Quantum mechanics

Modulation (music)

Energy (signal processing)

Economics

Energy–momentum relation

Acoustics

Classical mechanics

Band gap

Photonic crystal

Exponential growth

Position and momentum space

Photonics

Physics

Dielectric

Mathematics

Optics

Exponential decay

Exponential function

Mathematical analysis

Geometry

Momentum (technical analysis)

Point (geometry)

Finance

Optoelectronics

Optica

We study light propagation in spatiotemporal photonic crystals: dielectric media that vary periodically in both space and time. While photonic crystals (spatially periodic media) are well understood, the combination of periodic change in both time and space poses considerable challenges and requires new analysis methods. We find that the band structure of such systems contains energy gaps, momentum gaps, and mixed energy–momentum gaps in which both energy and momentum may attain complex values. We identify the unique interplay between the exponential growth induced by temporal modulation and the exponential decay caused by spatial modulation, and how these can completely counteract one another. Under proper conditions, these two opposing forces are exactly matched, causing the mixed energy–momentum gap to collapse to a single point, which is an exceptional point known from non-Hermitian dynamics. Such spatiotemporal photonic crystals possess unique properties that could pave the way to new ways of controlling the propagation of light.

Spatiotemporal photonic crystals