23.08091803

Progress In Electromagnetics Research, PIER 85, 425–438, 2008

ANALYSIS OF MULTILAYERED CYLINDRICALSTRUCTURES USING A FULL WAVE METHOD

N. Ammar and T. Aguili

Laboratory of System Communications in Engineering School ofTunis (ENIT)Belevedere, BP 37, Tunis 1002, Tunisia

H. Baudrand

Electronics Laboratory LEN7ENSEEIHT2 rue Camichel, Toulouse 31071, France

Abstract—In this paper, Wave Concept Iterative Procedure (WCIP)is used to investigate scattering by multilayered cylindrical structuresin free space and to calculate the diffracted far field by adopting acylindrical coordinate formulation. The WCIP principle consists ofalternating waves between the modal and space domains. Its iterativeresolution process is always convergent in lossless media case. Theproposed technique used for determining the electric far field diffractedby a multilayered cylindrical structure is validated and confronted toliterature results.

1. INTRODUCTION

The electromagnetic analysis of cylindrically layered media isimportant as it is widely used in many engineering applications.An understanding of the scattering by multilayered structures isimperative in order to take account of its interference and couplingeffect with other components. Several studies have been performedfor the analysis of electromagnetic and optical scattering of layeredcylinders composed of isotropic and anisotropic materials and alsoarrays of such structures in various configurations. Among thosetechniques are the integral equation formulation, partial differentialequation formulation, and hybrid techniques [1–7].

426 Ammar, Aguili, and Baudrand

In this paper, we employ the fullwave [8–10] iterative processtheory for the computation of the diffracted far field for multilayeredcircular cylindrical objects. The WCIP principle consists of expressingthe boundary and closing conditions in terms of incident and scatteredwaves related by a bounded diffraction operator. The WCIP avoidsthe undesired phenomenon of unbounded operators; relations betweencurrents and fields, obtained using unbounded impedance operators,are transposed to relations between waves, supplied by boundedscattering operators [11]. The modulus of the modal coefficientsof the scattering operators is less than unity, which is not thecase for the modal coefficients of the impedance operators. Themethod’s convergence is therefore always guaranteed [12]. The powerof the WCIP was first verified in the analysis of planar circuits byAkatimagool et al. [13] and Wane et al. [14]. The extension of theprocedure to scattering by small cylinders can be seen in [15–17].The results obtained from using the WCIP on planar circuits havebeen validated by comparison to measurements and the use of othersimulations methods [13, 14]. We will now attempt to extend the useof WCIP to cylindrical systems.

This paper is organized as follows: Section 2 provides a theoreticalformulation of the WCIP approach and outlines how the cylindricalformulation of the integral equations solves successfully the scatteringmultilayered cylindrical structure. In Section 3, the validation of theproposed approach is performed by comparing its results with those intechnical literature. Finally, the conclusive remarks are presented inSection 4.

2. THEORETICAL FORMULATION

Figure 1 shows an infinitely long conducting or dielectric circularcylinder of radius ‘R1’ with its axis coincident with the z axis. Thereare several coaxial cylindrical media layers with radii ‘R2’, ‘R3’, . . . ,Rp. The structure is placed in free space (ε0, µ0) and the timedependence is assumed as e−jwt. A TE (with Ez = 0) or TM (withHz = 0) polarized plane wave is normally incident on the multilayeredcylindrical structure.

2.1. WCIP Theory

The formulation of the iterative method has been described in [11–18].It is based on the wave concept, which is introduced by writing thetangential (in each interface Ωj) electric and magnetic fields, in termsof outgoing ( Aj

i ) and incoming ( Bji ) waves (Fig. 1). It leads to the

Progress In Electromagnetics Research, PIER 85, 2008 427

int

11

A

11

B

21

A

12

B

togg1

extA

extB

x

y

22

B

12

A21

B

22

A

Ω1 S2

ext^

Ω2

Ωp-1

Ωp Rp-1

R1R2

Incident plane

ϕ i

→

Γ

→

→

→

→

→

ΓΓ

→

→

→

→

^^

Figure 1. Multilayered cylindrical structure and definition of thewaves.

following set of equations:

Aji =

1

2√Zj

0i

(Ej

i + Zj0.iHj

i ∧ nj

)

Bji =

1

2√Zj

0i

(Ej

i − Zj0.iHj

i ∧ nj

) (1)

withthe subscript i refer to media 1 and 2 divided by the interface Ωj .

zj0i: the intrinsic impedance of medium i for each interface j.j: order of interface.nj : a unit vector normal to interface Ωj .

For reason of clarity, the surface current density is introduced:

J ji = HT ∧ nj (2)

By the same way, the transverse electromagnetic fields on each of thetwo sides of the interface Ωj can be deduced from the waves by:

Ej

i =√Zj

0.i

(Aj

i + Bji

)J ji =

1√Zj

0.i

(Aj

i − Bji

) (3)


Each homogenous layer of order j, in Fig. 1, (1 < j < P ),between interface j and j+1 is characterized by the toggling operatorΓ(j,j+1)

togg [19] relating the incoming and outgoing waves following:[Bj

2

Bj+11

]= Γ(j,j+1)

togg

[Aj

2

Aj+11

](4)

For the operators Γext and Γint, respectively, associated to the exteriorand interior layers, Equation (4) is rewritten as:

B11 = Γint A1

1

BP2 = Γext AP

2

(5)

withΓext: External diffraction operator modeling the wave behaviourin the outer region of the cylinder for radius Rp [15, 16, 19].

Γint: Internal diffraction operator modeling the wave behaviour inthe interior region of the cylinder for radius R1 [15, 16, 19].

The concept of toggling operator permits us to investigate the modelingof wave penetration into multilayered cylindrical media by generating anetwork representation of the wave formalism. This can be performedby defining a toggling operator for every two neighboring media besidesthe internal and external operators corresponding respectively to theinner and outer region of the global multilayered structure.

More generally, in the spectral domain, the incoming and theoutgoing waves are linked in each other by the following relation:

B = Γ A (6)

For the multilayered structure in Fig. 1, the condensed Equation (6)is developed as fellow:∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

B12

B11

B21......

Bj−12

Bj1...

B2P

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

Γint [Γ(1,2)

togg

]. . . [

Γ(j−1,j)togg

]. . .

Γext

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

A12

A11

A21......

Aj−12

Aj1...

A2P

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(7)


In the spectral domain, the modal expansion of the reflection operatorfollowing the Dirac notation is given by:

Γασ =

∑n

|fn,R1(ϕ, z)〉Γασ·n

⟨fn,R2

(ϕ′, z′

)∣∣ (8)

where:

(ϕ, z) the observation point.(ϕ′, z′) the excitation point.Γα

σ·n is the modal coefficient of the diffraction operator.

The subscript σ designate: togg, ext or int and the subscript α referto the TM or TE polarizations.

R1 and R2 are respectively cylindrical radius where are definedthe outgoing and the incoming waves A and B.

fn,Rk= 1√

2πRkejnϕ: The common modal basis function for both

TE and TM modes describing a cylinder of radius Rk.On the other hand, the space scattering operator is deduced

from the continuity of fields and the boundary conditions on eachsub domain of Ωj . At the discontinuity interface Ωj , enforcing thecontinuity conditions of the electromagnetic fields (Et1 = Et2 andJ1 + J2 = 0 on the dielectric domain and Et1 = Et2 = 0 on themetallic domain) and taking into account the definition of the wavesin (1), the outgoing and the incoming waves are linked in the spatialdomain by the following relation:[

Aj2

Aj1

]= Sj .

[Bj

2

Bj1

](9)

with

Sj =

−η2j − 1η2j + 1

HjD −Hj

M

2η2jη2j + 1

HjD

2η2jη2j + 1

HjD

η2j − 1η2j + 1

HjD −Hj

M

(10)

where

HjM = 1 on the metal of interface j and 0 elsewhere.

HjD = 1 on the dielectric of interface j and 0 elsewhere.

ηj =

√√√√zj0.1

zj0.2

(11)


One takes account of different interfaces (j) of the structure inFig. 1, the boundary conditions is expressed as fellows:∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

A11

A12

A21

A22...Ak

1

Ak2...Ap

1

Ap2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

[S1

][S2

]. . . [

Sk]

. . . [Sp

]

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

B11

B12

B21

B22...Bk

1

Bk2...Bp

1

Bp2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(12)

Equation (12) as rewritten in condensed form as fellow:

A = S B (13)

2.2. Iterative Process

Collecting the waves from (6) and (13), results in a set of coupledequations summarizing the continuity conditions in the spatial domainand integral relations in the spectral domain. In presence of the planewave excitation, the iterative scheme is initialized by the excitationswaves corresponding to the incident tangential electromagnetic fields.

At iteration k, the iterative solution is governed by the followingequation system:

A(k) = S B(k−1) + δjpf(A0, B0

)B(k) = Γ A(k)

(14)

where δjp is the kronecker delta,

f(A0, B0

)=

−η2j − 1η2j + 1

· B0HjD − B0H

jM − A0

2η2jη2j + 1

· B0HjD

(15)

A0 and B0 are the excitation incident and scattered waves, related tothe incident electric field Einc and J inc current density by using therelation (1).


The toggling between the spectral and the spatial domains isensured by the fast Fourier transform and this inverse.

The iterative process (14) is stopped when both the norms∥∥∥∆ Ak

∥∥∥and

∥∥∥∆ Bk

∥∥∥ tend to zero. In the convergence of the iterative process,the current density and the tangential electric field can be computedin each interface j.

2.3. Diffracted Far Field

The WCIP solution of the far-field scattered is carried out in two steps:-Step 1: Computation of the electromagnetic near field’s on the

exterior interface of the structure (courant density and the tangentialelectric field).

-Step2: When the convergence of current density or electric fieldsat the exterior interface is attained, the outgoing exterior waves Aext

are transformed at the far zone in order to calculate the tangentialdiffracted far field on a fictitious cylinder with electrically large radius(ρ λ) (Fig. 2).

extAext

B

ρ>>

inftogginf

B

x

y

εr.j,µr.j

ε.r.p,µr.p

λ

Γ→

→→

^

RR i

1

Figure 2. Schema of diffracted far field.

It is worth noting that this transformation is ensured by using theasymptotic development of the expressions of the toggling operatorΓtogg. We denoted by Γinf

togg this asymptotic value.

2.4. WCIP Method Versus the Method of Moments (MOM)

A numerical evaluation of the WCIP CPU run time is presented toshow the efficiency of the method when handling largely metalled


structures.Let N be the number of pixels and n it the number of iterations,

the total number of operations Nop-WCIP for a simulation is given byNop-WCIP = n it ∗ 4N(1 + 3 lnN).

n it, on the order of 50, to reach the convergence of the WCIPiterative process for 2D structure.

Since 4N operations are needed in the spatial domain becauseof the two components at each pixel and 12N logeN operations areperformed in the fast Fourier transform and their inverse (FFT andFFT−1) at each iteration.

P is the number of pixels used to describe the interface.In the method of moments (MOM), the number of operations

Nop-MOM can be calculated using [18]:

Nop-MOM =P 3

3(16)

To do comparison between the MOM method and the WCIP methodin terms of running time, P is also taken as the number of pixels usedto define the metallic domain of each interface j. Thus P can be relatedto N by:

N = KP (17)

with k being the ratio between the metallic part of the interface j andall the interface.

In Fig. 2, the number of multiplications for a different numbers ofiteration is compared, in the case of a single interface, to the evolutionof Nop-MOM against m.

One can conclude that for a large metalized structure (requiringa significant number of metal cells) k < 10, the Nop-WCIP is far lessthan the Nop-MOM. Therefore, in this case, the WCIP algorithm isobviously superior to the MoM implementation in terms of executiontime.

3. NUMERICAL RESULTS

In this section, some numerical examples are presented to demonstratethe applicability of the WCIP for analyzing largescale electromagneticscattering of multilayered cylindrical structures.

The first example, we consider a dielectric cylinder withpermittivity relative εr1 is covered of a dielectric coating of permittivityrelative εr2; the configuration is illustrated in Fig. 4.

The geometrical parameters are: R2 = 2.5λ; R1 = 2λ. Thedielectric permittivity are: εr1 = 9; εr2 = 4; εr3 = 1. The studied


Nu

mb

erof

op

erat

ion

s

m 2 3 4 5 6 7 8 9 1010

0

105

1010

1015

1020

50 iteration with WCIP





MOM with k =2

MOM with k =10

Figure 3. Computational cost.

Eincϕ εr1 εr2 ε r3

R1

R2

x

y

ϕ→

→

Figure 4. Circular dielectric coated cylinder.

structure excited by a TE plane wave under normal incidence. Wesimulate this structure with 27 resolution.

At the convergence of the exterior near electric field (Fig. 5), whenthe iterative process is achieved after 25 iterations, the induced currenton the external cylinder (at R = R2) is shown in Fig. 6. It is clear thatthe current density distribution is null on the dielectric interface. Infact, the boundary condition of the tangential current on the dielectricregion is respected, which proves the consistence of our approach.

The normalised scattering far field (to the incident field),calculated using the above WCIP formulation, is diagrammed in Fig. 7.


0 20 40 60 80 100 120-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Iterations numbers

Ele

ctri

c fi

eld

|E

ϕ|

on R

=R

2

Real parts

Imaginaire parts

Figure 5. Convergence of the iterative process.

Angle-ϕ(degrees)

Cu

rren

t d

ensi

ty |J

ϕ|

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

1.4x 10-12

Figure 6. Current density at ρ = R2.

The results obtained using WCIP is compared to the one calculatedby the integral equation in [20]. It can be seen that the agreementbetween the two results is good.

The second example we consider is a single dielectric cylinder withair core. The inner radius of the cylinder is a = 0.25λ and the outerradius is b = 0.3λ, having relative permittivity εr = 4, excited by aTM plane wave, with incidence angle ϕi = 0. The resolution numbersis 25.

The radar cross section represents a convenient way to describe


|Eϕ|

-50 0 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Ref

W.C. I.P

ϕ (degrees)

[20]

Figure 7. Normalized scattering far field.

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

3.5

4

4.5

ϕ (degrees)

SW/ λ

0

Ref [21]

WCIP

ε0

εr=4

ε0

R1

R2

x

y→

ϕ

Eincz

→

Figure 8. Plots of the normalized SW versus the observation anglefor a dielectric cylindrical shell.

the strength of scattered fields observed in the far fields. For 2-Dobjects like infinite cylinders, the RCS becomes the scattering width(SW) defined by:

SW = limρ→∞

[2πρ

|Esc|2

|Einc|2

](18)

In Fig. 8, the normalized SW (to the wave length λ0 of the incidentwave) plots versus the observation angle ϕ are given for a singlecylindrical shell. The result generated using the WCIP method iscompared to the SW data based on the method of moment solutionpresented in [21]. It is clear from the figure that the two solutions arein complete agreement.


0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7

8

Ref

WCIP method

Ref

WCIP method

ε0

(εr,µr)

ε0

R1

R2

x→

y

SW/ λ

0

ϕ (degrees)

[22]

[22]

Ezinc

ϕ

→

Figure 9. Plots of the normalized SW versus the observationangle of a single conducting cylinder coated with either dielectric ormetamaterial.

For the third example, we consider a single coated conductingcylinder, the coating layer was made of a dielectric material havingrelative permittivity of εr = 9.8 and µr = 1 or a metamaterial havingrelative permittivity εr = −9.8 and relative permeability µr = −1(Fig. 9).

The metamaterial parameters used in this paper are selectivelyused to provide confirmation of the validity of the WCIP formulationby comparison of special cases with published results. For comparison,the incident wave frequency is set to 1 GHz, the radius of theconducting cylinder is R1 = 0.05λ, the outer radius of the coatingcylinder is R2 = 0.1λ, and excited by a TMz plane wave with incidentangel ϕi = 0. Fig. 9 represents also the SW normalized to wave lengthλ0 of the incident plane wave; the results show a complete agreementwith the independently reported results in [22].

4. CONCLUSIONS

In this work, an original integral method based on a transverse waveformulation has been applied successfully to calculate the radar crosssection of 2D structure. A cylindrical local basis function is used in themodal expansion of the diffraction operators involved in the resolutionof scattering problem by multilayered cylinder. The computation ofthe scattered far field was performed thanks to an original conceptof fictitious cylinder. A good agreement with technical literature wasobserved which proves the efficiency of the present method.


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23.08091803

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