Electromagnetic Wave Scattering from Cylindrical Structure with Mixed-Impedance Boundary Conditions

ELECTROMAGNETIC WAVE SCATTERING FROM CY-LINDRICAL STRUCTURE WITH MIXED-IMPEDANCE BOUNDARY CONDITIONS

Mostafa Mashhadi*_{, Ali Abdolali, and Nader Komjani}

College of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

Abstract—Recently, a new boundary condition is introduced in which surface shows different impedances for TE and TM electromagnetic fields. This new boundary condition is called mixed-impedance (MI) boundary condition and can be expressed in terms of normal components of electromagnetic fields. In this paper, the cylindrical structures with MI boundary condition were investigated and the scattering of such structures was obtained for both normal and oblique incidence and both TEZ and TMZ polarizations. The interesting feature of MI boundary condition was that the boundary conditions of PEC, PMC, DB, D’B’, and isotropic impedance boundaries were special cases of the MI boundary. Therefore, by calculating the electromagnetic scattering from a MI boundary, scattering from various boundary conditions could be easily obtained. It was also demonstrated that, by proper choice of boundary conditions the forward or backward RCS (radar cross section) could be significantly increased or decreased.

1. INTRODUCTION

Electromagnetic boundary conditions are normally defined in terms of tangential components of electric and magnetic fields. These conventional boundary conditions are listed as follows [1–3].

• Perfect electric conductor (PEC) n×E=0

• Impedance boundary conditionEt= ¯¯Zs·n×H • Perfect magnetic conductor (PMC)n×H= 0

• Perfect electromagnetic conductor (PEMC)n×(H+ME) = 0

Received 28 December 2012, Accepted 4 March 2013, Scheduled 7 March 2013

• Soft and hard surfaceV·E= 0, V·H= 0 forV·n= 0.

wherenis the unit normal vector to the boundary surface,trepresents the tangential field components, ¯¯Zs denotes the two-dimensional surface-impedance dyadic, M denotes the admittance of the PEMC boundary andV is a real unit vector tangential to the surface.

Recently, a new class of boundary conditions has been introduced in which the boundary conditions are expressed in terms of normal components of the vectors D and B. These boundary conditions are named DB, D’B’, DB’ and D’B surfaces. The boundary conditions in these surfaces are stated as follows [4].

• DB surfacen·D= 0 andn·B= 0.

• D’B’ surface ∇ ·(nn·D) = 0 and ∇ ·(nn·B) = 0. • DB’ surfacen·D= 0 and ∇ ·(nn·B) = 0.

• D’B surface∇ ·(nn·D) = 0 and n·B= 0.

In [4], it was demonstrated that, if a given field can be decomposed to TE and TM polarizations with respect to the normal vector to the surface, then, these normal boundary conditions can be replaced with PEC or PMC surfaces for each of TE and TM polarizations. This expression is shown as a summary in Table 1.

Table 1. Expressing normal boundary conditions in terms of PEC or PMC surfaces according to the type of polarization.

TE TM

DB PEC PMC

D’B’ PMC PEC

DB’ PMC PMC

D’B PEC PEC

It can be seen that the DB’ surface acts similar to PMC for both polarization; then DB’ boundary condition can be considered corresponding to the PMC boundary. Similarly, D’B surface corresponds to PEC boundary. DB and D’B’ boundaries show different behaviors for TE and TM polarizations and can act similar to PEC or PMC surfaces depending on the type of polarization. So far, different methods have been introduced for the realizations of DB and D’B’ surfaces [5–10].

to the unit normal vector to the boundary surface was assumed and the boundary had two different surface impedances for TE and TM polarized fields.

ETE = ZTEn×HTE (1)

HTM = −_Z1

TMn×ETM (2)

whereZTE andZTM are surface impedances of TE and TM polarized

fields, respectively. It was indicated that MI boundary can be expressed as a combination of four normal components of electric and magnetic fields as follows [11].

jkηn·H−ZTE∇ ·(nn·H) = 0 (3)

jkZTMn·E−η∇ ·(nn·E) = 0 (4)

The interesting property of MI boundary condition is that PEC, PMC, DB and D’B’ surfaces are its specific cases. When ZTE = ZTM, MI

boundary is an isotropic impedance boundary that includes the specific cases of PEC (ZTE = ZTM = 0) and PMC (ZTE = ZTM = ∞).

Also, for isotropic and homogenous media, ZTE = 0 and ZTM = ∞

corresponds to DB boundary andZ_TE=∞ andZ_TM= 0 corresponds to D’B’ surface.

The two surface impedances were also expressed in terms of two other parameters ofs,a[11].

ZTE=η(s+a), ZTM= _(s−η _a) (5)

wheresandaare called self-dual and anti-self-dual surface-impedance parameters, respectively. The two parameters of s and a determine whether boundary is a dual boundary or not. The concept of self-dual is represented in [12]. [12] states that a medium and/or boundary condition are called self-dual when they remain invariant in the duality transformation. It is shown in [11] that when a= 0, MI boundary is a self-dual boundary for all values of s. Then s and a are called the self-dual and anti-self-dual surface-impedance parameters.

Until now, only scattering from a MI sphere has been studied [11] and scattering from cylindrical structure with MI boundary has not studied yet. Therefore, in this paper the scattering of cylindrical structures with MI boundary are investigated. Using (3)–(5), boundary conditions in cylindrical structures have the following form.

jkHρ−(s+a)_ρ1∂(ρH_∂ρρ) = 0 (6)

jkEρ−(s−a)1_ρ

∂(ρE_ρ)

whereH_ρandE_ρare normal components of electric and magnetic fields at the boundary surface.

In Section 2, scattering of a normally incident plane waves from a MI boundary cylinder is investigated. Scattering of an obliquely incident plane wave from a MI boundary cylinder is obtained in Section 3 and the conclusions are given in Section 4.

2. SCATTERING OF A NORMALLY INCIDENT PLANE WAVE FROM MI CYLINDER

We know that each plane wave can be expressed as the sum of two plane waves with TM and TE polarizations. Then, by obtaining the scattering fields of a cylinder for both polarizations, the scattering fields of any plane wave can be calculated. So in this section, the scattered fields from MI cylinder for both TMZ and TEZ polarizations are computed.

2.1. Normally Incident Plane Wave: TM_Z Polarization

Let us suppose that a TMZ plane wave is normally incident upon an MI boundary cylinder of radius ρ0, as shown in Fig. 1. According

to cylindrical wave transformation, the incident electric field can be expanded by an infinite sum of cylindrical wave functions [13]

E_Zi =E₀e−jkx=E₀

+∞

X

n=−∞

j−nJ_n(kρ)ejnϕ (8)

The scattered electric field from the cylinder can be considered as follows:

E_Zs =E0

+∞

X

n=−∞

Anj−nHn2(kρ)ejnϕ (9)

where H2

n(kρ) is the Hankel function of the second kind of order n, and An represents the unknown coefficient and is determined from the boundary conditions. Since the magnetic field along z-axis is zero (Hz = 0), it can be shown using Maxwell’s equations that only the components of EZ, Hρ and Hϕ have non-zero values and the other field components are null.

The radial component of incident and scattered magnetic fields are

H_ρi = −1

jωµρ ∂Ei

Z

∂ϕ =

−E0

ωµρ

+∞

X

n=−∞

nj−nJn(kρ)ejnϕ (10)

H_ρs = −1

jωµρ ∂Es

Z

∂ϕ =

−E0

ωµρ

+∞

X

n=−∞

nAnj−nHn2(kρ)ejnϕ (11)

The scattering amplitudes A_n are determined by (10) and (11) and applying the boundary conditions (6).

An=− Jn(kρ0) +j(s+a)J 0 n(kρ0)

H2

n(kρ0) +j(s+a)H02n(kρ0)

, n6= 0 (12)

Note that (12) is valid for n 6= 0. For n = 0 using (10) and (11); therefore, boundary condition is itself satisfied and A0 must be

determined in another way. Indeed, for n= 0, incident and scattered waves are TEMρ, whereas it was assumed that fields can be expressed as a sum of partial fields TMρ and TEρ. The coefficient A0 must be

determined from additional information. This situation is similar to the TEM wave incident normally to the planar boundary [4]. The additional information is obtained from the way of realization of MI boundary. If it is assumed that the MI cylinder acts like a dielectric cylinder for TEMρwave that have permittivity and permeability ofεd and µ_d, respectively,A₀ can be obtained from the following equation.

A0 =−

J0(kρ0)−ηd_η0JJ00(kdρ0) 0(kdρ0)J

0 0(kρ0)

H2

0(kρ0)−ηdη0

J0(kdρ0) J0

0(kdρ0)H 02

0(kρ0)

(13)

By comparing (12) and (13), it is observed that if the values ofε_dand

µdare chosen so thatj(s+a) =−ηd_η0J_J00(kdρ0)

0(kdρ0),A0 can also be calculated

After calculating the coefficientsA_n, RCS can be obtained similar to [13]. Assuming that all the coefficients A_n are computed by (12), then the normalized backward RCS (normalized scattering width for

ϕ= 180◦_{) and normalized forward RCS (normalized scattering width}

forϕ= 0◦_{) in terms of (s+a) are plotted in Fig. 2. It is observed that}

normalized forward and backward RCS change in the ranges of 5–35 and 1–5, respectively. The maximum RCS in two directions occurs approximately when s+a = −j. Normalized RCS in terms of ϕ for different boundary conditions is plotted in Fig. 3.

Also, by comparing (12) with the coefficients obtained from an impedance cylinder, it can be observed that by selectingZs=η0(s+a),

-1 0 -8 -6 -4 -2 0 2 4 6 8 1010 0

1 2 3 4 5

Im (s+a )

No

rm

a

liz

ed

back

wa

rd

RC

S

-10 -8 -6 -4 -2 0 2 4 6 8 10 5

10 15 20 25 30 35 40

Im (s+a )

No

rm

al

iz

ed

forw

ar

d RC

S

(a) (b)

Figure 2. Normalized backward and forward RCS as a function of Im(s+a). ρ0 = 0.6λ0, Re(s+a) = 0.02, is added to make the curves

smoother. (a) Normalized backward RCS. (b) Normalized forward RCS.

0 45 90 135 180 225 270 315 360360 0

5 10 15 20 25 30

0

s+a= s+a=0 s+a=-j

(deg)

ϕ

σ/λ

the coefficients A_n in both cylinders will be equal (Z_s is the surface impedance of cylinder). In other words, in normal incidence to the MI cylinder for TM_Z polarization, MI cylinder acts like an impedance cylinder with the surface impedance ofη0(s+a) (it is assumed that the

A0 is calculated from (12)). IfAn are computed in special cases of DB (s = 0, a= 0), D’B’ (s =∞, a= 0), PEC (s =−a=∞) and PMC (s=a=∞), it will be seen that the coefficients An for PEC and DB are the same; also,An are equal for D’B’ and PMC. Since, for normal incidence, TMZ polarization can be considered as TEρ polarization (unless n= 0). The obtained results are predictable from Table 1.

2.2. Normally Incident Plane Wave: TEZ Polarization

Incident and scattered fields for TEZ polarization can be expressed as follows.

H_Zi = H₀e−jkx=H₀

+∞

X

n=−∞

j−nJ_n(kρ)ejnϕ (14)

H_Zs = H₀

+∞

X

n=−∞

B_nj−nH_n2(kρ)ejnϕ (15)

whereBnrepresents the unknown coefficient. It is shown that, in this case, only the components ofHZ,EρandEϕhave non-zero values and the other field components are null. Therefore, the magnetic field has no radial component and the waves can be considered TMρ.

E_ρi = 1

jωερ ∂Hi

z

∂ϕ = H0

ωερ

+∞

X

n=−∞

nj−nJn(kρ)ejnϕ (16)

E_ρs = 1

jωερ ∂Hs

z

∂ϕ = H0

ωερ

+∞

X

n=−∞

nj−nBnHn2(kρ)ejnϕ (17)

Now, the coefficients Bn are obtained by applying the boundary condition (7).

Bn=− Jn(kρ0) +j(s−a)J 0 n(kρ0)

H2

n(kρ0) +j(s−a)H02n(kρ0)

, n6= 0 (18)

Similar to Section 2.1, B0 must be determined using additional

3. SCATTERING OF AN OBLIQUELY INCIDENT PLANE WAVE FROM MI CYLINDER

Similar to Section 2, oblique incidence to MI cylinder is investigated for two TE_Z and TM_Z polarizations.

3.1. Obliquely Incident Plane Wave: TM_Z Polarization

Assuming that a TMZ plane wave traveling parallel to the x-z plane is incident upon a cylinder of radiusρ0, which is shown in Fig. 4. The

incident electric field can be expressed as follows.

Ei=E₀(ˆa_xcosθ_i+ ˆa_zsinθ_i)ejk0zcosθi_e−jk0xsinθi ₍₁₉₎

Using (19) the z-axis component of the electric field can be obtained and expanded by an infinite sum of cylindrical wave functions in the following form

E_zi = E0sinθiejk0zcosθie−jk0xsinθi

= E0sinθiejk0zcosθi n=+_X∞

n=−∞

j−nJn(k0ρsinθi)ejnϕ (20)

By defining E0

0 = E0sinθi, G = ejk0zcosθi, k0ρ = k0sinθi, k0z =

k0cosθi Ezi (20) is simplified to

E_zi =E₀0G

n=+_X∞

n=−∞

j−nJn(k0ρρ)ejnϕ (21)

By using Maxwell equation, the radial components of electric and magnetic fields are equal

E_ρi = 1

k2

ρ

∂2_Ei z

∂ρ∂z =

jk0zE00G

k0ρ

n=+_X∞

n=−∞

j−nJ_n0(k_0ρρ)ejnϕ (22)

H_ρi = jωε

k2

ρ

∂Ei z

ρ∂ϕ =

−k₀E0

0G

η0k02ρ

n=+_X∞

n=−∞

nj−nJ_n(k_0ρρ)ejnϕ (23)

In this case, because variations of electric field along z-axis are not zero (∂E =∂z 6= 0), both radial components of electric and magnetic fields exist. So there are two equations (6) and (7) for boundary conditions which must be satisfied. This means that there must be two unknown coefficients which are determined from boundary conditions. Therefore, cross-polarization fields are created in the scattered waves to satisfy the MI boundary upon the cylinder.

Considering the special case of MI boundary whenZTE=ZTM(or

isotropic impedance boundary) is another way that prove the existence of cross-polarization fields. As we know, in the oblique incidence to impedance cylinder cross-polarization fields are created to satisfy the impedance boundary condition. Therefore, the cross-polarization fields must be also created in the general form of impedance boundary (or MI boundary).

If the scattered fields in the Z direction are expressed as

E_zs = E₀0G

n=+_X∞

n=−∞

AnHn2(k0ρρ)ejnϕ (24)

H_zs = E00G

η₀

n=+_X∞

n=−∞

BnHn2(k0ρρ)ejnϕ (25)

Then, the radial components of scattered electric and magnetic fields will be:

E_ρs = −jωµ

k2

ρ

∂H_zs ρ∂ϕ +

1

k2

ρ

∂2E_zs ∂ρ∂z =

k0E00G

k2

ρρ

n=+_X∞

n=−∞

nBnHn2(k0ρρ)ejnϕ

+jk0zE

0

0G

k0ρ

n=+_X∞

n=−∞

AnH02n(k0ρρ)ejnϕ (26)

H_ρs = jωε

k2

ρ

∂Es z

ρ∂ϕ +

1

k2

ρ

∂2_Hs z

∂ρ∂z =

−k0E00G

η₀k2 0ρρ

n=+_X∞

n=−∞

+jk0zE00G

k0ρη0

n=+_X∞

n=−∞

BnH02n(k0ρρ)ejnϕ (27)

For this case, the boundary conditions (6) and (7) can be written as

jk¡H_ρi+H_ρs¢−(s+a)1

ρ

∂¡ρ¡Hi ρ+Hρs

¢¢

∂ρ = 0 (28)

jk¡E_ρi +E_ρs¢−(s−a)1

ρ

∂¡ρ¡Ei ρ+Eρs

¢¢

∂ρ = 0 (29)

By substituting (22), (26), (26) and (27) in (28) and (29), the unknown coefficients An and Bn are obtained.

A_n= C3C5−C2C6

C1C5−C2C4

, B_n=η₀C1C6−C3C4 C1C5−C2C4

(30)

where the coefficientsC1 toC6 are defined as follows:

C1 =

jk_0zk_0ρ k2

0 µ

jk0H02n(k0ρρ0)−

(s−a)

ρ0 ³

H02_n(k0ρρ0)

+ρ0k0ρH002n(k0ρρ0) ´´

(31)

C2 = _knη0 0ρ0

³

jk0Hn2(k0ρρ0)−(s−a)k0ρH02n(k0ρρ0) ´

(32)

C3 = −jk0zk0ρ

k2 0

j−n

µ

jk0Jn0(k0ρρ0)−(s_ρ−a) 0

¡

J_n0(k0ρρ0)

+ρ0k0ρJ00n(k0ρρ0) ´´

(33)

C₄ = −n

k0η0ρ0 ³

jk₀H_n2(k_0ρρ₀)−(s+a)k_0ρH02_n(k_0ρρ₀)

´

(34)

C₅ = jk0zk0ρ

k2 0

µ

jk₀H02_n(k_0ρρ₀)−(s+a)

ρ0 ³

H02_n(k_0ρρ₀)

+ρ₀k_0ρH002_n(k_0ρρ₀)

´´

(35)

C6 = _k n 0η0ρ0

j−n¡jk0Jn(k0ρρ0)−(s+a)k0ρJn0(k0ρρ0) ¢

(36)

By obtaining A_n and B_n, the scattered far fields and then RCS can be calculated. By doing some simplification, the RCS in terms of the coefficients A_n and B_n are

= 4

k0ρ   ¯ ¯ ¯ ¯ ¯

n=+_X∞

n=−∞

Anjnejnϕ ¯ ¯ ¯ ¯ ¯ 2 + ¯ ¯ ¯ ¯ ¯

n=+_X∞

n=−∞

Bnjnejnϕ ¯ ¯ ¯ ¯ ¯ 2  ₍₃₇₎

Also, the ratio of cross-polarization to co-polarization fields at far field in terms of the coefficientsAnand Bn are

Rcross to co = lim_ρ→∞ ¯ ¯ ¯ ¯E s cross Es co ¯ ¯ ¯ ¯= ¯ ¯ ¯ ¯

n=+_P∞

n=−∞Bnj n_ejnϕ

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

n=+_P∞

n=−∞Anj n_ejnϕ

¯ ¯ ¯ ¯

(38)

3.2. Obliquely Incident Plane Wave: TEZ Polarization

Similar to Section 3.1, the scattered fields can be also calculated for TE_Z polarization. The incident and scattered fields are consider as

H_zi = H0₀G

n=+_X∞

n=−∞

j−nJ_n(k_0ρρ)ejnϕ (39)

H_zs = H00G

n=+_X∞

n=−∞

AnHn2(k0ρρ)ejnϕ (40)

E_zs = −η0H00G

n=+_X∞

n=−∞

BnHn2(k0ρρ)ejnϕ (41)

By obtaining the radial components of electric and magnetic fields and applying boundary conditions the coefficients An and Bn are calculated.

An= D_D3D5−D2D6

1D5−D2D4, Bn=

−1

η0

D1D6−D3D4

D1D5−D2D4 (42)

where the coefficientsD1 toD6 are defined as follows:

D1 = jk0zk0ρ

k2 0

µ

jk0H02n(k0ρρ0)− (s_ρ+a) 0

³

H02_n(k0ρρ0)

+ρ0k0ρH002n(k0ρρ0) ´´

(43)

D2 = _k−n 0η0ρ0

³

jk0Hn2(k0ρρ0)−(s+a)k0ρH02n(k0ρρ0) ´

(44)

D₃ = −jk0zk0ρ

k2 0

j−n

µ

jk₀J_n0(k_0ρρ₀)−(s+a)

ρ0 ¡

J_n0(k_0ρρ₀)

+ρ₀k_0ρJ00_n(k_0ρρ₀)

´´

D₄ = nη0

k0ρ0 ³

jk₀H_n2(k_0ρρ₀)−(s−a)k_0ρH02_n(k_0ρρ₀)

´

(46)

D₅ = jk0zk0ρ

k2 0

µ

jk₀H02_n(k_0ρρ₀)−(s−a)

ρ0 ³

H02_n(k_0ρρ₀)

+ρ₀k_0ρH002_n(k_0ρρ₀)

´´

(47)

D6 = −_knη0 0ρ0 j

−n¡_jk

0Jn(k0ρρ0)−(s−a)k0ρJn0(k0ρρ0) ¢

(48)

Now, using (37) and (38), RCS and cross to co-polarization fields can be determined.

By considering the similarity between the coefficients C1 to C6

withD1 toD6, there seems to be a relationship between the unknown

coefficients An and Bn in TEZ and TMZ polarizations. In TMZ polarization, if a is replaced with −a, it can be easily proved that the new coefficients of A_n and B_n are equal with coefficients of A_n

and B_n for TE_Z polarization respectively. In other words, to obtain the unknown coefficients of A_n and B_n for TE_Z polarization, it is only coefficient to change a to −a for the coefficients of An and Bn in TMZ polarization. In the special case of a = 0, the coefficients of An and Bn are equal for two polarizations. Normalized RCS and cross to co-polarization for boundary conditions of DB, D’B’, PEC and PMC for both TMZ and TEZ polarization are plotted in Figs. 5 and 6, respectively.

As can be observed, RCS and cross to co-polarization for DB

0 45 90 135 180 225 270 315 360 0

2 4 6 8 10 12

0

PEC PMC DB D'B'

0 45 90 135 180 225 270 315 360 0

2 4 6 8 10

cro

ss to

co

pola

ri

za

tio

n

PEC PMC DB D'B'

(deg) (deg)

ϕ

σ/λ

ϕ

(a) (b)

Figure 5. Normalized bistatic RCS and cross to co-polarization as a function of ϕ for different boundary conditions for TMZ polarization,

0 45 90 13 5 180 225 270 315 36 0 0

2 4 6 8 10 12 12

0

PEC PMC DB D'B'

0 45 90 135 180 225 270 315 360

0 2 4 6 8 10

cro

ss to

co

pola

ri

za

tio

n PE C

PMC DB D'B'

(deg) (deg)

ϕ

σ/λ

(a) (b)

ϕ

Figure 6. Normalized bistatic RCS and cross to co-polarization as a function of ϕ for different boundary conditions for TEZ polarization,

θi=π/4 andρ0= 0.6λ0.

and D’B’ boundaries are the same for both polarization. It happens since, for these boundaries, a = 0; therefore, RCS and cross to co-polarization are independent from type of co-polarizations. On the other hand, for PEC and PMC boundaries, a has a non-zero value and therefore RCS has different behaviors for two polarizations. Another property observed in Figs. 5 and 6 is that cross polarization is created for oblique incidence to DB and D’B’ cylinder. The cross polarization fields are functions of ϕand it can be proven in general case that, for all values ofsanda, cross-polarization is null in forward and backward directions.

3.3. The Optimum Choice of Boundary Conditions for Increasing or Decreasing the RCS

Considering that the aim is to minimize or maximize RCS in forward or backward directions and, if goal is desired for a specific polarization or both polarizations, the optimum boundary conditions can be chosen. If minimizing or maximizing the forward and backward RCS is desirable for both polarizations, a useful method is to select a = 0; therefore, RCS will be the same for both polarizations and the desired goal can be achieved by optimum choice of s. Normalized backward and forward RCS versus s are plotted in Figs. 7(a) and (b) respectively. It can be observed that Normalized RCS changes in the range 2.5 to 32 for forward and 0.5 to 2.5 in the backward directions.

polarization, the two parameters ofs and acan be chosen arbitrarily. For example, normalized forward and backward RCS for TE_Z polarization are plotted in Fig. 8 for different values ofsanda. In this case, Normalized RCS changes in the ranges of 2.15 to 38.2 and 0.08 to 7.2 in forward and backward directions, respectively. Normalized RCS and cross to co-polarization versusϕare plotted in Fig. 9 for different values ofsanda, which give the maximum and minimum values of the forward and backward RCS.

-10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3

Im(s )

No rm al ized back wa rd RCS

-100 -8 -6 -4 -2 0 2 4 6 8 10 5 10 15 20 25 30 35

Im (s)

No rm a liz ed forw ar d RCS (a) (b)

Figure 7. Normalized backward and forward RCS as a function of −10 < Ims < 10. Res = 0.02, a = 0, θi = π/4 and ρ0 = 0.6λ0.

(a) Normalized backward RCS. (b) Normalized forwardRCS.

Im (s )

Im

(

a

)

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5 1 2 3 4 5 6 7 Im (s) Im ( a )

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5 5 10 15 20 25 30 35 (a) (b)

Figure 8. Normalized forward and backward RCS as a function of −5 <Ims, Ima < 5 for TEZ polarization. Res = 0.02, θi = π/4 and

ρ0 = 0.6λ0. (a) Normalized backward RCS. (b) Normalized forward

0 45 90 135 180 225 270 315 360 0

5 10 15 20 25 30 35 40

0

s=3.46j , a=-1.81j s=-1.29j, a=0.17j s=1.4j , a=3.9j s=-1.47 j, a=0.42j

0 45 90 135 180 225 270 315 360

0 2 4 6 8 10

cro

ss to

co

-pol

ar

iz

a

tio

n

s=3.46j, a=-1.81j s= -1.29j, a=0.17j s=1.4j, a=3.9j s= -1.47j, a=0.42j

(deg) (deg)

ϕ

σ/λ

ϕ

(a) (b)

Figure 9. Normalized bistatic RCS and cross to co-polarization as a function of ϕ for different values of s and a for TE_Z polarization,

θ_i=π/4 andρ₀= 0.6λ₀.

4. CONCLUSIONS

This paper investigated the problem of scattering from cylindrical structures with MI boundary conditions. Initially, the normally incident uniform plan wave to the MI cylinder was studied, and it was shown that this structure acted like an impedance cylinder with different surface impedances for two TEZ and TMZ polarizations. Then, oblique incidence to MI cylinder was investigated and it is observed that cross polarization was created in this case. It was demonstrated that DB and D’B’ had cross polarization in oblique incidence and therefore could not act as PEC and PMC for TEZ and TMZ polarizations. This happened since, for both TEZ and TMZ polarizations, radial components of electric and magnetic fields existed. It was also shown that the forward or backward RCS can be significantly increased or decreased by selecting the optimum values of boundary conditions.

So far, various methods were presented for realizations of the DB and D’B’ boundary in [6–10], However the realization of the MI boundary is still a challenge and is a topic for future study.

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