Theory and applications of photonic time crystals: a tutorial

This tutorial offers a comprehensive overview of photonic time crystals: artificial materials whose electromagnetic properties are periodically modulated in time at scales comparable to the oscillation period of light while remaining spatially uniform. Being the temporal analogs to traditional photonic crystals, photonic time crystals differ in that they exhibit momentum bandgaps instead of energy bandgaps. The energy is not conserved within momentum bandgaps, and eigenmodes with exponentially growing amplitudes exist in the momentum bandgap. Such properties make photonic time crystals a fascinating novel class of artificial materials from a basic science and applied perspective. This tutorial gives an overview of the fundamental electromagnetic equations governing photonic time crystals and explores the ground-breaking physical phenomena they support. Based on these properties, we also oversee the diverse range of applications they unlock. Different material platforms suitable for creating photonic time crystals are discussed and compared. Furthermore, we elaborate on the connections between wave amplification in photonic time crystals and parametric amplification mechanisms in electrical circuits and nonlinear optics. Numerical codes for calculating the band structures of photonic time crystals using two approaches, the plane wave expansion method and the transfer matrix method, are provided. This tutorial will be helpful for readers with physics or engineering backgrounds. It is designed to serve as an introductory guide for beginners and to establish a reference baseline reflecting the current understanding for researchers in the field.