# Rigorous Solution to Plane Wave Scattering by an Arbitrary-Shaped Particle Embedded into a Cylindrical Cell of Similar Material

Authors: Constantine A. Valagiannopoulos
Journal: International Journal of Antennas and Propagation
 (2009)

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## Abstract

An infinite cylinder of arbitrary shape is embedded into a circular one, and the whole structure is illuminated by a plane wave. The electromagnetic scattering problem is solved rigorously under the condition that the materials of the two cylinders possess similar characteristics. The solution is based on a linear Taylor expansion of the scattering integral formula which can be useful in a variety of different configurations. For the specific structure, its own far field response is given in the form of a double series incorporating hypergeometric functions. The results are in good agreement with those obtained via eigenfunction expansion. Several numerical examples concerning various shape patterns are examined and discussed.


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## Body

## 1. Introduction

The electromagnetic scattering by arbitrary-shaped formations is a very intriguing issue examined by many researchers with obvious applicability to microwave and optic frequencies. In [1], an inhomogeneous and arbitrary-shaped scatterer is enclosed in a spherical cladding and scatters the incident field. The inclusion is treated as a nonspherical perturbation, and thus the equations of nonspherical potential are utilized, together with the well-known vector spherical harmonics. Also in [2], the scattering response of a chiral cylinder of arbitrary cross section is obtained through the generalized multipole technique. The accuracy of the method is demonstrated by comparison with the eigenfunction expansion method.In [3], the physical optics method is employed to solve the electromagnetic scattering by an inhomogeneous impedance cylinder with arbitrary bound. The induced electric and magnetic currents on the illuminated region are expressed by means of the inhomogeneous impedance boundary condition. Finally in [4], Tezel studies the case of an arbitrary-shaped object enclosed in a circular cylinder by taking the Taylor series expansion of fields with unknown coefficients on the circles that are inside and outside of object. The series equations are obtained through the boundary condition and by exploiting the orthogonality properties of Fourier expansion functions.In this work, we examine a two-dimensional structure comprised of two layers, the outer of which has a circular bound, while the inner one possesses an arbitrary one. We obtain rigorously the solution to the plane wave scattering from this complex rod via the scattering integral under the simplifying assumption that the materials the two regions are filled with, are similar. In particular, we  take the first-order Taylor expansion of the scattering formula around the wavenumber of the external cell medium. The initial approximation of the total field is obtained as function of the incident one. The final result is expressed as an easily calculable double series whose terms incorporate hypergeometric functions. To validate the followed approach, we compare the results of the approximate method with the eigenfunction sums for a simple example that possesses analytic solution. To this end, various structures with cardioid-, astroid-, and egg-shaped bounds for the internal formation are studied, and their far field response is represented as function of the incidence angle. The behavior of the curves is observed, and discussed and certain conclusions connecting the inclusion shapes with the variations are drawn and justified.


## 2. Problem Definition

We suppose the two-dimensional structure depicted in Figure1 where the used cylindrical coordinate system (ρ,φ,z) is also defined. An infinite cladding of circular cross section with radius A, made from a dielectric material with relative permittivity ε1 (region 1), engulfs a particle of arbitrary shape and dielectric constant ε2 (region 2). The formation is centralized at the origin O, and its bound is determined by the well-known polar formula ρ=R(φ) with 0≤R(φ)≤A. The considered device is surrounded by the vacuum region 0 with intrinsic parameters (ε0,μ0), and it is excited by a z-polarized plane wave incident at angle Θ with respect to the horizontal axis, whose electric field possesses the following expansion [5, page 596]:Figure 1
The physical configuration of the examined structure. A dielectric rod scatters the incident plane wave in the presence of an inclusion of arbitrary shape and similar texture.(1)E0,incprim(ρ,φ)=∑n=-∞+∞(-j)nJn(k0ρ)e-jn(φ-Θ).
The notation Jx(y) is used for the cylindrical Bessel function of order x and argument y, while k0=ωε0μ0 is the wavenumber into vacuum with a harmonic time dependence e+jωt. The superscript prim is used to remark that this quantity is the primary component added to the secondary one E0,incsec(ρ,φ), expressing the reaction of the circular dielectric rod, to give the incident field into vacuum. Needless to say that the first subscript indicates the referred area, and therefore, the incident field into region 1 is comprised of a single component denoted by E1,inc(ρ,φ). Both the shape and the excitation of the structure are invariant across the z axis. Accordingly, the only nonzero electric component is the axial one E=Ez, and the problem is reduced to a scalar one. All the participating materials are taken magnetically inert. The purpose of this work is to find an analytic solution for the scattered field of the described configuration, despite the arbitrary cross section of the internal formation, with the reductive assumption that the characteristics of the two materials do not differ significantly, that is, ε2≅ε1 [6].


## 3. Mathematical Formulation

One of the most useful formulas in electromagnetic theory is the scattering integral [7]. The scattered field, developed by a dielectric object, is defined as an integral of Green’s function multiplied by the unknown total field, over the volume of this inhomogeneity. If one denotes the wavenumbers inside the dielectric object and the host material with k2 and k1, respectively, the scattering integral in examining a two-dimensional case is particularized to give [8]:(2)ET(k1,k2)=EI(k1)+(k22-k12)∬(S)G(k1)ET(k1,k2)ds.
Mind that the dependencies on the geometrical parameters are suppressed and the volume integral has been replaced by a double integral on the cross section (S) of the scatterer. The notations G(k1), EI(k1) to Green’s function and the incident field which are computed in the absence of the inhomogeneity and thus are not related with its material k2. The total electric field throughout the configuration is denoted by ET(k1,k2).By considering the Taylor expansion of the functionET(k1,k2) with respect to k2 for values close to k1, one obtains the following simplified equation:(3)ET(k1,k2)≅EI(k1)+2k1(k2-k1)∬(S)G(k1)EI(k1)ds.
In this sense, the unknown electric field inside the scatterer (which equals the total field in this area) is computed by ET(k1,k2)≅(2k1/(k1+k2))EI(k1), which is a more sophisticated approximation than the classical Born formula [9]: ET(k1,k2)≅EI(k1), as it takes into account the texture of the scatterer. In particular, it covers the case of a conducting object whose field is inversely related to the material’s conductivity and implicitly with the magnitude |k2|.The aforementioned analysis is a general purpose one as it can cover any two-dimensional problem. If one wishes to use it in studying the considered device, one can define the corresponding case-oriented quantities. Green’s functionG(k1) equals to the axial electric field of current j/ωμ0 Amperes produced by a dipole source positioned along the axis (ρ=P,φ=Φ) at the observation point (ρ,φ). It is noted that P<A (as the inhomogeneity is contained into region 1) and ρ>A (as we are interested to the far field of the device into region 0). The analytic expression of G(ρ,φ,P,Φ) (where the geometrical variables are appeared) is given by [10](4a)G(ρ,φ,P,Φ)=∑n=0+∞CG(n)Hn(2)(k0ρ)Jn(k1P)cos[n(φ-Φ)]
with(4b)CG(n)=12πA·2-δn0k1Jn′(k1A)Hn(2)(k0A)-k0Jn(k1A)Hn(2)′(k0A).
The symbol δnm is used for Kronecker’s delta, while Hx(2)(y) is the cylindrical second type Hankel function of order x and argument y.Given the fact that the integral in (3) extends over the region 2, the quantity EI(k1) corresponds to the field E1,inc(P,Φ) defined through(5a)E1,inc(P,Φ)=∑m=0+∞C1I(m)Jm(k1P)sin[m(Φ-Θ)]
with(5b)C1I(m)=2jπA·(2-δm0)(-j)mk1Jm′(k1A)Hm(2)(k0A)-k0Jm(k1A)Hm(2)′(k0A).
The field EI(k1) in (3) outside the integral concerns the points with ρ>A and equals to(6a)E0,incsec(ρ,φ)=∑n=0+∞C0I(n)Hn(2)(k0ρ)cos[n(φ-Θ)]
with(6b)C0I(n)=(2-δn0)(-j)n·k0Jn(k1A)Jn′(k0A)-k1Jn′(k1A)Jn(k0A)k1Jn′(k1A)Hn(2)(k0A)-k0Jn(k1A)Hn(2)′(k0A).
The relations (5a)–(6b) are well known and have been carefully derived in [8, 10]. The total electric field ET(k1,k2) of (3) is referred as E0(ρ,φ), and we are mainly interested in the scattered field developed by the internal formation E0,scat(ρ,φ)=E0(ρ,φ)-E0,incprim(ρ,φ)-E0,incsec(ρ,φ), for an observer located into region 1.The double integral over the cross section of the arbitrary scatterer is written in polar coordinates [11] and can be analytically evaluated through nontrivial integrals of Bessel functions [12] as follows:(7a)E0,scat(ρ,φ)=2k1(k2-k1)∬(S)G(k1)EI(k1)ds=2k1(k2-k1)∑n=0+∞∑m=0+∞CG(n)CI1(m)Hn(2)(k0ρ)×∫02πcos[n(φ-Φ)]cos[m(Φ-Θ)]Ψ(n,m,Φ)dΦ,
where the auxiliary function Ψ(n,m,Φ) is defined via(7b)Ψ(n,m,Φ)=∫0R(Φ)Jn(k1P)Jm(k1P)PdP=[k1R(Φ)2]n+m×F(𝔚,𝔉,-k12R2(Φ))(n+m+2)n!m!×R2(Φ).
where 𝔚 denotes [(n+m+1)/2,(n+m+2)/2,  (n+m+2)/2] and 𝔉 denotes [m+1,(n+m+4)/2,n+1,n+m+1]. The notation F(u,v,z) corresponds to the hypergeometric function with input vectors (u,v) and argument z.


## 4. Numerical Results

A set of computer programs has been developed to implement the proposed technique for a variety of internal formations. Prior to presenting the results, we should validate the aforementioned analysis when applied to a simple example possessing analytic solution. In particular, we consider a concentric circular core of radiusb<A into the external rod, and we compute the radar cross section (RCS) σ(φ) via two alternative ways: (i) canonical solution and (ii) described method for R(Φ)=b. The RCS in two dimensions is defined by [5, page 578](8a)σ(φ)=limρ→+∞[2πρ|E0,incsec(ρ,φ)+E0,scat(ρ,φ)E0,incprim(ρ,φ)|2],
where the Hankel functions are replaced by their asymptotic expressions for large arguments [13]:(8b)Hn(2)(k0ρ)~2πk0ρe-j(k0ρ-(π/4))jn,ρ→+∞.
The double series in (7a) needs only a few terms to converge (for not electrically large scatterers) due to the two factorials in the denominator of (7b). Also, the azimuthal integration in (7a) is performed numerically with no problems—it concerns smooth integrand functions.In Figure2, the percent error of the method in computing RCS, averaged for φ∈(0,2π), is represented as function of the ratio k2/k1 for various radii b. Obviously, the error is nullified for k2=k1, that is, the case of a homogeneous cylindrical rod, where the term E0,scat(ρ,φ) equals to zero. The method’s error is increasing with increasing degree of inhomogeneity and with increasing size of the inclusion. However, within the range 0.93<k2/k1<1, the recorded error does not vary substantially with b. Note that the difference is kept overall below 5%, and therefore the results of the described technique are satisfactorily accurate when a ratio k2/k1 is close to unity as in the following examples.Figure 2
The average percent error of the method in computing RCS, as function of the wavenumber ratio for various sizes of the concentric cylinder. Plot parameters arek0=2π/λ0=2000π,k1=3k0,A=0.5λ0,Θ=0.In all the numerical simulations, we chose to examine the dependencies of the quantity defined as follows:(9)Q(φ)=limρ→+∞2π|E0,scat(ρ,φ)|∫02π|E0,incsec(ρ,φ)|dφ.
The function Q(φ) expresses the normalized effect of the arbitrary-shaped formation on the far field pattern of the cylindrical column. As the incidence angle Θ is a variable in the attached diagrams, it suffices to evaluate the far field response Q(φ) along the horizontal axis φ=0. In Figure 3(a) we present the variation of Q(φ=0) with respect to Θ for various cardioid-shape bounds of the inner particle appearing in Figure 3(b). The polar equations of the cardioid curves family are given by(a) The far field response along the horizontal direction, as function of the incidence angle for various cardioid shapes. (b) The polar plots of the corresponding cardioid shapes. Plot parameters arek0=2π/λ0=2000π,k1=3k0,k2=1.05k1,A=0.1λ0,a=0.5A.
(a)(b)(10a)Rcardioid(Φ,ν,a)=a2[1-cos(νΦ)],ν∈ℕ*.
Regardless of the shape, the response is maximized at Θ=180∘ as the source producing the plane wave is located opposite to the observation point, and thus we receive a direct ray. The larger output is recorded for ν=1 which means that the presence of the inner scatterer along the horizontal axis plays an amplifying role. It should be remarked that the higher the maximum is, the larger the fluctuation of the curve is; in other words, the received field at φ=0 is much less dependent on Θ for ν=2 than in case of ν=1. In addition, the minimum response is exhibited in all cases at Θ=0, because the formations are not dense enough to reflect the incident wave.In Figure4(a) we present the variation of Q(φ=0) with respect to Θ for various astroid-shape bounds of the inner particle appearing in Figure 4(b). The polar equations of the astroid curves family are given by(a) The far field response along the horizontal direction, as function of the incidence angle for various astroid shapes. (b) The polar plots of the corresponding astroid shapes. Plot parameters arek0=2π/λ0=2000π,k1=3k0,k2=1.05k1,A=0.1λ0,a=0.6A.
(a)(b)(10b)Rastroid(Φ,ν,a)=a[sin2νΦ+cos2νΦ]-1/2ν,ν∈(0,+∞).
Mind the symmetry of the curves reflecting the symmetry of the internal formation with respect to the horizontal axis. One can observe that the far field response is proportional to the size of the scatterer and thus is naturally maximized for the case of ν=2. The negligible received field from the scatterer with ν=0.2, despite its sharp edges (shown in Figure 4(b)), is clearly owed to the penetrable material they are constructed of; the results would be very different in case of a perfectly conducting core of the same shape. Furthermore, the curves for ν=0.5 and ν=2 coincide for Θ=45∘,315∘ where the latter one exhibits a local minimum due to the large front of the corresponding scatterer (otherwise the minimum would be recorded at Θ=0 as happens in the cases of ν=0.2,0.5).In Figure5(a) we present the variation of Q(φ=0) with respect to Θ for various egg-shape bounds of the inner particle appearing in Figure 5(b). The polar equations of the egg curves family are given by(a) The far field response along the horizontal direction, as function of the incidence angle for various egg shapes. (b) The polar plots of the corresponding egg shapes. Plot parameters:k0=2π/λ0=2000π,k1=3k0,k2=1.05k1,A=0.15λ0,a=0.33A.
(a)(b)(10c)Regg(Φ,ν,a)=a[1+sinνΦ+cosνΦ],ν∈ℕ*.
Now the diagrams are not symmetric with respect to a value of Θ, which is dictated by the corresponding asymmetry of the polar plots in Figure 5(b). The values of Q(φ=0) are again small as happens in all the previous figures due to the tiny deviation of the wavenumber ratio k2/k1 from unity. There is an almost common minimum at Θ=100∘ where the wave is reflected towards φ=0 in the same way for each of the three different boundaries of Figure 5(b). The distance between the curves gets substantial at Θ=200∘, to close again to a moderate extent at Θ=300∘.



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*Source: 301461-2009-09-27.xml*