Lateral shift and cloaking effect on grating cloaks

Because of the development of transformation optics, optical paths can be designed and it can turn virtual system into physical system to get material parameters. Therefore, light can travel through my designed optical path. Lots of optical devices had been developed by transformation optics. Cloaking is one of them.
Metamaterials are artificial materials created to have properties that may not be found in nature. Furthermore, humans usually obtain some special properties from designing artificial structure rather than composition, using small inhomogeneities to achieve effective macroscopic behavior.
However, broadband is confined in metamaterials. Under the circumstances, cloaking cannot be achieved in visible frequencies. Because resonance structures absorb electromagnetic radiation when increasing frequencies. In 2008, Pendry developed carpet cloaking which compressed cloaking region into a plane, instead of a point or a line. By coordinate transformations, appropriate material parameters can be obtained. Though carpet cloaking must lie in the plane, the broadband is wider. It can conduct in visible frequencies easily.
This thesis we present a function to design two-dimensional isotropic grating cloaks using conformal mappings. It has a good cloaking effect. There are two main characteristics that physical system is similar to virtual system when we take limit and the maximum of the bump in the metal carpet develops into a sharp tip and the refractive index becomes singular at the maximum ratio of height to width .When increasing height of the carpet, angular central gaussian distribution will be changed and ranges from 0 to 4.63o. The lateral beam displacements are similar to lifting carpets which ranges from -0.46 to 0.69 wavelengths.