Wave Propagation in Electric Periodic Structure in Space with Modulation in Time (2D+1). I. Theory

TL;DR

This paper analyzes electromagnetic wave propagation in a 2D space-time periodic structure, revealing complex band behaviors, phase transitions, and conical dispersion points akin to diabolos, with implications for wave control.

Contribution

It introduces a theoretical framework for understanding wave band structures and phase transitions in 2D space-time modulated transmission lines, highlighting the emergence of diabolic points and exotic forbidden bands.

Findings

Identification of forbidden and allowed frequency and wavevector bands.

Discovery of phase transitions at critical modulation parameters.

Observation of conical band touching points resembling diabolos.

Abstract

We studied electromagnetic wave propagation in a system that is periodic in both space and time, namely a discrete 2D transmission line (TL) with capacitors modulated in tandem externally. Kirchhoff's laws lead to an eigenvalue equation whose solutions yield a band structure (BS) for the circular frequency $\omega$ as function of the phase advances $k_{x}a$ and $k_{y}a$ in the plane of the TL. The surfaces $\omega(k_{x}a, k_{y}a)$ display exotic behavior like forbidden $\omega$ bands, forbidden $k$ bands, both, or neither. Certain critical combinations of the modulation strength $m_{c}$ and the modulation frequency $\Omega$ mark transitions from $\omega$ stop bands to forbidden $k$ bands, corresponding to phase transitions from no propagation to propagation of waves. Such behavior is found invariably at the high symmetry $\mathbf{X}$ and $\mathbf{M}$ points of the spatial Brillouin zone (BZ) and at the boundary $\omega=(1/2)\Omega$ of the temporal BZ. At such boundaries the $\omega(k_{x}a, k_{y}a)$ surfaces in neighboring BZs assume conical forms that just touch, resembling a South American toy "di\'abolo"; the point of contact is thus called a "diabolic point". Our investigation reveals interesting interplay between geometry, critical points, and phase transitions.