em scattering from a long dielectric w. z. yan ...in studying scattering and absorption of...

Progress In Electromagnetics Research, PIER 85, 39–67, 2008

EM SCATTERING FROM A LONG DIELECTRICCIRCULAR CYLINDER

W. Z. Yan, Y. Du, H. Wu, and D. W. Liu

The Electromagnetics Academy at Zhejiang UniversityZhejiang UniversityHangzhou, Zhejiang 310058, China

B. I. Wu

Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridge, MA 02139, USA

Abstract—A new iterative technique based on the T -matrix approachis proposed for the electromagnetic scattering by dielectric cylinders,in particular cylinders with large aspect ratios. For such cases theconventional T -matrix approach fails. We use hypothetic surfaces todivide a cylinder into a cluster of N identical sub-cylinder, for each theT matrix can be directly calculated. Since any two neighboring sub-cylinder are touching via the division interface, the conventional multi-scatterer equation method is not directly applicable. The couplingamong sub-cylinder and boundary conditions at the interfaces aretaken care of in our approach. The validity of the proposed method isdemonstrated through agreement between theoretical predictions andnumerical simulations as well as measurements for scattering fromdielectric circular cylinders with finite length. The results clearlydemonstrate that the new iterative technique can extend regular T -matrix approach to solve cylindrical cases with large aspect ratio.

1. INTRODUCTION

There has been growing interest in the investigation of vegetationusing polarimetric remote sensing techniques. During the past severaldecades, a number of theoretical models have been constructed tostudy the scattering mechanisms in the vegetation medium and arevery useful for forest stand or short crops [1–5]. Usually, these

40 Yan et al.

approaches treat the vegetation medium as a random collection ofdiscrete scatterers with different sizes, shapes and orientations. Forinstance, the branches and trunks are usually modelled as finitedielectric cylinders, and in coniferous vegetation needles are used tomodel leaves. Determining the electromagnetic properties of thosekey constituents such as branches and trunks requires knowledge ofthe scattering properties of dielectric cylinders [6–19]. In addition,in studying scattering and absorption of electromagnetic waves fromice needles in clouds, the dielectric cylinder are also used to modelthose needles [20]. Thus, finding an effective method to calculatethe electromagnetic scattering by dielectric finite cylinders motivatedmany authors. Because an exact analytical solution for the scatteringfrom finite cylinders does not exist, several approximations have beenproposed [21–24]. Among them is the generalized Rayleigh-Gans(GRG) approximation, which was widely applied in the studies of thevegetation samples [22, 23]. It approximates the induced current in afinite cylinder by assuming infinite length. Therefore, this method isvalid for a needle shaped scatterer with radius much smaller than thewavelength. Whereafter, Stiles and Sarabandi [24] proposed a moregeneral solution for long and thin dielectric cylinders of arbitrary crosssection, but still limited to small cross sections. Nevertheless, it shouldbe noted that the solutions of such approximate methods in generalfail to satisfy the reciprocity theorem.

In a more general setting, a semi-analytical method named T -matrix approach, originally introduced by Waterman [25] and is basedon the extended boundary condition method (EBCM), is one ofthe most powerful and widely used tools for rigorously computingvolume electromagnetic scattering based on Maxwell’s equations andhas been applied to particles of various shapes, such as spheroids,finite cylinders, Chebyshev particles, cubes, clusters of spheres, and soon [26–31]. In applying extended boundary condition to calculate theT matrix that relates the exciting field and scattered field, the excitingfield is assumed to be inside the inscribing sphere and the scattered fieldoutside the circumscribing sphere, respectively. However, for particleswith extreme geometries represented by very large aspect ratios,regular EBCM is reported to suffer from convergence problems [32].Physically, this ill-conditioning procedure stems from the fact that,since the exciting field is assumed to be inside the inscribing sphere,for cases of extreme geometries, the exciting fields will not be accuraterepresentative of surface currents. Nor will the scattered fields.

One approach for overcoming the problem of numerical instabilityin computing the T matrix for spheroids with large aspect ratio is theso-called iterative extended boundary condition method (IEBCM) [33].

Progress In Electromagnetics Research, PIER 85, 2008 41

The main feature of this technique is to represent the internalfield by several subregion spherical function expansions centeredalong the major axis of the prolate spheroid. The contiguoussubregional expansions are linked to each other by being matched inthe overlapping zones. The set of unknown expanded coefficients ofinternal field can be determined by using the point-matching method(PMM). It has been reported that in some spheroidal cases, the use ofIEBCM instead of the regular EBCM allows to more than quadruplethe maximum convergent size parameter. However, because the firststep in this procedure is to approximate the highly lossy dielectricobject with a perfectly conducting object of the same shape for itsinitial solution, it is restricted by the conductivities of the dielectricparticles and the maximum convergent size parameter of EBCM forsuch perfectly conducting object. Moreover, as pointed in [34], PMMis less flexible in terms of applications to different particle shapes dueto the fact that, the more the particle’s geometry departs from that ofa sphere, the more unsuitable the expansions of the fields in sphericalvector wave functions. Thus, elongated particles require the use ofspecially adapted PMM implementations with longer computationtime and higher computer-code complexity. Another similar techniqueusing PMM to solve scattering from particles enclosed by smoothsurfaces is the general multipole technique (GMT), which representselectromagnetic field vectors by multiple spherical expansions aboutseveral expansion origins which are located at appropriate positionsin the interior region [35]. The GMT has been successfully used forparticles with smooth surfaces, such as hemispherically or sphericallycapped cylinders, yet there are issues when it is used in the scatteringcomputations of finite cylinders with flat ends reported. Recently,null field method with discrete sources (NF-DS) is proposed to dealwith the instability of conventional EBCM [36–39]. Its numericalstability is achieved at the expense of considerable increase in computercomplexity, and the resolution of this method can be affected by thelocalization of the sources.

In this paper, we propose a new iterative technique applicablefor electromagnetic scattering by finite dielectric cylinders with largeaspect ratio. With the understanding that for such cylinders a directapplication of the EBCM often leads to numerical instability, theprocedure starts by dividing the cylinder into several identical sub-cylinder to reduce the aspect ratio for each part to which the EBCMcan be applied. A subtle technical issue arises here: since any twoneighboring sub-cylinder are touching via the division interface, theconventional multi-scatterer equation method is not directly applicablebecause it requires that the circumscribing spheres of the sub-cylinder

42 Yan et al.

exclude each other [40]. Rather, boundary conditions at the divisioninterface need to be satisfied and carefully incorporated into the EBCMformalism. The subtlety lies in the fact that boundary conditionsat the division interfaces are point-wise while the EBCM is in anintegral form. For such concern, we introduce some intermediatevariables that have specific meanings, where the boundary conditionsare incorporated. Moreover, since these variables are expressed interms of surface integrals, the drawbacks of PPM inherent in IEBCM orGMT is avoided. The intercoupling relations of multipole expansionsfor sub-cylinder are constructed with the help of translational additiontheorems and can be solved iteratively. The impact of translationaladdition theorem on the convergence property of the resulting linearsystem is also carefully treated in the iterative procedure.

2. T -MATRIX APPROACH

In T -matrix approach, the incident, scattered, and internal fields areexpressed in terms of the spherical harmonics, respectively

Einc(r′

)=

∑n,m

{ainc,M

mn RgMmn

(r′

)+ ainc,N

mn RgNmn

(r′

)}Esca

(r′

)=

∑n,m

{asca,M

mn Mmn

(r′

)+ asca,N

mn Nmn

(r′

)}Eint

(r′

)=

∑n,m

{aint,M

mn RgMmn

(r′

)+ aint,N

mn RgNmn

(r′

)}, (1)

where RgMmn, RgNmn, Mmn and Nmn are the vector spherical wavesrespectively as defined in [41]. A list of variables used in this paper isprovided at the end of the paper in Table 3 for referential convenience.Owing to the linearity of Maxwell’s equations and boundary conditions,the linear relation between the scattered field coefficients asca on onehand and the incident field coefficients ainc on the other hand is givenby the system transfer operator T as follows [25]:

asca = ¯T ainc. (2)

Eq. (2) is a cornerstone of the T -matrix formulation. Because the Tmatrix includes the full information about the wave scattering andabsorption properties of a particle at a given wavelength, all quantitiesof interest in remote sensing, such as the amplitude scattering matrix,the scattering cross section as well as the expansion coefficients of theStokes scattering matrix can be expressed in terms of the T matrix.This is a important advantage of the T -matrix approach.


The widely utilized scheme for computing the T matrix for simpleparticles is based on EBCM, which is also called the null field approach.As an alternative to solve the surface integral equation, EBCMassumes an inscribing sphere and a circumscribing sphere with theircenters at the origin contained within the scatterer, then applies theextended boundary condition inside the inscribing sphere and outsidethe circumscribing sphere, respectively. Rather than considering thecoupling of the incident and scattered fields directly, the couplingbetween the incident and internal represented by the RgQ matrix, andscattered and internal fields represented by the Q matrix is explicitlytreated. In the EBCM, expressions for matrices RgQ andQ are derivedfrom an integral equation approach which can be found in [41]. Thenthe T matrix can be easily obtained from the following relation betweenthe RgQ matrix and Q matrix

¯T = −Rg ¯Qt ·

(¯Q

t)−1

(3)

where the subscript t represents the transpose of matrix.An another important advantage of the T -matrix approach is that,

unlike many other methods of calculating scattering where the entirecalculation needs to be repeated for each new incident field, the Tmatrix only needs to be calculated once because it is independent ofany specific incident field. Moreover, utilizing geometrical symmetriesof particles can drastically reduce CPU-time requirements. Therefore,for a scatterer with moderate aspect ratio, T -matrix approach isa very effective technique for scattering computation, especially forcalculating the scattering properties of an ensemble of randomlyoriented particles. Fig. 1 clearly demonstrates the accuracy of T -matrixapproach predictions against experimental benchmarks reported byAllen and McCormick [42], where the parameters of two samples arelisted in Table 1 and the imaginary part of dielectric constant is 0.036i.The maximum diameter and maximum length of the samples are 2aand 2c, respectively.

Table 1. Parameters for the samples.

Sample � �

ka 0.305 0.609c/a 2.500 2.503

Re(ε) 3.16 3.14

44 Yan et al.

0 10 20 30 40 50 60 70 80 90-40

-35

-30

-25

-20

-15

-10

-5

0

θ i (Degree)

σ /

λ2 (

dB)

Sample1 (T-matrix )Sample1 (Measurement)Sample2 (T-matrix )Sample2 (Measurement)

Figure 1. Backscattering cross section comparisons between T -matrix approach and measurements of dielectric cylinders excited bya circularly polarized plane wave.

3. TRANSLATIONAL ADDITION THEOREM

Translational addition theorem is a powerful analytic tool to solvetheoretically for the scattering properties of multiple scatterers. Sincethe early studies of Friedman and Russek [43] in the 1950s, considerableefforts have been devoted to the formulation and evaluation of thescalar addition coefficients.

Consider now coordinate systems i and j having the same spatialorientation and denote by rji the vector pointing to the origin ofcoordinate system j from the origin of coordinate system i as illustratedin Fig. 2. Specifically, for rj = ri + rji, where (rj , θj , ϕj), (ri, θi, ϕi)and (rji, θji, ϕji) are their respective spherical coordinates, the scalartranslational addition theorem for the solid translation from thecoordinate system i to the coordinate system j can be expressed as:

ψmn(rj , θj , ϕj) =∑µ,ν

ψµν(ri, θi, ϕi)Cmnµν (rji, θji, ϕji), (4)

where the scalar wave function is

ψmn(r, θ, φ) = Pmn (cos θ)zn(kr)eimφ. (5)

Here a time dependence of e−iωt is assumed and Pmn is the associated

Legendre function. zn is appropiately selected between the following


Figure 2. Translation of coordinates from origin i to origin j.

two functions: the spherical Bessel function of the first kind jn or thespherical Hankel function of the first kind h(1)

n . The scalar translationalcoefficient is given as [43, 44]

Cmnµν (rji, θji, φji) = (−1)mi(ν−n)(2ν + 1)

∑p

ipa(m,n,−µ, ν, p)

×Pm−µp (cos θji)zp(krji)ei(m−µ)φji (6)

where a(m,n, µ, ν, p) is the Gaunt coefficient, which was firstintroduced by Gaunt [45] in his study of the atomic structure of heliumtriplets. In 1971, Bruning and Lo [46] derived a recursive schemeto calculate the Gaunt coefficients for the vector addition theoremefficiently for a special case of bi-sphere scattering problem in whichthe translation was along the z axis. However, the calculation ofthe required number of Gaunt coefficients represents a tremendousexpenditure of effort when the truncation number is large. Analternative is the recursive approach devised by Mackowski [47]. Byutilizing the recurrence relations for the scalar translational additioncoefficients [48] and the starting expression C00

µν , the scalar additioncoefficients can be obtained for all values of m, n, µ and ν.

For the case of vector spherical wave functions, the vector addition

46 Yan et al.

theorems have the form[RgMmn(rj)RgNmn(rj)

]=

∑µ,ν

{ [RgAmn

µν (rji)RgBmn

µν (rji)

]RgMµν(ri)

+[RgBmn

µν (rji)RgAmn

µν (rji)

]RgNµν(ri)

}(7)

and [Mmn(rj)Nmn(rj)

]=

∑µ,ν

{ [Amn

µν (rji)Bmn

µν (rji)

]RgMµν(ri)

+[Bmn

µν (rji)Amn

µν (rji)

]RgNµν(ri)

}(8)

for the condition ri < rji, where the vector spherical waves withand without the prefix Rg stand for regular and outgoing waves,respectively.

Cruzan formulated the vector translation coefficients using theWigner 3j symbols [49] as

Amnµν = (−1)µ iν−n 2ν + 1

2ν(ν + 1)

∑p

ip[ν(ν + 1) + n(n+ 1) − p(p+ 1)]

×a (m,n,−µ, ν, p) zp (krji)Pm−µp (cos θji) exp[i(m− µ)φji]

Bmnµν = (−1)µ+1 iν−n 2ν + 1

2ν(ν + 1)

×∑

p

ip√

[p2 − (n+ ν)2][(n+ ν + 1)2 − p2]

×a(m,n,−µ, ν, p, p− 1)zp(krji)×Pm−µ

p (cos θji) exp[i(m− µ)φji], (9)

where

a(m,n,−m, ν, p, p− 1) = (2p+ 1)[(n+m)!(ν −m)!(n−m)!(ν +m)!

]1/2

×(n ν pm µ 0

) (n ν p− 10 0 0

). (10)

Later Tsang and Kong [50] reported a sign error in Cruzan’s translationformulas.


In addition, both Stein [44] and Mackowski [47] expressed thevector translational addition coefficients in terms of seven and six scalartranslational addition coefficients, respectively. In fact, as Mackowskipointed out, these two formulations can be converted to each other [47].The expressions by Mackowski are given by

Amnµν =

(ν − µ)(ν + µ+ 1)Cm+1nµ+1ν

+2µmCmnµν + (n+m)(n−m+ 1)Cm−1n

µ−1ν

2ν(ν + 1)

Bmnµν =

i(2ν + 1)2ν(ν + 1)(2ν + 3)

[(ν + µ+ 1)(ν + µ+ 2)Cm+1n

µ+1ν+1

−2m(ν + µ+ 1)Cmnµν+1−(n+m)(n−m+ 1)Cm−1n

µ−1ν+1

]. (11)

It follows that this set of equations do not need geometrical informationof the volumes. This makes Mackowski’s formulations more convenientin practical computations, and we choose to use these formulations inour paper.

4. PROPOSED METHOD

Consider a dielectric cylinder with large aspect ratio. We use N − 1hypothetic surfaces to divide it into a cluster of N identical sub-cylinder, for each the T matrix can be directly calculated by usingthe conventional EBCM. The simple case of two sub-cylinder divisionis depicted in Fig. 3. Such case will serve as the basis of our analysis,since the extension to an N sub-cylinder case is straightforward. Nowthat these two sub-cylinder are touching via the division interface,

E i E i

Primary Surface (S2p )

Contact Surface (S12 )

Primary Surface (S1p )

Figure 3. Division of a cylinder into two identical sub-cylinder.

48 Yan et al.

a subtle technical issue arises here: the conventional multi-scattererequation method is not directly applicable because it requires that thecircumscribing spheres of the sub-cylinder exclude each other [40]. It isnecessary that boundary conditions at the division interface be satisfiedand carefully incorporated into the EBCM formalism. The subtlety liesin the fact that boundary conditions at the division interface are point-wise while the EBCM is in an integral form. For such concern, we shallintroduce some intermediate variables that have specific meanings,where the boundary conditions are incorporated.

Since the EBCM involves surface integrals and since cylinderdivision generates hypothetical interfaces, before we proceed we needto denote these surfaces carefully. Each sub-cylinder will contain aprimary surface and an interface. We shall start number orderingfrom the lowest sub-cylinder (see Fig. 3). The primary surface of sub-cylinder 1, S1p, includes its cylindrical surface and the lower surface,while that of sub-cylinder 2, S2p, includes its cylindrical surface andthe upper surface. The common interface is denoted by S12. Thecenter of sub-cylinder 1 is r1 and that of sub-cylinder 2 is r2. UtilizingT matrix, the expanded coefficients amn and as

mn for each part arerelated as follows[

as(M)(1)pmn

as(N)(1)pmn

]+

[a

s(M)(1)umn

as(N)(1)umn

]= ¯T ·

{[a

(M)(1)pmn

a(N)(1)pmn

]+

[a

(M)(1)umn

a(N)(1)umn

]}[a

s(M)(2)pmn

as(N)(2)pmn

]+

[a

s(M)(2)dmn

as(N)(2)dmn

]= ¯T ·

{[a

(M)(2)pmn

a(N)(2)pmn

]+

[a

(M)(2)dmn

a(N)(2)dmn

]}. (12)

Now for fields expanded in terms of vector spherical waves withdifferent origins, we introduce the following intermediate variables:[

a(M)(1)pmn

a(N)(1)pmn

]= −ik(−1)m

∫S1p

dS

{iωµn1 × H1(r) ·

[M−mn(rr1)N−mn(rr1)

]

+kn1 × E1(r) ·[N−mn(rr1)M−mn(rr1)

] }(13)[

a(M)(2)pmn

a(N)(2)pmn

]= −ik(−1)m

∫S2p

dS

{iωµn2 × H2(r) ·


]


] }(14)

Progress In Electromagnetics Research, PIER 85, 2008 49[a

s(M)(1)pmn

as(N)(1)pmn

]= ik(−1)m

∫S1p

dS

{iωµn1 × H1(r) ·

[RgM−mn(rr1)RgN−mn(rr1)

]

+kn1 × E1(r) ·[RgN−mn(rr1)RgM−mn(rr1)

] }(15)[

as(M)(2)pmn

as(N)(2)pmn

]= ik(−1)m

∫S2p

dS

{iωµn2 × H2(r) ·


]


] }(16)

where the superscript p denotes the primary part, and s the scatteredfield. These intermediate variables are not arbitrary quantitiesbut have specific physical meanings. They represent the expansioncoefficients of the exciting fields and scattered fields due to theprimary surfaces of these two sub-cylinder, respectively. Similarly, therespective expansion coefficients of the exciting fields and scatteredfields due to the interface of these two sub-cylinder are[

a(M)(1)umn

a(N)(1)umn

]= −ik(−1)m

∫S12

dS

{iωµn1 × H1(r) ·


]


] }(17)[

a(M)(2)dmn

a(N)(2)dmn

]= −ik(−1)m

∫S12

dS

{iωµn2 × H2(r) ·


]


] }(18)[

as(M)(1)umn

as(N)(1)umn

]= ik(−1)m

∫S12

dS

{iωµn1 × H1(r) ·


]


] }(19)[

as(M)(2)dmn

as(N)(2)dmn

]= ik(−1)m

∫S12

dS

{iωµn2 × H2(r) ·


]


] }. (20)

50 Yan et al.

In the above, ni is the outward pointing unit normal vectors on thesurface Si of sub-cylinder i. We have n2 = −n1 on the surface S12.

Since the terms a(1)umn and a

s(2)dmn are expressed in terms of the

tangential fields on the interface, it is natural to relate them in someway. Yet since each involves vector spherical waves with differentorigins, a transformation of origin must first be performed usingthe translational addition theorem. For the latter task we have theconvenience of reducing the double summation over m and ν to singlesummation over ν because the translation is along the z axis. Shiftingthe origin from r1 to r2, the term a

(1)umn can thus be expressed as[

a(M)(1)umn

a(N)(1)umn

]= −ik (−1)m

∫S1,u

dSiωµn1 × H1 (r)

·

∑ν

{A−mn

−mν (r1r2)RgM−mν(rr2) +B−mn−mν (r1r2)RgN−mν(rr2)

}∑ν

{A−mn

−mν (r1r2)RgN−mν(rr2) +B−mn−mν (r1r2)RgM−mν(rr2)

}

−ik(−1)m

∫S1,u

dSkn1 × E1(r)

·

∑ν

{A−mn

−mν (r1r2)RgN−mν(rr2) +B−mn−mν (r1r2)RgM−mν(rr2)

}∑ν

{A−mn

−mν (r1r2)RgM−mν(rr2) +B−mn−mν (r1r2)RgN−mν(rr2)

}

=∑

ν

A−mn−mν (r1r2)ik(−1)m

∫S2,d

dSiωµn2 × H2(r) ·[RgM−mn(rr2)RgN−mn(rr2)

]


]

+∑

v

B−mn−mν (r1r2)ik(−1)m

∫S2,d

dSiωµn2 × H2(r) ·[RgN−mn(rr2)RgM−mn(rr2)

]

+kn2 × E2(r) ·[RgM−mn(rr2)RgN−mn(rr2)

]. (21)

In the second equality of the above equation, we specifically make useof the boundary conditions

n1 × H1 = −n2 × H2, n1 × E1 = −n2 × E2. (22)

The intermediate variables of the second sub-cylinder (20) allow us to


further express (21) as[a

(M)(1)umn

a(N)(1)umn

]=

∑ν

A−mν−mn(r1r2)

[a

s(M)(2)dmv

as(N)(2)dmv

]

+∑

ν

B−mν−mn(r1r2)

[a

s(N)(2)dmv

as(M)(2)dmv

]. (23)

Note that for a single scatterer, the incident field is equal to theexciting field. In our case, the virtual partition shall not change thisproperty. Now if we let the global origin coincidence with r1, we have[

a(M)mn

a(N)mn

]= −ik(−1)m

∫S1,p

dSiωµn1 × H1(r) ·[M−mn(rr1)N−mn(rr1)

]


]

−ik(−1)m

∫S2,p

dSiωµn2 × H2(r) ·[M−mn(rr1)N−mn(rr1)

]


]. (24)

By applying transformation of origin on the vector spherical wavesin the second surface integral, and making use of the intermediatevariables (16), we obtain[

a(M)mn

a(N)mn

]=

[a

(M)(1)pmn

a(N)(1)pmn

]−

∑ν

A−mn−mν (r1r2)

[a

s(M)(2)pmν

as(N)(2)pmν

]

−∑

ν

B−mn−mν (r1r2)

[a

s(N)(2)pmν

as(M)(2)pmν

]. (25)

Combining (23) and (25), and using variable substitution, yields[a

(M)mn

a(N)mn

]=

[a

(M)(1)mn

a(N)(1)mn

]−

∑ν

A−mn−mν (r1r2)

[a

s(M)(2)mν

as(N)(2)mν

]

−∑

ν

B−mn−mν (r1r2)

[a

s(N)(2)mν

as(M)(2)mν

](26)

52 Yan et al.

where

a(M)(1)mn = a(M)(1)u

mn + a(M)(1)pmn (27)

a(N)(1)mn = a(N)(1)u

mn + a(N)(1)pmn (28)

as(M)(2)mn = as(M)(2)d

mn + as(M)(2)pmn (29)

as(N)(2)mn = as(N)(2)d

mn + as(N)(2)pmn . (30)

Similarly, we can also establish the following system of equationsby focusing on the upper part of cylinder as follows[

a(M)mn

′

a(N)mn

′

]=

[a

(M)(2)mn

a(N)(2)mn

]−

∑ν

A−mn−mν (r2r1)

[a

s(M)(1)mν

as(N)(1)mν

]

−∑

ν

B−mn−mν (r2r1)

[a

s(N)(1)mν

as(M)(1)mν

], (31)

where the multipole coefficients of the incident plane wave a(M)mn

′and

a(N)mn

′differ from a

(M)mn and a(N)

mn by a factor eikhcosθi . Here h is theheight of each sub-cylinder.

The scattered field can be treated similarly. Applying thetranslational addition theorems (7) to the (19) and the followingequation, respectively,[

as(M)mn

as(N)mn

]= ik(−1)m

∫S

dS

{iωµn× H(r) ·


]

+kn× E(r) ·[RgN−mn(rr1)RgM−mn(rr1)

] }

= ik(−1)m

∫S1,p

dS

{iωµn1 × H1(r) ·


]


] }

+ ik(−1)m

∫S2,p

dS

{iωµn2 × H2(r) ·


]


] }(32)


we can easily derive the following equations[a

s(M)(1)umn

as(N)(1)umn

]= −

∑ν

RgA−mn−mν (r1r2)

[a

s(M)(2)dmn

as(N)(2)dmn

]

−∑

ν

RgB−mn−mν (r1r2)

[a

s(N)(2)dmν

as(M)(2)dmν

](33)

and [a

s(M)mn

as(N)mn

]=

[a

s(M)(1)pmn

as(N)(1)pmn

]+

∑ν


[a

s(M)(2)pmν

as(N)(2)pmν

]

+∑

ν


[a

s(N)(2)pmν

as(M)(2)pmν

]. (34)

Combining the above two equations allows us to express thetotal scattered coefficients of whole cylinder in the primary coordinatesystem as[

as(M)mn

as(N)mn

]=

[a

s(M)(1)mn

as(N)(1)mn

]+

∑ν


[a

s(M)(2)mν

as(N)(2)mν

]

+∑

ν


[a

s(N)(2)mν

as(M)(2)mν

]. (35)

It is clear that the solutions of these coupled, linear, simultaneousequations can be easily obtained by iterative method. The iterationprocedure is summarized as follows: Starting with the initial solutionsa

s(M),2mn = a

s(N),2mn = 0. From (26) and by using T matrix we obtain

the new values of the scattering coefficients as,1mn in (31). In a similar

manner, we next use the new values to obtain the scattering coefficientsas,2

mn in (26). The procedure is repeated until all the coefficientsconverge.

For the specific case of far zone, the expansion coefficients of thetotal scattered field can be simply expressed as[

as(M)totalmn

as(N)totalmn

]=

[a

s(M)(1)mn

as(N)(1)mn

]+

[a

s(M)(2)mn

as(N)(2)mn

]· e−ikh cos θs (36)

If we are to use the widely used amplitude scattering matrix Fpq(ks, ki)to describe the relation between the amplitudes of the incident and

54 Yan et al.

the scattered fields and represent general properties of the scatterer,its expression is given by

Fqp(ks, ki) =∑m,n

{q · as(M)total

mn Mmn(kr, θ, φ)

+q · as(N)totalmn Nmn(kr, θ, φ)

}· p (37)

where p and q are unit polarizations for the incident and scatteredwave, respectively. The expressions of far-field solutions for theoutgoing vector spherical waves can be found in [41].

5. ERROR ANALYSIS

In the implementation of our proposed approach, since it involvesboth T matrix and translational addition theorem, and since it is wellknown that there are truncation errors and/or other types of errorsassociated with these two aspects, it is expected that any version ofimplementation will bear the impact of these errors to varying degree,depending on the nature of the problem at hand and the specifics ofthe implementation. In the following we shall discuss these issues inturn.

For T matrix computations, theoretically the multipole expansionsare of infinite length and the T matrix is of infinite size. Howeverin practical calculations, the summation over m and n has to betruncated, thus a special procedure should be used to check theconvergence of the resulting solution over the size of T matrix Nmax.For axially symmetric scatterers, the T matrix can be decomposedinto separate submatrices corresponding to different azimuthal modesm which are independently calculated. For a desired absolute accuracyof computing the expansion coefficients ∆ (∆ = 10−4), the followingsimple convergence criterion can be used to determine Nmax [52]

max[∣∣∣∣c1(Nmax) − c1(Nmax − 1)

c1(Nmax)

∣∣∣∣ ,∣∣∣∣c2(Nmax) − c2(Nmax − 1)

c2(Nmax)

∣∣∣∣]≤ ∆,

(38)

where

c1(Nmax) = −2πk2

ReNmax∑n=1

(2n+ 1)(T 11

0n0n + T 220n0n

)(39)

c2(Nmax) =2πk2

Nmax∑n=1

(2n+ 1)[∣∣T 11

0n0n

∣∣2 +∣∣T 22

0n0n

∣∣2] (40)


10 20 30 40 50 60 70 80 90 10010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Nmax

Rel

ativ

e E

rror

(a) ρ = 1

10 20 30 40 50 60 70 80 90 10010

-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Nmax

Rel

ativ

e E

rror

(b) ρ = 1 / 2

10 20 30 40 50 60 70 80 90 10010

-5

10-4

10-3

10-2

10-1

100

101

102

Nmax

Rel

ativ

e E

rror

(c) ρ = 1 / 4

Figure 4. Relative errors of the cylindrical cases, where rv is 5.

In Fig. 4, we plot the relative errors represented by the left side of (38)for three cylindrical cases, where the equal-volume sphere radius rv isidentically set to 5, and the ratio of the horizontal to rotational axesρ changes from 1 to 0.25. In this figure we clearly see that the relativeerrors do not monotonically decrease with increasing Nmax; rather,they demonstrate oscillating behaviors. Such behaviors indicate theneed to be very careful and specific when determining Nmax in applyingT -matrix approach.

For numerical computation of the translational addition coeffi-cients, the order of dipole moment has to be truncated as well whichresults in truncation error. In a truncation error analysis for the scalarspherical addition theorem [51], it was found that the truncation errordecreases as the truncation order Vmax increases. Yet in the presentanalysis, things become more complicated in that the truncation opera-

56 Yan et al.

tions in applying addition theorem and in applying T -matrix approachare convoluted; specifically, the truncation order Vmax in applying ad-dition theorem cannot be larger than Nmax in applying T -matrix ap-proach. Such constraint may lead to truncation errors that would notshow if these two operations were independent.

In order to examine the relative error behavior, we shall compareseveral cases with different translational scenarios, where varioustranslational distances rji are used and ri is identically set to 1. Weconsider the term M1,5 (rj) which is approximated by

M1,5 (rj) =Nmax∑n=1

{A1,5

1,n (rji)RgM1,n (ri) +B1,51,n (rji)RgN1,n (ri)

}.

(41)

Relative error values of M1,5 (rj) are shown in Fig. 5. As shown in thisfigure, the relative error can be appreciable when insufficient ordersare used in applying the addition theorem. We can make followingobservations: 1) the relative error decreases when the translationaldistance increases; 2) the spherical waves of higher order n requireslarger expansion orders; 3) the convergence can be improved whenthe expansion orders increase (from 5 to 35 in this case). Similarobservations can be made from Fig. 6, where ri is identically set to 3.

5 10 15 20 25 30 3510

-16

10 -14

10 -12

10 -10

10 -8

10 -6

10 -4

10 -2

100

102

Nmax

Rel

ativ

e E

rror

Vector spherical wave Mmn

for m=1, n=5

d=2*ri

d=3*ri

d=4*ri

Figure 5. Relative errors due totruncations in applying additiontheorem for various rji where riis 1.

10 15 20 25 30 35 40 45 5010

-16

10 -14

10 -12

10 -10

10 -8

10 -6

10 -4

10 -2

100

102

104

Nmax

Rel

ativ

e E

rror

Vector spherical wave Mmn

for m=1, n=10

d=2*ri

d=3*ri

d=4*ri

Figure 6. Relative errors due totruncations in applying additiontheorem for various rji where riis 3.

Bearing the impact of truncation errors in applying thetranslational addition theorems and T -matrix approach, when theeccentricity of the scatterer is large, the iterative procedure may


become an ill-conditioned one, for which caution needs to be taken. Weuse a method similar to the successive over-relaxation method (SOR),which takes the form of a weighted average between the previousiterate and the computed new iterate successively for each componentas reported in [53] where multi-sphere system was considered. Anextrapolation factor w (0 ≤ w ≤ 1) is used to speed up the convergencyprocedure.

6. NUMERICAL RESULTS

In this section, we compare the theoretical predictions of the proposediterative approach with numerical simulations, as well as measurementsfor scattering from dielectric circular cylinders with finite length. Forcases where the ratio of horizontal to rotational axes ρ, the equal-volume sphere radius rv and the refractive index fall in the region wherethe conventional EBCM is applicable, we expect that our proposedmethod should provide comparable results. To test this preposition,various size parameters of cylinders are used with rv ranging from 0.5to 3, ρ from 1 to 1/3, and the refractive index is 1.5+0.02i. For linearpolarization the scattering cross section is defined as

σpq = 4π∣∣∣Fpq(ks, ki)

∣∣∣2 , (42)

and for circular polarization

σ = π∣∣∣Fvv

(ks, ki

)± Fhh

(ks, ki

)∣∣∣2 (43)

where the + and − signs stand for left-hand and right-hand circularpolarization, respectively.

The results are shown in Figs. 7–12. One can observe from thefigures that both the vertically-polarized and horizontally-polarizedbistatic scattering cross sections obtained by our method are in perfectagreements with that of T -matrix approach when rv is smaller than 3.

Table 2. Parameters for the samples.

Sample � � � �

ka 0.1143 0.1904 0.2666 0.3428c/a 10.00 9.99 9.99 10.00

Re(ε) 3.13 3.13 3.15 3.14

58 Yan et al.

0 20 40 60 80 100 120 140 160 180-70

-60

-50

-40

-30

-20

-10

s (Degree)

vv (

dB)

Proposed MethodT-matrix Method

(a) Vertical polarization

0 20 40 60 80 100 120 140 160 180-18.8

-18.75

-18.7

-18.65

-18.6

-18.55

-18.5

-18.45

-18.4

s (Degree)

hh (

dB)


(b) Horizontal polarization

θ θ

σ σ

Figure 7. The bistatic scattering cross section of a cylinder under ourproposed method and EBCM at broadside incidence where rv is 0.5and ρ is 1/2.

0 20 40 60 80 100 120 140 160 180-3.2

-3

-2.8

-2.6

-2.4

-2.2

-2

-1.8

-1.6

s (Degree)

hh (

dB)


0 20 40 60 80 100 120 140 160 180-60

-50

-40

-30

-20

-10

0

10

s (Degree)

vv (

dB)


θ θ

σ σ

(a) Vertical polarization (b) Horizontal polarization

Figure 8. The bistatic scattering cross section of a cylinder under ourproposed method and EBCM at broadside incidence where rv is 1 andρ is 1/2.

Comparison between the theoretical predictions of our methodand experimental data is shown in Fig. 13, where the experimentaldata are taken from [42] for different samples with the parameterslisted in Table 2. The imaginary part of dielectric constant is 0.036i.The overall agreement is good except at locations near the null wherethe experimental results are much higher. Similar discrepancy with theGRG approximation was also reported by Karam et al. [22] where theysuggested that the discrepancy may be due to the finite equivalent


0 20 40 60 80 100 120 140 160 180-15

-10

-5

0

5

10

15

20

s (Degree)

vv (

dB)


0 20 40 60 80 100 120 140 160 1805

6

7

8

9

10

11

12

13

s (Degree)

hh (

dB)


θ θ

σ σ


Figure 9. The bistatic scattering cross section of a cylinder under ourproposed method and EBCM at broadside incidence where rv is 3 andρ is 1/2.

0 20 40 60 80 100 120 140 160 180-65

-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

s (Degree)

vv (

dB)


0 20 40 60 80 100 120 140 160 180-24

-23

-22

-21

-20

-19

-18

-17

s (Degree)

hh (

dB)


θ θ

σ σ


Figure 10. The bistatic scattering cross section of a cylinder underour proposed method and EBCM at end-fire incidence where rv is 0.5and ρ is 1/3.

bandwidth of the measurement system. In addition, they expectedthis difference not to be important when dealing with scattering fromrandomly oriented cylinders where returns from volumes are summedincoherently. It is clear from Fig. 13 that our method outperforms theGRG approximation.

Finally we compare the backscattering coefficients obtained fromour proposed method for both vertical and horizontal polarizations

60 Yan et al.

0 20 40 60 80 100 120 140 160 180-60

-50

-40

-30

-20

-10

0

10

s (Degree)

vv (

dB)


0 20 40 60 80 100 120 140 160 180-35

-30

-25

-20

-15

-10

-5

0

5

s (Degree)

hh (

dB)


θ θ

σ σ


Figure 11. The bistatic scattering cross section of a cylinder underour proposed method and EBCM at end-fire incidence where rv is 1and ρ is 1/3.

0 20 40 60 80 100 120 140 160 180-10

-5

0

5

10

15

20

25

30

35

s (Degree)

vv (

dB)


0 20 40 60 80 100 120 140 160 180-10

-5

0

5

10

15

20

25

30

35

s (Degree)

hh (

dB)


θ θ

σ σ


Figure 12. The bistatic scattering cross section of a cylinder underour proposed method and EBCM at end-fire incidence where rv is 3and ρ is 1/3.

with that by GRG approximation when it is applicable. Fig. 14illustrates the results for a needle case, where ka is 0.7, ρ is 1/25 andthe refractive index is 1.5 + 0.02i. The results show good agreementbetween the two methods.


(a) (b) (c)

σλ

θ

Sample1

Sample2

Sample3

Sample4

0

-10

-20

-30

-40

-50

-600 30 60 90

i (Degree)

/2

(dB

)

σλ

θ0 30 60 90

i (Degree)

/2

(dB

)

σλ

θ0 30 60 90

i (Degree)

/2

(dB

)

Sample1

Sample2

Sample3

Sample4

Sample1

Sample2

Sample3

Sample4

0

-10

-20

-30

-40

-50

-60

0

-10

-20

-30

-40

-50

-60

Figure 13. Comparisons among our proposed method, GRGapproximation and the measured backscattering cross section ofdielectric cylinders excited by a circularly polarized plane wave. (a)Measurement datas. (b) Prediction by our method. (c) Predictions byKaram.

0 10 20 30 40 50 60 70 80 90-110

-100

-90

-80

-70

-60

-50

-40

i (Degree)

hh

GRGA VVGRGA HHProposed Method VVProposed Method HH

θ

σ

Figure 14. Comparison of backscattering cross sections of a cylinderbetween our proposed method and GRG approximation.

62 Yan et al.

Table 3. This list includes the variables in this paper.

Variable DefinitionRgMmn, RgNmn regular vector spherical wavesMmn, Nmn outgoing vector spherical wavesainc incident field coefficientsasca scattered field coefficients

mn scalar wave functionPm

n associated Legendre functionzn spherical Bessel function or spherical Hankel functiona(m,n, , , p) Gaunt coefficientCmn scalar translational coefficientAmn, Bmn vector translational coefficientSip primary surface of sub-cylinder iSij interface between sub-cylinder i and j

a(i)pmn primary coefficients of sub-cylinder i

as(i)pmn primary scattered field coefficients of sub-cylinder i

a(i)umn upper tangential field coefficients on the interface of sub-cylinder i

as(i)umn upper tangential scattered field coefficients on the interface of sub-cylinder i

a(i)dmn lower tangential field coefficients on the interface of sub-cylinder i

as(i)dmn lower tangential scattered field coefficients on the interface of sub-cylinder iFpq(ks, ki) amplitude scattering matrixp unit polarizations for the incident waveq unit polarizations for the scattered wave

ratio of horizontal to rotational axesrv equal-volume sphere radius

scattering cross section

¯

ψ

µ ν

σ

ρ

¯¯ ¯

¯¯

µν µν

¯¯¯¯

¯

7. CONCLUSION

In this paper, we proposed a new iterative technique to analyzescattering from dielectric circular cylinders with finite length. Itwas demonstrated that the new method can extend regular T -matrixapproach to solve cylindrical cases with large aspect ratio. The newmethod can also be used for finite dielectric cylinders with arbitrarycross section as long as the T matrix of each sub-cylinder can beaccurately obtained. Moreover, the new method holds the potential formulti-cylinder problems, especially for the cases of closely positionedcylinders.

ACKNOWLEDGMENT

This work was partly supported by China National Science Foundationunder Grant No. 40571114 and Zhejiang Science Foundation underGrant No. Y106443.


REFERENCES

1. Ulaby, F. T., K. Sarabandi, K. McDonald, M. Whitt, andM. C. Dobson, “Michigan microwave canopy scattering model,”Int. J. Remote Sensing, Vol. 38, No. 7, 2097–2128, 2000.

2. Yueh, S. H., J. A. Kong, J. K. Jao, R. T. Shin, and T. L. Toan,“Branching model for vegetation,” IEEE Trans. Geosci. RemoteSensing, Vol. 30, 390–402, 1992.

3. Chen, Z., L. Tsang, and G. Zhang, “Appication of stochasticlindenmayer systems to the study of collective and clusterscattering in microwave remote sensing of vegetation,” ProgressIn Electromagnetics Research, PIER 14, 233–277, 1996.

4. Chiu, T. and K. Sarabandi, “Electromagnetic scattering fromshort branching vegetation,” IEEE Trans. Geosci. RemoteSensing, Vol. 38, No. 2, 911–925, 2000.

5. Du, Y., Y. L. Luo, W. Z. Yan, and J. A. Kong, “Anelectromagnetic scattering model for soybean canopy,” ProgressIn Electromagnetics Research, Vol. 79, 209–223, 2008.

6. Jin, J. M., V. V. Liepa, and C. T. Tai, “A volume-surfaceintegral equation for electromagnetic scattering by inhomogeneouscylinders,” J. Electromagn. Waves and Appl., Vol. 2, 573–588,1988.

7. Elsherbeni, A. Z., M. Hamid, and G. Tian, “Iterative scattering ofa Gaussian beam by an array of circular conducting and dielectriccylinders,” J. Electromagn. Waves and Appl., Vol. 7, 1323–1342,1993.

8. Konistis, K. and J. L. Tsalamengas, “Plane wave scatteringby an array of bianisotropic cylinders enclosed by another onein an unbounded bianisotropic space: Oblique incidence,” J.Electromagn. Waves and Appl., Vol. 11, 1073–1090, 1997.

9. Naqvi, Q. A. and A. A. Rizvi, “Low contrast circular cylinderburied in a grounded dielectric layer,” J. Electromagn. Waves andAppl., Vol. 12, 1527–1536, 1998.

10. Wang, L. F., J. A. Kong, K. H. Ding, T. Le Toan, F. Ribbes,and N. Floury, “Electromagnetic scattering model for rice canopybased on Monte Carlo simulation,” Progress In ElectromagneticsResearch, PIER 52, 153–171, 2005.

11. Arslan, A. N., J. Pulliainen, and M. Hallikainen, “Observationsof L- and C-band backscatter and a semi-empirical backscatteringmodel approach from a forest-snow-ground system,” Progress InElectromagnetics Research, PIER 56, 263–281, 2006.

12. Ruppin, R., “Scattering of electromagnetic radiation by a perfect

64 Yan et al.

electromagnetic conductor cylinder,” J. Electromagn. Waves andAppl., Vol. 20, 1853–1860, 2006.

13. Hamid, A. K., “Multi-dielectric loaded axially slotted antenna oncircular or elliptic cylinder,” J. Electromagn. Waves and Appl.,Vol. 20, 1259–1271, 2007.

14. Henin, B. H., A. Z. Elsherbeni, and M. H. Al Sharkawy, “Obliqueincidence plane wave scattering from an array of circular dielectriccylinders,” Progress In Electromagnetics Research, PIER 68, 261–279, 2007.

15. Zhang, Y. J. and E. P. Li, “Fast multipole accelerated scatteringmatrix method for multiple scattering of a large number ofcylinders,” Progress In Electromagnetics Research, PIER 72, 105–126, 2007.

16. Valagiannopoulos, C. A., “Electromagnetic scattering from twoeccentric metamaterial cylinders with frequency-dependent per-mittivities differing slightly each other,” Progress In Electromag-netics Research B, Vol. 3, 23–34, 2008.

17. Illahi, A., M. Afzaal, and Q. A. Naqvi, “Scattering of dipolefield by a perfect electromagnetic conductor cylinder,” ProgressIn Electromagnetics Research Letters, Vol. 4, 43–53, 2008.

18. Svezhentsev, A. Y., “Some far field features of cylindricalmicrostrip antenna on an electrically small cylinder,” Progress InElectromagnetics Research B, Vol. 7, 223–244, 2008.

19. Ahmed, S. and Q. A. Naqvi, “Electromagnetic scattering from aperfect electromagnetic conductor cylinder buried in a dielectrichalf-space,” Progress In Electromagnetics Research, PIER 78, 25–38, 2008.

20. Yeh, C., R. Woo, A. Ishimaru, and J. Armstrong, “Scatteringby single ice needles and plates at 30 GHz,” Radio Sci., Vol. 17,1503–1510, 1982.

21. Schiffer, R. and K. O. Thielheim, “Light scattering by dielectricneedles and disks,” J. Appl. Phys., Vol. 50, No. 4, 2476–2483,1979.

22. Karam, M. A. and A. K. Fung, “Electromagnetic wave scatteringfrom some vegetation samples,” IEEE Trans. Geosci. RemoteSensing, Vol. 26, 799–808, 1988.

23. Karam, M. A. and A. K. Fung, “Electromagnetic wave scatteringfrom some vegetation samples,” Int. J. Remote Sensing, Vol. 9,1109–1134, 1988.

24. Stiles, J. M. and K. Sarabandi, “A scattering model for thindielectric cylinders of arbitrary crosssection and electrical length,”


IEEE Trans. Antennas Propag., Vol. 44, 260–266, 1996.25. Waterman, P. C., “Matrix formulation of electromagnetic

scattering,” Proc. IEEE, Vol. 53, 805–811, 1956.26. Mishchenko, M. I. and L. D. Travis, “T -matrix computations of

light scattering by large spheroidal particles,” Opt. Commun.,Vol. 109, 16–21, 1994.

27. Roussel, H., W. C. Chew, F. Jouvie, and W. Tabbara,“Electromagnetic scattering from dielectric and magnetic gratingsof fibers: A T-matrix solution,” J. Electromagn. Waves and Appl.,Vol. 10, 109–127, 1996.

28. Mishchenko, M. I., L. D. Travis, and A. Macke, “Scattering oflight by polydisperse, randomly oriented, finite circular cylinders,”Appl. Opt., Vol. 35, 4927–4940, 1996.

29. Mishchenko, M. I., L. D. Travis, and D. W. Mackowski, “T -matrix computations of light scattering by nonspherical particles:A review,” J. Quant. Spectrosc. Radiat. Transfer, Vol. 55, 535–575, 1996.

30. Wielaard, D. J., M. I. Mishchenko, A. Macke, and B. E. Carlson,“Improved T -matrix computations for large, nonabsorbing andweakly absorbing nonspherical particles and comparison withgeometrical optics approximation,” Appl. Opt., Vol. 36, 4305–4313, 1997.

31. Mishchenko, M. I., “Calculation of the amplitude matrix for anonspherical particle in a fixed orientation,” Appl. Opt., Vol. 39,1026–1031, 2000.

32. Barber, P. W., “Resonance electromagnetic absorption by non-spherical dielectric objects,” IEEE Trans. Microwave TheoryTech., Vol. 25, 373–371, 1954.

33. Iskander, M., A. Lakhtakia, and C. Durney, “A new procedurefor improving the solution stability and extending the frequencyrange of the EBCM,” IEEE Trans. Antennas Propag., Vol. 31,317–324, 1983.

34. Kahnert, F. M., “Numerical methods in electromagneticscattering theory,” J. Quant. Spectrosc. Radiat. Transfer, Vol. 79–80, No. 2, 755–824, 2003.

35. Al-Rizzo, H. M. and J. M. Tranquilla, “Electromagnetic wavescattering by highly elongated and geometrically compositeobjects of large size parameters: The generalized multipoletechnique,” Appl. Opt., Vol. 34, 3502–3521, 1995.

36. Doicu, A. and T. Wriedt, “Calculation of the T matrix in the null-field method with discrete sources,” J. Opt. Soc. Am., Vol. 16,

66 Yan et al.

2539–2544, 1999.37. Eremina, E., Y. Eremin, and T. Wriedt, “Extension of the discrete

sources method to light scattering by highly elongated finitecylinders,” J. Modern Opt., Vol. 51, No. 3, 423–435, 2004.

38. Pulbere, S. and T. Wriedt, “Light scattering by cylindrical fiberswith high aspect ratio using the null-field method with discretesources,” Part. Part. Syst. Charact., Vol. 21, 213–218, 2004.

39. Wriedt, T., R. Schuh, and A. Doicu, “Scattering by aggregatedfibres using a multiple scattering T-matrix approach,” Part. Part.Syst. Charact., Vol. 25, 74–83, 2008.

40. Tsang, L., J. A. Kong, and R. T. Shin, Theory of MicrowaveRemote Sensing, Wiley Interscience, New York, 1985.

41. Tsang, L., J. A. Kong, K. H. Ding, and C. O. Ao, Scattering ofElectromagnetic Waves — Numerical Simulations, John Wiley &Sons, Inc., New York, 2001.

42. Allan, L. E. and G. M. McCormick, “Measurements of thebackscatter matrix of dielectric bodies,” IEEE Trans. AntennasPropag., Vol. 28, No. 2, 166–169, 1980.

43. Friedman, B. and J. Russek, “Addition theorems for sphericalwaves,” Q. Appl. Math., Vol. 12, 13–23, 1954.

44. Stein, S., “Addition theorems for spherical ware functions,” Q.Appl. Math., Vol. 19, 15–24, 1961.

45. Gaunt, J. A., “On the triplets of helium,” Philos. Trans. R. Soc.Lond. A, Vol. 228, 151–196, 1929.

46. Brunning, J. H. and Y. T. Lo, “Multiple scattering of EM wavesby spheres. Part I and II,” IEEE Trans. Antennas Propag., Vol. 19,No. 3, 378–400, 1971.

47. Mackowski, D. W., “Analysis of radiative scattering for multiplesphere configurations,” Proc. R. Soc. Lond. A, 599–614, 1991.

48. Chew, W. C., “Recurrence relations for three-dimensional scalaraddition theorem,” J. Electromagnetic Waves Appl., Vol. 6, 133–142, 1992.

49. Cruzan, O. R., “Translational addition theorems for sphericalvector ware functions,” Q. Appl. Math., Vol. 20, 33–39, 1962.

50. Tsang, L. and J. A. Kong, “Effective propagation constant forcoherent electromagnetic waves in media embedded with dielectricscatterers,” J. Appl. Phys., Vol. 11, 7162–7173, 1982.

51. Koc, S., J. M. Song, and W. C. Chew, “Error analysis for thenumerical evaluation of the diagonal forms of the scalar sphericaladdition theorem,” SIAM J. Numer. Anal., Vol. 36, No. 3, 906–921, 1999.


52. Mishchenko, M. I., “Light scattering by size-shape distributions ofrandomly oriented axially symmetric particles of size comparableto a wavelength,” Appl. Opt., Vol. 32, 4652–4666, 1993.

53. Xu, Y. L., “Electromagnetic scattering by an aggregate ofspheres,” Appl. Opt., Vol. 34, 4573–4588, 1995.

em scattering from a long dielectric w. z. yan ...in studying scattering and absorption of...

Documents