In this paper, we report a fabrication of a three-dimensional (3D) carpet cloak that works for any polarization in free space. Two-dimensional (2D) conformal mapping is first employed and the 3D structure is generated by a rotation of the 2D cloak. The structure of the cloak is hole-in-dielectric. The triangular invisible region has a height of 36 mm (one third of the height of the whole device) and a width of 240 mm. The cloaking effect is examined in free space by measuring the scattering parameters. The results show our device has very good cloaking performance in a wide frequency range from 4 to10 GHz.
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Attaining invisibility cloaking has long been a dream for scientists. Recently, some breakthrough has been attained, due to the development of the optical transformation theory and the discovery of metamaterials [1–13]. Transformation optics is able to design an invisible cloak, by guiding the flow of light around the object [1,2,14]. This method has its foundation on the invariance of Maxwell’s equations under coordinate transformation . In the optical transformation approach, empty space geometry in virtual space is mapped into a real material parameterized by a set of anisotropic dielectric tensors. In particular, if this transformation is the conformal mapping, the obtained invisible cloaks have isotropic in-plane dielectric parameters [2,16,17]. The usual stringent requirements and large contrast of anisotropic dielectric tensors make the cloak implementation very difficult [3,18]. For practical application an invisible cloak should be designed to meet two basic requirements: (1) it should have a reasonable size for the invisible region compared with the operating wavelength and the whole size itself, and (2) it should have a 3D response capability and work under any polarization. In this paper, we report our progress in fabricating 3D microwave carpet-type cloaking devices that have practical broadband performance and work in free space at arbitrary incident and polarization angles.
A 3D invisible cloak was first experimentally demonstrated by Cui et al . Their 3D cloaking structure was configured through a rotation of a 2D carpet cloak whose index profile was obtained through a quasi-conformal transformation . In this work, similar method has been adopted to build the 3D invisible device. Our major differences compared to Cui’s work are: (1) here we used exact conformal mapping method rather than the quasi one  and this helped us obtain an isotropic dielectric invisible device; (2) to the authors’ best knowledge we achieved the largest value (1/3) for the height ratio of the cloaked region to the whole device.
2. 2D model design and simulation
In this work the 3D cloak is built by simply rotating a 2D isotropic cloak due to the missing of a directional 3D conformal transformation. The initial 2D cloak is conformally constructed so that it has isotropic dielectric properties. The conformal cloak in principle has infinite size. But in practice we could truncate it into a manageable dimension with the outer part of index n close to unit replaced simply by air. The reason behind is due to comparable operation wavelength with respect to the sizes of our device and cloaked region, which incurs small wavefront distortion by the outer layer approximation. Here we choose a rectangular geometry for the device with a triangular bump at the bottom. Things put inside the empty bump are invisible to an outside observer. Saw-tooth like boundaries is used to approximate the straight bump edges. Figure 1(a) gives the calculated index profile. Their values vary from 0.6 to 2.2. But only two unit blocks at the bottom apexes of the triangular bump have an index less than 1. Thus we approximate them by air. In addition the region far away from the cloaked bump with index close to 1 is also approximated by empty air. Consequently, only a relatively small region needs to be filled by dielectric media with indices all larger than unity. In implementation we discretize the cloaks into blocks of square unit cells. Each square has a dimension of 6 × 6 mm2. As shown in Fig. 1(b), the whole 2D cloaking device was divided into 41 × 18 unit squares and their local indices change from 1 to 1.7.
To evaluate the device performance designed, far-field electromagnetic (EM) scattering patterns are calculated by COMSOL at several frequencies for four different cases: (1) only a flat ground, (2) a triangular bottom bump without cloak, (3) a triangular bottom bump with an ideal cloak (continuous inhomogeneous medium) and (4) a triangular bump with a discrete cloak (homogeneous multilayers). The calculated scattering curves are plotted in Figs. 2(a) to 2(d) where the insets give the corresponding snapshots of the simulated electric field patterns.
The polar angle θ starts clockwise from the y axis as schematically shown in the middle inset of Fig. 2. A Gaussian beam is incident from the top left at a fixed angle of 45° from the y axis. The scattering patterns are calculated at 10 GHz. For the case of a flat conducting ground as shown in Fig. 2(a), simulation shows a typical mirror reflection. For the case of a naked triangle metal bump as shown in Fig. 2(b), the incident beam was irregularly scattered and had a very wide outward reflection angles. Two reflection peaks around 15°and 70° can be discriminated as also evidenced from the inset field profile. For the case with an ideal continuous cloak as shown in Fig. 2(c), simulation shows a single reflection beam at 45° and proves a perfect carpet cloaking. Discretization is necessary to implement the cloaking device. This may cause structural imperfection and degrade the overall cloaking performance. As shown in Fig. 2(d), simulation did show a broadened reflection peak around 45°. But consider the finite sample size with the wavelength reaching almost 5 times of the unit cell size, such a cloaking performance is still reasonable acceptable. The increased noise level in this case is mainly caused by the internal boundary scattering after discretizing the device.
3. Implementation of the 3D cloak
The 3D carpet cloak is achieved by rotating the 2D cloak around the symmetric axis by 360°. Each horizontal line on 2D plane forms a circular disc in 3D space. The refractive index on each disc is a function of radial position (r). Also, each 6 × 6 mm2 unit cell approximately becomes a 6 × 6 × 6 mm3 cubic unit cell in the 3D space. Transverse electric (TE) and transverse magnetic (TM) polarization are both considered in our experiment, respectively corresponding to E-field parallel and perpendicular to the axe of empty holes in the unit cells used in our structure. A software HFSS is used to calculate the hole diameters for each refractive index for both polarizations. As the refractive index is required from 1 to 1.7, we use Teflon (n = 1.45) and Delrin (n = 1.92) as the embedding materials.
In the microstructures two types of unit cell are used. One has a dimension of 6 × 6 × 6 mm3, where hole in dielectric elements occupies the whole cubic unit cell element. In order to achieve the refractive index close to 1 for regions near the edge of cloak, a 6 × 6 × 2 mm3 brick is also used with a hole in each dielectric element. The calculated refractive indices as a function of hole radius of the three types of unit cells are presented in Figs. 3(a) to 3(c). The unit cell of 6 × 6 × 2 mm3 Teflon is used for regions with n close to 1, 6 × 6 × 6 mm3 Teflon is used for regions with both n > 1.15 and n < 1.45; 6 × 6 × 6 mm3 Delrin is used for n >1.45. Figures 3(a) to 3(c) show that all the three unit cells have almost the same profile for both polarizations with the variation less than 5%. Figures 4(a) to 4(d) show the different views of the assembled 3D cloaking device. In assembling thin foam with ε = 1.05 is used to separate each dielectric disk layer.
4. Far-field free space measurement
The experimental set up is shown in Figs. 5(a) and 5(b). The whole measurement system contains two movable arms, two antennas mounted on each arm, a sample holder in the middle and a pair of microwave focusing lenses .The distance from the sample to the antenna is fixed at 1.7 m. A vector network analyzer (model Agilent N5244A) is utilized to emit, receive and process the measurement signals. We perform the measurement at two incidence angles of 35° and 45°for both TE and TM polarizations. For the case of 35° incidence, we measure the reflection angle from 20° to 90° in a step of 5° and from 15°to 90° for the 45° incidence. The polarization of the incident beam is changed by rotating both antenna horns by 90°.
Figures 6 and 7 give the measured far-field reflection wave intensity profiles as a function of reflection angle θ at incident angles of 35° and 45°, respectively. Our measurement frequency is from 4 to 10 GHz. Here we selectively show the results at four frequency points, i.e., 4 GHz, 7 GHz, 8 GHz and 10 GHz. Each figure gives both TE and TM polarization results together with that for a flat ground for comparison purpose. From these reflection curves, we can see our cloaking device performs very well in the measured frequency range. The reflected wave profiles for both TE and TM polarizations can nearly agree with that reflected from a flat ground. However discrepancies are also observed in particular at higher frequencies (8-10 GHz). This could be due to the large unit size employed in our structures. At 7 GHz, the wavelength (4.5 cm) is 7.5 times the dimension of the unit cell used. This touches the limit of the effective medium theory based in our medium design, which usually requires a wavelength-to-unit length ratio larger than 10. In addition, the broadened reflection peaks are consistent with the above simulations for a real device made of discrete index profile, which causes impedance-mismatched internal boundary scattering. Such negative influence can be minimized using smaller unit cells.
In this paper, we demonstrate that it is feasible to construct a large 3D carpet cloak with the cloaked region height one third of the height of the cloak for all polarizations in free space. Far-field experimental results show that the cloak performs well in a wide frequency range from 4 to 10 GHz. The cloaking effect and the frequency range studied can be improved by adopting smaller unit cells. There is a tradeoff between good performance broadband cloak and the size of the unit cell used in the fabrication.
The project is funded by DIRP project INKHEART. MYG is partially funded by the NSFC 61271085 and 91130004, NSF of Zhejiang Province (LY12F05005), NCET and MOE SRFDP of China.
References and links
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