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

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.