Making waves round a structured cloak: lattices, negative refraction and fringes

Using the framework of transformation optics, this paper presents a detailed analysis of a non-singular square cloak for acoustic, out-of-plane shear elastic and electromagnetic waves. Analysis of wave propagation through the cloak is presented and accompanied by numerical illustrations. The efficacy of the regularized cloak is demonstrated and an objective numerical measure of the quality of the cloaking effect is provided. It is demonstrated that the cloaking effect persists over a wide range of frequencies. As a demonstration of the effectiveness of the regularized cloak, a Young's double slit experiment is presented. The stability of the interference pattern is examined when a cloaked and uncloaked obstacle are successively placed in front of one of the apertures. This novel link with a well-known quantum mechanical experiment provides an additional method through which the quality of cloaks may be examined. In the second half of the paper, it is shown that an approximate cloak may be constructed using a discrete lattice structure. The efficiency of the approximate lattice cloak is analysed and a series of illustrative simulations presented. It is demonstrated that effective cloaking may be obtained by using a relatively simple lattice structure, particularly, in the low-frequency regime.