Dispersive Diffraction in a Two-Dimensional Hexagonal Transmission Lattice

Consider a hexagonal LC lattice of inductors and capacitors. For the lattice diagram shown in Fig. 1, we assume that each edge consists of an inductor that connects two nodes, and that there is a capacitor connecting each node to a ground plane that is not pictured. This medium is a twodimensional generalization of the standard transmission line structure. We take the inductances L and capacitances C to be identical at each edge and node. We assume the lattice is divided into two halves by a barrier with a thin slit. Waves of lattice voltage incident on this aperture will diffract. In this work, we seek to understand the effect that dispersion will have on the observed diffraction patterns, especially at wavelengths close to the Bragg cutoff for the lattice. We are interested in this problem mainly because the lattice serves as a useful model for many two-dimensional, linear, dispersive media. The numerical solution of high-frequency diffraction problems in such media involves a computational discretization of the domain. For sufficiently high frequencies, waves propagating through the discrete domain will have wavelength comparable to the lattice spacing, at which point dispersive effects will be prominent. A full understanding of the influence of dispersion on diffraction in discrete domains is important for further advancement of high-frequency computational methods. Additionally, we note that this work extends prior work that has analyzed diffraction in a square LC lattice while ignoring the effects of dispersion [1].