Invisibility cloaks are a favorite device used by science-fiction and fantasy authors. Yet, despite the fact that invisibility fascinates many people, its practical realization has rarely been treated as a serious research topic in science and technology. Very recently, researchers noticed that one can deduce the exact material requirement for making a completely passive cloaking device using the theory of transformation optics.^{1, 2} The basic principle is illustrated in Figure 1. Even for the simplest cylindrical cloak structure, however, the ideal material parameters prove difficult to match with current metamaterial fabrication technology. Therefore, one would prefer to simplify the material required for a cloak. Here we describe two promising approaches for constructing a simplified cylindrical invisibility cloak that should ease implementation while retaining good performance.

**Figure 1.** Formatting a cloak of an arbitrary shape by transformation optics (a): and (b) depict correlated coordinates' lines in electromagnetic space and physical space, respectively. A void region *S*_{2}in (b) has appeared, which is obtained by blowing up a point (or a line, if the structure is invariant along the paper-normal direction) of zero size in (a). The annular shape bounded by *S*_{1}and *S*_{2} in (b) is electromagnetically equivalent to the region bounded by *S*_{1} in (a). However the regions' material parameters are different owing to a change of coordinate. The region bounded by *S*_{1}and *S*_{2} in (b) constitutes a cloak, whose interior can conceal objects. Notice that the coordinate changes only for a finite spatial region bounded by *S*_{1}in both panels. As a consequence, the material outside *S*_{1}(which we assume is air) remains the same in both panels.

In a cylindrical coordinate system (*r*, θ, *z*), one can simplify the cloak's materials parameters ε_{z}, μ_{θ}, and μ_{r} for normal wave incidence—with a single transverse electric (TE) polarization (in which the electric field is solely aligned along the axial or z direction of the cloak)—by keeping intact the products ε_{z}μ_{θ} and ε_{z}μ_{r}.^{3} In the simplest case, a cylindrical cloak can be obtained through linear radial spatial mapping, i.e. ,

where *a* and *b* are the cloak's inner and outer radii, respectively, and *r*′ and *r* are the radial coordinates in electromagnetic and physical spaces, respectively. The corresponding ideal material parameters are shown in the left column of Table 1. Based on this ideal cloak, a simplified version, as proposed by John Pentry's group^{3}, has material parameters shown in the middle column in Table 1. However, such a simplified cloak still scatters electromagnetic fields considerably, largely due to its unmatched impedance to free space.^{4}

**Table 1.** Material parameters for linear cloaks

We propose a new version of a simplified cylindrical cloak that restores impedance matching to the space outside the cloak. We do not vary the products ε_{z}μ_{θ} and ε_{z}μ_{r}.^{5} This improved set of material parameters is shown in the right column of Table 1. Using the full-wave finite element method, we numerically compare the invisibility performance between a previously proposed simplified cloak and the improved cloak (see Figure 2). Both cloaks have the same geometry as *a* = 0:1m, *b* = 0:3m, and a perfectly electrically conductive lining is imposed at the inner cloak surface. The wavelength is fixed at 0:15m. From Figure 2, it is apparent that a plane wave is less perturbed by the improved simplified cloak. This cloak also scatters less light, and the scattering is dominated by the zeroth-order cylindrical component.

**Figure 2.** Snapshots of the electric field components in the *Z* direction (*E*_{z}) around (a) a previously proposed simplified linear cloak, and (b) an improved cloak. Both originate from the linearly coordinate transformed ideal cloak. The scattered *E*_{z} fields of the two cloaks in (a) and (b) are shown respectively in (c) and (d).

The second approach for constructing a simplified cloak is a further improvement on our previous proposal. Here we would like to eliminate the zeroth-order scattering. In addition, we try to simplify the cloak medium to non-magnetic material. To achieve the two objectives, we use a quadratic radial coordinate transformation function

After simplifying, as discussed above, we obtain a cloak with impedance matching to free space.^{6} If the incident wave consists of only transverse magnetic (TM) polarization, we need only non-magnetic material for constructing the cloak. Unlike the simplified cloak based on another nonlinear coordinate transformation function proposed by Vladimir Shalaev's group,^{7} our designed cloak does not suffer from the minimum shell thickness restriction. To eliminate the zeroth-order cylindrical scattering wave, we deploy a layer of air at the inner surface of the cylindrical cloak, which is further terminated by a perfect magnetically conducting layer. When the air gap is at resonance, the zeroth-order cylindrical wave tunnels through the structure without experiencing any scattering.

In Figure 3, we present the interaction of a plane wave with such a quadratic cloak, both with the extra air gap—Figure 3(a)—and without the gap—Figure 3(b). The cloak has *a* = 0:3m and *b* = 0:6m, and the wavelength is 0.3m. The plots in Figure 3 show that the extra air gap introduced at the inner surface of the cloak has almost completely canceled out the zeroth-order scattered field. The remaining scattered field in Figure 3(d) is comprised of mostly high-order cylindrical wave components with significantly lower amplitudes. The final cloak preserves the plane wave profile better as compared to the case without the resonant gap.

**Figure 3.** Snapshots of magnetic field components in the *Z* direction (*H*_{z}) for a plane wave incident on (a) a quadratic cloak without a gap, and (b) the same cloak but with the resonant gap. The corresponding scattered *H*_{z} fields for (a) and (b) are shown in (c) and (d), respectively.

Based on the theory of transformation optics, we have shown how one can realize practical cylindrical invisibility cloaks with near-perfect performance. Such invisibility cloaks can be used for military and civilian applications, e.g. for stealth technology, invisible electric wires, and electromagnetic radiation protection of either people or sensitive electronic devices.

Certainly we are still far from the flexible invisibility cloaks depicted in science-fiction novels. One of the main limitations yet to be overcome is the operation bandwidth. Due to inevitable material dispersion, invisibility cloaks designed so far can perform well at only a single wavelength. Further theoretical efforts are necessary to extend the operation bandwidth. In addition, fabrication of a fully 3D metamaterial whose individual material tensor components are all tailored independently remains a technical challenge. Cloaks with better invisibility performance and more flexible shapes can be realized only after complex metamaterial fabrication technology matures.

*This work is supported by the Swedish Foundation for Strategic Research (SSF) through the Future Research Leader program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR).*

Min Qiu, Wei Yan

Department of Microelectronics and Applied Physics

Royal Institute of Technology

Stockholm, Sweden

Min Qiu is an associate professor at the Department of Microelectronics and Applied Physics, Royal Institute of Technology, Sweden. He received an Individual Grant for the Advancement of Research Leaders from the Swedish Foundation for Strategic Research, as well as a senior researcher fellowship from the Swedish Research Council.

Min Yan

Department of Photonics Engineering

Technical University of Denmark

Lyngby, Denmark

References:

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D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, et al., Metamaterial electromagnetic cloak at microwave frequencies, *Science* 314, no. 5801, pp. 977-980, 2006.doi:10.1126/science.1133628