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.
