1. Introduction

Cloaking an object with metamaterials and artificial structures has recently attracted a great deal of attention because of potential applications in various scientific fields. Different approaches have been proposed to suppress the scattering from a given object, involving coordinate transformation techniques [1–3], anomalous localized resonances [4], scattering cancellation [5–8] and other novel concepts [9,10]. These techniques are usually designed to work at a single operating frequency.

Obviously, extending these schemes to multifrequency (MF) operation would invest them with considerable practicability. Recently, several possibilities have been put forward in order to achieve the MF cloaking [11–14]. Gao et al. have reported a MF cloak with multiple shells based on coordinate transformation method [11]. Alu and Engheta [12], extended their scattering cancellation cloaking (SCC) theory to multi-layered geometries and suggested that suitably designed multiple homogeneous and isotropic plasmonic layers may drastically reduce the total scattering cross section simultaneously at several distinct frequencies. Serebryannikov et al. have proposed a MF cloak with a single cloaking shell with high-index media based on the Fabry-Perot type radial resonances [13,14]. This latter cloak with high-index medium would be very feasible to implement in reality, since proper materials for the cloaking shell are easy to obtain, for instance, polar dielectrics at THz frequencies and ferroelectrics at microwave and THz frequencies [15], as well as Drude-Lorentz composites in a wide frequency range. Among these candidates, ferroelectrics may be a good one whose relative dielectric constant is effectively upon 100 at microwave frequencies and THz [16].

The dielectric response of ferroelectric materials is well tuned by temperature or external DC (or low-frequency) electric field. This tunable property has been proven successful in describing the performance of tunable microwave and THz devices [16]. Here we extend the high-index MF cloak to the frequency-tunable operation by taking into account the fact that the dielectric constant of ferroelectric materials can be well tuned external DC electric field. Firstly, we analytically study the cloaking mechanism of the high-index concentric shell based on the well-known Mie scattering theory and obtain the condition that frequencies of the minima of the scattering cross section satisfy, which is consistent with results based on the conventional Fabry-Perot resonators [13,14]. In the following section, we brief introduce the electrical tunability of the dielectric constant of ferroelectrics and present a cloaking scheme with bulk Ba_0.5Sr_0.5TiO₃ (BST-0.5) material. In Section 4, we show that the cloaking frequencies of this cloak with a shell of BST-0.5 can be effective controlled by external electric field, and demonstrate the cloaking effect by numerical simulations based on the finite element method. The final section summarizes our results.

2. Basic theory and the minimum scattering condition

In [13,14] it was shown that, under suitable conditions, it is possible to drastically reduce the scattering cross section of cylindrical objects using single-layer high-index shell. This phenomenon can be heuristically understood by using the analogy with between the zero reflection regime in the planar resonators and near-zero scattering cross section regime in the cylindrical resonators. Although this used analogy is quite justified, since the location of the corresponding frequencies can be estimated with a high accuracy by using a mode of planar resonators, an analytical proof is still necessary to better understand the physical mechanism of this high-index cloaking. Below we would deal with this problem. In this work, the fields are assumed to be TM polarized (electric field parallel to the axis of the cylindrical object) but similar steps could be carried out in the TE polarization. Moreover, all the materials are assumed to be nonmagnetic.

We start by considering the dielectric obstacle is an infinitely long cylinder along z with radius a and the symmetry of z-axis. The dielectric constant of the obstacle is assumed to be ε. The object is covered with a concentric shell with radius a_c and high dielectric constant ε_c. This combined system is surrounded by air as shown in the inset of Fig. 1 . The e^-jωt time convention is assumed throughout. A plane wave with the unit amplitude $E_z^i=\stackrel{\wedge }{z}e{}^{i{k}_{0}x}$is incident on the system along x direction depicted in Fig. 1, where k₀ is the wave number in free space. The cylinder symmetry of the problem allows us to analytically solve the electric fields inside and outside our structure from the Mie scattering theory. By applying the continuity at the inner (r = a) and outer (r = a_c) interface of the cloak, the scattering coefficients $c_n^{TM}$ can be determined

 

Fig. 1 The dependence of the dielectric permittivity of BST(x = 0.5) on the applied electric field. The inset is schematic diagram of a cylindrical object covered with a ferroelectric shell.

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It is expected that some interconnects would exist between the approach discussed in this work and the theory of scattering cancellation in plasmonic cloaking [5,6]. The purpose of this high-index cloaking shell is also to cancel the electric dipole moment of the system, since which contributes the most to the scattering properties for a relative small object [6,17]. Therefore, the locations of minimum $c_0^{TM}$ are the case of interest in this work, which also correspond to the minimum of the total scattering. Considering a limit of a relatively small shell (λ>>a_c) with high dielectric constant (ε_c>>1), the minimum scattering condition $U_0^{TM}$ = 0, which ensures $c_0^{TM}$ = 0, may be obtained analytically. This is because the expression $U_n^{TM}$ for 0th order is reduced to the following expression,

The above condition is also expressed as $a_c-a=m{\lambda }_c/2$, where ${\lambda }_c=\lambda /\sqrt{{\epsilon }_c}$ is the wavelength in the shell. This equation exactly coincides with the model of multiple zeros of the refection coefficient of the conventional planar Fabry-Perot resonators, and well supports the cloaking theory based on Fabry-Perot type radial resonances [13,14]. As is shown in [13,14], it is easy to estimate the cloaking frequencies with a high accuracy by using above equation. It is an obvious advantage of this method in designing cloaking frequencies over other recently proposed techniques.

Let’s come back to the main purpose of this work. It can be easily deduced from Eq. (4) that externally changing the dielectric constant of the shell would naturally lead to externally control cloaking frequencies when the dimensions have been determined. From this important feature, the ferroelectrics provide a possibility of designing a tunable MF cloak, since whose dielectric constant of is well tuned by the applied electric field.

3. The electrical tunability of the dielectric constant of ferroelectrics

The main attraction of ferroelectric materials is the strong dependence of their high dielectric permittivity on the applied bias electric field. This dependence would lead to considerably tunable application in our cloaking model. Typical representative ferroelectric material is Ba_xSr_1-xTiO₃ (BST) that can be synthesized in polycrystalline, ceramic layer, and bulk forms [16,18,19]. Solid solution BST is a very suitable material for tunable cloak in practice because the real part of the relative dielectric constant exceeds several hundred depending on the barium concentration, whereas the loss tangent can be less than 10⁻². For this material, the aforementioned dependence has an approximated form [16],

The tunability is obvious that the dielectric constant decreases from 200 to 150 when the bias field varies from 0 to 77V/μm. The dielectric constant approximately varies linearly with the applied electric field in the range of 20-80V/μm. Evidently, the cloaking frequencies would be externally controlled by applying this bias-field-dependent property to the high-index cloaking.

4. The performance of the tunable cloaking with a BST shell

In this section, we present the results for tuning cloaking frequencies of the cloak with ferroelectric shell through the biased electric field. As an example, the dielectric obstacle is characterized by given material parameters ε = 6, which is covered with a BST-0.5 layer discussed in Sec. 3. And the structure of the system is set as a_c/a = 2. In Fig. 2(a) , we report the contour plot of the variation of Q_s of our cloaked system as a function of frequency and the external field on the BST-0.5 shell. The orange regions represent the small Q_s regions. It is obvious that there are two cloaking frequencies for the minimum Q_s in the range of interest when the applied field is absent: a_c/λ = 0.07 and 0.14, corresponding to mλ = 2(a_c−a)ε_c^1/2 at m = 1 and 2, respectively. The higher dielectric constant of the shell, the more the cloaking frequencies in the same frequency range. This is because the cloaking frequency interval is inversely proportional to ε_c^1/2. It is clear that these cloaking frequencies change with the variations of the external field. When the electric field of 70 V/μm is applied (ε_c = 155.8 under this field), cloaking frequencies vary to the high frequencies, 0.08 for m = 1and 0.16 for m = 2. The cloaking frequency for large m model are more sensitive to the applied electric field, since the variance ratio ΔE/Δ(a_c/λ) is proportional with m.

 

Fig. 2 (a) Contour plot of the variation of total scattering cross section Q_s as a function of frequency and the applied electric field. (b)The total scattering cross section Q_s as a function of frequency for three cases: No cover (green), with cover and zero applied electric field (red), with cover and an applied electric field of 70 V/μm (blue).

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For implementing such a cloak in reality, the loss of shell material should be under consideration. For bulk BST-0.5, the loss tangent is tanδ = 0.01 [16] for both in the absence of a bias field and under it. This choice relies on the fact that the increase of the loss induced by the bias field is not substantial in this BST with intermediate concentrations of barium [16]. Figure 2(b) reports the total scattering cross section Q_s for three cases of uncovered cylinder object, the cloaked system in the absence and in existence of the applied electric field. It can be seen that the reduction effect of scattering is still very obvious. The total scattering cross section have been reduced to about 27% (at a_c/λ = 0.07) and 11% (at a_c/λ = 0.14) with respect to the uncloaked scenario without the applied field, and to about 16% (at a_c/λ = 0.08) and 8% (at a_c/λ = 0.16) with an applied field E = 70 V/μm. Actually, as is shown in [14], the presence of the loss only affects the minima locations in a very minor way, and affects the corresponding Q_s values significantly. For the lossless case, the total scattering cross section can be reduced to about 8% at a_c/λ = 0.07 with respect to the uncloaked scenario without the applied field.

To demonstrate the cancelling of scattering and frequency-tuning effect of the cloak, we perform the numerical simulations of the propagation of plane electromagnetic wave by the finite element method. Figure 3 show the modulus of the axial electric field in the three cases of uncovered cylinder object (corresponding to green line in Fig. 2(b)), cloaked system without the applied electric field (corresponding to red line) and under an applied field (corresponding to blue line) at two frequencies of a_c/λ = 0.14 and a_c/λ = 0.16 (the mode m = 2). In all the cases the structure is excited by a uniform plane wave impinging from the left of the figure with electric field amplitude equal to 1 V/m. For the case of the cloaked system without the applied field, it can be seen that the field in the air region is nearly uniform at the cloaking frequency a_c/λ = 0.14, showing that the scattering field is very weak, whilst the scattering field is very strong in the case of uncover cylinder object. This confirms that the scattering is effectively suppressed by the cloaking shell. If an electric field of E = 70 V/μm is applied in the shell, the cloaking frequency becomes a_c/λ = 0.16, where the field is nearly uniform in the air region, while the scattering interaction is very strong at the non-cloaking frequencies a_c/λ = 0.14. The frequency-tuning effect is very obvious when the electric field is applied on the shell. The simulated results based on the finite element method demonstrate the minimum scattering condition and the electrically frequency-tunable cloaking from the Mie scattering theory.

 

Fig. 3 The modulus of the axial electric field for the three cases in Fig. 2(b) with reasonable loss at two frequencies: (a) a_c/λ = 0.14, (b) a_c/λ = 0.16.

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Figure 4 further shows simulated modulus of the total magnetic field in the orthogonal plane of polarization (x-y plane), which is dominated by high-order multipoles [12]. Although the cloak has been originally designed to cancel the electric dipole moment, it is clear that the higher-order moments are also reduced at cloaking frequencies by this BST-0.5 shell. And the frequency-tuning effect is valid for the higher-order moments in the same way. This polarization independence is another advantage of this cloak.

 

Fig. 4 The modulus of the total magnetic field in the orthogonal plane of polarization for the three cases in Fig. 2 (b) with reasonable loss at two frequencies: (a) a_c/λ = 0.14, (b) a_c/λ = 0.16.

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5. Conclusions

In conclusion, we have theoretically proposed the possibility of designing an electrically controlled multifrequency cloak with a single shell of ferroelectric materials. The proposed scheme depends on the tuning of the dielectric permittivity of ferroelectric material with the applied field. The calculated and simulated results have demonstrated that such cloak can drastically reduce the total scattering cross section at multiple frequencies and these cloaking frequencies can be controlled by the applied field. We also analytically obtain the minimum scattering condition based on the Mie scattering theory, which may provide better understanding of the high-index cloak.