBB, which is 0.76λ. The bispheres isalong the z-axis or parallel to the z-axis. The coordinates ofthe bispheres relative to the curves in Fig. 8 are shown inTable 3.

In Fig. 8, the characters “BB” and “GS” indicated Besselbeam and Gaussian beam, respectively. In addition, theRCS of two close-packed SiO2 spheres along the z-axisilluminated by an on-axis Gaussian beam is calculated. It isthe curve denoted by “x10¼x20¼0(GS)”. The beam waistwidth of the Gaussian beam is also assumed as 0.76λ. Thecurve denoted by “x10¼x20¼0(BB-Si)” indicates the angu-lar distribution of the RCS of two silicon (Si) spheres withεt¼εzE2.10135, namely refractive index n¼1.4496 illu-minated by a ZOBB. The discrepancy of the RCS betweentwo silicon spheres and two silicon dioxide spheres is notvery obvious due to the little discrepancy of the perme-ability. In fact, the bispheres is illuminated by an off-axisZOBB since they moves along the x-axis together. And the

Fig. 9. Normalized RCS in E-plane versus the scattering angle and conicalangle for two SiO2 spheres along the z-axis illuminated by a ZOBB.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–6564

position offset of the beam centre decreases the RCS andbreaks the symmetry of the RCS in E-plane. A comparisonbetween a ZOBB incidence and Gaussian beam with thesame central spot size indicates that the RCS for a ZOBBincidence is obviously larger than that for a Gaussian beamincidence. Moreover, the maximum RCS for the ZOBBincidence occurs in the direction of the conical angle sincethe radii of the bispheres equal the central spot size.

In Fig. 9, the RCS of two close-packed SiO2 spheresversus the scattering angle and conical angle is calculated.The radii of the spheres are a1¼a2¼2.5 μm, and thecoordinates are (0, 0, 0) and (0, 0, 5 μm), respectively. Itcan be found that position of the extremum point increaseswith the increasing of the conical angle when α4101 dueto the larger size of the particle. Such result also accordswith foregoing analysis. Li et al. [15] also show similarphenomenon for a larger isotropic sphere illuminated by aZOBB. For the single sphere, the extremum point locates inthe direction of neighboring direction of the conical anglewhen αo421, it will be smaller than the conical anglewhen α4421. However, the variation of the extremumpoint is always located in the direction of neighboringdirection of the conical angle when the conical angle islarge for bispheres resulting from the effect of two spheres.

5. Conclusion

In summary, an analytical solution of the scattering of aZOBB by homogeneous uniaxial anisotropic bispheres withparallel primary optical axes and arbitrary configuration isderived. The accuracy of the theory is verified by compar-ing the numerical results reduced to the special cases of aplane wave incidence given by references. The effects ofthe conical angle, beam centre position, sphere separationdistance, and anisotropic parameters on the field distribu-tions are numerically discussed in detail. Some results foran isotropic bispheres are also given. A comparison withthose results for a Gaussian beam and plane wave inci-dence indicates that the position of the extremum point isinfluenced by the ratio of the particle sizes to the centrespot size of the beam. Moreover, a comparison withscattering of single sphere indicates that the position ofthe extremum point is also influenced by the sphereseparation distance and configuration. In our future work,the interaction between high-order Bessel beam and anaggregate of anisotropic sphere with arbitrary configura-tion will be considered.

Acknowledgment

The authors gratefully acknowledge support from theNational Natural Science Foundation of China under Grantnos. 61308025, 61172031, and 61475123, the NaturalScience Foundation in Shaanxi Province of China(2013JQ8018) and the Fundamental Research Funds forthe Central Universities.

References

[1] Durnin J. Exact solution for nondiffracting beams. I. The scalartheory. J Opt Soc Am A 1987;4:651–4.

[2] Garcés CV, Roskey D, Summers MD, Melville H, McGloin D, WrightEM, et al. Optical levitation in a Bessel light beam. Appl Phys Lett2004;85:4001–4.

[3] Marston PL. Axial radiation force of a Bessel beam on a sphere anddirection reversal of the force. J Acoust Soc Am 2006;120:3518–24.

[4] Zhang LK, Marston PL. Axial radiation force exerted by general non-diffracting beams. J. Acoust Soc Am 2012;131:EL329–35.

[5] Milne G, Kishan D, David M, Karen VS, Pavel Z. Transverse particledynamics in a Bessel beam. Opt Express 2007;15:13972–87.

[6] Vahimaa P, Ville K, Markku K, Jari T. Electromagnetic analysis ofnonparaxial Bessel beams generated by diffractive axicons. J Opt SocAm A 1997;14:1817–24.

[7] Ding D, Liu XJ. Approximate description for Bessel, Bessel–Gauss,and Gaussian beams with finite aperture. J Opt Soc Am A 1999;16:1286–93.

[8] Cizmar T, Kollarova V, Bouchal Z, Zemanek P. Sub-micron particleorganization by self-imaging of non-diffracting beams. New J Phys2006;8:1–24.

[9] Taylor JM, Love GD. Multipole expansion of Bessel and Gaussianbeams for Mie scattering calculations. J Opt Soc Am A 2009;26:278–82.

[10] Ambrosio LA, Hugo EH. Integral localized approximation descriptionof ordinary Bessel beams and application to optical trapping forces.Biomed Opt Express 2011;2:1893–906.

[11] Mishra SR. A vector vave analysis of a Bessel beam. Opt Commun1991;85:159–61.

[12] Marston PL. Scattering of a Bessel beam by a sphere. J Acoust Soc Am2007;121:753–8.

[13] Ma XB, Li EB. Scattering of an unpolarized Bessel beam by spheres.Chin Opt Lett 2010;8:1195–8.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 162 (2015) 56–65 65

[14] Gouesbet G, Maheu B, Grehan G. Light scattering from a spherearbitrarily located in a Gaussian beam, using a Bromwich formula-tion. J Opt Soc Am A 1988;5:1427–43.

[15] Li RX, Guo LX, Ding CY, Wu ZS. Scattering of an axicon-generatedBessel beam by a sphere. Opt Commun 2013;307:25–31.

[16] Mitri FG. Arbitrary scattering of an electromagnetic zero-orderBessel beam by a dielectric sphere. Opt Lett 2011;36:766–8.

[17] Mitri Farid G. Electromagnetic wave scattering of a higher-orderBessel vortex beam by a dielectric sphere. IEEE Trans AntennasPropag 2011;59:4375–9.

[18] Chen ZY, Han YP, Cui ZW, Shi XW. Scattering of a zero-order Besselbeam by a concentric sphere. J Opt 2014;16:055701.

[19] Cui ZW, Han YP, Han L. Scattering of a zero-order Bessel beam byarbitrarily shaped homogeneous dielectric particles. J Opt Soc Am A2013;30:1913–20.

[20] Qu T, Wu ZS, Shang QC, Li ZJ, Bai L. Electromagnetic scattering by auniaxial anisotropic sphere located in an off-axis Bessel beam. J OptSoc Am A 2013;30:1661–9.

[21] Brian S, Michel N, Evgeny P. Mie scattering by an anisotropic object.Part I. Homogeneous sphere. J Opt Soc Am A 2006;23:1111–23.

[22] Wong KL, Chen HT. Electromagnetic scattering by a uniaxiallyanisotropic sphere. IEE Proc.—H 1992;139:314–8.

[23] Qiu CW, Li LW, Yeo TS. Scattering by rotationally symmetricanisotropic spheres: potential formulation and parametric studies.Phys Rev E 2007;75:026609.

[24] Geng YL, Wu XB, Li LW. Mie scattering by a uniaxial anisotropicsphere. Phys Rev E 2004;70:056609.

[25] Yuan QK, Wu ZS, Li ZJ. Electromagnetic scattering for a uniaxialanisotropic sphere in an off-axis obliquely incident Gaussian beam. JOpt Soc Am A 2010;27:1457–65.

[26] Wu ZS, Yuan QK, Peng Y, Li ZJ. Internal and external electromagneticfields for on-axis Gaussian beam scattering from a uniaxial aniso-tropic sphere. J Opt Soc Am A 2009;26:1779–88.

[27] Brunding JH, Lo YT. Multiple scattering of EM waves by spheresPart I—multipole expansion and ray-optical solutions. IEEE TransAntennas Propag 1971;19:378–90.

[28] Fuller KA, Kattawar GW. Consummate solution to the problem ofclassical electromagnetic scattering by an ensemble of spheres I:linear chains. Opt Lett 1988;13:90–2.

[29] Mackowski DW. Calculation of the T matrix and the scatteringmatrix for ensembles of spheres. J Opt Soc Am A 1996;13:2266–77.

[30] Mishchenko MI, Mackowski DW. Light scattering by randomlyoriented bispheres. Opt Lett 1994;19:1606–904.

[31] Thomas W, Schuh R. Decompositioning method to compute scatter-ing by complex shaped particles using a multiple scattering T-matrixapproach. J Quant Spectrosc Radiat Transf 2008;109:2315–28.

[32] Xu YL. Electromagnetic scattering by an aggregate of spheres. ApplOpt 1995;34:4573–88.

[33] Xu YL. Electromagnetic scattering by an aggregate of spheres: farfield. Appl Opt 1997;36:9496–508.

[34] Li ZJ, Wu ZS, Shi YE, Bai L, Li HY. Multiple scattering of electro-magnetic waves by an aggregate of uniaxial anisotropic spheres. JOpt Soc Am A 2012;29:22–31.

[35] Li ZJ, Wu ZS, Li HY. Analysis of electromagnetic scattering byuniaxial anisotropic bispheres. J Opt Soc Am A 2011;28:118–25.

[36] Li ZJ, Wu ZS, Bai L. Electromagnetic scattering from uniaxialanisotropic bispheres located in a Gaussian beam. J Quant SpectroscRadiat Transf 2013;126:25–30.

[37] Barton JP, Alexander DR, Schaub SA. Internal and near-surfaceelectromagnetic fields for a spherical particle irradiated by a focusedlaser beam. J Appl Phys 1988;64:1632–9.

[38] Wiscombe WJ. Improved Mie scattering algorithms. Appl Opt1980;19:1505–9.

[39] Marston PL. Negative axial radiation forces on solid spheres andshells in a Bessel beam. J Acoust Soc Am 2007;122:3162–5.