Problem formulation

To formulate the problem, two concentric spheres of different radii are taken into account. The inner sphere is assumed to occupy region 1 whose relative permittivity and relative permeability are taken to be ε _r1 and μ _r1, respectively. The propagation constant for an inner sphere is k ₁ = $k_{o}\sqrt {\mu _{r1} \epsilon _{r1}}$ where k _o is the free space propagation constant. The region 2 consists of a metamaterial coating having the relative permittivity as ε _r2 and the relative permeability as μ _r2. The propagation constant in this region 2 is k ₂ = $k_{o}\sqrt {\mu _{r2} \epsilon _{r2}}$. The inner radius of sphere is taken to be a whereas its outer radius is b. Thus, coating thickness is Δ = b − a. The background region 3 is taken to be the free space. This metamaterial coated dielectric sphere is covered with a spherical MS which has almost zero thickness and is characterized by an isotropic and homogeneous averaged surface impedance of Z _s Fig. 1. This surface impedance is given by Z _s = R _s − iX _s. Here the surface resistance R _s represents MS losses and X _s represents the surface reactance of the MS which can be positive or negative. For a lossless MS, the surface impedance Z _s is purely imaginary. Here, we have assumed lossless MS, i.e. Z _s = iχ_s with the surface reactance of χ_s as adopted by Alù. It is also shown and discussed in the numerical results section.

Fig. 1.

The geometrical configuration of a metamaterial coated dielectric sphere covered by an MS having a homogeneous and isotropic surface impedance of Z _s.

It is assumed that an x-polarized plane wave is incident upon an MS covered meta-material coated dielectric sphere whose electric field vector E^inc can be expressed in terms of spherical vector wave functions (SVWFs) as below [Reference Bohren and Huffman4],

(1)$${\bf E}^{inc} = \sum_{n = 1}^{\infty}E_{n}\left[{\bf M}^{( 1) }_{o1n}( k_or) - i{\bf N}^{( 1) }_{e1n}( k_or) \right]$$

Here M⁽¹⁾( · ) and N⁽¹⁾( · ) are the SVWFs with nth order spherical Bessel function j _n( · ) [Reference Bohren and Huffman4]. By using Maxwell's equations, the corresponding incident magnetic field vector H^inc can be obtained as follows:

(2)$${\bf H}^{inc} = -{1\over \eta_{o}}\sum_{n = 1}^{\infty}E_{n}\left[{\bf M}^{( 1) }_{e1n}( k_{o}r) + i{\bf N}^{( 1) }_{o1n}( k_or) \right]$$

where η _o is the intrinsic impedance of the free space and E _n = i ⁿE _o[(2n + 1)/(n(n + 1))]. The scattered electromagnetic field vectors (E^sc, H^sc) in the background region can be expanded as follows,

(3)$${\bf E}^{sc} = \sum_{n = 1}^{\infty}E_{n}\left[ia_{n}{\bf N}^{( 3) }_{e1n}( k_{o}r) -b_{n}{\bf M}^{( 3) }_{o1n}( k_{o}r) \right]$$

(4)$${\bf H}^{sc} = {1\over \eta_{o}}\sum_{n = 1}^{\infty}E_{n}\left[ib_{n}{\bf N}^{( 3) }_{o1n}( k_{o}r) + a_{n}{\bf M}^{( 3) }_{e1n}( k_{o}r) \right]$$

where superscript (3) referred to the spherical harmonics for which the generating function is specified by spherical Hankel function of first kind, i.e. ${h_{n}}^{( 1) }({\cdot})$. Here a _n and b _n are the unknown scattering coefficients and needed to be determined. Likewise, the electric and magnetic field vectors (E₁, H₁) inside the dielectric sphere can be written as,

(5)$${\bf E}_{1} = \sum_{n = 1}^{\infty}E_{n}\left[c_n {\bf M}^{( 1) }_{o1n} ( k_{1}r) - id_n{\bf N}^{( 1) }_{e1n}( k_{1}r) \right]$$

(6)$${\bf H}_{1} = -{1\over \eta_{1}}\sum_{n = 1}^{\infty}E_{n}\left[d_n {\bf M}^{( 1) }_{e1n}( k_{1}r) + i c_n {\bf N}^{( 1) }_{o1n}( k_{1}r) \right]$$

where η ₁ is the intrinsic impedance of the inner dielectric sphere. The factors c _n and d _n are unknown expansion coefficients. The electromagnetic field vectors (E₂, H₂) in the metamaterial coating region can be expanded in terms of SVWFs as below,

(7)$$\eqalign{{\bf E}_{2} = & \sum_{n = 1}^{\infty} E_{n} \left[\,f_{n} {\bf M}_{o1n}^{( 1) }( k_{2}r) - ig_{n} {\bf N}_{e1n}^{( 1) }( k_{2}r)\right. \cr & \left. +\, v_{n} {\bf M}_{o1n}^{( 2) }( k_{2}r) -iw_{n} {\bf N}_{e1n}^{( 2) }( k_{2}r) \right]}$$

(8)$$\eqalign{{\bf H}_{2} = & - {1\over \eta_{2}}\sum_{n = 1}^{\infty} E_{n}\left[g_{n} {\bf M}_{e1n}^{( 1) }( k_{2}r) + if_{n} {\bf N}_{o1n}^{( 1) }( k_{2}r) \right. \cr & \left. +\, w_{n} {\bf M}_{e1n}^{( 2) }( k_{2}r) + iv_{n} {\bf N}_{o1n}^{( 2) }( k_{2}r) \right]}$$

where η ₂ is the intrinsic impedance of the metamaterial coating region and f _n, g _n, v _n, w _n are unknown expansion coefficients. Here M⁽²⁾( · ) and N⁽²⁾( · ) are the SVWFs with nth order spherical Neumann function y _n( · ). It is noticed that the eight unknowns a _n, b _n, c _n, d _n, f _n, g _n, v _n, and w _n can be found by applying the tangential and impedance boundary conditions at the interfaces r = a and r = b, respectively and using the orthogonality property of angular functions. These boundary conditions are given in [Reference Bohren and Huffman4, Reference Alù34, Reference Chen and Alù35, Reference Awan and Seetharamdoo37].

Using equations (1)–(8) in these boundary conditions, the unknown coefficients can be determined. Once these unknowns are known then the electromagnetic fields in each region can be found from equations (1) to (8). For present study, the scattering coefficients a _n and b _n are of an interest. After some manipulations, they are given below,

(9)$$a_{n} = {\eta_{o}Z_{s}\left[\psi_{n}^{( 1) }( k_{o}b) X_{1}\right] + \left[Z_{s}j_{n}( k_{o}b) + i\eta_{o}\psi_{n}^{( 1) }( k_{o}b) \right]\eta_{2}X_{3}\over \eta_{o}Z_{s}\left[\psi_{n}^{( 3) }( k_{o}b) X_{1}\right] + \left[Z_{s}h_{n}^{( 1) }( k_{o}b) + i\eta_{o}\psi_{n}^{( 3) }( k_{o}b) \right]\eta_{2}X_{3}}$$

(10)$$b_{n} = {\eta_{o}Z_{s}\left[\,j_{n}( k_{o}b) X_{4}\right] + \left[Z_{s}\psi_{n}^{( 1) }( k_{o}b) - i\eta_{o}j_{n}( k_{o}b) \right]\eta_{2}X_{2}\over \eta_{o}Z_{s}\left[h_{n}^{( 1) }( k_{o}b) X_{4}\right] + \left[Z_{s}\psi_{n}^{( 3) }( k_{o}b) -i\eta_{o}h_{n}^{( 1) }( k_{o}b) \right]\eta_{2}X_{2}}$$

(11)$$X_{1} = \eta_{1}A_{1}-\eta_{2}B_{1},\; \qquad X_{2} = \eta_{2}A_{1}-\eta_{1}B_{1}$$

(12)$$X_{3} = \eta_{2}C_{1}-\eta_{1}D_{1},\; \qquad X_{4} = \eta_{1}C_{1}-\eta_{2}D_{1}$$

(13)$$A_{1} = \psi_{n}^{( 1) }( k_{1}a) \left[y_{n}( k_{2}a) j_{n}( k_{2}b) -j_{n}( k_{2}a) y_{n}( k_{2}b) \right]$$

(14)$$B_{1} = j_{n}( k_{1}a) \left[\psi_{n}^{( 2) }( k_{2}a) j_{n}( k_{2}b) -\psi_{n}^{( 1) }( k_{2}a) y_{n}( k_{2}b) \right]$$

(15)$$C_{1} = j_{n}( k_{1}a) \left[\psi_{n}^{( 2) }( k_{2}a) \psi_{n}^{( 1) }( k_{2}b) -\psi_{n}^{( 1) }( k_{2}a) \psi_{n}^{( 2) }( k_{2}b) \right]$$

(16)$$D_{1} = \psi_{n}^{( 1) }( k_{1}a) \left[y_{n}( k_{2}a) \psi_{n}^{( 1) }( k_{2}b) -j_{n}( k_{2}a) \psi_{n}^{( 2) }( k_{2}b) \right]$$

where $\psi _{n}^{( 1) }( z) = [ zj_{n}( z) ] '/z$, $\psi _{n}^{( 2) }( z) = [ zy_{n}( z) ] '/z$ and $\psi _{n}^{( 3) }( z) =$ $[ z h_{n}^{( 1) }( z) ] '/z$ and prime (′) shows the derivative. For completeness, we have also derived the mathematical expressions of these scattering coefficients a _n, b _n for a metamaterial coated dielectric sphere without MS. In this case, they can be written as,

(17)$$a_{n} = {\eta_{o}\psi_{n}^{( 1) }( k_{o}b) X_{1} + \eta_{2}j_{n}( k_{o}b) X_{3}\over \eta_{o}\psi_{n}^{( 3) }( k_{o}b) X_{1} + \eta_{2}h_{n}^{( 1) }( k_{o}b) X_{3}}$$

(18)$$b_{n} = {\eta_{2}\psi_{n}^{( 1) }( k_{o}b) X_{2} + \eta_{o}j_{n}( k_{o}b) X_{4}\over \eta_{2}\psi_{n}^{( 3) }( k_{o}b) X_{2} + \eta_{o}h_{n}^{( 1) }( k_{o}b) X_{4}}$$

It can be shown that if we take k ₁ = k ₂ = k _o, a = b, η ₁ = η ₂ = η _o and Z _s = 0 in equations (9)–(16) then the scattering coefficients a _n and b _n become same as that of a hollow perfectly electric conducting (PEC) sphere placed in the free space background which are given in [Reference Balanis38, Reference Jin39]. Likewise, if we take k ₁ = k ₂ = k, a = b, η ₁ = η ₂ = η in equations (17) and (18) using equations (11)–(16) then the the scattering coefficients a _n and b _n become consistent with that of a dielectric sphere having propagation constant of k and placed in the free space background medium and are given in [Reference Bohren and Huffman4, Reference Jin39]. This is also discussed and shown in the section of numerical results.

As we are interested in the radar cross-section (RCS), therefore, it is required to find the scattered electric field vector in the far zone. In this case, it is desired to use the asymptotic expressions for spherical Hankel function and derivative of spherical Hankel function for large argument as given by Bohren and Huffman, i.e.

(19)$$h_{n}^{( 1) } ( k_{o}r) \approx ( -i) ^{n + 1} {e^{i k_{o} r}\over k_{o}r}$$

(20)$$\psi_{n}^{( 3) }( k_{o}r) \approx {( -i) ^{n}e^{i k_{o} r}\over k_{o}r}$$

Using equations (19)–(20) in equation (3), we can find the far zone scattered electric field vector. Once it is known then the normalized RCS σ_n = $\sigma _{3D}/\lambda ^{2}_{o}$ of a metamaterial coated dielectric sphere covered with a MS can be found using definition of RCS σ_3D as given by Balanis, i.e.

(21)$$\sigma_{n} = {1\over \pi}\left[ \left\vert \sum_{n = 1}^{\infty} {2n + 1\over n( n + 1) }\cos\phi S_{a}\right\vert ^{2} +\, \left\vert \sum_{n = 1}^{\infty} {2n + 1\over n( n + 1) }\sin\phi S_{b}\right\vert ^{2} \right] $$

(22)$$S_{a} = a_{n} \pi_{n} + b_{n} \tau_{n}$$

(23)$$S_{b} = a_{n} \tau_{n} + b_{n} \pi_{n}.$$

(24)$$\pi_n = {P_n^{1}( \cos\theta) \over \sin\theta},\; \qquad \tau_n = {d\over d\theta}\left(P_n^{1}( \cos\theta) \right)$$

where $P_n^1({\cdot})$ is an nth order associated Legendre polynomial of the first kind. It should be noted from equations (21)–(24) that if sin ϕ = 0 then σ _n corresponds to an E-plane pattern whereas for cos ϕ = 0 it represents an H-plane pattern. For practical problems related to radar engineering, we are interested in the backscattering cross-section. The backscattering cross-section σ _back can be calculated by substituting ϕ = 0 and θ = π in equations (21)–(24) for an E-plane pattern. For cloaking related applications, we are often interested in the total scattering cross-section σ _T which can be calculated using the procedure as given in [Reference Bohren and Huffman4, Reference Alù34], i.e.

(25)$$\sigma_{T} = {2\pi\over k^{2}_{o}}\sum_{n = 1}^{\infty}( 2n + 1) \left(\vert a_{n}\vert ^{2} + \vert b_{n}\vert ^{2}\right)$$

Some of the authors have used ordinary metals and dielectric to cloak a spherical scatterer without MS [Reference Hayran, Kurt, Herrero, Botey, Staliunas and Staliunas40, Reference Sheverdin and Valagiannopoulos41]. They argued that their proposed theory has applications in the recent paradigm shift of photonic inverse design [Reference Sheverdin and Valagiannopoulos41]. It should be noted that if the effective size of the considered spherical scatterer increases and its radius becomes electrically large then the proposed formulation in this section can not be used. In this case, for example, one can use the Watson transformation [Reference Valagiannopoulos42] and the technique given in [Reference Wu, Shang and Li43].

Numerical results

Numerical results along with their discussions have been presented in this section. As we are interested in the E-plane backscattering cross-section, that is why only E-plane backscattering cross-sections have been shown for all figures except for Figs. 2, 10 and 11(a). For all numerical results, we have assumed a lossless MS, i.e. Z _s = iχ_s with the surface reactance of χ_s as adopted by Chen and Alù. In order to have the convergent results for summations encountered in equations (21) and (25), it is required to truncate the summation index n appearing in these equations to the highest value of N. The truncation criteria for these summations has been taken from [Reference Wu, Shang and Li43].

Fig. 2.

The normalized RCS σ _n for dielectric and perfectly electric conducting (PEC) spheres based upon the proposed formulation using electrical and geometrical parameters as given by Jin [Reference Jin39]. These normalized RCSs are found to be in good agreement with those reported by Jin which validates the proposed formulation.

In order to validate the proposed formulation of a metamaterial coated dielectric sphere covered without MS and with MS as presented in Section ‘Problem formulation’, the normalized scattering cross sections σ _n for dielectric and PEC spheres are shown in Fig. 2. For a dielectric sphere, it is assumed that ε _r1 = ε _r2 = 2.56, μ _r1 = μ _r2 = 1 and a = b = λ _o in the proposed formulation of a metamaterial coated dielectric sphere without MS as adopted by Jin in Fig. 7.14(a) of [Reference Jin39]. It is found that the normalized scattering cross section is in good agrrement with those reported by Jin. Likewise, if we take ε _r1 = ε _r2 = 1, μ _r1 = μ _r2 = 1, a = b = λ _o and Z _s = 0 in the proposed formulation of a metamaterial coated dielectric sphere covered with MS then the σ _n become same as that of PEC sphere given in Fig. 7.10(a) of [Reference Jin39]. These agreements with the reported works validate the proposed formulation. It is important to note that the inner dielectric core for Figs. 3–11 has been assumed to be made of a realistic material, i.e. MD except Fig. 10. This MD material is commercially available in the form of aluminum doped ferrite and has electromagnetic parameters of ε _r1 = 13.8 + i0.1 and μ _r1 = 11 [Reference Shore and Yaghjian36, Reference Awan, Ullah, Ullah and Ashraf44].

Fig. 3.

The influences of (a) ENG, MNG, and DNG coatings (b) ENZ, MNZ, and DNZ coatings upon the backscattering cross-section of a metamaterial coated magnetodielectric sphere covered by various lossless metasurfaces.

Fig. 4.

The backscattering cross sections of an ENG metamaterial coated magneto-dielectric sphere covered by a metasurface having (a) χ_s > 0 and (b) χ_s < 0 as a function of ratio (b/a) and their comparisons with σ _back of an ENG metamaterial coated magneto-dielectric sphere without metasurface.

Fig. 5.

A comparative study of the backscattering cross-sections of an MNG metamaterial coated MD sphere covered by a MS having (a) χ_s > 0 and (b) χ_s < 0 as a function of ratio (b/a) and the backscattering cross-section of an MNG metamaterial coated MD sphere without MS.

Fig. 6.

The backscattering cross sections of a DNG metamaterial coated MD sphere covered by a MS having (a) χ_s > 0 and (b) χ_s < 0 as a function of ratio (b/a) and their comparisons with σ _back of a DNG metamaterial coated MD sphere without MS.

Fig. 7.

The σ _back of an ENZ metamaterial coated MD sphere covered by a MS having (a) χ_s > 0 and (b) χ_s < 0 for various values of ratio (b/a) and its comparison with σ _back of an ENZ metamaterial coated MD sphere without MS.

Fig. 8.

The backscattering cross-sections of an MNZ metamaterial coated MD sphere covered by a MS having (a) χ_s > 0 and (b) χ_s < 0 as a function of ratio (b/a) and their comparisons with σ _back of an MNZ metamaterial coated MD sphere without MS.

Fig. 9.

The backscattering cross-sections of a DNZ metamaterial coated MD sphere covered by an MS having (a) χ_s > 0 and (b) χ_s < 0 and their comparisons with σ _back of a DNZ metamaterial coated MD sphere having no MS.

Fig. 10.

The total scattering cross-section σ _T of a metamaterial coated dielectric sphere covered with MS as a function of χ_s for (a) ENG, MNG, and DNG coatings (b) ENZ, MNZ, and DNZ coatings and its comparison with a MS coated dielectric sphere. Here b = 0.1λ _o and a = 0.08λ _o with Δ = 0.02λ _o which represents a thin coating layer. For a MS covered dielectric sphere case, it is assumed that ε _r1 = ε _r2 = 10, μ _r1 = μ _r2 = 1 and a = b = 0.1λ _o as adopted by Chen and Alù. Using proposed formulation and taking all parameters as given in [Reference Alù34, Reference Chen and Alù35], it is found that the total scattering cross-section is in good agreement with [Reference Alù34, Reference Chen and Alù35].

Fig. 11.

(a) The relative permittivity and permeability of a lossy DNG coating based upon the frequency dispersive model given in equations (26) and (27). (b) The backscattering cross-sections of realistic DNG coated MD sphere without MS and with MS having χ_s = ±1 kΩ as a function of frequency. Here geometrical parameters are taken to be b = 100 mm and a = 50 mm.

In Fig. 3, the influences of various types of metamaterial coatings upon the backscattering cross section σ _back as a function of χ_s have been shown. In this case, it is assumed that a = λ _o and b = 1.2λ _o at an operating wavelength of λ _o. It should be noted that the σ _back = −10.24 dB for a bare MD sphere having radius of 1.2λ _o. Here a bare MD sphere is taken to be a sphere made up of MD material having no coating and not covered by a MS. The electromagnetic parameters of ENG, MNG, and DNG coatings are selected in such a way that their impedance magnitudes are same as that of the free space. Therefore, it is assumed that ε _r2 = −1, μ _r2 = 1 for ENG, ε _r2 = 1, μ _r2 = −1 for MNG and ε _r2 = −1, μ _r2 = −1 for DNG coatings. Likewise, it is assumed that for an ENZ coating, ε _r2 = 0.001 and μ _r2 = 1 as given by Basharin et al. [Reference Basharin, Mavidis, Kafesaki, Economou and Soukoulis45]. Using this information, it is assumed that ε _r2 = 1, μ _r2 = 0.001 for an MNZ coating and ε _r2 = 0.001, μ _r2 = 0.001 for a DNZ coating. The backscattering cross sections for ENG, MNG, and DNG coated MD sphere covered with MS have been shown in Fig. 3(a). From Fig. 3(a), it is found that the maximum value of σ _back = 15.8 dB and is observed for an ENG coated MD sphere having χ_s = 857 Ω. Likewise, the minimum value of σ _back = −16.15 dB and is found for an MNG coating with χ_s = 224 Ω. The Fig. 3(b) deals with σ _back of ENZ, MNZ, and DNZ coated MD as a function of χ_s. In this case, the maximum value of backscattering cross section is observed for an ENZ coated MD sphere having χ_s = 383 Ω. This maximum value of σ _back is 18 dB. On the other hand, the minimum value of σ _back = −16.5 dB and is again found for an ENZ coated MD sphere with χ_s = −478 Ω. From the above analysis, it is concluded that an ENZ coated MD sphere having some finite coating thickness and χ_s = 383 Ω can be used to enhance the σ _back by 28 dB as compared to σ _back of a bare MD sphere. In this case, both spheres have same size. Thus an ENZ coating having some fixed coating thickness and paired with a specific type of MS can be used to enhance the backscattering cross section as compared to a bare MD sphere. This enhancement in backscattering cross section is desirable for radar detection problems.

In Figs. 4–6, the backscattering cross sections as a function of ratio (b/a) for ENG, MNG, and DNG coated MD spheres covered with MS having χ_s > 0 or χ_s < 0 have been shown. These backscattering cross-sections are also compared with ENG, MNG, and DNG coated MD sphere without MS. Here the radius of inner MD core is taken to be one wavelength, i.e. a = λ _o. It is studied that σ _back linearly increases with the increase in the coating thickness of ENG and MNG coated MD spheres without MS. Likewise, if ENG and MNG coated MD spheres are coated with MS having χ_s > 0 then the σ _back represents a fluctuating behavior with the increase in coating thickness except for an MNG coated MD sphere with χ_s = 1 kΩ. On the other hand, for ENG and MNG coated MD spheres having χ_s < 0, the backscattering cross-sections are fluctuating for very thin coating and become almost same for relatively thicker coatings, i.e. 1.35 ≤b/a≤3.1. Thus, it is concluded that the backscattering cross section is almost independent of MS having χ_s < 0 for a relatively thicker coatings of ENG and MNG coated MD spheres. The maximum value of σ _back is 25.6 dB and observed for an MNG coated MD sphere having χ_s = 0.5 kΩ for b = 2.68a. For the same geometrical parameters, the σ _back for an MNG coated MD sphere without MS is 13.5 dB. This shows that there exists an enhancement of 12.1 dB in σ _back for this specific MS covered MNG coated MD sphere as compared to the σ _back of an MNG coated MD without MS. Likewise, the minimum value of σ _back is -11.6 dB and found for an ENG coated MD having χ_s = 1 kΩ for b = 1.37a. In this case, if there is no MS then we have σ _back of 7.57 dB for an ENG coated MD sphere having b = 1.37a. From here, it is argued that a MS having χ_s = 1 kΩ can be used to reduce σ _back by 19 dB for an ENG coated MD sphere having no MS. Such type of reduced backscattering may find applications in the stealth design. From Fig. 6, it is seen that MS with χ_s > 0 or χ_s < can be used to enhance the σ _back for a DNG coated MD sphere by increasing the coating thickness except at some specific ranges of coating thicknesses where σ _back diminishes. In this case, the maximum value of σ _back is 25.35 dB and observed for a DNG coated MD sphere having χ_s = −0.1 kΩ with b = 2.88a. For the same value of b, the value of σ _back becomes −9.55 dB for a DNG coated MD sphere without MS. Thus, it is argued that a MS with χ_s = −0.1 kΩ can be used to enhance σ _back of almost 34 dB for a DNG coated MD having some specific coating thickness.

The Figs. 7–9 deal with the backscattering cross sections of ENZ, MNZ, and DNZ coated MD sphere with and without MS as a function of increasing coating thickness. It is studied from Figs. 7 and 8 that for relatively smaller value of χ_s, i.e. 0.1 kΩ, the backscattering cross sections of ENZ and MNZ coated MD spheres represent fluctuating behaviour with an increase in the value of b. The maximum value of σ _back is 21.52 dB and observed for an ENZ coated MD sphere having χ_s = 0.5 kΩ for b = 2.19a. For the same geometrical parameters, an ENZ coated MD sphere without MS has σ _back = 10.36 dB. Thus, it is observed that this specific type of MS can be used to enhance the σ _back by 11.16 dB as compared to the σ _back of an ENZ coated MD sphere. As the value of χ_s approaches to ±1 kΩ for an ENZ coated MD sphere, the value of σ _back is almost independent of coating thickness and become nearly same as that of σ _back of an ENZ coated MD sphere without MS for 1.35 ≤ b/a ≤ 3.1. The minimum value of σ _back is − 27.21 dB and observed for an ENZ coated MD sphere having χ_s = −0.5 kΩ for b = 1.21a. If the same ENZ coated MD sphere is covered with no MS then we have σ _back = −0.583 dB for b = 1.21a. This shows that a MS having χ_s = −0.5 kΩ can be used to enhance σ _back by 26.63 dB as compared to σ _back of same ENZ coated MD sphere without MS. It is observed from Fig. 9 that the significant decrease in σ _back is found in case of DNZ coated MD sphere. It is also studied that with the increase in the coating thickness of DNZ coating, σ _back further decreases. Thus, it is concluded that a DNZ coating can be used to hide a finite size MD sphere. On the other hand if the same DNZ coated MD sphere is covered with MSs having χ_s = ±0.1 kΩ, ±0.5 kΩ and ±1 kΩ then a significant enhancement in the σ _back is observed as compared to σ _back of a DNZ coated MD sphere without MS. This is an interesting result which shows that specific types of MSs with either χ_s < 0 or χ_s > 0 can be used to significantly enhance the backscattering cross section. For example in case of χ_s = 0.5 kΩ with b = 2.19a this enhancement in the σ _back approaches to almost 122 dB and such type of enhanced backscattering cross section finds application in radar detection related problems.

The total scattering cross-section σ _T of a metamaterial coated dielectric sphere covered with MS as a function of χ_s has been shown in Fig. 10. In fact, Alù and co-workers [Reference Alù34, Reference Chen and Alù35] have shown that a non-magnetic dielectric sphere with the relative permittivity of 10 and radius of 0.1λ _o can be cloaked using a lossless MS having χ_s = 70 Ω. Based upon the proposed formulation given in the present study, we reproduce the total scattering cross-section result as reported in [Reference Alù34, Reference Chen and Alù35] which further validate the proposed theory. In order to highlight the effects of MS reactance, very thin metamaterial coating layers have been assumed. This is because our spherical scatterer has metamaterial coating and covered with a lossless MS. That is why we have inserted a very small layer of metamaterial coating having thickness Δ = 0.02λ _o between MS and dielectric sphere as considered by Alù et al. and want to study how this very thin layer of considered metamaterial affects the cloaking phenomenon. It is observed that this thin layer of considered metamaterials weakens the cloaking phenomenon. It is further argued that a thin layered metamaterial coated dielectric sphere can not be apparently cloaked using lossless MS having χ_s lying in the range − 0.5 k Ω ≤ χ_s ≤0.5 kΩ.

In the previous analysis, it is assumed that all considered metamaterials are lossless, but these metamaterials are inherently lossy and frequency dispersive [Reference Pendry, Holden, Robbins and Stewart46–Reference Smith, Schultz, Markos and Soukoulis49]. Therefore, in order to show the effects of losses in these metamaterial coatings, we have considered only DNG coating. In this case, it is assumed that the corresponding frequency dispersive relative permittivity ε _r2(ω) and relative permeability μ _r2(ω) are governed by the following equations:

(26)$$\epsilon_{r2}( \omega) = 1 - {\omega^{2}_{\,pe}\over \omega^{2}-\omega^{2}_{oe} + i\Gamma_{e}\omega}$$

(27)$$\mu_{r2}( \omega) = 1 - {\omega^{2}_{\,pm}\over \omega^{2}-\omega^{2}_{om} + i\Gamma_{m}\omega}$$

In order to have lossy DNG metamaterial coating, we have chosen ω _pe/2π = 3.2 GHz, ω _pm/2π = 3.3 GHz, ω _oe/2π = 8.4 GHz, ω _om/2π = 8.5 GHz with Γ_m = Γ_e = 2 GHz, see for example [Reference Ziolkowski and Heyman48, Reference Smith, Schultz, Markos and Soukoulis49]. It is seen form Fig. 11(a) that a DNG behavior is observed from f _r = 8.5 GHz to f _p = 8.98 GHz. Thus, there exists a DNG bandwidth of 0.48 GHz. From closer look of Fig. 11(b), it is observed that there exists a specific DNG range of frequencies, i.e. 8.540GHz≤ f ≤8.695GHz for which MS having χ_s = ± 1kΩ can be used to reduce the backscattering cross section as compared to the σ _back of a DNG coated MD without MS. In this case, an optimum case exists at f = 8.55 GHz where we have σ _back = 43.524 dB for a lossy DNG coated MD sphere without MS. On the other hand, at the same optimum frequency, if the same DNG coated sphere is covered with (i) χ_s = 1 kΩ and (ii) χ_s = −1 kΩ then we have σ _back = 22.6 dB and σ _back = 21.8 dB respectively. Thus, it is concluded that metsurfaces with χ_s = ±1  kΩ can be used to reduce the backscattering cross section of almost 20 dB for a lossy DNG coated MD sphere at a specific DNG frequency of 8.55 of GHz.

Concluding remarks

The scattering characteristics of a metamaterial coated MD sphere covered by a MS has been studied. The problem has been solved analytically based upon the Mie theory and with the application of boundary conditions. Two types of boundary conditions, i.e. tangential and impedance boundary conditions have been used to solve the problem. The main focus of the current study is to investigate the backscattering cross-section of a metamaterial coated MD sphere covered by a MS. The influences of various types of metamaterial coatings including ENG, MNG, DNG, ENZ, MNZ, and DNZ and lossless MSs upon the backscattering cross-section have been explored. During the study, it is found that a DNZ coating can be used to hide a finite size MD sphere in the backscattered direction. If the same DNZ coated MD sphere is covered with the considered types of lossless MSs having positive or negative surface reactances then these MSs can be used to significantly enhance the backscattering cross-section. For example, in the case of a DNZ coated MD sphere covered with the surface reactance of 0.5 kΩ and has a coating thickness of 1.19a then this enhancement in the backscattering cross-section approaches to almost 122 dB. This is an interesting result that finds applications in the radar detection problems. The proposed formulation is also extended to the lossy DNG metamaterial coated MD sphere covered with a lossless MS. It is found that for a specific part of the DNG bandwidth, a lossless MS having surface reactance of ± 1 kΩ can be used to reduce the backscattering cross-section as compared to the backscattering cross-section of a lossy DNG coated MD sphere without MS. It is also studied that for the specific DNG frequency of 8.55 GHz, the metasurfaces with surface reactances of ±1 kΩ can be used to reduce the backscattering cross-section of almost 20 dB for the lossy DNG coated MD sphere.