Lateral Shift Makes a Ground-Plane Cloak Detectable

We examine the effectiveness of the ground-plane invisibility cloak generated from quasiconformal mapping of electromagnetic space. This cloak without anisotropy will generally lead to a lateral shift of the scattered wave, whose value is comparable to the height of the cloaked object, making the object detectable. This can be explained by the fact that the corresponding virtual space is thinner and wider than it should be. Ray tracing on a concrete model shows that, for a bump with a maximum height of 0.2 units to be hidden, the lateral shift of a ray with 45° incidence is around 0.15 units.

Figure 1

An example of creating an invisibility cloak by coordinate transformation. In general the transformation is not a conformal mapping and the transformed cloak is anisotropic.

Figure 2

Left column: Ray tracing diagrams of a ray with incident angle of ${\theta }_i$. Right column: A Gaussian beam with incident angle of ${\theta }_i$. (a),(b) A PEC ground plane in free space with no cloak. (c),(d) A perfect anisotropic cloak squeezing space between $z=0$ and $z=d$ into space between $z=\delta d$ and $z=d$. (e),(f) Replace the anisotropic cloak with isotropic medium. In the illustration, $\delta =0.5$ and ${\theta }_i=45°$.

Figure 3

$k$ vectors used in phase matching between the incident and cloak media. The solid circle represents the incident medium and the solid ellipse represents the cloak medium. The dotted circle represents the isotropic medium that replaces the cloak medium.

Figure 4

A ray (blue solid line) incident at 45° to the normal onto an isotropic ground-plane cloak. The bump has a maximum height of 0.2 units [5]. The red dotted line represents the trajectory of an undistorted ray in the absence of the cloak and the bump. The background represents $n^2$. Notice the distorted ray has been shifted on top of the cloak toward the incident point by 0.15 units.

Figure 5

Procedure of quasiconformal mapping. (a) A rectangle of $1\times m$ in virtual free space. (b) Compressed rectangle of $1\times M$ in the intermediate space. (c) An anisotropic cloak created from quasiconformal mapping.