Recently, the concept of electromagnetic cloaking has drawn considerable attention concerning theoretical, numerical and experimental aspects [1-8]. One approach to achieve electromagnetic cloaking is to deflect the rays that would have struck the object, guide them around the object, and return them to their original trajectory, thus no waves are scattered from the body [1]. In the coordinate transformation method for cloaking, the body to be hidden is transformed virtually into a point (3D or spherical configuration) or a line (2D or cylindrical configuration), and this transformation leads to the profile of ε, µ in the cloaking coating. Some components of the electrical parameters of the cloaking material (ε, µ) are required to have infinite or zero value at the boundary of the hidden object. This requires the use of metamaterials which can produce such values, however, they are narrow band since they rely on using array of resonant elements (as split ring resonators) [9-12].

Approximate cloaking can be achieved by transforming the hidden body virtually into a small object rather than a point or a line as shown in Figure 1, which eliminates the zero (point transformed) or infinite (line transformed) values of the electrical parameters [13,14]. This, however, leads to some scattering, since the hidden body is virtually transformed into a small object rather than a point or a line, and the scattering decreases as the transformed sphere radius is smaller.

The radially-dependent spherical cloaking shell can be approximately discretized into many homogeneous anisotropic layers, provided that the thickness of each layer is much less than the wavelength, and this discretization decreases the level of scattering as the number of layers increases. Each anisotropic layer can be replaced by a pair of equivalent isotropic sub-layers, where the effective medium approximation is used to find the parameters of these two equivalent sub-layers [15,16]. Near the boundary of the hidden sphere, the values of the radial components of ε, µ of the cloaking material are nearly zero. This makes the values of ε, µ in one of the pair of isotropic layers to be very small, and can be implemented using metamaterials [11,17,18]. Two approaches are possible in choosing the material properties in the pair of sub-layers. The first is to take the smaller values of ε, µ to be in one layer and the larger values to be in the second layers [19], whereas the other possibility is to take the smaller value of ε together with the larger value of µ in one sub-layer, and the opposite combination in the other sub-layer [20]. The scattering properties of cloaked conducting sphere [21] and conducting cylinder [22] were investigated using a combination of approximate cloaking, where the conducting sphere is transformed into a small sphere rather than a point and the conducting cylinder is transformed into small cylinder rather than a line, together with discretizing the cloaking material using pairs of homogeneous isotropic sub-layers.

In this work, the scattering properties of cloaked dielectric sphere is investigated using a combination of approximate cloaking, where the dielectric sphere is transformed into a small sphere rather than a point, together with discretizing the cloaking material using pairs of homogeneous isotropic sub-layers. The solution is obtained by rigorously solving Maxwell equations using Mie series expansion. The back-scattering normalized radar cross section, the scattering cross section patterns are studied and the total scattering cross section against the frequency for different numbers of layers and the transformed radius.

Perfect spherical cloak can be constructed by compressing the electromagnetic fields in a spherical region into a spherical shell as shown in Figure 1. The coordinate transformation is , with f(R₁) = 0 for perfect cloaking or f(R₁) = c for approximate cloaking and f(R₂) = R₂ [14]. The radial and transverse permittivity and permeability of the spherical cloak, depending on r, are given as [23]:

A linear transformation is usually used, given for approximate cloaking by (for ideal cloaking c =0) [14], [24]:

Thus, the permittivity and permeability of the approximate spherical cloak are given from the above equations by:

at r = R₁,

Thus, the permittivity at r = R₁ decreases as the cloaked radius c and the shell thickness decrease with values less than unity. At r = R₂,

We first discretize the cloaking anisotropic and nonhomogeneous shell into 2M homogenous layers with identical thickness (d_B = d_A = d) as shown in Figure 2, and then the outer radius of each sub-layer is:

An anisotropic layer can be replaced by a pair of equivalent isotropic sub-layers, where the effective medium approximation is used to find the parameters of these two equivalent sub-layers, Figure 2 [19,20]. The radial parameter in the anisotropic layer can be considered as a series combination of the parameters of the equivalent sublayers (considered as capacitors) related by:

The tangential parameters in the anisotropic layer can be considered as a parallel combination of the parameters of the equivalent sub-layers related by:

One can obtain the equivalent medium parameters for the isotropic sub-layers as:

where are given by Equations (3) and (4). is small, particularly near the conducting sphere where is nearly zero. The value of r for a discretized layer is taken at the average radius of this layer i.e. at the interface of layers A and B.

Two approaches are possible in choosing the material properties in the pair of sub-layers. The first is to take the larger values of ε, µ to be in one sub-layer, i.e. given by Equation (11), and the smaller values in the other sublayer from Equation (10) (case I). For this case, the impedances in the layers are the same, but the refractive index (the square root of the product of the relative ε, µ), suffers from strong jumps at the successive layers. The other possibility is to take the smaller value of µ, Equation (11), together with the larger value of ε, Equation (10), in one sub-layer, and the opposite combination in the other sub-layer (case II) [20]. In this case, the impedances of the successive layers suffer from jumps, but the refractive index profile changes continuously.

The configuration for electromagnetic scattering by a dielectric sphere coated by 2M layers is shown in Figure 2. The external radius, permittivity, and permeability of the core and the layers are denoted by and µ_i (i = 1, 2,···, 2M + 1), respectively. Figure 2 shows an E_x polarized plane wave with amplitude E₀, , incident upon the coated sphere along the direction. is the wave number in free space. The time dependence is suppressed.

The fields in the different regions are expanded in terms of spherical harmonics of TE_r and TM_r modes w.r.t. the radial directions. The field (E or H) with only transverse components (θ, Ø) is expressed by the harmonics m, whereas the other field having the three components is expressed by the harmonics n [25,26].

where J_n(kr) is a spherical Bessel function of order n of any kind and is a Legendre polynomial of the first kind, first order and n^th degree. The Ø-dependnce is taken as sinØ or cosØ (odd, o, and even, e) to conform with the incident plane wave.

The incident plane wave can be expressed in terms of spherical harmonics with Bessel functions of the first kind as:

where

and .

The scattered fields can be expanded in terms of spherical harmonics with spherical Hankel functions representing scattered outgoing waves with unknown coefficients e^s, d^s of the TE, TM parts, respectively.

We can write the fields in the i^th Layer as:

where the modes in the cloak region are represented by Hankel functions of the first and second kinds with arguments (k_ir_i), where , and unknown coefficients d, e. The boundary conditions at the interface between layers i, i + 1 leads to two equations relating the TM coefficients of the two layers, and two equations relating the TE coefficients of the two layers. The finiteness of the field in the dielectric core leads to the following ratios in the dielectric core [27]:

The ratios and in the successive larger layers can be obtained iteratively from the following equations [27]:

where and are the Riccati-Hankel functions.

Finally, the boundary conditions between the outer layer and air lead to the scattering coefficients b_n = d^s(TM part) and a_n = e^s(TE part):

The scattering cross section and the normalized radar cross sections are given by [28,29]:

where and are defined by :

In the above two equations and describe the angular scattering patterns of the spherical harmonics used to describe S₁ and S₂ and follow from the recurrence relations [29,30]:

starting with the initial values: π₀ = 0, π₁ = 1.

The total scattering normalized cross section Q_sca follows from the integration of the scattered power over all directions θ, φ, given by [31]:

The backscattering normalized radar cross section Q_b, applicable to monostatic radar, is given by [31]:

The mode series is truncated at the mode number n_m = K₀R₂ + 4 ×+ 2 [29,30].

To check the above analysis, the scattering pattern and the total scattering cross section are calculated for scattering by a dielectric sphere and compared with References [27,32], and for a lossy dielectric sphere coated by a lossy dielectric layer and compared with Reference [28], leading to identical results.

The scattering properties of cloaked dielectric spheres (ε_d = 2, 5) coated with isotropic homogenous layers are investigated concerning the normalized total scattering cross section versus the normalized frequency k₀ and for the bistatic radar cross section (RCS) with varying the reduced radius c and the number of isotropic cloaking layers with R₂ = 2R₁. 

The scattering properties of a cloaked dielectric sphere

with isotropic homogenous cloaking layers are investigated concerning the normalized total scattering cross section versus the normalized frequency k₀R₁ and for the bistatic RCS for both case I and case II and with varying the transformed radius, where the dielectric sphere is transformed into a small sphere rather than to a point, together with discretizing the cloaking material using pairs of homogeneous isotropic sub-layers with different numbers. The solution is obtained by rigorously solving Maxwell equations using Mie series expansion. When the number of layers increases (each isotropic layer is thinner), the scattered field decreases. As the transformed radius decreases the total scattering decreases, and approaches the ideal layered cloaking as the transformed radius tends to zero. The discretization has an effect on the scattering at the higher frequencies. The results are compared with the case of cloaked conducting sphere, showing a number of similarities, particularly for small transformed radius.