Wave Propagation in Electric Periodic Structure in Space with Modulation in Time (2D

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 stopbands to forbidden k bands, corresponding to phase transitions from no propagation to propagation of waves. Such behavior is found invariably at the high symmetry X and 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 "diabolo"; the point of contact is thus called a "diabolic point". Our investigation reveals interesting interplay among geometry, critical points, and phase transitions.