Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries

Abstract

Recently, there are emerging demands for isotropic material parameters, arising from the broadband requirement of the functional devices. Since inverse Laplace's equation with sliding boundary condition will determine a quasi-conformal mapping, and a quasi-conformal mapping will minimize the transformation material anisotropy, so in this work, the inverse Laplace's equation with sliding boundary condition is proposed for quasi-isotropic transformation material design. Examples of quasi-isotropic arbitrary carpet cloak and waveguide with arbitrary cross sections are provided to validate the proposed method. The proposed method is very simple compared with other quasi-conformal methods based on grid generation tools.

A Planar Focusing Antenna Design With the Quasi-Conformal Mapping

“…Similar results have been obtained by Kong et al in the planar antenna design [26]. We note that for the given virtual and physical spaces, there exist infinite number of transformations that can realize the corresponding mapping, among which some can be expressed analytically, while others can only be calculated numerically [1,19]. However, if we optimize the transformation function between the two spaces, we can then minimize the anisotropy of the metamaterial and treat it as an isotropic one.…”

“…Without going further into the technical detail, we borrow the results given by [5] that the minimal for this functional occurs at the quasi-conformal map, which can be numerically calculated using grid generation techniques or through solving the Laplace's equation [19,34,35]. In this paper, the mapping is obtained through the following two steps [36].…”

“…Their success has aroused great interests, and the method, transformation optics, coupled with the emerging metamaterial technology, has been intensively explored in various designs such as cloaks [3][4][5][6][7][8][9][10][11][12][13], perfect lenses [14][15][16], concentrators [17,18], and other novel devices [19][20][21]. Being an indispensable part in telecommunication systems, many new types of antennas have also been proposed based on this powerful tool [22][23][24][25][26][27][28][29].…”

Integrated photonic systems based on transformation optics enabled gradient index devices

“…14,30 Such QC mappings are numerically generated orthonormal grids as shown in Figure 2, which can be implemented through a variety of different techniques. [29][30][31] These mappings approximately satisfy the Cauchy-Riemann equations, and can be regarded as possessing all the properties of a conformal mapping for the purposes of the TO approach. [32][33][34] Here, the QC mappings are obtained by solving Poisson's equation with NeumannDirichlet boundary conditions using the Poisson transform algorithm.…”

Design of a Pyramidal Horn Antenna With Low E-Plane Sidelobes Using Transformation Optics

“…A wise combination of TO and metamaterials offers great opportunities to design novel electromagnetic devices. Cloaks [17][18][19][20][21], bends and waveguide connectors [22][23][24][25][26], concentrators [27][28][29], and new types of lenses and antennas [30][31][32][33][34][35][36] are some examples of such innovative components. Anisotropic, inhomogeneous [37] and even materials with negative constitutive parameters [38] are among media, which can be realized using metamaterials.…”