Band Structure and Temporal Topological Edge State of Continuous Photonic Time Crystals

This article introduces the propagation simulation of photonic time crystals (PTCs) using continuous time. In theory, it can simulate the transmission (
$T_{\text {r}}$
) map of any refractive index (RI) function n(
${t}$
). To demonstrate this, we have selected the RI of the sine function, cosine function, and exponential function as examples and studied the characteristics of
$T_{\text {r}}$
and band structures systematically by creating maps. The dimension of the slow wave angle has also been added to analogize the abovementioned graphs at 0°, 30°, 45°, and 60°. By calculating the Zak phase of two different PTCs systems, it is predicted that the interface state will appear. It can be observed that these properties are very similar to the conventional photonic crystals (PCs). Additionally, the continuous PTCs have a topology similar to that of a topological insulator, producing the effect of a temporal topological edge state. The temporal topological edge state is represented by a local peak near the amplitude and verified by plotting the electric field variation. This work advances the theoretical propagation study of PTCs and accelerates theoretical engineering applications with extraordinary potential applications.