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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]

0022-40

http://d

n Corr

E-m

wuzhs@

PleasGaus

journal homepage: www.elsevier.com/locate/jqsrt

Electromagnetic scattering from uniaxial anisotropic bisphereslocated in a Gaussian beam

Z.J. Li, Z.S. Wu n, L. Bai

Xidian University, School of Science, Taibai south road 2#, Xi’an, Shaanxi 710071, China

a r t i c l e i n f o

Article history:

Received 11 June 2012

Received in revised form

14 September 2012

Accepted 16 September 2012

Keywords:

Gaussian beam

Multiple scattering

Anisotropic medium

Spherical vector wave functions

73/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.jqsrt.2012.09.011

esponding author.

ail addresses: lizhengjuncsoabc@yahoo.com.c

mail.xidian.edu.cn (Z.S. Wu).

e cite this article as: Li ZJ, et al. Esian beam. J Quant Spectrosc Radiat

a b s t r a c t

Based on the Generalized Lorenz–Mie Theory (GLMT) and generalized multiparticle Mie-

solution (GMM), scattering of two interacting homogeneous uniaxial anisotropic spheres

with parallel primary optical axes and arbitrary configuration from a Gaussian beam is

investigated. By introducing the Fourier transformation, the electromagnetic fields in the

uniaxial anisotropic spheres are expanded in terms of the spherical vector wave functions

(SVWFs). The interactive scattering coefficients and the expansion coefficients of the

internal fields are derived through the continuous boundary conditions on which the

interaction of the bispheres is considered. The influences of the beam waist widths, beam

center position, and spheres separation distance on scattering characteristics of uniaxial

anisotropic bispheres located in a Gaussian beam are numerically analyzed in detail.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Due to recent advances in materials science and tech-nology, and the enhancement and reduction of radarcross section (RCS) of various scatterers, the interactionbetween EM wave and anisotropic media is growning.Many scholars have investigated the scattering of a singleanisotropic object by using analytical methods. Usingadifferential theory, Stout et al [1] gave the solution ofthe scattering of an arbitrary-shaped body made of arbi-trary anisotropic medium, but he did not give any numer-ical results. Wong and Chen [2], and Qiu [3] also devotedtheir endeavors to the scattering of a uniaxial anisotropicdielectric sphere through the expansion in terms of scalareigen-functions method. By introducing the Fourier trans-formation method, Geng [4] and Wu [5] studied thescattering of a uniaxial anisotropic sphere illuminated bya plane wave and Gaussian beam, respectively. However,the scatterer in all these studies is limited to a single

ll rights reserved.

n (Z.J. Li),

lectromagnetic scattTransfer (2012), htt

anisotropic object. The published works on the scatteringfrom multiple anisotropic objects are exiguous.

Since the first presentation of a comprehensive solutionof scattering by a two-sphere chain by Bruning and Lo [6],the study of scattering of multiple isotropic spheres hasbeen developed quickly. Fuller and Kattawar [7] 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 [8,9]. Xu introduced the generalizedmultiparticle Mie-solution (GMM) to explore EM scatteringby an aggregate of isotropic spheres [10–12]. Recently, thepresent authors have studied the EM scattering by uniaxialbispheres [13] and multiple uniaxial anisotropic spheres[14]. However, in most of the articles that we have referred,the investigations focus on a plane wave scattering bymultiple isotropic spheres.

With the advent of lasers and their growing use in thefields of particle sizing, biomedicine, laser fusion, opticallevitation, and so on, the scattering problem from shapedbeams has drawn considerable attention. Gouesbetet al. [15] derived three methods to calculate the beamshape coefficient (BSC) based on several relatively simple

ering from uniaxial anisotropic bispheres located in ap://dx.doi.org/10.1016/j.jqsrt.2012.09.011

Fig. 1. Geometry for uniaxial anisotropic bispheres illuminated by a

Gaussian beam.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]2

approximate models of Gaussian beam introduced byDavis [16] and put forward the Generalized Lorenz–MieTheory (GLMT) to solve scattering of spherical particlesilluminated by Gaussian beam [17]. Utilizing the additiontheorem of the SVWFs, Doicu presented the BSC throughdifferent methods [18]. Subsequently, many authors havefurther studied various cases on particle scattering to aGaussian beam [19]. But most of the above referencesconsidered only a single object scattering to a Gaussianbeam. In 1999, Gouesbet and Grehan studied the scatter-ing of an assembly of spherical particles located in anarbitrary electromagnetic shaped beam, but the numer-ical analysis are not given [20].

In this paper, based on GLMT and GMM, we investigatethe scattering of two interacting homogeneous uniaxialanisotropic spheres with parallel primary optical axesalong the axis of propagation of a Gaussian beam. Theeffects of the beam waist widths, spheres separationdistance, and beam center positioning on the angulardistributions of the radar cross section (RCS) are numeri-cally analyzed in detail.

2. Formulation

Considering two uniaxial anisotropic spheres withradius aj (j¼1,2) and the primary optical axes coincidentwith the z-axis in a global coordinate system Oxyz, asshown in Fig. 1, the centers of the spheres are located at(x1, y1, z1) and (x2, y2, z2) . The dashed lines indicateprimary optical axes of the uniaxial anisotropic bispheres.The particles are illuminated by a z-propagating andx-polarized Gaussian beam.

In terms of spherical vector wave functions (SVWFs), theincident fields can be expanded in the particle coordinatesystem Ojxjyjzj as

Eij ¼

X1n ¼ 1

Xn

m ¼ �n

Emn½aijmnMð1Þmnðr,k0Þþbi

jmnNð1Þmnðr,k0Þ�

Hij ¼

k0

iom0

X1n ¼ 1

Xn

m ¼ �n

Emn½aijmnNð1Þmnðr,k0Þþbi

jmnMð1Þmnðr,k0Þ�

ð1Þ

where Emn¼E0in{(2nþ1)(n�m)!/[n(nþ1)(nþm)!]}1/2, andE0 is the amplitude of electric field at the beam waist center.The expansion coefficients can be derived by introducingthe localized approximation of the BSC [18]

aijmn

bijmn

24

35¼ Cmnð�1Þm�1Knmc

0

0e�ik0zj1

2eiðm�1Þjj Jm�1 2

Qrjrn

w20

!"

8eiðmþ1Þjj Jmþ1 2Qrjrn

w20

!#=Emn ð2Þ

where

cj ¼ iQ jexp ð�iQ jr2j =w2

0Þexp ð�iQ jðnþ0:5Þ2=k20w2

0Þ,

Qj ¼ ði�2zj=ðk0w20ÞÞ�1rj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

j þy2j

q, rn ¼ ðnþ0:5Þ=k0,

jj ¼ arctan ðxj=yjÞ, k0 ¼ 2p=l ð3Þ

Please cite this article as: Li ZJ, et al. Electromagnetic scattGaussian beam. J Quant Spectrosc Radiat Transfer (2012), htt

Knm ¼

ð�iÞ9m9 iðnþ0:5Þ9m9�1 , ma0

nðnþ1Þnþ0:5 , m¼ 0

8<: ð4Þ

In Eqs. (1)–(4), j¼1, 2 denotes the correlative para-meter of jth sphere.

The scattered fields of the jth (j¼1, 2) uniaxial aniso-tropic sphere can also be expanded in terms of SVWFs inthe jth sphere coordinate system Ojxjyjzj as

Esj ¼

X1n ¼ 1

Xn

m ¼ �n

Emn½asjmnMð3Þmnðrj,k0Þþbs

jmnNð3Þmnðrj,k0Þ�

Hsj ¼

k0

iom0

X1n ¼ 1

Xn

m ¼ �n

Emn½asjmnNð3Þmnðrj,k0Þþbs

jmnMð3Þmnðrj,k0Þ�

ð5Þ

The uniaxial anisotropic medium is characterized by apermittivity and permeability with tensor ej andmjðj¼ 1, 2Þ, respectively, which can be expressed as:

ej ¼ e0

ejt 0 0

0 ejt 0

0 0 ejz

264

375, mj ¼ m0

mjt 0 0

0 mjt 0

0 0 mjz

264

375 ð6Þ

The EM fields in the jth uniaxial anisotropic sphere canbe expanded in terms of the SVWFs as [13]

EIjðrjÞ ¼

X2

q ¼ 1

X1n ¼ 1

Xn

m ¼ �n

X1n0 ¼ 1

2pGjmn0q

Z p

0½Ae

jmnqMð1Þmnðrj,kjqÞ

þBejmnqNð1Þmnðrj,kjqÞþCe

jmnqLð1Þmnðrj,kjqÞ�Pmn0 ðcosykj

Þk2jqsinykj

dykj

HIjðrjÞ ¼

X2

q ¼ 1

X1n ¼ 1

Xn

m ¼ �n

X1n0 ¼ 1

2pGjmn0q

Z p

0½Ah

jmnqMð1Þmnðrj,kjqÞ

þBhjmnqNð1Þmnðrj,kjqÞþCh

jmnqLð1Þmnðrj,kjqÞ�Pmn0 ðcosykj

Þk2jqsinykj

dykj

ð7Þ

ering from uniaxial anisotropic bispheres located in ap://dx.doi.org/10.1016/j.jqsrt.2012.09.011

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]] 3

where Aejmnq,Be

jmnq,Cejmnq,Ah

jmnq,Bhjmnq and Ch

jmnq are theexpansion coefficients and their expressions can be foundin [3], but the expressions should be coincident with thecorresponding permittivity tensor ej and permeabilitymj for each uniaxial anisotropic sphere. Gjmn0q is theunknown expansion coefficient relative to the internalfields and is determined by the boundary conditions ofthe jth sphere.

On the spherical boundary at rj¼aj, the tangentialcomponents (designated by the subscript t) of the EMfields continue as:

EIj9t ¼ Eit

j 9tþEsj 9t , HI

j9t ¼Hitj 9tþHs

j 9t ðrj ¼ ajÞ ð8Þ

where Eitj and Hit

j represent the total EM fields incident onthe jth(j¼1, 2) uniaxial anisotropic sphere. The totalincident field can be derived applying the addition theoremof the SVWFs, then substituting the expressions of theincident, internal and scattered fields into Eq. (8), we canderive the scattering coefficients as [13]:

asjmn ¼

1

hð1Þn ðk0ajÞ

1

Emn

X2

q ¼ 1

X1n0 ¼ 1

2pGjmn0q

Z p

0Ae

jmnqjnðkjqrjÞPmn0 ðcosyjkÞk

2jqsinyjkdyjk

"

�f itjmnjnðk0ajÞ

ið9Þ

Fig. 2. Angular distribution of the RCS of two uniaxial anisotropic

spheres placed along z-axis illuminated by a Gaussian beam with

different beam waist widths.

Please cite this article as: Li ZJ, et al. Electromagnetic scattGaussian beam. J Quant Spectrosc Radiat Transfer (2012), htt

bsjmn ¼

1

hð1Þn ðk0ajÞ

iom0

k0

1

Emn

X2

q ¼ 1

X1n0 ¼ 1

2pGjmn0q

Z p

0Ah

jmnqjnðkqrjÞPmn0 ðcosyjkÞk

2jqsinyjkdyjk

"

�gitjmnjnðk0ajÞ

ið10Þ

where f itjmn and git

jmn are the expansion coefficients of thetotal incident field illuminating the jth (j¼1, 2) uniaxialanisotropic bispheres, they can be derived by applying theaddition theorem of the SVWFs. [13].

Based on the scattering coefficients above, the totalscattered fields which are the vector superposition of thescattered field of every uniaxial anisotropic sphere can bederived by applying the addition theorem of the SVWFsagain. Then according to the definition of the RCS, we cancalculate the angular distribution of the RCS of theuniaxial anisotropic bispheres illuminated by a Gaussianbeam.

3. Numerical results and discussion

The effect of the beam waist width on the RCS ofuniaxial anisotropic bispheres with both electric andmagnetic anisotropy in the E- and H- planes is shown by

Fig. 3. Effect of beam center position on the RCS of two uniaxial

anisotropic spheres placed along x-axis.

ering from uniaxial anisotropic bispheres located in ap://dx.doi.org/10.1016/j.jqsrt.2012.09.011

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]4

Fig. 2(a) and (b), respectively. It can be found that the RCSfor Gaussian beams is smaller than that for a plane wavebecause of the influence of the beam shape coefficients. Asthe beam waist width increases, the RCS approaches thatfor plane wave incidence, but the angular distribution hasa little subtle change. As expected, our results in the caseof Gaussian beam incidence with a relatively large waistradius of w0¼20l are in excellent agreement with theresults in the case of a plane wave incidence provided in[13]. This agreement can verify the validity and correct-ness of our theory and codes. Note that the E-plane andH-plane indicate the xOz plane and yOz plane, respectively.

Fig. 3 shows the angular distributions of the RCS ofclosely packed uniaxial anisotropic bispheres placed alongx-axis with different separation distances from the beamcenter. The other coordinates of the sphere centers arey1¼z1¼0, y2¼z2¼0. As bispheres offset from the beam-center along x-axis,the RCS decreases because the beamwaist illuminate on the objects finitely. And the offsetsymmetrical axes of the Gaussian beam and the bispheresalso cause the angular distributions of the RCS to beasymmetrical. But the angular distributions of the RCSin 0–1801 when the offset of the bispheres is along thepositive x-axis from the beam center are the same thosein 180–3601 when the offset of the bispheres is along the

Fig. 4. Effect of beam center position on the RCS of two uniaxial

anisotropic spheres placed along z-axis.

Please cite this article as: Li ZJ, et al. Electromagnetic scattGaussian beam. J Quant Spectrosc Radiat Transfer (2012), htt

negative x-axis from the beam center. The distribution ofamplitude is also the same.

As in Fig. 3, Fig. 4 shows that, the angular distri-butions of the RCS of closely packed uniaxial anisotropicbispheres placed along z-axis with different separationdistances from the beam center. The other coordinatesof the sphere centers are x1¼y1¼0, x2¼y2¼0. As thebispheres move away from the beam center along z-axis,the RCS decreases integrally but the form of the curveexhibits little change. In Fig. 3 the angular distribution ofRCS show obvious difference when bispheres move awayfrom the beam center along x-axis. This may be attributedto the fact that the incident beam also propagates alongz-axis. Moreover, the angular distributions of the RCS whenthe bias of the bispheres is along the positive z-axis fromthe beam center are the same with those when the biasof the bispheres is along the negative z-axis from the beamcenter and the bias is the same. This property can beobserved from the BSC in Eq. (2).

TiO2 characterized by et¼5.913 and ez¼7.197 is atypical uniaxial anisotropic medium. In Fig. 5, the angulardistributions of the RCS of uniaxial anisotropic bispheresplaced along x-axis with different spheres separationdistances illuminated by an off-axis Gaussian are shown.The other coordinates of the sphere centers are y1¼z1¼0,

Fig. 5. Effect of the spheres separation distance on the RCS of two

uniaxial anisotropic spheres placed along x-axis.

ering from uniaxial anisotropic bispheres located in ap://dx.doi.org/10.1016/j.jqsrt.2012.09.011

Fig. 6. Effect of the spheres separation distance on the RCS of two

uniaxial anisotropic spheres placed along z-axis.

Z.J. Li et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]] 5

y2¼z2¼0. Because the second sphere moves away fromthe first sphere along positive x-axis, the RCS in 180–3601in E-plane oscillates more drastically with the increase ofthe spheres separation distance. The effect of the spheresseparation distance on the RCS in E-plane is more sensi-tive than that in H-plane due to the fact that the uniaxialanisotropic bispheres are placed along x-axis. Moreover, itcan be found that as the spheres separation distanceincreases, the angular distribution of the RCS tends to beconsistent. It is obvious that the curves of the RCS inH-plane are almost the same for x2¼5l and 6l. This isbecause the beam waist width is finite and the Gaussianbeam can be incident on only some part of the bispheres,which is different from plane wave incidence.

The effect of the spheres separation distance on the RCSof uniaxial anisotropic bispheres placed along z-axis isshown in Fig. 6. The other parameters are the same asinFig. 5. The other coordinates of the sphere centers arex1¼y1¼0, x2¼y2¼0. It can be observed that the effect ofthe separation distance on the RCS when the bispheresplaced along z-axis is distinct from that when the bispheresare placed along x-axis. The angular distributions ofthe RCS of the bispheres with larger spheres separationdistance oscillate sharper than that of bispheres with

Please cite this article as: Li ZJ, et al. Electromagnetic scattGaussian beam. J Quant Spectrosc Radiat Transfer (2012), htt

smaller spheres separation distance, resulting from the factthat both the interaction scattering and the interferencescattering act.

4. Conclusion

In summary, an analytical solution of the scattering ofa Gaussian beam by homogeneous uniaxial anisotropicbispheres with parallel primary optical axes and arbitraryconfiguration is derived. The accuracy of the theory isverified by comparing the numerical results reduced tosome special cases of a plane wave incidence given byexisting references. The effects of the beam waist widths,beam center position, and spheres separation distance onthe RCS are numerically analyzed. Due to the lengthrestriction, the scattering characteristics of the uniaxialanisotropic bispheres placed along y-axis are not calcu-lated, but the characteristics may be similar to those ofthe uniaxial anisotropic bispheres placed along y-axisdue to the symmetry. The scattering of an aggregate ofuniaxial anisotropic spheres illuminated by an arbitrarydirection Gaussian beam may be investigated in ourfuture work.

Acknowledgement

This work was supported by the National NaturalScience Foundation of China under Grant Nos. 61172031and 60971065, and the Fundamental Research Funds forthe Central Universities under Grant No. K5051207001.

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