## Abstract

An original all-dielectric design that performs cloaking at 0.58 THz is demonstrated. The cloak consists of radially positioned micrometer-sized ferroelectric cylinders which exhibit under Mie theory a strong magnetic resonance. Full-wave simulations coupled with a field-summation retrieval technique were employed to adjust the rods magnetic plasma frequency; hence, the radial distribution in the permeability of the cloak. The behavior of the complete micro-structured device was simulated and results unambiguously show good reconstruction of the E-field wavefronts behind the cloak with high power transmission. This all-dielectric configuration provides an attractive route for designing cloaking devices at microwave and terahertz frequencies.

©2008 Optical Society of America

## 1. Introduction

Conformal transformation of electromagnetic domains has been proposed as an exciting approach to control the flow of propagating waves [1, 2]. It enables the design of objects displaying unprecedented functionalities [3] with the requirement of space gradients and anisotropy of the constitutive parameters (permittivity and permeability). Unfortunately, the fabrication of structures with such properties is challenging for the simple reason that they can not be found in Nature or synthesized in bulk. On the other hand, it is well established that strong anisotropy is obtained by structuring dielectric [4, 5] or metallic matter. Also, the plasma electric [6] and magnetic frequencies [7] can be tailored. Originally introduced for the design of single- or double-negative media (see reviews [8, 9]), this artificial matter has the advantage that the desired properties can be spatially localized. This fundamental concept enables to build complex electromagnetic objects from spatially distributed discrete elements.

The magnetic cloak of invisibility demonstrated at microwave frequencies was the first experimental breakthrough under these guidelines [10]. The visible spectrum is without any question the frequency range at which most impact is expected but the fabrication of nanometer-scaled elements remains a technological challenge [11]. This constraint can be alleviated by investigating magnetic cloaking devices at THz frequencies. In 2002, O’Brien and Pendry investigated the origin of the magnetic activity in high-κ (i.e. high permittivity) ferroelectric rods for photonic band gap applications [12]. They reported that for a TE polarization (H along the rod axis), a zeroth-order Mie resonance is created from resonant displacement currents in the scatterers. Consequently, negative values in permeability were predicted within a frequency range dependent on the geometry and material parameters. A left-handed behavior (permittivity and permeability are both negative) was also observed in an ordered and random medium constructed from rectangular Mie rods [13].

In this work, Mie theory is applied to engineer the magnetic plasma frequency of high-κ ferroelectric rods at THz frequencies. Full-wave simulations coupled with a field-summation retrieval technique [14, 15] were employed to assess the effective parameters (permittivity *ε _{eff}* and permeability

*µ*) of ferroelectric Ba

_{eff}_{x}Sr

_{1-x}TiO

_{3}(BST) rods excited by an H-field along the rod axis. From the results of this study, a device that performs cloaking at terahertz frequencies was carefully built from individual cylinders. Here, the idea is to superimpose the electromagnetic response of cylinders with different radii in such a way that the microstructured cloak shell displays a strong radial distribution in permeability and falls into a class of cylindrical objects derived from the conformal transformation theory. The performance of the magnetic cloak was assessed through full-wave simulations and the results are discussed in the last Section.

## 2. Permeability engineering in high-κ ferroelectrics rods

In this Section, we describe the electromagnetic response of micro-cut BST rods using the 3D full-wave solver HFSS (high-frequency structure simulator). The dielectric rods were excited by a plane wave along the *y*-axis, as the inset in Fig. 1(a) depicts. The dispersion of the investigated material was neglected and the permittivity *ε*′ was set to 200 with a dielectric loss tangent *ε*″/*ε*′ = 2.10^{−2} for all frequencies. This assumption relies on the fact that the presented micro-structured device operates, by definition, within a narrow frequency range [2] where the bulk parameters can be assumed constant. Perfect magnetic conductors and perfect electric conductors were used for the lateral and top/bottom faces, respectively. Hence, the incident wave injected by the input port propagates through a unique rod along the propagation direction *x* and infinite set of rods in the *y-z* plane. Then, the scattering parameters (reflection S_{11} and transmission S_{21}) were computed from the input/output ports to determine the Mie resonance frequency.

Figure 1(a) presents the frequency dependence on the reflection and transmission parameters of a plane wave impinging onto a BST cylinder with a 34 µm diameter and a height of 30 µm and for frequency values ranging from 0.36 to 0.66 THz. These frequencies are within the range where the rod electromagnetic behavior can be described by an effective medium. Here, the simulation domain was 50 µm×40 µm×50 µm. The 5 µm spacing between the top/bottom surfaces of the rod and the lateral faces insures that the incident wave does not impinge onto an infinitely long rod along the y axis. Although this aspect is not discussed in this work, note that the Mie resonance is rejected to lower frequencies with an infinite rod. For this example, the Mie resonance is predicted to occur at 0.527 THz; frequency at which the rod displays the most significant magnetic response.

The complex effective permittivity and permeability were retrieved using a field-summation technique over the simulation domain and are presented in Fig. 1(b). The data indicate that the real part of the permeability displays a Lorentz-like behavior around the Mie resonance frequency. In contrast, the real part of the permittivity remains almost constant with a value of 1.86 over the frequency range of interest. The imaginary part of the permittivity and permeability is ~1.10^{−5} and ~8.10^{−2}, respectively. Finally, the data show that the magnetic plasma frequency at which the permeability vanishes is 0.597 THz. This information is critical because it enables the design of metamaterials with permeability values ranging from 0 to 1.

In order to tailor the magnetic activity of the dielectric rods, the simplest approach would be to shift upward or downward the Mie resonance frequency by changing the dielectric function of the material. However, this solution is technologically impractical since it would require mastering the rod stoichiometry with high accuracy. Another alternative would be to increase or decrease the path length that the displacement currents undergo at the resonance. Physically, this means adjusting the radius of the rod. If one increases the radius, the resonance goes to lower frequencies and vice-versa. For instance, Fig. 2(a) presents the frequency dependence on S_{11} and S_{21} for BST rods with diameter values ranging from 34 to 40 µm by steps of 1 µm. The Mie resonance frequency is clearly red-shifted with increased radii, thus confirming our hypothesis. Consequently, one immediately grasps the advantage of geometrical modifications in high-κ rods to adjust the magnetic plasma frequency. Figure 2(b) presents the results of the retrieval technique for the investigated designs. The magnetic plasma frequency is predicted to shift from 0.597 THz to 0.543 THz. Note that we also observe a slight increase in the real part of the permittivity from ~1.9 to ~2.5.

## 3. Cloak design

Now that we have identified and quantified the magnetic plasma frequency in individual BST rods, we turn our attention to the design of a full cloak using Mie rods and excited under plane wave conditions. In the case of an annular cloak defined by its inner radius *a* and outer radius *b* [see Fig. 3(a)], the effective parameters of the cloak shell (permittivity *and* permeability) must be independently engineered to satisfy a set of equations derived from the conformal transformation theory [3]. On the other hand, it was shown that this design burden can be overcome by using a reduced set of equations [16]. This allows one parameter only (permittivity *or* permeability) to be varied with the cloak radius. In our case, a progressive variation in the permeability is obtained by positioning the Mie rods radially [11]. Under this configuration, a transverse-electric (TE) polarized plane wave (*H _{r}*,

*H*,

_{θ}*E*in cylindrical coordinates) has to be employed to illuminate the magnetic cloak and the original set of equations can be reduced to:

_{z}Similarly, the first cloaking experiment was demonstrated under these conditions with metallic split-ring-resonators at 8.5 GHz. Here, we simply substitute the metallic loops with BST rods whose electromagnetic response matches closely Eqs. 1 to 3. In an arbitrarily configuration where *b* = 2*a*, the radial component in the permeability is zero at the inner surface of the cloak where *r* = *a*. In contrast, a permeability value of 0.25 is obtained at the outer surface of the cloak where *r* = 2*b*.

Here, the cloak consists of 7 levels of 30-µm-high BST rods with diameter values increasing from 34 to 40 µm by steps of 1 µm and 10 µm radial pitch separating each rod, as shown in Fig. 3(a). Finally, the inner radius of the cloak was arbitrarily set to *a* = 280 µm and outer radius *b* = 560 µm to limit the size of the computational cell.

Figure 3(b) presents the cloak effective parameters (*μ _{r}*(

*r*),

*μ*(

_{θ}*r*), and

*ε*(

_{z}*r*)) dependence on the normalized radius r/

*a*calculated (i) for the investigated BST rods at 0.58 THz and (ii) from Eq. 1. This operating frequency was selected by matching the discrete distribution in permeability from the rods with Eq. 1. The data also show that the permittivity value is about half of the one computed from Eq. 3. In the following Section, we will see that this permittivity mismatch does not fundamentally perturb the cloaking ability.

## 4. Results and discussion

In order to achieve an efficient cloaking device, key points must be satisfied. First, the device itself must not be detected. Consequently, the field pattern should not present wavefront distortions around the cloak nor shadow regions behind it. This particular point is critical because it is understandable that the electromagnetic signature of the device must not be revealed. Hence, the outer surface of the device must be well impedance matched with its surrounding environment, typically air, to minimize reflection. In addition to this impedance matching condition, the distribution of the effective parameters inside the cloak should fit the reduced set of cloak parameters given by Eqs. 1 to 3 for TE-polarized incident waves. Finally, the electromagnetic activity at the innerside of the cloak should be in principle close to zero, to minimize scattering on the hidden object.

The efficiency of the designed cloak in terms of phase front distortion (Fig. 3(a)) was evaluated, based on these criteria, with HFSS by simulating the full three-dimensional micro-structured cloak at 0.58 THz (*λ* = 517 µm) with a TE plane wave (E-field perpendicular to the cloak plane). This means that each elementary cell is taken into account in contrast to previous works based on a homogeneous piecewise region [16]. With this configuration, approximately half a wavelength is encompassed into the body of the cloak. This does not impose a geometrical size condition on the cloaked object (copper cylinder located at the center). However, it should be emphasized that our computational resource was limited. As a consequence the cloak and the simulation box including surrounding air were designed in such a way that the memory does not exceed the maximum memory available on our workstation. Perfect electric conductors were assigned to the bottom and top faces of the simulation cell. A radiation boundary condition was applied to the sidewalls to avoid the influence of virtual cloaking devices introduced by perfect magnetic boundaries. Finally, input and output waveports were assigned to the remaining faces. This boundary setup intends to model a parallel-plate waveguide configuration whereby the incident E-field is constantly guided between the two metallic planes throughout the simulation domain, thus close to an experimental characterization with a vectorial network analyzer. Also, note that to the knowledge of the authors, this is the first time that the simulation of a micro-structured cloaking device is performed and reported by modeling the elementary cells.

We first studied the scattering of a plane wave at 0. 58 THz (λ = 517µm) for an uncloaked copper rod. The results of the E-field distribution achieved in this case are displayed in Fig. 4 (a). Strong distortions of the E-field are clearly observed behind and in front of the metal scatterer. Fig. 4 (b) shows the E-field *z* component for the metallic object now surrounded by the microstructured cloak. These results unambiguously demonstrate that cloaking is achieved at the targeted frequency. First, the E-field wavefronts are well reconstructed behind the cloak with minimal scattering. A high power transmission value of ~66% and low reflection value of ~13% were computed, thus indicating that ~20% of the total energy is either absorbed within the BST rods or reflected toward the radiation boundaries. Also few back-scattered ripples are observed at the front and side of the cloak.

Fig. 4 (c) shows a zoomed view of the cloak region. For clarity the field magnitude was magnified in order to have further insight into the field map within the cloak. In contrast with recently reported simulations of piecewise homogeneous annular cloaks, the wavefronts do not clearly follow the annular geometry of the cloak. Thus, the E-field is mostly concentrated within the first outer concentric rings of rods and does not penetrate deeply within the annular shell. This latter observation may be a consequence of the slightly negative permeability values displayed by the two inner-side set of rods. This effect, which results from the linear variation in the diameter shrinking, contributes to isolate the inner object by taking advantage of single negative inner layers.

## 5. Conclusion

Mie theory was applied to high-κ micrometer-sized BST cylinders and results showed that the ferroelectric rods exhibited a strong magnetic resonance dependent on the cylinder radii. This artificial magnetism was exploited to build an original all-dielectric design that performs cloaking at THz frequencies. Full-wave simulations coupled with a field-summation retrieval technique were employed to adjust the electromagnetic response of individual ferroelectrics rods. The rods magnetic plasma frequency was engineered such that the cloak shell displays a piece-wise variation in the radial component of the permeability; hence satisfying, for a TE polarization, the reduced equations derived from the conformal transformation theory. The cloaking performance was assessed by modeling, for the first time to our knowledge, the complete micro-structured device. Results clearly show that cloaking of any wavelengthscaled object located at the interior is achieved at 0.58 THz for the present device. In particular, the wavefronts of the electric field behind the device are well reconstructed. Additionally, a relatively high transmissivity level was obtained. These results show the potential of this all-dielectric solution for cloaking applications at microwave and terahertz frequencies.

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