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Achieving Invisibility of Homogeneous Radially Anisotropic Cylinders by Effective Medium Theory

In this paper, we establish the full-wave electromagnetic scattering theory to study the electromagnetic scattering from infinitely long cylinders with radially anisotropic coatings. We show that the total effective scattering width can be dramatically reduced by the suitable adjustment of the dielectric anisotropy of the shell, while it is not the case for tuning the dielectric anisotropy of the core. Furthermore, we could make the cylindrical objects invisible when both dielectric and magnetic anisotropies are adjusted. In the long wavelength limit, we develop effective medium theory to derive the effective isotropic permittivity and permeability for the anisotropic coated cylinders, and the invisibility radius ratio derived from the full-wave theory for small coated cylinders can be well described within the effective medium theory.


I. INTRODUCTION
In recent years, designing optical and electromagnetic (EM) invisibility cloaking devices has attracted great attention from both physics and engineering societies. Scientists have developed coordinate transformation method [1] and conformal mapping method [2], which can protect the cloaked object of arbitrary shape from electromagnetic radiation. This idea has been verified by full-wave simulation of the cylindrical cloaking structure based on both finite element method (FEM) [3] and finite-difference time-domain method (FDTD) [4].
The experimental demonstration at microwave frequencies [5] and even at optical frequencies [6] for non-magnetic material has been realized lately. They have investigated the transformation based cloaks possessing spherical/cylindrical geometries with rotational symmetry [1,5,7,8,9] and geometries with reduced symmetries, such as eccentric elliptic cylinders [10], square cloaks [11], and arbitrary shapes [12]. In addition to the method of spatial transformation, Alu et. al. have proposed to use isotropic plasmonic coatings to render objects invisible [13] based on the dipolar cancellation, and proposed a parallel-plate metamaterial cloak [14] that significantly reduces the total scattering cross section of a given two-dimensional dielectric obstacle in some frequency band.
Due to the spatial compression, the required parameters should be anisotropic and dependent on the position [1]. To alleviate these contraints, EM invisibility cloaks have been realized by isotropic coatings based on the effective medium theory [15,16]. However, it still requires sufficient layers of alternating isotropic media in order to maintain the validity of that effective medium theory. Although recent development in metamaterial may allow the control of material parameters, such strict requirements for the parameters of a perfect cloak are still too difficult to realize in practice. As a way to reduce the number of required constitutive parameters to realize a manufacturable cloak, the incidence can be decomposed into transverse electric (TE) and transverse magnetic (TM) components so that 2D cloaks with simplified parameters [17,18] can be used. Unfortunately, the simplified cylindrical cloak will cause large reflection at the cloak's outer boundary due to the impedance mismatch. From the discussion above, one can see that: (1) ideal cylindrical cloaks require 6 parameters, some of which are infinite values; (2) simplified cylindrical cloaks requires 3 parameters but still some value tends to the infinity at the inner boundary; (3) both ideal and simplified cylindrical cloaks need certain parameters to be inhomogeneous i.e., a func-tion of position. This motivates our current work on achieving cylindrical cloaks, which is based on only one single coating by a homogeneous and radially anisotropic medium. The invisibility condition can be established by the effective medium theory. It is obvious that the contraints in material fabrication are greatly alleviated.
A lot of theoretical and numerical approaches have been developed to deal with the scattering problem of homogeneous anisotropic or even gyrotropic cylinders, e.g., integrodifferential equation [19,20], volumetric integral equation method [21], combined field surface integral equation method [23], finite difference method with measured equation of invariance (FD-MEI) [21], and dyadic Green's functions [24,25,26]. However, it should be noted that all these anisotropic tensors in those mentioned papers are defined in Cartesian coordinates, which is different from the radially anisotropy defined in the cylindrical coordinates as discussed in this paper. Although the latter can be mapped into Cartesian anisotropy, then the anisotropic cylinder will be inhomogeneous and angle-dependent.
Following our discussed motivation, in order to investigate the EM invisibility of radially anisotropic coated cylinders, we develop the compact scattering theorem for a coated cylinder with homogeneous radial anisotropy, by extending the idea of embedding radial anisotropy in the orders of Bessel/Hankel functions in spherical case [27]. We focus our analysis on the invisibility characteristics of a homogeneous anisotropic coated cylinder.
EM field components and scattering width of such coated cylinders are formulated. We discuss the roles of anisotropic parameters as well as the core-shell ratio on the reduction of the total scattering section. From the scattering algorithm, the effective permeability and permittivity of the core-shell system are also established so that the required condition for invisibility performance can be exactly determined. The numerical results are given.

II. FULL-WAVE ELECTROMAGNETIC SCATTERING THEORY
Let us consider the electromagnetic scattering from a radially-anisotropic coated cylinder of infinite length (see Fig. 1).
For the simplicity, we assume a normally incident plane wave with the magnetic field along the z direction (i.e., the transverse magnetic (TM) wave). We assume that the coated cylinder is composed of the core with the radius a and the shell with the radius b, and it is surrounded by free space (ε 0 , µ 0 ). As for radial anisotropy, we indicate that the permeabil- ity and the permittivity tensors for the core and the shell can be expressed in cylindrical coordinates (r,θ,ẑ), where ε p (or µ p ) represents the permittivity (or permeability) tensor of the core by p = c and/or the shell by p = s.
In Fig. 1, the incident magnetic field with unit amplitude can be written as In what follows, e −iωt will be omitted. In this connection, Maxwell equations in the core and shell are written as, In cylindrical coordinates, Eq. (2) can be decomposed into After some algebraic manipulations, we can obtain the governing equation of H z Inserting the solution in the form of H z = Ψ(r)Θ(θ) and introducing a seperation term m 2 , one obtains the ordinary differential equation for Ψ(r) For the part Θ(θ), we have the form e imθ . The general solution to Eq. (5) is AJ pm (k p r) + BN pm (k p r), where J pm and N pm respectively denote the mth-order Bessel and Neumann functions with the argument of k p r, k 2 p = ω 2 ε pθ µ pz , and (pm) 2 = m 2 (εpr/ε pθ ) . Note that if it is an isotropic case (i.e., ε pr = ε pθ ), pm reduces to be an integer.
In order to match the boundary conditions at cylindrical interfaces, the incident magnetic field can be expanded as follows, The scattered magnetic field in each region can thus be formulated where A m , B m , C m , and D m are the unknown coefficients to be determined, and H (1) m represents the mth-order Hankel function of the first kind.
Applying the boundary conditions of E θ and H z being continuous at r = a and r = b, we can derive the scattering coefficient D m to compute the farfield parttern where the prime denotes the derivative with respect to the argument. Note that other coefficients can also be solved simultaneously. and hence the electric and magnetic fields in each region are obtained. The scattering problem for the transverse electric (TE) case, can be solved in a similar way and the corresponding scattering coefficients can be obtained by the duality of ε → µ and µ → ε.
Scattering and extinction efficiencies are expressed through scattering amplitudes [28],

III. EFFECTIVE MEDIUM THEORY IN LONG WAVELENGTH LIMIT
In this section, we present the formulation of our effective medium theory for the coated cylinders in the long wavelength limit, i.e., k 0 b ≪ 1 and k s b ≪ 1. As a result, the higherorder moments proportional to (k i b) 2l+1 (i = o, c, s) are expected to be negligible and the effective scattering width of the coated cylinder is dominated by m = 0 and m = 1 terms [29,30]. Thus, we set the conditions for an effective medium as D 0 = 0 and D 1 = 0 [29], corresponding to and where the subscripts s1 = ε sθ /ε sr and c1 = ε cθ /ε cr . In the limit of k 0 b ≪ 1 and k s b ≪ 1, we can use the following approximations J 0 ( [31]. Substituting these approximations into Eqs. (10) and (11) and replacing ε 0 , µ 0 by ε ef f , µ ef f respectively, one can obtain and It is evident that in the long wavelength limit, the electric and magnetic fields are decoupled.
For an isotropic case ε pr = ε pθ (p = c, s), they reduce to the results in Ref. [29]. As a consequence, the coated anisotropic cylinder in the long wavelength limit can be viewed as an effective homogeneous cylinder, and the scattering efficiency is expressed in a simpler form as follows, Furthermore, the invisibility condition implies that ε ef f and µ ef f in Eqs. (12) and (13) should be identical to the parameters of the host medium, i.e., ε 0 and µ 0 . Thus, the relation between core-shell ratio and the anisotropic parameters in the core and the shell can be , purely nonmagnetic. (16) In fact, the invisibility conditions above possess a physical constraint, i.e., 0 ≤ a/b ≤ 1 according to the set-up in Fig. 1, which implies that given a certain set of anisotropic parameters for the shell and core, there may be no invisibility no matter how the core-shell ratio is tuned. It is also worth noting that if the coated cylinder is isotropic, Eqs. (15) and (16) respectively become which are in accordance with the results in Ref. [13].

IV. NUMERICAL RESULTS
Based on our theoretical results, we provide numerical calculations of the scattering efficiencies under different radial anisotropies and physical insights into the invisibility phenomena. Fig. 2 shows the full-wave scattering efficiency of a plasmonic cylinder coated with dielectric anisotropic shell versus the core-shell ratio a/b at different sizes. First, for small coated cylinder, the long wavelength limit is valid, and hence one can resort to effective medium theory. In this connection, we discuss the effective permittivity (effective permeability is unity because of nonmagnetic components) in Fig. 2(a). We find that for certain radius ratios a/b, the effective permittivity equals ε 0 and the scattering cross section of the coated cylinder is almost zero correspondingly (see the dips in Fig. 2(b)). As a consequence, the coated cylinder is invisible or transparent. On the contrary, when the effective permittivity equals −ε 0 in Fig. 2(a), the surface plasmon resonance takes place and strong scattering cross section is obtained (see the corresponding peaks in Fig. 2(b)). Then, we calculate the scattering efficiency for larger objects, which are shown in Fig. 2(c) and 2(d). In this situation, the long wavelength limit is not valid, and effective medium model cannot be used. In fact, with the object's size being increased, the scattering efficiency can still be reduced significantly (see the dotted lines in Fig. 2(c) and 2(d)) at the certain radius ratio (we call "near-zero scattering" ratio), though these values are larger than those within the long wavelength limit. Of particular interest is that significant reduction arises if we tune the value of ε sθ in the shell. In Fig. 2(c), with decreasing ε sθ (i.e., be reduced to 2.5ε 0 ), the scattering efficiency can be reduced considerably, and thus the transparency or nearly "invisible" is attained for large objects. Fig. 2(d) have verified this tendency at an even larger size. In other words, through adjusting the dielectric anisotropy of the shell, it is helpful for us to realize much lower scattering cross section and better electromagnetic invisibility. Moreover, the nearzero scattering radius ratio can be tuned at the same time. We note that a resonant peak is also presented in Fig. 2, in the long wavelength limit (or for small size), the maximum of scattering is due to the resonance of D T M ±1 . In comparison with the small objects, the additional small peaks in Fig. 2(c) and (d) result from the contributions of D T M ±2 because the higher terms of the scattering coefficient cannot be neglected due to the increasing size of the objects. Next, we consider a coated cylinder with radially anisotropic core but isotropic shell as shown in Fig. 3. Again, for a small coated cylinder, it is evident that the minimum of the scattering cross section takes place at the core-shell radius ratio determined by the condition ε ef f = ε 0 . Hence the results based on the full-wave theory should be in accordance with those from effective medium theory. We can observed from Figs. 3(b-d) that the isotropic core results in the smallest scattering efficiency, i.e., better electromagnetic invisibility (e.g., red dashed curves in Figs. 3 representing isotropic cores in each case). This fact is further proved for larger sizes (see Fig. 3(c) and Fig. 3(d)), and even for either plasmonic core with dielectric shell (see Fig. 3(c)) or dielectric core with plasmonic shell (see Fig. 3(d)). It reveals that in the core-shell system incorprating radial anisotropy, the isotropy in the core is a better choice to minimize the scattering width, resulting in "good" invisibility. It is known that for large coated objects, multipolar terms contribute to the scattering cross section, and the coated cylinder is visible for almost all the radius ratio (see Fig. (4)). In order to make the coated cylinder transparent or at least less visbile, one can further adjust the magnetic anisotropy to reduce the scattering cross section. From Fig. 4, we can conclude that the invisibility effectiveness of large objects can be greatly improved through suitable adjustment of both the dielectric and magnetic anisotropy and the scattering efficiency can be reduced for nearly one order in magnitude.
If the size is sufficiently small, the core-shell ratio determined by the cloaking condition definitely results in better invisibility performance. In order to demonstrate the invisibility performance for a large size b = 0.2λ 0 improved by the adjustment of the magnetodielectric anisotropy, we present the patterns of magnetic field in the xy plane, calculated by finiteelement solver of the Comsol Multiphysics. In Fig. 5(a), for the isotropic case (the same parameters as the red dashed line in Fig. 2(d)), there exists noticeable perturbation caused by the scattered field. Comparing Fig. 5(b) with 5(a), one can see that the perturbation is reduced through adjusting the anisotropic permittivity of the shell. If we take the permeability into account and properly tune its value, the near "invisibility" could be achieved for even relatively larger objects. In Fig. 5(c), the incident wave is almost unaltered outside the shelled cylinder, as if there was no scatterer. The specific core-shell ratios chosen in Fig. 5 are based on the given material parameters, and obtained in numerical calculation for the 2.5ε 0 , ε sr = 5ε 0 , µ sz = µ cz = µ 0 , a = 0.155b. (c) Parameters are: ε cθ = ε cr = −3ε 0 , ε sθ = 2.5ε 0 , ε sr = 5ε 0 , µ sz = 0.5µ 0 , µ cz = −7µ 0 , a = 0.62b.
lowest scattering efficiency correspondingly instead of theoretical effective medium theory (due to the size constraint). It should be mentioned that the transparency mechanism in this paper is different from the ideal cloak proposed by Pendry [1] via spatical compression, where there is no EM field in the core cylinder. Here the coated cylinder is penetrable.

V. CONCLUSION
In summary, we have demonstrated the full-wave scattering theory by magnetodielectric anisotropic coated cylinder of infinite length normally illuminated by a TM polarized plane wave. The effective permittivity and permeability of the anisotropic coated cylinder have also been derived in long wavelength limit. The non-scattering radius ratio obtained from the full-wave theory for small objects can be well described within the effective medium theory. Numerical results have shown that the effective scattering width c sca = 2bQ sca can be significantly reduced by adjusting the anisotropic permittivity of the shell, so as to achieve better transparency. However it is found that tuning the core anisotropy is not able to improve the efficiency of transparency. Furthermore, we could further improve the invisibility performance of the cylindrical objects by the adjustment of both electric and magnetic anisotropies. Although we have only considered the 2D case and losses materials, our research may be useful for the design of low-observability cylindrical targets. Based on the above investigation, we may extend our investigation to more general situations, such as obliquely incidence with arbitrary polarization, and finite-long cylinders. We could scale well towards the research of making multiple cylindrical objects "invisible".