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We studied electromagnetic wave propagation in a system that is periodic inboth space and time, namely a discrete 2D transmission line (TL) withcapacitors modulated in tandem externally. Kirchhoff's laws lead to aneigenvalue equation whose solutions yield a band structure (BS) for thecircular frequency $\omega$ as function of the phase advances $k_{x}a$ and$k_{y}a$ in the plane of the TL. The surfaces $\omega(k_{x}a, k_{y}a)$ displayexotic behavior like forbidden $\omega$ bands, forbidden $k$ bands, both, orneither. Certain critical combinations of the modulation strength $m_{c}$ andthe modulation frequency $\Omega$ mark transitions from $\omega$ stop bands toforbidden $k$ bands, corresponding to phase transitions from no propagation topropagation of waves. Such behavior is found invariably at the high symmetry$\mathbf{X}$ and $\mathbf{M}$ points of the spatial Brillouin zone (BZ) and atthe 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 thatjust touch, resembling a South American toy "di\'abolo"; the point of contactis thus called a "diabolic point". Our investigation reveals interestinginterplay between geometry, critical points, and phase transitions.

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