4740 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

Off-Axis Gaussian Beam Scattering by an AnisotropicCoated Sphere

Zhen-Sen Wu, Senior Member, IEEE, Zheng-Jun Li, Huan Li, Qiong-Kun Yuan, and Hai-Ying Li

Abstract—An analytical solution to the scattering of an off-axisGaussian beam incident on an anisotropic coated sphere is pro-posed. Based on the local approximation of the off-axis beam shapecoefficients, the field of the incident Gaussian beam is expandedusing first spherical vector wave functions. By introducing theFourier transform, the electromagnetic fields in the anisotropiclayer are expressed as the addition of the first and the secondspherical vector wave functions. The expansion coefficients are an-alytically derived by applying the continuous tangential boundaryconditions to each interface among the internal isotropic dielectricor conducting sphere, the anisotropic shell, and the free space. Theinfluence of the beam widths, the beam waist center positioning, andthe size parameters of the spherical structure on the field distribu-tions are analyzed. The applications of this theoretical developmentin the fields of biomedicine, target shielding, and anti-radar coatingare numerically discussed. The accuracy of the theory is verifiedby comparing the numerical results reduced to the special cases ofa plane wave incidence and the case of a homogeneous anisotropicsphere with results from a CST simulation and references.

Index Terms—Anisotropic layered, electromagnetic scattering,Gaussian beam, off-axis.

I. INTRODUCTION

T HE scattering of a plane electromagnetic wave by a coatedspherical particle has been extensively discussed in many

areas such as combustion, biomedicine, chemical engineering,remote sensing, etc. Within a theoretical framework similar tothe classic Lorenz-Mie theory [1], [2], rigorous formulationswere first developed for scattering using two concentric spheresbased on the properties of Bessel functions by Aden and Kerker[3]. This general solution was subsequently specialized for alossless or lossy dielectric-coated conducting sphere [4]–[8].Wu further presented numerical results for plane wave scatteringby a multilayered isotropic sphere [9].

In recent years, interest in the scattering characteristics ofanisotropic media has grown because of their wide applicationsin optical signal processing, radar cross section (RCS) control-ling, microwave device fabrication, etc. Particularly, researchon layered anisotropic spheres is of special importance in thetarget shielding field. In addition to studies on the homogeneous

Manuscript received October 15, 2010; revised March 21, 2011; acceptedJune 02, 2011. Date of publication August 22, 2011; date of current versionDecember 02, 2011. This work was supported National Natural Science Foun-dation of China under Grant 60771038 and Fundamental Research Funds forthe Central Universities.

The authors are with the School of Science, Xidian University, Xi’an Shaanxi710071, China (e-mail: wuzhs@mail.xidian.edu.cn; lizhengjuncosabc@yahoo.com.cn).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAP.2011.2165489

anisotropic sphere [10]–[14] and other shaped anisotropic par-ticles [15], [16], numerous investigations have focused on theinteraction between a plane wave and an anisotropic coatedsphere. Using the boundary integral method, Baker studied theelectromagnetic scattering by arbitrarily-shaped, two-dimen-sional, perfectly conducting objects coated with homogeneousanisotropic materials [17]. Subsequently, the plane wave ex-pansion, along with the Fourier transform and the vector wavefunctions (VWFs), was widely employed in the analysis of uni-axial or plasma coated spheres [18], [19]. The dyadic Green’sfunctions based on modified spherical VWFs were constructedto investigate multilayered radial anisotropic spheres [20].However, the excitation source in all these studies is limited toa plane wave.

With the advent of lasers and their growing use in the fieldsof particle sizing, biomedicine, laser fusion, Raman scatteringdiagnostics, optical levitation, aerosol cloud penetration andnear filed scattering and calibration, the scattering problemfrom shaped beams has drawn considerable attention. Examplesof beam applications are the use of radiation pressure to ma-nipulate biological cells and the analysis of the phase Dopplertechnique. Theories regarding Gaussian beam scattering havebeen well established on the basis of the decomposition of theincident beam into an infinite series of elementary constituents,with amplitudes and phases given by a set of beam-shapecoefficients [1]–[10]. Using the first-order approximation ofa Gaussian beam developed by Davis [21] as basis, Bartonand Alexander derived the high-order approximate expres-sion for a TEM Gaussian beam, and calculated the internaland scattered fields of a homogeneous isotropic sphere [22].The generalized Lorenz-Mie theory (GLMT) developed byGouesbet et al., concerns the expansion of the incident shapedbeam as a series of spherical VWFs, which effectively describesthe electromagnetic scattering of a beam by a spherical particle;the beam shape coefficients are obtained by applying localizedapproximation [23], [24]. In the approach presented by Doicuet al., the translational addition theorem for spherical VWFswas used to calculate the beam shape coefficients of an off-axisbeam [25]. Khaled et al. used the angular spectrum of planewaves to model a shaped beam and compute the fields insideand outside an isotropic sphere [26]. Although past effortswere primarily spent on the interaction of a Gaussian beamwith a homogenous isotropic medium, we have recently madeprogress in the interactions of an on-axis Gaussian beam (thepropagation direction of the Gaussian beam are coincident withthe z-axis of particle coordinate system) with a multilayeredisotropic sphere [27] and a homogenous uniaxial anisotropicsphere [28].

0018-926X/$26.00 © 2011 IEEE

WU et al.: OFF-AXIS GAUSSIAN BEAM SCATTERING BY AN ANISOTROPIC COATED SPHERE 4741

Fig. 1. Anisotropic coated isotropic sphere illuminated by an off-axis incidentGaussian Beam.

However, the EM scattering of a Gaussian beam by ananisotropic coated sphere has not yet been studied. Beam prop-agation and scattering in inhomogeneous anisotropic media arecomplex, making analysis difficult. Moreover, the expansionterms m are not only equivalent zero for off-axis Gaussianbeam (the propagation direction of the Gaussian beam areparallel with the z-axis of particle coordinate system but thecenter of the particle may not in the propagation direction)scattering, which also enhances the difficulty of the solutionand numerical calculations of the anisotropic medium scatteredfrom an off-axis Gaussian beam. Previous theories are nolonger applicable to current problems, but provide a referableframework and an effective calibration. This paper aims to de-velop a precise solution to the general case of off-axis Gaussianbeam scattering by an anisotropic coated isotropic dielectric orconducting sphere, study the role of the anisotropy and beamwaist center positioning in far-field scattering diagrams, andunderstand the mechanism of wave-medium interaction. Theaccuracy of the theory is verified by comparing the numericalresults reduced to the special cases of a plane wave incidenceand the case of a homogeneous anisotropic sphere with resultsfrom a CST simulation and references. CST is a three-di-mensional electromagnetic simulation software based on thenumerical method of finite integration. This high-performancecommercial software was developed by the German CST com-pany in 1992; our laboratory is the Sino-Germany Joint CSTTraining Centre in Northwestern China. The applications ofthis theoretical development in the fields of biomedicine, targetshielding, and anti-radar coating are numerically discussed.

In the subsequent analysis, a time dependence of the formis assumed for all the EM fields, but disregarded

throughout the treatment.

II. THEORETICAL FORMULATIONS

A. Off-Axis Gaussian Beam Expansions

Consider an anisotropic coated isotropic dielectric or con-ducting sphere center located in a Cartesian coordinate system

. As Fig. 1 shows, its outer and inner radii are and, respectively. The inner isotropic dielectric or conducting

sphere is coated with an anisotropic material characterized bypermittivity tensor and permeability tensor with thickness

in the primary coordinate system . Three

distinct regions are thus defined, namely, region 0 for the freespace with and , region 1 for the anisotropic medium, andregion 2 for the isotropic dielectric or conducting sphere withand . This composite structure is illuminated by an -polar-ized Gaussian beam that propagates in the direction, and thecenter of the beam waist is located at .

The spatial distribution of the incident electric field (desig-nated by the superscript inc) in the plane is given by

(1)

where is the beam waist radius, and is the amplitude ofthe electric field at the center of the beam waist, and taken asunity for simplicity.

In terms of spherical VWFs, the electromagnetic field of theincident Gaussian beam can be written as [24], [25]

(2)

(3)

where

(4)

in which the expansion coefficients and are known

as the beam shape coefficients. Spherical VWFs andare defined as [12]

(5)

(6)

where represents the appropriate kind of sphericalBessel functions , and , for

and , respectively. Applying the local approxima-tion of the GLMT [24], [25], the beam shape coefficients canbe determined as follows:

(7)

4742 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

where

.(8)

B. Internal and Scattered Fields

The E-field vector wave equation in the source-freeanisotropic shell in region 1 can be written as follows:

(9)

where permittivity and permeability tensors and are charac-terized in primary system by

(10)

It is very difficult to solve the characteristic equations to ob-tain the angular spectral representations of the wave fields ifnone of the 18 parameters in these two tensors are zero. Souniaxial anisotropic medium and plasma relatively simple andthe most common anisotropic medium are considered here. Foruniaxial anisotropic medium, the components in (10) have fol-lowing relations:

and. For plasma, the components in (10) have following

relations:and . Then using the Fourier transform and spher-

ical VWFs, the fields on coated uniaxial anisotropic medium orplasma in region 1 ( , designated by the superscript1) can be expanded as [18], [19]

(11)

(12)

where the expressions of eigenvalues and co-efficients for the uniaxialanisotropic medium and plasma can be found in [13] and [14],respectively. The EM fields are finite at the origin; thus, usingthe spherical VWFs, the internal fields of the isotropic sphere inregion 2 can be expanded as (13)-(14), shown at thebottom of the page, where is the wave numberin region 2, and denotes a conducting sphere. Further-more, the scattered fields (designated by the superscript ) canbe expanded thus [13]:

(15)

In (11)–(15), the unknown expansion coefficientsand can later be deter-

mined by working out the matrix equations derived from theboundary conditions.

C. Scattering Coefficients

On the spherical boundary at and , the tangen-tial components (designated by the subscript t) of the EM fieldscontinue as

(16)

(17)

Substituting (2), (3), (11), and (12) into (16) yields the followingrelationships at

(18)

(13)

(14)

WU et al.: OFF-AXIS GAUSSIAN BEAM SCATTERING BY AN ANISOTROPIC COATED SPHERE 4743

(19)

(20)

(21)

Similarly, the substitution of (11), (12), (13), and (14) into theboundary condition at leads to

(22)

(23)

(24)

(25)

Combining the above mentioned equations, we obtain(26)–(29), shown at the bottom of the following page. From(26)–(29), expansion coefficient is very hard to ana-lytically derive, but can be numerically calculated throughprograms. Then, substituting back into (18) and (20),we obtain the following expressions for scattering coefficients

and

(30)

(31)

Taking as the amplitude of the incident electricfield, the radar cross section (RCS) for the far-region scatteredfield can be calculated as

(32)

III. NUMERICAL RESULTS AND DISCUSSION

The equations derived in the previous sections for the scat-tering coefficients are analytically solved. In this section, somenumerical solutions to off-axis Gaussian beam scattering byan anisotropic coated sphere are provided. The E-plane corre-sponds to the -plane and the H-plane corresponds to the

-plane.To verify the accuracy of our theory, we initially make

three comparisons. As Figs. 2 and 3 show, the results ob-tained from our codes reduced to a plane wave incidence for aplasma coated spherical shell and a uniaxial anisotropic coatedisotropic sphere are coincident with the results generated in [18]and the CST simulation, respectively. Moreover, the numerical

4744 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

Fig. 2. Results reduced to the case of a plasma anisotropic coated dielectricsphere in a plane wave compared with those in [18]. �� � ����� � ������ � � � � ���� � � � ���� � � � �� � �� � � �� �� � � � � � � � � � � �� � ����� ���� � ���.

results reduced to a homogenous uniaxial anisotropic sphereagree well with those in [28], as shown in Fig. 4.

Fig. 3. Results reduced to the case of a uniaxial anisotropic coated dielectricsphere in a plane wave compared with those by the CST simulation. �� ��� � � ����� � � � � ������� � � � ���� �� � � � � �� � ����� � � � ���� � � � ���.

The effects of the beam width on the RCS in the E- andH-planes are shown by Fig. 5(a) and (b), respectively. As the

(26)

(27)

(28)

(29)

WU et al.: OFF-AXIS GAUSSIAN BEAM SCATTERING BY AN ANISOTROPIC COATED SPHERE 4745

Fig. 4. Results reduced to the case of a homogeneous uniaxial anisotropicsphere in a Gaussian beam compared with those in [28]. �� � �� � ���������� � � � � �� � � � �� � � � � � � �� ������ �� � � � � ����� ����.

Fig. 5. Effects of the beam width on RCS. �� � �� � � ���� � �� � ����� � � � ������� � � � � � � � ���� � � ����� � �� � � � � ��� �� ����.

beam width increases, the intensity of the RCS rises, but the an-gular distribution exhibits minimal change.

The role of the beam waist center positioning in the RCS dis-tributions is shown in Figs. 6 and 7(a), (b), (c), and (d), which areplots for a uniaxial anisotropic coated sphere and plasma coatedsphere, respectively. The position offset of the beam waist centerweakens the scattering intensity and deflects the scattering anglecorresponding to the largest RCS from . Along both and

axes, the bias of the beam waist center in the positive and neg-ative directions imposes the same effects on the RCS. This can

Fig. 6. Effects of the beam waist center positioning along the �-axis on theRCS �� � �� � � ���� � � � �� � � ���� � � � ���� � � ����. (a) E-plane, � � ����� � � � ������� , (b) H-plane, � ������ � � � ������� , (c) E-plane, � � � � ����� � � ��� � � � �� � ��� , (d) H-plane, � � � � ����� � � ��� � � � �� � ��� .

also be indicated from the derived expression of the beam fac-tors. Fig. 6 also shows that the value of influences the E-planeRCS more visibly than it does the H-plane. As the value of in-creases, the scattering angle corresponding to the largest RCS in

4746 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011

Fig. 7. Effects of the beam waist center positioning along the �-axis on theRCS. (a) E-plane, � � ������� � � � ������� , (b) H-plane, � �������� � � � ������� , (c) E-plane, � � � � ������� � � ��� � � � �� � ��� , (d) H-plane, � � � � ������� � � ��� � � � �� � ��� .

the E-plane increases as well, whereas the angular distribution inthe H-plane exhibits minimal change and remains symmetrical.Conversely, produces contrasting results (Fig. 7).

In the field of biology, we may encounter roughly sphericalbut inhomogeneous cells, essentially composed of nuclei sur-rounded by a liquid solution and a shell. Shown in Fig. 8 are cal-

Fig. 8. Results reduced to the case of two concentric isotropic spheres fora nucleated blood cell. (a) on-axis, � � � � � � �� �� �� (b) off-axis,� � � � � � ������������. � ������ � ����� � � � �� � ����� � � �� � � ����� ������� � � � � � � ��.

culations made for the reduced case of two concentric isotropicspheres for a nucleated blood cell. Fig. 8(a) is for an on-axisGaussian beam, which is compared with the GLMT results,whereas Fig. 8(b) is for an off-axis beam. The influence of thebeam waist center positioning on the intensity and angular dis-tributions of the scattering fields are further witnessed.

The effects of the thickness of the anisotropic coating uponthe E-plane RCS are characterized in Fig. 9 for a plasmaanisotropic coated conducting sphere when the radius of theinner conducting sphere is constant. Compared with the ho-mogeneous conducting sphere , the presence of theplasma anisotropic coating enhances the scattering intensity inthe region near the forward and backward directions. Note thatthe RCS for the coated sphere is highly oscillating and produceblind areas at certain scattering angles, which is of specialimportance in radar target shielding. The angular distributionis dependent to a great extent on the thickness and the permit-tivity tensors of the coating; thus, in practical applications, theparameters of the plasma layer should be optimized to generatea better absorption effect.

Fig. 10 presents the H-plane scattering of a uniaxialanisotropic coated spherical shell by an off-axis Gaussianbeam. The thickness of the anisotropic coating is constant;hence, understanding and characterizing the effects of theinner radius of the shell upon the scattering performance isstraightforward. The RCS in the region near the forward direc-tion always increases with the inner radius of the shell. Such

WU et al.: OFF-AXIS GAUSSIAN BEAM SCATTERING BY AN ANISOTROPIC COATED SPHERE 4747

Fig. 9. Effect of the thickness of the anisotropic coating upon the E-planeRCS for a plasma anisotropic coated conducting sphere when the radiusof the inner conducting sphere is constant. �� � ����� � � � ��� � � � �� � � � �� � �� � ��� � � ���� � � �� � � � � � ������ ����� � ��.

Fig. 10. Effects of the inner radius upon the scattering characteristics for a uni-axial anisotropic coated spherical shell when the thickness of the coating is con-stant. � � ����� � � � � ����� � � � ����� � ��� � � ���� � �� � � � � � � � � � � � ���������������� � ��.

anisotropic coated shells have been widely used as phantomtargets in the anti-radar technique.

IV. CONCLUSION

On the basis of the local approximation of the off-axis beamshape coefficients, the incident Gaussian beam is expandedusing first spherical VWFs. By introducing the Fourier trans-form, the electromagnetic fields in the anisotropic layer areexpressed as the addition of the first and the second sphericalVWFs. Matching the continuous tangential boundary condi-tions at each interface among the internal isotropic sphere, theanisotropic shell, and the free space, the expansion coefficientsof an off-axis Gaussian beam incident on an anisotropic coatedsphere are proposed. The accuracy of the theory is verified bycomparing the numerical results reduced to the special casesof a plane wave incidence and the case of a homogeneousanisotropic sphere with results from the CST simulation andreferences. The influence of the beam widths, the beam waistcenter positioning, and the size parameters of the sphericalstructure on the field distributions are analyzed. The beam

width considerably influences the intensity of the RCS, buthas little effect on the angular distribution. The position offsetof the beam waist center weakens the scattering intensity anddeflects the scattering angle corresponding to the largest RCS.The bias of the beam waist center along the axes in the positiveand negative directions has the same effect on the RCS. Theapplications of this theoretical development in the fields ofbiomedicine, target shielding, and anti-radar coating are alsonumerically discussed. Only the transverse electric (TE) polar-ization is considered in this paper, while a similar formulationof the transverse magnetic (TM) polarization can be obtainedvia duality. The results presented in this paper are hopeful toprovide an effective calibration for research on the scatteringproperties of multilayered anisotropic targets. In the paper,we only consider the uniaxial anisotropic medium and plasmacoated sphere. Scattering for a general anisotropic-coatedsphere in an obliquely incident off-axis Gaussian beam will bediscussed in future research endeavors.

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Zhen-Sen Wu (M’97–SM’04) received the B.Sc.degree in applied physics from Xi’an Jiaotong Uni-versity, Xi’an, China, in 1969 and the M.Sc. degreein space physics from Wuhan University, Wuhan,China, in 1981.

He is currently a Professor at Xidian University,Xi’an, China. From 1995 to 2001, he was invitedmultiple times as a Visiting Professor to Rouen Uni-versity, France, for implementing joint study of twoprojects supported by the Sino-France Program forAdvanced Research. His research interests include

electromagnetic and optical waves in random media, optical wave propagationand scattering, and ionospheric radio propagation.

Zheng-Jun Li received the B.Sc. degree in appliedphysics, from Xidian University, in 2007, where heis currently working towards the Ph.D. degree.

His current work concerns the electromagnetic andoptic scattering by single and multiple anisotropicspheres.

Huan Li, photograph and biography not available at the time of publication.

Qiong-Kun Yuan, photograph and biography not available at the time ofpublication.

Hai-Ying Li, photograph and biography not available at the time of publication.