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We propose, design, and implement a two-dimensional dc carpet cloak for steady electric field using the transformation optics (TO) method. Based on the circuit theory, we introduce a resistor network to mimic the resulting anisotropic conducting medium. The experimental prototype is fabricated using metal film resistors, and the measured results agree perfectly well with theoretical predictions. This study gives the first experimental verification of a dc carpet cloak, which expands the application of TO theory, and has potential applications in related areas.
© 2012 Optical Society of America
1. Introduction
Transformation optics (TO), the theory of manipulating electromagnetic (EM) fields based on invariance of Maxwell’s equations, has long been recognized [1, 2]. In 2006, Pendry et al. presented an elegant design of invisibility cloaks using TO [3, 4], which was later verified in the microwave band [5]. Invisibility cloaks have then received great attentions from all over the world [6–12]. Nowadays, researches on invisibility cloaks have gradually expanded to various fields, including optics [13, 14], acoustics [15–17], thermodynamics [18, 19], surface plasmon polaritons [20, 21], and temporal cloaking [22, 23], etc. Cloaking for static EM fields has also been proposed, numerically proved and/or experimentally verified [24–31]. Actually, Greenleaf et al. discovered the conductivity cloaking in electric impedance tomography (EIT) even before Pendry’s design [30].
Among various cloaking devices proposed so far, carpet cloaks have attracted huge interests in recent years [13,14,32–39]. These cloaks are used to cover objects on the ground and mimic a half-infinite vacuum space. Carpet cloaks can be designed using the quasi-conformal mapping technique, leading to cloaks with approximately isotropic and inhomogeneous materials, which are thus immediately realizable and generally broadband [32–36]. They can also be designed using homogeneous and anisotropic materials with linear transforation functions, and are easily realized using natural materials near the optical bands [13, 14, 37–39]. Like other invisibility cloaks, carpet cloaks have also been introduced in acoustic waves [40, 41], elastic waves [42], and even plasmonics too [43, 44].
In this paper, we design a dc carpet cloak for steady electric field based on the TO method, and realize it using an anisotropic resistor network. Our measurements clearly confirm correctness of the proposed method. The dc carpet cloak is also valid in the three-dimensional scenario and can be extended to have illusion functions [8–10], such as virtual shifting, geometry change [45–48], etc. Hence it has potential applications in medical imaging, noninvasive detection, and underwater/underground explorations.
2. Theoretical model
The proposed dc carpet cloak is illustrated in Fig. 1, in which panel (a) depicts the virtual space in TO theory, representing a half-infinite conducting material (with conductivity profile σ) on a perfectly-conducting plane. The conductivity profile can be accurately obtained using the EIT technology [30, 31], which measures voltage/current distributions around a specific target region. Now we introduce an object with a different conductivity on the ground. The change of conductivity will certainly distort voltage distributions, leading to the detection of the object. However, by covering the object with a ‘carpet’, as shown in Fig. 1(b), i.e. the physical space in TO theory, no distortions will occur outside the device, and hence the object will be invisible from the EIT measurement.
Fig. 1 Schematics of the proposed dc carpet cloak. (a) The virtual space: a half-infinite conducting material on a perfectly conducting plane. (b) A triangular perfectly conducting bump (used to hide the object) covered by the carpet cloak in the physical space, which is equivalent to the virtual space. (c) Transformation used in the design, where AOB is stretched to AC’B and ACB is kept unchanged.
Design of the dc carpet cloak has no difference from its time-varying EM counterpart, as illustrated in Fig. 1(c). Here, the x-axis represents an infinite PEC plane, and a line segment AOB on the ground is pulled to AC’B while another line ACB is kept unchanged. This transformation produces a protruded metallic bump on the ground and a compressed space around it, represented by polygon AC’BC. By filling appropriate materials in this area, the protruded triangular bump will be invisible in the background medium, which can be used to hide certain objects from the outside detectors. According to TO theory [6, 7], the required electric conductivity is
where is the Jacobian matrix, and primed variables refer to the physical space.Since
in the two-dimensional (2D) transformation, a simple calculation gives [13, 37] where σ represents conductivity of the background medium, k = (tanα − tanβ)/tanα, τ = tanβ, and other parameters are shown in Fig. 1(c). Note that due to the symmetry of the structure, only the right part of the device is considered. Apparently, realization of the carpet cloak requires anisotropic conductivities. For the 2D case, only in-plane parameters are relevant to the problem. However, they form a symmetric 2 × 2 matrix, which is difficult to realize. The problem is readily solved in the principal axis system, where the ‘new’ components of the conductivity tensor are and the rotation angle between the two systems is [13, 37] Equations (4) and (5) show that the material inside the carpet cloak is homogeneous and anisotropic (diagonalized matrix), which can greatly reduce the difficulties in implementation.In our recent work [28, 29], we have shown that anisotropic conductivities can be emulated using different resistors in two orthogonal directions, where
Then the bulk conducting medium can be conveniently mimicked using a periodic resistor network.A typical unit cell for the resistor network is given in Fig. 2(a), in which the local principal axis system is utilized. In our design, the background medium is uniformly divided into 16 × 31 small bricks with Δx = Δy = 1 cm, as shown in Fig. 2(b), and then emulated with the resistors from Eq. (6). For materials inside the carpet shell, whose principal axes are different from those of the background medium, Eqs. (4)–(6) are used to calculate the equivalent resistors and determine the rotation angle. In Fig. 2(b), blue dots inside the device represent node positions of unit cells and two insets show the relative position of the two coordinate systems. Note that in the primed coordinate system, Δx′ and Δy′ are not equal,
Fig. 2 (a) Unit cell of the periodic resistor network to mimic an anisotropic conducting medium. (b) Actual grids used for the carpet cloak fabrication. (c) Rear view of the fabricated carpet cloak.
3. Simulation, measurement and discussion
Based on the above theory, we design and fabricate a dc carpet cloak with the resistor network, as shown in Fig. 2(c). In our design, the background medium has a conductivity of σ = 1 S/m, and the ground is represented by a perfect electric conductor. The carpet cloak has a width of 2a = 20 cm, a hight of 10 cm and a maximum thickness about 5 cm. Then, it is easy to see that tanα = 0.5 and tanβ = 1. In simulations and measurements, a dc power supply with a magnitude of 10 V is used to excite the circuit (Row 12 and Column 4). To emulate a half-infinite background, matching resistors are used at the top, left and right sides of the device [28], while the bottom side (Row 0) is grounded. All resistors are commercially-available metal film resistors except for those inside the device along the x′ directions, for which we use surface-mounted resistors due to their small volumes. The carpet cloak is fabricated using the printed circuit board (PCB) technology and manually welded. In the experiment, we use a 4-1/2 multimeter to measure the voltage at each node. The measured data are then imported into Matlab for visualization. Figure 2(c) shows details of the fabricated carpet cloak.
The simulated and measured voltage distributions of the proposed dc carpet cloak are demonstrated in Fig. 3, where the contour plots are given to visualize the data. Note that all simulations are made using the commercial software, Agilent Advanced Design System (ADS). In Fig. 3(a), four sides of the studied area are matched to the ambient environment, mimicking an infinite conducting medium. The isolines in this figure are concentric circles centered at the voltage source, which firmly validates the matching resistors. In Fig. 3(b), the bottom side is grounded, mimicking a half-infinite conducting medium. We observe that the voltage distribution alters dramatically to reflect this change. Note that this distribution actually represents the voltage profile in the virtual space and used as the control in our experiment. Figure 3(c) illustrates the voltage distribution when a triangular bump is introduced on the ground, which is bounded by the black solid line. Comparing Figs. 3(c) with 3(b), the voltage profiles are distorted significantly, suggesting the existence of the bump on the ground if measured using the EIT technique.
Fig. 3 Simulated and measured voltage distributions. (a) The simulation result when four sides of the resistor network are matched to the background, emulating the case of an infinite conducting medium. (b) The simulation result when the top, left and right sides of the resistor network are matched to the background, while the bottom side is grounded, emulating the case of a half-infinite conducting medium. (c) The simulation result when a triangular perfectly conducting bump sticks out from the ground. (d) The simulation result when the triangular bump is covered by the proposed carpet cloak. (e) Similar to (d) but with commercially-available resistors. (f) The measured voltage distribution. Note that the dashed/solid black lines refer to position of the carpet cloak for the sake of comparison. In all figures, the contour plots are given.
We then cover the triangular bump with the designed dc carpet cloak in Fig. 3(d), expecting that the triangular bump will be invisible from the outside detection. Comparing Figs. 3(d) with 3(b), we observe that after filling the designed anisotropic resistors between the outer and inner walls of the invisibility cloak, denoted by the black solid lines in the figure, the field distribution outside the carpet cloak (in the background) looks exactly the same as that in the virtual space shown in Fig. 3(b). Hence a perfect dc carpet cloak is realized.
Since ideal-valued resistors may be unavailable in practical situations, we also give the simulation results using actual-valued resistors, as demonstrated in Fig. 3(e). Comparing Figs. 3(e) with 3(d), we notice that the voltage distribution is distorted slightly when actual-valued resistors are used in the simulation. However it is apparent that the distortion is very small and hence will not affect the invisibility performance when the carpet cloak is physically realized. This fact is also reflected in the later experiment.
Figure 3(f) illustrates the measured voltage distribution. A comparison between Figs. 3(e) and 3(f) shows that the measurement data agree perfectly with the actual-valued simulation results. The negligible discrepancy mainly comes from the following parts: (1) the theoretical resistor model in Eq. (6), where current directions are assumed to be along two orthogonal directions and be uniform on each block. However, when the conducting medium is discretized into small blocks, actual currents on them may slightly differ from ideal assumptions; (2) approximations between ideal and actual resistors; (3) errors of resistors with their nominal values, which is less than 1% for metal film resistors; and (4) the manual welding process in the fabrication. The above observation suggests that the use of anisotropic resistor network can successfully accomplish the goal of ground invisibility for the steady electric field.
Generally speaking, the proposed dc carpet cloaks are simple, compact and easy to fabricate. They can also be seamlessly integrated with current IC technology. Moreover, their realization is not constrained by singular conductivities, i.e. zero or infinite conductivities. Actually, even negative conductivities can be mimicked with negative resistors realized using active elements. The main disadvantage lies in the fact that they only work for the dc case or at low frequencies, which may limit their practical applications.
4. Conclusions
To summarize, we have designed and fabricated a dc carpet cloak for the steady electric field. The simulation and measurement data confirm its excellent performance. Our study suggests that the device has potential applications in defence, underwater/underground detection, and EIT related areas. Other dc illusion optical devices [8–10, 45–48], such as exterior cloaks [9], virtual shifting devices, and geometry-changing devices [45, 46], can also be designed and emulated using the same method.
Acknowledgments
This work is supported in part by a Major Project of the National Science Foundation of China under Grant Nos. 60990320 and 60990324, in part by the 111 Project under Grant No. 111-2-05, and in part by National High Tech (863) Projects under Grant Nos. 2011AA010202 and 2012AA030702. Z. L. Mei acknowledges the Open Research Program Funds from the State Key Laboratory of Millimeter Waves (No. K201115), Natural Science Foundation of Gansu Province (No. 1107RJZA181), the Chunhui Project (No. Z2010081), and the Fundamental Research Funds for the Central Universities (No. LZUJBKY-2012-49).
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