In the optical cloak design, an m-sided polygon is used in a virtual coordinate space (m
= 6 in ), which is filled with isotropic material of permittivity
= 1). In its center is a smaller polygon rotated at an angle of
compared to the outside polygon. The space between the two polygons is divided by several triangular segments. Due to the symmetric pattern shown in , the triangular segments in the cloak can be grouped into two types, which we marked as Segment I and Segment II with their local coordinate axes, (
) and (
), respectively. A linear homogeneous transformation method (see Methods Summary) is applied in all segments, along their local coordinate axes, so that the center polygon with radius
in the virtual coordinate space is transformed to a bigger polygon with radius
in the physical coordinate space, while the outer polygon with radius
in the virtual coordinate space remains the same as that in the physical coordinate space (). If we only consider the trajectory of the light3,17,24
, we can get the following nonmagnetic constitutive parameters for parallel polarized light (electric field is parallel to the polygon plane):
, for Segment I, and
, for Segment II, where
are the compression or extension ratios of the triangular segments (see Methods Summary). The cloak is formed by lots of such segments, whose materials are homogenous and anisotropic. No extreme values of the material parameters are involved. Different from the two-folded transformation scheme26
, the linear transformation applied here is very simple, yielding simple constitutive parameters of the cloak. The hidden object, here the hexagon with radius
, is mapped to a hexagon with much smaller radius,
, and therefore, becomes much more difficult to be revealed to an outside observer. We can set
to be small enough so that it is invisible to the naked eye. In the virtual coordinate space, the trajectories of the rays marked as blue, red, and green are straight lines (), while in the physical coordinate space, the rays are guided around the hidden object, and appear in the other region without any deflection (). If we increase the number of sides of the polygon cloak (see Supplementary Information
for a 20-sides polygon cloak), it would be very close to a cylindrical cloak but still, would be composed of homogeneous materials in each segment.
Illustration of the transformation optics based cloak design.
In order to conceal a large object, the calculated electromagnetic parameters show that the materials in Segment I should have a big anisotropic degree (
). Natural birefringent crystals are good candidates to construct the cloak for experimental demonstration10,11
, but the anisotropic degree of most of these crystals (e.g. calcite is around 1.26) may be not big enough to squeeze a large space into a much smaller one. In order to observe a clear phenomenon of the cloaking effect in the experimental demonstration, we simplify the cloak design based on the trajectory and the refraction behavior of the horizontal ray marked in red in . A simplified hexagonal cloak with only six segments is designed by replacing the Segment I materials in the original cloak with Segment II materials (see Supplementary Information
for more details). This simplification sacrifices the performance of the cloak for other part of the rays, but can still approximate some properties of the original cloak. We can use natural birefringent crystal for experimental demonstration. The simplified hexagon cloak is fabricated by using six calcite trapezoids glued together (). The calcite has a refractive index
= 1.66 for ordinary light and
= 1.49 for extraordinary light. The optic axis in each calcite trapezoid is perpendicular to its sides, i.e. along the
direction. The inner radius of the cloak is
= 1.5 mm, and the outer radius of the cloak is
= 13 cm. The height of the cloak is
= 13 mm. The inner six surfaces of the cloak are coated with silver. Because of fabrication errors in the calcite crystal, the outer radius and the height of the cloak may have an error up to 1 mm, while the gap between two glued pieces of the cloak may have an error up to 0.3 mm.
In the experimental setup (), a yellow column with a radius of 1.3 mm is put inside the cloak as a hidden object. The upper part of the column is unwrapped by the cloak for comparison. The cloak with the hidden object is immersed in a glass tank filled with a transparent, light yellow, liquid with a refractive index of 1.72 at 589.3 nm. The letters and the logo of Zhejiang University, printed on a paper placed behind the tank in the yz plane, are used as an image. A polarizer is attached in front of the tank to ensure parallel wave polarization. A camera is also placed in front of the tank to capture the image. The image captured by the camera () is exactly what an observer sees through the tank and the cloak. As the upper part of the hidden column hasn’t been covered by the cloak, in the image captured by the camera, the letters ‘g’ and ‘U’ in the upper “Zhejiang University”, and the upper part of the logo are blocked by the column and cannot be fully seen. The lower part of the column is covered by the cloak, and we find that the lower part of the logo and all the letters in the lower “Zhejiang University” become visible. There is a small stripe in the center of the lower part of the image, which is introduced by the imperfect small gap in the gluing of the six calcite trapezoids. The logo and the letters in the captured image show no distortion and are located in the same position as the real objects, as if the hidden column were not there.
Experimental characterization of the optical cloak.
Due to the symmetric pattern of the cloak, the simplified cloak can also work for rays from the other five directions. In the second experiment, we use laser beams to demonstrate how the cloak manipulates the incident light to flow around the hidden object. Three lasers, two for green light (561 nm) and one for red light (650 nm), are used in the demonstration. The red beam from the laser is incident along the
direction, while the two green beams from the lasers are incident along the
directions, respectively (). In order to make sure the beams haven’t changed their directions when propagating from air into the liquids inside the tank, a hexagonal tank is fabricated and the cloak is put in the center of the tank. The laser beams are normally incident onto the three sides of the tank at different positions (). The original path of the red beam (Ray 1) is normally incident onto the hidden column. When it hits the cloak, the beam is split into two. The two split beams are smoothly guided around the hidden object, and emerge back as one beam at the other side of the cloak without any deviation. The cylindrical object at the center of the cloak is therefore invisible for this beam. The original paths of the two green beams (Ray 2 and Ray 3) are very close to the hidden object, although not normally headed to it. The cloak pushes the two beams further from the hidden object compared with the two split beams of Ray 1, and guides them back to their original path again. We see the cloak is able to manipulate the rays flow around the hidden object as if it were not there. A ray tracing model is also applied to demonstrate the path of the rays inside the cloak (the inset in ), verifying the performance of the cloak.
The propagation of the laser beams through the cloak.
We next show the ability of the cloak in manipulating the beams that are obliquely incident onto the cloak. We change
, the angle between the wave vector of the incident wave and the horizontal
plane while keeping the electric field to be horizontal by using a linear polarizer. Three cases are studied: no object, bare object, and cloaked object. The beam with green light propagates through these three cases before projecting onto a black screen (, first row). We use a camera to capture the image of the laser pattern on the screen. As the maximum
that we can measure depends on the height of the cloak and the width of the laser beam, three different incident angles,
, are selected for measurements. The images of the laser beam for the no object case are shown in the second column in , which are used as references. For the bare object case, the beam is totally blocked by the hidden object, resulting in a big shadow in the image (, third column). For the cloaked object case, the images of the beam in the screen are restored, indicating that the hidden object is well concealed by the cloak (, right column). Because the six pieces of the cloak cannot be glued perfectly during fabrication, there is a small gap in the center of the pattern. Compared with the big shadow in the bare object case, this gap is relatively small, without altering our conclusion on the cloaking effect. In previous literature, both the ray tracing model27
and the full wave electromagnetic scattering model28
have shown that the transformation-based cylindrical cloak with ideal constitutive parameters has a three-dimensional behavior, i.e. it can hide an object from oblique incident waves, notwithstanding its transformation function being performed only in a two dimensional xy
plane. In our experimental demonstration, the captured images of the transmitted beam at oblique incident angles clearly show the hidden object is well concealed, providing a complementary experimental evidence to support previous theoretical predictions27,28
Characterization of the cloak for oblique incidence.