When the cloak is irradiated with "light" from a distance, the light starts to meander greatly from the vicinity of radius 3 and passes through a surface with a discontinuous refractive index n off into the distance again (Fig. 2). At the surfaces with discontinuous index n, light is either refracted in the shape of a left angle bracket or is not refracted at all.

The core portion of the cloak is impenetrable to light coming from any angle. Thus, anything placed in that portion becomes invisible when seen from a distance. The radius of the core region in the figure is approximately 0.12, although it differs depending on the maximum value of n and the distribution of index. The cloak is considerably thick, assuming that the radius 5 is the outer edge of the cloak.

Here, the distribution of refractive index n plays an important role for the cloak to be invisible. Light goes around the core portion because the distribution of refractive index in the media is designed such that the squared value of index n becomes greater in the direction of the core.

To the incident light, the squared value of the refractive index n is like bumps on a green to a golf ball. When a ball is hit with the same amount of force, there will be some places where the ball cannot reach depending on the height of bumps on the green.

No reflection or phase delay

The researchers set the following conditions for designing the media. (1) No light reflections occur at the boundaries between the media, (2) No phase delay generated in the light that passes through the media, with regard to the light that passes in the distance and (3) No variation between the direction of light approaching the cloak and the direction of light going away from the cloak.

Omitting the detailed description of how to satisfy these requirements, conditions (1)-(3) are met simultaneously by formulating the structure shown in Fig. 1, which consists of general materials combined with left-handed metamaterials with a negative index n.

The outermost medium w1 has a positive refractive index n, and the medium w1 merges continuously with the general space as it goes farther away from the core, that is, n =1. In contrast, the index n of the medium w1 slowly changes from 1 when it comes closer in the direction of the cloak core from the distance.

The deviation from n = 1 is smaller at positions distant from the cloak core, but suddenly varies at positions somewhat closer to the core.

The other four media are located inside radius 1. Media R1 and R3 both have a positive value for the index n, whereas media R2 and R4 have a negative value for the index n. The surfaces obtained by attaching the media R1-R4 with one another all have the same absolute value for the index n, only with the inverted signs.

The media R1 and w1 are attached together in such a manner that the index n of both media is continuous, whereas the medium R4 is attached with the medium w1 so that the signs of the indices n are inverted with respect to each other.

Easier to produce than existing cape?

The invisible cloak has been studied by a number of researchers since around 2006. A practical prototype targeted at the microwave has already been developed. But the existing cloaks are not "perfect" because they are designed without giving due consideration to the reflection.

Moreover, the refractive indices n are positive values. According to a mathematical theory called "Nachman's theory," it has been known that, when only using materials with a positive refractive index, "a perfect invisible cloak cannot be produced unless the materials that induce birefringence are used."

Birefringence is a phenomenon where the refractive index at one point of a medium varies depending on the incident direction of light and the wavefront direction. It is extremely complicated to artificially design such a medium.

Meanwhile, the latest cloak overcame the mathematical constraint by combining the left-handed metamaterial media with a negative refractive index n. Thus, it is no longer necessary to develop a birefringent medium. It is highly possible to develop a truly perfect invisible cloak if a left-handed metamaterial whose refractive index n becomes a negative value in the entire visible light range may be produced and combined in accordance with the latest design.