Synthesis of spatially variant lattices

Abstract

It is often desired to functionally grade and/or spatially vary a periodic structure like a photonic crystal or metamaterial, yet no general method for doing this has been offered in the literature. A straightforward procedure is described here that allows many properties of the lattice to be spatially varied at the same time while producing a final lattice that is still smooth and continuous. Properties include unit cell orientation, lattice spacing, fill fraction, and more. This adds many degrees of freedom to a design such as spatially varying the orientation to exploit directional phenomena. The method is not a coordinate transformation technique so it can more easily produce complicated and arbitrary spatial variance. To demonstrate, the algorithm is used to synthesize a spatially variant self-collimating photonic crystal to flow a Gaussian beam around a 90° bend. The performance of the structure was confirmed through simulation and it showed virtually no scattering around the bend that would have arisen if the lattice had defects or discontinuities.

Spatially variant periodic structures in electromagnetics

“…There are not enough degrees of freedom to do this perfectly so the solution is a best fit and is calculated using a method such as least squares [44,97]. Given the grating phase, the permittivity is calculated directly from it using equation (A 3), best fit where the errors in the period and orientation of the grating are minimized with equal preference throughout the entire lattice.…”

3d Printed Lattices With Spatially Variant Self-Collimation

“…This description was chosen to speed the procedure of spatially varying the lattice [56]. The grating vectors describing the three planar gratings were chosen to produce a lattice with simple cubic symmetry as follows: K 1 = a x 2π/Λ, K 2 = a y 2π/Λ, and K 3 = a z 2π/Λ.…”

“…To do this, we used a novel algorithm capable of spatially varying all of the attributes of a lattice independently and simultaneously while still rendering the overall lattice smooth and continuous [56]. Attributes include unit cell orientation, lattice spacing, fill fraction, geometry, material composition, and more.…”

Electromagnetic Isolation of a Microstrip by Embedding in a Spatially Variant Anisotropic Metamaterial

“…In order to spatially vary the orientation of the unit cells throughout a lattice without changing the size and shape of the unit cells, we used a novel algorithm to synthesize spatially variant lattices [25]. The algorithm is capable of spatially varying any Figure 11.…”