1. Introduction

In the past few years, invisibility cloaks have attracted great attention due to their exciting properties of deflecting incoming electromagnetic (EM) waves and directing them to propagate around the cloaked objects without any scatterings. Coordinate transformation was proposed by Pendry et al [1] to deform an empty space into a shell region, which makes the object inside the shell invisible to incident EM waves. According to this concept, many different kinds of cloaks have been proposed, including elliptical cloak without singularity [2], cloaks of twisted domains [3], petal-shaped cloak [4], toroids cloak [5], and arbitrary shaped cloaks [6]. Analytical methods based on Maxwell’s equations have been used to reveal the nature of invisibility cloaks and characterize the wave interaction [7,8]. More recently, the inverse mechanism of cloaking designs without specifying the needed coordinate transformation beforehand has been developed [9]. Different from these aforementioned approaches, in which the coordinate transformation is operated along the radial direction, one-directional transformation is proposed [10] to manipulate EM wave by layered systems [11]. Recently, a conical cloak is investigated in detail using plane transformation [12]. Different from the complete cloaks [1–12], a “carpet cloak” has been proposed to mimic a flat ground plane [13] and has been experimentally verified [14]. Such a carpet cloak could avoid both material and geometry singularities. However, the material for the carpet cloak is still inhomogeneous and needs to be carefully designed using quasi-conformal transformation.

In this paper, we propose a general trapeziform cloak that can be realized by isotropic nonmagnetic materials. The cloak could degenerate into a triangular cloak as well as flat cloak by particularly setting the geometry of the trapeziform cloak. Note that the degenerated flat cloak only requires homogeneous uniaxial medium, when the size of the cloaked object is much smaller than the cloak or the trapeziform’s slopes tend to zero. The flat cloak can be a large-scale device and easily realized due to the simplicity in the parameters of the medium above the PEC. Meanwhile, the position of the PEC cover can be adjusted based on the practical sizes of military targets which can be perfectly concealed and protected from the changes in external environments. Moreover, since the space below the PEC cover is blind, one can make the PEC cover thicker and use hard pillars between the PEC and ground as firm supports. The proposed isotropic trapeziform cloaks and its degenerated cloaks suggest insightful scenarios for exciting applications, and this concept may nurture further degrees of freedom in the design of large-scale cloaks.

2. Design scheme and analysis

A schematic diagram illustrating a trapeziform cloak is shown in Fig. 1 . The region within the dashed trapezoid in the virtual space [Fig. 1(a)] is compressed into two triangles and one rectangle regions in the physical space [Fig. 1(b)] with inner PEC boundaries, resulting in a trapeziform region blind to external illumination. Thus the spatial deformation is only along y axis in a uniformly linear manner within the cloak area.

 

Fig. 1 The scheme of coordinate transformation from the virtual space (a) to physical space (b). $k_1$ represents the slope of the inner PEC boundary in Region I (identical to that in Region II).

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Due to the form-invariance property of Maxwell’s equations throughout the transformation, the permittivity and permeability tensors of the medium in the physical space can be expressed [1]

We can find that the trapeziform cloak is composed of three regions in Fig. 1(b). The transformation equations for region I are

The coordinate transformations of region I and II compress two triangular regions into another two triangular regions, and the coordinate transformation of region III deforms one rectangular region into another in accordance with the deformations of the triangles on both sides, which can be expressed as

It is interesting to note that the constitutive parameters in Eqs. (3), (5) and (7) are spatially invariant, which facilitates the experimental verification by periodic metamaterial structures and could also be easily realized through an alternating layered system of two homogeneous isotropic materials.

3. Simulation and representative results

In order to verify the designed formulae, we make full-wave simulations based on the finite element method (FEM). The working frequency is set to be 2 GHz under TE polarization. We create a trapeziform cloak with $k_1=1$ and $d=b=h=0.5\text{\hspace{0.17em} m}$ illuminated by a Gaussian beam incident from the right at ${45}^{\circ }$ as in Fig. 2 . Figure 2(a) demonstrates the electric field distribution of a perfectly reflective flat boundary corresponding to Fig. 1(a), and Fig. 2(b) shows the electric field distribution of a trapeziform cloak with PEC inner boundary corresponding to Fig. 1(b). Comparing Figs. 2(a) to 2(b), the fields remain unperturbed in the presence of the cloak, as if no scatterers were present above the PEC ground.

 

Fig. 2 Snapshot of the electric field for a Gaussian beam incident from the right at an oblique direction of ${45}^{\circ }$. (a) A perfectly reflective flat boundary. (b) A trapeziform cloak with inner PEC boundaries as in Fig. 1(b). (c) The triangular cloak degenerated from a trapeziform cloak when Region III vanishes. (d) The degenerate flat cloak when $b=h$.

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Assuming $d\to 0$, region III in Fig. 1(b) vanishes and the trapeziform cloak then degenerates into a triangular cloak as Fig. 2(c) illustrates. The distribution of the reflected beam by the triangular cloak demonstrates the perfect cloaking performance.

Assuming $k_1\to 0$ while h and b being equal and invariant, the trapeziform cloak degenerates into a flat cloak only composed of region III ${\epsilon }_{\text{III}}={\mu }_{\text{III}}=\text{diag}[2,1/2,2]$. Figure 2(d) presents the electric field distribution of the flat cloak. Compared Fig. 2(a) to Fig. 2(d), it can be seen that the directions of reflected waves are identical. Obviously, the flat cloak maintains good performance as long as the width of the cloak is large enough. Hence, the flat cloak with uniaxial materials could be extensively utilized to cloak groups of static objects in the cloaked space from $y=0$ to $y=h$ beneath the PEC.

The flat cloak in Fig. 2(d) composed of uniaxial materials is less demanding on the requirement of parameters and design complexity compared to [14] and it could be easily realized through an alternating layered isotropic mediums [11,15,16]. The permittivity and permeability of the flat cloak could be expressed as ${\epsilon }_{\text{III}}={\mu }_{\text{III}}=\text{diag}[{\epsilon }_x,{\epsilon }_y,{\epsilon }_x]$. According to effective medium theory, the flat cloak is equally discretized into M layers, and each layer consists of medium-A and medium-B. The old set of material parameters of isotropic medium-A and medium-B is represented by

Using old and new sets of material parameters respectively, the flat cloak with homogeneous uniaxial materials could be easily imitated by multilayered homogeneous isotropic materials. The electric field distributions of the flat cloak with the old (left column) and new (right column) sets of isotropic materials are shown in Fig. 3 , respectively. It is obvious that the cloaking effect is quite pronounced when the discretization increases.

 

Fig. 3 Electric field distributions of the flat cloak with M-layered isotropic materials based on two sets of effective medium theories. (a-c) correspond to the old set [Eq. (8)] while (d-f) correspond to the new set [Eq. (9)]. (a, d) M = 20. (b, e) M = 40. (c, f) M = 60. The dashed semi-circle is the observation radius for measuring the average power flow for the reflective beam.

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We simulated the average power flow at the observation circle with the radius of 1.8m whose origin is at (2m, 0). In Fig. 3(c), from the angles $0^{\circ }$ to ${35}^{\circ }$, the dashed circle is located below and inside the flat cloak, so the observation angle starts from ${35}^{\circ }$. In Fig. 4(a) , it is obvious that the flat cloak with the new set of isotropic parameters behaves much better than that with the old set of isotropic material parameters. The reflective beam by the flat cloak with the new set is very close to the ideal case, while the old set still leads to measurable deviation even though the layer number M is up to 60 [Fig. 4(b)]. It can be attributed to the fact that the relative index of the new set of isotropic materials equal to 1 resulting in the improved invisibility, which is less oscillative and varied more smoothly than the old set of isotropic materials with step-index profile, as shown in Fig. 5 . Although the impedance mismatch of the new set will lead to slightly increased backscattering, the index matching will bring down the forward scattering more significantly, resulting in improved overall scattering reduction. This feature in turn provides us more degrees of freedom to make the trade-offs between perfect invisibility and fabrication cost-effectiveness. Then near-perfect flat cloak with restored invisibility performance based on the new set of isotropic parameters can be an economic alternative, especially when the coating process of layered isotropic dielectrics has to be terminated at a certain medium value of M (e.g., M = 40 or even smaller).

 

Fig. 4 Comparison of the average power flow of the isotropic flat cloak in Fig. 3(c) and Fig. 3(f), mimicked by old [Eq. (8)] and new [Eq. (9)] effective medium theories. (a) M = 40, (b) M = 60. The same incidence is used as Fig. 2.

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Fig. 5 The relative index profiles of the isotropic flat cloak in Fig. 2(d) mimicked by old [Eq. (8)] and new [Eq. (9)] effective medium theories with M = 5.

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The examples discussed above are with complete EM parameters, which can be realized by metamaterial. For application at optical wavelength, we should reduce the material parameters to a nonmagnetic form. By keeping the products of ${\epsilon }_x{\mu }_z$and ${\epsilon }_y{\mu }_z$ unchanged, the permittivity and permeability of the flat cloak can be simplified as ${\epsilon }_{\text{III}}=\text{diag}[{\left(\frac{b+h}{b}\right)}^2,1,1]$ and ${\mu }_{\text{III}}=\text{diag}[1,1,1]$, respectively. Figure 6 shows the magnetic field distributions for four different cases. Figure 6(a) corresponds to the magnetic field of a flat reflective mirror. When a triangular PEC is placed on the flat boundary, the magnetic field scattered by the obstacle is quiet irregular shown in Fig. 6(b). In order to make the obstacle invisible, a nonmagnetic flat cloak with $h=0.2\text{\hspace{0.17em} m}$ and $b=0.8\text{\hspace{0.17em} m}$is placed above the obstacle, as shown in Fig. 6(c). Clearly, the reflective magnetic field is confined highly in the specular direction, which is in consistence with the case of Fig. 6(a). Based on the effective medium theory, the nonmagnetic flat cloak could be realized through an alternating layered isotropic mediums with ${\epsilon }_{\text{A}}=0.625$ and ${\epsilon }_{\text{B}}=2.5$, as shown in Fig. 6(d). It can be seen that the isotropic nonmagnetic flat cloak perfectly mimics the case of a flat reflective mirror [Fig. 6(a)]. In order to design a realizable isotropic nonmagnetic flat cloak, assuming the cloak is embedded in a composite background medium with permittivity of 1.6, the alternating isotropic dielectrics proportionally become air (${\epsilon }_{\text{A}}=1$) and silicon (${\epsilon }_{\text{B}}=4$), respectively.

 

Fig. 6 (Color online) Magnetic field distributions for a Gaussian beam incident from the right at ${45}^{\circ }$on (a) a perfectly reflective flat boundary, (b) a bumped reflective boundary, (c) a nonmagnetic flat cloak placed above the bumped boundary, (d) an isotropic nonmagnetic flat cloak placed above the bumped boundary with M = 40.

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4. Conclusion

In this paper, a trapeziform cloak that only requires homogeneous anisotropic material is proposed, which can degenerate into flat or triangular. The flat cloak is constructed by isotropic mediums based on new set of effective medium theory, which furthermore improves the performance of actual flat cloak. We also design an isotropic nonmagnetic flat cloak that is able to work at optical frequency and could be easily realized through alternating layered dielectrics. This general cloaking concept may pave a realizable way to the practical applications in large scales and in higher frequency.