Optics Express 18, 12646-12652(2010)

Electrically controlled multifrequencyferroelectric cloakPeining Li, Youwen Liu*, and Yunji MengCollege of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China*ywliu@Abstract:We have suggested an electrically controlled multifrequencycloak with a single shell of ferroelectric material for the first time to the bestof our knowledge. The theoretical and simulated results have demonstratedthat this cloak with high-index ferroelectrics can reduce the total scatteringcross section of the cloaked system at multiple frequencies. These cloakingfrequencies of our cloak can be externally controlled since the dielectricconstant of ferroelectrics is well tuned with the applied electric field. It mayprovide a potential way to design a tunable multifrequency cloak withconsiderable flexibility.©2010 Optical Society of AmericaOCIS codes: (260.2110) Electromagnetic optics; (290.5839) Scattering, invisibility; (160.2260)Ferroelectrics.References and links1. J. B. 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Ozbay, “Multifrequency invisibility and masking of cylindrical dielectric objectsusing double-positive and double-negative metamaterials,” J. Opt. A, Pure Appl. Opt. 11(11), 114020 (2009). 14. A. E. Serebryannikov, P. V. Usik, and E. Ozbay, “Non-ideal cloaking based on Fabry-Perot resonances in single-layer high-index,” Opt. Express 17(19), 16869–16876 (2009).15. D. P. Gaillot, C. Croënne, and D. Lippens, “An all-dielectric route for terahertz cloaking,” Opt. Express 16(6),3986–3992 (2008).16. A. K. Tagantsev, V. O. Sherman, K. F. Astafiev, J. Venkatesh, and N. Setter, “Ferroelectric materials formicrowave tunable applications,” J. Electroceram. 11(1/2), 5–66 (2003).17. K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-typemetamaterials at optical frequencies,” Phys. Rev. Lett. 102(13), 133901 (2009).18. G. A. Smolensky, Ferroelectrics and Related Materials, (New York: Academic Press 1981).19. O. Vendik, Ferroelectrics at Microwave technology, (Moscow: Sov. Radio 1979).#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 126461. IntroductionCloaking 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 Ba0.5Sr0.5TiO3(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 conditionIn [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 εc. This combined system is surrounded by air as shown in the inset of Fig. 1. The e-jωt time #123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12647convention is assumed throughout. A plane wave with the unit amplitude 0ik x i z E z e ∧=is incident on the system along x direction depicted in Fig. 1, where k 0 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 scatteringcoefficients TM n c can be determined ,TM TMn n TM TM n n U c U iV =−+ (1)where TM nU , TM n V are available in the literature [6]. The total SCS of the combined system denoted as Q s , is given by 2,004(2).TM s n n n Q c k δ∞=−∑ (2)where δ is the Kronecker delta.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 0TM c are the case of interest in this work, which alsocorrespond to the minimum of the total scattering. Considering a limit of a relatively smallshell (λ>>a c ) with high dielectric constant (εc >>1), the minimum scattering condition 0TM U =0, which ensures 0TM c = 0, may be obtained analytically. This is because the expression TM n Ufor 0thorder is reduced to the following expression,2021/4)/4)0/23/4)3/4)0/4)/4)103/4)3/4)/2c c c c TM c c c c c c c c o c k a k a k a k k a k k a U k a k a k k a k k a k a ππππππππ−−−−−−−=−−−−−−− 2204sin[()]().2c c c k k a a o k a π=−+ (3)If we set 0TM U = 0, the following minimum scattering condition is obtained,λ= (4) where m = 1, 2, 3…, and the max of integer m ensures λ>>a c .The above condition is also expressed as 2c c a a m λ−=,where c λ= is thewavelength 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 #123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010(C) 2010 OSA 7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12648the 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 ferroelectricsThe 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 x Sr1-x TiO3 (BST) that can be synthesized in polycrystalline, ceramic layer, and bulk forms 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],(V/ µm),0.7≈<(5)E xwhere n= ε(0)/εr defined as the ratio of the dielectric permittivity of the material at zero electric field to its permittivity at some non-zero electric field. In this work, we consider that the cloaking shell consists of bulk ferroelectric BST with barium concentration x = 0.5, and ε(0) = 200 [15]. The electric field dependence for this material is shown in the Fig. 1.Fig. 1. The dependence of the dielectric permittivity of BST(x= 0.5) on the applied electricfield. The inset is schematic diagram of a cylindrical object covered with a ferroelectric shell.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 shellIn 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)εc1/2 at m#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12649= 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 εc1/2. 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 (εc = 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 ∆E/∆(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 offrequency and the applied electric field. (b)The total scattering cross section Q s as a function offrequency 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).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 #123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12650minimum 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 reasonableloss at two frequencies: (a) a c/λ = 0.14, (b) a c/λ = 0.16.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 thethree cases in Fig. 2 (b) with reasonable loss at two frequencies: (a) a c/λ = 0.14, (b) a c/λ =0.16.5. ConclusionsIn 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.#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12651AcknowledgmentsThis work was supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Specialized Research Fund for the Doctoral Program of Higher Education (200802871028), and the Natural Science Foundation of Jiangsu Province (BK2009366).#123241 - $15.00 USD Received 26 Jan 2010; revised 7 May 2010; accepted 11 May 2010; published 28 May 2010 (C) 2010 OSA7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12652。